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Proceedings of the 8 lnternational Sym posium
on OPERATIONAL RESEARCH

Nova Gorica, Slovenia
September 28-30, 2005

Edited by:
L. Zadnik Stirn • S. Drobne

l

,
The 8h lnfernalional Symptnium 011 Operafional Rosearch in S/ovenla
Nova Gorica, SLOVENIA, Soptem/Jor 28 - 3('1 2005
Edited by:
L Zadnik Stirn and S. Drobne

Slovenian Society Informatika (SDI)
Section for Operational Research (SOR)

lho glh lnlernallonal Symposium 011 Operallonal Research in S/011011/a - SOH
Nova Gorica, SLOVENIA, September 28 - JO, 2005

© 2005 Lidija Zadnik Stirn - Samo Drobne

Program Committee:

th

Proceedings of the 8 lnternational Symposium on Operational Research
SOR'0S in Slovenia, Nova Gorica, September 28 - 30, 2005.

Organiser Slovenian Soc1ety Informatika - Section for Operational Research, SI 1000 Ljubljana,
Vožarski pot 12, Slovenia (www.dmstvo-informatika.si/sekcije/sor/)
Under the auspices of the Slovenian Research Agency
First published in Slovenia 111 200" by Sloveman Society Infonnatika - Section for Operational Research,
SI i 000 Ljubljana, \ ožarsk1 pot l L Sios;ema (www,dmstvo-informatika si/,,,krii,,/enr/\

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INTERNATIONAI Svmposium on Operational Research in Slovema 18 2 1JO'i \Jova Gorica)
SOR 'O'i proceedings The 8th lnterna!Jonal Symposium on Operational Research m Slovema.
'-lova Gorica. Slovenia. September 28-30. 2005 · edited hy L Zadnik Stirn and S. Drobne .
1organiser Sloveman Sociel\ Informatika, Section for Operational Researchj
1._1ubiJana Sloveman Soc1et:, Informatika ,SDI), Sectton for Operattonal Research !SORJ. 200"

Organizing Committee:
B Nemec. HIT, Nova Gorica, Slovenia, Chairman
M Bastič, Faculty of Business and Economics, Maribor, Slovenia
V, čančer. Faculty of Business and Economics, Maribor. Slovenia
S. Drobne. Faculty of Civil Engineering and Geodesy, Ljubljana, Slovenia
L, Ferbar. Faculty of Economics, Ljubljana. Slovenia
8, Peček, School of Public Administralion, Ljubljana, Slovenia
L, Zadnik Stirn, Biotechical Faculty, Ljubljana, Slovenia

1 Zadmk Stim, L1d1.1a 2 Slovensko društvc [nformatika, Sekcija za operacijske raziskave
2220:_I l 104
- -

L. Zadnik Stirn, Biotechical Faculty, Ljubljana. Slovenia, Chairwoman
Baslič, Faculty of Business and Economics. Maribor, Slovenia
L. Bogataj. Faculty of Economics, Ljubljana Slovenia
M. Bogataj, Faculty of Marilime Studies and Transport, Portorož, Slovenia
K. Cherlarova, Faculty of Science. Kašice. Slovakia
A Čizman, Faculty of Organising Sciences Kranj, Kranj Slovenia
S, Drobne, Faculty of Civil Engineering and Geodesy, Ljubljana, Slovenia
Ferbar. Faculty of Economics, Ljubljana Slovenia
S, lndihar, Faculty of Business and Economics, Maribor. Slovenia
P Koechel, Chemnitz University of Technology, Chemnitz, Germany
J Kušar, Faculty of Mechanical Engineering. Ljubljana, Slovenia
L, Lenart, Institute Jožef Stefan, Ljubljana Slovenia
M, Marinovic, Faculty of Philosophy, Rijeka Croatia
L. Neralic, Faculty of Economics, Zagreb. Croatia
M, Simončič. institute tor Economic Research. Ljubljana. Slovenia
K. Šoric Faculty of Economics, Zagreb Croatia
K, Zimmerman. Faculty of Mathematics and Physics, Prague, Czech Republic
Žerovnik. Institute tor Mathematics, Physics and Mechanics in Ljubljana. Slovenia & Faculty of
Mechanical Engineering, Maribor, Slovenia

Chairmen and Chairladies:

____J

Ali rights reserved. No part of th1s book may be reproducecl. stored in a retrieval
system or transmitted by any other means without the prior wntten perm1ssion of
the copyright holder.
Proceedings of the 8th International Symposium on Operational Research in Slovenia (SOR'0S)
is cited in: ISI (Index to Scientific & Technical Proceedings on CD-ROM and ISI/ISTP&B
online database), Current Mathematica] Publications, Mathematical Review. MathSci,
Zentralblatt flir Mathematic / Mathematics Abstracts, MATH on STN International,
CompactMath, INSPEC, Journal ofEconomic Literature

Technical editor : Samo Drobne
Designed by : Studio LUMINA and Samo Drobne
Printed by: MIGRAF, Ljubljana, Slovenia

Bastič,

Faculty of Business and Economics, Maribor, Slovenia
L, Bogataj. Faculty of Economics,,Ljubljana, Slovenia
M. Bogataj, Faculty of Economics, Ljubljana, Slovenia
B. Bohm, University of Technology - Vienna, Vienna, Austria
K, Cechlarova, P J. Šafarik University. Faculty of Science, Kašice, Slovakia
V Čančer, Faculty of Economics and Business, Maribor, Slovenia
L. Ferbar, Faculty of Economics, Ljubljana, Slovenia
S. lndihar, Faculty of Business and Economics, Maribor, Slovenia
H. Th. Jongen, Aachen University, Aachen, Germany
P. Kochel, Faculty of lntormatics, Germany
L. Lenart, Institute Jožef Stefan, Ljubljana, Slovenia
R. Manger, University of Zagreb, Department of Mathematics, Zagreb, Croatia
M. Marinovic, Faculty of Philosophy, Rijeka, Croatia
B. Nemec, HIT d.d., Nova Gorica, Slovenia
L. Neralic, Faculty of Economics, Zagreb, Croatia
B. Peček, School of Public Administration, Ljubljana, Slovenia
K. šoric, Faculty of Economics, Zagreb, Croalia
W. Ukovich, University of Trieste, Trieste, ltaly
L. Zadnik Stirn, Biotechical Faculty, Ljubljana, Slovenia
J. Žerovnik, Institute tor Mathematics, Physics and Mechanics in Ljubljana, Slovenia & Faculty of
Mechanical Engineering, Maribor, Slovenia
M.

Preface
The purpose o/ this volt1111e is tu rejlecr the scientijic activit1es d11ri11g The 8th I11ternatio11al
Sy111posi11111 011 Operarions Research, which was held in Nova Gorica, Slove11ia, J,-0111
Septemher 2H through September 30, 2005. The sy111posi11m 011 Operations Research is one
of the 1raditio11a/ series
the bia111111al i11ternatio11al co11fere11ces organized by Sloveman
Societv INFORMATIKA., Section of Operational Research, and represents the co11ti1111ity of
seven previous symposia. At this symposi11111 the scientists, researchers and practitioners
f,'0111 differe11t areas. like mathelllatics, ecu110111ics, stotistics, computer science e11gi11eeri11g,
environll/ent and sysrem theurv. ufte11 working together on cm11111011 projects, ca111e together
The 8th !11ternatio11al Sv111posi11lll 011 Opemtions Research SDR'05 stood l//Uler the high
auspices oj the Slove11im1 Research Agency, and was granted by sponsors cited in these
Proceedings The ope11i11g address was given by lvfr. N Schlamberger, the Pres1de11t o(
Slove11im1 Society lNFORMATJKA, Proj. Dr
Zad111k Srirn. the President of the Slove11im1
Sectio11 of Operations Research, and the representatives of 111i11istries, representatives of
HIT, Nova Gorica, dijferent professional i11stitutio11s wul Upemtions Research Societies
from other countries.
Operations Research colllprises a large variety of 111athematica/, stalistical and
i11forlllatio11al theories and methods to analvze complex sit11atio11s and to contrihute to
responsible decision making, pla1111i11g and the efficient use of the resources. In a worlcl of
increasing colllplexity and scarce natura/ resources we believe that there will he a growing
11eed far such approaches m ll/a11yfields ofour society.
As traditionalfr, a!so this sylllposi11111 was cm i11ternatio11al forum far scientific exchange at
the frontiers of Operations Research in 111athe111atics, statistics, econolllics, e11gi11eeri11g,
education, e11viro11111ent a11d computer science. We helieve that the prese11tatio11s rejlected
the state of the ari in Operations Research as well as the actual challenges. Besides
co11trib11tio11s on recent advances in the classical fields, the prese11tatio11s, 011 new
i11teractio11s ,vith related fields as well as an i11te11se dia/ogue hetwee11 theorv mul the
1111111ero11s applications. were delivered at the sy111posiu111. Thus, we hope that the divisio11
into Invited lectures and 1O sections. rejlects 011 the one hand the variety offields e11gaged,
011 the other hand separahng too many subjects which could belo11g together. The scientiflc
program was divided i11to the following sections (the 11umber of papers in each section is
given in parentheses). Ple11ary section (6). Scheduling a11d Control (4), Stochastic and
Combinatorial Optimization (4). Algorith111s (7). E11virom11e11t and Human Resources (7),
Locatio11 The01y and Transport (10), Finance and Invest111e11t (8), Multicriteria Decisio11
Making (5), Networks (4). Product/011 and I11ve11t01y (5), Ed11catio11 and Statistics (3;.
The first part of the Proceedings includes invited papers, prese11ted by 6 pro111i11e11t
scientists: Bemhard Bo/1111, Vienna University of Teclmology, Vie1111a, Austria; Hubertus Th.
Jo11ge11, Aachen University, Aachen, Germany; Robert Manger, University of Zagreb, Zagreb,
Croatia; Viljem Rupnik, Interacta, Ltd, Business I11formatio11 Processing, Ljubljana,
Slovenia; Dragomir Sundac, I11ter11ational Business Co11s11lti11g Center, Ltd, Rijeka, Croatia;
and Walter Ukovich, University of Trieste, Trieste, Italy. The second part of the Proceeclings
includes 57 contributed and refereed papers written by 92 authors and co-authors. Most of
the authors ofthe co11trib11ted papers camefrom Slovenia (41), the11fro111 Croatia (30), Italy
(4), Slovak Republic (3), Czech Republic (3), Austria (2), Belgi11111 (2), Germany (1),
Australia (1), Hungmy (1), Israel (1), Latvia (1), UK (1), and Ukraine (1).

o 1<1w1d 1ah/ei (j/ohulizutwn - Cupital
11rrher rhe ~,111pos111111 as lughl1gl11ed
l Jo11u1wwm uwl /)eve!np111e111 mul Fd11cu11n1111! Proh!ems w Fru11orslw Reg um with Regard
SUR '05
greut rnteres/
F11ropea11 /'rel/{/s und JJe11w11ds The·, hoth atnacred
participants. The discussion led tn re11wrkahle C<mc!11sio11s concenung regiona! cooperation
(III(/ ussociations witlt other FL co1111tries, us wel! as deve!up111e11t and use o/ econometric
·
111odels in regiona/ and internatwna/ eco110111ies.
Tite Proceec/111gs of the prevw11s seven J111ernatlu11a/ ~J'lllposw 011 Upera1w11s Research
organizecl by Slove11ia11 Sectiun of Operaf1011s Research are cited In the followi11g seco11dmy
and /erliwy p11h/ica1io11s C11rre111 Mathemalical P11b/icatio11s. mathe111at1cal Review.
vfarltSci, Zentralhlaufuer Ma1he111mik1Mathematics Abstracts, MATfI 011 STN J11ternational.
C.u111pact!vla1h, IVSPEC Alsu t!te presen/ Proceedings wi/1 be s11b1111ttecl mul ,s supposed to
he c1red 111 1/te Slll/le p11bl1ca11011s
We wo11lc/ nor ltave succeeded in uttracti11g so 111a11y distinguislted speakersfrom ali over tlte
world i_vitltout the e11gage111e111 and lite aclnce oj active members of Slove11ia11 Sectim1 oj
Ope1_·at1011s Researclt !vlany thanks to 1he111. F11rtlter we would like to express our deepest
graftt11de to the 111e111bers of tlte Program mul Urganizmg Co111111ittees, to tlte referees,
/Je /,sred ltere i11dividually cluurpersons cmd tn ali rhe 1111mero11s people fi'II' 100 11wnv
wlw ltelpec/ ill cwrving u111 Tite Rth J11remu!to11al ():v111pusu1111 011 Operw1011s Research
SUR '()5 a11d 111 p11tt111g /Ogether 1hese Pruceec/ings A1 /List, we uppreciate rite authors ejforts
III prepanng and presel/fillg 1he papers. wluch made The 81h SvmpOSIU/11 U/1 Operational
R.esearc/1 SOR 05 s11Gcessf11f.

Nova (innca September 2R. 200.'i
l.idi;a Zadnik Stirn
Samo Drobne
(Editors)

Foreword
Mathematics with all its theoretical and applied disciplines. in particular with operations
research, is afield where people are active to discover 11ew ulgoritlt111s and approacltesfrom
the p/easure uffinding things out. lts purpose is not to explain phe110111e11a or to discover
rules; this is a mission of uther natura/ sciences. There, however. exists a close mul firm
relationship hetween those and mathematics/operatio11s research. without its tools, methods,
a11d i11sights ali other sciences would be 111ore or less descriptive a11d narrative and they
would he able to explain hardly a11y ofthe things that they have discovered. Moreover.
ever higher degree they depend on 111athe111atics in general a11d on its bra11ches
today,
where we find alsu operatio11s research, social and humanistic sciences. Without
111athematical methods they would not be what they are now. Let usjust remind 011 the
frequencv analysis of letters in Slovenian /a11guage. Wlti/e is true that this is not tlte /atest
achieve111ent it is sure/v an illustrative one.
Mathematics in general has seen an i11te11se development in the 11i11eteentlt and twentietlt
centurv. The description ofelectro111ag11eticfield mul irs pruperties tltat were able to he
predictedfrom solutions ufpartial differential equatio11s are ltardl.v imaginable without
vector analysis and quaternions. Had Rie111w111 not jiJu11ded differential geometn of curved
spaces which was a result of developing ufpurely abstract ideas, it should have been
i11vented anew, as without it an analytical approach to Einstein 's themy of gravity is
i111possihle. In the twenties the 111atrix calculus has been developed, so to speak to wait on the
shelf, 011/y he used soon afterwards as an i11dispensahle 100/ far the first descriptio11 of
qua11tu111 mechanics. An overview of quantw11 states of elementarv particles that has bee11
painstakinglv put together in a form of tables by W Heisenberg that are hard to u11dersta11d
even today and which are thefoundation of matrix mechanics is now possible to present as
matrices that are readily possible to calculate with. Asa co11seque11ce of 11eed tu so/ve
complex problems in technology, economy, and social scie11ces, a new field of operations
research has heen invented, Today is an apparatus rhat is hardlv imaginable to do
without in the areas that at thefirst sight seem to be too complex to be accessible bv
mathematical tools - such as medicine, hiology or ecology Topology that has /Jecome of age
in the second halj of twentieth centw) is a discipline that /Jroadens our insight 011 properties
of 11111/tidimensional spaces. Last but not any less important. nearly at the end ofthe past
century the solution the Fermat great theorem that has 1110/ested the greatest mindsfor a
three h undred years, has been found.
What is the agent of the development of mathematics and in particular of operations
research?
A simple a11swer in a form of one co11dition that is at the same time necessary and sufficie11t
does probably not exist. There are at least two conditions that must be met. The necessary
condition is that there exists curiosity, a pleasure offindi11g things out, as the Nobel laureate
R. Feynman has put it. We need not worry about this one; as long as man exists also the
human curiosity will exist. The other condition is that new problems arise a11d that new
questions are asked to which no existing knowledge can provide answers. Disciplines wltere
no such problems come up any more that require search far new solutions have completed
their evolution. That ca11 even be measured; the index is the number of 11ew publications. By
this standard hardly anything else is going 011 in mechanics and thermodynamics, they both
are solving only practical technical problems.

At the begi1111ing ofthisforeword a thought has heen e.,pressed that mathematics and
operations research have seen an intensive development in the last century and a half llas it
already lived past its golden age? This is hardly to helieve. Asidej,'om the questions to
which there are stili no answers there are also new challenges that arisefrom the
developme11t ofnew technologies. Computers mul infonnation science arejust one of them.
The alliance of mathematical methods combined with an immense processing power offers a
wholly new perspective and possibilities far solving of special kinds ofproblems. Operations
research is a typical field where such a combination promises an intensive progress. The
preselit vohune bears witness that such a belief is realistic.
Nevertheless there is one reservation and one hope. The reservatio11 is thatfor the progress
a motor a11d afl1el are 11eeded. The motor ofthe progress are the before described
conditio11s, the curiosity and 11ew problems that bear 11ew challenges. Both will be available
in thefuture as they were avmlable so far The fl1el is the co11cern of those that are to use the
results ofthe research. Regardi11g this we may be worried by the dwi11dling i11terest of users
far solving their problems using methods of operations research. Somewlwt disappointed we
see that in practice cases have begun to repeat fi·om the time when results of research and
developmelll were able to compe11sate ge11erously mul profitably hy primitive economy
methods. The swprise is genuine as we 110w that in the long run only k11owledge provides a
solid basis far adding new value. Nevertheless there is a hope that the decline of motivation
to use the methods of operatio11s research is only a passing episode mul that the users will
understand the importa11ce of this discipli11e to their advantage a11d the benefit of ali. The
1111111ber of contributzons in this volumen( bolh domesric and foreign authors bears witness
that such hope 1s not in vail1.

Nova Gorica . September 2R. 2005
Niko Schlamberger
(President
Slovenian Society lNFORMATIKA)

Sponsors of the 8th International Symposium on
Operational Research in Slovenia (SOR '(}5)
The 8th Jnternational Snnposium on Operational Research in S!ovenia was organized under
the auspices of the Researh Agency, Repuh!ic oF\'lovenia.
Thefollowing institutions and companies are gratefullr acknowledgedfor theirfina11cial
support of the symposium.
Austrian Science wui Research Liaison O{fice. Departmenl of L;11hl;a11a. L111bljana
HIT, Nova Gorica

Contents
Invited Lectures

1

Bernhard Bohm (keynote speaker)
Approaches to the Measurement ofEco-Effic1ency

3

Hubertus Th Jongen (keynote speaker;
On Gradient Flows in Globa! Optm11zat10n

ll

Robert Manger (keynote speaker;
Some Results Dealing with the Algebra1c Approach to Path Problems m Graphs

lJ

Viljem Rupnik (keynote speaker)
On Some Trappmg States to OR

23

Viljem Rupnik and Dragomir Sundai (keynote speakers)
Domination of Capi tal = The Trappmg S tate to Mankmd

l,5

Lorenza Castelli, Rajfaele Pesenti and Walter lkovich (keynote speaker;
A Review of DEA Models when the Interna! Structure the Dec1s10n Makmg
Units is ( 'ons1dered

37

Section 1: Scheduling and Control
Janez Abram, Lidija Zadnik Stim, Mirko Tratnik and Leon Ohlak
Optimisation of Scheduling: A Case of Industrial Chair Manufacturing

49
51

Matej Borovinšek and Janez Žerovnik
Local Optimizat10n Applied to a Volunteer Timetabling Problem
Lado Lenart and Jan Babič
Dynamic Programming in Control
Jan Pelikan
Scheduling Serial - Parallel Processors

63
A ( ~ase Study

Section 2: Stochastic and Combinatorial Optimization

69

75

Katarf11a Cechlarowi, Tamas Fleiner and David Ma11/ove
The Kidney Exchange Game

77

Vladimir I. Ka!ika
Modeling Stock Buying-Selling Using a New MCDM Methodology Accounting for
Uncertainty

85

Lada Rokov
Using Game Theory in Analysis ofMobile Telecommunication Market

93

Damir Vukičevi/: and Janez Žerovnik
An Example of a Game and its Combinatorial Analysis

99

F
Section 3: Algorithms
A,fajda

105

Bastič

The Scale oflnnovativeness with Structural l:..juation Model mg

07

Igor Dukanovic and Franz Rendf
Practical Bounds m Error-Free Commumcat1on

3

Samo Dmh11e and
Intermurncipal

BogaWJ
Model of Sloverna

Samo Drobne, Marija Bogata;, Dejan Paliska and Oafo Fah;an
Will the Future Highway Network lmprove the Access1n1lity to Admirnstrmive
Centres in Sloverna')

213

Peler Koche/
Some Cons1derat1ons on Opt1misation m Logistlc Systems

219
227

Liljana Ferbar, David Čreslovnik, Blaž Mojškerc and Martin Rajgelj
Convergence in the C'ase of the Bullwh1p Effect with Centralized Demand Informat10n

19

Uljana Ferbar, David fres/nvnik Blaž MoJi.'kerc and Martin Rajgelj
( onvergence in ihe 'ase of the Bullwh1p Lffect with Decentrahzed Demand lnformat1on

Andrej Lisec and Marija Bogalaj
Travelmg Salesman Problem at Post of Slovenia

25

Ana Osvald, Raffae/e Pesenti and Waher l'kon, II
Deaimg w1th the Fluctuat1ons of the Trave! Tunes

Martin uavalec and Jan Plavka
On the Eigenspace Structure of a Matrix in Max-Min Algebra

1 ]

Luka Neralii

Preservation of Effic1ency and lneffinency Class1ficat10n m Data Envelopment Analys1s

137

Janez Pavl, and Renata Sotirov
Recent Approaches to the Quadratlc Ass1gnment Problem

139

Section 4: Euvironment and Human Resource
Management
Josip Americ and Elza Junm
Cross-Tabulation Analyses ufthe Survey Research on the Moral Values
Daria

145

Na</ja Dami) and Janez Grad
Simulation of the Business Process Salcs Claim
Andreas Ehren111a11n, lvlarek Kočan and Yves Smeers
Boam and Bust Phenomena in Electricity Market

Bojan Peček
Simulation Analysis During Law Adoption
Pe/ra Ulamec, Lidija Zadnik S!im and Kalja Vadnal
Distribution Channels far Organic Produced Agricultural Products

Section 5: Location Theory and Transport
Tomaž Berlec, Janez Kušar, Lado Lenari and Marko Starbek
Hidden Logistic Potentials in the Company
Ludvik Boga/aj and Marija Bogataj
Reverse Logistics in Value Chain

235

A ( 'ase Sudy for lntervcntion

Pave/s Pa!lins
Lmversal Routmg Algonthm for ( '1t1es and Other Buil,-l

241

r Areas

247

Jvlarko Urh and A111011 Čižman
Pohce Patrol Route Optmusat10n

253

259

Section 6: Finance and Investment
.
.
Zdravka Alji11ovii', Branka Marasovii' and ,Veli To111ic-Plaziha1
Mult1-Cnterion Approach Versus Markuwitz m Seleclion uf the Optimal Portfol!o

261

153

Draženka Č'izmir
fhe Ch01ce of Pnce !ndex Formulae for Flementar, Ind1ces

267

159

Ksenija Dumičii, Mirjana Č'ižmeš(ia and Anila Pavkovic
Sample Survey Research on Fmancial R1sk Management: A Case

165

Nataša Rrjavec and Boris C'ota
Modeli mg Exchange Rate

Špela Malovrh

Evaluation of Parameters which Influence the Association of Pri vate Forest Owners
by the Use of AHP and DEXi

Dejan Paliska, naša Fahfan and Samo Drobne
Allocat1on of Specially Equ1pped Fire Rngade l
in Tunnels on Sloveman Road Neiwork

Real-\\ orid )1stnbut10n Problems

147

Bor"fič

Cointegration and Purchasmg Power Paritv m Selected New Members ofE1

207

ot Croatian C:ompames

Croatia

273

279

Elza Jurun and Snježana Pivac
Macroeconom1c Model!ing Relaxmg Theoret1c Assumpt10ns in the Croatian
Financial Sphere

285

179

Alenka Kavkler and Bemhard Bolim
A Nonlinear Monetary Model oflnflation

291

185

,
Dubravko Mojsinovic
Five Croatian Banks Foreign Currency Position Analysis Using Value at R1sk

297

Ivan Šmid and Valerija Bublic
Analysis and Risk Management by Investment Projects Using Method Mante Carla
Simulation

303

73

191
193

Section 7: Multiaiteria Decision Making

199

Zoran Babic and Tihomir Hunjak
Multiple Prices and De Novo Programming

309
311

i es11a (\mcer
( omparison olthe Applicab1ln:,
Dec1s1on-Makmg Methods

Mile Pmlil,
Model ofBasic

omputer Supported 'v1ult1-( 'nteria
17

JosefJablonski
An Interval AHP Model for Effic1ency Evaluation of Product1on

mts

Igor Lipušc'ek, Lidija Zadnik Stim m1d Leon Oh!ak
Development of a Multi-Criteria Evalua1Jon Model for Class1fymg Wood Producrs
Anka Lisec and Lidija Zadnik Stim
Multi-Attribute Utiltty Theory m Sustainable Rural Land Management

323
33
337

•
,
Jvlarko Powkar, (iregor lvliklav6{ umi :vlir1a11a Rakamarii' Segli
Calcu!atwn of the Optimal Nurnber o,
the Queuemg Theory
Apphcation
Employees ofthe Call-Centre

APPENDIX
Authors' addresses
Sponsors' notices

Section 8: Networks

343

Nfatko Botinc'an mul Goranka Nogo
the ( :apac1tated Veh1cle Rouung Problem \Hth
On Distributed Solnng
Branch-and-Cut Algorithms
Dušan Hvalica
Sensitivity Analysis of the Mrnimal ( ost Solut10n hy Andi( >r ( rraphs
Tomaž Kramberger and Janez Žernvmk
Chmese Postman Problem w:th Priorit,es

351
J57

Krunoslav P11lj1i and Robert Manger
An Irnproved Evolutionary Algorithm for Soi,,ng the Vehichle Routmg Problem

Section 9: Production and lnventmy
Anton Čižman and Marko Uril
A PC-Based Dec1s10n Support Systern for lallorrng ,Jf Lugs m

345

363

369
eneers Product1on

Matjaž Fe/trin, Lidija Zadnik Stim and Jasna Hmvatin
ConJomt Analys1s :n Wooa Products Des1gn
Zrinka Lukač, Kristina Šoric and Višnja Vojvodic Rosenzwe1g
L TU Heuristics for Capacitated Lot S1zmg Problem\\ 1th Sequence )ependent Setups
and Overtimes
Aleš Silič, Lidija Zadnik Stim, Janez Kušar and Marko Starbek
Markov Chains in MRP Model of a Multi-Leve! Producnon System

383

389

Ly11d111yla Zahvoyska
Irnprovement Design and Management ofWood Processmg Manufacture Systems

Section 1 O: Education and Statistics
Vlasta Bahovec and Miljana Čižmešija
Tests of no Assotiation between Variables-Components of Consumer Confidence
Indicator and Stratification Variables for Croation Consumer Survey

397

403

405

u111i Su11111 (a11drih
for lnfr1rmat1on System )evelopers

41

419

427

Autlwr inde.ic
A
Abram Janez
Aljinovic Zdravka„
Arneric

... 5
.26

47

B
... 63
Jan
Babic Zoran ..
..405
Bahovec Vlasta..
107
Bastič Majda ......
93
Berlec Tomaž
199
Bogataj Ludvik
Bogataj Marija ....... 99. 207, 2 3,227
29
Bi:ihm Bernhard
........... 57
Borovinšek Matej .
153
Boršič Darja ...... .
.. 345
Botinčan Matko
.303
Bublic Valerija„

Babič

C
Castelli Lorenzo
77
Cechlarova Katarina
Cota Boris ...................................... 279

č

..317
Čančer Vesna......
Čandrlic Sanja ........................... ..411
37
Čižman Anton.
405
273,
......
Mirjana
Čižmešija
..... 267
Čizmic Draženka
Čreslovnik Igor ..................... 19, 25

D
159
Damij Nadja
Drobne Samo
Dumičic Ksenija ............................ 273

D
Dukanovic Igor.. ............................ 113
E
165
Ehrenmann Andreas
Eijavec Nataša ............................... 279

f
Fabjan Daša ..... .
Feltrin Matjaž .. .
Ferbar Liljana .. .
Fleiner Tamas

... 213, 241
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119. 125
................. 77

G
Gavalec Martin
Grad Janez .....

,,.,

l.1

59

H
Hrovatin Jasna .....
HunJak Tihomir.. .
Hvalica Dušan ..

J
Jablonsky Joser
Jongen Hubertus Th.
Jurun Elza

.,

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...... 85
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Kavkler Alenka ............................ 291
...... 219
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. 165
Kočan Marek.
357
Kramberger Tomaž
l 93, 389
Kušar Janez
L
Lenart Lado
Lipušček Igor.
Lisec Andrej
Lisec Anka
Lukač Zrinka ......

331
227
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.................. 383

M
Malovrh Špela.. ...................... 173
Manger Robert ........................ 13, 363
Manlove David .............................. 77
Marasovic Branka .................. ,. ..... 261
Marinovic Marija .......................... 411
Miklavčič Gregor .......................... 419
Mojsinovic Dubravko ................... 297
Mojškerc Blaž ....................... 119, 125

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u

Nera!ic Luka
'\Jogo Goranka

37
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Oblak Leon
Osvald Ana ....

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... 235

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Paliska DeJan
Patlms Pavels.
Pavkovic Anita.
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Pelikan Jan ..
Pesenti Raffaele
Pivac Snježana ..
Plavka Jan

... 21:i. 24
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411
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131

Povh Janez
Puljic Krunosia\

... 363

R
RaJgelj Martin ..
Rakamaric Šegic Mirjana
Rendl Franz
Rokov Lada.
Rupnik Viljem.

. 119. 125

...419
13

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s
Silič Aleš
Smeers Yves ...
Sotirov Renata
Starbek Marko
Sundac Dragomir..

..... 389
65

139
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Šmid Ivan .......... .
.,. .......... 303
Šoric Kristina ................................. 3 83

T
Tomic-Plazibat Neli ...................... 261
Tratnik Mirko .................................. 51

Lkovich \Valter..
Llamec Petra ..
Lrh Marko ..

235
85
253. 37

v
Vadnal Katja
V vodic Rosenzweig Višnja.
Vukičevic Damir

85
383
99

The 8th lnternalional Symposium on
Operotional Rese0rch in Slovenia

z

S@R 'OS

Zadnik Stirn Lidija ..
51, 85.33
Zahvoyska Lyudmyla

ž
Žerornik Janez

57.99. 357

Nova Gorico, SLOVENIA
September 28 • 30, 2005

Apprnaches to Hm measurement of eco-effidency
Bernhard Bohm
Institute for Mathemat1cal Methods Ecunom1cs
Gnn-ersit:- ofTechnoiogy \"1enna, AustTia
ibemhard.boehmičutuw1en.ac.at)

Abstract
The present paper presents a selection of approaches to measure eco-efficiency at different levels
considering the mult1-obJective nature of th1s concept. It shortly reviews the leve! of implementation
in mdustnes and discusses maJor e!ements and processes, D1fferen: measurement concepts at the
industry level will be rev1ewed, Aspects of eco-efficiency bevond ihe individual finn are considered
next M1cro- and macroeconom11; aspects wiil be integrated by a discuss1on of the multiple objective
the macroeconom1..: problem and an attempt to measure macro-eco-efficiency
nafure

Keywords: Eco-effic1ency. data ern,e!opmem analys1s, input-output analys1s. distance functlon
1. Introduction

During recent years the temi "eco-efficiency" has surfaced in business economics as new
criterion among the objecfr,es of a firm. It is now playing an mcreasing role m fim1
management where sustainable deYelopment has been included among the set of firm
targets. White forma! decision analysis ts often applied to complex economic problems, the
development of the eco-efficiency concept and its appiication in practical management only
indirectly relates to such analysis. While looking for an answer as to how the objective of
sustainable development can be reached by the firm, the concept itself is often understood as
an indirect objective or even an instrument or tool to achieve the sustainability objective. In
the definition of the World Business Council for Sustainable Development we read that
"eco-efficiency is reached by the delivery of competitively priced goods and services that
satisfy human needs and bring quality of life, while progressively reducing ecological
impacts and resource intensity throughout the life cycle. to a level at least in line with the
earth's carrying capacity".
Approaches differ among countries, industries, and firms. The paper shortly reviews some
measurement concepts at the industry leve! and compares eco-efficiency across individual
fim1s. Micro- and macroeconomic aspects will be integrated by a discussion of the multiple
objective nature ofthe macroeconomic problem and a macro-eco-efficiency measure.

2. Eco-efficiency at the industry level
"Eco-efficiency" is considered an important tool ofbusiness practice and management where
innovations in technology, production, processes, product design and business organisation
and practices can lead to lower unit costs, improved product quality, lower environmentrelated liability, less material usage and less adverse impact on the environment. The main
message in promoting this "tool" is that economy and ecology do not exclude each other but
can be made to cooperate profitably for both the firm and for society: more value can be
generated with less use ofresources and less damages for the environment in general.
Measurement issues of the degree of eco-efficiency achieved do not stand in the foreground
of interest. Incorporating the four required strategic elements, de-materialisation, closure of
production loops, service extension, enhancing functionality, the identification of indicators

3

BCSD
the
and the construction of a umfied measure are hard 1.ssues. The approach
product or service rnlue' where hoth. nurnerator
,
- •,
.. .
.
concentrates on a measure defined b:, ,
Lennronmenta l mfluence J
and denominatoL ha\·e to be derived from indicators Because of large differences due
rele\ ant
products, branches, techni4ues. customers etc. firms may need to define their
mav
mdicators
Thus
indicators appropriate for extemai communicati on and decision making.
eco·An
ones.
specific
belo~g either to the group of generally applicable or business
both.
incorporate
will
profile
effic1ency performance
A more demanding concept to assess eco-efficienc y is provided by life cycle analysis (LCA)
of a product or process. It should help to assess potential effects of a product or process on
env1ronment startmg from the inception of the idea and the use of the raw material through
the user phase u~til the end in a waste deposit or a recycling process. focuses specifically
on energy matenals used and wastes released to the environment. Multi-criteria decision
the application of LCA
techniques can generate dramatic productivity improvement s
(Brunn and Rentz 998 Results from LC A will contribute to other measuremen t concepts.
A promment one m Germany is MIPS (Material Input Per Senice unit), developed at the
Wuppe1tal Institute. It calculates the input of total material resources required for the
production of goods and services, measured in service units. Material inputs cover all
re~~wabl~ or exhaustible resources, land. water and air. while a senice unit represents the
uhhty denved from the use of a particular good.
While many of_ the concepts discussed so far provide usefu! steps for the concemed firm,
there ~re many mstances when comparisons between firms are desired to udge either policy
effechveness or to design policy instruments appropriately. For such objectives the tool of
data envelopment analysis (DEA) has been recommende d. For the evaluation of economic
efficiency this technique already enjoys widespread applicability as is witnessed by the large
number of available publications (cf Chames et aL (1994)) To a large extent it avoids
problems of valuations of inputs and outputs. or better. provides even valuation estimates.
However. applying DEA to ecological and econom1c efticienc:, requires reconsideration of
r weighted sum ol outputs 7
.
, in order to adapt it to
'J .
.
the conventional effic1ency measure ,
L wezghted sum oj inputs j
environment al magnitudes. The WBCSD definition, the MIPS, and similar concept that have
been proposed, ali relate environment al magnitudes (value of damages, emissions, resources)
to _some economic magnitude (service unit, utility. product value) Especially the
env1ronmental magnitudes cannot be easily classified as inputs or outputs as many of them
are ~ndesirable from the point of view of production. If one characterises ecological
effic1ency as a way of 'producing with the least energy, resources, waste and emissions '
then the Jack of prices for emissions and waste or other undesirable outputs makes it difficult
to construct measures of ecologically efficient production. But some of these problems can
be overcome by using DEA.

3. The DEA approach to eco-efficiency
The use of a nonparametr ic method for productivity comparisons with non-desirable outputs
seems to have started with a paper by Fare et al. ( 1989). This approach has later been applied
to evaluate enviromnent al performance of firms (cf. the review by Tyteca (1996)). A recent
paper by Korhonen and Luptačik (2004) contrasts two types of approaches. The first

4

decomposes the problem in two parts and measures first technical efficiency along
conventional lines and then measures ecological efficiency by using a ratio of a weighted
sum of desirable outputs to the weighted sum of undesirable outputs. The efficiency
indicators of both models are used as output variables in a new DEA model with inputs
equal 1, yielding the eco-efficienc y indicator.
The second approach specifies ratios which simultaneously take into account desirable and
undesirable outputs. ln this case at least four different ratios can be constructed:
Model A maximises the ratio of the difference between a weighted sum of k desirable (yr)
and (p-k) undesirable outputs (y,) to the weighted sum ofm inputs (x,) for each unitj=l, .. ,n.
k

max hA

-!

I

= ~,_=1,___ _ _._=~h+;

i

S.{.

.,_,=.,-l_ _ _ ~=A •_!

( 1)

III

"vx

,x

~!i,

(" 11011 - Archimedean")
m. & >
p. 1 =
and maximises the ratio ofthe weighted sum of
inputs
as
outputs
undesirable
Model B treats
desirable outputs to the weighted sum of inputs and undesirab!e outputs

J =L .. , n, r =

k

Lµ,v,.o
max h8

= ,,,

,=;

_Lv,x,,+
i=:

P

s.t.

- - - " ' - ~ · - - - - '.So

L

µ ' v i 2'.&

s=k··

("non -Archimedea n"
m, & >
p, 1 =
j = l, ... ,11, r =
Model C maximises the ratio of the difference between a weighted sum of k desirable
outputs and the weighted sum ofm inputs the weighted sum of (p-k) undesirable outputs
m

k

"µv.-"v
~
n
LJ
!-

==

r=l _

i-J

s.t.

.:.=;,_ _ _,;,:i-:,,,1 - - -

$;

µ.

•=k+,

("non -Archimedea n")
m. & >
i=
j = l, ... ,n, r = 1, ... ,
Model D is the reciprocal of model B and minimises the ratio of the sum of weighted
undesirable outputs and inputs to weighted desirable outputs.
The eco-efficienc v indicator of model B cannot be smaller than the one of model A because
both inputs and ~ndesirable outputs have to be decreased to increase eco-efficiency. In
model C only undesirable outputs have to be decreased, so the indicator of C cannot be
greater than that of B.
Applications of such models to firms of the same industry will yield deeper insights into the
causes of eco-inefficie ncy and can indicate potential improvements in inputs and outputs.
They are also very useful ,to identify achievements of emission reduction programs when one
compares eco-efficienc y indicators before and after the implementati on of such programs. A
particularly interesting application of this technique can also help to solve allocation
problems in the context of preparations for the trading of emission certificates. An
application to the French cement industry can be found in Patnaik (2004). He presents
alternative calculations of proposed initial allocations of emission rights (allowances), ali
based on the solution of a variable retums to scale DEA problem. It tums out that it is not so
easy to determine how eco-inefficiency should be reflected in the allocation of allowances.
The reason why an efficiency approach is desirable at ali is given by the fact that a
traditional allocation procedure based on historical emission values (so called

5

"grandfathe1ing'') would h1as m favour of those firms that have contributed to the highest
enussion reductions. The DEA
levels of ernissions mstead ,,f those that have contributed
results of eco-efficiency can therefore be used to distnbute allowances according to the ecoefficiency of a plan t. As the various models discussed abo\ e all show shght di fferences in
may be left to the
the inefficiency indicators the efficient units are always the same
policy maker to select the appropriate one. Information requirements are higher when
additionally to outputs and emissions also input factors are considered. A DEA based ecoefficiency calculation will have a motivating effect on competing companies because
information about the direction of efficiency improvement is provided as welL

r

th1s partni0ned svstem by treating the
Applying the concept of the distance functwn
mputs. a proportional measure of eco-efficiency can be
undesirable outputs similar
derived from the following model fomrnlations where the m x \ ector of primary mputs is
given by z.
l. Minimise the use of primary factors for a given leve! of final demand and tolerated
pollution:
s.t.
mm

Let us now turu to the whole econom) and the problem to characterise its eco-efficiency
Due to ·'rebound effects" firm ·s eco-efficiency not necessarily implies macro-eco-efficiency.
Growth effects may more than compensate efficiency gains. The ecological balance may
deteriorate due to a larger increase in sales of a product that itself is resource efficienL One
of the most useful tools for analysing production relations within an economy considering
the mterrelatedness ofproducts and sectors is the so-called input-output approach. An inputoutput table relates all goods and services produced and used in a country to all economic
branches of the economy. Because of its detailed structure of the inter-industry relations such
a model reveals the role of key sectors or industnes, the supplier links and the delivery
growth
chains throughout the economy. With the beginning concem about the limits
around the seventies of last century and the public awareness of existing environmental
problems the input-output approach has been amended to take into account also
environmental dimensions. We shall use this framework to obtain measures of ecoefficiency. degrees of achievements of both. economic and environmental, objectives for the
whole economy.
Let the production possibility frontier of the economy be determined by the input-output
model. primary inputs, pollution generation and abatement, and fina! demand. The degree by
which a net-output , ectoL for given primary inputs and environmental standards, could be
extended, can be considered a measure of eco-inefficiency. Equivalently this could also be
achieved by a reduction of primary inputs for given environmental standards and given fina!
demand. We have to take into account that desirable outputs are strongly disposable while
undesirable ones are only weakly disposable meaning that their reduction can only be
achieved by a reduction of desirable outputs oran increase ofprimary inputs.
The augmented Leontiefmodel known from the literature (e.g. Luptacik and Bohm(1994))
can be written in partitioned form
(I - A, 1 )x, - A, 2 x 2 ;::: y,
(4)
-A2,x, + (I -A22)X2 ;;=: -y2
with A 11 the k x k input-coefficient matrix of the economic subsystem, A12 the k x (p-k)
input-coefficient matrix of the abatement system, A 21 the (p-k) x k emission-coefficient
matrix of the economic system, and A 22 the (p-k x p-k) emission-coefficient matrix of the
abatement system. x 1 and x 2 are the (k x 1) gross production vector and the (p-k x 1)
abatement vector respectively, y 1 is the k x 1) fina! demand vector and Y2 the p-k x 1 vector
of tolerated pollutants (or pollution standards) vector.

6

A1,X2

;;=: .•·1

-A. 2

:s,

B,x, + B,;c - yz

SO

A2

4. Macro Eco-efficiency

-

(/ - A,1
-(/

(5)

It is easily seen that this model corresponds to model A of Korhonen and Luptač1k (2004)
This input distance function considers the minimal proportional contraction of the mput
vector, gi\·en the output vector and tolerated pollution.
2. Maximise proportional expansion of fina] demand for given le\els of tolerated pollut1on
and primary factors:
s.t.
max a
- A12x, - l0'1 ;:::

(I - A11
A2,X1 -

-Ai2)X= S

(6)

Y,

B1x, + B,x, s z
Xi,X 2

,a ;;=: 0

This is the output oriented model D. We note that due to the presence of the pollution
subsystem representing undesirable outputs, the optimal values of a and "( are not the
reciprocal of each other. However, by treating these undesirable outputs like inputs in the
model, i.e. by changing the problem formulation into a proportional reduction of primary
inputs and undesirable outputs for given fina! demand, the reciprocal property ofthe distance
function can be established. This happens in the following version which corresponds to
model B:
3. Minimise the use of primary factors ond tolernted pollution for a given level of fina!
demand:
s.t.
mmy
X

(/ -A1JX1

-A12X2 2'._

A21x1 - (I - Ai2 )x2 -

Yi

rv, :s, O

(7)

B1x1 +B2x 2 -yz :o, O
XpX2,Y ;;=:

0

Still another version corresponding to model C can be formulated by minimising tolerated
pollution for a given level of fina! demand and primary factors
W e note that efficiency indicators derived from the distance function approach are based on
optimisation without the possibility to alter the proportions among net outputs or primary
inputs. If its slack variable is positive it is still possible to reduce a specific input without
reducing any of the net-outputs. By defining a new slack based measure as in Luptačik and
Bolim (2005) we can go beyond the proportional approaches and take into account of
changes in the structure of fina! demand, pollution sources and the composition of primary

7

v"'<::1

mputs. 1he problem 1s fonnulated as a goaJ prograrnmmg model treating the (p-k)
undesirable outputs like inputs:

fs. + f
min~ p

.. k)+ III

s.

·=· Z.

2.

= -------'--,:-.- - - 1
1+
k

IS:.

z
i \,

This efficient envelope is used to evaluate the relative inefficiency of the economy given by
11
the actual output and input data (y 1
2 ) in the following input oriented DEA problem
0
S.[.

suh7ec1 to

°. ,

= .1

µ

Piµ
(8
B

=z

+B ., +s

Xi,XnS: S 2 .s, 2'. 0
where we usek desirable outputs, (p-k) undesirable ones, and rn mputs. Values on the right
hand side are g1ven. This fractional program can be linearised by a simple transformation.

The macro-eco-etlic1ency models (5), (6), and (7) can be related to the micro-eco-efficienc.~
concept by reformulating them as DEA models under a particular assumption: Construct a
fully eco-efficient economy that can serve asa standard against which the actual economy is
evaluated. While the microeconomic application of DEA uses inputs and outputs of different
independent decision making units, the macroeconomic I-O model uses data of usually only
one economy composed of interrelated sectors. To have these sectors performing the role of
independent fim1s one needs to explore the potential output of each sector given ali other
outputs and mputs. This generates as many efficient artificial economies as there are outputs
and inputs. These will establish the production possibility set (or the input requirement set).
Therefore, a multi-objective optimisation problem is fonnulated in which each output is
separately maximised subject to constraints on the production of other outputs and required
inputs, and each input is minimised under the same constraints. Denoting hy s the vector of
k slack variables of the k sectors, by s2 the slacks in the (p-k) undesirable output relations
and bv s 3 the slacks in the m input relations, the following model is solved p+m times for
given ·values of outputs and inputs to obtain the maximal values of each slack variable:
max s
s.t.

(I -A11JX1
Aw'X"1

-(I

-

A12X2

-A22)X2

B1x 1 + B2 x 2 + s 3
XpX2,SpS2,S3

-s1

= Y1

+ S2 = Y2

(9)

=z

2'. 0

j = l, ... ,k,k + 1, ... ,p,p + l, ... ,p + m
Individually optimal outpul and input values are calculated from y' = Y1 + si, y\= Y2 - s2.
and z\= z 3 - s 3 , and are affanged to forma payoffmatrix P.

l

Y1 + s:

P = Y2

-

s~

z - s~

Y1 + s~
Y2

-

s;

z - s~

···

Y1 + si+m ]

"·

Y2

.. ·

z - Si+m

-

Si+m

This matrix is used to establish the frontier of the production possibility set ( or the input
requirement set) and thus, the efficient envelope.

8

2'.

P.,µ - Br ::; O
-Bz ::;o

(1 O)

µ2'.0,02'.

The efficiency score. 8. gives the proportions of ali input of the economy which must be
achieYe the given output le\ els. -0)
sufficient - compared to the production frontier
describes the necessary reduction of ali inputs to achieYe the efficiency frontier. Vector µ
provides the weighting pattem in the projection poinl of the efficient surface deri \ed frnm a
radia! input contraction.
It has been shown in Luptačik and Bi:ihm (2005) that the minimum rnlue of 0 is the same as
the minimal rnlue of'Y in the LP model version B ofproblem

References
Brunn H. and Rentz O. 998), LCA as a decision-support tool for product optimization,
ch.17 in J.E.M. Klostermann and A. Tukker (eds.), Product lnnovation mul Ecoefficiency, Kluwer, 1998
Chames A., Cooper W.W., Levin A., Seiford L (eds.). 994), [)ara Envelopme111 Analvsis.
Theon, Methodology, and Applications, Kluwer Academic Publishers. Boston.
Fare R„ Gro~skopf S., Lovell K, Pasurka C. ( 989), Multilateral productivity comparisons
when some outputs are undesirable: A nonparametric approach, The Review of
Economics and Statistics 71 1)
Korhonen P .. M. Luptačik (2004 ), "Eco-Efficiency Analysis of Power Plants An extension
of data envelopment analysis" European Journal of Operational Research 54
(2004), 437-446
Luptačik M.. Bi:ihm B. ( 1994 ), "Reconsideration of Non-ne~ative solutions for the
Augrnented LeontiefModel", Economic Systems Research 6, No.2, 167-170
Luptačik M., Bi:ihm B. (2005), The Analysis of Eco-Efficiency in an Input-Output Framework, 9 th European Workshop on Productivity and Efficiency Analysis, Brussels, 29.
June - 2. July 2005, Belgium
Patnaik S. (2004), Emissions Allowance Trading, with a special investigation of initial
allocation mechanisms. Master Thesis, University ofTechnology, Vienna
Tyteca D. (1996), On the measurement of the enviromnental performance of firms - a
literature review and a productive efficiency perspective. Journal of Environmental
Management 46, 281-308
WBCSD (2000), Measuring eco-efficiency - a guide to reporting company performance,
June
WBCSD (1995), Achieving Eco-Efficiency in Business. Report prepared for the Second
Antwerp EfficiencyWorkshop, March 14-15,

9

ON GRADIENT FLOWS [l\J GLOBA! OPTIMIZATION

Hubertus Th. Jongen
RWTH Aachen University (D) and Maastricht Liniversity (NL)
Department of Mathematics. D-52056 Aachen. Germany
jongen(0rwth-aachen.de
Keywords: nonconvex optimization, globa! optimization. transition points of first ordeL
gradient flows. min-max digraph. adapted metric. globa! interior point approach

Extended Abstact \Ve consider smooth finite dimensional optimization problems w1th a
compacL connected feasible set M and objective function The basic problem, on which we
focus, is: how to get from one local minimum to ali others. Transition points of first order
of index play a crucial role this context.
(i.e .. Karnsh-Kuhn-Tucker

Firstly. we consider the unconstrained case with moderate asymptotic behaviour of the
is shown that (generically) any two local minima can be
objective function at infinity.
connected via an altemating sequence of local rninima and transition points of first order. ln
particular, the graph with local minima as its nodes and first order trans1tion points
representing the edges tums out to be connected. On the other hand, any connected ( firn te J
graph can be realized · the abo\e sense by means of a smooth fonction of three \ ariables
having a rninimal number of stationary points.
Secondly. we introduce a bipartite graph r as follows. lts nodes are formed by the set of
respectively. Given a smooth
local minima and maxima of the objecive function
Riemannian (Le. variable) metric, there is an are from a local minimum x to a local
max1mum if the ascent (semi-) flow induced by the projected gradients offconnects points
from a neighborhood of x with points from a neighborhood of y The existence of an are
from y to is defined with the aid of the descent (semi-) f1ow. Strong connectedness of the
digraph r ensures that, starting from one local minimum, we may reach any other one using
an altemating way. In case that no inequality constraints
ascent and descent traJectories
are present or active, is well known that for a generic Riemannian metric the resulting minmax digraph ris indeed strongly connected.
However, if inequality constraints are active, then there might appear obstructions. The latter
phenomenon is due to active set stategy In particular, we show that r may contain absorbing
two-cycles. If one enters such a cycle, one cannot leave it anymore via ascent and descent
trajectories. Moreover, the appearance of such cycles is stable with respect to small
perturbations of the Riemannian metric.
By means of a globa! adaptation of the metric involved, the appearance of absorbing cycles
may be prevented. This adaptation makes the metric singular at the boundary of the feasible
set and it is automatically performed by means of the constraints. In paiiicular, the interior
of the feasible set is now invariant under the corresponding ascent (descent) flow. The
resulting bipartite graph [' becomes (generically) connected. Finally, the underlying ideas
can be interpreted as a global interior point approach for 11011convex constrained
optimization.

10

11

References
H.ThJongen and A.Ruiz Jhones: So11li11ear Opti111izatio11: On 1he 111111-111ax digraph wul
'alculus of Variations and Differential h1uations (Eds. A.loffe,
globa/ smoothing.
S.Reich. LShafrir), Chapman and Hall' CRC Research Notes in Mathematics Series.
pp. 1 9-135 (1999).
CRC Press, Boca Raton.
Vol 4

SOME RESlJLTS DEALING VVITH THE ALGEBRAIC
APPROACH TO PATH PROBLEMS IN GRAPHS
R,,bert :\langer
Matherna11cs. l 'niversr \ ,)f Zagreb
BiJenička cesta 30, 10000 Zagreb, Croatia
E-mail manger@math.lu

Departmen:

Ci-lobaf Opumization: On Pathlengths in ,v!in-i'vlax
2) H.Guenzel and H.Th.Jongen
Graphs. Joumal ofGlobal Optimization, VoLl 7, pp,161- 65 (2000).
H.Th.Jongen, P.Jonker and FTwilt:
K!uwer Academic Puhlishers

No11li11ear Opri111izatio11 in Finite Di111ensio11s.

1-LGuenzel and H.Th.Jongen: On ahsorh111g ncles in mi11-111ax digraphs.
Joumai of Globa! Optimization, Vol. 31, pp. 85-92 (2005).
(5) H.Th.Jongen and O.Stein: 011 the complexitr of equalizi11g inequaliries.Joumal of Globa!
Optimization. Vol27. pp.367374 (2003).
l6) H.Th.Jongen and O.Stein: Co11stra/11ed Globa/ Opti111izatio11 Adapiive <Jrad1ent Flows.
In: Frontiers in Globa! Optimization (Eds„ C.A.Floudas, P.Pardalos). Kluwer Academic
Publishers, Boston, pp. 223-236 (2004 ).
(7) C.A.Floudas, H.Th.Jongen: Globa! Optimization /,ocal Minima a11d Tra11sirio11 Poi11ts.
Joumal of Global Optimization (to appear).

in grapl1s cai, ge1,erajj1· Le fori,,111atec 1 and su, 1,,c
µat
Abstract. A wide \·anet.\
algebraH stn;n 11re is introcl11ccd wi1,JSC
abstracl
mi
p1irpose.
d1is
Fur
hy algehraif' 1nea11s.
pat L problem 1s charactcrizcd bv a
Eacli pan 1rnlar t.vpe
instances are , allecl
paper presents :,wL recelll. rcsu!ts dcalmg w1t ! , he
clifferent ii,s;,ance of t l1e S1rlld 1:re.
l he hrnt part of the paper imroduccs
algebraic approacL Lo µath problerns basecl on
The ubtair,ecl cowpos1te
proble1us in voivmg exphcit 1dcnt1ficaL1011
semirings 1,)rrespornl te, relativeh
of optimal pad1s or rnulti-c1iteria uptimizaLiou. The seconcl pan ,)t the paper ,icscnbes a
in,pieu,enLecl
disuil,uted algui!lrn, for solvi1,g µaLii
{a~
of <,i11qF1!,ers Tlianks
1, differen1 t,µes of
be
l

imizac101.

Keywords: direcr,ed grnphs, pat
distributecl alg,,ritlrnis

l. Introduction
rccl.:,:e
arnl ei;:u:1era: 1, pr,J.ilen,s.
, upi 1m1za1
Patl1
iirech"d graplis Sou,e r:xa:11p:es
dirececl,
of paths
to geHerat
paths u!'
are d1ecki1,g pa1h ex1stence, fincLng shortest or 1wJs: rehaiiie patl1s.
all pat hs. e1,
maxmu m, , apa1.
()f path prohlem < a11 be t reated separatelv au soivcd in.
EacL pan icnlar
hms LJ,. TloweYer. a u:ure effident approa< h 1s to estabLsh a geueral
declicated
ot prnulems. and H, use general algonLlw1s. Tlie lat1,cr
framework tur tl1e wlwle
int roducing a smtable absLract algebraK strncture.
can be ad11eYed
Few vanaHLs , he algebraic approacb for soldng patli problerns have IJee1, prnposed
[1,4,8,~ 1. Our favorite variant fo,rn 'l' m;es a structure whose mstances are ,;allccl
semirings. The approach frorn il' relies heav1ly on matnces and on analogies with
ordinary lmear algebra. Each type of path problem 1s formulatccl by using a different
semiring. Solving a concrete problem reduces tu computing with matrices over Lhc
corresponding semiring.
The aim of this paper is to present two recent results dealing with the algebra1c
approach to path problems. The first result is a method for building more complex
semirings from simpler semirings. The obtained composite semirings can be appliccl
to salve relatively complex but still meaningful path problems. The second result
presented in the paper is a specification of a distributed algorithm for solving path
problems. Since it works over an arbitrary semiring, the algorithm can be applied to
salve different types of simple or complex path problems.

12

13

r
1es aliuul Se.llid,1,gs a1 d
l.J' paper ,; ,,ga:,zed as
algehraH appt1,aci1 based ,
a,
exanq;1es
graphs Scc1 ,uL ; s1 ,ws
semirings can be used tu sol ve si111ple pat,h problerns. Sen 1011 4 introdnces eornposH,e
scnmings. Section 5 illusLrates agam by exarnples how composiLe senmings ca11 be apideEt 1Ecal iou
wit
plied to sol ve more JmplicaLed I asks. sud1 as qpt muzal
algodescnlies
6
Sechon
multi-eriteria
nr
, opt1rnal paths
ion
at
1,Pl
Lual
3(
all
reports
7
Seet10u
probiems.
r1thm for soivlllg paLh
1

,)f the algorithm. The fina! Seetion 8 gives a eonclusion.

2. Serniri.ngs and graphs
cyur algei!l'aH si. rw 11,re A ~em, n ny 1s a set r eq1
\,Ve start w1cl1 thc ueimn ,'Jll
, wluch liave Lhe follow
w,th two bmary operat1011s. v unrn 1 awi o
ive c is assc,( mt 1ve. left.assc,ciai
ancl
mg propcrties: v is 1dempotent. eomnmtat1,:e
elernem riJ a11d a J.lllt elezen,
a
drntributive and right-cl1stnbutJ ve over v tltere ex1st
ment c such that I for any x, ep v x = :r ep x = cp = x o <P E 0 x = x = x E V\ihen
cvaluating algebra1c expressi011s over P we will always assume that o takes preeedence
,wer v unless otherwise regtdated bv parentheses. Concrete ms;,ances , our algebrail
structurc can be found m fl.6,8,91. and some of them are a!sc, repeated m Table l

notation

- --~p~

f---- - - - ~ - - -

r
D

J 3

ER

n

/ ,j

1'5
n

1

6

'---I's

E:

R,

>

:_

o

<

ry
{ J.

{ CX.

X~,iJ

!ix*Uy

uas\.1uy_

{1c:,.-*ll·y

X

o
!J}

1
0C.

yE:y}

(/!

li'xE:_.,__yE:!J}

0

UxE,l

3. Solving simple path problems
u,1r algehraH approach t-u pat prohlems ft,r a ccr a11:
Now we are readY 1
we chnose a s:lltal)le se1t1ll'mg P and ass1gL ap 1;r,JIJI ,a:P
a
problem p,Jsed
are labeis so tltat t l1e adJacern \ rLatrix A of O is stable m·er P v\ie comput.P t liP s: r\111g
or weak l iosure A* or A. respecfr;eh arni read fo;rn rt 1 l1e soluL11m to t Le ,
i losnres 1s that 1 he first
problern. Tite diiiere1;ce between tile tw" L\pes
pal lis. while the secowl one restrwts t nut,-111.111 paths. l\,;1,P r Lat eal
into accz,nlit
type of problem reqnires a different sernin11g. althougli t lte uvernll prnblern strw trn·e
. l' co11 1 ams \·alues t haL are used w describe ar( µr,.,pen 1es
remams t he same.
are cowlimed to <lesen Le t hat
a
o defines Low the values assigned tu arcs
mides are
µa1 Ls eou11cn wg t he san,e pall'
specihes how l he \-alues
patL wliile
1

X
-

X+ y

l}

P.I:*
BI::*
.
'

y

.r

,iJ}

,ax {.L. y

X.,

<
1

X.

X

o

rniri{r y}--

:Hax{ :r. y}

R~ \
R {-

T'~

cp __E_I

_;;oy

xV .iJ

Illat nx E 1111 1ne diagonal, ep elsewhere), X 2 stands for X o X.
Here. J;,
3
he mat fr, opcrat10Ls v and o are denved from the corresponding
X for 1· . \ ~
as m ordman lmear algebra. provided that V is analogous
sealar oµerat 1011s
to the conve11twnal addition and o t the c011vent1011al muluplication.
Te 8Xpressions i , and ( 2 i are called t he str1my and ·u·eak riosure ot X, respectively.
is s table. rnealilng Lhat : l ancl (2) become
lu most sit um ions t l1e imuived mauix
X can usuall,,.
Tln,s t ite z l'lsmes *
p(Jwers
sal urated afl er arl.diug
algel1raiz operauons. T\Jote Lhat X* and X are
a fiuite mm1ber
be
ca11 eas1h he uhtamed frorn the other one
t 118 seuse t liat am· of
eqnivale11t
:mp11ti11g t he stn,::g
Indeed. X* = F X. whil8 X=_\ X* Thus a1n
a:
dns,uP.
weak
the
zompllle
t
rnodilted
be
<ali
closurP
whose arcs are laiieleci witL 11u1:-zer,
,HsHler directed
this paper
Jesenbed hy ,ts 11 x
e,,· :s f1
labelecl
11-nude
AH
T'.
elemeHts from a
1
1i1e an fron: ,,;,•
lahel,
ltiP
is
of
e1l11T
J
:1
The
!
adJn,:er!/y mol1'i1·
herw1se.
ut
1_1
to node .J sud1 an ex1s1s. ur
patl1s m graµlis. i.e. seq11PLc·es , 1 011sec: jyp ar, s. A
\\'8
Iu pat li
ltoes
A pat is eienwn r·y
eircular pat
a grapL 1s (, r:1uectPd
llode
ead1
La1
;
more 1 han ouce. Fur co11\·eniern:e. is snpposecl
pat
a nuLI path co11taiirn1g
to itsell

{

}

l'aule l 'v\ieii lmuWll sellill'll,gs

finalh ( ornhilled t oget her
The first five examples in Table l are "extremal" sernirings dealing wifo real nnmhers, mfin1ty symbols and conventional arithmetic operarions The semirings P5 ancl
P8 usc "linguist1e" concepts, namel:v I: denotes a fini te alphahet and I;* is t he set
of all words (finite sequenees of letters\ over I:. C'unsequent1v. PiI;*) is the set of all
1s basecl on t he set umon lJ. and o
languages (set s of words) over I; The operation
011 word eoneatenat1011 * The symbol ,\ stands for the empty word. and 0 is the empty
la11guage. The semiring P8 deals with basic languages. vVe define an abbreviation of a
worcl w as any word that can be obtained from w by removing at least one of its letters.
For any language L C I;*, bas(L) is the basis of L, i.e. the language eonsisting of all
words from L that do not have abbreviations in L If bas(L) = L then L is a basie
language. B(I;*) clenotes the set of all basie languages over I;.
Let X be a square matrix whose entries belong to a semiring P. Then we eonsider
the following two expressions:

x*
X

E VX VX

2

VX

3

V ... ,

xvx vx v ....
2

14

3

(1)
(2)

8. ()_ \.

C

n,lde 2 ~ - - - - ~ n o c l e 4
•
/
-"i. 1.0. g
.....
/
-1 U 7

ty//

~
....

.<::

node 1

{

U.3. e

O.S. h •. -~- 0.6. j

C

':.L~

3, O.!:l, a.
____..,,,-,-~
. . . ___
-------------- node 3
9, 04, d

U

87

node 5

-3. 0.2. i

Figure l • A graph G with given are lengths, are reliabilities and are identifiers.
To illustrate our algebraie approaeh, let us eonsider the graph G in Figure 1, whose
ares are assigned triplets of values. For each are, the first assigned value (an integer)
is interpreted as its length, the second value (a real number) as its reliability, and the
third value ( a let ter from an alphabet I; = {a, b, c, , .. } ) as an identifier.

15

(; .e. we wa11:
'-iuppose t:,at WP want t,. SOJ\TP ;P :::aXl!LlUL :·ei:al
a1 µair
:,; detcrmme · Le rciiabiln\ •Ji a pa· w1:1 ::,ax1mun• re!Jai
w1Ll1
labeled
as
C,
aeal
arnl
lahels
are
as
rclialni1t1es
aI<
use
should
we
uoclcs. Tben
are
*
A
losure
l
strong
1ts
aud
A
matnx
aciJacenc,
lic scrnir1ng 1-'.1 lhe corrcspc,ndmg

o

u

,_,

A=

1) ~

II

l

u

U.;J
l.O

0.4

o

Jl2

lJ
1)

O.S
U.2

u U8
o u
u l,

A* =

U.4
07
1142

U.48
l.O

O.fi

IJ.4:32
O.~
IJ6:l
O oU24
048
1
0.6~l
Ul
1)378

U.72
O .SlJ4

eLt
: ne 11·Pak dosl1rc A ,iver P8 ,1sts ali non-null elcAs we an see. tne 1 .J
uode I to n,icle
me1il an µat lis
CorreCl ness uf 1 he abuve exawµles 1s g11ara11t.eeci lw sou1e 1 lieorems from [l i\lorc
precisek. 1hose theorerns assure that the mvolved adjacenn matrices are stable. so tlrnt
t hus giYJng the desired soluhons SLability in 1 he case
their dosmes can be
C: has nmmegative
the fart that ea(h eleme111.an
slH,r1est
lengt

U.8

O 504

4. Composite semirings
Se(\
Lave sLown si ·ellgl lis a11d ,vcak1,esses "f rl1e dassical scm:r
The
iS
exrrernal seuwmgs SLWli as F'2 I'3 . P4 nr ['.
l. B.\
iugs
i)ljP
1deHtifY paths 1\here
possil 1le tu
JL li1,gust1, sc1:
iier 1m11d. i
those "vt illlal ,alues are acltieYed.
are not necessan;1
pachs.
ings sud1 as l '6 or F8 oae ca11
! 1,e
si
Tal,le
Yahes. tlie seniJr11tgs fr,
op1 irnal. IH order to ge1 1u1ll'e
cornhiaed. :\ow we will shtJw how 1his ,an be ,1,e
a11J ;c;. ,et d1e zer,, elenie11L Jf
Let P be a serniri11g wli,ise hir;ar\ ,,perat 1011s are
hat
t
Suppose
E.
elemenl
unit
the
P lie o and
1

\. { a pa1 frun1 11ude 1.
A* 1s 8qt.J.al u 1l1e t1Laxi1n1
c 11:r\·
c;
uucic J !\',iti! that aualogously we could also sc,lve 1 he shonest distance
he
:
arni
lis
lengt
an
Tliu; 111steacl ui an relialdi11es a1,d : Le sen:iring P.1 we shciuld use
SUllll'lllg P2
he11 we ca11
ali eiemeHtaiy paths
\cx: let dS suµpose Liia1 wc walli :1, 1deHt
,et I:. aud one• lener words
iccrprct are 1dcu11hers as "ue-lener wrn ds tr,er l he
as uuc•v,)rr! languages. h1 il11s way. we ca11 cons1der c; as labeled w1th tile seminug !'8
'll ,1Lijacc11L.' rnatrix A alld 11s weak d,)snre A are : he1; : he followmg
;1c

I

j

·t

A

1

0

0

a

(/)

i/J

{b}
d}

0

0
0

{C}

0

0

{f}

{h}

1/J

i/J
i/J

0
0

1f

l

{e}
{g}

0

}

t

ad.
aeb.
afJgb

A

{ bae
bafjg.
cg.
che

d, }
{ eb,
fjgb

rb,}
hd,
heb

Jf

ae.
afJg

J

f b. 1,
l chd J

{. 1

l

(
/

l

1

J aec ~

\

l
J

f

J

l

da.
eba,
ech.

fjgba.
f1,
fjh

C

{ baf }
chf

J

11

Lhe uperatio11 o lias the "cancella1 iun proper: \.

J

{;~}

{ g~a,}

Jhe

16

gc, }
{ gbafj,
hec,
hfj

{ g~:f,}

For all

:x x' ry.f);

E

Pz;,I'

(x, xJ

v

iy. f})

i

{

if, }
jgbaf,
jhf

f=y

{

X. i:)

(y,f})
'x y x

x. !J z E' f'
Z=

i:Vy=x=ffj
x\/y=fjf x
~1

f);

if

x= y

(fl, f)l

It has been proved in [7] that the proposed v and o are correctly defined binary

operations on the set P 0 P, and that P 0 P with these operations really constitutes
and unit element is E = (E', i). The newly
a semiring whose zero element is q; = 1~'
semiring. The paper [7] also gives some
composite
a
constructed P = P 0 P is called
in practicc, and which can assure
checked
be
easily
sufficient conditions, which can
examples in the next section will
The
stable.
is
P
0
P
that an adjacency matrix over
previously mentioned drawbacks
the
eliminate
can
illustrate how composite semirings
of simple semirings.

J,

{ i~c,}

1.e. for all

Let P he auy other serniriug whose hinan opera, 1uns. zern aj 1'rnt eiement are cle1 wt,ccl
o - E. respecLJ\'elv. The11 we ( a11 111sLru(, a set P = P Q<'. P au l w1
wiL11
aEd s. i11 l he
operations.

{ f 1

'

•

ZOT=Z"Y'

®

{:~ }

aJi f. y E f

,.e.

1 he

(x. x) o

l'd, pq {jiH
~eb,
Jgb,
jhd,
jheb

f baf j. \

1

(Jperation v is a ·, h01ce opera, 1,

•

{ af}

t afJ)

{ ba [
, eh

j

{ f~g}

a

1

17

r

5. Solving complex path µroblems
cxpl1, 1: 1de1,; :'.1-.at H•L of 1!pt1rnai patlts iit graphs
Cuwpos,te se111i11l!gs ,;aL ;Je app;;eei
sem1n11gs 1aL l,e
Aii1,:i1er applicaLWL rn 11,u;L;-u,:c:m ,,ptu1,1La,1,,i; Cumpos1,iull
1tera:ed :11 LillS ,vay olle ca.ll acc111uplisL u,lii1 ;-( nu,ria upllm1zauu11 wi1li explicit ide11; 1hcatw11 oi pat!ls tltat are "!JU;;al anonlmg tu several cnteria
St,I:lt' co1,1rPl,P appi1ca" 1,,1:s arP Lsred 111 Table 2. .\ll proposed comp<)SJi 1or1s are
basecl on simple semurngs from Tal;le 1 There are of course manv utlier poss1hilities
fur , umbi11111g s1mple semJrmgs wlrn h are not listed due t<, space limit at iuns. Table 2
1lesu1bcs lH,,v a g, aph shc,i,!d bc labeled w11 li a pan i( ular semiring. and what results ,au
1
,e ui au,cd 'n curnµ:tu11:!, •.Le s1n,rtg ,;r weak dosure uf 1he ltJ1Tesp01Hlmg adJace11n·
n,a:., :x m rhat scrun,,g. 1f 1 he g,veu as:,mmptwns ahou1 t hc graph are satisJiecL t Leli
fcasib1l:ty and correctncss of 1 he mvolved computar 10ns ( an he guaraHeecl lJy r7

more detail, let us consider again
lo dl11st ra: e 1 Le appi;( aL1ous fw,;: la' ,;e 2
the grapL c; rn Figure l \\ bose arcs are ass1g11ed le1,gt hs, reliabilities and identifiers.
an leHgt hs and ar, 1dentihers. and treat G as labelcd with
Take fast iuto an uullt
P 2 ,gi P6 . The11 l he adjace11n matn:, A arni 1Ls strung closure A* are the following:

:x..O

.:x,.V1;
i-1. {b}:

A

X..~)!

rg {d}1
(rx.

! l. {e} i
!-5. {g} i

0·

1

(

(

' }
'1)
'. JtA

\-LtbJ
Composite
scm1ri1,g P

1

'(\,mpr,nenLs ,Jf the
, 1 J ,-th e11t.ry of the
dosure matrix A* ,,r A

Assl,mptwns
about the
grapli G

Componems uf 1 he
1. 7 rth entn ,Ji t he
acljacency matnx A

a1n ele1ac111 arv
cyue l,as
positive
lengr,L

- le:tgth of are ·,,. J •
:de1111her u! an , 1 J ·,

Ll1ere are

~gt h 1 ;f a11 · 1 . 1
- 1den:ifier of an :. 1.;

llCl ,

·,rr

!8S

1e1

1

- shortest d1stauce
frum Hode ? to nude .J
- 1dentifiers of all
sliorLest pat hs fo ,;11
nurl.e I t,, nocle J
- --1
lo1;ges1 distance
frorn 1iode i to 11< ,cte ,;
- irielll ifrers uf all

lci1,gest ('rJt1cal; patl1s
frurn ;10de l to 1,1 ,de J
1
re::a n:i,\ "' ar, '1 j' t--::- maxirnurn reliahilit,: -1
am c.e, 1e111 arv
1
of a path from
- 1clell' ifier 0f ar< 1? j ,
1, vele ccintains
uode 1 to node ;
an are wn h
- 1dentihers of all
most reliahle pat hs
frc,m node 1 to node J
1--------~+- ---- short,est distance
P2 (8) · R1 @ P8 \ any elementarv - lcngth of an -i .?
from node z tu node
reliability of ar< i. j ,
1 cycle has
- 1dentifier of are 'i. j \ - maximum rehab11it.v
non-negative
of a shortest path
:
length
i from node i to node ./
1 - identifiers of all
shortest paths from
node i to node j
1
achieving maximum
reliability

A*

(

\ {h} 1

i'X)0i
! 2. {J} !

rr)

X),

01
{f},
'4X. 0,
X 0
i

1)
fJg J

\\ X.

)

(

{ b a f.J )~ \ 16
\ '

baf

) )

c

(
1

!)

r,

-'i

)

( 6. { fcJ '

,ll.{,\},

i , ·•f g b a } ·)
f\ g f) '-,).
( .. "f
i
( ·J., 1
l { Jg})

)

17

;

/'

,J

ll. {,\}
!i''

(2 { J

/

l

af 'r \ i

(

g . Il af -'1 r:.

3 { a}
! 2 fl ba 'lJ
; .

i

0i

'X)

1

11.

~.

i

1

'·

f J;

)i

gbaf

1

1

J

i 1. { \}

1

'

In accordance with Table 2. the iz.J.1-r!i entrv uf A* g1vcs tl1e shortest d1s1,a11ce in1m
node i to node ; . t ogether w1th the identifiers of all corresponding shortest pa, hs
As another iliustratioIL let us cor,sider once more the graph (; m F1g 1_u-e l bu1 :10w
take iutu accourn ali values assignecl to its arcs. i.e. an leng: hs. arc rehab1l1t ies artd arr
ident iliers. Each triplet of values ca11 be regarded as a label from P 2 131 P4 >;i 1-'8 , The
matnces A arni A * 11uw look as follllWS
r

A

x.. O. 0,
1. 0.7. {b}

g U4.{d}
(,X

o \/)1

( ixi. (J

l,'XJQ.1/J,
, x.il. 0
UU {e}1
i-5. 1 O. {g}

0)

(\~;} )
(~{t)
()

1

18

;x.0,

e

{

\ 6. l gb})
(-4. {( Jgb} J1

1 -

Table 2: Some applications of composite semirings.

(o {f~:b})
[

0'
{c},

X.

8.

:x;\J)!

( 4 { ae
afJg
\

,\ f ;

!

l

:-3. {i} 1

ix.0

r1

{a}

l,X.1/Ji

1

{x 01
;U.

')

1 d.

\

i

\

U4:-l2.
{afjg}
0

J

)
io,
{,,\}

(

1

( 0336,

)

{fjgb}

(

0,s )
{fjg}

(~i;~} ) (~i~l l
( ~~2, )
{jgb}

'

, )C. U. 0
, x..ll,01

:s. 0.5, {h}

XJ.

u, 0
(

!

XJ. 1),

1
1

7

:cU 1/J

{d
0;
'XJ, U. 0·,

X !l 1

'-'.1.0.2. {i}

O.0r

( 4

0

A*

OG.

3.lJ9.{a},

2.06.{j}

X

.IJ.0

4 il X { f}
'X'

11 (/)1

)C.

O. 0j

1

J

l r;)72 l
l (6' 1
(\~1 l ( ciAs, ) ( ~s. l

r

\

00.9.

1

)
(ci63,)
'

{a}

{ba}

(;,m

{afj}

( ~3024,
{bafj}

{fj}

( ~~3, )
{gba}

(~~. ) (~i,)
{i}

{jg}

19

,

{af} ,

0.504, )

{baf}

{f}

(\~\)

( ci 504,

( ci 6, )

( o; o.

{j}

{,,\}

)

l

{gbaf}

T: acr,)rdau:e wit1, TahlP 2. rP 1
LP
u :r
n(,dP I
SLort,cst rlista1wP fn
bv r1011sicl<'ri11g ,mh shortest patls fnnll nudP
those shortcst pa1 hs fr, ,rn nndP 1 11) w,r!e

*

ams r i,ree vat,1es : LP
1 ha! , ai: 1e achfeved
rP :at
Hlent1tiers nf
t 11or'!P / all(l t he sel
t bat achiPYP t he maximuni
1)

I

6. A distributed algorithm
sokiug paLh pruhlems is
Accordi11g to the adoptcd algebra1e approad1, an algurithrn
hm sLoulr! be geHeraI
s1mply a pr<Jccclure t llat co111µuLes a mauix dosure. Such
1o
awl laH 1herefure be
t Le seuse l i,at it w, ,rks u\·er an arl!ll ran
( urnputes
01 pan1cuia1
S'll1µle ,ir ( uH:plex pa1
u1ffcrc11t 1,ypes
A over a semiriug P. The express.io11
: lie s:roug closure A* , a gi\·e:: n x n wa1
uy uf a srntahle matnx
,icahJe sq'uuril!y unJ
tlie fc,rn1 11 1s Pvaluated
Thc alg,irit lrn, s: ans w1ti1 lJ = C A am! sh ,ps whe11 JJ srnl >1li7,es
cow urrelll
Our algor:t Lm 1s d,stf·lbuted , t ae se11se 1liat it cuusisLs , a ring
µroccsscs, wbcrr l S m S n 2. EacL process c<;mrnurncaLes wi! h ns predec:essur awl
each process has 1ts uwu
its suc,1essor aiong the rmg. T 1,ere 1s m, shared rnemorv. i
µrocesses
4
=
m
private memon F 1g1tre 2 refers ti, a ring
of the l!,al nx
lm; mairit ains 1w,; ,,
mg 1 he alg,
T,, mrn 1 le distnh;,ted c,
B deuoted h, R = ,r aw! C= r:; The L,aai:,; n iS cfr,1ded iutu 2m blucks. se, that
(' is divided
ad:a,:em rnws.
cvery block consists of a roughl\ equal ,urnber
intu 2m blocks of colurnns. The range of r(ilumn ind1ces ass1gned t one panicular
1 he rnrrespondiny l,lock
block of C 1s the same as the range nf rnw id1ces ass1gned
of R The blocks of R ai1d are r!1stnhnted amcmg processes. su that uae prucess keeps
cxact ly two blocks ilf R and the cc,rresponding two hiorks of ('

n.

uiocks ll1 01rle1 l'; ,urm 11ew combinations
Af1,e1 a11., plase 1ue
C rw,\·es ugPll,Pr w;t lLe corresponding
piiase.
tile
hii,cks
F1gure 2. for L11e case whcre
!{ ThP exact bluck excha1ige µrucedure 1s
blulk
111 = 4. Rules l and II are altemateh apµlied. Ead1 mdex rn Figure 2 Lie11otes one block
Figure 2 mdicate block
blork of C. Arrows
H togetlter 111th the
5 tliat for a stable matnx A d1e algunthm termmates
1w;\·es. lt has hee11 prnYed
h d,e lrnal rualrn:es R all(l ( equal to A*
Pratiuns.
ier
e
afrer a

7. Network implementation
Pd as a dis:rilmted C' program by us1ng
, a11 eas1 h ilC
cude bas liee11 r!esigued su
performaw;e
j(l
li1
soh'e Yarious I ypes
ncrat :uLs aui rec,irus 1rs \J\\
t he µrogram Las heell Pxtendecl s,, 1Lat
time.
execul
ucesses. wllld1 ca1, lie al
c·i:)lLZ·111·re11l
dist rilmted prngraw rm,s as a set
,rk m a
t he ne1
PxchaHgP da, a
difforern cumputers. and
luca1,ed
ual pa;aliel machme· asscn,
a
rrng·like fasliiun The program Las Lieen 1,es1.ed
e !ia\'e heeu ahie tu 1u11 tl1e prograw w11 h up t
L>i!X compui;ers.
bled uf
concurrent processes.
few l11mdreds oi µa, proD1t Ii,S
Lhe prugrarn Las beeu rneas1:red
Performance
\>l d1ffereut type and size. The 1tamed periurrnauce measmewe11Ls are g;vell m 5
small par1 { 1 l1e measuremellLS rs also prese111 ed l,Pre

r

de11si1y

H11ie

l
f---1

- --, 7

2

"i

f+-----1

:l

o

4

L

X

l

F1g1.:re 2 Block exchallge rnles

Til=

2

4.

As previously explamed, the algoritlim consists uf iterat1011s. However due tu distributed computing, each iteration is further cb ided intu 2rn - l smaller parts called
phases. In one phase a process generates row-column pairs from its locally available
blocks. In each phase, all possible pairs from 11011-corresponding blocks are generated
In the first phase within an iteration, additionally. all possible pairs from the corresponding blocks are also generated. For each generated pair. consisting of, say, the i-th
row of R and the j-th column of C, the process computes Lhe "inner product". The
obtaincd value is used to update both the j-th element of the i-th row of R and the
i-th element of the j-th column of C; thus:
:=

C;j · -

v

10%

30%

X2. i

46.:1
+---316.7
164.7

8

49.9
324 8
167.2

4

8\J 9

8

51.0
374.3
191.6
101.9
57.7

4

8

T;koCkj·

(3)

k=l

20

l il!l
l

4

1
2

1u

l"

Lel

88 ::1

1
2

sµeedt,p cumpared

::š.'56
6.J2
---1.00

21

-1

l.\12

3F:i9
o.35
100
l.94
J.61
6.37
1.00
1.95
3.67
6.49

Table 3: Performance for shortest distance problems with size n

n

1';j

111

2

f---1
f---1
f+-----1 8 h
~

f---1

prucesses

1utal execut 1m1
t iH1e rn secornls
2\)2.7

= 256.

Tarie I s1:rr1:narzes t>ie per 1,,rn,au:e data for 411 ra1,d,
geaerai.eci sLonest rl1s0
t au;e pruhlcms uf the same s1ze 11 = 2'ifi. TLe prohlems lia\·e l,eell furt her rl.1ndeei
gr,li1ps of lil accc,rding n thelT graph Jens1t\ percemage
mm-zeros
A, wlud1 cau
be J%. 3r;1c.. 109i,. or 30% Each partu uiar problem Las beeu solvecl four t imes. 1.e. w11 h
1 hc rn rn,h<'r uf c,1rn 111Tcnt prncesses
1u 1 2. 4 ur /\. Em ries
Table are mean
\·alaes r·rimp1,tcd 1,\·er a wlwlf! grnnp
10 similar µr"hlems A table row shows the
a; f'XCC1ttin11 tnnc. exprcssed as an absolme value secouds, a11d as well asa relatiYe
value \spcedup ,:omparecl tu Lhe seque1,tial case 1.
Tahle 1, lcarl,
licatcs 1liat the alg"ri1Lm 1s
efticient whe11 appiiecl Lo larger
sLurtcsi. Ll,s:a;;ce tin iler:1s. Sa\lslact
speedups are ad11eYed 8\'8.ll
X processes
'crfon1,alH'" 1s ictter t,ir deJJscr graphs
8. Conclusion

111 t his papcr wc Lave prcsented two results clealwg wil h lie algehraH approad1 l o
seminugs intu
pa: h prublems lhe ilrst resull 1s a med1a111srn lur
fur soivillg pa1
cc,111pos1Lc senurings. Thc second result is a d1st rihut.ed algorn
;;rnblems
Both resuits extend the appiicab1hLv of the adopted algebraic aµproach :\ amely.
w1th cc,rnposite semirings it is poss1hle tu use the same algebraic formula! 10n nut unl:v
for srn:µic pruhlen1s tJUL also tur rwJre
tasks. SU(ll as explicit identi±icatior1
ui optirnal paths or multi-critena optimization Un he other iiaud. the described
d;stnbuted algonthm allows solving 1,he whole rauge nf sm1ple and cowplex problerns
a J'anct.; uf ,:on,p.,: e! 11etworks
The 1c.ica ,JI ext-.cr1cl1Lg : 11e appb ai
ul Le aigehra1t appruad1 Las an nl;\ 10us
aesthct,c a1-1pea1. i\.ioreover. the descnbed exte11S1ULS hrn1g some yrall 1cal he11eiits Fur
.,1s\,au:e.
becnmcs possi11le h, sol\'e a wir!er dass,
>len,s hY alreadv k11uwu all(i
tesiccl algoritl1;11s. nr the same dass ,f pr, ih'.ems ca.t1 he so!ved more effo 1ern
emplovi11g rm,rc C<Jn:pu,,ers

ON SOME TRAPPJNG STATES TO OR
by Viljem Rupnik
!NTERACTA, LTD. Business !nfonnat1011 Processing
Pannova 53. Ljubljana
e-mail V1ljem.Rupruk@siol.net

Abstract
There can always be some hand1caps to OR to be successful. but to remam honest, we are not
persistently entitled to blame OR users for not bemg effic1ent We are first obliged to recog111ze the
problem in its intrins1c scope and essence: consequently. we also have to fmd an efficient solut1on for
any OR trapping state .. This very last task 1s in our focus in th1s paper

Keywotds decision theory, stochastic and chaotic processes, parameter activated
optimisation, flexible economic modelling, entropy measurement

An lntroduction
There are two significant and crucial scientific disciplines which influence pragmatical
reputation of OR gnoseology and hermeneutics Both of them should be taken into account
by OR problem solver: the first one determines his problem solving procedure and the latter
one shapes the scope and depth of its impiementation in real business lives Thev internet
strongly, ifwe are not paying enough attention to such an interdependence, it ~ay well
happen that our OR produce will not be accepted by practitioners/managers/users lf such an
event persists again and again, we are due to check it asa trapping state to OR We assume
that resposibility is imposed on OR designer; we picked up some most chalenging reasons for
such trapping states.

References
l ( :arre. TL C1apils and Netwurks Oxfurd L mversi\\ Press. Oxfurd. 1979.
P\'M. Paral/el \/,rfoa/ Muchuie A r sers Gwde awi Tuforrnl
1
Vetwurked I'arai:e: Cornpu.t-mg. The MlT Press. Camlmdge MA. 1994
3 Hu, T.C Shing. M T, Combinator1al Algonthms Duver Lon<lon. 2002.
4. Huckenbeck l1 E:rtremaL Paths in Graphs. Akademie Verlag. Berlin, 1997.
5. l\langer R "Solving path problems on a network of computers" fojnrmat'lLO. \'ol

Embedience problem

2. Gc1st. A et a]

26 (2002) 1 91--100.
6. l'vlangcr, R, "A new path algebra for finding paths in graphs". in Lužar-Stiffier.
V., Hljuz Dobric, V. (editors), Proceedings oj the 26th International Conjerence on
Injormation Technology Interjaces - ITI 2004, Cavtat, Croatia, lune 1-10. 2004.
University Computing Centre, Zagreb, 2004, 657-662
7. l\langer, R., "Composite semirings and their applications dealing with path problems
in graphs", Grazer Mathematische Berichte (to appear in 2005).
8. i'dohri, M., "Semiring frarneworks and algorithms for shortest-distance problems",
Journal oj Automata, Languages and Combinatorics, Vol 7 (2002), 321-350.
9. Rote, G., ''Path problems in graphs". Cornputing Supplement, Vol 7 (1990), 155-·189.

22

When solving some well defined problem, it may appear that the solution obtained Just not
satisfy the management, although it is perfectly defined from the mathematical point of
view. As we shall consider various potential reasons ofit later on, it is safe to start with most
common issue on so called embedience phenomenon
To illuminate it, let us start with general decision theory and its fundamental equation, whose
constituents are as follows:
let X 11 be an arbitrary finite dimensional vector space, representing ali conceivable
decision variables

X E

xn,

let ~ be an object of decision making process (which, in general, is not a problem
itselt);
there is always at least one consequence y E Y"', again from finite dimensional vector
space, corresponding to input x E X" ;
we also introduce an operator Y = SX producing output from input; let us call it as
decision-generator;

23

r
we usuaHy simplify »our« space of consequences to be Z=(LS)X and thus
1
X=(l
Z (if possible"! and consequenth S({ 'S) it means that a) we neglect
any alternative solutions, b) we neglect admissible solutions. c) we are too bold to
assume that ali inputs could be solutions:
1s really known us 9
is U really known to us' 1
the two operators and S in FED are tacitly assumed to be uncertain: do we use it as
deterministic operators'7

X should serves asa space of adm1ss1ble decision variables, X"

"
let z E Z be an estimate of a consequence of decision variable

E

f,,,

an

arbitrar\ dimension)_
there is a mapping { ' of
E Y,,, into z E Z ,
let q = (y .y,,,) E (2 be a primary (backwards) construction of a problem, based on
,

1

\TT, under the conditions of certainty:
as an analogy, let q = (z I z,,,) E Q be a secondary (backwards) construction of the

are we sure that /1Q is sufficient for our decision on X• J )
a similar doubt as to the operators and S to hold true for all other operators in FED ,
a transition from (2
Q has not been e:xamined whatsoe,eL 1s a reflect1on of ow
dangerous oversimplification of decision proces,
a bridge between hard and soft sciences is often demolished bv usmg 11= identit\
operator which is. in general, very far from being realistic and adequate approach_ it
may weil hapen that an/the solution E }/,, is acceptable from technical aspects oni\
but from the others:
are we not worried about making some adequate snapshots of \TT
get problem
space Q, oc even worse. Q0

same problem, based on 9t under the conditions of uncertainty,
.
.
let <1> is a mappping of Y into (2 ( a primary generator of problem constructions ); m
real situations. this mapping reduces (2 into some part .:1(2
for the case of uncertaintv. let 4' maps Z imo
produce some shrinkage L1 of

Q

here again, this mapping may

Q,

the shrinkage of Q is then projected onto z• c Z via operator n.
S is an operator uf induced subspace „d of alternative admisible decisions, being
mapped from Z' c Z
Based on these minimum categories of decision making process the following graph might be
useful ( see/ 1/):

From this we see that there many reasons to fall into trapping state when solving real life
problems to meet the demands of managers What can we do anyway')
l when fighting against Y=,')X as input-output mapping, we have to consider three possible
distintions is there our problem deterministic. stohastic or chaotic {ultrastohastic) The
corresponding tool may bring less disappointments to both OR-specialists and managers
2.looking at evaluation mapping Z=UY the main target of ours is that the definition of utilitv
which should be paid to our OR clients: what are their measures of satisfaction, what are their
criteria of our solution quality, ect should be told by them to us in advance

(

n

········•,..

H······•....

r

·- .....•..

4. since the mapping y
"•·

Q

3. problem mapping Q= 'PZ detects our ability of perception haYe we passed through a
sufficiently deep information analysis of our problem9 As a whole. we are to relay on
information sciences as a confident and reliable assistant.

..........

?

X *c1 = (30Ll'I'US)X

A series ofpotential standard failures is as follows:

24

Y,,, practice is largely used (regardless of its quality ), the explicit use

of mapping Q • Q is almost totally neglected, although promising much richer space of
problems under uncertainty Here, we are in doubt, whether the reversed procedure is more
fruitful, such a dilemma reduces to our decision which principle of causality should be used
first.
S. the space

For the case ofuncertainty we derive the fundamental equation of decision theory (FED):

E

.!1Q of

reduced problems is very often an origin of misunderstanding between

managers and OR specialists: it lies deeply in a dualism of perception (its treasure being

Q=

'I'Z) and aperception (its treasure being LlQ) of uncertainty-conditioned problems.
Consequently, operator Ll is most obscure factor in FED; ifwe choose it as identity operator,
we exposed ourselves to a lot ofblames from the side ofusers.

25

6 mrhe above h1ghly simplified formulation of l:< ED an operator n represents an evaluation
solution to a problem under uncertaintv the reversed operator would describe our
conceptualisation of problem(s) in an indirect wav this might fil! in the space iiQ in the
sense of its gradual refinement (like auto-adaptive control processes do)
be the expert modelling whose task is

7 a straightforward by-pass of FED is trying
the mapping

/iQ • x•d

find

T

number of am arbitrary probabtlitv distributions Consequently a chaotic model can not be
processed in its full extem. but in its probabilistic representati, e (see
Let .1 =. T be the tirne domain of stohastJc proces s F( t ), streched o ver its empirical
realisatlons f(t ); let us furthermore introduce corresponding differential probability function
being the future ofthe same process (where F(t) appears as
(DPF) p(t) = plF(t)_t
a concatenmion from the past). and

we have to cherish it asa potential way to reduce OR failures

8 furthermore. an artificial intelligence looks for a mapping

/iQ •

( n, S) , where for

iiQ

as a subspace of Q a pair ( '¼', H) is applied (meta-mathematics)

An example let ITT=firm"s gross revenue per emplovee, then Q= {Qj, {2 6

=N

ll lF(t) =

economic problem, {J, 2 = administrative problem, (J, =psychological

problems, either Simplifying the case by taking Q

= Q. our promise to manager to run a

project on object 91 has then at least six dimensional criteria of consequences Z and S is
E Y"', the same nature is
six-component operator and Y is 6-tuple »criteriased« output
featured by 6-dimensional vector z E Z
how to find an/the optimal vector z E Z in case of its component having different
dimensions (it calls for multidimensional ranking procedure).
how to find an/the optimal vectorial solution to 91 problem, if dim 91 is different from
ali six dimensions of z E Z components (it calls for non-formal optimisation
modelling of \rt);
if we avoid the above two approaches, what is the mapping of 6-dimensional
»dimensionally non-congruent« components of z E Z into dim 91.

The example above (see for the details /2/) has shown that the embedience phenomena are
different and numerous: so are the potential reasons ofOR being trapped.

2. Stochastic/chaotic modelling problem

26

being anv fini te set of arbitrarv differential

"'.t=
L,':> . n •

x(t = F(

:=N

I;c; = 1
iEf,

rhere are the two exstrerns of'\1andelbroot dilemma
independent,
are
p(t) and
= p( t which is found as traditional approach of nai, e
they are identicaL
business modelling
= '¼'( p(t )) Thus. the point
Thus. our task isto discover the existence of mapping
serves as switching point from p(t) to TT(t) ln order to »palpitate« the future behaviour we
insert or implant some points offuture process and. according to balance between the two
extrema . we concatenate all the past points and the future points »chosen« The production of
future points might be leaned upon Monte Carlo simulation procedure and then ~e mix the
two sets ofpoints In short, the whole procedure may be called as i111plantatio11 what a set of
points should be accepted as a operandum of impiantation process? The simpliest way 1s to
use mjimal and supremal of processes both in the past and the future; in most of the pract1cal
cases, an additional information is useful, namely that of modus process as a turnpike curve.
In such a case we deal with 3-parameter implantation process. To simplyfy our exposition, we
confine ourselves to the above presented »3-bone skeleton« throughout the implantation
procedure. Such a choice says we assume that »a majority« of process lies inside the cone
build up by its skeleton, inside ofwhich we have no further skeleton-wise information ( see
the details. 14/)
To generalize our approach we allow to assure the prediction [F(t). IT(t)] for any t>T based
on implantation described above and obeying a constraint [F(t) -111 ~F(t) ~ F(t) +M], m
and M being arbitrary finite limits. The corresponding cone is expected at probability
ITI: (m

A transition from stochastic to chaotic modelling is not a trivial task of modelling in general.
To start with, we propose a rather non-conventional definition of chaos: the domain of a
variable is said to be a chaos, if and anly if there does not exist any probability distribution
(we shall sharpen this definition later on). Basic problem, then, when building upa model, is
to recognize a/the chaotic nature ofvariables in a model. From pragmatic point ofview, we
look for its approximation via some finite series composed by convex composition offinite

lf, ( x)

probability functions, in case, when this series is finite or infinite. but convergent the
the opposite case such a process is said to be
process F(t) is said to be stochastic.
11/trastochasric or chaot1c f'onsequenth in practice we ha\'e 10 deal with some c· approximation of chaos

where Q 1 =

problem, (J, 4 =technological problem, Q 5 =organisational problem and Q 6 = problem of social
policy Our object thus embeded six (induced) problems; is likely that each of these six
problems is going to be treated as an object again: embedience phenomenon streches over

I:c; = 1

= 1:fn;(x) , x(t) = F(t).

being the corresponding DPF and

to decrease the frequency of OR falling into trapping state
9 finally, of course. we do not dare to call for help a mapping

TT

Q and thus helps us

'¼'. H) •

~ x(t >T) ~ M)

or risk being p

=

fIT(t)dt + fIT/ t)dt

A series of prediction

.1.H

-c{I

impulses 11, 12, ... .. is matched to a series of Fo(l1), Fo(l2), ....... , m(t 1 ), m(t2), ........ and
M(t1), M(t2), ....... , thus creating a prediction skele ton. In practice, it is thoughtful to assume
it to be finite, having a length /; we shall call it a prediction window and denote as

·0;"(ti&r:'!Y.

27

Our task has been to discover whether prediction window tends to qualify the process
observed as stohastic or chaotic one In addition. ifwe simplvfy our discussion bv assuming
p=lm(I,) Fo(liJi= lFo(i) M(li).then 0,;(r~'!i}!)isachaolicwmdow,if 11.(/) for t
> T is divergent: in such a case a sequence of\ocali~~d probability distributions represent
.,(!)- approximation of chaos. Both parameters /1 an / are heavily dependent on p. Thus.
we have to
0
construct apriori probability space, out of which we slice-out prediction probability
subspace embracing the modus turn-pike trajectory of width 211 ,
0

find the corresponding space of apriori random variable. streched over / pairs of
supremal values F{-r;, IT(,,, + //) an infimal values F(, .TI(,,) - µ) respectively for
each , inside chaotic window

To resolve Mandelbroot dilemma we have to observe the rwo pairs [F, (, J, TT,(,)] and
[/, ft) p,{t)].

It is convenient to define '-11 = ::

IT - Pi

!! as a measure of prediction quality

The issue of prediction needs one more operator, say <l>, which measures a dependence
J-, ( , ).

After assuring the same domain we have <l> =

it stil! may not be satisfactorv The approach wh1ch promises some additional increase of
output quality (like accuracy, optimalitv. ect) could be rendered bv the concept of flexible
inputs They are allowed to change its content and not to keep them fixed over tirne. Ifwe
insist on traditional econometric approach, for example, the error acruing from the two
possible approaches may beto large to be endured in practice Following the scheme of
solution proposed below we can easily detect all dangers to OR ofbeing stuck in its practical
aplications
To introduce the improvement oftoday's econometrics we shall refer to simbols in Ch. 2. The
inverse of 3 is in question, but not in a direct way following traditional econometric modeL
the space X of decision variables effects (indirectlv) the problem space Q.· thus, we look for
mapping J:· • Q and consequently, it effects z• as our output space. For this purpose, let
P(t) ~ .Y-(t) be a maximum set of economic categories, assumed as stochastic processes of an
arbitrary type and having been defined through x(O) == m, and covariance matrix R0 lfwe
follow the concept offluid modelling (FM) (see /5/), we shall. in serace ofeconom1c
ii

in terms of probability It is computed inside the chaotic window provided we have shifted
the corresponding »past slice« onto the chaotic window domain thus, the recording of '-11
supports information on 1he relationship between the pasi andjuture stochast11;1ty (a
measure of Mandelbroot's dilemma) The left Mandelbroot has value zero, the right one
refers to infinity

between ji (, ) and

r

IV: -/III,

which could be considered asa measure oj predichon o.f realizahons Consequently, a pair Q
= ( <l> , 'l') is a measure of prediction q11ality, operator <l> being a criterion of risk ignorant
and operator 'l' being a criterion ofrisk relevant decision maker Finally, a complete
information needed by decision maker is given by 3 = [[F 1 ( r ), II.(,)], Ql
Fram this formulation, it is obvious what reasons for OR to be stuck in practice. In short, in
general the past stochasticity does not hold true in the future: moreover, ifthe future is
chaotic (being ., -approximated), the error produced by stochastic modelhng might be
disastrous.
For example, traditional statistics rests on aposteriori stohastics, since it deals
with regression analysis as a tool for extrapolation/forecasting/prediction of the random
process.

modelting, have l'(r) ==

u

l'. (t ),I

E

J, where 1'0 (!) is being called a »kernei«. if it

1~0

containsn0

š N

linearly independent (or »almost-independent«) processes, where

11 0

1s

maximum number of non-correlated stochastic processes The rest of processes is then
located to a finite number n of »satelytes« 1'1• 1~, having prescribed their levels of
dependancy on the kernel P0 , say oj, .011 , which are known to be the values of functional
determinant s computed for each satelyte set of processes. Here, statistical theory of
classification has been applied in order to get clusters as desired. A question on minimum
number of satelytes has been left open, but interesting All satelytes are thus dependent at
some arbitrary predetermined levels
and are algebraically expressed through the kernel (as
done in linear case) For the service of economic modelling we are »happv« to use the same
kernel asa set ofindepenedent »variables« to act as »hard« part ofthe econometric model
and a set of satelytes is »swinging« part of it Apart from the control vector 11( t) in case of

o,

control oriented econometric model we vary
functional y( t) as you see below

oj,

8" to optimize/improve criteria/goals

Thus the state-space operator of control oriented econometric model is a system of
differential/difference equations

x == A(t)x(t) + B(t)u(t),

(1)

where ali satelyte variables are algebraically modelled, e.g. in linear case

3. Flexible versus stiff category input application in economic modelling
; == 1,

x 1(t) ==L1 [x(t)],
The traditional economic models, mostly econometrics, deal with a some input, which
determines some output, both defined by some pre-chosen economic categories. Apart from
the issue on how and why these inputs and outputs were selected (by the way, we may
seriously argue about it), there is a variation ofoutput asa consequence ofvariation ofinput,
where the latter one reflects its data-history. Such an inaccuracy
may be decreased by
using wavelets, which brings the quality ofregression procedure at arbitrary leve!. However,

28

(2)

x(t) E P0
X; (t) E

.n

I'; (t)

As partition of P(t) depends on t, (1) and (2) represent afluid model (see /5 /), which is now
a substitute of econometric ( or any other economic model). Functions chosen within a kernel
as well as those within satelytes vary over tirne: input categories are thus flexible and the
kernel is being determined by input for each t. For this reason we may call it as »one-way«

29

fluid model, m the contrast to »two-way« fluid model, where the kernel 1s input-iutput
determined. In the latter case we need some criterion functional
y(t)

= Y[x(t ), u(t)]

x(t) E P0

which qualifies (1 ), (2) and (3) as a control oriented econometric model (see /6/) Even in
the case of 11(1) kept fixed, the two-way fluid model could be improved through adjusting
8 0 > 1- s0 ,81 , ... 8" so thaty(t) improves. Evidently, the same procedure is wellcome
oneway fluid model.
Any conomic ppolicy should be as good as possible, why not to replace stiff input categories
with flexible ones. varying over tirne, provided the policy goals are improved. We can see
ftom a composition offluid model, what reasons for OR to get stuck are worth while to be
paid our attention.

4. Swapping (interchangeable) optimisation
When forma! modelling some problem, there are two distinct groups ofmodel constituents
variables and parameters. In practice, we hit upon the situation when variables become
constants and parameters call for variation. If we take such a situation into account, the so
called swapping procedure takes place.
Let us discuss it in an abstract way first. According to Ch. l we have Y=SX as a forma!
presentation ofproblem. A swapping procedure needs to reshape it in a way Y= S(t)X,
where t represents a space of ali relevant model parameters. Ifwe confine ourselves to finite
dimensional vector spaces appearing in (1 ), we have
y=S(a)x
<l>(x) s; O

(1)

where x, y and a are vectors. Let ( l ') be Hicksian-sense dynamic model of a problem in
consideration; its total operating horizon T= u 1; is thus being split by the »swapping«
behaviour of a and x vectors. Let us simplyfy our discussion by assuming that S is
independent of i ; optimizing across 1; we get a functional
opt y(T;)=y/(a/or x;°)

Vi

(2)

subject to some constraints to either x or a .
Ifwecannotendure either Llopty(J;)= y;°(a;°or x/)-y/( a/,xi =const) or
Ll opt y( I;) = y;° (a/ or x;°) - y;° ( ai = const, x;°), then, refusing to deal with swapping
phenomenon, a serious damage may be caused to the problem. This damage may increase in
case, when y;° ( I;) = opt y( I;) is a criterion aditive o ver tirne. Following FED ftom Ch.l
this damage may be demonstrated stili stronger when introducing space Z (expressing

30

economic categories. e.g However. the swapping procedure extends to l ( y) which may
p!av role simultaneously with
a triple swapping procedure
The trapping state stemming from swapping phenomenon might be today verv difficult to
overcome. For example, if S(a = const) is linear operator with constant coefficients and z=
r being linear function ofy, then the corresponding problem is bi-linear version of
Wolfe's program in variables x and r, provided we added some linear constraints on y. Asa
whole, any swapping analogue to ( l) as a mathematical programming problem might be a
trapping state per se, since the corresponding existence theorems may not exist; a short
review of swapping allowed problems is given in /7 1 However. the simulation and nonformal modelling (see /8/) may be ofuse in such cases

It is worth while to illuminate the simulation of problem ( ) and its swapping analogue in
their discrete version through the following
EXAMPLE when running business, manager is obliged to respect balance sheet and income
statement which represent very simple model, mostlv used as an aposteriori information
source. It is very seldom applied asa managerial optimization tooL one ofthe reasons is a fact
that it does not contain many functional relationships, instead, is more a set of»soft
mapping« between economic/financial/technical/technological categories one onto another
To show how this tool could be used for optimal decision making along the strict line of FED
structure shown in Ch. l, we shall combine both sheets in one
-space X contains the following decision variables x. volume ofinterets, nominal price of
share, EBIT/total assets, sales volume;
-space X contains the following decision parameters a (X) interest rate, non-risk rate of
return, risk rate of return, beta coefficient, expected return of assets, reinvestment coefficient.
coefficients of ordinary/priority shares, tax rate:
-space Y consists of: leverage, total assets, number of outstanding shares, fixed costs, variable
costs, selling price(s), in adition, it contains also induced (=not primary criteria) vector a( !
ROA,ROE, dividend paid, market price of share, the ratio between market price of share and
its return, the ratio between market price of share and its book price, WACC, debt capitaL
equity capital,
-space Zof consequences z= EPS (earning per share)
The policy goal here is a single one, EPS, being able to be expressed analytically through
y E Y (e.i. operator U has forma! description), whereas S does not have such a possibility
We face a hybrid offormal and non-formal part ofa EPS- optimizing model. To derive z E Z
we apply a finite (but arbitrary) step simulation over the union of spaces X (ofboth parts)
and non-induced part ofY; the simulation procedure is always finite, regardless to what
swapping parameters a (X) and a(Y) were used (even both ofthem). For certain subperiod
oftime a firm is not able to vary some component of x; ifthe corresponding leve! ofEPS is
below the manager's expectation, then he activates some component(s) of a(.); for the details
you may see /2/. It is also worth to stress that the traditional balance sheet u income statement
can always be extended towards non-financial spaces, e. i. to the spaces a manager feels as an
additional tool to improve EPS. Such enlargements are not vulnerable by category dimension
involved neither they are prohibited, when space Z is a vector space; in the latter case we deal
with multicriteria swapping procedue, exercised over (in general) a hybrid managerial
decision making problem: it stili obeys FED philosophy.

31

T

6. Stat1stical and non-statistical nohon of uncertamty
The uncertainty is. simply saying, a complement to expectation To deal with uncertainty of
some object in a quantitative way we have to define a) the object, b) the domain of object and
c) a measure ofuncertainty.Ad a): as an object each ofthe variables, mappings and
properties ofFED could be used. Ad b): depending on the nature ofthe object we define its
domain (universum U). Ad c): a measure ofuncertainty/expectation may not only be
dependent on the nature of an object, but also/or on the aspect (criterion) of our analysis of
the object. Ifthis analysis is a quantitative one, a measure ofuncertainty/expectation should
be a uniform and real function/functional/operator There is a significant dichotomy of ali
possible measures. i) it is either a measure in the sense of the set theory or ii) it is a measure
of existence of the object. In the first case we may speak about non-statistical (or nonprobability) uncertainty resp expectation, while in the latter case uncertainty/expectation is of
statistical type. However, we s hali need some additional condition for modification of the
above distinction
Let

(!

= vU, be a set of II objects

U, and µ([}) and µ(U,) set theory measures; we labei it

as absolute measure. Thus the relative uncertainty or enthropy of U, is

which we treat as obJects in a way that each ofthem has its non-probabilitv uncertaintv I J ).
Thus we dea! with n-dimensional optimisation where optimisation algorithm assures the
optimal direction
optimal direction of F = ( 0'
0Xl •
. .

.?H
gradH= 1 -

which helps us to compare different objects. ln this case µ(ll ) is one-dimensional
information on object rI, (for the sake of simplicity we disregard an issue of moredimensional non-probability entropies.
lfwe replace µ(U,) by its probability p(U,), (1) becomes a measure ofrisk, if a set
f p(U,)} is complete.It besomes a measure ofthe probability entropy, if a set { p(U,)} is
incomplete An analogue to ( 1) as a relative measure of probability entropy can not be used,
unless
p(U)-p(U)
·

'

p(U)

=

provided i = 1, . .n' < n

probability entropy = H(p)

u (x) is then defined on vector field

?H
-J

(4)

rx„
8x1
\\e are interested in the increase ofthis uncertaintv m the direction ofr e.1 11s proJection on
·
·
optimal direction (3)
?v.

-·-) grad H
r1x
II
'

w here

811
ox

=" ?lx?lv· rlxrH
L.,

-

(5)

II

.
b ..
are non-pro ab1hty uncertamties of individual optimizing variables x .

II

(1)

non-probability entropy = H ( µ)

µ([ !)

(3)

X,,

The uncertainty H (µ) [ U (x) J of the universurn
having its gradient

Thus, the non-probabili1y uncertainty ofthe optimal solution of any forma! problem
increases along whh the scope of universum and the responsiveness of optimal solution w
each variable; when optimisation procedure is carried out through simulation it increases too
if simulation step is decreasing From (5), it follows that one and the same d~cision space X '
cha~ges _1ts type( 1) - entropy, if it serves as an optimisation domain Tt is an important
findmg, if X serves at the same tirne to some other optimisation problem, bringing a
new/different gradient (4 ). If there is a need for simultaneous optimisation ( e.i in the sense
of Balza control problem) based on fini te set of criteria, it might be a problem of detecting
»an overall measure of uncertainty« belonging to the given set of optimisation tasks
1

µ(U)- µ(U;)

0)

= l, . .n

On the other hand, when applying definition (2), an entire different handicaps mav arise As
known, there exist various probability based entropies A measure of probabilitv-based
entropy havily depends on the type of probability distribution we know only a part of which
Tfthere 1s a severe doubt about some particular probability distribution, the question is on
how to find some »better«one There are only two »boundary« conditions for a H(p) to be
(2)

where 11 is the number ofall objects. The distinction between (1) and (2) is crucial when
dealing along FED umbrella. Consequently, in case of (2) we can not expect to have a
complete description of probability distribution: we have only an incomplete information on
probability space belonging to U. A notion of probability based entropy rests on (2) only.
Apart from the ignorance ofthe difference between (I) and (2) we may commit an
additional severe error by not immersing deeply into various types of(l) as well as (2).

found:
1) if the existence of an object is certain, thenH (p) =O; 2) if the existence of an object is
uncertain, H (p) =- oo The remaining behaviour is rather simple it is monotonically
increasing with the increasing number of events. All of its finer properties heaviliy depend on
the problem we study no apriori steps are assured to be the correct ones.
A review of OR possible trapping states has not been finished by now. Let us look at most
;~N

simple Shanon entropy, H(X)

= - Lp(X)log 2 p(X); it is very sensible to small
1~1

To show that it is not a fairy tale, let us first turn our attention to one single type of nonprobability entropy, namely that one which refers to optimisation processes, based on formal
modelling (see /9/). Asa consequence (1) asa definition of non-probability entropy we face a
fact that the only measure of entropy is based on the existence ofthe object. In case offormal
optimisation modelling we look for uncertainty of an/the optimal value of criterion
function/functional/operator. Let us have scalar field y = F(xw.x 11 ) of optimising variables

32

~robabiliti_es of an~ type.The way of getting out lies in the redefinition of such an entropy,
hke adophng Tsalhs entropy as a generalisation of Shanon's entropy and being convenient
for distributions not from exponential family; another way might be Bayesian priors.

33

DOJ\tllNATION OF CAPHAL

There 1s a strong JUstification of our care to be much mvolved the classification of
probabihty based entropies. Before we start fulfilling spaces and Y in Ch . 1s very
important to discover inter-relationships between them, e.g. may be of great help to use
interaction analysis which is essentially based on probability-entropy-measure shown in
interaction graphs (see / 0/).
References

= TRAPPING STATE OF MANKIND

\"il3em Rupmk
Prnfessor ementus of \lathematics.
Econormcs and Engineering
Faculty of Econonucs Ljubl_1ana.
mversity
LJublJana
Kardeljeva ploscad 17, !000 Ljubljana, Slovenia

/ 1/ Rupnik V., Generalised Economic Modelling and its Applications; Third conference on
Operational Research, , Oct. 5-7. 1993, Rovinj. Croatia, pp- 33-49

&

\ 1l1em.rupmk(a)siul.net

Dragomir Sundac
Tenured professor of Economics
Faculty of Economics Rijeka,
·mversity of Rijeka,
Filipm 1Ca 4, 51000 Rijeka,
Croatia
dragomir.sundac(cilefri.hr

12/http: 1lwww.atnet.si/interacta/ssop. html

/3/ Volcjak R., Applicability Limits to the Quality ofEconomic Modelling; PhD thesis,
Department ofEconomics, University of Ljubljana, 2000

Keywords: neol1beral globahsat1on = ant1globalisa110n,
colomalism".
modelmg pru1ect. mtellectual
dec1si(m poweL E! · capital al!ocat1on strateg;.

transit10n

c:ountries.

Jualism

"corporatton

<.:apllal power vs

A/ Rupnik V . Prediction Chaotic Modei!ing; unpublished ,Interneta Ltd. 2003
/5/ Rupnik V, Some Experiences with Fluid Modelling ofEconomic Systems; Systems
Research, The Official Journal ofthe lnternational Federation for System Research, Vol 2,
No.3, 1985, pp.2 1-219
/6/ Mičkovic S , RupnikV., The Building up of a Functional Optimal Control Oriented Fluid
Model; XII lntenational Conference on Dynamic Modelling, Nova Gorica, Yli, 989
1

71 Rupnik V., The Limit s of Structural Control o ver Production; Proceedings of the Seventh
lnternationalConference on Operational Research, . Rovinj, Croatia, Sept- 30-Oct.2, 998,
pp- 219-227.
/8/ Rupnik V., An Attempt to Non-Formal Modelling, Proceedings ofthe Fourth International
Symposium on Operational Research, Preddvor Slovenia, Oct. 1-3 1997, pp 207-213.
/9/ Rupnik V., Research report on uncertainty ofmathematicai programming solutions;
InterActa Ltd, Parmova 53, Ljubljana, 2004, unpublished
/10/ Jakulin A., Bratko I., Quantifying and Visualising Attribute lnteractions; Faculty of
Computer and Information Science, University ofLjubljana, August, 2003, unpublished.

Extended. Ahslact

Ali the books talking about globalisation. an<l in connection with that abom dominalion of
capitaL remain on a description of causes and consequences. This book is currenth unique
since is simulates the functioning of globaiisation for the next U-15 years. This 1s done by
using metamathematics2
The created mathematical model sho"'s that the world's economy is gorng towards
economic downfall unless correctives are implemented, that is
globa!isat1on is spread
through liberalisation and i f domestic economies are not protected from negative
consequences of globalisal1on.
This particuiary applies
a small coumry aml a c,nmtrv transition such as the Repuhlic
ofC'roatia.
The authors support their clanns with mathemal1cal accurac\ whicb presents a great
scientific step in shaping the process of globalisation.
The book consists of three parts l) a description of the legahties of the process of
globalisation. 2) mathematical simulation and 3) proof of economic downfall; as well as a
list of needed correctives which should be implementecl in order to prevent econom1c crises
ofthe world economy of unseen proportions.
In the first chapter, titled THE GLOBALIZATION CRITIQUE ~ WHY ARE PEOPLE
FORGOTTEN IN THE MODELS OF ECONOMIC DEVELOPMENT?, authors analyse
and describe the consequences of the cmrent process of globalisation, paiiicularly focusing
on the destiny of transition countries (among which is the Republic of Croatia) and
developing countries, that following a model of"fast integration" into globa! economic system
(forced by world's leading financial institutions: WB, IMF i WTO) experienced a downfall
oftheir own economic, social and political systems.
In
the
second chapter,
GLOBALIZATION MODELLING
PROJECT
MATHEMATICAL PROOF OF GLOBALIZATION BREAKDOWN, authors present a
mathematical model which they built (based on metamathematics) and by which they project
the fitture and discover scientifically exact possible ways of the development of
globalisation.
1

2

34

Last Internet research was completed on 10th of June, 2005.
Metamathematical apparatus which was created for the concrete case.

35

--------------iilliiililllinl----------------·······-·--··-They are especially retrospective on the transition countries.
set formally-consistent
According to the words of authors, the goal of the model was
soc10-econom1c
vanous
to
groumi
training
potential
open
to
,
globalisation
of
doctrine
detem1ine
aspects and above ali, by disc!osing negative consequence s of g!obalisation
globalisation for national economies.
possible defence against negative consequence s
Resu!ts which the model gives are daunting ami bring to an understandin g that the present
globalisation works against itself and leads;guides the mankind into an economic selfdistruction, as well as any other.
The third part of the book, APPENDICE S, contains mathematica l evidences of what has
been said.

A

REVIEW OF

DEA

STRUCTU RE OF THE DECISION MAKING lJNITS IS
CONSIDE RED
Lorenzo Castelli
a.

References

b

-FORMAL lvJODELLliv0: !Vth lntemat1onal Symposium on
Rupmk. V • Al\ A fTEMPT TO
Operations Research, Preddvor. Slovema, Oct 997.

2r Rupnik, M. and Rupmk, V RKLR me/hod: lnterActa. LttLBusmess !nformat10n Proccessing,
Ljubljana. l 995
/3/ Rupnik, V.: THE SHADOWED DECJMONS; The Fourth lnternat10nal Symposium

Operat10ns

Research, October 997. Preddvor, Slovema
/4/

Rupnik, V.: THE RELATJONSHIP BETWEEN r'ONSVMPTJ ON AND TNVESTMENTS ASA
THE THEOR OF ECOVOMf(' CONTROL Economic
METHODOL OGWAL ASPECT
Review, Vol. 25. No. l., 1974.

MODELS WHEN THE INTERNA L

DEEI

DIAI

a ,

Raffaele Pesenti

b,

vValter Ukovich

a

Trieste, 34127 Via A. Valerio 10, Trieste, ltaly
Cniversita. degli Studi tli Palermo, 90128 Viale delle Scienze, Palermo, Italy
castelli@units.it, pesenti@unipa.it, nkovich@nnits.it
Universita. degli Studi

Abstract
Classical Data Envelopment Analysis (DEA) models consider each Decision Making
Unit (DMU), whose relative efficiency they evaluate, as a "black box", i.e., its internal
structure is disregarded. The paper presents a comprehensi ve framework of the most
advanced theoretical findings in DEA when t.he internal structure of the DMUs is t.aken
into account, thus giving directions for novel applications of such methodology and
introducing it as a powerful tool for complex processes performance evaluation.
Keywords: Efficiency evaluation, Data Envelopment Analysis, Internal structure.

15! Rupnik, V .. Strmčnik. S., Karba, R„ Bremsak, F. · A NON - TERMTNAL CONTROL APPROAC'H
COMPACT SPACE. IIl IFAC IFORS
TO DYNAMI(' Er'ONOM!C MODELJNG II\ \O."!\

Introdu ction

Conference, Warsaw, June 16-19, 1980.
/6/ Rupnik, V„ AN ATTEMPT TO NON - FORMAL MODELJNG. Proceedings to l ourth
Intemational Symposium on Operations Research. Predd,or, October l-3. Slovema.
/7/ Sundac, D., Pulic, A.: INTELEKTUA LNI KAPITAL - KLJUČNI RESURS 21. STOUEr'A.

Intemational Business Consulting Center, Rijeka, 2001
/8/ Sundac, D., Nikolovska, N„ SCENARIJ ZA EKONOMSKI SLOM ZEMALJA V TRANZICIJI,
International Business Consulting Center, Rijeka, 2001
/9/ Sundac, D., Nikolovska, N.: GLOBALIZAT ION AND ECONOMIC DOWNFALL OF
COUNTRJES IN TRANSITION, MAGOR, Skopje, 2002.
/10/ Sundac, D.: ESENCIJALN I UZROCI (NE)ZAPOSL ENOSTI SA STAJALIŠTA EFIKASNOST I

VPOTREBE KAPITALA, Zbornik radova Ekonomskog fakulteta Rijeka, br. 14, sv. 2, 1996.

/11/ Sundac, D.: PRJLOZI ZA NOVU EKONOMSK U ZNANOST, Verza!, Rijeka, 1992.

36

Data Envelopment Analysis (DEA) has been a standard tool for evaluating the relative
efficiencies of Decision Making Units (DMUs) since the seminal paper by Charnes et.
al. [1]. Throughout the years, different variations on the original model and interpretation have been proposed (see, e.g., [2] and in particular [3]). However, some underlying
assumptions are common to classical DEA models. Thc efficiency of a DMU is defined
as the weighted ratio of the outputs (products or outcomes) yielded by the DMU over
the inputs (resources used or consumed). All the considered DMUs are homogeneous,
i.e., they all have the same types of inputs and outputs, and independent, i.e., no constraint binds input and output levels of a DMU with thc inputs and outputs of the other
DMUs. F\trthermore , DMUs are seen as black boxes, i.e., thcir intcrnal structure is not
considered. As a consequence, generally there is no clear evidence of thc transformatio ns
the inputs are subject to within the considered units.
In the last two decades, various authors have explored the possibility of measuring
efficiency relative to subprocesses or components of the DMUs within the DEA framework. These authors abandon the black box perspective in the assumption that, in some
particular contexts, the knowledge of the internal structure of the DMUs can give further
insights for the DMU performance evaluation. As an example, such knowledge allows
to detcrmine whether better performances can be theoretically obtained by merging thc

37

Lechnologies of some substructnrcs of thc obscrved DMlJs. In addition, assessing the
efficiency of each of the processes or subunits might prevcnt the inefficiency of some of
them being compensated by the efficiency of other ones. In this framework, the aim
of this paper is to review the models proposed in the literature that consider interual
structures or processes of the DMUs.

1

The basic model

Thc common building block of the models cousidering DMU interna! processes is represented by a set of N DMUs, each composed of a set of K Decision Making SubUnits
(DMSUs). The DMUs are assumed to be homogeneous (i.e., each DMU is composed
of thc same set of K DMSUs) and independent (i.e., the output of a DMU cannot be
thc input of anothcr DMU or re-enter thc same DMU). In addition, the basic model
assnmes that:
l. no intcrmcdiatc flows among DMSUs exist,

the DMSUs are paralleL In other
words, thc output of a DMSU cannot be thc input of another DMSU (and also
cannot re-enter the same DMSU), bnt it must be an output of the whole DMU;

2. any input (output) of the DMU is also an input (output) of one of its subunits;
3. the amount of any input (output) of any DMSU is a-priori fixed.
When a set of N homogeneous and independent DMUs composed of K subunits
according with Assumptions 1 + 3 is evaluated, "the overall efficient production system can also be improved in technical or scalar efficiency with the aid of information
from othcr DMUs" [5]. In particular, other NK - N non-observed homogeneous units
are added as terms of comparison: they are all the DMUs whose interna! structure is
composed of a set of J( observed DMSUs [6]. As an example, consider U1 = (a 1 , bi)
and U2 = (a 2 , b2) as two homogeneous and independent DMUs to be evaluated, where
(a;, b;) are the DMSUs of unit U;. Since the internal structure is known, the non-existing
DMUs (a1,b2) and (a2,b 1 ) can also be added to the comparison set (Figure 1).
Observed DMUs

Non • Observed DMUs

®

®

weak efficient if and only if thcrc exists at lcast one of its suhmdts which is wcak
efficient relative to the corresponding subunits of other DMt:s;
CCR-efficient if and only if cach of its subunits is CCR-efficient relative to thc
corresponding subunits of other DMCs;

where, according to [4], we define as CCR-efficient a DMU whose optimal objective value
of the dual linear problem is equal to 1 and ali the slacks are equal to zero. When only
the first of these two conclitions is achieved, the DMU is referred
as radial, technical
or weak efficient.
The introdnction of returns
scale of t he DMSCs does not affect the abovc results.
Dy assuming that each of the DMSl:s inside a DMU exhibits constant returns to scale
independently
the other one. the maximum re]ativc efficiency (Waluation of cach
existing DMU shonld be assessecl by comparing it with all the infinitely many DMUs
that can be obtained merging the DMSCs whose inputs aud uutputs are scalcd by a
(generaily) different positive constant. Accordiug to thc above arguments, cach DMU
shonld he compared with an infinite nnmber of other DMUs. Nevertheless. the problem
may be reduced to the comparison of the DMt: with all its subunits [6].
When Assumptions 1 + 3 are dropped, more complex models are produced. They
are going to be discussed in the next section. In particular. threc main partially overlapping categories emerge form the literature. The models in the first category deal with
DMUs that are still independent and homogeneous but perform several diffcrcnt and
clearly identifiable functions, or can be separated into different components f7]. Thc
models in both the second and in the third category consider DMUs that are themselves
components of greater structures. The rest of this paper presents a review of results
dcaling with models of the first type.

2

Multicomponent models

The models described in this section deal with DMUs that are independent and homogeneous but perform severa! different and clearly identifiable functions. or can be
separated into different components. The literature refers to such modcls as multicomponent [7], joint efficiency [8], or multi-activity [9] models. Formally, according to [7],
we define as component the bundle of outputs and inputs that charactcrizcs a function
of a DMU (see Fig. 2).

It can hc easily shown ([5] and [6]) that the maximum relative efficiency of a DMU
whose internal structure fulfils Assumptions 1- 3 is equal to the maximum of the relative
efficiency of its subunits. In particular, such a DMU is [5]:

In [10], Beasley introduced one of thc first examples of a multicomponent DEA
model. It is applied to university departments concerning the same disciplines. It was
not originally referred as a multicomponent model, but it is nowadays acknowlcdged
as part of the literature on the subject. The considered departments are homogeneous
and independent DMUs. However, within them, the teaching and the research activities
define two different clearly separable functions. DMU outputs are split: the number
of undergraduates and of taught postgraduates are outputs of the teaching function;
the number of research postgraduates, research income, and research rating are outputs
of the research function. One input, again research income, is specifically dedicated to
the research function. The other inputs, general and equipment expenditure, are shared

38

39

Figure 1: Comparison Set for DMUs U1 and U2

component
dedicat1;d oulputs

shared input
funclion

function

dedicated inputs

shared optpu!

a department k aggregate efficiency ek = qiel +qie~ is dcfined as thc weighted
In
combination of the partial efficiencies of its two components, where the weight qi:, of each
the proportion of the total inputs it receives. Tn the spirit of
component r equals
DEA, Beasley proposes that each DMU can allocate the shared inputs among its different
components so that its aggregate efficiency is maximized. Under such a hypothesis, he
proves that, for each DMl:, the expression of the weighted combination of thc component
partial efficiencies is equal the usual efficiency considcred in DEA models, i.e., ek =
ek, Beasley proves that such a result. generalizes to the case in which more than two
components are cousidered. The general Beasley's model is

Figure 2: A multicomponen t DMU.
(joined) bctwccn thc two functions.

In the following, a notation similar to the one proposed in
ular, for each DMU k, define

< l

e~

<

[lb)
(le)

\/k,

l

Vi

I:a~

Note that Roll and Cook in [11] introduced earlier the concept of partial efficiency
measures for a DMU. However, in [11], each component is a bundle of independent
outputs and inputs. No shared inputs exist. A similar model was later also proposed in
[5] by Yang et al.

(la)

maxe 0

e~
eo

r

(µr, \/r)

E

(11s,11r,vr)

E

O;n

r

?

f;

r
S
II; , II; , O'. i ,

r

µj

le)

nout

(lg)

Vi,j,r.

is adopted. In particwhere c > O is the usual non-Archimedea n constant, and the sets Oin aud f2 ut are
a8.mrance regions as defined in [12]. Differently form the classicai DEA models, Beasley
includes conditions (le) imposing that even the cross efficiency 13] of each DMU components cannot exceed l. Moreover, Beasley points out that it may turn useful to
incorporate the additional constraints (lf) and (le) involving value judgments concerning the proportions ar and the weights µr and vr of the different DMU components,
These constraints are not strictly necessary for the definition of a multicomponent DEA
model, but prevent the model from yielding unreasonable results. In this contest, the
author provides an example where, in absence of constraints (lf)-(le), one research
postgraduate was worth about 880,000 undergraduates for a given department.
0

• i, j, r: t.he indices of rcspectively the generic input, output, and component,

• xr, = {xrk}·
• Xf
@

the vector of the dedicated inputs of component r,

= {xr,.,}: the vector of shared inputs,

Y[ = {y;,J: thc vector of the outputs of component r,

• vr

= {11[}: the vector of weights of the dedicated inputs of component r,

•

vS

= {111}: the vector of weights of shared inputs,

@

µr

= {µ;}: thc vector of weights of the outputs of component r,

• ar = {ar}: the vector of proportions of the shared inputs allocated to component r.

With a little abuse of notation, ar Xf is also defined as the column vector whose
generic entry is a';Xii. In this context, a';Xii is the amount of shared input i allocated
to component r by DMU k to maximize its efficiency. However, when a shared input
cannot be divided, then a'; can be seen as the proportion of the (virtual) value of the
input i allotted to component r.

As for the classical DEA models, (1) can be rewritten as follows
e•o

L
r

lir X~+

L

lis (ar xt)

maxI:µrYo"
r

(2a)

1

(2b)

r

µrY{

<

I:a~
r

lir X~+ vs(ar X{)

1 Vi

(2c)
(2d)
(2e)

(µr, Vr)

E

flout

(vs, vr, \/r)

E

O;n

>

f;

\/k,r

(2f)
(2g)

By using thc above notation, t.he aggregated efficiency of the whole DMU k is exE r}".r
pressed as ek = E . .,. x;;+•E)s(a• xf>' and the partial efficiency of the single component r
µ•}".•
as er k - 11 r x;; +vS(ar Xf) •

Even provided that the assurance regions are expressed in terms of linear constraints,
model (2) is not a linear programming problem. More precisely, inequalities (2b)-(2c)

40

41

r

S

1/i ' II; ,

r

r

a;' µj

\/i,j, r.

includc thc rectangular terms involving variablcs vf, a[ Below it is shown how different
authors have proposed possible ways of overcoming such a difficulty.
Many authors havc introduccd variants of model (2) cithcr changing some of the
assumptions madc by Beasley or generalizing them. A common feature of thc different
variants is that in ali of them the aggregate efficiency of the generic DMU cannot exceed
the unity and a DMU is efficient if and only if it is efficient in ali its componeuts.
Molinero and Tsai, in [8, 14), propose an approach dual to (1). In addition, the
authors include shared outputs, i.e., outputs yielded syncrgically by two or more cmnponents. Thcir output, oriented model can be written as

maxI:qo0~

e~

L ,\~x;k

< Xio

r
Vi,r

< X;os

Vi

(3a)

k

I: I:r "~(arxfic)
k

L,\''kYjkr

> 0oY';o

(3c)

Vj,r

(3d)

k

L L,. ,\~(fJ';YJk)
k

>

Lr 0;; (fJS YJo)

Vi

1 Vi

(3f)

I:fJ;

l

(3g)

r
r

I:q;;

1

(3h)

r

,\~J

Qo, a; fJ; >
l

€

Vi,j,r,k,

(3i)

whcre Y]k are the shared outputs of DMU k, fJ'; are the proportions of the shared
outputs attributcd to component r by DMU k, q0 are positive numbcrs representing
thc relativc importance of cach component r for DMU O, and 00 are measures of the
inefficiencies of the DMU O components. Actually, 00 are the reciprocals of the distance
functions [15] from the frontier of the production set defined by the conic combination
of thc components of the observed DMUs in the hypothesis that the assumptions on
free disposal and minimum extrapolation hold [16].
Notice that the values (J'j, differently from ar, should not be seen as the proportion
of the amount of output j yieldcd by the component r. Actually, no component can
produce a shared output by itself but nceds synergy with other components. Instead, (J'j
can be seen as the proportion of the (virtual) value of output j that can be attributed
to component r.
If the values ar, fJ';, and 00 are hold as constants, model (3) is a linear programming
problem and its dual can be determined. The model obtained in this way is the output
oriented version of (2). However, two main differences occur. The overall weighted sum
of the outputs of the component r of the generic DMU k becomes now

+ µ 8 ((FY/),

(4)
8
where Y = {yfk} and (š = {(3';}. Also, additional constraints on the outputs of
µry{

DMU O and the numbers representing on the relative importance of each component are

42

µrYo

+ µs(/3''Y/)
I:q;;

q~

Vr

(5a)
(5b)

\/r.

(5c)

r

q~

>

€

Conditions (5) are particularly interesting. Thcy mathematically state a precise
relationships between the relative importance attributed to a component and the optimal
amount of outputs allocatcd to ( respectively, the optimal amount of allocatcd inputs in
an input oriented version of the model as
Then, conditions (5) give a mathcmatical
justification to the choice in
of expressing thc weight qk of the generic component r.
As already mentioned, q); is fixed equal to the proportion of the total inputs component
r receives. Without (5), such a choice might appear arbitrary, altltough reasonable.
Molinero and Tsai,
[8], prave that the
set and (3a) is a convex function. In l9,
reference example, they introduce and discuss
The efficiency of each component r of DMU k

feasible solutions of (3) define a convex
considcring university departments as a
a variable returns to scale version of (3).
is then defined as

(3e)

I:a;

Vj

present

(6)

where the variable 8k is unrestricted and its optimal value defines the components returns to scale status. Depending on 8k being negative, null, or positive, a component
homogeneous to r, but with proportionally smaller inputs and outputs, results to be
more, equally, or less efficient than component r, hence component ris in an area of dccreasing, constant, or increasing returns to scale, respectively. The aggregate efficiency
ofDMU kis

(7)
Note that the optimal value of I:r J;; may be zero even if some orali elements in the sum
are different form zero. In this case, as Tsai and Molinero point out, DMU k may appear
to be operating under constant returns to scale and technically efficient when analyzed
asa single activity DMU, but when its individual activities are analyzed it may be foun<l
that the DMU is scale inefficient in each activity rg] Tsai and Molinero stress that in
generala DMU, that turns efficient when considered as performing a singlc activity, may
result inefficient when its different components are taken into account, indcpcndcntly of
its returns to scale status.
Tsai and Molinero, in [9], introduce the performance evaluation of National Health
Service (NHS) trusts in England as a case study. They analyze the trusts from two
different points of view. The so called system perspective considcrs the trusts as single
activity DMUs. The so called trust perspective considers the trusts as mutlticomponent
DMUs. The system perspective is the one of a general authority, such as the Department
of Health, supervising the trusts. On the other hand, the second perspective could be
the one of an inefficient trust interested in determining which are its more efficient
components. Sixteen of the considered trusts are system efficient, but only one of them
results efficient from a trust perspective.

43

r
In l18], Diez-Ticio and Mance bon report and comment in detail an application of a
multicomponent DEA model to evaluate the efliciency of Spanish Police Service. The
authors use the variable returns to scale model proposed
[17]. In 19], Arcelus and
Coleman introduce a DEA application to review the efficiency of the departments of
thc University of New Brunswick. The authors use dassical DEA models, but cite
thc modcls in [10, 14] as possible ways to overcome the dichotomy observed between
thc scientific-technical departments and the remaining ones. In particular, the authors
mention the probable existence of different production functions per DMU.

Cook et al., in [7], allow a same shared input i to he weighted differently hy the
components of the same DMU. The rationale behind such a choice is that different
components may disagrcc on the importance of a same input. Consequently, the multicomponent model as in '.7] inclndcs a set of vectors v 8 r, one for cach component r,
instcad of a single one in (2b)-(2c). Thc rectangular terms are uo more vf n; bnt vfro{
Also, a change of variablcs is proposed. In particular, let i = l, .. , s be the index of the
shared inpu ts, then vfr = vfr a'; for i = 1,
, 8 - l and vfr = vf'' (1 - 1:r:t ar). Thanks
to the new variables, Cook et al. obtain a linear model, when constraints (2e)-(2f) are
disregarded. The rectaugular terms vf (a"Xf) in the conditions (2b )-(2c) become D8 r Xf
and v,~:::: c in (2g) turns Df" ::=:car. Unfortunately, non linearity may arise again when
additional constraints concerning valuc judgements as (2e)-(2f) are necessary. If such
jndgcmcnts are expressed also in terms of
the variable substitution may not lead to
a linear model. In fact, in i7], the authors present a case study on 20 Canadian banks
branches. They take into account two components for each DMU, the first one related
to service specific activities, t,he second one related to product specific sales activities.
They obtain (and solve) a non linear model, since they impose ratio constraints on the
weights of the shared inputs of type vir : =: avt.

v;,

Also Cook and Hababou deal with Cauadian banks branchcs in [20]. The model
proposed presents variables and constraints as in [7] but it is an additive one. The
authors discuss how to formulate an additive objective function that represents an aggregare measure the efliciencies of ali the DMU components. In the classical additive
DEA models, a possible measure of inefficiency for a generic DMU k is given by the
differencc bctwcen the weighted sum of the inputs minus the weighted outputs of DMU
k. Then thc authors suggest a multiobjective approach where the partial inefliciencies of
ali the components are considered. For each component, the weighted sum of its inputs
minus the wcightcd sum of its outpnts is considered, In particular, Cook and Hababou
minimize thc maximum partial inefliciency in ordcr to give equal importance to cach
component, i.e., their objective function is
minmax{vr X~+ v 8 r(ar Xf) - µ'Y{: \/r component of DMU0 }.

(8)

Finally, the authors linearize their model with the same variable changes proposed in [7].
The results concerning the Canadian banking industry in [7, 20] are cited in [21, 22, 23].
Cook and Green in [24] deal with a manufacturing multi-plant company. They point
out that in such a context some outputs of different components of a same DMU can
partially overlap, i.e., some outputs may be common to different components. Note that
thc overlapping outputs are different form the shared outputs considered in [8, 14, 25].
In [24] any component can yield a given amount of each overlapping output j, with no
need of synergy with the other components and with no possibility of attributing the

44

considered amount to other components. Then, Cook and Green cannot approach what
thev call the overlap problem
introduciug variables Br as
to determine which
pro~ortions of shared outputs are attrihuted to each componenL
24], the efficiency
· ei: = .,,·xr+.,s'(a'"Xs)
~~
of a single component r remams
and, consequent ly, t hc aggrega t e
L /tryr
. [ ]
d
efficiencv
a whole D MU k remains ek = s;,·
xr L r V t._\ Q' ~-:,i.,F. However, m 24 , share
,
"
L...ir v1 k+
~ k )
inputs are no more allocated to the components, as such task could bc hardly performed
without introducing some ambiguities _due to the component overlapping, Shared inputs
are allocated directly to the outputs. In part.icular, consider model ( l) and the extension
propose<l in
, Cook and Green introduce a new set of variables ai as the proportions
of the share<l inputs i allocated for outputs j In additiou, they replace condition (ld)
with Lj a{ = 1, for all i, Finally, they define
as
= I;,EOr , whcrc or is thc set
of shared outputs of component r. Note that now in general 2:i aI ? l.

. .
'

ar ar

In '24], the anthors also address the problem of determiuing in which areas a generic
DMU k would perform better. Such areas form the DMU k core business. In a perspective of a multi-plant company general mauager, the core husiness areas of the different
DMUs should be privileged even at the cost of possibly forcing some DMUs to abandon
the components with less satisfactory performances. With this aim, Cook and Green
further modify the original model (1). In this case, they also introducc as objective function e Le d,;µrY{
whcre di: are binarv variables which assume valuc l
k - Lr d,vr x; +I:r d;;vSr(ar Xf )'
.
"
'
.
.
if component r is attributed to DMU k, O otherw1se. When such an obJect1ve funct10n
is considered, a DMU is assigned only its most efficient components. Some constramts
are also added to the mnlticomponent model. In fact, each DMU must have assigned at
least a component and each component must be assigned to at least a DMU.
Jahanshahloo et al., in [25], extend the model proposed in [7) They introduce
sharcd outputs and consider panel <lata. The same way _25, rclates to (t as ·s] rclatcs
to [10]. Also, Jahanshahloo et al. prove that the aggr~gate efficiency of th_c wholc DMU
is a convex combination of the efficiency of the DMU components even m presence of
shared output. In [26], the same authors further extend their model introducing nondiscretionary inputs as something different from the normal inputs. The former inputs
are not under decision making control, then they are considered as negative terms in
the numerator of thc fraction that describe a DMU efficiency value. The efficiency of
each component r of a generic DMU k is then defined as

ek =

µrY{

+ µs'(f3"Y/) _ PNr(,r X{:)
yr Xf; + ySr(ar XkS)

(9)

where X N = {x{,;} is the vectm· of the non-discretionary inputs, ''( = {,[} is the vector
of the proportions of the non-discretionary inputs allocated to component r by DMU k,
and pNr = {pfr} is the vector of the weights of the non-discretionary inputs. :"'eig~ts
may differ in the different components of a same DMU. Jahanshahloo et al._ lmeanze
the models proposed in [25, 26] changing variables as in [7]. The case study m [25, 26]
deals with Iranian commercial bank branches. Panel <lata are considered to measure
possible progress and regress (see [27]) of the bank sectors.

45

151 R. Shephard, Theory
1970.
Princeton,

References
[1] A. Charnes, W. Cooper, E. Rhodes, Measuring the efficiency of decision making
units, European ,Journal of Operational Research 2 (6) (1978) 429-444.

[2] W. Cooper, L. Seiford, J. Zhu, Handbook on Data Envelopment Analysis, Kluwer
Academic Publishers, Boston, Ma, 2004.

[3] W. Cooper, L. Seiford, J. Zhu, Data envelopment analysis: History, models and

Costami Producticm F'rontier Princeton Cniversity Press,

16] R. Banker, A. Charues, W. Cooper, Some models for estimating tcchnical and scalc
inefficiencies
DEA, :Vlanagement Science 30 (9) (1984) 1078-1092.
P Tsai,
:Vlolinero, The joint determination of cfficicncics in DEA: An application
to the UK health service, Department of Management Discussion Paper, University
of Southampton .

interpretations, in: W. Cooper, L. Seiford, .J. Zhu (Eds.), Handbook on Data Envelopment Analysis, Kluwer Academic Publishers, Boston, 2004, Ch. 1, pp. 1-39.

(18] A. Diez-Ticio, :vr. :Vlancebon, The efficiency of thc Spanish police service: an application nf the multiactivity DEA model, Applied Econornics 34
(2002) 351-362.

[4] W. Cooper, L. Seiford, K Tone, Data Envelopment Analysis, Kluwer Academic

F Arcelus, D. Coleman, An efficiency review of uuiversity departments, Interuational Journal of System Sciences 28 (7) (1997) 721 729.

Publishers, Boston, Ma, 2000.

[5] Y. Yang, B. Ma, M. Koike, Efficiency-measuring DEA model for production system
with k independent subsystcms, Journal of the Operations Research Society of
Japan 43 (3) (2000) 343-354.

[6] L. Castelli, R. Pesenti, W. Ukovich, DEA-like models for the efficiency evaluation of
hierarchically structured units, European Journal of Operational Research 154 (2)
(2004) 465-476.

[7] W. Cook, M. Hababou, H. Tuenter, Multicomponent efficiency measurement and
shared inputs in data envelopment analysis: An application to sales and service
performance in bank branches, .Journal of Productivity Analysis 14 (3) (2000) 209224.

[8] C. Molinero, P. Tsai, Some mathematical properties of a DEA model for the joint
determination of efficiencies, .Journal of the Operations Research Society 48 (1)
(1997) 51-56.

[9] P. Tsai, C. Molinero, A variable returns to scale <lata envelopment analysis model
for t.he joint det.erminat,ion of efficiencies with an example of the UK health service,
European Journal of Operational Research 141 (1) (2002) 21-38.

[10] .J. Beasley, Determining tcaching and research efficiencies, Journal of the Operational R.eseach Society 46 (4) (1995) 441-452.

[11] W. Cook, Y. Roll, Partial efficiencies in data envelopment analysis, Socio-Economic
Planning Sciences 37 (3) (1993) 171-179.

[12] R.. Thompson, L. Langemeier, C. Lee, E. Lee, R. Thrall, The role of multiplier
bounds in efficiency analysis with application to Kansas farming, Journal of Econometrics 46 (1-2) (1990) 93-108.

W. Cook, \1. Hababou, Sales performance measurement bauk branches, OmegaInternational Journal of Management Science 29 (4) (2001) 299-307

:21] K. Bala, \V. Cook, Performance measurement with dassification information: an
enhanced additive DEA rnodeL Omega International Journal of Mauagement Science 31 (6) (2003) 439-450.
22] M, Asmild, J, Paradi, V, Aggarwall, C. Schaffnit, Combining dea window analysis
with the malmquist index approach in a stu<ly of the Canadian banking industrv,
Journal of Productivity Analysis 21 (1) (2004) 67-89.
i23] J. Paradi, C. Schaffnit, Cornmercial branch performance evaluation and results
communication in a Canadian bank: a DEA application, European Journal of
Operational Research 156 (3) (2004) 719-735.
r241
)
J W. Cook, R. Green, Multicomponent efficiency measurement and core business

identification in multiplant firms: A DEA model, European .Journal of Operational
Research 157 (3) (2004) 540-551.

[25] G. Jahanshahloo, A, Amirteimoori, S. Kordrostami, Multi-component performance, progress and regress measurement and shared inputs and outputs in DEA
for panel <lata: an application in commercial bank branches, Applied Mathematics
and Computation 151 (1) (2004) 1-16.
[26] G. Jahanshahloo, A. Amirteimoori, S. Kordrostami, Measuring thc multicomponent efficiency with shared inputs and outputs in data envelopment analysis,
Applied Mathematics and Cornputation 155 (1) (2004) 283-293.
[27] H. Tulkens, P. Eeckaut, Non-parametric efficiency progress and regress measures
for panel <lata: methodological aspects, European .Journal of Operational Research
80 (1995) 474-499.

[13] .J. Doyle, R. Green, Efficiency and cross efficiency in DEA: Derivations, meanings
and uses, Journal of thc Operations Research Society 45 (5) (1994) 567-578.
[14] C. Molinero, On the joint determination of efficiencies in a <lata envelopment analysis context, Journal of the Operations Research Society 47 (10) (1996) 1273-1279.

46

47

+
r
i
1

The 811, lnternalional Symposium on
Operalional Researoh in Slovenla

S@R 'OS
Nova Gorica, SlOI/ENIA
September 28 • 30, 2005

Section 1:

48

v

1

OPTIMISATION OF SCHEDULING: A CASE OF
INDUSTRIAL CHAIR MANUF ACTlJRING
Janez Abram
Novoles d.d .. Na žago 6. 835 Straža, Slovenija
Janez.Abram@novoles.s1
Lidija Zadnik Stirn, Mirko Tratnik. Leon Oblak
Univerza v Ljubljani, Biotehniška fakulteta
Jamnikarjeva l O l 000 Ljubljana
lidija.zadnik@bf. uni-lJ .s1
rrmko.trntnik@bLum-lJ .s1
leon.oblak@hfuni-U.si

Abstract
fndustrial chair 111a1111jact11ri11g can be looked at as a complex production system requmng
orga11isatio11 a11d teclmical skills. Heller-Logema1111 algorithm was tested jor better s11rveyi11g and
orga11isa1iu11 uf prod11ctio11. The productiun limes far each opera/ion were time co-ordinated. A
ge11e1ic algurithm based computer programme tu uptimise mamtfacturing cycle limes wm developed.
The 9de limes jiJr 1he selected products were reduced anJ Jorecast i11for111atiu11 /iJr leading
production were ohtained

Keywords: schedu!ing, I lel!er-Logemmm !1lgori1hm. genetic algori1/1111. chair 111amtfac!11ri11g

L

INTRODUCTION

Because of the market demands, gone are the good o!d days when companies held high stocks
ofraw materials and use inflexible production planning techniques based on stock control and
offer more options in products with greatly
long range forecasting. Companies have
reduced hfe cycle and product complexity. All this has had a great effect on !he production
organisation and the manner of scheduling and launching products into the manufacturing
process.
Production scheduling is the activity perfmmed in manufacturing companies to manage and
control the execution of the production process. The basic task is to perform the production as
planned while at the same tirne trying to satisfy the overall goals of the company. Production
scheduling involves master production scheduling (MPS ), materials requirements planning
and shop floor scheduling. Each of these levels of scheduling impact on the others.
There exists much interest from industry in using software systems to support the scheduling
process but the application of such systems has shown to be problematic. This paper reviews
area of production scheduling and outlines the Heller-Logemann and genetic algorithm
implemented in complex production system.

2.

SCHEDULING METHODS

The scheduling approach and methods that are suitable for a production environment depend
on the characteristics ofthe environment, the complexity, uncertainties and randomness ofthe

51

production system, the scheduling objectives and the orgamsation around the schedulmg
function. Scheduling methods can range from simple rules for choosing which job to execute
sophisticated
next often called dispatching rules or individual heuristic approaches,
optimising methods [6].
For solving the problems of individual product operation schedule on an individual machine,
where scheduling is perfo1med on a fixed set of orders, there exist many methods. For the
tirne scale planning, we can use the method of linear programming, dispatching mles, the
Johnson algorithm, the method branch and bound ... For dynamic scheduling where new
orders are continuously added during scheduling, we should use other method such as the
the production
is up
Heller-Logemann algorithm or heuristic priority mles [2].
characteristics as to which method to use.
The most basic scheduling method isto use dispatching mles (also called priority sequencing
rules) to determine which order to run next at a work centre. These rules are applied when
jobs arrive at a work centre or when fixed set of orders are planed at a work centre, to choose
the next task to be executed. Since dispatching mles only use infomrntion that is available at
the moment when the next activity shall be selected, they work equally well in systems with a
high degree of uncertainty as in more stable environments. When there are high levels of
randomness and uncertainty in the production environment, dispatching mles may be the only
viable way to schedule the production. There exist many dispatching rules, some of the most
common are (6]:

First come, jirst se1ved (FCFS). Jobs are processed in the order they arrive at the work
centre.
Shortest processing !ime (SPT). The job with the shortest processing tirne is processed
first.
Earliest due date (EDD). The job with the earliest due date is processed first.
Critical ratio (CR). A priority index is calculated using (tirne remaining/work remaining).
A ratio less Ihan 1 means that the Job is late. The job with the lowest ratio is processed
first.
Least work remaining (LWR). Priority based on ali processing tirne remaining until job is
completed.
Fewest operations re111ai11i11g (FOR). Priority based on number ofremaining operations.
Slack time (ST). Jobs run in the order of the smallest amount of slack.
Slack time per operation (ST/O). Slack tirne is divided by the number of remaining
operations. Jobs are sequenced in order ofsmallest value.
Next queue (NQ). The queues in front of successive work centres are measured (in hours
or number of jobs). The job that is going to the smallest queue is processed first.
Least set-up (LS). The job with the least set-up tirne is processed first.
The general properties of these rules are different. SPT, and its variations LWR and FOR,
reduces work in process inventory, average job completion tirne and average job lateness but
can cause starvation of jobs with long processing limes and thus cause missed due date. EDD,
and its variations ST and ST/O, reduce job lateness but result in higher average tirne in the
system. NQ and LSU maximise machine utilisation. There exist many other dispatching rules
and also variations of the above rnles. To combine rules, for instance using different rules for
different work centres is also possible. Scheduling using these rules can, depending on the
scheduling problem, give good results but there is a risk of sub-optimisation since the
information used is local and no consideration is given to the globa! stale of the production
system.

52

Johnson's algorithms. have been developed for single machine, two or three machine and jobshop scheduling problems. These algorithms could be used in the multi-machine set-up in the
circumstance that one ofthe machines is a bottleneck to production for example, or where two
machines are very closely associated based on product routings. The algorithms could then be
applied to the appropriate machine and the rest of the schedule developed around this core by
forward and backward scheduling [5].
In a tlowshop scheduling problem there is a set of n jobs to be processed in a set of m
machines in the same order. First in machine 1 then on machine 2 and so on until machine m.
find a sequence for the processing of the jobs in the machines so that a
The objective is
in the chair manufacturing process there is a set of n jobs to
given criterion is optimised [
be processed in a different set of machines in the different order. Assembling of a product
influences the manufacturing and the choice of algorithm (e.g. we cannot assemble the fina!
chair joint if we do not have all joint elements, which is typical of connected working
assignments). The scheduling problem is solvable with the Heller-Logemann algorithm in
which we should also consider a shifts utilisation.
The Heller-Logemann algorithm is a derivative of network planning, allowing scheduling of
parallel network plans where operations are dealt with as knots in a network, the relations
among them or the technological sequence of operations is shown with aITows. Ali other
characteristics of a classical network planning are true also for the Heller-Logemann
algorithtn. The algorithm not only considers the production tirne but also the inte1iwining of
operations and inter-operational deadlocks and ailows scheduling ahead (running to the right)
and backwards (running to the left) f2].
In future account ahead (running to the right) schedules are defined from the anticipated
starting deadlines of work orders onwards. What is formed are tables with operations needing
to be performed in a term unit. The table is then arranged according to codes of working
places in the way that ali the operations performed in the same work place are joined in
groups, iITespective of the work order they appear in. The calculation of the initial and the
fina! te1m of operation is performed in many transitions through the table of operations. When
operations perfmmed in the same work place at the same tirne, external - costumer and
interna! - scheduling priority rules should be considered. Interna! priority rules are set by
assemble technological characteristics of products (figure 1) and basic scheduling methods
like first come, first served or shortest processing tiine. External p1iority rules influence on
completion tirne of the ordered products. With a job permutation we could search a schedule
where a given criterion is optimised.

element3

@

@D

@)

@)

@)

@)

@)

@)

@)

(working day

1

Figure 1. Influence of assembling of sub-joint on scheduling process

53

Elements must be produced before assemhlmg mto sub-Jomt Element 1s processed first and
have highest priority because completion tirne of elements 2 and is shorter. Critical path of
assembling product determines the completion tirne and include element and sub-joint.

some authors, such as Bernik [3 Crossover is not usually applied to all pairs of individuals
selected far mating. A random choice is made. where the likelihood of crossover being
applied, it is typica!ly between 0.6 and
If the crossover is not applied, offspring are
produced
by duplicating the parents.

2.1 Genetic algorithms for scheduling

Fitness function is responsible for evaluation of quality of given code string. In the considered
cases the cycle time or completion tirne of the last job is a value, which has to be minimised.
This tirne is calculated for specified number of jobs, consisting of a large number of
operations.

Genetic algorithms are a general methodology for searching in a discrete solution space m a
way that is similar to process of natura! selection procedure in biological systems. The
algorithm can be applied to different problems [7].
Genetic algorithms have many variants, many of the basic ideas exist, and imliv1dual
applications may be highly different. Standard genetic algorithms at random or heuristicaHy
generate an initial population of individuals (chromosomes). At every evolutionary step.
known as a generation, the individuals in the current population are evaluated according to
some predefined quality criterion, referred to as the '·fitness'" function. To form a new
population ( the next generation), individuals are selected according to their fitness. Then
some or all of the existing members of the current solution pool is replaced with the newly
created members. Creation of new members are done by crossover and mutation operations
l9J. Genetic algorithm strings encoding the solutions are often binary coded. ln ,mr case a
chromosome is represented as a hst of ordered products ]. whose production should be
scheduled according to prescribed criteria and restrictions in the production process.

3.

The genetic algorithm obJect determines which individuals should survive, which should
reproduce, and which shouid die. also reconJs statistics and decides how long the evolution
should continue. Typically a genetic algorithm has no obvious stopping criterion. One must
tel1 the algorithm when to stop. Often the number of generations is used as a stopping
measure, but goodness-of-best-solution, convergence-of-population, or any problem-specific
criterion can be used [8].

Comparing the actual schedule of products in the manufacturing process with optimal
computer schedule, we can observe that in the latter results are better. The lowest cycle times
for individual products are achieved in the schedule found with the genetic algorithm. By
looking for the optimal schedule, the average cycle tirne for manufacturing products
compared to a planned schedule determined by a monthly plan is shortened by up to 30 % [4J.

When creating new population by crossmer and mutation, we have a big chance that we will
loose the best chromosome. Elitism is a method. which first copies the best chromosome (or a
few best chromosomes) to new population. The rest is done in classical way. E1itism can very
rapidly increase performance of GA, because it prevents losing the best found solution [9].
The mutation operator defines the procedure for mutating each genome. Mutation means
different things for different data types. For example, a typical mutator for a binary string
genome flips the bits in the string with a given probability. A typical mutator for a tree, on the
other hand, would swap sub trees with a given probability. Mutation should be able to
introduce new genetic material as well as modify existing material [8].
The crossover operator defines the procedure for generating a child from two parent genomes.
The standard crossover operator called simple crossover has numerous variants such as
partially-mapped, position-based, order-based, sub tour chunking, cyclic, acyclic, inversion,
and edge-recombination crossovers. Ali of these involve two parents [9]. Better results have
been obtained by rejecting the conventional binary representations and using more direct
encoding (a schedule genotype is a list specifying the order). But simple crossover applied to
such strings would nearly always result in illegal offspring with some orders missing, while
other orders presented twice. Hence more sophisticated crossover operations are needed. In
our scheduling problem we used operator called liner order crossover which was suggested by

54

RESULTS

In the "hand made'" schedule different dispatching or priority sequencing rules were used to
determine which order to run next at a work machine, where utilisation level is the highest.
Because of the complex production scheduling without computer support, we focus on
bottlenecks in the production system. The bottleneck resource determines the throughput of
the whole system and therefore its utilisation should be maximised. A buffer is put before the
bottleneck resource to ensure that never has to wait for jobs to execute. Resource-based
schedule requires that there exist just a few bottleneck resources and that they should be well
defined. In reality happens that the bottleneck resource shifts over tirne and thus, there appear
problems.

The biggest savings in cycle tirne of operation in manufacturing chairs is achieved by
rearranging certain working operations to optimal terms, considering the utilisation of
individual, i.e., key working places. Success in searching for good schedule depends also on
the type of product being simultaneously manufactured. Similar products produced in the
same working places in a similar sequence of operations burden only certain machines, which
therefore become overloaded. By planning mutually different products, we can partly
reallocate the burdening ofproduction to different working places.
The results of the application are shown in Gantt charts and graphic displays of the capacity
burden of the machines and workers.

4.

CONCLUSIONS

Due to the fact that the sequence of products influences the cycle tirne, we tried to find a
better schedule with the help ofHeller-Logemann algorithm and stochastic techniques such as
genetic algorithms. Comparing the "hand made - manual" schedule with optimal computer
schedule, we can observe that in the latter results are better.
The next step is modified Heller-Logemann algorithm to consider machine and worker
utilisation. With further comparison of various scheduling methods and quality function we

55

could achieve better production results. The planned deadlines of work operations vary due to
the breakdown of machines, the variable quality of input raw material and the cancellation of
orders. Therefore, the organisation model should be supplemented with the reallocation of
activities in the manufacturing process.

REFERENCES

T
LOCAL OPTIMIZATION APPLIED TO A VOLl.l\ITEER
TIMETABLING PROBLEM
\lateJ Boro,mšek 1 • Janez Žerovmk' ·
t·,m·er,n, ,,t \lanbor. Facult\ ofMechanical Engmeermg. Smetanova !7. 20()[1 Maribor Slovenija
. lMFM/DTCS. Jadranska 19. !000 L1ubljana. Slovenija
!:mails: mateJ.bon,qnsek@uru-mh.si. 1anez.zerm lll k@1ml 111.um-l 1.s1

Abstract

!2j LJLBIČ, T.: Plamranje in vodenje proizvodnje. Moderna organizacip. Kranj, 2000. 443 str.

The tirne table assernbly is problem in which the feasibilit\ llf solutions 1nvoives solution of a graph
C(1l()r111g pn,J,lern while the qualit\ nf solution 1, ohtai11ed h, upt1mizat1on ,,f a fitness function. In
this paper the time table assemhh was done us111g the method \lf local opt11mzat1011. For this pmpose
a ,peL·11tl fit11ess funct1on \\as de\eluped w!th regard to the suft cpnstramts ol the part1cular proble111

[31 BERNIK, l.: Večkriterijsko razporejanje z genetskimi algoritmi in vizuelno simulacijo s Petri mrežami,
Doktorska disertacija, Kranj, FOV, 2001. 75 str.

Ke:n~ords: ume table. local opt11rnzat1on

[ll BRESKVAR, lJ., KLJAJTC'. M.: Prednosti planiranja proizvodnje s simulacijo in genetskimi algoritmi.
Organizacija, 200 L letn. 34, št. 1O, str. 670 675

[4] ABRAM, J.: Optimizacija termmskega in kapacitetnega
Ljubljana, BF, 2005. 82 str.

načrtovanja

pri proizvodnji stolov, Magistrsko delo.

[5J TOAL, D., COFFEY, T., SMITH, P.: Expert systems and simulation in scheduling, Department of
Electromcs and Computer Engineering, umversity of Limerick, http:,,www.ul.ie1~toald!Publications/lmcl les.pdf, 8 str.
[6] KJELL, O.: An investigation into production scheduling systems, Doctoral dissertation, Karlstad University,
2004, !08 str.

l71

ZDANSKY. M„ POZlVIL, J.: Combination genetic/tabu search algorithm for hybride flowshop
optimization, Conference on Scientific Computing. Proceedings of ALGORITMY. 2002. str. 230 - 236

l8] MATTHEW, L„ GAlib:A c, • Library of Genetic Algorithm Components, Mechanical Engineering
Department, Massachusetts Institute of Tecnology , http;;,']ancet.mit.eduiga/galib-reguest@nut.edu, 1996,
101 str.
[9] SANDIKCI, B.: Genetic Algorithms, Department of fndustrial Engineering, Bilkent Cniversity,
http://www.bilkent.edu.tr1~burhan, 2000, 9 str.
[I0]RUBEN, R., CONCEPCI6N,M.: A comprehensive review and evaluation of permutation flowshop
heuristics, European Joumal ofOperational Research, 165 (2005), str. 479- 494

56

l. lntrodudion
It 1s \\eli kno\\ n that e\en in 1ts s1rnplest form limetabling is NP-hard. As. 011 the other
hand. there are many praclical l!metabling and scheduling prohlems. there 1s a Hlst literature
1m th1s topic. and many forms of the problerns hm·e been studied rncluding course ami
school ll!netahl111g. examination timetabling, sports timetabling, emplovee 11metabli11g. etc.
For relerences. see the suney [Schaerf. 1999I and a weh page [Tnck].
The Ernplnvee Timetabling Problem consists of a periodical assignment ot employees to
tasks 111 shifts with fi.xed start and end limes. A simple exarnple of a real world prnbiern 1s
that o! timetablrng nurses in a departmem of a large hospital. Formally. 1here are 111
emp!ol'ees. 11 slzif'ts, and t tusks. We search for an assignment. wh1ch can be defined as a 3dimensronal binary matnx X. such that X(i,).k!= 1 if employee i is assigned to t.ask j in sh1ft
k. All\ solu1ion must fulfill aJditional constraints, which can be briefly g1ve11 as foilows
ReCJuiremem.1: Each shift is composeJ of a number of tasks. some of lhem mu!t1ple tlmes.
An ernplmee is needed to be assigned for each task beionging a shift. Abilitv: Each
ernpJO\ee has 4ualifications that enable her/him to fulfill certain types of tasks: that 1s. each
emplovee has a sel of tasks that can be assigned to. Availabi/it'.: There are personal
preferences uf employees. which restrict them to be assigned only to subsets of the shifts.
Cw;/lin,1: Obviously. an employee cannot be assigned to more than one task in the same
shift. ln addition. employees cannot be assigned to two shifts that are in cmifliu with each
other. Sources of conflicts could be different: overlap in tirne, consecutive, or combinations
that are forbidden by organizational rules. Wnrkload: There is an upper and lower limit on
the number of tasks that each emp!oyee can be assigned to. There are actually a set of limits,
because employees can be assigned to a limited total number of tasks per schedule and also
to a limited (smaller) number of specific assignments. For more details, see [Meisels and
Schaerf, 2003 J and the references there.
A feasible solution is any assignment that satisfies ali of the above constraints. General
defimtions mclude aiso soft constraints that constitute the fitness function (to be minimized).
Soft constraints generally regard fair distribution of loads for employees. For example, when
a nurse is assigned to two night shifts, it is considered much better to have them spread
evenly over the week (e.g., at least two free nights in between). In employee timetabling, a
search problem is sometimes considered, meaning the goal is to find any feasible solution
because already a feasible solution may be very difficult to find.
In this paper we consider a problem which may be regarded as an example of volunteer
timetabling. We use the name because of the particular example which motivates this work

57

and hecause we beheve thal lor vuiunteer llrnetablmg, the constramts ,,!II 1:,p1call) be eas\
t(1 sallsl\
wh1le un lhe other band vanous mterpretations of lairness are \ery important
Uearlv. d volumeer that leels that he 1s trealed irnproperly would easily quit the associmion.
Thcrefore. 1t 1s \erv 1mporta!ll 10 des1gn a fitness functiun which \,lil force the solut1ons to
sat1sl\ sume la1rness requirements. In other words. the forma! defirntion of the volunteer
t1rneL1blmg is the same as for ernployee tirnetabling. but in the typical instances of \olunteer
timetabling the d1fficult part of the problem is expected to he in the optimization of the
fitness luncuon.
Thc paper 1s nrgarnzed as follows. A motivation with the practical problem description is
given :md a simple procedure for genernlion of feasible solutions is explained. Then the
fitness funct1on 1s defined. wh1ch is used for a iocal search improvement algorithm. Results
are b11efh d1scussed.

y
Star wHh lhe first
day of lhe monlh

. )•

, - - ____J

Nextday

1~

"NC

>\l~:~~n~~~;re

)4

··e~

s the monlh over

sortec
Arrange assigned duty
tums

1

-----,;;:-

1s lhe number of duty
turns be!low lhe limit'i

Remove redundar:
duly '.Jm~

3. Problem description
In tirne nf ski season on Pohorje there is a mountmn rescue service responsible for the
safe!v' of sk1 tourists. The safel\- i~ assured with dail\ dutv., turns on ali ski trails during da)
ancl rnght ski111g. The tirne table for duty turns has to be assembled.
Day ancl rnght cluty turns have to be distributeci along the members uf the rnountain rescue
service. In respect to the month of the ski season and on lhe spec:ific da) of the week there
have tu be 4, 6, or 8 duty turns per Jay.

~s lhe number of ducy "'turns above the limit~

/e~

~

Add missing duty :'.,,;ns

~

Table 1: Duty turns J1vision and theff symbols
r>uty turns per day

4
6
8

Daily duty

Nightly duty turns

4( ,2
4)
6(1,2,3,4,A,B)
/
6
(
,
2
,3,
4,
A,
B)
2
i_N
L__:_'-C_'--=-=2--'-"----'--'------'--'--'---

t_. _ _ _ _.:::__ _ . _ _

Daily cluty turns take place on two different ski areas of Bellevue and Areh. On the Areh
sk1 area there are four rescuers needed to fulfill the safety regulations (symhols from l to 4J
while on the Bellevue ski area there are only two rescuers needed (symhols A and B J. In case
of clays with night skiing in the area of Bellevue two additional rescuers are needed (both
marked with symbol N).
Each member of the mountain rescue service has to fulfili the norm of ten to twelve duty
turns per ski season. Because the tirne table is assembled only for one following month this
means the norm is from three to five duty turns per month. In lhe beginning of the new
month each member may choose days of his duty turns (using symbols 1 - 4. A. B. and N)
and clays when he will be absent and can not do his duty (symbol X) Finally, wishes of
members with longer membership are considered of greater importance then wishes of new
members.

4. The algorithm for the generation of a feasible solution
The first solution is obtained using the algorithrn shown in figure l. As mentioned before it
consiclers only the condition about the number of dernanded duty turns per day. The number
of demandecl duty turns can be defined separately for each day or the default values are used.
The algorithm first determines what kind of day is selected. It can be the day with more,
equal or less entered duty turns per day as demanded.

58

/

Ne
~

Figure 1: Diagram ni the algorithm for the first soiut1on
[11 case ol more entered duty turns per dav as demanded it 1s necessarv for redundant drnv
turns to he renwved. Redundant Juty turns could be selected randornly but 111 order fen the
firsl solution to be as goocl as possible the sorted list was used. The 11st consists only of
members whose wish was to do duty on selected day and is sorted usmg two criteria. The
first cnierion 1s lhe number of monlhly duty turns of the member ln case of more than one
member with tile same numher of mornhly Juty turns they are sorted with regard to their
rnernbership tnne. So when the redundant duty turn is removed it is first removed frorn the
member \\ 1th the largest number of dut\ turns per month. !! there are more tlian one such
members the Jut\ turn is remo\ed frnm the member with the shortest mernbership ume. This
rerncl\al process ·is then repeated until ali redundant duty turns are removed.
ln the oppos1te case the duty turns have to be added. The adding of duty turns is also basecl
on a list smtec! similarly as described above but the list consists only of mernbers which do
not have a w1shed duty turn on selected day. The duty turn 1s first added to the rnember w1th
the least dut\ turns per month. The process of adding is repeated until the demanded nurnber
of clut\ turns is reached.
The;e dutv 1urns have to be subsequently reordered with respect to the ski area and the tirne
of dut\ (T~ble I l This reordering is done using another sorted list which consists only of
rnernb~rs whose wish was to perforrn duty on selected day. In case of two rnembers having
the same wish for the selected day the wish of the member with longer membership tirne is
accepted while the other member gets the next available duty turn.

S. The cost function and the optimization algorithm
A cost or fitness function that takes into account the duration of the rnembership, number
of duty turns per month and the number of duty turns changes with respect to expressed
wishes. The optirnal solution will be the one with the lowest fitness value.
First the evaluation of each mountain rescuer is done by comparison of changes between
wished and assigned duty turns. The state of each day in the rnonth is compared and number

59

T

1

(lf removed N .,,,

addecl

tv,,,,.t

1ind

changed A' '"' ,!ut;, lurns 1s counte,i. ! he changed dut)

turn re1°resellls lhe change ,ii dut, turn area ur tirne, Tu evaluate these changes three weight
par,uneters were 1ntroduced. Parameter h,rn, to account for rem1l\ ed duty turns. h„J,1 lo
accllun: tor &!ded Juty turns and h ,,,, for changed duty turns. With those parameters definecl
the mernber's i1tness s can be written as

( 2.1)

Thč' llč'\\ ;ipproad1 tur arrangmg remmed Juty turns rncluded the mformation about
mt'mber' s \\ 1shes. Tile solutwn changtng was started the same \\ a;, as before w1th clut\ turns
re1rn)\ al and then for each dm uf the munth tile following procedure was repeated. For ali
members wllh duty turns remO\ed it was first checked 1f an\ of them e\pressed a w1sh for
the dut, tm11 011 the selected da\. ln case ol existing clutv turn wish dut\ turn was ass1gned to
lhe rnernbers. The membership list was taken into account to give alh antage to w1shes of
lllt'mbers \\ 1th lunger membersh1p tirne. Lastly tile remaining dut) turns lor the selected clav
,vere randoml\' assigned to the rest of the members. With this approach the increase of
4uality of generated solutions was achieved (Figure 2)

Add1t1011a!h the number of ass1gned clut) turns per month has IO be e,aluateJ in respect w
presen bed ii11ws. The mm1murn number of Jul:, turns per munth is three v, h!le the maxirnurn
m1111he1 uf
turns :s fiH· If N„ 1s the numher 1)f some member·s
tu1ns per munth
then the d1ffcrence

111

- "•

N It ;:c:

F: st ar,rvoacr

.,,,. -Nev, ari :oach

9 - 4 - \ - - - - - - - - - - - - - - - - - - - - --\

between the number of dur, turns and 1t's lim1ts can be \\ntten as

o.,

12.2)

nI=
' !\,

~

_:::J

<;

~

---~--

-----~ --------------

U'

______,

~
E
- 5 -i---->-----""'=-~---------------

The member's fitness can m1w be 1mprmed co11sidering this difference as

---,

""
D

B o,4

s I' =lh

.!\.· .,,,

~

+ h„,1,1 · N,,, 1,, + h .,,,, N .,,,, )+ c"' .

1

•_ 3

+---4=--------'>-==-'k-~--- _ _ _ _ _ _ _ _ __,
1

--til-

8--- - . .

whcre the c stancls for the weight parameter of the difference in dutv turns numher.
The 111embersh1p duration is accounted for with the weight parameter a and the position
1 index) of a member in the membership list. The membership list is a list consisting of all
rnembers sorted hv their mernbership duration w that the member w1th the longest
membership 1s 111 the first place nf the list. The solution fitness rnlue can he calculatecl as:

S=

L

N
i::::I

(/

1

,v
[

(br,·111 N

re,n

+h„,1,1 N,,,,t,I +brl,a N,,c1,,,)+c111j.

t24)

where N is the number of ali members ancl I is an index of a member in the rnembership list.
Using this function any ttme table configurat1on can now be evaluated 1f the values of
weight parameters a, br,.,,,, h,,,1,1 , bc1,,, and c are defined. Based on nmltiple randomly generated
sets of cluty turn wishes the values of weight parameters were determmed to achieve the
convergencc as fast as possible. Selected values of weight parameters equal to a = 1.2.

hrna

= 0.3 , h,,,/,1 = 2,0,

h„J,,,

= 0,1

and c

= 1000 .

2ocn

400L

600fl

=i
~---4

--e----._._

ID-

'~ L

_ , , __ _ _

8000

1000('

()rti n1izati 011 ster

Figure 2 Cornparison between the first and impnned approach for arranging remcn:ed dutv
turns
The nurnber (lf mernbers whose duty turns were rernoved 111 each optnnization step was
fixecl. The number was hased on some optimization tirne companson bet,,een tesrs of
ranclomly generatecl sets of tirne tables and was set to 10% of ali mountain rescue sen 1ce
mernbers. E,en faster optirnization could be achieved with additional comparisons between
optimization times and solution qualities.

6. Results

The optirnization was done using the method of local optimization ln order to get the
optimal solution with thts method the existing solution is ranclomly changed on a small
(local) area and these changes are then evaluated using the fitnes s function. The generated
sol uti on is accepted only if it is better than the existing one.
The solution in the described problem is represented by the duty turns of mountain rescue
service members. To change such a solution on a local area some members were randomly
selected and their duty turn entries completely removed. These duty turns were then arrangecl
back to those members in a random manner. With this process the working algorithm was
developed but a lot of generated solutions were necessary to improve the fitness. Because of
that a different approach for arranging removed duty turns was developed.

As a test e\arnple the w1shes of mountain rescue service members for Februar, were used.
The data was entered _iust for members that expressed their wishes on tirne. For other
members the data was left empty. At the moment there are sixty-seven active members of the
mountain rescue service. Because of the high number of expected visitors the number of
clemanded duty turns per day was set to eight for ali days in February
The first step of the algorithm assures that each day of the month has the clemanded number
of duty turns. In our case after the first step ali days have eight cluty turns. But big
differences m number of members rnonthly duty turns are observable. They range from zero
to eight duty turns per month and are shown in figure 3. There are thirty-two members with
duty turn numbers within limits ancl the rest of the members is whether under or over the
limit. These nurnbers should be reduced by the optimization step.

60

61

DYNAMIC PROGRAMMING IN CONTROL

5[

------------------

4L
~

-----------

:35 + - - - - - - - - - - - -

--

Lado Lenart. Jan Babič
"Jožef Stefan" Institute, Jamova 39. LjublJana
jan.bab1c@ijs.si
iaciu j:_•g1 '-'-'- '~-~

--------,

- - - - - ---·-~~

.0

~

3[

E

Abstract: The use of dynam1c programmmg (DP) 111 control systems 1s based ona relation between the
Hamilton-Jacoby-Bellman equation (HJBE) and the DP. W1thout calculating costate variables in
canonical HJBE system equations, the iteration algorithm 1s shown to get solut10n in minimizing a series
of control variables.

b '251:i;
.C

2[1

E

~

5

+

5+

Keywords : Hamilton-Jacobi-Bellman equation. opllmal controL direct 1terative procedure
3

4

6

8

Duty tums per month

hgure

1·

Tile Jistnbu,1on uf the rnonthly dut) turns

To nptllrnze tl11s case approxirnatelv 100.000 optirnization steps were necessar) t(1 fulfill
tile rcquirements lor the number of duty turns per month of each member :\umber oi
opt11111zal!o11 steps ,s d1fferent for eacl! optm11zation nm because of randornness in used
optm11zctt1ll11 method. It can be noticed tha! the nurnbers ol duty turns o! sing!e mernber per
rnonth c1re ali wnhm iimils and \ery evenly distributed. From tile fina! time table
c()nfiguratH'!l high concentratH'll of assigned Juty turns is ohsened h members that entered
a iarger nurnbe1 of dut, turn w,shes. Th1s is because the Jay select1011 rnutine in method used
lur ct1Tangmg removecl duty turns Juring opl!irnzation is not random bul stra1ght. The code
coulcl be changed to use rallClom dav selecuon but it was not necessan because the reason
t't,r emermg such large amciunts \lf Ju:, turns was rernm·ed w1lh the rntrnduclwn of coniputer
mded vers1on. namely such large numbers ol dut) mrns were entered only to ease tl1e time
tahie assernhl\ rnalie b\ hand.
7. The software
As lhe bas1s for the computer aided vers1on a Microsoft Office. specifically Microsoft
Excel was selected. Excel suppons adding and running a code written in software language
V1suai Bas1c for Appiical!on. Despite the fact that such code runs slower the Excel platform
was chosen because of ils ex1sting user :nterface and cornpatibility It has a huiit in interface
for work wirh files. for <lata entry and for priming. Such solution is includecl in an Excel file
.xls and can be used on every computer with Exce! installed.

LINTRODUCTION
The theoretical basis for the optimal control prohlem is the Pontryagin's principle. The
simplified proof can be found in [l]. The field of non-numerical solutions of the problem is
very restricted. Linear systems with quadratic functional already effectively use the analytic
solutions of Riccati matrix equation. A broad field of these equation application areas can be
found in [2]. The newest development of the general problem theory, connected with HJBE,
can be found in [3]. Numerical solutions are often based on two point boundary problems in
ordinary differential equations [4] or are presented in the programming environment [5J.
Our intention was to prove the behaviour of some algorithms, which work on a direct iteration
schema. The method proposed by [6] appeared to be perspective for some problems where the
problem of switching surfaces does not appear, or at least is not prevailing. The method is
partially modified for the reason of effective programming.
The contribution in the second section presents the connection between DP and HJBE, whereby
the connection with optimal p1inciple is systematically avoided. The third, central chapter,
shows the possibility to use the direct method of rninimizing control sequence for 'smooth'
problems.

2. HAMILTON EQUATION BASED ON DYNAMIC PROGRAMMING
In this section, the relation between dynamic programming and optimal control is shown for
continuous control systems. The dynamic system is given with (l)

References

dx,
[Schaertl A.Schaerf, A Survey of Automated Timetabling, Artif. Intel l. Rev.13 (l 999) 87-127.
[Meisels and Schaertl A.Meisels and A.Schae1f, Modelling and Solving Employee Timetabling
Problems. Annals of Mathematics and Artificial Intellingence 39(2003)41-59.

dt

=f

'

(1)

(x1,···,x,,,u1,···,u,.,t)

or in his vector form (2)

[Trick] Michael Trick's Operations Research Page, http://mat.gsia.cmu.edu/

62

63

Ti
(2)

x = f(x(t),u(t).t)
Performance function for terminal tirne T is :

*

l
T

J=h(x(T),T)+

*

(T)
J .,
L

s

ff (x(r),u(r),r)dr

(3)

0

Q
1../_

s

u1

'~I

<//

s1

1'i

Using embedding princip le we get:

~

s

T

J(x(t),t,u(t)) = h(x(T),T) +

ff (x(r),u(r), r)dr
0

(4)

Time

The construction of the optimal cost functional can be read from fig. l, , formula (5) results
( the suffix '*' always means the optimal value - under given restrictions),

t

T

t+ d.t

fig. l: Graphic representation of continuous dynamic programming

r (x(t), t) =

1

min

u(r);/.,; r .,;1+/1/

{

11

~f fodT + j' (x(t + /'J.t,

f

+ /'J.t )}

(5)

1

3. MINIMIZING CONTROL SEQUENCE

Assuming that (5) can be expanded into Taylor series, this is done and after abbreviation
follows (6).

Say we simply start with any adrnissible control u 1 (x,t) transferring the sys1em from state x 0
at time O to any state at tirne T Then, by definition the sequence u 1 , u?,.

'}
aJ'T
(
O= J,' (x(t), t) + minJ / 0 (x(t ), u(t).t) + -;::-(x(t ),t).[f(x(t), u(t), t )]
u(1)

1

(6)

UX

To find the boundary value for this PDE, at t=T, one has

(7'
' J

= h(x(T),T)

lf Hamiltonian H is defined as

H

aJ•T

= f 0 (x(t), u(t), t) + -(x(t),t ).[f{x(t), u(t), t)]
ax

(8)

will be called
...

2:J(u'). Now

av

f, av

•

.

1
1
([0)
L,-J,(x,u 1 ,t) + } 0 = O
-+
s=] ax,
at
V1 in (10) is analogue to J in (6) if V replaces J, the renaming shall make the point, that V1 1s
not optimal, but it is built along the trajectory, which is defined with u 1 • Now (2) shall run with
u 1 and its solution is dependent from initial vector x 0 .

x=x(t,x 0 )

(6) is written for optimal control, then one gets the Hamilton-Jacobi-Bellman (HJB) equation
in its usual form.
(9)

O= J,'(x(t),t)+ H(x(t),u' (x(t),J;,t),J:,1)

do not increase and J(u;)2:J(u 2 )

(3) with u 1 is valid and ali subsequent equations remain the same with the exception, that
minimization cannot be made, as u 1 is not optimaL The most important, dynamic system
obeys the HJB equation (9), which shall now be written in the slightly changed form.

1

J'(x(T),T)

minimizing sequence, if J(u 1 ),J(u 2

).,

(11)

From (4) one gets
dV
- 1 = -/0 (x, u,{t),t)
dt
If ( 12) is integrated, the boundary condition must be again
V1 =h(x(T,x 0 )) if t=T

(12)

(13)

It follows for V1 :
T

~ =fl;(t,x 0 )=h(x(T,x 0 ))+

ff (x(r,x ),u (r),r)dr
0

0

From (11) one gets the inverse function x 0

1

= <D(x,t)

one gets the identity:

64

65

(14)

This function can be inserted into (14) and

T
1

T

Vl(x,t)

= i7;(t,<l>(x,t)) = h(x(T,<l>(x,t))) +

J!o(X(r,<l>(x,t),ul(r),i)di

(15)

propose, that the generated system has the soiution ( 19). Any proper solution can be found here
and is then shifted because V1 -V1 (x,t) is constant (16).

1
1

x appears as function in ( 15) and as argument in <D and on the left side of equation. The recent
means, that any solution of ( 11) can be inserted for x, even with x 0 changed. But, the
difference V1 -V1 (x,t) remains constant for x asa fixed argument

(19)

1

Fram u 2 (x,t) the function V2 (x,t) can be constructed in the same manner as V was
constructed from u 1 etc. One gets the three sequences: u,,x 1 ,v;; i = 1,2, ... , and it remains to

Proof:

prove, that sequence u1 is the minimizing sequence.

The solution of(l2) is:

Sketch of the proof:

1

V,= V/

0

First from (15) one can see:

fj~{x(r,x ),u (r),r)dr

l -

1

0

o

Jfo(X( r, Xo), ul ( r), r)dr + h(x(T, <D(x,t))) -

(20)

f

vi - vi (x, t) = V/

0
) -

o

(16)

Then l41i(xi,ui,t) is identically equal O by (10). lt is then

T

- f/ (x(r,<l>(x,t)), u ("r), r)dr =
0

(21)

1

Before integrating (21) some preparations must be made. The critical step in integration is the
term V1 {x 1+ 1( 0),0), where x 1, 1 (O) is the point in process space and by proposition for

The difference above is only functional of u 1 , therefore a constant, q.e.d.
Now it is possible to begin with the construction ofminimizing control set. Function V1 (x,t) 1s
extended over the whole interval with the new function ( 17)

constructing functions u 1 (19) is the process initial point x 0 . This again is the initial point for
the first sol uti on of ( 15), so one can write:
(22)
V1 (x 1+1 (0),0) = V1 (x 0 ,0) = Vi(x I (0),0) = V1 (xi,O)
The last inequality in (21) can be integrated:

( 17)

f

T

o

o

T

Wi(x 1+P u 1+i,t)dt = V1 (x 1+i,(T),T)-Vi(x 0 ,0) +

Function above is the 'measure' for the distance between actual point ( x, t) and the end surface
S at t=T. Then the optimal control oftransition process ( see Annex) can be applied to find the
optimal control u2 {x,t) . This control (see (28)) will set the function
W1 (x, u,t)

" av
av + I
- f,(x, u,t) + f

=-

8t

1

8x,

0

T

= h(x 1+1(T)) + f/

0

1+1'

u 1-i,t)dt =
(23)

(x 1, i , U 1+i,t)dt-V1 (xi,O) ~ O

o

From (23) and (22) one can see that:

1

s=l

f/ (x

0

(18)

(u,x,t)

(24)

to its minimum (i.e. the negative maximum) along each point of the trajectory. Surely finding
u2 (x, t) is coupled with some numerical work, although when the partial derivations of V1 are
given, the space of u2 (x, t) must be thoroughly checked - this can already be made often in
quite direct manner and the procedure like dynamic programming must only be applied
occasionally. When obtaining the numerical solution, the fact, that the optimal control of
transition process is very similar to the minimal tirne problem can be used thoroughly. We then

66

and from (15) it follows directly:
(25)
Putting together (24) and (25) generates

67

T
i

(26)

Scheduling Serial - ParaUel Processors - A Case Study

1

Conseq uentl y, the seq uence J ( u n x 1

,

l

=

Jan Pelikan
Vysoka škola ekonomicka v Praze, Czech Republic
pelikan@vse . cz

is not- increasing and the control sequence

uP u,, ... is minimizing q.d.e.

Abstract

APPENDIX : OPTIMAL CONTROL OF TRANSITION PROCESS
If the dynamic process is given with
control u E G induces the dynamics:

l) and initial condition is x,J at tirne

then every

(27)
Let S be the surface in the process space
task is to choose u E G in the way that

Xi,.

x

x n) given by S(t,

= x(t, u, x0

,x,, = O The control

is crossing S at tirne t, Additionally

the function V (t, Xi, , x,,) shall exist, as weighing function of the distance between moving
point x and surface S. The task ofthe control shall beto lessen this distance.
0
Definition. The control u

= (u~,

. . ,u~ is transition optimal for V, if V decreases along the

O

trajectory x(t, u , x 0 ) in the fastest possible way. One has to find the values for V along the
path (27) and the total differential for V can be written in as:
dV

"av =W(t,x,u)
av I-f,

-=-+
8t ,= 1 8x,
dt

(28)

·

It is obviously, that the optimal transition control gives the function W x, u) the minimal
(negative) value - it means the fastest decreasing of V - for all possible controls u, which are ali
limited of course.

REFERENCES
[l] J.Macki, A.Strauss, Introduction to Optimal Control Theory, Springer, New-York,1983
[2] D.P.Bertsekas, Dynamic Programming, Prentice-Hall,Englewood Cliffs, New-York,1987
[3] M.Bardi,I.C.Dolcetta, Optimal Control and Viscosity Solution ofHamilton-Jacobi-Bellman
equations, Birkhauser, Boston, 1997
[4] J.Fox, L.Leslie, The numerical solution oftwo-point boundary problems in ordinary
differential equations, Dover Publications, New-York, 1990
[5] F.Lewis,L.Vassilis,L.Syrmos, Optimal Control, Wiley, New-York, 1995
[6] V.I.Zubov, Lekcii po teorii upravlenija, Nauka, Moskva, 1980

68

Methods of Operations Research can play an important role in the process of increasing production
productivity and decreasing production costs. We show in this case study how discrete optimization
models can be used to find the optimal ordering production batches on lathes. The aim of the
optimization is to suggest a processing order of production batches, which 1s connected with the
minimum total processing tirne of the corresponding production order For solving the problem, an
original mathematical model including discrete variables is proposed. The model 1s solved hy using
of the system LINGO 8.0. Obtained results are very mteresting smce they lead to a s1gmficant
decrease ofthe total processing tirne in comparison with the present pract1ce.
Keywords: operations research, mathemat1cal modeling, discrete models, scheduling models

1 Introduction
The subject of this case study is the production scheduling designed for the enterprise KBass Brandys nad Orlici. The scheduling consists in ordering batches. which are processed
on lathes in the order satisfying given technological requirements. The lathes are considered
as parallel processors. A production batch is a prescribed amount of a certain product Each
production order consists of a finite number of production batches. The aim of the
scheduling is to find such processing schedule of batches, which minimizes the total
processing tirne of the whole production order. lt is assumed that before the processing of
any batch ona lathe can begin, the corresponding lathe must be adjusted for the batch to be
processed. Therefore the adjustment of a machine is the first phase of batch processing and
the proper production is its second phase. We shall assume only one worker for adjusting the
machines at our disposaL Therefore the adjustment process will be considered as a serial
processing of batches on a unique processor. On the other hand, the proper production run,
during which each batch is processed on different machines can be considered as the parallel
processing of batches on parallel processors. Therefore, delays can occur for the following
reasons: (a) ali machines are working so that no machine can be adjusted for some next batch
(a delay in the adjustment process); (b) a machine must wait for adjustment while another
machine is being adjusted, since the adjustment is carried out only by one worker (a delay in
the production process). The minimization of the total processing tirne of all batches of a
given production order is carried out in such a way that the resulting schedule leads at the
same tirne to the maximum exploitation of the machines, minimum delays, and minimum
production costs.
2 Description of technological conditions of the case study

The production sector consists of ten lathes, six from which are of the same type. The case
study will find a schedule of batches for a one-week production. This production order could
not be managed within the capacity of the two-shift working tirne, i.e. within 75 hours. The
production order consists of 27 products. Types of products in production batches, numbers
of pieces of the corresponding products as well as the corresponding processing and
adjustment times are included in the Table l.

69

------\
Table

The production batches and their assignment to machines
Product
number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

17
18
19
20
21
22
23
24
25
26

'--- _I1__

Machine

Amount in
_f)ieces

2
2
2
3*)
4')
5*)
6*)
7*)
7*)
7*)
8*)
9
9
9
10
10
10
10
10
10
10
10
10

1000
50
1000
50
50
1500
2000
2000
667
100
300
100
200
1000
100
100
100
2000
1500
1000
100
100
300
2000
100
667
200

10
10

Adjustment ProcessTng tirne
(f!l)nut~
!i_m~iminutes)
8,16
8,16
10,20
10,20
10,20
61,20
91,80
61,20
61,20
61,20
61,20
61,20
91,80
91,80
91,80
91,80
6,00
6,00
6,00
6,00
6,00
6,00
6,00
6,00
6,00
6,00
6.!__0__Q_

100,00
5,50
260,00
16,50 I
20,50
2370,00
2440,00
1380,00
1280,64
40,00
54,00
82,00
172,00
2800,00
11,00
168,00
14,00
460,00
795,00
140,00
14,00
45,00
18,00
820,00
28,00
426,88
_§8,00 _

machines for them. The adjustment order is limited by the fact that we have only one worker
adjusting the machines and it is assumed that he cannot adjust two or more different
machines simultaneously so that he prepares the machines for the production of different
products on a sequence similarly as in the one-processor case. In the other words, he can
begin to adjust a machine only after the adjustment of a preceding machine had been
finished. Another limitation for the adjustment order follows the assumption that the
adjustment of any machine can be commenced only after the machine have finished the
production and it is idle.

xu ,

which will provide the information about the
We will introduce binary variables
adjustment order in the following sense: xu = if the adjustment for product i precedes the
adjustment for productj, xiJ = O otherwise.

Model parameters:
11 the amount ofproducts (batches) in pieces;
m - the number of machines;
the processing tirne ofthe i-th batch minutes;
d1 the adjustment tirne for the i-th batch in minutes;
M >> O - a large positive number;
the set ofindices ofthose batches, which are assigned to machine k.
Model variables:
T - the tirne, at which all batches are finished;
x;- --- binary variables determining the adjustment order;
c~ - the tirne, at which the adjustment for product is finished in minutes and starts the
producing.

T • min

*) the batch can be assigned to any ofmachines 3 - 8.

+ x1;= l
- d1 ;::: c1 + t 1 - M(l - xij)
crd1,2c;-M(l-xij)

3 A model with a priori assignment of batches to machines

d;SC;

XiJ

First we shall assume that each production batch is assigned to a machine. This assignment
mostly follows the fact that certain types of products can be produced o_n one certain
machine (lathes), except the products 6 - 13, which can be produced on an arb1trary from the
machines 3 - 8 (see Table 1). These products will be firstly assigned to lathes in such a way
that all machines (lathes) are approximately equally loaded as far as the processing tirne is
concemed. The result is given in Table l.
We can assume that a machine will begin to produce the product immediately after it is
adjusted to its production. Otherwise, the worker adjusting the machines will have to wait
until the processed product is finished so that a delay in the adjustment process co_uld occur.
Further we will assume that the worker will immediately start to adjust the machme for the
next pr~duct as soon as the machine is free whenever he is not busy with ~djusti~g some
other machine. Since the products cannot be produced without the correspondmg adJustment
of the machines, which must precede the proper production process, the order, in which the
adjustment is carried out, determines the order, in which the products are processe~. It_ means
that the products are processed in the order, corresponding to the order of\dJustmg the

70

c;

+ t1 sT

Xij E
C; ,2()

(0,1}

i, j = J, .. , 11, < j
iJ=f, ... ,n, ;r:j,iES1u
iJ = 1, . ,11. ;ej;
i = 1, . n.
... , 11.
i =
iJ = 1, ... ,11, i ;ej;

= 1, ,,.,n.

(3)

(4)
(5)
(6)
(7)
(8)

Equation (2) ensures that for each pair i, j either the adjustment for batch i precedes the
adjustment for batch j or the other way round , i.e, the adjustment for batchj precedes the
adjustment for batch i.
Inequality ( 4) ensures that the beginning tirne of the adjustment for batchj, i.e, Cj - d1, _must
follow after the moment c;, at which a preceding adjustment for batch i had been fimshed
under the assumption that the adjustment for batch i precedes the adjustment for batch j
(regardless on which machine the proper production process is carried out). This constraint
follows the assumption that the worker cannot adjust more than one machine at the same
tirne. Constraint (5) shifts the tirne moment, at which the adjustment is finished according to
the corresponding adjustment. Inequality (6) determines the total processing tirne of all
batches, which is (according to (1)) minimized. The model was solved using the program

71

LINGO 8.0 (special professional integer programming software) for a one-week production
program in Table 1, a number ofproduction batches was n = 27 and a number ofmachines m
3,4,5 ,
= 10. The sets Sk are given in Table 1, from which it follows that = 1, 2},
S3= {6},S4= 7},S5= {8},S6= {9},S7= {10,11, 2 ,Ss= {13},.59= {14,15,16 },S10.=
{17,18,19,20,21,22,23,24,25,26,27} The mathematical model included 757 variables (729 of
them were binary variables) and 1238 constraints. The computations took 5.4 minutes (PC
.3 GHz). The resulting order, in which the machines are adjusted, is given in Table 2.
4 A model with a partial a priori assignment of batcbes to machines
Since a priori assignment of products to machines, which was used in the previous section, is
not necessary to be optimal in general, we shall try in the sequel to optimize this assignment
together with the total processing tirne of the schedule. As we have already mentioned, the
l which can be processed on any of
assignment can be chosen only at products 6
machines 3 - 8. The assignment ofthe other products to machines is fixed.

Model l.
To optimize at the same tirne also the assignment, we shall introduce binary variables Yik ,
which wi!l determine the assignment in the sense that Yik = 1, if batch i is assigned to
machine k and Y1k = O otherwise. Further we set D = {6,7, .. , 3} and S= {3, ... , 8}, i.e. D
contains the indices of those products and S the indices of those machines, where the
assignment is not fixed. The remaining products are assigned a priori to corresponding
fixed technological requirements. The mathematical model is based
machines according
(8) with the following additional constraints
on the model (
)

= 1,

-J,k)-

'

11 '

i

,CJ,

i ED' J ED,
(9)
k ES

(10)
Table 2. Optimal ordering ofproduction batches
order

-·

1
2
3
4
5
6
· - - -7
8 -9

o
--

--

11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27

machine

i

k
9
-

14
20
4
8
18
2
7
27
19
6
26
5
10
12
-9
22
24

Start adj.
c,-d1

o

10
---

-

2
5

10
1
-4
10
10
3
10
-~-2
1-7
7
6
-10
-10
- ~ _3___ ,__i __
13
8
7
11
15
9
9
16
1
1
10
21
10
17
10
23
10
25
-

----

batch

-

~

--

-~

_!!1,_!3_
166,4
176,6
237,8
603,84
- - 612
703,8
_767,8
_!3_23,2
1568,8
1574,8
1585
1686,2
1912,56
2277,4
2328,4
2728,6
2738,8
2830,6
2891,8
2994,6
3146,24
3154,4
3174,4
3194,4
3218,4

End adi.

Finish tirne

C;

C;+f;

~1&
~7,_!3_
176,6
237,8 _

-

--

143,8
612
703,8
I._09,8
773,8
884,4
1-_Q_74,_!3_
1585
_!~46,2
1J1],4
1973,76
2283,4
2334,4
2738,8
2830,6
2891,8
2983,6
3086,4
3154,4
3160,4
3180,4
3200,4
3224,4

~

_?891,8 _
237,_!3_
193,_1_ _
1617,8 703._!3__
617,5
_1-143&___
767,8
1568,8
3254,4
20Q1,68
1605,~
1686,2
1829,4
3254,4
2328,4
3154,4
2998,8
3002,6
2945,8
2994,6
3254,4
3254,4
3174,4
3194,4
3218,4
3252,4

Constraint (9) replaces constraint (3) for machines k E S and the corresponding sets of
products Sk, It ensures that the adjustment process for a batch can be started only after
includes
finishing the production of the preceding batch. Constraint (9) (unlike to
variables )';k and )}k, which are both equal to , if both products are produced on the same
machine k. Equation ( 1O) ensures that each product from D is assigned to exactly one
This extension of the model (1) - (8) included 805 variables (777 of them
machine from
were binary variables) and 1576 constraints. The computation tirne increased to 1 hour and
35 minutes. The resulting total processing tirne T was equal to 3254.4 minutes, i.e. the total
processing tirne was the same as in the previous model, which means that a priori assignment
in the previous model was optimal. The substantial increase of the computational tirne is
remarkable in the case of extended model, which included the optimization of assignment
lathes to machines.
Model 2.
The assignment ofproducts to machines have an alternative solution, since the machines, for
which the assignment is not fixed are completely identicaL Therefore, it is not necessary to
assign products to these machines. is sufficient to find out, which products can be assigned
to an identical machine. In this way, we obtain a simpler model with a lower number of
variables.
In Model 2, binary variable YiJ = , ifproduct is assigned to the same machine as productj
and YiJ = O otherwise. For the indices of these variables always i, E D and i < j holds.
Further, we shall replace constraint (9) from Model with the following constraints 11) and
(12), which ensure (similarly as constraint (9)) that the adjustment process for any batch can
be commenced only after the production of a preceding batch had been finished.
C;- 0:? C;

+ l; - M(] -

Xij) -

M( 1 - YiJ ), i < j, i ED, j ED

(11)

(12)
Since we deal with an assignment of 8 products to 6 machines from the set S, it is necessary
to assign at least 2 products to the same machine, i.e. at least 2 variables YiJ must be equal to
1, which is ensured by the following inequality (13):

72

73

L L

Yij ~2,

3)

ED.

iED jED,i<_i

Therefore inequality ( 3) repiaces relation (10) from Model . Model 2 includes, in
comparison with Model 1, 24 variables and 280 constraints less than Model
5 Conclusion

The obtained schedule, in comparison with the schedule used in the practice, leads
the
total processing tirne, which is 30% shorter. This result can substantially increase the
productivity, exploiting the machines and lead to lower production costs. Nevertheless, it
requires on the other hand the usage of sophisticated modeiing techniques and a professional
optimization sofiware. It seems that an interconnection of these models with the in formati on
system would be possible too, because the program system LINGO 8.0 supports this
interconnection. On the other side, disadvantage of this approach consists in the fact that we
have to solve NP-hard problems and the increase of the size of problems we have to sol ve
may cause a substantial increase of the necessary computational tirne. This happens also
when we passed from Model 1 with a priori assignment to Model 2, which includes also the
this case a larger
optimization of the assignment of products to machines. We obtained
problem and the necessary computational tirne was higher too.

The 8th lnternational Symposium on
Operotional Resecm:h in Slovenia

R'OS
Nova Gorica, SLO\IENIA
September 28 • 30, 2005

Section 2:

References
[1] PELIKAN, Jan, An Optimization of the Production System - Case Study. New Delhi
08.12.2003-11 12.2003. In: APORS 2003. New Delhi: ORSI,2003,p. 48.

l2] PELIKAN, Jan, Diskretni modely v operačnim vyzkumu.
PROFESSIONAL PUBLISHING, 2001. 64 p., ISBN 80-86419-17-7.

vydani,

Praha:

[3] PELIKAN. Jan, A minimax parallel processors scheduling problem. Edinburgh
08.07 2002-12.07.2002. In: IFORS 2002. OR in a globalized network world economy.
Edinburgh: IFORS, 2002,155 p.

74

r
THE KIDNEY EXCHANGE GAME*
Katarina Cechlarova1 . Tama.s Fleiner 2 and David Manlovea
1

lust it ute

Bndapest
0

\lathematics. P.J Šafarik l:niversitY. Jesen na 5. 041 54 Kašice. Slovakia
ernai:: cechlarova©science upj s sk
2
Department of 'ompuLer ScieJ1ce arnl lnformat ion Theory,
of Tedrnology arn! Economics. '.1Hiegyetcrn rkp. :J.Y. H·lil l Budapcst Hungarv

Depart mern of

email: fleiner©cs bme hu
i1,g Sciellce. l'ui1·ersitv of c;;asgm, C!asgu,·. Cl2 SQQ. l
ernail: davidm©dcs gla, ac uk

know1: is t a1csp!aL
Abstract. Tlte mos1 effecfr:e trealmcnt for kidne:: failme tl1at is
tation. As tile nnrnber of caclaveric donors is not sufficieut aHd kidneys frorn iiving r\01:ors are
ofteu liOt suitabie for imm,.wological reaso11s. 1 here are at ,,empts t.o organize exchanges betwee11
asa cnopm ai ivc game and p, nposc so1:1c
sit na,
l his paper we model
patient· donor pairs
fhdiHg a solutio:1.
algoritLrns

Keywords. h.idne,v ,ranspla11tation. cooperative game, Paretu opti1:1ai solution. core. algori: L,,

1

Introduction

Renal failure is a ven· serious illness fnr which the most effec:tive treatment rliat is <·nr•
rentlv kuown is kidne:v transplantation. ideallv, a kidney frorn a deceased donor could be
spite of JC,illt effor 1,s of national and even international
nf those
used, lmt the
urgauisatiuns :for example Eurntranspla11t Fom1datiun lY aml t,he 1."nited l\e1,work fr)l'
Organ Sharillg in the U,A 120]: is not sufficient for the growi11g demand and the waiting
a patie11t is n:tpredictable. As the operation techui4ues irnproved and Lhe risk for a
la ge11et1c uran el!H1ti01ial relative uf the patiellt I was 111i11iu,ized.
a
t he rn1mber of live-donor transplautations inc:reased. \/loreO':cr. some stuclics · 171 shnw
t ltat grafts from living donors have a highcr survival rate.
Fur a transplautation to be successful, some imrnunulogical requirements 11rnst be fuifillcd, Basically ABO incompatibilitv and a positive cruss-match are an absolute contraindication. moreover the greater the nurnber of HLA mismatches between the donor
and the recipient, the greater the chance of rejection [9). Hence, it often happens that a
willing donor cannot donate his/her kidney to the intended recipient. Therefore in severni
countries systematic kidney exchange programs have been established: in Romania [10],
the Netherlands [18], USA [13, 14, 15), in other cases there are isolated examples [7], e.g.
in the Middle East.
Kidney exchange (KE for short) is stil! a controversial issue, however the aim of this
paper is not to discuss the ethical and legal aspects of this concept. In spite of some

/

'This work was supported by the VEGA grant 1/0425/03, Science and Teclmology Assistance Agency
contract No. APVT-20-004104 (Ccchlarova), OTKA F 037301 rcscarch grant (Flcincr), EPSRC grant
GR/R84597 /01 (!vianlove) and the Royal Society of Edinburgh International Exchange Programme (Fleiner
and Manlove).

77

pessirnistic expectatio1Js tl1e Dritisli Tra11splamatiuu Societ\ eslimmed poteutial beHefas
Romauia the
au iust
be arouwl
douors· excha11ges
from
111ca11 1mmber of transplantations incrcascd frorn 4.2 to 6.1 sincc thc J<E program startcd
ancl a simulation studv using real r·sA statistical data revealed 1hat the utilizatiou of
·
douors wit hcmt exchanges was armmd 55%. whilst wit e:,chaHges
kidlle~rs frnrn
increased to 91 % when the size of the simulated population was 300 fl3
t liat we re;re~eut KE as a ,oWe follow the approach started in /13; and
Since all
exchaHges
whid1 patieut-donor pairs seek
opcrati\'e gau,e.
avuid the risk tliat one
uperat.iui1s 011 a cyde shouid be perforrlied simultaueously
donors will withdraw his or her commitment after the others lia,·e urnlergrn;e
.t I n des slwllld be as short, as possible for logistical reasous. Tltereiore we direct
the obtaiued n de lengt hs
pora te cyde lengths iHto preference models (notice that ill
rlte
wcre Just observed aft.er sirnulations of the algorithrn used. and
of t he rnaximurn allowed
11,atchcd paLicms was deri ved as a f1mc1
core
solntions we also ,,msider the core and the sl
tlier. besides Pareto
!<E game. \Ve coudude witb computational 1. omplexiLy cousiderat ions of t he
algoriLluns aud sorne opell questions.

2

The KE game

Au instance of the KE game is represented hv a directed graph G = 1V. AJ wl 1ere each
donors 1
vcrtf'x 1 E \ corrcsponds l a pa! icnt aud his imen ded
the patient correspondiug to vertex i call accept a kidaeY from a
J) E A
A pair
· 1)· E A :
d0110r correspomliag Lo vert.ex 1· 1.llence. patient 1 bas a
Moreover. for each vertex i there is a linear orderi11g Si 011 the se1 of endvertices of arcs
rncidem from i. Tliis orderiug is represeuted by a preference list of i and we assurne that
we sm 1 liat
1 is the last entry in each preference list. If J <; 1 k and simultaneously k <,
z is ind~tferent betwcen j aHd k. or that J and k are tzed in i·s pref~reu; list. and. write
J ~; k; if J Se; k but not k Se; J, then i strictly prefers to k. and we wriLe <i k Hence.
iu general, preference lists may contain ties.
Definition 1 A kiduey exchauge game ( KE game far short J is a triple r =
where V ·is the set of piayers. G zs n digmph with a 1•ertex set V and O = {:S;; i E

Oi
}

such that
Definition 2 A solution oj the KE game r = rt· G. O) is a permutatw11 1r 0 t
i # 1r(i.) implies ,i.1r(i)J E A far each i EV. lf iJ,1r(i)) E A, we say that i ~s covered by
C" (z))
7r, otheiwise i is uncovered. A player ·i is assigned in a sol-ution 1r the pair
·
where C"(i) denotes the cycle oj 1r containing i.

A player e':'aluates a permutation not only according to the player he is assigned to. buL
he also takcs mto account the cycle length. This is expressed by the following extension
of preferences from O to preferences over (player,cycle) pairs. The same symbol is used
for preferences over players as well as over pairs and permutations.

Definition 3 A player i prefers pair (j, M) to pair (k, N) if
(i} j <; k or
(ii} j ~i k and IMI Se INI.

Definition 4 A coalilwn
a of S such tlial

~

\ · weakl_Y blocks a sol-utwu

1f fi1ere exis+s n perm:uta/:im1

eac/1 1 E S and
at least one j E S
there e:rw!s 11 pern;11/ufw11 a of ,'-,' such that
a solution 7r
each i ES
,C"(

Jar

~ow we def-ine tlte st udied solin ion cm1cepts.

r

(belmigs to PO(r) far short
game
ima]
7T is Pare1o
does not block: it is strough Pare1 o opl in1al (1r E .',' PO• l
the gmnd coalition
does no/ ,ceakly l!lod·.
s/wrf)

Definition 5 A

Definition 6 A permulatwn 7r -ts w //:ernre ('if; ufgame !
uf f' if no coal-dum U'eakly blocks.
is the stroag core

w

1

I

oaittion biocks

7T

ancl

7T

sfro11.gly Pareto optimal a!id l'areto uptmiai sobtiom; rnr~otice tltai our nmcept.s
and weakly Pare/o optimal solutim,s
Pare/o
called
to what are
HmveYer. we nse the first two 1,erms to make t Le l,erminoiogy consistcll1
respect i ,,eJy
ll1is paper.
wit
As each blockiHg 1 oalition is also weakly blocki11g, wc l,a:e
and SC 1r, <;;; SPO(f) <;;; PO fj
<;;; C(f; C
generaL but 1he inclusion C,r; <;;; SI'O,f; is not always trne
even in the case without indifferences. Consider the KE game
r given b,v the digraph to the left. where the first e11tries i11 the
prefereuce lists are denoted by solid li11es. whilc thc sccoml cntrics
correspond to dott.ed lines. Permutation 7r = (1. 'l.. 3)(4. CJ. 61 is
C(I'). hut 7r <t S'PO/f) as the grarnl rnaliti011 weakl1 l,locks 1r
a = (1.2.4/5.6.:1;
h.Y
the practical kidney exchange applicatio11 it is 11ot
HoweYer.
dear whid1 sol ut ion coucept is the most Sllitable ime

3

The case with preferences

With no indifferences. the famous Top Trading Cycles (TTC for short) algorithm /Figure
1) ca11 be used for the KE game. The TTC algoriLhrn was originally proposed by Gale in
"16] for housing markets, where cycle lengths were not taken into account. It was shown
that the TTC algorithm outputs a permutation i11 the core of the housing market also
in the case with indifferences (ties are broken arbitarily). In r12] Roth and Postlewaite
proved that if there are no indifferences. the strong corc of the housing market is noncmpty
and contains a unique permutation. Further, Roth [11] proved that the TTC algoritlun is
strategy-proof. However, a detailed consideration of algorithmic questions connected with
the TTC algorithm is quite recent; in [l] its implementation with O(m) tirne complexity
was proposed, where m is the number of arcs in G. In [6], the TTC algorithm is called
Algorithm B-stable and it was shown that in the case with no indifferences, its output is
in SC(f).

/

78

7r

79

r

r

Input. A KE game = i\.G.O,. Output. A penrmtatiou 1r = C;
Step O. N := V. round r := ()
Step l. C'hoosc an arbitranr player io
m .Y eh.
in .T\: i I poillts 10 his favourite
Step 2. Player i 0 points to his favourite
A rycle arises or player ik l'a1111ot point.
Step 3. r := r + l. If a cycle C was obtained. then C,. := C. otherwise C = (ik),y := N-C,..
Step 4. If ]\. -/= 0 go to Step l. otherwise end.
Figure 1: The Top Trading Cycles (TTC) Algorithm.

4

QJ

3

C

Otherwise go to Step 2.

However. the TTC algorithrn may fail to fiml a perm111atiun
iu the C(r) and also ill T'O(f) if iHdiffereHces are present
fo the example to the left. the TTC algoritlrn1 may. depeudiHg on how the ties in vertices l au<l :J are hrokell.
output the permmation ( l 2. ::1. 4). wliile liere S PO{r! =
PO(I') = Cif) = SC:(I'; a11d the: ccmtaill a u11ique permutatiou (1.4;(2.:J,.

Moreover. in [!'jl it was proved that iu the case wit h iudiffere11ces it is :-/l'-cqwplet,e
to decide whether C(r) -/= 0 and it is also NP-complete to decide whether S'CilJ f 0
Notice that it is always thc case that SPO(I' 1 =f. 0: if we define a partial order 011 tlie
set of all permutations of V by setting a :S 1r if (a!i). C" i :S, (n(i;. C"(i); for each
player, then the minimal pennutations in this partial order are strongly Pareto optimal
(and hence also Pareto optimal) We show later however. tliat it is NP-liard to find a
permut.ation in SPO(r) (Theorern 5;.

4

.
Input: A siwple h:E game r = ( i ·. G J Output: A permutation 7r. of V
Step 1. Set k 1= 2 aud create the 2-reduce<l graph C2 = i V2. E2; ot d1gi:aph? by lettmg:
\ 2 = {c EV· 1· lies ona 2-cyde in G'}. E2 = {e = i:;: (i.j) E A & i,J 'l) E A1
Find a maximal rnaLchiug J12 in G2 and set n(i) = J for eaclt lJ E .Vh.
Delet,e from G all ,·ertices coverecl by M2.
Step 2. k := k + 1.
bv lettiug: . .
Step k. Create the k-reclncecl digrapli Gk = (Vk. Ak) of
\ k = {c E \ : 1· lies on a k-cn,le in G}. Ak = {e E A : e lies on a k-ivcle lil G}.
Find a maximal k-cycle packing .\h of Gk and set 1r(i1 =.) for each are 1.1 J' E 1\ih
Delete from C all 1·e:-tices covered h_1· \!h. If C rnutains no (·1·de of lei,gth k+ 1, end.

Figure 2 Algorithm CoreSirnpleGame
E\·eu witl10ut :.Le restricliou oll cyde lellgths. we are able to forrnulat,e a uecessar, ancl
sufficiellt ccmdition for a penmnation to be in C,f':

Theorem 1 1r E C'(f') if ar1d on/y ,f G wntnms no ryrle ( ·
) = -i1 for ead, J = l. 2. . . k
1 > k or
Proof. Sucl, a cyde dearly blocks rr
coalition has a form of C !II

Theorem 1 suggests au iterative approach for fotding a pen11utation ir: Lhe cq,·e uf the
simple h.E game. The algorithrn is depicted in Figure 2

Theorem 2 Algorilhl!I CoreSim.pleGame correctly finds a perm ufatim1
sirnple I<E game in polynomial tirne.
For the stroag core. Theornrn l

The Simple KE game

Let us now suppose that all compatible kidne:vs are eqnalh snitahle for t ransplaut.at icm. as
We will say that now the players have dichotomous preferences
suggested in [141 and
(we follow the terminology of [21). or that all acceptable vertices are tied. Hence patieuts
with a compatible donor are not considered and the KE game is identified simply witli a
digraph G = (V, A) without loops.

Definition 7 A simple KE game is the pair r
G = (V, A) is a digraph without loops.

=

(V, G). where V is the set oj players and

Similarly as for the KE game from Definition 1, the solution of a simple KE game is
a permutation 1r of V such that i f n(i) implies (i, n(i)) E A for each i E V and the
notation from Definition 2 can be used. Further, preferences of players over (player, cycle)
pairs now reduce to saying that a player prefers to be covered to being uncovered, and if
covered, he prefers to be in a shorter cycle.
In [14], where dichotomous preferences were considered, the authors allowed only cycles
of length 2. Bence the KE game was represented by an undirected graph G = (V, E) with
ij E E if patient i can accept kidney from donor j and also conversely. A strongly Pareto
optimal solution was identified asa maximum cardinality matching.

\

80

On the other lmud. if 1r </:. C,I'J tLen a hl(Jdcillg

ca.J1

·i11 ( ·,

f' ! uf fhe

be easih n:odified:

. ik J suci, ihat
Theorem 3 1r E S'C(r) ,f and onlµ if G contains no r.ycle C = (i1. i2
k}. and either C":11) 2. k
either ,C"(i1 ) >kor· n!i 11 = i1 Jor at least one .7 E {l.2.
or n(i 1) = i 1 for the rema-ining vertices i, oj r.ycle C.

Corollary 1 Jf 1r E SC (I') then 1r restricted to the 2-reduced gmph G'2 '!s a perfect match-ing oj Vi and far each k > 2. 'IT restricted to the k-red·uced digmph Gk is a partit.wn of
1-1,; intu directed k-cycles Consequently. ·if either C2 admits no perfect matching or Gk
admits no parlition oj Vk into directed k-cycles far some k, then SC(l) = 0.
Bence Algoritlun CoreSimpleGame could be used to find a permutation in SC(r:, if in
Step 2 a perfect matching and later a partition of Vk into directed k-cycles can be found.
Bowever. this problem already becomes difficult.

Theorem 4 The problem oj deciding whether SC(I')
complete.

= 0 for

a simple KE game is NP-

Proof. Membership in the class NP follows from Theorem 3. In [8, Theorem 3.7] the
NP-completeness of UNDIRECTED TRIANGLE PACKING is established. It is easily possible
to obtain NP-completeness for DIRECTED TRIANGLE PACKING in graphs with no bidirected

81

T
construct,ed
ar,;s_ where eacii verLex lie::; ona directed triangle, hy ( 0Hsideri11g tLe rednl-,
[8, Figure :38,
the grapb shown
t here and directing appropriatel, t he arcs
a!J installCC of
An instance(;= (VA1 of !J!RECTED TR!A0GLE PACK!NC givcs rise
adrnits a directecl triaugle
a sin1ple I-<F game_ It is then straightforward to verify that
packing if aml only if S('(I'; =p 0 II
However, in spite of the fact that S'PO(I') =p
Theorern 5 It ·ts

rvp hai-d to

0 for

4

Cechlaro\ a ai,d ,J
SO!f99, Preddnir
( 1999),

T Game Theorv :31 (2002; :L 353 :_l64

all KE games_ we have:

find u perni-utation m S PO(I' !

61

a simple KE game.

gam.es, lnter-

Cecltlarma and A_ H.omero-;-,leqi11a, Siub-ihty
29, 200 l _ 487-494
Came
11at iollal ,Journal

/\-ansplantaiwn, J\ew

7· F.L Delrnouilo, Exchangi-ng }(zdneys Adniw:es w
Eaglaud J_ :\[ed :J-501812-1814 r2004).
111,d

8 Ivl. _ Garey aud DS Jolms011(l 979J

As::;rn ia,
Eiilbe et aL Guuie!we~
_Pebrnan 20(J:l.
l'
http://www_ uroweb. org/f iles/uploaded f iles/ guidelines/renal transplant -pdf

Lucan.
alternatice

Rotarin, !)_ :\ec1iloiu aud c;_ lal'oh, I<id11ey ex1hn.:1ge progm111 a ,ria/J/e
muntries U'dh /01c rate of cad11uer lwr"l·estmg, TrausplaJJ!,aLiou Proceedi,,gs

Conclusion and open questions
l

start.ecL So
The research on efficient algorithms for kidney exchange prohiems has
there are still rnauy questions to be answered aud 11:an,v problems to be solved. Let us
meution Just a few direc-tions for possi ole further resean h _

6

sfoiJle partitfo1;s with

-'i' K. Ced1larma a!l(l
B-preferern:es,

a pernllltat
A
Proo[ S11ppose we l,ave a polyuornial-time algorit
\Jow let '= i V ki be a direc:tec! grapli with uo bidirect.ed arcs !give11 as a11
PO' l
7r
instaHce of DlREC'TED TRJA,,\:LE PAl'KINc:) Ru11 A on G to fiml 7f E
\'ow s11ppose
,;onr.ai11::; oni, 'l-z ydes t hcn clearly 7r gives a directed triangle packiug
'l-c,des_ Snppose for a
that it is not t he case rlrnt 7f contaius
rr. ('cmsicter am· venex
admits a directed tria11gle packing. T!iis defines a
c is
is at least one sitz-h
belong to a :1-cyde in 7f
- that does
1- E
a
rr. O,Lerwise c belungs
rr_ siHce 1· is covered
7f t i1en ,- irnproves
uncovered
i11 rr. lleace
rr_ sim:e c belougs
cycle of lengtli > 3 iu 7r and v also improves
t hat 7f E
1 weak!v blocks 7f v ia rT, cc,11tradid ing t he assurnpt

5

coo!;t 101
gumes, 'r0c
Zaduik-Stirn_ S lJrnbue, lli-llG

., A::; the basic priority is to treat as maw: pat ients as possible, lincl a sol dior:
l(E game wiLh he 1

the

" A::; patients emer and '.eave the wani:ig 'i:,es
of on-line aignrith111s for kidney exchauge

:es

Acknow ledgement

We would like to thank to Dr. Uuboslav Bena, Transplarn centrum Košice. for detailed
explauatious of medica! and ethical problems connectecl with transplalltations of organs.

References
[1] D. Abraham, K. Cechlarova, D. Mantove and K Mehlhorn, Pareto optimality in house
allocation problems, Lecture Notes in Comp. Sci. 3341, Algorithms and Computation.
ISAAC 2004, Hong Kong, December 2004, Eds, R. Fleischer, G. Trippen, 3-15 (2004).
[2] A. Bogomolnaia and H. Moulin, Random matching ·under dichotommts preferences,
Econometrica 72: 257-279, 2004,

A_ Roth, lncent11e
9127 1:i2 (1982),

u1 a market u>ith indiuis1bil'i/1,es, E, z;1wrnics Let 1.ers

12] A. K Roth and A. Postlewaite, V.eak uersus strong domination in a 1riarket with
1\lath. Econ.4l'll-l:l711977)
indiuisible goods.
A. Roth. T. Somnez aud
4fi7-488 (2004
;14] A, Holh, T. Somuez alld
appear.

Kidney hJxchunge. Qtrarterh J_

Ecmi.

1

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m·er. l'airwise Kulney Exchunge. J_ Ec01•_ Theon

u

A. Roth, T. Si:inmez and U. frrn·er_ Effirient [(idney Ex, hange.- Coincideru;e oj Wanfs
in u Strudured Nlarket. marmscript. May 2005
16] L. Shapley all(l II. Srnrf. On cores und indivisibility,

J_ \/la1.h Econ. 12'l-:l7 1974

117] P_ LTcrasaki_ J M. Cccka, D. W. Gjertson and S. Takcmoto, High Surv-ival Rates oj
Kidney Transplants Jrom Spousal and Living Unrelated Donors, New England J Med,
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:19] The Eurotransplant International Foundation, http://www. eurotransplant. org/
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MODELING STOCK BUYING-SELLING liSING A NEW MCDM
METHODOLOGY ACCOUNTING FOR UNCERTAINTY
Vladimir I. Kalika
Natura! Resources and Environmental Research Center.
Haifa University, Mount Carmel, Haifa 3 905, Israel,
E-mail: kalika@econ.haifa.ac_il

Ahstract
A new methodology is proposed select a predeterrnined number
"reasonable" altematives from
thelf considerable mitial set according an arbllrar; number of optmuzation cnteria. fh1s mtu1t1ve
methodology, based on accounting for uncertamty factors, mherem m real problems, by perfonning
the multi,level lnerarchical system of mult1 vanan! computat10ns and of findmg their "stab[e„
optlma]" solutions, is appiied for modelmg the stocks' buymg„sellmg on stock exchange. The
"bottleneck"
this modeling - the construction of adequace optim1zat10n cntena, 1s the mam obJect
for ccms1deration.

Key words: multi-criteria decision makmg; six-level h1erarch1cal system of muhi-variant
computation series; scenarios; statistica! data on stock deal sums and stock prices; prognos1s trends;
sinusoidal character of statistical <lata on stock prices.

1. Introduction
The paper reflects the attempt of applying the new multi-criteria decision making (MCDM) approach
accounting for uncertainty factors to solve a problem of stock buying-selling on stock exchange,
where this applied problem is considered mainly as an object suitable for th1s approach applicatlon
(for now the practical aspects ofbehavior on stock exchange are considered by us as less important).
This proposed quite general MCDM approach (its methodology and some applications are
reflected in [1-5]) is oriented on solving various real problems, connected with decision making,
characterized by selecting the "reasonable" (the best in certain sense) altematives from their
considerable (maybe vast) initial set according to many objectives, expressed by multiple criteria
including an arbitrary number of optimization criteria. The main specifics of this proposed approach
consists in the methodology, developed to take into account uncertainty factors, inherent in such
problems, where these factors could have anyone nature. According to this methodology, the
following ways are proposed to meet various uncertainty aspects, inherent in such real MCDM
problems:
, A two-phase process is considered to reach afina/ solution of the whole problem, where the first
phase of this process should reflect a computer calculation process to select a predetermined number
of the resulting "reasonable" altematives (their set is named RAS) from their initial set (ISA)
according to the assigned multiple criteria. On the process second phase, the fina! solution is found,
basing on the derived RAS analysis. It will be a non-formal process to account for uncertainty
factors by considering additional qualitative/quantitative criteria and other <lata, expert estimates, the
current observations of last situations, etc.
2. 171e methods to reach the RAS, concerning the first phase of this solution process, may include
various ways of accounting for uncertainty factors: (1) considering different versions of conditions
and parameters of the problem solved as well as of the appropriate calculation process intended to
solve this problem; (2) forming various combinations of such versions to provide performance of
mu/ti-variant computation series and finding their "stable-optimal" solutions; (3) modification of
used multi-criteria optimization techniques, other calculation models and procedures to account for

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85

some uncertainty factors; (4) usmg random variables (e.g., to use Mante Carla simulatio11), expert or
hypothetic (in situations close to full uncertainty) estimates in various calculation operations. etc.
Thus, thisfonnalizedfirst phase of the solution process, intended to solve the suitable real MCDM
problems by reaching their resulting RAS. should include the following basic stages:
l) formi11g a totality of JSA versio11s, where such versions should reflect ali possible cases of initial
alternatives' combinations accounting for availabihty of appropriate uncertainty factors;
2) co11structi11g the versions of i11itial set of criteria assessment vectors (ISCA ~' where each ISCA V
version, interrelated with the appropriate ISA version, includes a totality of criteria assessment
vectors, determined for each alternative from the ISA using the criteria calculation models developed
a priori for ali considered criteria. Thus, each criteria assessment vector, corresponding to one initial
alternative and conversely, is represented for this alternative by one numerical value, determined for
each considered criterion using its calculation model;
3) multi-criteria optimizatio11 011 the space of lSCA V (ISA) accounting for 1111certainty factors
achieve the resulti11g RASby decreasing the ISA.
Performance of stages 1-2, consisting in constructing (often, simultaneously) the ISA and ISCAV,
presents the processes, specific for each considered MCDM problem. However, the approach to
perf01m stage 3 was developed (by us) as a quite universal one. This approach is based on
performing a mu/ti-leve! hierarchical system of MVC series, where each leve] of the system should
include a totality of sce11arios, having the same nature specific only for this leve!. The scenarios of
different levels reflect various versions of: the ISA/ISCAV (they correspond
the first/second
levels of this multi-level system, /=1/2), the used multi-criteria optimization techniques, modified to
account for uncertainty (the third leveL /=3); some conditions and parameter values, involved in the
problem (such levels, since with fourth one, !>3, are specific for various MCDM problems as well
as for the appropriate types ofmulti-level hierarchical systems used to solve these problems).
We considered various versions of multi-level hierarchical system of MVC series, differing by
various aspects: (a) the number of levels; (b) the specifics of scenarios, accepted for various levels:
(c) the peculiarities of calculation process performance, etc.

The process of forming such six-Ievel system to solve the real MCDM problems could be
connected with necessity to overcome the "bottlenecks" in forming the 1/2-scenarios, i.e. the
ISA/ISCAV creation. In the considered problem we have the s1mple case of ISA construction (an
initial alternative concept is very implicit ~ a stock buying-selling operation is such an alternative
and it's not needed to develop the special calculation procedures to construct a vast ISA, etc,).
However, the main "bottleneck" of this problem modeling is connected with the ISCA V creation,
where we have the analytic case: a necessity to deve/op 11011-implicit artijicia/ criteria. This case is
differed from many other suitable MCDM problems (e.g. [1,21) with the opportunities to form
natura! optimization criteria (economic, environmental, technical, etc.). This is precisely the main
aspect, considered further in this paper.
The calculation process to reach the resulting RAS (e.g.[4,5]) is perfo1med by fixing the
combinations of /-scenarios. where such combination (fi1!l Sce11ario) includes /-scenarios, taken on at
a tirne for each /-leve! (/=1-6). Eachfi1l/ Scenario forms a mono-optimization problem by fixing for
the random {W1 *, ... ,W1 *} in ( 1) their determined values {w 1, . .• w,} using J Monte-Carlo simulations
for the appropriate J intervals (3). This problem solution 1s a subset of sub-opti111a/ alternatives.
Further, we forma group ofji1ll Scenarios. changing ali 6-scenarios with the unchanged /-scenarios
for ali upper 1--levels (/=l-5). that allows derive a set ofsub-optimal subsets for this group as well as
its subset of stable-optimal alternatives. constructed using a special analysis of sub-optimal
alternatives' entering quantities in their set. Varying such groups by changing the fixed 5-scenarios
only. we derive a set of stable-opti111al subsets (of 5-level) and its subset of stable-opti111al
altematives (of 4-level), corresponding to the fixed combination of /-scenarios for the four upper /levels (/=l-4). Continuing this process with the variation in fixing 4,3,2, -scenarios, we reach the
resulting RAS as a stable-optimal subset (of O-leve!), deri ved by the set of stab/e-optimal subsets ( of
1-level) analysis.
Thus, the calculation process to reach the RAS is a moving from this scenarios' system "bottom"
(/=6) to its "top" (/=1), using ali its /-scenarios for ali its /-levels (/=1-6).

3. Some principles of stock buying-selling modeling using tbis new MCDM approacb
2. Tbe multi (six)-level bierarcbical system, used for tbe problem of stock buying-selling

3.1. The co11te11ts a11d pwposes ofsuch modeli11g

We use to solve the considered MCDM problem of stock buying-selling the six-level hierarchical
system including the following totalities of scenarios for each /~leve! (/=1-6):
/=1/2) ISA/ISCAV versions - 1/2-scenarios, having the numbers klm (k=l, .. ,K/m=l, .. ,M);
/=3) the multi-criteria optimization technique version (11=N=l), presenting our modification [3] of
TOPSIS method [6], based on the consideration ofthe following scalar goal function:
min {cu"W1*+c;1"Wi'1<+ ... +cu"W1*)
(l)
{i= l , .. ,I}
where: c;/ (i=l, .. ,I;j=l, .. ,J) is a normalized relative value ofj-criterion for i-alternative, calcula!ed
for eachj-criterion (i=l, .. ,J) with respect to their maximal/minimal (on i=l, .. ,I) values {cf'..'/c/""};
thus, the values cij" are calculated according to the formula:
cij" = (cij- cf';") l(cJ'"" - ct"), i=l, ... ,I; j=l, ... ,J;
.
(2)
111
W/(i= l ,.. ,J) is j-criterion random weight, reflected by its possible values interval [w/' ,wf'..'];
/=4/5) the intervals {[wt",wf'..'],j=l, ... ,J} versions (p=l, .. ,Ph=l, .. ,R), each ofwhich is a pair of 4/5scenarios, taken one at a tirne for each /-leve! (/=4/5) and corresponding to p-version of "central
point" values {wj", j=l, ... ,J} for ali these j-intervals (it's 4-scenario) as well as to r- version of ali
thesej-interval bounds (5-scenario), calculated around the values w/:
wt" = Wj" (1-g_J, wt"' = Wj" (l+g), j=l, ... ,J
(3)
where 0<gj<<l is a relative value, assigned a priori to determine these bounds values which might be
derived using the equal grvalues for all j---criteria, e.g., grg, g=0.05, 0.1, 0.15, ... ).
/=6) Ali 6-scenarios (q=l, .. ,Q) reflect various combinations ofvalues {wj,j=l, ... ,J}, corresponding
to ali j-intervals, formed on the 4/5-levels (/=4,5). Each such q-combination includes J values Wj,
taken one at a tirne inside ofeachj-interval [wt" ,wf'1 using J Monte Carlo simulations.

The following problem is considered: to selec/ (in the current T-day) a holdi11g of stocks, including a
predetermined number of stocks, proper far buyi11g and selli11g the111 011 the stock excha11ge. It's
proposed to perform stocks' buying for the next prognosis (T+ 1)-day to seli them for any subsequent
(T+S+ 1)-day, where the integer number S>0 is assigned a priori.
This selection of proper stocks (the operations with them are the "reasonable" alternatives in
accordance with our proposed MCDM approach) is performed ona basis of the appropriate statistic
data processing, where these <lata are considered for the period l l , .. ,T] of T days, as well as of the
expert estimates' use. All this concerns two parameters of stock exchange process: deal s11111s (DS),
stock prices (SP). Thus, the problem sol11tio11 p111pose consists in se!ecti11g fi'0/11 a fi1ll i11itial set of
stocks (the ISA in our MCDM approach) a predetermi11ed 1111111ber of stocks (their set is the RAS),
proper to be bought far the prognosis (T+ 1)-day as well as to be salt far the prog11osis (T +S+ 1)days.
The accent on the statistic dala processi11g in comparison with the use of expert estimates for ali
considered stocks is more convenient for the problem of selecting the proper stocks among their
great quantity (selecting the small RAS from the considerable ISA). Usually, the best way to select
the proper stocks for their buying-selling is based, first of ali, on using the expert estimates of
production and financial conditions for the enterprises, whose stocks are bought and salt. However,
the principal peculiarity of the considered problem, connected with the competition of great quantity
of stocks, seriously limits the possibility to take such expert estimates for ali considered stocks.
Accounting for it, now we will be oriented on using the statistic <lata for the quite long period,
especially if this period is characterized by "a stable behavior" of the considered stock exchange. In
this situation it's possible to expect that these observed statistic <lata as if accumulate various aspects,

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1
i

affecting on "stock exchange behavior of each considered stock" (among such aspects: the state
appropriate objects and productions, the psychology and interaction of stock exchange buyers and
sellers, etc. ).
Thus, the pwpose of such ow· approach with the practical positions might be treated as to help a
stock exchange expert ("broker") or any exchange "player" in the following aspect: to make the
preliminary "limitation" for the stocks holding, which this "player" might be analyze further using
his regular intuitive methods. For instance, the full totality of stocks in our real sample, see Section
4, includes 97 stocks, but it's possible to select from them, using our approach, a predetermined
small number (e.g. 5-1 O) of "proper" stocks to continue their analysis using the traditional intuitive
methods. This case is very convenient for a new unknown situation on the stock exchange
considered.
3.2. Peculiarities of this MCDM approac/1 applyi11g to the stock buyi11g-selli11g mode!ing
This problem choice to apply this MCDM approach is caused, first of all, by availability of required
initial (statistic) data as well as of specific methodological dijjiculties to realize such approach for
this real situation. Such dijjiculties ("bottlenecks") are co1111ected mainly with the criteria modeling
and the ISCAV construction.
The ISA co11struction has not been of our main methodological interest, since: (a) an initial
alternative concept is ve,y implicit - a stock buying-selling itself is such altemative; (b) the possible
number of stocks isn't vast usually, and the measurable quantity of stocks are possible to be
considered on each existing stock exchange. Thus, the ISA (each its version) may be presented as the
set {i=l, ... ,l} of i-stocks' (initial altematives') numbers.
The ISCAV creation is based on using 2 groups of Criteria calculation mode/s, reflecting:
Bn) stock buying for the prognosis (T+ 1)-day, including 6 criteria types (Criteria 1-5,9);
SI) stock selling for the prognosis (T +S+ 1)-day, including 3 criteria types ( Criteria 6-8).
These criteria values for each considered i-stock (i= , ... ,1) are determined on basis of two
principles of statistical dala processing: (a) finding tile trend-prognosis values ( Criteria 1,2, 6-9),
(b) using the sinusoidal clzaracter of these statistica/ data ( Criteria 3-5). The Criteria 1,2, 6, 7 reflect
the deal sums (DS) prognosis characteristics, the Criteria 3-5,8,9 - such characteristics for stock
prices (SP).

3.3. The Criteria ca!culation mode/s
The main principle of Criteria modeling, based on finding the trend-prognosis values is as: any
such criteria value {Cj[i],j=l,2,6-9; i=l, .. ,I} is determined according to the formula:
(4)
Cj[i}=cr [i]w(l)+cr [ij1/1J+cr [ ij1/1J+cr [i]w(4)+cp; [ijw(JJ+C/6) [i}w(6J+c/R<) [i]w(fa)
where: q(kJ [i] is the k-trend (k= 1, .. ,6) prog11osis value far i-stock, corresponding to J-criterion;
c/ErJ[i} is the prognosis expert value; the weight values {w(kJ,1J1J,w(J),1l4J,w(51 , 1/6J,w!ErJ}, assigned by
experts or calculated (such method for w(ErJ=o was developed [4,5]).
In the appropriate calculations we use the following 6 types of trends: ( 1) Linear (Lin), (2)
Expo11ential (Exp), (3) Logarithmic (Log), (4) Polynomial (Pol) 3-rd order, (5) Power (Pow), (6)
Hyperbolic (Hpr). Further, we consider the sense and calculation methods for ali Criteria, using the
trend-prognosis values and connected with the formula (4) realization.
The i-stocks, having the greatest expected prognosis (for (T+ 1)-day) absolute values
{C 1[i]=AP' (i, T + !), i= 1, ... ,1} far deal sums (DS), are preferable with the position of stocks' buying for
(T+l)-day, since the stocks with the greatest values AP'(i,T+I) might be in great demandfor (T+l)day and some next days. Thus, we use the maximization (on ali i-stocks, i=l, ... ,I) of the prog11osis
(for (T+ 1)-day) absolute values {AP'(i, T+ 1),i=l, ... ,I} far deal sums as the Criterion 1:
(5)
max (C 1[i] =AP'(i,T+I), i=l, .. ,l)
where the prognosis (for any prognosis t-day) values AP'(i,t) are calculated, according to (4), using
the trend-prognosis values {A(JJ (i,t),A(1J(i,t),A(3J(i,t),A(•J (i,t), A(5J(i,t),A(6J(i,t)} as follows:
(6)
AP'(i,t)=A(JJ (i,t)1llJ+Ar1J(i,t)w(1J+ ... +A(JJ(i,t)w(JJ+A(6J(i,t)w(6J+A(R<J (i,t)w(E.tJ

k=l. .. 6} are calculated for ail i-stocks (i=l, .. J) ona
Here, ali trend-prognos1s values
basis of DS stat1st1cal data processing for the penod L.,t-1 For our case t=T+l, the k-trendare calculated for the period l ,.. ,T].
prognosis DS ubsolute values {A 1''(i, + l;,k=l
If ali these prognosis values are obtamed and the expert values A 1E<'(i,T+l) are added., the
reqmredprognosis values AP'(i,t) of Criterion 1 are calculated for ali i-stocks (i=l, .. ,l) using formula
4
where the weight values {1l/J.w(:\w 13',w( 1,w1·\1vf61 .w1Et'l are assigned by experts or calculated.
The last case we have [4,5] when the expert values A!Et'(tJ aren't used (w!ErJ=O).
The Criterion 9 model is the same one, but we prognosis the SP values. Thus, we should change
the parameters offormulas (5),(6) on the SP parameters: BP'(i,T+l); {B!kJ(i,t),k=l,.,6}, reflecting the
/\ in (2)) tre11d-prog11osis SP va!ues: B!E<J(i,t).
relative (according to
The Criterion 6 realizes the same ( 5),( 6) to de fine the DS prognosis va!ues AP'(i, T+S+ 1), but for
the (T+S+ l )-selling day, using the trend-prognosis values {A"J(i, T+S+ 1;,k= 1,.,6}. Since our
prognosis computat10ns are perf01med in T-day. we have not the DS fact-statistlcal data for the days
fT+ !, ... ,T+S]. To overcome this "bottleneck", we propose 2 approaches:
are "as 1fprolonged" to the
A) The statistical k-trend-prognosis, determined for the period [L .,
these k-t1·end-prognos1s formulas.
(T+S+ )-day, i.e. we accept x= T+S+ l
B) We "extend" the statistical basis for the determination of such k-trend-prognosis to the penod
using the method: l) we find the k-trend-prognosis values for the (T, )-day and add them
l
T]; 2) the k-trend-prognos1s values
(as any new "fact" data) the statistical data for the period
for the (T+2)-day are defined by processing "fact-data" for the period , .. ,T+ lJ, and such "statistical
we determine the required k-rrendbasis" is extended up to the period ['], .. , T+Sj, and for
prognosis values {A(kJ(i,T+S+IJ, k= ,.,6; i=I
Thus, we form the calculations models for dete1mining the Criteria 6A/6B.
Comparing the DS prognosis values {AP'(i,T+l),i=l, ... ,I} with the DS fact values {,4lact(i,T),
, taken for the last T-day from the statistical data, it's poss1ble take their d1fferences as the
i=
estimates of quality of these DS prognosis values, deri ved using trends. If the absolute value of such
difference is lesser for the considered i*-stock as compared with other i-stocks, then the DS absolute
prognosis value AP'(i*,T + 1) may be considered as more reliable. With such positions, th1s i*-stock 1s
considered as more preferable. However, such estimates are actual only for a very short-term period.
Thus, we accept the following Criterion 2: the mi11imizatio11 of the relative va!ues {Dor(i,T + 1),
of cha11gi11g the DS absolute prognosis values in co111pariso11 with their fact (statistic)
i=l
values of the preceding T-day, where the Criterion 2 are presented as follows:
(6)
min {Ci[i} =1Y''(i,T+l)=J(AP'(i,T+l)-,4lact(i,TJ}i / ,4fact(i,T), i=l, .. ,l}
The same principle is realized for the Criteria 7/8 models, reflecting tlze maximization of the
of cha11gi11g (for S days) the prognosis (for tlze selling
relative values {V'(i.S)IP"(i,SJ!,i=
(T+S+l)-day) DS absolute / SP relative values in co111pariso11 with the same type ofvaluesfor the
buying (T+ l)-day (both a.m. AJB versions are considered for these Criteria):
(7)
max {C7[i] =V'(i,SJ=W'(i,T+S+l)-AP'(i,T+l)i/ AP'(i,T+l), i=l, .. ,I)
(8)
max (C8[i] =P"'(i,S)=jBP'(i,T+S+l)-BP'(i,T+l)J/ BP'(i,T+I), i=l, .. ,l}
The Criteria 3-5 models, making the prognosis (for the buying (T+ 1)-day) of SP values, take into
account the sinusoidal character of changing the SP values during the period (the days [l, .. ,T]) of
these SP statistical data observation. This sinusoidal character is dictated by the nature itself of stock
buying-selling on stock exchange: when the SP fall, the volumes of such stocks buying increase, and
this tendency is remained up to the days of reaching such SP minimum. After this, another pic ture is
observed, when the SP are growing up, and the holders of these stocks begin to seli such stocks, that
leads again to SP fall, and so on. Accounting for it, it's expediently to buy those stocks, far which are
expected the current 111i11im11m of their SP si1111soids are closed to the prognosis (T+ 1)-day. Using
this principle, the Criteria 3-5 values are determined, based on the calculation of some such SP
sinusoids' parameters. Their values for each i-stock, calculated ona basis of constructing its set of ''jsinusoidal Hills" {H[iJJJ=l, .. ,J[i]}, allow to select i-stocks with the a.m. minimums [4,5].
11
3.4. The Solutio11 Mode/s far the problem of 11reaso11able i-stocks' selection to buy-sell them

I
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89

In a framework of the proposed MCDM methodology applying, we form 4 Solution Mode/s. based
on using various combinations ofusing the a.m. Criteria calculation models:
Model 1, oriented only on buying i-stocks at the (T+l)-day, takes into account the Criteria 1-5. It's
proposed the resulting "reasonable" i-stocks (RAS), which are found accounting for the prognosis
maximal DS/minimal SP values in (T+ 1)-day, might be freely used (to seli, to accumulate) further by
their holders using any additional, mainly non-formal, estimations.
Model 2, oriented on the "reasonable" i-stocks' accumulation for the future to sell them for a
maximal profit obtaining or their holders' capital increasing, takes into account the Criteria 1-5,9 or
1,2,9 only. Here, finding the prognosis on the buying (T+ 1)-day, we attempt also to estimate the
increasing tendency for SP values (Criterion 9) to maximize the holder capital.
Models 3/4 consider the prognosis operations of i-stocks buying at the (T+ l )-day jointly with
their selling at the (T+S+ 1)-days, where S are the predetermined natura! numbers {e.g., 1,3, .. ). Both
these models account for various combinations of Criteria l-5,6AJB-8A/B (e.g.,{l-5,6A-8A},{l5,6B-8B},{l,2,6A-8A},etc.). The distinction between these models consists in the following: Model
3 prognosis only one S"-day oj stocks' selling with the fixed such (T+S"+ 1)-day: Model 4 - severa/
(> 1) such S"-days (e.g. S"= l , .. ,Smax) and its result should reflect the mean resulting estimations,
reached for ali considered S"-days. Thus, Model 3 corresponds to a rare situation when it's needed to
seli the "reasonable" stocks only in one fixed (T+S"+ 1)-day (e.g. to accumulate money by the profit
obtaining for a duty payment at the next day); Model 4 c01responds to the more natura! situation of
speculations on the stock exchange during severa! nearest S-days to increase the holders' capitaL
Here, we could use the mean (for ali considered S-days) results, when we buy i-stocks at (T+l)-day.

4. Testing the proposed MCDM approach and Models of stock buying-selling
We make this testing as a "debug-regime", based on using the jact dala jor the past period
[l, .. ,T,T+l, .. ,T+ l+Smax], that allows: (a) to reach the RAS versions by the ISA/ISCAV decreasing
applying the sL"(-level hierarcllical system oj MVC series; (b) to compare the "reasonable" i-stocks
from the RAS with the ''fact-good" i-stocks, obtained by analysis of fact data from the whole past
period [l, .. ,T+ 1+Smax]; this analysis allows to estimate the quality of the prognosis results, obtained
by processing the fact-statistical data for the period [1, .. ,T]. On this way, we met the dijficulties in
the exposes oj ''fact-good" i-stocks, where these difficulties are initiated by a contradiction between
the SP increasing and the DS low levels, observed for the same i-stocks. This contradiction might
overcome, on our opinion, using any intuitive compromises: the acceptance of limiting levels and
adequate weights for the Criteria values, etc.
The real sample, considered early for the testing of Model 1 [4,5], presented the fact (statistic)
data, reflecting the "moving" of deal sums (DS) and stock prices (SP) for ali "Tel-Aviv-100" i-stocks
{i=l, ... ,97} of the Israeli stock exchange for the past period (11/11/01-02/01/02) (in ali for 33
days). We considered the day 01/01/02 as T-day, and 02/01/02 as the prognosis (T+l)-day. Thus, the
period [l, .. ,T], reflecting the statistic data needed to construct ali Criteria 1-5, included in the
Model 1, is 11/11/01-01/01/02 (T=32). The statistic data for the prognosis 33-day were not used in a
framework of six-level system calculations, but they were used further to estimate the deri ved results
(in the debug-process). To illustrate this six-level system perf01mance, we accepted the 1-scenarios
for ali 1-levels (/=1, .. ,6) ofthis six-level system (see this paper Section 2 as well as [5]):
/=l) the ISA (1 version, K=l), presenting the part ofthe "Tel-Aviv-100" i-stocks {i=l, ... ,20};
/=2) the ISCAV (M=l), including Criteria 1-5 (according to Model 1) values, calculated by the
DS/SP processing for the data, taken for the period [l, .. ,T] (T=32) and for ali 20 i-stocks;
/=3) the TOPSIS modification (1)-(2) as the multi-criteria optimization technique used (N=l);
/=4/5) 24 versions of the random weight possible value intervals (3) to realize the function (1), that
reflects ali possible combinations of 6 versions (p=l, .. ,6; P=6) of interval (3) "central points" (4scenarios) and 4 versions (R=4) ofinterval (3) bound values (5-scenarios);
/=6) 2 versions (q=l,2) of Monte Carlo simulation (MCs) series (6-scenarios). Each MCs series
\
includes 5 MCs, each ofwhich allows fix the interval (3) interior value.

90

the proposed M< "DM methodology and the a.m. /rhe cakulations performed according
scenarios system (l=l , .. ,6) usmg th1s real sample data allowed to find a set of RAS versions. Their
ana!ys1s led to findmg a final solution, mcludmg 5 best i-stocks. The1r comparison with the "goodfact" stocks. found using a specific analysis, showed their comcidence on 40-60%.
At present, the tesiing is performed for the full lSA of this sample (i= ... ,97). the extended period
statistical <lata (up to 40 days with varying values T and Smax) and for Models 314.
ln the future. such testing will be performed on-line. for the current data of exchange.

Conciusions
The performed work showed the proposed MClJM approach might he app/ied far the
considered problem solution. Further we will accent on practical aspects ofsuch applying

References
Kalika,V., Frant, S. (2000) "Multicritena Optimizalion Accouming for Uncertainty m Dynam1c
Problem of Power Generation Expansion Plannmg"Research mul Pmctice in Mullip!e Criteria
Decision Making. Haimes YY; Steuer RE. (eds.). Lecture Notes in Economics and Mathemallcal
Systems, Springer-Verlag 487.409-420.
2. Kalika, .. Frant, S. (2001) "Multi-criteria Approach for Power Generation Expans10n Plannmg"
Multiple Criteria Decision in the New Millennium. Koksalan M. Zuioms S. (eds,).Lecture Notes m
Economics and Mathematical Systems, Springer, 507, 458-468.
3. Kalika, V. (200 "Multi-Level Hierarchical System to Take into Account Uncertamty m MultiCriteria Decision Making Process" Proceedings of the Sixth lntemational Symposium on the
Analytic Jiierarchy Process, ISAHP 200 l - TSAHP Switzerland 200 l, 173-180.
4 Kalika,V. and Rossinsky, G. (2003) 'Methodology of multi-criteria decision making accounting
for uncertainty and some its application'. IJMDM (l11ternatio11al Journa/ oj Managemem and
Decision Making), 4, Nos. 2/3, 240-272.
5. Kalika,V. (2004) 'A new methodology of accounting for uncertainty factors in multiple criteria
decision making problems'. JTTT (Joumaf of Teleco111m1111ica1io11s and I11jormatio11 Teclmoiogy).
3/2004, 44-56.
6. Massam, B.N. (1988). 'Multi-criteria decision making (MCDM): Techniques in planning'
Progress in Plmming 30, 1-84.

91

··1
1

LSING GAME THEORY 11\/ Al\J"ALYSIS OF MORILE
TELECOM:VIL'NlCA TIONS MARKET
Student of postgraduate

l,ada Roko,
Operations Research. Facuit) of Econom1cs. l
e--11rn1l rola4@net.hr ( or rola4@Yahoo.com)

lll\

erstt, of Zagreb

Abstract
This is analysis of telecomrnunicat1011s market where two symmetric mobtle operators compete for
access to each other's customers. J
final prepaid customers but cooperate rn setting the price
presemed model of four-stages game using Hotelling specification and Bertrand price competition
twO-\\ a::- access and terminat1on-basec! pnce d1scrirninat10n.
under assumptions

Ke:1-words: Game theor1 . '-et,1 ork
discrimination.

Te]ecommunications. Network externaliues. Price

Introduction

A distinctive feature of the telecommunications industry is the need for competing networks
prn\ ide each customer "'ith access to the en tire consumer
access) This leads to au interesting coexistence of \·ertical
population (so called
and horizontal relationships. where each net\\ork is the monopolistic supplier of an
intermediate input - the terrnination of calls directed towards its own subscribers to its
turn relies on them to provide access to their customers. H
rivals in the retail market and
be socially inefficient if each firm sets its
is generally agreed that the uutcome
termination fee independently. because a unilateral increase by any firn1 raises its riYals'
costs without affecting its own.
local competition is deYeloped, two-way access in telecommunications were mainly
confined to two types of interconnection (as mentioned by Laffont and Tirole):
international calls: telecommunications carries have traditionally enjoyed a monopoly
position in their respective countries and cooperated in the dete1mination of
termination access charges. These settlement charges are determined in private
negotiations between the carriers. They are set above the marginal cost of terminating
calls. Such high termination charges may be quite inefficient even from the point of
view ofprofit maximization.
fixed-mobile interconnection: the offering of mobile services by companies other
than the incumbent fixed-link operator has given rise to a second set of two-way
termination access charges. Charges for termination on a mobile network are often
set quite high.
Cooperation among competitors in setting access charges is relatively new in the
telecommunications industry and the telecommunications industry and regulators are
currently defining principles for interconnection.
Assumptions oftwo-way access price economic theory are:

92

93

T

calling compan::, pa:,s terrnination charge the call rece1ver s ne1work charges an
amount a per minute to the calier·s network for termination. The call receiver pays

Finally. consumers place the1r a,.:tual calls accordmg to the!f calling pattems and tariffs. and
payoffs are realized. As usual. the model 1s soh ed back\, ards.

nothing.
unregulated retail markets: telecomrnunications carriers are free to set the prices they
wish to fina! consurners.

Stage 4 (consumption

l!nder two-way access. we distinguish between two kinds of calls:
An on-net call is a call that origi11ates and terminates on the same netvvork. That is, the caller
and the receiver belong to the same network. In contrast, an off-net call occurs when the two
parties belong to different networks.
Retail tariffs offered by operators in most mobile-telephony markets often discriminate
between on-net and off-net calls (so called termination-based price discrimination).
A regulator can be interested in this type of pricing because the traffic-dependem costs of
on-net and off-net calls may be different due to cost reasons, such as:
differe11t efficie11cy le vel s of the operators' networks:
terminating access prices set aboYe the traffic-dependent cost of the local loop (so
that the traffic-dependent cost of an on-11et call is higher than the cost of an off-net
call)

demwu!;

Under termination-based price d1scrimination (if on-net and off-net calls have different
prices), a consumer makes longer or more frequent calls to those who are subscribed to the
same network.
6
Given a price per minute equal to p ( p " for on-net or p JJ for off-net). each co11sumer has
0

an indi\ idual demand

of x[p]

call mimnes. and deri ves utility

u[x]

from calli11g for x

minutes. Consumers deri\e uti!ity frorn calls made to consumers subscribed to the same
net\vork (their share is 1· ) and from calls made to consumers subscribed to the nther
network.
The indirect utility that a consumer deri,es from a call at a per-minute price of p can bc
expressed as:

v[p]=u[x[p]]- px[p]

Accordingly. the idea is that terrnination-based price discrimination may result in per-rninute
prices that reflect the underlying costs a11d access fees to a better extent than nondiscriminatory prices.
To the extent that interconnection fees and costs are publicly observable, it seems highly
doubtful that most networks terminate each others' off-net calls at a discount.
When off-net calls are more expensive than on-net calls, then consumers choose the same
network as those whom they call the most, so the majority of calls is made at the on-net rate
(so called positive network externality). For this reason, the networks tend to prefer a
positive markup on interconnection.

Model
Here is presented model based on Hotelling specification and Be1irand competition (which is
price competition with the comminnent to supply whatever demand is forthcoming at the set
price). The model is similar to the one discussed by Peitz, but I focused my analysis on
telecommu11ication operators with prepaid customers.

where x[p,] = arg max.

:u[x] · xp

; is the uptimal call length g1ven p (irnJindual demand 1s

deri\ed by solving the first-order cundition for utility maximization u [x]= p ).
ln addition to the utility from calling. each costumer derives a fixed utilit:,. leYel U from
subscribing to network i. mdependent of the nurnber of telephone calls that are made. ,\ b1ch
may come from, for example:
brand preference
senices offered in addition to HJtce telephony (SMSes, VAS)
the quality ofthe network and senices deli\·ered
ha, ing a telephone connection in the case of unforeseen evems
the possibility to make free calls (ambulance. police, ... ).
It is assumed that the fixed utility L is the same for all cousumers of network i
The total net utility of a subscriber to network i, who optimally chooses its caliing tirne_ is
composed of a traffic-dependent (utility derived from on-net cails plus the utility derived
from off-net callsJ and a traffic-independent part U,, and is written as:

lpt'' P!'JJ

j = U, + s,vlpt' j+ (l

s )v[p,~/T

j'

(1)

The point of departure for this model are two operators practising termination-based price

V;

discrimination.
Operators i and j offer telephone services to prepaid end users. Both operators use the

Traffic dependent part only depends 011 the calls made and not on the calls received. Clearly,
net utility is decreasing in prices.
To guarantee that consumers want to participate in the market, net utility leve! v1
p;'1l

same technology, so they have symmetric cost structures. Bach operator has a full-coverage
network, and ali consumers (total mass of consumers is n ) are subscribed to either one of
the two operators. Consumers of both operators do not pay monthly subscriber charge (so

lPt,

j

has to be positive.

called prepaid customers).

Stage 3: Subscription
The game has four stages, and the order of moves is as follows:
.
First, in the cooperative stage, the networks jointly detennine the access charges. Second, m
the competitive stage, they independently and non-cooperatively set their tariffs.
Third, each costumer subscribes to exactly one network.
.,...---

The subscription stage is best described as a market for network membership with unit
demand and positive network extemalities (positive network externality is when product
becomes more valuable as more customers use them) 011 the consumption side .

94

95

--1
1

off-net calls: cails that term mate on another operatur's network ( k = 2) with
associated costs c. a for operator i !on-net costs
wholesa\e access price paid

It is assumed that networks are horizontally d1fferentiated and Hotelling spec1fication 1s

used.
Consumers are assumed tu be

the

distributed

l ]-intenaL Operator

lS

to the other operalllr):

1

- 0[/ z:
l] incurs a
Consumer z E
located at /_ = O and operator 2 at / 2
which is linear in the distance between consumer location and the location of the operator
The parameter {) expresses the substitutability between networks: if fJ =O. networks are
perfect substitutes; the larger {), the more differentiated networks are (larger 0 makes it
more difficult to gain market share).
Each consumer subscribes to exactly one operator. The consumer \\ ho is identified by its
11
location z subscribes to operator l if vlp~".p~11 ]-0z>v 2[p~ p;1f 0(1-z). where

v, lPt. P? j denotes the net utility of a network at the ideal location z
The realized market share of operator i is equal to:
s

1

'

=-

2

v,[p"",p/I]
+

vJ[p~",p;o]
_or
-

20

+0
S,:::

2-

1

~ (v[p;"' ]+ v[p"" ]- v[p'.'
0

1

1

Before profit functions can be deri\ed. we ha\e to specify calling pattern.
We make the assumption that calling patterns are balanced in the following sense:
consumers' share of on-net call minutes c01Tesponds to the market share of the operator to
which a consumer is subscribed to. after c01Tecting for differences in per-rninute prices. That
is. the total \olume of call rninutes that originates and terminates on network i is 11s xlp"" j
and the demand for off-nel calls is ns,xlp °ff J1

I

usmg 1):

Operator · s profits are:
lf,

Clearly, it is presumed that s

j

[P,011. P,off po" p~ff

=

+ nu- s. )s;x[p ff

]-v[p D
0

0

0

"

"

'

Stage 2. Co111petitio11 in tar(tfs

la, -c, i

s, )xlp,off poff -

.-a

nsJ

By Bertrand competition_ operators set their on-net per-minute price equal to on-net
marginal costs. that is:
011*

P,

=C!

and their off-net per-minute price equal to off-net percei\ed marginal costs, that is
o«•

P/'

[p;"' ,p;ff' ,p;" ,p;ff] = n(l - s, )s;x[p;-8.

of type k and let a; denote terminating access prices paid to operator i. We distinguish
three types of telephone calls under assumption that both operators use the same technology:
on-net calls: calls that originate and terminate on a single operator's network (k= 1)
with associated costs c 1 for operator i;

96

ka, - c

3 ) -

nsJ

Accordingly. operators have only one source ofprofit - revenue from incoming calls.

Stage 1: Determination oj access charge
This stage can be cooperative - when two networks jointly dete,mine access charges a_ (or
a 1 ) and corresponding access markup (difference between access charge and marginal cost
of terminating call per minute), in which case operators usually mutually set reciprocal price
(a;

denote operator i 's traffic-dependent cost per-minute associated with a telephone call

=c2 +a1

This follows directly from the first-order conditions Dlf, = O and alf, = o
auff
o""
'P,
'P,
profit-maximizing perits
chooses
operator
which
in
profits
·s
i
operator
Consequently.
minute prices can be written as:
lf;

In order to specify profit functions and corresponding retail prices. costs must be defined.
The fixed costs include ali costs that don't depend on traffic (these costs are denoted by f )_
Such costs include, for instance, the costs of building a backbone, the maintenance cost of
the local loop, loans repayments, cost of billing, etc. It was implicitly assumed that in the
calculation of marginal costs, no traffic-independent costs are included.
Total traffic-dependent costs include an operator's marginal cost (real costs of originating
and tenninating calls) plus per-minute charges paid to other operator for interconnection and
access denoted by a; (or a 1 ).
c;k

k

POII - c, ) + ns

l

depend on ali retail prices.
Consumers base their decision on which network to join on market share because, for
instance, at a iow on-net price an operator is attractive if has a large market share. This
means that consurners' choices are affected by the market share. Therefore. terminationbased price discrimmation gives rise to a positive network extemality.

Let

l= m.
0

01

The market shares resulting from competition and consumers' choices are denoted by
p 2 ' p 2ff] ' and as we see ' market shares
p 21rj cand s 2 = s 2Llp l ' p~rr
p'_ 1r p""
S 1 -- s 1
1
1, ,
1 ,

lp""

It is assumed that c = c _ + c so that society's rnarginal cost of an off-net call is the same as
the one of an on-net calL

·r .

11,f p"11 ]-v[p"" D
- L J )- + 0\L,
J
11

calls: calls that originate from another operator's network /k= 3) with
associated costs c and wholesale access price u, for operator i

= a 1 ).

There are some cases when access charges are determined by regulator:
when operators can not make an agreernent
when regulator wants to stimulate competition - this asymmetric access pncmg
exists as long as there is one dominant player in the market (especially in the recently
liberalized markets)

97

Operators' cooperat1ve agreement on access charge immediately raises the question of
whether an unregulated access charge can be an instrument of collusion the retail market

Conclusion
This paper has been focussed on a telecommunications market where two operators compete
for retail prepaid customers under two-way access pricing and terrnination-based price
discrirnination.
An interesting feature of this new form of competition is that terrninmion-based price
discrimination reintroduces network extemalities among consumers, Interconnection
together with uniform (non termination-based) pricing implies that consumers, when
choosing a network, do not take into account the choice of network by the people they want
to call. In contrast if on-net calls are cheaper than off-net calls, consumers are better off if
the people they want to call selec! the same network. More generally. an access markup
(access charge above marginal cost) generates positive network externalities among
consumers.

AN EXA:\'IPLE OF A GAME AND ITS COMBINATORIAL ANALYSIS
l lanm Vukiče,
Janez Žerovnik:
l Je1•,1nrnent "! '.\lathema11cs. L'rnversil> uf Split. NtkPle Tesle l 2, HR-21000 Srlil, Croalia
1 111ve1
Faculh of Vlechanical Engineern1g. Smetanova 17, 2000 Marihor, Slovenija
!MFM/DTCS. Jadranska 19. 1000 L1uhi1ana. Slovenija
Emails: rnk1cevic@prnfst.hL Janez zerovnik@11nfmuni~lj,si

Abslrad.

e g1\e an
slw\1rng hov,, graph
,iluatl(lns
econunm:s, The
mformallon. ancl known rules
a coalit1on
g1ven

mm be used
model and anahse
1s a game
tliree plavers. perlec1
An 1'•pt1mal strateg\ iur the third plaver 1s

Keywords: cPrnh111atonal game the11n. lOC1perallve game,

Limitation of marginal cost pricing
Social welfare is achieved by setting prices equal to marginal cost But there are some
problems with marginal cost pricing as discussed by Courcoubetis and Weber tirni follO\\:
Marginal cost prices can be d1fficult to compute and they can be etose to either zero or
infinity. This is a problem since telecommunication networks typically have fixed costs
which must somehow be recovered,
Think of a network that is built to car1y
cal!s. The main costs of running the network are
fixed costs (such as maintenance. loans repayments. cost ofbilling, subsidy on handsets and
staff salaries), i.e, invariant to the leve] of usage, Thus when less than x calls are present. the
short-run marginal cost that is incurred by canying another call is near zero, If the network is
critically loaded (so that all x circuits are busy). the cost of expanding the network to
accommodate another call could be huge,
Another difficulty in basing charges on marginal cosl is that even if we know the marginal
cost and use it as price. it can be difficu!t to predict the demand and to dimension the
network accordingly. There is a risk that we will build a network that is either too big or too
srna!!. Prices are used to signal the need to expand the network.
However, welfare maxirnization is not the only thing that matters, A firm 's prices must
ensure that it is profitable, orat least that it covers its costs.
In reality, operators typically impute fixed costs to telephony traffic, enabling them to define
a reference point for prices.

1.lntroduction
ears
thanks
the 1994 >Jobe! '.'vlernorial pri,::e
Nash, Selten and Harsa11y1,
game Lheor\ and its applicat1011 to economics are hecommg increasingly popuiar liame
theon 1s a u1sti11cl aml interd1sciplinary approach to the study of human hehav1or The
d1sciplrnes most imoh'ed in game theory are mathematics. econurnics and the other social
anJ beha, ioral sciences, liame theory (like computatJonal theory and so man\ other
comnbutiPns, was founded hy John \011 Neumann [Neumann and Morgenstern 1
The airn of 1h1s paper is to show how comhinatorics.
particular graph theory can be used
for anahsis of a game Although our example is not very comp!ex. we claim that the analysts
given shows the potential of graph theory Wilson and Watkinsj to prov1de a simple and
effic1ent method for such tasks We will model the game by a directed graph G . The set of
will represent the sel of ali possible states of husiness
(; uenuted b\ V (
vert1ces
affairs anct the set of arcs A ( c;) of graph G will represent changes from one state of
the other state of husiness affair which can. in the example outlined below.
be causect
hy one of our actions anJ predicted reactions from lhe opposn
There is
also the function
which teli s how rnuch monev we earn during this transition i of course
/ can take negative \alues if we spend or loose mane\ in such transition). Our aim 1s to
maximize (in long terms) our profit.

References
Courcoubetis, Costas; Weber, Richard, 2003.: "Pricing Communication Networks", Wiley,
130.
Laffont, Jean-Jacques; Tirole, Jean, 2000.: "Competition in Telecommunications", MIT
Press, 179-180.
Peitz, Martin, 2004.: "Assymetric access price regulation in telecommunicationsmarkets",
European Economic Review, 3-8.

98

We will define a game, which may be motivated by a practical situation where we have a
small number of competing businesses. We assume the players have agreed on some rules
and play fair in the sense that they tend to increase their profit but do not attempt to decrease
the profit of others. Cooperative games are particularly important in economics. The idea of
cooperative games is illustrated by the following example [McCain].

99

Example. We suppose lhat Joey has a b1cvcle. Joev woulct rather haH' a game machrne than
a b1cycle, ancl he could buy a game machine for $80. but Joey doesn t ha\ e any money We
S,l\mg that
npress th1s
ouid rather lia\e a
sayrng that M1kcv ,alues a

\alues h1s
cle than
at $ 00

ike) llas 'i,I00 and
cle at $80.
else he can hU\

give or to keep. That is. Joey can gi\e his
The strateg1cs a\ ailable to Joey and Mikey are
b1cycle to M1key or keep It, and M1key can g1ve some of this money to Joey or keep alL
is suggested lhat Mikev give Joey $90 and that Joey gi\'e Mikey the bicycle. This is what we
cal 1 · exchangc. Here are I he payoffs

fifteen
R and
fifteen
4
h\ A and
ones
51 ,\fler a11ol1ler fifteen
11

nes

61

arn iou1t1()!1 clifferelll form
A can relocate h1s \ehiclr
( A can choose , Sla\ ar h1s present l,icatwn
LLfferenl form
am local
mrnutes 1-J can relocate h1s \el11cle
( . B ca11 choose to stay at h1s present locatmn i.
all\ locmwn different form
rnrnutes C can relocate h1s \eh1cle
sla\ at h1s presenl location i.
. ( · can choose
repem

Remark. lf the Step 2
si

a \ ariant

lhe agreement sounds to favour player A too strongly. one can
the game where the first player and the nrder 1s detennmed

Table 1

keep

give
!\like)

gt\e

O. 90

_ ,

: keep ' 200.

170

00. 80

Explanation: At the upper left. Mikey has a hicvcle he values at $ 00. plus $1 O extra. while
At the luwer left Mikey has
Joev has a game machine he \alues at $80. plus an extra $
the bicvcle he values at $100. plus $100 extra. At the upper lefL foey has a game machine
$ O
and a h1ke. each uf which he values at $80. plus $ l O extra. and Mikey 1s left wnh
hike.
a
and
$100
Mikey
wilh
begin
they
what
have
simplv
At the lower right. thev

1f we think nf this as a noncooperative game. it is rnuch like a Prisoners Di!emma. To keep
is a dommant strategv and (keep. keep) 1s a dominant strategy equilihrium. Howe,er (giYe.
give1 makes both better off Being children thev may distrust one another and fail to make
the exchange tirni will make them better off. Bul market societies have a range of institutions
that allow adults to commit thernselves to mutually beneficial transactions. Thus. we would
expect a cooperative solution. and we suspect that it would be the one in the upper left.
In the next section we give a motivation and a definition of the game that we will analyse. A
model using graphs is given in section 3 while section 4 gives an efficient method to fine\
optimal strategies In the last section the rnethod is applied to the example given in section 2.

d1stnhuted and that 1he\
lile heach are
Lan !Je assumed that
the clnsest se>ller of the ice•cream. ,el 11s e\plrnn th1s on one s1mple
1ce-creams
th
the
the 15" iocallon, and C
the 4 locat1oi1. B
Suppose that A 1s located
the peop!e on the beach near locations 1 2. 3, 4,
locat1011. A wil seli the 1ce-creams
the
ali ice-creams. B wlil seli the 1ce-crearns
I seli 36o/c
'>. 6. 7 X. and l/. lience he
the half
14. i "· 16. ! 7 and 18 and also
the heach near ioca! 1uns l
hence he will seli
s ice-cream from
19 !the nther
the peurle 11ear the locat
will seli the 1ce-creams to the people on the beach near locations
~):\'31, of ali 1ce creams.
1he half of the peopie near the iocallon 19 hence he wi! 1
20. 2 1 22. 2'.? 24 and 25. and also
al 1 1ce-creams.
sel 1 26%

c,.

such p!ace that he rnakes
Suppose that ··s strategy 1s known He alwms places h,s \'eh1cle
fifteen rninutes. Tf there are !lllXe than one most profitable
in the
the greatest
locatlon C chooses localion that will make the greatest prnfil of B in the next 15 rnmutes. B
uses the facl that R relocates his ,;ehicie after (' anrl that he knuws the r·s strategv. Hence.
h1s next 30 minutes. Tf there are
optimize his pwfit
he Lall locate his vehicle lll orcle1
more than une rnPst profitable locathm. B chooses locallon that will make the greatest profit
1 the ne.\t 1.'i rnmutes. lf su!i the locatmn is not urnguely defined B chooses locat1011
and are "in the eak u.Jalition" Th1s increases iheir
w11 tile srnaller number \i(lle thal
profit anJ reduces the profit of A But since their strategies are k11own. A can adopt h1s
strateg\ in nrder 10 obtarn the greatesl profil.
!ere 1he strne
Hence.

business affa.rs is descnhed b) lhe locauun uf the veh1cles B and C

\.

!=

2. Example
Let us illustrate our ideas by using a very simple example. Suppose that three people. A. B
and C sel! ice-crearn on the beach and suppose that there are 25 convenient locations for
selling ice-crearn that are equally distributed along the beach. In order not to fight among
themselves the following agreement is obtained:
1) No two sellers will seli ice-cream on the same location.
2) First, A chooses any location and sells ice-cream, after a quarter of an our B chooses any
chooses any
location different then one chosen by A, and after another fifteen minutes C ,----location (different frorn ones occupied by A and B).

100

The ordered pair (

.\ / means that B is located at iocat10n x and C is located at the

locat1on v whtle 0 denotes the position at the verv beginning of their business, when no
vehicle is located. After A relocates his vehicle, it is uniquely defined (since both strategies
will be located. Hence, the reaction of B and C to any
are knownJ where vehicles B and
action of A 1s uniquely defined. The function f assigns the profit of A (i.e. the percentage of
the market taken by A) m the next 45 minutes. Our aim isto fine\ the cycle with the greatest
average value of f on its edges which can be reached from the initial state.

r

101

---............--

! .e

3. Definitions

1s ll1e ma\.1rnal sum o! edge v.e1ghts o,er ali paths from ./

u

\\1,rds.

1epresents the pr"fit ol the 11,ost protitable palh irorn

11

t,, k
1

!(i

iength

,;

k

ni

I

ie11gth

In nrher
I

Now,

(1

De1wte b\

/

A i( i)

----i

1s1 (

R

G i the vertex of (,

which represents the mitial state \ertex.

N. 1s the set uf real numbers) be the function called

I where

Let

den1,te 1)\ /,

1Si. J S II the pair of the numbers ·

;1mfit j u11ctio11.

Let H be the subgraph uf ( 1 v. hich consists ol ali vert1ces thal can be reached frorn the
vertex 1sv (C)

Let ( 1.

1,

1 he tile pair of numbers such that h,

the lex1cograph1cal order lf h

Let (' he any ,1riented C)Cle 111 !! Denote
)

;_;

uv(('J=

1,

\I

ill

?.1c, h

for ead1 1S

1.

.1 S n, where

1

= 1,

s,,.,

1s

rhen there is the cycle C E SetOpt of length

v,,, in order

. Denote \ertices of this cycle bv v. v...

such 11a\ that

!(11eac·h ;i=l.2. ... 1,
!l1

( - 00 • i0 ).

11 111

\Ve nrn 111Juct11el\ recnnstruct the rerna1111ng

1 ertJces b\ the fol lowmg procedure

111

where rn 1d i A (C)) 1s the cardinality i or

>1...

hich passes through the venex

of the1r appearance

![C')

\'
E'-i,:
(llid ( A I C))

-/

u._

=q suchthat

!.let r.

+

+a

,tf !o

=ll

lo /o /o

the finiie case the nurnher of elements l of the set

'\ole 1hat tile L'1JP1ce ot C/ Joes not necessarih lia\e to IJe U!i1que. but hs. ch1iosii1g d1fferen1
A(Ci

The a1111 of th1s analvsis 1s to find an effic1ent algorithm that finds one cycle C· such that
11,

\ alues ot 11 . \\ e ah\ a\ s get the c ', cle \\ ilh the reqrnred prnpert1es.
It ,s clt'ar J1()Jl1 the defirnt1on that this algorithrn can be executed 1n poi\norrnal 11me ;n the
number ol \ert1ces anJ that this polvnomial 1s 01 degree 4. Hence. there 1s an effic1enl
algonthrn that s,11\es th1s problem.

1C) S uvi (' ) for each oriented cycle C in (j

r·

does not ha\e w be u111que. In this case the algorithrn (tube explamed in
Note that cvcle
the nexl sect1onJ returns one of the possihle shortest cycles that satisfies the relation 12)
Denore ri) SeiO[!i the set uf ali such cycles.

5. The solution of the example
the solution of the example described

'.\ovv. we use our algorithrn to find

sect1011 Recal I that \ · ( (; 1= {( 1. r) l S .L

.1

S 25 . .\

* v] __; •[ 0}

111

the second

has 60 l vert1ces.

Hence. 0

For 1h1s narnrle. we ha\e created graµh (, b\ computer. it has 12869 arcs. After thaL we
ha I e lTeated graph H [i !:as 59 1 ert1ces anJ 1298 edges. The opt1rnal , \ cle reads as

4. The algorithm
One can eastl) obtain the graph H when (J and isv ( (J) are known. Denote by n the
number of \'ert1ces m H Let us mductl, el:, define the numbers "

1S i.

!. k

a='·

119Ji1 ----i ilU~))

S 11 r,\:

, =98 '

a=8

11=19

,20.81

~•

----i

17.2!J1 •
· =98/"

· =98

· =il6 '

11=19

i18.71

----i

,6,181 •

, =106. '·

119.61.

=98,

where o denotes the place where A shall re!ocate his rnhic!e. Nore thal the a1erage prnfit 1s
j()(J
,02
> -,- hence us1ng 'h1s strateg'\ A earns more than the a\'erage wh1ch 1s interest1g
·
'
9
forma weak coalition agamst him On the other hand,
beanng in mmd that players B and
it is oh\lousl'\ a maJor advantage for A that the agreement between players B and C is
known to him.

jkE A(H)

/k 'l. A(H).

r

a,.•

,,k

=maxla
:~x~n l ,

+a,.; f\

It can be easily seen that

u;.;.k

= max ~ I f
llff~~

( e): P is path fromj to k of length

A,~

In order to cornpletely dernonstrate the strategy for A. one has to explain how to enter the
optimal cycle. In fact we can enter in this cycle right from the initial position by the
fol10wing step
t1=7

•

11=7 =
a=l9
a:::::8
a=l9
{/;:::c.7
c1=l8
=
(6,18) • (19,6).
(20,8) • (7,20) • (18,7) •
(19,6) • (8,19) •

·=17(1/3~/=98/3

102

./=10613

1=98/3

103

/=98/3

f=I06/3

/=9813=

Remark.

r we

u>11s1der a , anant with random urder at the start. then things

signirieantly Analyzrng our mitial graph with 600 vertiees.

change

can be shown iby exhausti\e

seareh. for example) that for eaeh vertex. there is a path that leads from that vertex to the
,Jptimal cvcle and then the game rernarns in the same cycle. Since we are interested

The 8th lnternalional Symposium on
Operalional Research in Slovenia

111

th1s cycle ,md 1wt in the first few 1niual moves everything remains essenllall: the same.
the long run earn more than a verage despite
Hence. m, 1naller who starts the game. A
the weak cu,tl

,,f

S@R'OS

B and ( ·

Nova Gorica, SLOVENIA
September 28 • 30, 2005
References
Neumann and Morgenstern] John \011 Neumann and Oskar Morgenstern. Theor, of games
Jnd ecnnorn1c hchav1or ."\rcl ed .. Prineetl1 n Cnivers1ty Press. Princeton. 195'.\.

Section3:

1Me( ·arn I Roger A. Me( ·a111. An lntroductory Sketch of Game Theorv. lecture nntes at

http:/ /w i l l1am-king. www.drexel.edu/top/eeo/game/game.html
[Wilson ancl Watkins Robin J. Wilson and John J. Watkins. Graphs. An Introducrionary
Approaeh. John Wile\ & Sons. New York. 990.

1

104

!

\

---.--',

THE SCALE OF lNNOVATIVENESS WITH STRllCTURAL
EQlATION MODELING
Majda Bastič
university of Maribor
Econorn1cs and Business
Facult,
Razlagova 14. 2000 Maribor
111aJda.bastic@uni-mb.s1

Abstract
rhe structural equation modeling is rhe best knm,n methodology for determining the scale for
th is paper is tu presen! the application of structural
interest. The
thec,retical C(lllStructs
organisational i1movative11ess on the sample "f 2: 4
equm1011 model ing on determining the sc>ale
Sk,venian organisations. lnnU\ation capabilit, is namely one of tile ke~ competitive success factors
tile mtense giubal c,,mpel1lion

Keywords: Structural equation

ati,·eness. Dimensiuns

L Introduction
Today. the abilit) to develop ne\\ ideas and innmati,ms is one of the top pri,irities of
organisations and 1he key competitive success factor. Therefore. many researches were
focused on faciors influencing the mnovation capability of firms. In these researches one of
the main research 4uestions v:as associated with the scale uf the urgarnsationai
1nnmatiYeness.
The purpose of measurement in theory of testing and Jevelopment research is tu pruv1de
an empirical estimate of each theoretical construct uf interest L:S j. The irnportance of
unidimensiuuaiit) has heen staced succinctl\ b\ [attie "that a set of items formrnrs an
instrument all measure just one thing in common is a most critical and basic assumption of
measurement theory" [4].
The scale development process must include an assessment of whether ,he multiple
measures that define a scale can be acceptably regarded as alternative inclicators of the same
construct. The need for unidimensiona]it:, has first been established by Nunnaly by stating,
"ltems within a measure are useful only to the extent that they share a common core - the
attribute which is to be measured. 'V j.

2. Structural equation model

/

Structurai equation modeling can perhaps best be defined as a class of methodologies that
seeks to represent hypotheses about the means, variances, and covariances of observed clata
in terms of a smaller number of structural parameters defined by a hypothesized underlying
model. The general structural equation model consists oftwo paiis: a) the measurement part,
linking observed variables to latent variables via a confirmatory factor model and b) the
structural part, linking latent variables to each other via systems of simultaneous equations.
The estimation of the model utilizes maximum likelihood estimation. In the case where it is
assumed that there is no measurement error in the observed variables, the general model
reduces to the system of simultaneous equations model developed in econometrics [5].
The structural equations, as represented in a path diagram, are seen as a one-to-one
representation ofthe theory. They are defined by

107

{

}

mdicative o! gooc.i fit. tlwse between iHl" and under 0.8 show reasonable fit, values between
08 and U.
are indicati \ e of mediocre fit and values larger than O. !O are the sign of poor
fit ECVI focuses
overall erroL i.e. the discrepancy between the population covariance
matrix and the model fitted
the sample (implied covariance matrix). It is a useful indicator
of a model· s overall fit because it measures the discrepancy between fitted covariance rnatrix
in the analysed sarnple. and the expected CO\ ariance matrix that would be obtained in
assess the mode!'s
its \·alue must be compared
another sample of equi\alent size.
against the ECVl values of other models: the model witb the smallest ECVl value is then
chosen as representing the greates1 potential for replication. RMR is a summary rneasure of
fitted residuals.
represents the average \alue of the element in matrix obtained as the
difference between the sample co\ariance matrix and implied matrix. A problem w1th
interpreting this statistic is that their size rnries with the unit of measurement and the iatter
\ ariable.
prohlern can be a\ oided h, using rhe standardized
rnn \ an from variahle
residuals A summar: measure of standardi,::ed residuals 1s the standardized RMR: \alues
behi\\ 0.05 are indicali\e uf acceptable fiL &FI is an rndtcator of the relevant amount uf
\ ariances and cm ariances accounted for by the mudel and thus shows huw closeiv the model
cumes
perfectly reprnducing the ,,bsened co\ ariance
Values of GFI should range
between and and , alues greater than O. 9 are usually taken as reflecting acceptable fit
CFI is a relative fit index which shows how much better the model fits compared to a
baseline model. usually the independence model. has a range between O and l and values
duse
represent good fit.
ln eYalualing the measurement pan of the mudeL we focus on the relationship between
the latent variables and their indicators The ann is to determine the validity and reliability.
\ alidit:, reflects the extent tu which an indicator actuall\ measures what it is supposed to
measme. while reliabili 1y refers to the consistenc y of measurement
The aim of evaluating the structural pmi of the model is to determine whether the
theoretical relationships specified in the conceptualisation phase are indeed supported by the
Jata Three issues are of releYance here. The signs ,,f the parameters indicate whether the
d!l'ections of hypothesized relationship are as h\ pothesized The magnitudes of rhe estimated
parameters pro\ ide important information on the strength of the hvpothesized relationships,
and the squared multiple correlations for the structural equations indicate the amount of
variance in each endogenous latent variable that is accounted for by the independent latent
variables.
i!

here 1; is an mx \ector of emlogenous latent Yariab!es. <; is a kx Yector
exogenous
latenl \ ariahles. B is an mxm matrix of regression coefficients relating the latent endogenous
\ anahle to each other. ris an mxk mairix of regression coefficients relating endogenous
\ ariahies to exPgenous \ ariahles. and .; is an mx l vector of disturbance terms.
rhe latent \ariahles are !mked
obser,able \ariahles Yia measurement equations for the
endogenous variables and exogenous variables. The model used
re!ate obserYed measures
factors is the linear factor analysis model defined hy the fo!lowing equations
(2)

X=

1,<; +

Č'

where
1s a px \·ector of observed responses
p questions assurned to measure an
Cf questions that are
e11dogenous latent \ariahles. is a LJX \·ector of obsened responses
assumed
measure exogenous latent \ ariables.
and
are pxm and qx; matrices of
factor loadings. respecti \·ely and & and !i are a px 1 and a qx , ectors of uniqueuess.
respectivel\ The number uf endogenous ariahles is expressed by p and the number
exogenous variabies is expressed b: q, kis the number of the important factor' s dimensions.
ln cases, when the research objective is to find the appropriate indicaturs to measure a
factor of interest then the following model can be applied

X=

(5)

1,<; + 6

Assessment of modei fit refers to the extent
wh1ch a hypothesized model is consistent
wirh the data.
is usualh done in three stages. i!l\
mg the al assessment ofthe moders
, iverall fit b l the assessn;ent of the measurement part of the modeL and c) the assessment of
the structurai part of the model
The purpose of assessing a moder s uverall fit is to determine the degree to \\hich the
model as a whole is consistent with the empirical data in hand. Over the years, a wide range
of goodness-of-fit indices have been developed. yet none of them is unequivocally superior
co the rest in all circumstances. Particular indices hm·e been shown to operate somewhat
differently given the sample size. estimation procedure. model complexity, violation of the
underlying assumption of multivariate normality and \ ariable independence. For practical
purposes, the results ofthe chi-square test used in conjunction with root mean square error of
approximation (RMSEA), expected cross-validation index (ECVI), standardized root mean
square residual (RMR), the goodness-of-fit index (GFI), and the comparative fit index (CFI)
should be more than sufficient to reach an informed decision concerning the model's overall
fit [2].
The chi-square statistic is the traditional measure for evaluating overall model fit in
covariance structure models and provides a test of perfect fit in which the nuli hypothesis is
that the model fits the population data perfectly. A statistically significant chi-square causes
rejection of the nuli hypothesis, implying imperfect model fit an~ possibl~ rejecJ:ion _of the
model. RMSEA focuses on the discrepancy between the populatlon covananct! matnx and
rnodel-based covariance matrix per degree of freedom taking model complexity into account.
It is generally regarded as one of the most informative fit indices. Values less than 0.05 are

108

3. Dimensions of innorntiveness on the case of Slovenian firms
In the past. different dimensions of innovation and their importance were emphasised by
different authors [8], [6], [11. [9]. Taking into account their findings, five dimensions and a
theoretical model of an organisation · s overall innovati veness was established. According to
theory organisational innovativeness can be measured by product, marketing, process,
behavioural and managerial innovativeness.
Product innovativeness is often referred to as perceived newness, novelty, originality,
or uniqueness of products. Marketing innovativeness emphasises the novelty of marketoriented approaches. It refers to innovations related to market research, advertising and
promotion as well as identification of new market opportunities and entry into new markets
[9]. Process innovativeness captures the introduction of new production methods and new
technology as well as new management approaches that can be applied to improve
production and management processes. The primary focus of managerial innovativeness is
to measure an organisation's ability to manage ambitious organisational objectives and
identify a mismatch of those ambitions and existing resources in order to stretch or leverage

109

lumted resources creativity. Behavioural innovativeness iS a fundamemal factor that
underlies innovative outcomes because 1t enables the formation of an inncnati,e culture. the
is demonstrated through
overall interna! recept1vity to new ideas and innovation.
mdividuals, teams and management.
1 aking into account all five dimensions revealed in previousl) mentioned researches we
tested the following hypotheses:

each set 1)f 111easures. The rel iability
, aiidation procedure refers
,if the scales using Cronbach · s aipha as a measure uf interna! consistenc\ was encouraging
with all rhe scales adequately meeting standards for suc.h research 17 J The behavioural
innovativeness scale achie\ed an alpha of 0.722. product innovativeness 0.716, process
mnornliveness 0.716 and marketing innorntiveness 0.886
anal:, sis

Table . Loadings of the firsHirder

H 1.
H2.

H3.

The covariance among the items included in the model can be accounted for by one
general factor, i.e. organisational innovativeness.
The covariance among the items included in the model can be accounted for b\
restricted rnulti-factor model where each factor presents one of the conceptual
components describing the organisational innorntiveness. Each item loads onlv on one
·
factor.
Responses to each item are reflective of two factors: a general organisalional
innovativeness factor and a specific component factor.

--------- ---Standardized first-order loading"
Process
Behaviour_a_l_ . J . - -_ _M_ac_r_ket

~ariati~

R·
~~havi(;u-ra~I__,__ _
f--- -----+----+
O..'i92'
(U50
V1

Product

0.284

o ·' 78

---+---0-=.2:.::6.:_9_

---+_c_)_.'i9_o--+_o 768 _ 16.94_3J~·+ - - - - - _0._4_0'7_.+--0_.6_3_8_ (65~9)
-------+---\~ 868).
0.538
O 289
~
+\larketrng
0.9~~
0.9i2
+ iJ8.,3
1)693
Process
0.783
O 613
\8
\-'}
!1'718
o.~ 1s

-1

f-v2

1-\,

-+

i-Yi
2

4. Research methodology
To test a hypothesised factor structure a two-step approach developed by (jerbing and
Anderson was used [3l The measurement model defined by (5) is first developed and
evaluated separately from the full structural equation model defined by (4) and ( 5) which
simultaneously models measurement and structural relations. In the approach described.
confirmatory factor analysis was applied using computer program AMOS and the Maximum
Likelihood estimation method.
A sample of 000 Sloveman manufacturing organisations was randornl) selected from
the IPIS database. A total of 254 completed questionnaires were retumed. representing a
25.4 per cent response rate which is a normal response rate for most surveys. The rate of
usable responses was 21.4 per cent ln the questionnaire. 3 questions referred to the
organisational innovativeness. ltems V l, V2. V3. V4, V5 referred to behavioural
innovativeness, items V6 and V7 to marketing innovativeness. V8 and V9 to process
innovativeness, items V 1O and V 11 to product innovativeness, and items V 12 and V 13 to
managerial innovativeness. The questionnaire used a seven-point Likert scale with verbal
anchors of scale. CEOs were chosen as informants as they were most likely tu observe and
analyse the characteristics of the organization. A pilot study was conducted including 5
experts to aid questionnaire wording and design.
In the first step, ali 13 items were included in the first-order measurement model ( 5) for
organisational innovativeness. The initial model fit indices were assessed. They are
z2=173.765, df=61, p<0.05, GFI = 0.882, RMSEA = 0.093, RMR=0.142 and CFI = 0.895.
Their values showed that the original model had to be respecified to obtain better fit with
sample data. The pattern of normalized residuals analysis confirmed the need for a
respecification. Item V5 had small squared multiple conelation (0.323) and large error
variance (2.29) and was therefore removed. ltems V12 and V13 had large error covariance
(38.754). These two items were thus deleted because each estimated construct is defined by
at least two indicators.
Having eliminated 3 items, the modified first-order confirmatory factor analysis model fit
2
indices were: x =33.062, df=28, p>0.l, GFI=0.972, RMSEA=0.029, RMR=0.085,
CFI=0.993. Ali fit indices show that model fits data very well. The standardized7egression
weights of ali variables loadings onto their respective factors were between 0.538 and 0.955,
with ali critical ratios above 1.96 (which means that ali the regressions are statistically
significant at the 0.95 per cent confidence leve!). Their values are given in Table l. The

11 O

1---·-·

1.206

"

~!

L_\11

0.580
(1.543

-

-------1-

_ ___,_____

+

0761
I 0. 7 ~ 7

(9221)

'.'sote· • Standardised first-order loading is the standarclised regression weight of the 111clividual
variables· loading on to one ofthe component factor. Numbers in parentheses are critical rattos frnm
the unstandardised solutions: b Standardised first-order luading for component factors 1s the
covariance between any two of these component factors; ' Critical ratio is not available, because the
regression weight ofthe first rnriable ofeach component factor is fixed at one.
To assess the multidimensional structure of the organisational innovati \ eness construct
the second-order confirmatory factor analysis using the systern of equations (4 J and (5) was
carried out. rhe fit indices obtained for this model showed similar results as the first-order
x2=37.27'. df=30. p>0,l, Gfl=0.968,
confirmator~ factor analysis and were
RMSEA=0.034. RMR=0.097, CFl=U.99. The slight difference in the first-order and secondorder estimates occuned due to different degrees of freedom. The standardised regression
weights of ali component factors loadings onto the general factor organisational
inncl\ati\eness are giYen in Table 2. They ranged from 0.385 to 0.972.
Table 2. Loadings of the second-order confirmatory factor analysis
Critical
S tandardized
Ratio 1
Regression Weight
1
3.977 1
0.385
0.148
Behavioural innovativeness
1
8.169 i
0.972
0.945 '
Product im10vativeness
'
1
7.769
0.916
0.840
1
Process innovativeness
!
0.692a
0.478
i
1
Marketing innovativeness
1
Note: a This critical ratio is not available, because the regression weight for the component factor
marketing innovativeness is fixed at one.

Rl

Factors

1

.

1

1

To assess hypotheses Hl and H2 the hypothesized multidimensional model was
compared with a competing unidimensional model. As shown in Table 3, the one-factor

111

significam chimodel which loaded ali O md1cators to one factor. y1elded slatisticail:
square of 228.617 while the four-factor mode! resulted in the statistica!l) significant chisquare of 33 062. suggesting a s1gnificant imprO\ement. Furthermore, the imprU\ernents in
GFL RMSEA. CFl and NFI were substantiaL indicating that the four-factur model presents a
better fit to the data. Thus the convergent validity of the constructs is also supported.
Table :; Results for nested model
Model

Description
One general factor
1
One general factor · four
-'-compone_!l~

2

1

J

1

1

2

X

df

1
1

288.617

35 1

33 062

28

GFl

p
000
O
0.233

1

0.758

RMSEA
0.184

0.972

0.029

1

CFl

NFI

0.664

O 640

0.993

0.959

Practical bounds m error-free communication
lg,,r D11kaum,i<'Maribor, El-'F
Gniversity
Razlagova 14. 20(10 Maribor. Slovonia

Ali hypotheses tested were confirmed The results obtained showed that inmnati\eness
of Slovenian organisations is also the rnuitidimensional categor) described b: four
dimensions The importance uf indi \ idual dimensions and their indicators also reYealed
order to stimulate innO\ ative
those features in organizations that may need improvements
capability of organizations.

References
Capon, N., Farley, J.U. Hulbert, J., Lehman, DR. (1992) l'rofiles of product
57innovators among large US manufacturers. Managemen! Science. 38 (February
69.
[2] Diamantopoulos. A. and JA Siguaw (2000) Introducing LlSREL - A Guide for the
Uninitiated. SAGE Publications London.
[3] Gerbing, D.W. and J.C. Anderson (l 988 An Updated Paradigm for Scale
Development Incorporating Unidimensionalitv and Its Assessment. Journal of
Marketing Research Vol. XXV (May) 186-92.
l4 J Hattie, John ( 1985), Methodology Review Assessing Unidimensionaiity of Tests and
ltems, Applied Psychological Measurement 9 (June) 139-64.
151 Kaplan, D. (2000) Structural Equation ModeUing · Foundations and [xtensions Sage
Publications, International Educational and Professional Publisher. London.
l6] Miller, D., Friesen, P.H. (1983), Strategy-making and environment the third link.
Strategic Management Journal, 4(3): 221-35.
17] Nunnaly, J. C. (1978), "Psyhometric Theory, 2nd ed" New York McGraw-Hill Book
Company.
[8] Schumpeter, J.A. (1934.J, The Theo,y of Economic Development. Harvard University
Press, Cambridge, MA.
[9] Wang, C.L., Ahmed, P.K. (2004), The development and validation of the
organisational innovativeness construct using confirmatory factor analysis, European
Jo urna! of Innovation Management, 7( 4): 303-13.

!lj

l:nivcrsiLiit KlagcnlurL lnsr,itul fitr Mathcmatik
l.'11iversiiitsslralle 65-67, A-9020 Klagenfurt, Austria
franz.rendl@uni-klu.ac.at

igur.1aat@u11i-n1b.si

Abstract
is an upper bound on the Shamwn capacitv of a graph ddiucd
Celehrat.ed Lmasz theta
= supn> 1 1;/n( Gn). lH this paper we suggest copositive-programming-motivated upper
hounds on rnm1l)ers in the corresponding sequence.
Keywords: Error-free cornmnnication. Sha1mcm capacit_,·

1 Introtluction
Consider 1
decoding

5. Conclusions

Franz Hcndl

dma nmsisting of cocling data. lraHsmit
coding

channcl

decoding

Occasionall, errors occur. Fur exampie while transmit t iag binary data digit 1 can change
ur vice versa. Tlte standard remedy is te, add a parit_, bit after transmitting every
illto
8) bits. The aclded bit equals L t he number of imes in the seven bits is odel,
seven
ancl otherwise. This makes 1t possible to detect, if one error (or an odel number of
errors) has been made during transmission. Since double errors are unlikcly, this is a
very popular ayproach. Another popular approach which with extrcmcly high probability
detects errors is computing checksum consisting of (weighted) sum of the transmittecl
clata.
Tn this paper we wil! address a similar but tougher task posed by Claude Shannon,
the founder of information theory [2( Assume that during ttansmission of a coded
message some pairs of symbols occasionally get confused while others do not vVhat
is the maJcimum rat,e of Lrausmission such that the rcceiver may (always) rccover the
original message withont errors?
Such channel is modeled by a so called confuswn graph G(V, E). The vertex set V
consists of all possible symhuls. The edge set E consists of all pairs of s~ mbols wluch
can be ronfusecL i.e. symbols B and P in rnmmunicating over a phone (voiceJ
Since we want to communicate error-free we can - if we only senci single symbols - use
one letter from each pair that might he confused. In the language of graph theory,
this means that the best we can do is to use the symbols from the largest stable subset
S <;;;: V. Recall that 8 s;; V is stable. if
1

1.j ES==> (iJ)

rJ_

E.

The cardinality of the largest stable set, the stability number, is denotcd by a(G).
For example. if the confusion graph is a 5-cycle

1

5~/'
/3

\4

I

112

it ovcr a chaunel aud

113

·,............--1
1
ali 5. i.e.
wc can il! crror-frec commumcation use only cljJ to tW(1 symhols instcad
maximal possiblc, rat.e fur .'j
ni ( ·5 1 = 2 ancl thc mfnrmafinn rnte is log, 2 instcacl
symbols log 2 5.
The obvious idea how to increase information rate is tu use larger strings in place of
on each position
lin can be coufused.
single symbols. Two striugs u 1 u 2 . Un and
her "confusable". i.e. u,i- 1 E E. So we
ar0 dthcr
1 thc two s.vmbols ·u, aucl
deti ne tlw n -th ronfus11m gruph ()" wit h vertt,x set i n and edges as defiued above. The
is so
correspomling ratc of information pcr

H S is a stablc set in G

r hen

oh, ic,usl.v 8" is a s1,ahle set

en

Therefore

0(G)

=

ma,x1X. 11
trace X =c=

= OV(
>-

(2)

E E

:19. r

(G") 2'.

using loHger striugs iustead of
rat,e
mcaning that wc ncvr•r decrease thc informa,
si11glc sy111bols. Also "' piaciug shurt siriags of size k wii strings wil size uk.11 E
can also only mcrease information rate. This is the basic idea of coding theorY (see. i.e.
'f) which rnotivates the following Shannon·s fundamental definilion 24'
1s

at lcast one of the numbern
t.J E
Also.
>- and l race X =
13, c1mst
( \s Ji and ( s) 1 an" zeru since snd1 1 and J can not both bc in thc stabic set S. Therefore
:c,J = U \-/(1.1) E L.
Cunsider t he maxinmm over ali such matrices

Since all matrices l) are feasible for the above sernidefinite program, 0( G) 2 a( G).
Lod.sz f-1 mimber is also well-knowu upper bound ou the Shannon capacity of the graph
and 1t for all alternative
(see
seems all roads lead to (F And indeecl (J( c;) is one of the best
Goernannsi 1:J'
11
puhnumial b;ml!ds saHdwid1ed bPtween two >il'-liard

and

Definition The zero-error capac-ity of a graph G

.

1

SfJ(C:J S
((:) is the chrnmatic numl ,er of the eomplcrnZ'!ll graph C with wrtex
where \
set V and edge set E defined hv

E= {(

gwen ny

E-)(G'J := sup t/a(G")
n~;

Notice that the Shannon capacit:v of the graph describing transmissic,u of digital data
G({O, l}, f(O, 1)}) has Shannon capacity O, i.e. it is impossible to to be absolutelv sure
that a transmitted message is correct. Of course. we are more interested in computing
the first fcw numhers in this sequence rather than the "theoretical" limit as the strings
in prnctical apphcations cau not be arbitrarily large.
For examplc consider the product of two 5-cycles C 5 X C5 . It is easy to check that the set
(l 1 1), (2, 3), (3, 5), (4 2), (5, 4)} is stable. Hence. by using strings of length 2 we have
increasecl the lower hound for the capacity to 0( C5 ) 2 /5
Can wc irnprove this number hv furthcr increasing the number of the symbols ill a string?
Though already Shannon found an upper bound on his error-free capacity uf the graph
r24J and thus computecl 0 of ali graphs on up tu 5 vertices. this question remamed
unsolved up to the break-through paper of Lovasz[l9].

=I= /. (

rf. E}

Graphs for whid1 equalit, a( JI) = y( H J holcls for H = (; and all incluced subgraphs are
called perfect or lJerge graphs. Berge conjectured "3i that a graph U is perfect if and only
if neither ' not C contain an odel z-vele with length at least 5. This strong perfcct graph
conjectme has recentlY been proved (5 .. Lovasz fi nurnber of any graph can be computecl
in polvnomial tirne to arbitrar, fixed precision since it is a sermdcfinite program (see
survey '25:) But stability number of a perfect graph cau be checked in a polvuornial t.ime.
too [14 as a sequence uf compntations of (J uumbers of induced subgraphs. Therefore
the determination of Shannon capacity of perfcct graphs sandwiched between two equal
and 0(G) is trivial, too. So the maximum information rate ona perfcct
numbers
G 2) =
graph is alreath- achievecl hv transmitting single symbols, since od G i =

/n (

ijo:(G 3 )=
C 5 is incleed /5 achievecl on
While already Lovasz shuwecl that the Shannoll capacin
size two 119. l the Shannon capacity of U1 is stili an open problem :see ,2t
striugs
See also i{ for bounds on compiements of odd <.'veles.

2 Lovasz theta number

3 Strengthening Lovasz 0 number toward stability number

Lovasz theta number can be formulated as a semidefinite program, so we first introduce
some standard notation. Let us denote by e E R" the vector of ali ones, and by J := ee 1
the n x n matrix of ali oncs. The trace inner product
(X. Y) = trace X
is a well-known inner product in the space of symmetric matrices. Obviously (X. Y) =
~i,j X;jYij so (X. J) is just the sum of ali entries in the matrix J. NotatJon A c B stands
for A - B c O is a positive semidefinite (symmetric) matrix.
Set S be a stablc set in the graph G(V E) Its characteristic vector ,\s is defined by

ln the following we will be interested in bounding a first few numbers in the Shannon's
sequence
a(G), Ja(G 2 ), y'a(G 3 ), ...

1 i ES
(X s )'. -- { O
otherwise
Since

ISI = xixs

where the matrix X is defined by

(X, J) =

114

Xžii1

!SI

T
1
X := -r-XsXs·
XsXs

hoping that the limit 0(G) is already achieved, and also since coding very long strings
is not practical. Solving semidefinite program (2) does not only give an upper bound on
the stability number of the corresponding graph, but also motivates severa! heuristics
for extracting large stable sets (see [15, 2, 26]). Ali of them are motivated by the bclicf
that the computed near optimal semidefinite matrix variable X of (2) approximates well
a matrix X in (1) defined by a single stable set or that X of (2) approximates a convex
combination of such solutions. In both cases a large diagonal entry X;; suggests that the
vertex i is with a high probability contained in one (or severa!) large stable sets. So one
can order the vertices of G so that

!

(1)

2

Xi2h

2 ... 2

Xinin

and build large stable sets by a randomized greedy algorithm. A safer (but more expensive) heuristics recompute 0 number of the induced graph G \ {v} after vertex v has been
included in a stable set. For both tasks we could benefit by a matrix variable X which

115

!
bet ter approximates X of l l; To this cnd Il(!ticc that the elements of matrices ( l) are
non-negative. This motivates introducing Schnjver's nurnber 23
0- (G)

= max{ IX. J 1 . tr

X= l x,;

= UV(i.i JE E. x,J 2 OV(

EE.X>::O}

Notice further tlrnt anv matrix (l) is also completely positive. A matrix is completeiy
pos·1.twe I f

wherc cach vector X; has only non„negative entries. Vi/e Jenote the cone of all cornpletely
positivc matrices by C* and its Jual cone of copositive matrices hv C. So Schrijver·s
numhcr can fmthcr bc stn'ngthened lo

= max{(X. J) • trace X=

l x,J

=

O\/(iJ) EH. X E C*}.

(4)

The dual of the (4) is

0-r• ~G)

=

min{t • c;;

= Ni, c;i = O\/('lJ)

E

.E, tC -

(5)

JE C}

wherc the set of copositive matrices C is characterized bv
C E C ~ \fx 2 on xTCx 2 O
Since therc are easy to find strictl,v feasible solutions of (4) an<l (5) by Slater's theorern
therc is no duality gap, i.e. 0-c• = 0-c. So we could apply a primal-dual algorithm.
However this number is again NP-hard to compute. so a chain of rela'<ations

P
has bcen found i6]. Define
0(i)(G)

C

Ko

C K,L C .. C K,n

=C

o:,(i)

(3)

which is sandwiched betweeu o:(C:) and 0(G). See 10. 8] for practical issues in computing
these bounds.
A near optirnal matrix variable of (3) resemhles X of (1: mure thmi does a near optimal
solution of (2). Of coursc, 0-(G) is also an upper bound on the stability number o(GJ
by the same rcasoning as for 0( G) while it is at the same tirne smaller then 0( G) since
the fcasibk set in (3) is a subsct of thc fcasihle set in (2).

g-rrG)

The group of all autom(Jrphisms is denoted hv Anti U;
Definition C:raph c; ( V. E), V = { l. ... n} is vertex transil'ive d there exist automorplusms 02, ... , nn s·uch thnt

(6)

=

(9)

l.

All odd-c_H"les. their complements, ali their strong products CL'
esting graphs are vertex transitive. So we suggest simplifyiug (81

.\' E A.:i

~

M>- O
W E V:3M : X
mu =0mii -t- 2m1a ,,n
m,J

The rnatrix J\;l in (10) is defiued
1 1
' ,

h\·

=o

Vi > 1

·+ m,a.1(i/a.11,, + ma,!;1a,,,

and many other intcrto

(10)

> n \/1 < -i < J.

•= ,!: LaEA P; !vJa(i Pa wherc subgroup A <
see 9, t for de1ails. Notice that in the abovc

M

1
satish {8J.
.. ,\f n
Aut(GJ and l\;f
characteriza1ion the rnatrix X is "s11pported" hy onlv one matrix I'vl. 'vVe use a primaldual interior point method tu compnte g("(C) hv sc,h·iug (7) C'haractcrizatioH I l(J
reduces the computational effort to solve one step of the interior point method from
O( m 9 ) to 0( m. 6 ) where m is the nurnber of vertices of our graph. For example. ca
has /II = na ver1ices 1 This makes even the sirnpiiiied model hard tu computc. llnwevcr
notice that this model has a lot of syrnmetry. If u is any automorphism which fixes tlw
first vertex. i.e. o(i) = l (and there are at least n 1 such automorphisms in na. see alsc,
20;) theu P;[ .vi Pa also sat isfies ( lO) This rnll be exploiwcl as in a sirmlar bound uu
chromatic number (see. !9J to dramaticallv decrease the munber of frce variables. This
method follows recent successes in exploiting symmetries in semidefinite programs and
u11derlying comhinatorial optimization problems 17. 18, 1:2J.

1 1

6 Conclusions
The powers of graph G appearing in the Jcfi11it10u uf the Shannon capacity of a graph
have ver\' rich automorphism groups which can be efficieutly exploited to compute boun<ls
on the stabilitY numbers n( G;. o(G 2 ). .. n( ca) for medimn powers a.

G!

= min{t.

c;;

=

lVz. c;1

=

(7)

0\/(ij J E E. tC - JE Ki}

an<l notice that (6) implies
0(G) 2". 0( 0 )((}) 2". 0\lJ(G)

2 ... 2 0 c(G) 2 o:(G).
It is also known that 0( ) ( G) = 0-( G) while (j(l) ( G) = o:( G) for ali o<ld cycles [61. For
0

properties of g(r) in general see also [16, 22]. Each 0(r) can be described as a positive semi<lefinite program of quickly increasing size. For example. in [21] the following
characterization of set K, 1 is proved

X E K.1 ~ Vi E

V::JM(i) •

X - M(i) >- O
m(i) =0 -

G. \L Ziegler. Proofs frorn the book. Sµringer, 1998.

r21 S. Benson and Y. Ye. Approximating maximum stable set an<l minimum graph
coloring problems with the positive semidefinite rela'<ation. In Applications and
Algorithms of Complementarity. Kluwer Aca<lemic Publishers, 2000.

[31 C. Berge. Fiirbung von graphen deren siimtliche beziehungensweise dere ungerade
kreise starr sind (zusammenfassungJ. In Wissenschafliche Zeitschrift, Martin Luther
Univ. Halle-Wittenberg, Math-Naturwiss Reihe, 114-115, 1961.

f4] T. Bohman and R. Holzman. A nontrivial bound on the Shannon capacities of

= O Vi T-1- y·
+ 2m(i)
l)
m(k) + mU> + m(i) > O Vi< J < k
1k ik
'J
m\J)
1.t

5 Computing 0(ll(G) of a vertex transitive graph
Definition Permutation o:: V---+ V is an automorphism oj the graph G, if
(ij) E E ~ (o:(I)o:(j)) E E.

(

116

References
ll ),l. Aigner aud

(8)

complements of o<ld cycles In JEEE Transactions on Informat-ion Theory, 49:721-722, 2003.
[5] M. Chudnovsky, N. Robertson, P. D. Seymour and R. Thomas. The strong perfcct
graph theorem. Manuscript, 2003.
[6] E. De Klerk and D. Pasechnik. Approximation of the stability number of a graph
via copositive programming In SIAM J. Optim., 12:875-892, 2002.
[7] I. Dukanovic. Semidefinite programming applied to graph coloring problem. PhD
thesis. University of Klagenfurt, Austria, 2005 (forthcoming).

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F
r
181 L Dukanovic. ZgornJa meja za neodvisno število grafa. ln
Slovenian Society informatika, 240-246. 2003.

J),','J

2008. Ljubl]anu.

:91 L Dukanovic. j, Povh. F. !{endl all(! A. Wiegclc. Heductiuus symmetric rnpositive
programrning motivated bouncls on l hc dic.iuc ami chrumatic number VVurking
paper, Gniversity of Klagenfurt Austria, 2005.
[10] l. Dukanovic ancl F. RendL Semidefiuite programming rdaxatious for graph
ancl ma,"Ximal cliquo problems. Working paper Uptimization onltne, 2005.

CONVERGENCE IN THE CASE OF THE BLLLWHIP EFFECT WITH
CENTRALIZED DEMAND INFOR!\IIATION
Liljana Ferbar. David Čreslovnik Blaž Mojškerc, Martin RaJgelJ
Universitv of Ljubljana, Facultv ofEconomics
Kardeljeva pl. 17. 1000 Ljubljana, Slovenia
hljana.ferbar@ef.uni-lj.si, david.creslovnik(n]ef.uni-lj.si.
blaz.mojškerc@ef.uni-h.si, martin.rajgelj@cf.uni-lj.si

111] M.R. Garey ancl D.S, Johnson. Comp·uters and Intractability. A Guide to the Theory
of NP-Completeness. WJI. Freman ancl Company, :\ew York. 1979.

i12j K Gatennann ancl P.A. Parrilo. Symmetry groups. semidefiniLe programs, aud
sums of squares. ln Journal of .Pure and Appl. Algebra, 192 l<l. 915- 128. 2004.
13; M. X Goemans. Semiclefinite programrning iu combinatorial uptimization. Jvlathe
matica[ Programming, 79143-162. 1997.
;14] M. Gri:itsehcl, L. Lovasz and A. Sehrijver. Polynornial algorithms for perfect graphs.
In Annals of Discrete lvlathematics. 125 356. 1984.
)5] G. Gruber ancl F. Rencll. Computational experienee with stahle set relaxatiuns. In
SIAM Journal on Optimization, l.1(4):1014 10~8. 2003.

16] N. Gvozclenovic aucl M. Laurent. Semidefinite bouncls for the stability number of a
graph via sums of squares of polynomials, In Proceedings oj IPCO Xl. Berlin. 2005.
1
interior-point
;17 Y. I<anno, M. Ohsah K Murota and N, Katoh: Group symmetrv
methocls for sern1-definite program In Optimizatfon and F:ngineeriuy. 2:293-820.
2001.

[18] E.de I<lerk, D. V. Paseehnik and A. Sehrijver Reduetion of symmetric sernidefinite
programs using the regular *-rcpresentation Priprint. 2005.
,191 L. Lovasz. On the Shannon capaeity of a graph.
25:1-7. 1979.
120i

JEEF: Trmis.

Theory.

B. D MeKay. Praetical graph isomorphisrn. Congressus ,Vumerantium. 3045-87
1981.

i21] P. A. Parrilo. Structured Semidefinite Programs and Semialgebraic Geo11ietry lvlethods in Rob·ustness and Optimization. PhD thesis, California Institute of TeclrnulogY.
•
2000.

[22] J. Pena, J. Vera and L. F. Zuluaga. Computing the stability numher of a graph via
linear ancl semidefinite programming. Working paper. 2005.
[23] A. Schrijver. A comparison of the Delsarte and Lovasz bounds. IEEE T'ransactions
on lnformation Theory, IT-25:425-429, 1979.
[24] C. E. Shannon. The zero-error capacity of a noisy ehannel. In IRE Trans. Information Theory , 3:3-15, 1956.
[25] M.J. Todd. Semidefinite optimization. Acta Numer., 10:515-560, 2001.
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Tcchnieal Report, Stony Brook University, 2003.
[27] A. Vesci and Janez Žerovnik. Improved lower bound on the Shannon capacity of
C7. Inf. Process. Lett., 81:5, 277-282. Elsevier Nort-Holland, 2002.

)

118

Abstract
Our aim 1s to improve an algorithm for seeking convergence m the case of the bullwh1p effect m a
supplv chain with centralized demand mformation We mveshgate the conditions necded for
production and rnventory to converge in the sense that we examme the number of iterations needed
for stab!lization ofthese two quantities Aditionaly we stud\ the mfluence ofthc sizc of perturbation
in extemal demand on to tal mventon costs of supplv chain
Kewords: supply cham, bullwh1p effect, perturbed extemal demand, centrahzed mformation

1 INTRODUCTION
According to Lee et al. ( 19976, p 93 ). the term "bullwhip effect" was first used by
Procter & Gam6le, when they experienced extensive demand amplifications for their diaper
product "Pampers" Lee et al (1997a, 19976) descri6e the bullwhip effect as the result of
information distortion in a supply chain, where companies upstream do not have information
on actual consumer demand Consequently. their ordering decisions are based on the
incoming orders from the next downstream company This may lead to amplified order
varia6ility demand coming in from a downstream company has a lower variability than
demand to an upstream companv. The bullwhip effect can have dramatic effects on
companies that are removed from the end-user (Metters, 997. p. 89).
The bullwhip effect has been noted and assigned various causes across a range of
academic disciplines The first academic description ofthe 6ullwhip phenomenon is usually
ascri6ed to Forrester in 196 in his seminal book Jndustrial Dynamics. Forrester indicates
that it is empirically common for the variance of perceived demand to the manufacturer to
far exceed the variance of consumer demand and for seasonality to be larger for
manufacturers than for retailers Furthermore, he notes that the effect is amplified at each
stage in the supply chain He states that the principal cause of this is difficulties involving
the information feedback loop between companies and that such systems are too complex for
managerial intuition alone to ameliorate Consequently, Forrester's remedy lies in
understanding the system as a whole, and modeling that system with specific "system
dynamics" simulation models, so that managers can determine appropriate action. Perhaps
the most well-known description of the bullwhip effect emanates directly from Forrester's
work and has come from the system dynamics discipline. Sterman (1989) has used a
simulation to demonstrate the 6ullwhip effect: a classroom game known as the "beer
distri6ution game" where pariticipants simulate a supply chain consisting of a beer retailer,
wholesaler, distri6utor and 6rewery. The decision-making task is straightforward: subjects
seek to minimize total costs by managing their inventories appropriately in the face of
uncertain demand. As the game proceeds, a small change of a consumer demand invaria61y
is translated into wild swings in 60th orders and inventory upstream. Sterman (1989, p. 322)
concluded that the interaction ofindividual decisions with the structure ofthe simulated firm
produced aggregate dynamics which diverged significantly and systematically from optimal

119

behavior. He interpreted the phenomenon as a consequence of players' systematic irrational
behavior, or "misperceptions of feedbackll On the other hand, Lee et al 997a, p 558)
daim that demand distortion may ari se as a result of optimizing behaviors by players in the
supply chain. In a practical study Fransoo and Wouters (2000) have experienced that the
theory of measurement of the bullwhip effect in a practical setting has received very limited
attention so far In their paper, they have made a first start with a method to accurately
document and define various ways of measuring the bullwhip effect. However, the bullwhip
effect can be measured as the ratio of demand-variance at a given supplier to the variance of
external demand, assuming that expected values of each stage's demand are the same, i. e ,
(Simchi-Levi, Kaminsky and Simchi-Levi, 2000, p 88)
u = vm(e,J
Var(D)
i

Many investigations (e g., Lee et aL, 997a, 997b; Chen et aL, 2000) have shown that
providing the supplier upstream with centralized dala can significantly reduce the bullwhip
effect. Such information cuts short all kinds of information distortions in a supply chain.
Other improvements that can reduce the bullwhip effect include the reduction of lead times,
revising reorder procedures, limiting price fluctuations, and the integration of planning and
performance measurement (Lee et al., 1997a, 997b ). Chen et al (2000) have demonstrated
that the bullwhip effect is due. in part, to the effects of demand forecasting. They have drawn
the conclusion that centralizing customer demand information can significantly reduce the
bullwhip effect However, centralizing customer demand information does not completely
eliminate the bullwhip effect. In addition, they have shown that the difference between the
variability in the centralized and decentralized supply chains increases as we move up the
supply chain, i.e., as we move from the first stage to the second and third stages of the
supply chain.
As a practical matter, the bullwhip effect is a well-documented problem that affects many
businesses in serial supply chains across a variety of industries. Although it may seem an
obvious inefficiency that is easy to correct, discovery of the bullwhip effect does not
automatically lead to its solution case studies (e.g , Metters, 1997) demonstrate that, despite
significant effort, the bullwhip effect can persist Although the precise causes ofthe bullwhip
effect remain under debate, it is generally agreed that a lack of inter-company
communication combined with large tirne lags between receipt and transmittal of
information are at the root of the problem. Simchi-Levi, Kaminsky and Simchi-Levi (2000,
p. 91-92) have listed a number of suggeestions for reducing the bullwhip effect or for
eliminating its impact. These include reducing uncertainty, reducing the variability of the
customer demand process, reducing lead times, and engaging in strategic partnerships.
2

1

We study the model

2

a case where two more assumptions have to hold

External Demand ( ) changes its value bv .i.\
The perturbation is negative Lil< O

:'ix

)=x

assumptions teli us that consequences of only one negative change of external
These
demand are examined and that the state following this change remains unchanged. We
suppose that these two assumptions can be very real: in many real-world situations we note
the stabilization of the external state after single perturbation. The perturbation could. of
course. also be positive - in that case the same analysis would follow
=O, since the production can not be negative The
<O, we set J
Furthermore if

=mx + n& =mx ni&!

production can be expressed in a form

so the condition

mx- nifui < O implies
m

l&I,

11

X

.

- < -'-( = relati ve change
2.1 Algorithm

l&I <

Since for any negative perturbation,
obtain production, P1

and any period,

and suppliec

[which equals final stock,

and initial stock, JS1 +

we can

l

the

model can be expressed with the following algorithm
function centralized_model
// r - period
supplrer
/ / J
l;
i
while TRUE

for j = l te fina~_J do
lf (2 * Dii,
else P(l 1 ji = 2
end if

= P
D(i, J + l
IS(i + 1 1 j) = P

j

IS

* D

1

1.

1

1.

~i

1

-

< Q then
IS(l 1 j

+ TS(.1

1

-j} -

P'1.
;

(.1.

1

j

Q

_;;

end for
:f

THEMODEL

1

iP(ir

final_J

==X+ deltaX)J

then return

i;

end if

The focus of our research is the centralized supply chain described above. The assumptions
ofthe model are:
1.
2.
3.

1

Production ( P) + Initial Stock (IS) = (Interna[) Demand ( D) + Fina! Stock (_FS)
Fina! Stock (_FS) converges to Balanced Stock (BS)
Balanced Stock (BS) = External Demand (ED)

Assuming this we get (i - period,./ - stage in a supply chain from retailer toward suppliers):

l. Pj(i) = ED+ D1 (i)-IS1 (i) (=D1+1 (i))
2. IS 1 (i + 1) = P1 (i)+ IS1 (i)- D1(i)

120

i = i + l;
end while
end funct1.or:

By analysis of the model we can calculate production, Pii), and initial stock, ISJi), for
any supplier at any period in the future with given perturbation, ~x. Using following
equations we eliminate tirne consuming step by step algorithm presented above:

\

121

l'

o

.::!:.L.>..!...

1 X

•~; S 1l,1 and

x+&

i:..J.L:.L<_.L

Proposition 2: '.\umber of iterations = max 11

i&I

•

',/1,/

> 1l<1

+ 2 where / is the number of

links of a chain

'x+&
1+(i-i)

As we can see from Proposition 2. convergence of the algorithm is dependent on the relati ve
change of external demand ( and on the number of links) These results of the algorithm can
be illustrated by graphs. Figure 1 shows the impact ofthe relative change of external demand
on balance number for two different lengths of supplv chain

;r:

' -,-.-> l&J

x
-( i - 1) · X -

itxi

··· i"'xl

,, ,·,

IS/1 + 1) = FS/1) =

·i ,

> ( _L_ -

·

(i

+ j -· 1) · fu

.:±l.> .2:... and
i&I
;

i+(J-I)

;

< 2-.
-i&I

(a)

Fram these explicit equations we can get the following table

Model\',ith cenlralizedinforrnalior

Modelwilh centralized informat1on 9 suppl!ers

suppliers

---

5C

Table 1: Consequences in production, initial and fina/ stock when relatlve change uf
external demand hes on an interval¾~ <¾

1:

1

Supplicr 4

!

o

Prod

1
1

X

1

i

1

l !

1

o

1

Supplicr 3

1

!

Stock

i

X
X

i

X

1

'

Prod

-·

X

1

i
o

2

1

1

o

1

3 1

i4

o
4x+9t,,·

1

X

1 X

1

1 X

1

5 1

x+m

'

1

!

X

' x+m

i

1

Supplicr 2

l

1

1

Prod

1 X

X

X

o

i

o

1 X

X

4x+8&

i

1

x+L\x
x+L\x

1

!
'

x+L\x

1

ll 2. 5½

X

1

X
1

X

i

i

-2x-6L\x

U 25-rl,

Supplicr 1
Prod

1

1

X

1

X

' X

o

··X·

1

3x+61\x
1

1

x+L\x

1

1
1
1

'i
!

:

2x+4m

x+L\x

U

4&

-x-4&
x+&
x+ .n
x+L\x
x+L\x
x+ .n

24-fo-

x+m
1

x+&
1

1

x+L\x

1
1

1

x+L\x
x+-11:
x+L\x

U >3.ll
20
-

1

1

X

x+2fil i X
x+ n

!

x+m
x+m
x+m

1

x+ fil

x+L\x

1

x+L\x

1

x+&
x+m
x+L\x
x+ 1x

x+m

1
1

l
1

x+n

i

! x+m
1
1

BO

i

X

X

X

-2&
x+m
x+m
x+m

4G /

----

1

!

X

-2&

X

90

Dcman
i Manufacturcr
Prod ! Sto4. ~

X

X

X
1

Stock

X

! o

-2, 6&·

x+m
x+m
x+L\x

Stock

1

X

X

1

Stock

45

1

i

1

i

i

U=2f

x+n

'

'

./

r·J,

20

x+ 1x

f
1

1r:i;

x+ .n
X

o. 1

0.7
0.6
0.5
0.4
0,3
0.2
Rela!iR cham~e of eltemal demand !6. Ml '

O8

0.9

O.~
0.6
0.5
0.4
0.3
0.2
Relalil/ll chariueofeMlemaldernaridlll. xl

O."

U.8

0.9

i 1_x

X+ .'u:

1

It is obvious that the fina! stock equals the initial stock in the next period.
J•S1 (1) = JS 1 (1 + l), and that the production (order) of supplier J equals the interna! demand
for supplier j + 1, P/ i) = D 1 1(i) Another confirming property seen from this table is that
was
the measure of the bullwhip effect, U, increases moving up the supply chain as
expected
Further analysis requests the definition oftwo terms.
Definition 1: Balance number. k, is a number of iterations needed far the algorithm to
converge.

=

Figure 1: The dependence ofthe balance number on the relative change of
external demand far two different lengths oj supply chain

Graphs show that up to a certain point the convergence is stable But as the relative change
of external demand approaches towards l the balance number rises very quickly E g., in the
case of a manufacturer and 4 suppliers and at relative change of O 9 the algorithm converges
in 46 iterations.
2.2 Inventory costs
We observe inventory costs ofthe whole chain from the beginning to the balance number, k.
Perturbed costs are tata! inventory costs that are attained through our algorithm. Ideal cosls
are total inventory costs that would emerge in the case of immediate adaptation. Relative
costs are defined as the ratio between the perturbed and ideal costs. Considering this we set:
5

Perturbed costs C/i+ l) =C1 (i)+ 1S/i+1) =}C/k)=

I:r (k)
1

J=I

As it is seen from table above, the balance number in this case is k= 5

5

Ideal costs
Definition 2: Splitting point is a point on an interval

[o,

from positive to negative.

L; J

Proposition 1: Splitting points are exactly the set
..

i

1&1 .

cond1tion - - < -

i+ j

(k)=k x+(k-l)·fu=}Co(k)= I:Clk)=5k x+S(k-1) fu
j=l

1] such that P/i) changes its sign
far indices that correspond to

Figure 2 presents the impact of relative change of external demand on relative costs. It is
obvious that total inventory costs increase very quickly with increased relative change of
It is also evident that relative changes up to O.S do not increase costs so
external demand, l&J.
X
drastically.

X

122

123

(a)

CONVERGEl\/CE 11\i THE CASE OF THE BlLLV\IHUP EFFECT WITH
DECENTRALIZED DEMAND !NFORMATION

(b)
',c'mp,uiscnofcosts

---~

ce11tralized info,,.l suppLiers

,---- --

'1

:omparisor ofco.,:> cenlral1zed info,

c:.upp!iers

~

Liljana Ferbar. David Creslovmk, Blaž Mojškerc, Martin Rajgelj
University of Ljubljana, Faculty of Economics
Kardeljeva pL 17. 1000 Ljubljana, Slovenia
liljana.ferbar@ef.uni-lj_s1,david.creslovnik0l.ef.uni-U.si.
blaz.mojškerc_@~f.JrnHJ-.~_!, martin.rajg~li@~f.J!.@3J.~_j

tf

1

sl
J i

1

5

a

~

t

Abstract

!

15 ~

1

'---095L

0

0.05

t 1

--------- -

os L___,___~~,-~--~--~--0,15

0.25

O.J

O 35

0.4

::: 45

0

O.C

C1

J,5

J.6

D?

0.8

O.S

In our contribution we examine consequences of a single perturbat10n in the case of the bullwhip
effect in a supph cham with decentralized demand mformat10n. We mvestigate the conditions
needed to stabilize production and inventory levels. The correlat10n between the s1ze of perturbation
m external demand and total inventory costs of supply cham is also studied.

Relalirn chanae of,,atemal demand\ll. 111'

Figure 2: The dependence qf relauve costs on the re/at1ve change o.f externa/
demandfor two d1.fferent interva/s, 'O. O S] and

Kewords: supply chams, perturbed external demand, bullwh1p effect, decentrahzed mformation

1 INTRODUCTION
3

LI The bullwhip effect

CONCLUSIONS AND FURTHER RESEARCH

In this paper we have shown that recursion of the algorithm can be replaced by expressing
production and stock explicitly. We have also found out that the balance number and relative
costs increase very rapidly as relative change of external demand tends to l
Further research could include elaboration with positive changes of external demand and
also introduce more perturbations.

References
1. Chen et al (2000): Quantifying the Bullwhip Effect in a Simple Supply Chain The
lmpact ofForecasting, Lead Times and lnformation Management Sc1ence, Linthicum (MD),
Vol 46, No. 3, pp 436--443
2. Fransoo C. Jan, Wouters J. F. Marc (2000) Measuring the Bullwhip Effect in the Supply
Chain Supply Chain Management. Bradford, Vol 5, No 2, pp 78-89
3. Lee L. Hau, Padmanabhan V, Whang Seungjin (1997a) lnformation Distortion in a
Supply Chain: The Bullwhip Effect. Management Science, Linthicum (MD), Vol. 43, No 4,
pp .. 546-558.
4. Lee L. Hau, Padmanabhan V, Whang Seungjin ( 19976) The bullwhip effect in supply
chains, Sloa,1 Management Review, Cambridge (MA), Vol 38, No. 3, pp. 93-102 ..
5. Metters Richard (1997): Quantifying the Bullwhip Effect in Supply Chains. Journal oj
OperationsManagement, Washington, D.C., Vol. 15, No 2, pp 89-100.
6. Simchi-Levi David, Kaminsky Philip, Simchi-Levi Edith (2000)
Managing the Supply Chain. Boston: McGraw-Hill

Designing and

7. Sterman J. D. (1989): Modelling Managerial Behavior: Misperceptions of Feedback in a
Dynamic Decision Making Experiment, Management Science, Linthicum (MD), \Vol. 35,
No. 3, pp. 321-339.

124

Today customers are getting more and more demanding, looking for a customized
solution delivered within days, and no !onger willing to accept a commodity product in
weeks At the same tirne, most firms' manufacturing processes are becoming increasingly
dispersed and globa! These conditions make it difficult for firms to run their supply chains
more efficiently and effectively Why is it, for instance, that Procter & Gamble had to deal
with widely fluctuating orders for its nappies, when babies' consumption was generally quite
steady 9 And why was it, that Procter & Gamble's orders to its suppliers fluctuated even
more? (Lee et al., 19976, p. 93) The reason is that each retailer bases his orders on his own,
slightly exaggerated, forecast, thus increasingly distorting the information about real
consumer demand (Simchi-Levi, Kaminsky, Simchi-Levi, 2000, p. 82). This is one of the
most important causes of inefficiency in a supply chain.
This increase in variability as we trave! up in the supply chain is referred to as the
bullwhip effect (Lee et al., 1997a, p. 546). This implies that the variance of the orders
becomes larger as we move up the supply chain, so that the orders placed by the second
stage of the supply chain are more variable than the orders placed by the first stage (the
retailer) and the orders placed by the third stage of the supply chain will be more variable
than the orders placed by the second stage, and so on. In other words, the distortion
propagates upstream in an amplified form (ie., variance amplification). Formally: suppose
that the expected value of each stage's demand is the same. Then the bullwhip effect occurs
when the ratio of demand-variance at a given supplier to the variance of extemal demand,
u = v..-(r,; increases as we trave! up the supply chain (Simchi-Levi, Kaminsky and Simchi'

Vmc(D)'

Levi, 2000, p. 87).
The distortion of demand information implies that the manufacturer who only observes its
immediate order <lata will be misled by the amplified demand patterns, and this has serious
cost implications. For instance, the manufacturer incurs excess raw materials cost due to
unplanned purchases of supplies, additional manufacturing expenses created by excess
capacity, inefficient utilization and overtime, excess warehousing expenses and additional
transportation costs due to inefficient scheduling and premium shipping rates (Lee et al.,
1997a, p. 547).

125

information sharing. channel alignment. and operational efficiencv (Lee et al. 1997h. p. 98).
With information sharing_ demand information al a downstream site is transmitted upstream
in a tirnelv fashion Channel alignment is the coordination of pricing, transpo11ation,
inventorv planning, and ownership between the upstream and downstream sites in a supply
chain Operational efficiency refers to activities that improve performance. such as reduced
costs and lead time ln his v.ork Metters ( 1997) has sought to identify the magnitude of the
problem by establishing an empirical lower bound on the profitability impact of the bullwhip
effect. Results indicate that the importance of the bullwhip effect to a firm differs greatly
depending on the specific business environment Given appropriate conditions, however,
eliminating the bullwhip effect can increase product profitability by 10-30%

L2 Causes of the Bullwhip Effect
Lee et al ( 1997a, 1997b) have identified four major causes ofthe bullwhip effect
( ! ) Demandforecast updahng Links in the supply chain base their expectations about future
demand on orders they receive from the succeeding link An increase in orders leads to
higher demand forecasts, which is transferred to the next link by increased order quantities
That link also sees an increase in demand, updates its forecasts and distorts information for
the subsequent link. It works in the reverse way when end customer demand decreases A
possible solution for this is to make <lata on consumer demand directly available to
companies further upstream in the supply chain Further, a single source of forecasting can
be determined for the entire supply chain
Chen et aL (2000) have shown that if a retailer periodically updates the mean and
variance of demand based on observed customer demand data then the variance ofthe orders
placed by the retailer will be greater than the variance of demand More importantly. they
have shown that providing each stage of the supply chain with complete access to customer
demand information can significantlv reduce this increase in variabilitv However, they have
also shown that the bullwhip effect will exist even when demand information is shared by all
stages of the supply chain and all stages use the same forecasting technique and inventory
policy

2 MODEL AND ALGORITHM
The focus of our research is the decentralized supplv chain described abo\·e
assumptions ofthe model are
1
2
3

Production ( P) + lnitial Stock (!S)= (lnternal) Demand ( n) + Fina! Stock W'0
Final Stock (F'-;) converges to Balanced Stock ( HS)
Balanced Stock (R'-0 = (!nternal) Demand ( /J))

Assuming this we get (1 - period,/

(2) Order batching Demands come in, depleting inventories but the company may not
immediately place an order with its supplier It often accumulates demands before issuing an
order, for instance because of fixed order costs or distribution efficiency. Consider a
company that receives daily orders but places orders with its suppliers once a week
Variability of orders placed with the suppliers is higher than the demands the company itself
faces. In addition to making consumer demand data available throughout the chain, reducing
batch sizes and increasing order frequencies assist in decreasing this effect.

stage in a supply chain from retailer toward suppliers)

~(i)=2Di(i)-[Si(i) (=lJi l(i))
2

D 1 (i)

1S1 (i+l)=P1 (i)+ISJ(i)

We study the model in a case where two more assumptions have to hold

(3) Price f/11ct11ations Because of promotions and trade deals, the price of a product
fluctuates, which increases variability of demand. When the price of a product is low, a
customer buys in bigger quantities than needed When the price returns to normal, the
customer buys less than needed to deplete its inventory. Stabilizing prices and decreasing the
number of promotions is a way of reducing this effect

2

External Demand ( x) changes its value by Cl.x: 1'~1 J = x + & ,
The perturbation is negative & < O

These two assumptions tel1 us that consequences of only one negative change of external
demand are examined and that the state following this change remains unchanged We
suppose that these two assumptions can be very real in many real-world situations we note
the stabilization of the external state after single perturbation The perturbation could, of
course, also be positive - in that case the same analysis would follow.
Furthermore if P1 (i) < O , we set Pj (i) = O, since the production can not be negative. The

(4) Rationing and shortage gaming. When product demand exceeds supply, a supplier needs
to ration its product to customers. Knowing that, customers may order more than they really
need. Later, when there are no shortages, orders disappear Introducing rationing methods
based on past sales rather than on orders placed takes away the incentive for customers to
inflate order sizes.

production can be expressed in a form P = mx + 11& = mx -

mx -

nf&I,

so the condition

ni&! < O implies

l&I = relat1.ve change \J
- < --(

1.3 How to Counteract the Bullwhip Effect

111

Measuring the total bullwhip effect does not teli which of the different causes contributes
most and which solutions are most relevant (Fransoo, Wouters, 2000). For example, to
assess the possible benefits of exchanging demand information, it is important to bea:ble to
measure which part of the bullwhip effect is due to incomplete demand information in a
particular supply chain.
Understanding the causes of the bullwhip effect can help managers find strategies to
mitigate it. Indeed, many companies have begun to implement innovative programs that
partially address the effect whereby they tackle each of the four causes. Various initiatives
and other possible remedies based on the underlying coordination mechanism are:

126

The

11

X

Table below shows us consequences in production, initial and fina! stock when relative
change of external demand lies on an interval ,&- ~ ~ < ½. At this point it is convenient to
define the term balance number.

Definition 1: Balance number, k, is a number of iterations needed for the algorithm to
converge.

127

J

Table 1: Consequences in prod11ct10n, 1111twl and fina/ stock in the wse
change oj external demand i s; ill.>.i <

relat1ve

t

Supplier 3

1

r
1

:

Prod

l

oj

X

i
2

:

5

X

1

X

! o

X

i

X

X

i

i

i

-2x-8&
-2&
-2&

Y+.1..\"

i

1

Manufacturer
Prod

!

X

X

X

X

o

1

-2&

-21.lx

2x+4i,x

I
1

i

Dema
nd

2, lnventory costs

X

We observe inventory costs of the whole chain from the beginning to the balance number, k
l'erturhed costs are total inventory costs that are attained through our algorithm. Ideal costs
are total mventory costs that would emerge in the case of immediate adaptation. Relative
costs are defined as the ratio between tl}e perturbed and ideal costs C'onsidering this we set:

X

X

2x+4m

Stock

x+2i,x

X

x+&

x+m

x+m
x+&
x+m

x+m

1

1

x+i,x

4

1

: x+ ..lx

i

: Stock

X

X

-2x-8i,x ,

1

2x+4l1X

1

1

3x+Mx

1

1

Prod

X

1

1

2x+4m

1

1

Supplier 1

Stock

X

1 X

:
:

J

4
1

1

X

1

::

o

:

1

Prod

X

1

o

Supplicr 2

Stock

1

conJecture that tbis
emerges due
slow transfer of mformation about small relative
changes of external demand higher-level suppliers

x+i,,·
x+l)_x

u < 1li,\

1

i 2x+4i,x

-2LlX

i

1

1
1

i x+
x+.:u:

1

i
1

i

!

x+ll-,

;

i
1

1

x+m

i

'IX

! X+ill

x+&
X+ll-X
x+i,,

1

i

1

x+,'1X

!

x+.'lx

1

i

U<7½

1

x+m
x+m
x+m
x+m
x+i,x
x+i,x

x+ill
x+:u

1

i

1

X+ill

1

X+ lx

i

l

U=2i

fu=}

i Balance

Ideal costs

numbeL

As it is seen the balance number in this case is k = 5,

C0 (k)

k

whereby that is also the balance number for ali cases of relative change of external demand
from the interval [O, ½) Another confirming property seen from this table is that the

1.

5

~

m
,. ,,r'l,

/

70

"25

:

1

; 20

J

! 15

,

/
r,-'

10

]
60
E

~

!

50

40

/

)

30

r
oLI-~~-~~-~-~~-~_J1
o
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9

)

', i \

20

. _ __ _ _ _ _ ____J, ....

RelalMJ chanCJH of edemal demand f.6. xl /

10

o
o

ll

< l~x!

0.1

0,2
0.3
0.4
0.5
0.6
0.7
Relattt'B chanQB ofEu:lerna\dllmandlA liti•

0,8

0.9

Figure 1: The dependence oj the balance number on the relative change oj
extemal demandjor two di.fferent lengths oj supply chain

Perturbed
costs

Costs increase
at least for (in

Cp(k)

%)

1

-1.98

19x+6Lix

-l.98

s

20x + l6ru-

i 2lx+l8m

2.5
2.5

8 91

8.93

11.25

25

2"i

16x+2Lix

s

20x+ l6Lix

19x+ lOru-

< 1&1 < .l

s

20x+ l6Li.Y

l4x ... 2ru-

2

1.39

'.U25 1i
- .. ---1
3.125

20x + l6Lix

X

1

--+-------- --i

39

5

2...

o

20x+ !4ru-

1

+i

~

As seen the relative changes of external demand by less than k decrease the relative costs
but not more than for l. 98%. The decrease of the relative costs occurs again on the interval
[¼ f.i However, other relative perturbations that do not exceed ½mostly increase relative
costs but not more than for 25% Results presented in table 2 can refer us to expectation of a
prevalence of rapid increase of relative costs for the relative changes of external demand
bigger than ½
This expectation is confirmed by figure 2 that presents the impact of relative change of
external demand on relative costs for the whole interval used in table 2 and also for a broader
interval: figure 2 (a) reflects the relationship on [o, 0.5] and figure 2 (b) shows the same on

[o,

0.9] Reader can observe a very rapid increase oftotal inventory costs as relative change

of external demand tends to 1

The reader can observe an anomaly that occurs on the graph: a "bump" unexpectedly arises
at the first third ofthe interval [o, 1] and from then on the graph has the expected shape. We

128

Costs increase
at most for (in

i %)

20x+22Lix

x < s3
s3 -< xl~-'I < u5
12 -

100
90

16

X

T6-

1

Modetwi1h no information • 9 suppliers

< 1t.xl < .1..

4 -

(b)

e'

(k)=LCj(k)=4k x+4(k-l) fu-

20x + l6ru-

5

C'onvergence ofthe algorithm is dependent on the relative change of external demand. This
correlation can be illustrated by graphs Figure shows the impact of the relative change of
external demand on balance number, for two different lengths of supply chain (for 3
suppliers + a manufacturer, and for 9 suppliers + a manufacturer)

:-f- - - - - - ~)

'0

---1

measure ofthe bullwhip effect, ( T, increases moving up the supply chain as it was expected.

Model with no mformation • 3 supp!iers

'p(k)=LCj(k)

Table 2: 'omparison hetween perturbed and ideal costs for relative changes of
.,,
external demandfrom the interval

U=l

From this table it is obvious that the fina! stock equals the initial stock in the next period,
(1 + , and that the production of supplier
equals the interna! demand for

(a)

=}

4

ldealcosts Cj(k)=k x+(k-

/'S,

supplier j + I, ~ (i) = D1+1 (i)

+

+IS

i

1

x+& 1: x+m
: x+.'lx
x+m
X+ll-,
x+m
1 x+i,,

U<3f

+

Perturbed costs

X"t" :i,:

129

(a)

r

(b)

25,--------~-----------~
Compan<J.on of cos1s no info, 3 suppl1ers

c~--_-_.om_p~an~_,1_,o_~,_n_oi_~_.~_'"_P~_im
_ _ __

/i

12

ON THE EIGENSPACE STRUCTURE OF A MATRJX IN MAX-MIN
ALGEBRA
11

; 5

MARTIN CAYALEC'. JAN PLA\KA
C

()5

-----

CL

u.9-5~o 0.05 O 1

/

0.15
0.2
0.25
0.3
03!5
0.4
Relalr:a drnnQa of 1idemal dwrnnd 1.0. lll ,

J
__
·
LJ ~-~-~~-~-~-~

i
.J
045

0.5

O

Cl

O .L
O.J
DA
J,!i
0.6
0.7
Relalt\'e ch.-.nae of elllemal dsmand lli. lil/ ,

ABSTRACT For a given

x

matrix A in a max-min algebra. the structure o! the eigenspace

F( A) is studied. A necessary and snf!icient condition is described. under which the maxirnum
cigcnvector is nrnstanr FurtheJ res:1lts show tliat :F( A) ca11 cm1tai21 exponc1nially 111auy 1101i.
0.8

0.9

empty clisjoint intervals of strictly monotone eigenvectors. with distinct endpoints As a
rnnseqnence, the eigenspace structnre cannot be completely described
polvnornial tirne.

Figure 2: The depe11de11ce of relat1ve costs on the re/alive i;hcmge of extemal
demandjor two dtfferent inten1a/s, [o, O 5] and [o, O 9]
l

3 CONCLUSIONS AND FURTHER RESEARCH
In this paper we have examined consequences of a single perturbation in the case of the bullwhip
effect in a supply chain with decentralized demand information. We have found out that the
balance number and relative costs increase very rapidly as relative change of external
demand tends to 1
Further research could include elaboration with positive changes of extemal demand and
also introduce more perturbations.

References
Chen et al. (2000) Quantifying the Bullwhip Effect in a Simple Supply Chain The
Impact ofForecasting, Lead Times and Information Management Science, Linthicum (MD),
Vol. 46, No. 3, pp 436-443.
2. Fransoo C Jan, Wouters l F. Marc (2000) Measuring the Bullwhip Effect in the Supply
Cham. Supply Chain Management, Bradford, Vol 5, No. 2, pp 78-89
3. Lee L. Hau, Padmanabhan V., Whang Seungjin ( 997a) Information Distortion in a
Supply Chain: The Bullwhip Effect Management Science, Linthicum (MD), Vol 43, No 4,
pp .. 546-558
4. _Lee L. Hau, Padmanabhan V., Whang Seungjin (1997b): The bullwhip effect in supply
chams, SloanManagement Review, Cambridge (MA), Vol 38, No 3, pp. 93-102
5. Metters Richard (1997): Quantifying the Bullwhip Effect in Supply Chains. Joumal of
OperationsManagement, Washington, D.C., Vol. 15, No. 2, pp. 89-100.
6. Simchi-Levi David, Kaminsky Philip, Simchi-Levi Edith (2000): Designingand
Managing the Supply Chain. Boston : McGraw-Hill.

lČ\TRODLCTION

.\lax-min algebras have wide app!i(1ations
the fuzzy set theor-v (the max-min algebra
on the unit real interval is one of the most important fuzz:v algebras) The eigenvectors ot
max-miu matrices are usefol in t luster aualysis (see [4]J 1Jr
fuzz:v reasoning (see
J The
eigenproblem in max-min algebra and its connections to paths in digraphs were investigated
in 1, 4. 5, fi, A procednre for computing the grcatest cigenvcctor nf a givcn rnax-mi11 rnatrix
was proposed
and an efficient algorithm was described in f2
The eigenprnblem
distributive lattices was studied
18]
·
The aim of this paper is to answer severa! question connected w the strudure of Lhe
eigenspace .r(A) of a given max-min square matrix A. In [3] the structure of .r(A) was
described as a union of intervals containing permuted monotone eigenvectors of various types.
As the uumber of possible permutations is exponentiall:v large_ it is natura! t ask whether
exponentially man:v of these intervals can simultaneousk be nonemptv. The positive answer
is presented in Theorern 4.2 and it indicates that the structure of the eigenspace F( A) can
be rather complex. Further results concerning consta11t eigenvectors and perrnuted monotone
eigenspaces are presented in the paper.
2. MONO'T'ONE EIGENVEC'T'ORS
Bv a max-min algebra we ,mderstand a linearly ordered set (B, .S) with the binarv operations
of maxirnum and minimum. denoted by EC and ®· The matrix r>pcrations over B are dcfiucd
with respect to EB, ®, formally in the same manner as matrix operations over any field. For a
given natural n > O. we use the notation N = {L 2,. . 11}, and the notation B(n) (B(n, n))
for the set of all n-dimensional column vectors (square matrices) over B- We say that a vector
b E B( n) is increasing, if bi .S bj holds for any i, j E N. i .S j Vector b is strictly increasing,
if b; < bj whenever i < j. The set of all increasing (strictly increasing) vectors in B(n) is
denoted by B~(n) (by B<(n)). For x,y E B(n), we write x S y, if Xi .S Yi holds for all 'i EN,
and we write x < y. if x .S y and x /= y. In other words, x < y if Xi .S Yi for ali i EN, but the
strong inequality x; < Yi holds trne for at least one i E N.
For given A E B(n, n), h E B, the threshold digraph Q(A, h) is the digraph 9 = (N, E), with
the vertex set N and with the are set E = {(i, j); i, j E N, aij 2: h}. The strict threshold
digraph 9(A, h+) has the are set {(i,j); i,j E N, % > h}. The set of all permutations on
Date: Angnst 15, 2005.
1991 Mathemalics Subjecl Classification. Primary 04A72; Secondary 05C50, 15A33.
I<ey words and phrases. eigenvector, max-min algebra, fuzzy algebra.
This work was partly supported by Czech Science Foundation,

130

131

# 406/03/0115.

N will be <lcnoted by Pn. If A E 8(11, 11) and h E B(n.) .p E Pn, then we denote hy Aip,i,, the
matrix neated hv applying permutation '-P to the rows aud pernmtation 1p to the colurnns of
k and hv b'P we denote the vector created by applying the pennutatiou {J to vector b.
For any square rnatrix A E B(n.n). the eigenspace of A is defined hv

n;J

irnplies that there is .JE 1\' such that
> M(A) Hence. ihere exisLs a doscd path p from j
> jf(A). i.e. p is a rycle in 9

•

to./ sud1 that c{p)

F(A) ={bEB(n): A®b=b}

The vectors in F( A) are called eigenvectors of matrix A. The set of all increasing eigenvectors
is denoted by J:". ( A) and the set of all strictly increasi11g eigenvectors of A is denoted l,y
:F<(A). AH aHv V<\Ctor b E B(n) c:m be pernmted to aa irnrcasiug vector, the next theorern
says t.l1at tlw strudurP of the eigeuspace :F(A) of a give!l n X n max-min matrix A call be
descrihed by investigating the strudure of monotone eigenspaces :F:".(A) and :F<(A).

Theorem 2.L [3] Let A E B(n. n). b E B(n) and ep E Pn
b'P E :F!Acpip)

Then b E :F(A) if and only ,j

Far A E B(n n). we define vectors m*(AJ. M*(A) E B(nJ in the following way For any
1

EN. we put

m''

(A) := llHI.XaJk
k>J

M*(A) = min JvJU1(AJ

rn*,. (A) := maxrr/1!(Al
j~i

'

i

'

3.

=

okk

for euery i. J. k E .\'. 1
i= ~) for ei•ery ,r E l'n

i=

j. 1

i=

,a;

i.e„ the ith entry of the maximum eigenvector equals the maximum capacity c(p) of a path
beginning at i and ending by a cycle at j, where the capacity c(p) is assigned to each path p
as
c(p) = _min akt
\k,l)Ep

In the following theorem, au interval for constant eigenvectors is described. For A E 13( n. n).
we define the value M(A) E Bas M(A) := minmax%.
,EN JEN

Theorem 3.1. [3] Let A E B(n. n) and let b E B(n) be a r;onstant vector. Then b E :F(A) if
and only if O :S; b1 :'.S: 1\J(A).

=

M(A) holds far

Theorem 3.3. The maximum eigenvector x* is constant if and only if the digraph
is acyclic.

Pmof. (i) •
Le1 sLaLemeIH (i) be fulfillecL aud lei '{ E /!11 be a11 arb1:rnry permutat1u11 on
.V. Clearlv. Lhe maLrices A. A'P'P have thc same Rcts of diagoual aud 11011-diagoual dc•11J<'Hts.
!Vforeover. ai,v two ele1nents belcmging t" a r:omnwn row (columu) in A lic in a, ornm011 row
(columnJ in Aip<p· as well Tlms. statemeut (i) is invariant with respect to permutations on N.
and it is suflkient to consider only tlte case ,c = id\ i.e. A = A'P'P
Clearly. the iuequalitv
m

(A) = maxau S maxu,1 = :vŠ''(A)
l?.i

holds trne for even- ·, EN. Funher, by the assumption

J

we get a strict inequalnv

G>M(A)

= M(A) holds trne for all i E N. and 9 = G>M(A) contains a cycle
i1, i2, ... , ik. By the definition of the maximum eigenvector x* we have xf > lvl(A) for
every i E {i1, i2, ... , ik}, which is a contradiction.
For the proof of the converse implication, let us suppose that there exists i E N such that
xt =f M(A). As x* is the maximum eigenvector, in view of Theorem 3.1 we have xf > M(A).
The formula

(4.:l)

ru1d. for i < 1 we ltave
(4.4)

According to Theorem !'i.3 in :3:, tt1e inequalities (4.:1). (4.4) implv tha, :F< A1 te
,(iiJ • ,{ij Let us assume that statemenl (i) 1s not satisfied. i.e.
i. j. k E N. 1 =f J. 1 =f k such that

0

there exist indices

(4.5)
We shall coustruct a permutation ,p E Pn with :F<,Aipip) = 0 in the following way. We start
witlya pernmtation wbich interchanges indices k. n E ]\;. i.e. we put ep(k) = n. sp( n) = k aud
ep(l) = 1 for all l i= k. ri By assumption I i= k we have ep(1J i= ep(k). i.e. ep(i) < n.
Further. the assumption i =f J implies 1p(-i) =f ,p(j). If ep(j) < ep( i). then ep(j) < n = ep( k)
and the mutual interchanging of indices ep(i),ep(j) does not influence the vaiue ,p(k) = n.
Hence, without any loss of generaiity, we may assume that ep(i) < ep(j) and ep(k) = n. Let us
denote the matrix A'P'P by A'. The assumption (4.5) can now be written in the form

(4.6)

i
i
a<p(i)ip(j)
-> ann

O'=

Then we have

(4.7)
what implies

(4.8)

132

k

for everv J, k E 1V, j < k.
By Lhe i11equalil1es 1.4.lJ. (4.2) we get

It was stated in [1] that the greatest eigenvector x* exists for every matrix A, and its entries
are given by the formula
x~ = maxmin{af1 1 }

Proof. Suppose that xf

<

(42)

CONSTANT EIGENVECTORS

Corollary 3.2. The maxirnurn eigenvector x* 1.s constant if and only if xf
all i EN.

a,J

(ii' :F<(Aipipl

i>1

(m*(A;,M*(A); r B<(n)

J

Theorem 4.1. Lel B be a dense max-rnrn algebra and let A E B(n, n). Tli.en thc following
statements are eqmvalent

(4. l;

j2i

Theorem 2.2. [3] Let A E B(n, n) and let b E B(n) be a strictly increasing vector. Then
b E :F(A) tf and only if m*(A) :'.S: b :S; M*(A). In formal notatwn we can write

:F<(AJ

4. !'ER '.\ll TED El(;ENSPAI 'F, STRl ( 'T1 RE

133

In view of Theorem 5.3 in j3j the inequality (4.8). gives
:F< rA

1

J

Y -- {O. 1} wit h fixerl ,·aiues c' 1, -l = c TI) = O,
h nms nYe: all 2 11 -' ol possihle mappir;gs
theli the correspcmding penu1uat.iuns 'f' create 2n 2 uf perm,;t,ed u,a,rices A'P'P with distinct
lower bounds in intervals

= :F< l A,p,p) = 0

i.e. the statement (iiJ does not hold trne.
Theorem 4.2. Let B be n dense max-min algebra. lf
B( n, n J s·uch that :F< ( AP'P) #- 0 for every <P E Pn.

IBI > 1.

t/!en there is

11

rrwtrix A E

Proof. Let us choose x. y E B snch that x < y. We define a matrix A E B(rL n) by putting

if 1 =/= }
if i = j
It is <)asy to veri!\; \ lml matrix A sat ishes i he rnwlit ion ( i) iu Theorern 4.1 lleuce :F< ( A'P'P) =/= f/J
for c,very <P E Pn
Remark 4.1. The matrix A in Theorem 4.2 is invariant with respect to all penrnnations in
Pn. As a consequence, we have

m*(A)

=

m*(A,p,p)

J1J* ( A J = }'vi* ( A'P'P)

= (x. x.
= (y. y. .

. x)

.y!

Tlms. all iutervals :F< ( A'P'P). <p E Pn have the same endpoints

Theorem 4.3. Let B be a dense max-min algebra with !B 1 > l. let n > 2. Then there is a
matrix A E B( n. n) with at least 2n- 2 distinct interna/s uf the form :F< (A'P'P) #- f/J. 'P E Pn
Proof. Let the assumptions of the theorern be fulfilled. As B is dense. there is an in(Teasing
sequence of length 2ri + l in B. The elements of the sequence will be denoted hy numbers
O. L 2 ..... 2TI. The matrix A E B(TI. n) is defined by

if i
iln

= { ~+ l
j

.,

o

Corollary 4.4. Let B be a dense max-min algebra with !BI > l. let n > 2. Then the eigenspace
slructure oj a matnx m B( n. n; nrnnot be. ,11 yenem( cr;mpletely des, nbed 1.n polynomiai !ime.
unth respect to 11.
BEFERE;\/ ES
! K ( Jechlarova, Eigenvectors in boltleneck algebra. Lin. Algebra Aµpl 175 / !992). 63-73
·2 T< ( lechlaru,·a, Rffi11ent compul.ation oj th.e greale,,t eigenveclor ,r, fuzzy algebra.. ra,ra \it Vladi i''l'il 12
, l997;. 73-79
.'f \L ( ;avalec. lvlonoione eigenspace slructure in max· min algebra, !.iu Algebra Appl. 345 : 20021. l49· 167
4 M. f ;ondran. \'aleurs prnpres e! uec!eurs pmpre,, en classificatirm hufrurchique. R A l. R /) lliforrnat1que
·1 hec,rique 10 ( 1976). 39-46
·;; M. Uondran and M. :vlinoux. Eigenvalues and e,genveciors in semimodules and their inlerpretation in graph
theory, Proc. 9th Prog Symp. (H!76/ :33-148
i6. M. iJondrau and M. Minoux. Vaieurs propre:, ei vecieurs propres en theorie de, gra.phes. ( •ilioques !nler·
natwnaux. (' '.\f RS .. Paris. 1978. 18 l83
·71 E. Sanchez. Resolution oj eigen fuzzy sets equations, Fuzz,v Sets and Svsterns 1 i. 1978) 69-74
18· Yi-Jia l'an. Eigenvalues and eigenveclors Jor matrices over distributive lati.ices, Lin. Algebra Appl 283
( 1998) :!57 272
DEPARTMENT OF lNFOR.M::\TION TECHNOU)CTES. FAc~·LTY OF i>FORtv1ATICS A~D rvlANAI ,ErvIEl';T
VERS!T', ilRADEC' l(RALOVE. ROKITANSKEHO 6;1. ,50003 HRADEC KRALOVE. C:ZECH REPl.Bi!f'

E- mail arldress martin. gavalec©uhk. cz

=)
~- lJEPARTMENT OF MATHEMATICS. FMTLTY OF ELECTRiCAl. ENGJNEERING AND TNFURMATJ!;S TE<'HNI·
! NI\'ERSITY IN KoŠICE. B. NEMCOVE.' 32. 04200 KošICE. SLOVAK'A

if 1 < n = J
if J < fl = 1
otherwise

{JA(

E-mail address Jan.plavka©tuke sk

It is easy UJ see that A satisfies the statement (i) in Theorem 4.L hence :F< (A,p,p) =/= 0 holds
true for everv <P E I'n.
l) =
Now. let c be an arbitrary mappiug c N -> {O, l} with the two last values c(n
c(nJ = O. The unit values iu the sequence (c(lJ, d2), .... c(n)) occur in m ,m 2: O) disjoiut
intervals ~c(ik·), .. . c(jkj). k= 1.2 .... m. separated by zeros. Each interval is of the form
1 Ui with indices l S lk < Jk S TI -1 and )k< 1k. • Thus. for every 1 EN we have
(l. l.
1

c(i)

={

~

if ik Si < jk for some k= L 2 1
otherwise

We assign to c a permutation <P,

= :p E Pn,

• , ,

m

defined for i E N. as follows

if ik S i < jk, k= l. 2, .... m
if i = jk
otherwise
By the above definition, the permutation preserves the value n = <p(n). Therefore, when
applied to the rows and the columns of the matrix A, <p permutes only the elements in the
last row and in the last column, and the elements on the diagonal, while the remaining entries
of A remain equal to zero. Further, it is clear that <p consists of m disjoint cycles ( cyclic
permutations) (i1,:, ik + 1, ... , jk), Having this in mind, we can easily verify that

*(A
mi

'P'P

)) _ { i + c(i)
n- 1
-

134

( .NI,

if 1 :S: i
if i = n

<n

135

Preservation of Efficiency and Inefficiency
Classification in Data Envelopment Analysis•
Zagreb. Faculn of Economics
Luka :\'erafa·. uiversity
Trg J F Keimedva 6. lUUUO Zagr0b. 'roatia
F-mail: l11eralic@cfzg.hr

Dala Fuvelopmcnt Anaiysis
Extended Abstract. Seusitivits Analysis
1
(ahhreviation: DEA) fur the Additive muJcl (see Charncs et aL 1 .' was
Charnes and \erah{· ·2j for thc case of thc simultancous chaugc
studied
or/ami of ali outµuts of au arbitrarv efficient Decision Makir,g
of all
µresen iug its effkiew y S1dficiem conditions foJ:.
( abbreYiatiou
l.
µreserving etticiency of D \ll. uader t hese changes were obtained. Sensih,·i,\·
DEA for arbitrarv perturbations (and for non-negativc pcrturbations) uf
1
.\eralir 4 . For some reccnt
the Additive model was stmlied
all data
developments of sensitivit.v and stability analysis in DEA sce Cooper et al.

:-r
t he regiun of efficiencv around an
The airn of this paper is
effident DMl: 0 according to the Additive model in DEA, which is projcction
of an inefficient DM1' 9 tu the efficiency frontieL under the simultaneous nonnegative perturbations of all <lata of all DMUs preserving efficiency of DMC 0 ,
Based un snfficient conditions for preserving efficiency of DMU under thcsc
.54) regiou arouncl DMC and the corresponcling
changes (see Theorem
rcgion around DMC 9 are ohtainecl. Considering relationship between these
regions, conditions for simultaneous preservation of efficiency of DM1: 0 and
inefficiency of DMU 9 are given, SeC'ondly, using the result on simultaneous
preservation of efficiency of DMU and inefficiency of DMU 9 (see Theorcm 3.
p, 55) the case of region of joint efficiency around cvery efficient DMC with
the corresponding region around each inefficient DMP is studicd under thc
non-negative perturbations of all data of ali DMUs. Sufficient conditions for
preserving efficiency of ali efficient DMUs and inefficiency of ali inefficicnt
DMCs simultaneously are obtained (see Thcorem 4, p. 56). An illustrative
example with data from Seiford and Thrall [5] is provided.
0

0

0

*This is an extended abstract of the paper L. NERALIC, Preservation of Efficiency and
Inefficiency Classification in Data Envelopment Analysis, Mathcmatical Communications
9 (2004) 51-62. This research was partly supported by the MZT grant 0067010 of thc
Republic of Croatia.

136

137

RECENT APPROACHES TO THE QUADRATIC
ASSIGNMENT PROBLEM
Janez Povha Renata Sotirovb
aschooi for business and management, l\a Loko :2. 8000 ;'-/ovo mestu, Slovenia
1
'The Universily uf '.\[elhourne. Deparnnent of :'llathematics an<l Statistics, Parkville, Australia
email Janez.povh©guest arnes si, rsotirov©ms unimelb.edu.au

I3ccausc coll(litious in Thcorem 1 are sufficient but not necessary so are
cornlitions in Theorem 3 and Theorem 4. Au open question isto find necesordcr to apply the results in
sarv aml sufficient conclitions for thesc cases.
practicc it is ncccssary to make an algurithm aml compnter code. which is a
chalkngr· for thc rcscareh in thc futme.
suffi('ient conditions for
An open question is also to find (necessary
and iuefficiency of
simultaneous preservation of efficienc_v of efficient D~Il
·R.l model
inefficient D:Vll:(sl according to the Charnes-Cooper-Rhodes
and thc Bankcr-Chancs-Coopcr (BCC) model nnder the nou-negative
of all DMCs. The same holds
puts
arbitrarv) chaugcs of
and BCC
for tho pwportionat0 changc of ali data for the Additive. C(
modds. Thc rcsults for thcsc cascs will be discussed clscwhere.
Key words: Data Envelopment Analysis, Efficiencv. Additive Model,
Seusitivity Analysis.
Preservation of Efficiency and Tnefficiern y Clasificat
Lincar Programrning.

References
): A. CHARNES. W. W COOPER, B. GOLANY. L. :VI. S.EIFORD.
J. STl!TZ. Foundatzons oj Data Enuelopment Analysis Jvr ParetoKoopmans Efficient Ernpirical Productwn Functwns. Journal of Economctrics 30 (198.5) 91-107.

:2) A. CHARNES, L. NERALlC, Sensitivity Anaiysts of the Additwe lvlodel
in Data Envelopment Analysis. Europeau Journal of Operational Rescarch 48 (1990) 332-341
:VI. THRALL.
[3] W. W. COOPER, S. LI. L. M. SEIFORD, K TONE.
J. ZilG, Sensitivity and Stability Analysis in DEA · Some Recent Deve/opments. Journal of Productivity Analysis 15 (2001) 217-246.

[41 L. NERALIC, Sensit-ivity in Data Envelopment Analysis far Arbitrary
Perturbations oj Data, Glasnik

Matematički 32

(1997) 315-335.

'.5] L. M. SEIFORD. R. M. THRALL. Recent Developments in DEA. The
Mathematical Programming Approach to Frontier Analysis, Journal of
Economctrics 46 (1990) 7-38.

138

Abstract In this paper we present severa! recent approaches for solving the Quadratic Assignmell! Problem (QAP J. \Ve first recall se\icral sernidefini10 rdaxal ions of QAP. and thcn prescnt
copositive programming. \\e discuss on the strength of the new
new QAP relaxatious basecl
relaxations and compare them wit SDP ones.
Key words: qnadratic assignment problem. semidcfinite programrnmg. copositive program
ming.

1

Introduction

The Quadratic Assignment Problem (QAPJ ca11 be stated in lhe following way. Let IT be the
set of X n perrnutation matrices. For gi\·en A and B real symrnetric n x n matrices ancl G'
real n X n matrix. find a permutation matrix X E TT which miuimizes
(QAP)

, l)

This formulation is calied the trace formulation of QAP. fhe QAP is nowadays wideiy considerecl asa dassical nJmbiuatorial optimization problem. hut it is also known asa generic model
for various real-life problems.
The QAP is well known to be ~P-hard. and even finding au c approximation of QAP is
a 1\'P hard problem. The romputational effon to solve the QAP is very likely \1, grow expouemially wilh the problem size. and problems of size r1 2 25 are currently considered as huge
algorithms as well as i11 computational pla1forms have
problems. The recent developments
resulted in a large improvement in the capability to solve QAPs exactly. In 2002, Anstreicher
et al. [2j made a hreak-lhruugh by soh-iug a 1mwber of pre,.-iousl_v unsolved large QAPs . Their
computations are considered to be among the most extensive computations ever performed to
solve discrete optimization problems. Anstreicher et al. [2] incorporated the Branch & Bound
(B&B) algorithm ona supercomputer. For the summary of recent advances in the solution of
QAP by B&B see [1].
The study of lower bounds for QAP is very important for the development of B&B algorithms. Namely. the performance of B&B algorithms depends on the computational quality
and efficiency of the lower bounds. The most recent and promising trends of research for the
bounding methods for QAP are based on semidefinite programming [8, 10] and lift--and -project
techniques [4]. Bounding procedure that is used in [2] is known asa convex quadratic programming bound (QPB), and it combines orthogonal, semidefinite and convex quadratic relaxations
in a nontrivial way. The bounds computed in [4] and [8] are stronger than QPBs. SDP bounds
were also tested in B&B settings and the promising results are reported, see [8].
In this paper, we first recall semidefinite programming (SDP) relaxations of QAP. Semidefinite programming is an extension of linear programming where the nonnegativity constraints
are replaced by positive semidefiniteness constraints on matrix variables. SDP has been shown
to be a very powerful tool for providing tight relaxations for hard combinatorial problems,
notably QAP. In Section 2, we recall and summarize the approach from (8, 10] to derive SDP
relaxations for QAP. For deriving SDP relaxations we use new matrix representations derived in
[9]. Ali SDP relaxations are formulated in the space of symmetric matrices of order (n-1) 2 + l.
The strongest SDP relaxation of QAP contains O(n 3 ) equations and O(n 4 ) sign constraints.

139

l{eudi and Sotimv [8' sol ve t hat relaxation using the B11ndle 'Vlelhod aud report ven s1 rong
bmmds. ln Section 3 we present severa! new QAP relaxations based on 1 he copositive program~
miug. We give first a reformulation <>f the QAP as lmear problem over the nme uf completelv
positive maLrices. Since cupositive prograrnrning is not tractable. we approximat,e the cone of
1he syrnmetric cornpletely positive matrices with the cone of symmetric doubly uonnegative
rnatrices. and obtain a new relaxation. The relaxation ohtained in this way is relaxed further
by ornitting the nomregativity constraints and adding extra O(n 3 ) coustraints. In Seclion 4 we
discuss on the strength of the new relaxaLions and cornpare them with SDP relaxations.

II, order to derive the SDP reiaxatiuns we exploiL the fact that Lhe row and colurnn snms of
perrnutation matrices are une. In .9i is proved l hat t he set 1Jt' n x II matrices with row ancl
column sum equal to l is equivalent to the following set:
- 1. l

n

where
V:= (

1.1

\

N otation

!11 this paper we ('Onsider followiug convex coues: symmetric n

X II

maLrices are deuo1,ed by

Sn = {X E rn:nxn: X= X 7 } the n x II positive semidefinite matrices s;;= {X E Sn y7 Xy 2:
ll.'dy E rn:n}. the n x n copositive maLrices Cn = {X E Sn.. y 7 Xy 2: O.'dy E IR+}- the n x 11
cor11pletely posiLive rnatrices C~= {X= I:7=l !Jt!J/. k 2: L !J, E IR:+. 1 < i < k}. and the TI X n
symrnetric nonnegative rnatrices Nn = {X E Sn: X,.J 2: O. 'di. :i}.
For X. Y E Sn. X >- O (resp. X >- Oj denotes positive semidefiuiteness (resp. positive
clefiniteness). For two rnatrices X. Y E rn:mxn_ X 2: Y. (X> Y) means x,J 2: y,i' (x;'.l > y; 1 )
for all i. j The Kronecker product of rnatrices X = (x 1J) and Y = (Yktl is X® Y = (x,JYktl =
(x,JY). 'di.j.k,/. We clenote by ( .. ·) the standard scalar product. For u.v E rn:n is therefore
7
(1t. v)= u 7 v and for X,Y E rn:mxn is (X. Y) = trace(X Y) For X E rn:mxn_ vec(X) denotes
the vector in rn:mn that is fonnec! frorn the colurnns of the matrix .'(
We denote the ·ti,h standard unit vec:Lor with e,. the vec:Lors of all <mes is Un E IR: a uuit
rnatrix is E;j = e;ef The square matrix of ail ones we deuote b_\· Jn (or simply J when 11 is
obvious) and the ic!entity matrix bv In or I. For matrix cohmms and rnws we use MATLAB
notaLion, hence X(:, i) and X(i. :) stan<ls for ith colmm1 anc! row. respecLively.

:\ow is easv to prove (see [8. 9]) that for any }' E P exists a symmetric rnaLrix R of 1Jrder
2
(n - l) + l. indexecl from () lo (11 - 1)2. such that

l ;2 f.
(n
lf }. is an extrerue poim of P then R,J E {O. l }. ot herwise H;1 E iO. 1j for 1. J E {O.
see [9]. Bv e:xploring Lhat fact. we can forrnulate SDi' relaxation t.hat irnpose constraints on
the set of smaller dimensional space than P an<l explorc sparsity cfliciently. sec /J ,
From ccmditions (4i anc! (.5) we derive the following set P containing P

p

=

{v E Sn2. 3R

1
\ 11

s. L Roo = l.1 = WRW1.11 = <liagff). (

11

,F

\/l >y .

olj

•

s;;

is called a semic!efinite program. while a linear program
A linear program over the cone
over Cn or c; is called a copositive program.

\Ve cau shuw t hal l he interior of P is not emptv hy fiwiing a barv center of P Tu what it follows.
we e:xplore further a strncture of the eiemems frorn P. To express the zern pattern. we iac!ex
. 7l} X {l. . . /1 }. f • .J. k. i E
the elements of the matrix }' Ep by !Jr,s = lr1,n:k.li for r. SE { l.
{ l. .. n}. The zero paLtern is coverec! by the following equalities:
Yrs = O for r = ('i.

2

SDP Relaxations of QAP

ln order to derive a SDP relaxation of QAP we rewrite the objecLive function in the following
way
trace (AXB + C)XT = (x. vec(AX B +C:))= xT(B ® A)x + xTc.
where x

= vec(X) and c = vec(C). Therefore QAP becornes
rnin{x 7 (B ® A)x + x 7 c: x = vec(X), X E II}.

Since c7 x = trace(Diag(c)(xx 7 )), for X
rnin{trace(B ® A

E

(2)

II, the equivalent forrnulation to (2) is

+ Diag(c))xx7

:

x = vec(X). X ETI}.

(3)

:

Note that the relaxation is formulatec! in the space of symmetric rnatrices of order (n - 1) 2 +
l. Relaxation QAPR 2 contains O(n 3 ) equations (n 3 - 11 2 to be precise) Model QAPR 2 is
introducec! in [10] as the Gangster m()del. The results in [8] show that QAP R2 provides very
tight approxirnations of OPTQAP· Because of the large nurnber of the constraints in QAPn 2
the interior--point rnethod is not appropriate for solving this relaxation. anc! therefore Rendl
and Sotirov [9] salve it hy the Bundle Method.
The relaxation QAP R2 can be further tightened b:v adding nonnegativity constraints
(WRW 7 )rs

In order to obtain tractable relaxations for QAP which will give good lower bounds for QAP, we
need to approxirnate the set P by larger sets containing P. We first irnpose a sernidefiniteness
constraint on elements Y EP, Le., Y - diag(Y)diag(Yf '.'.:: O. This condition is well known to
be equivalent to the convex constraint
1

(

y ~

~T )

'.'.:: O, 17 = diag(Y).

140

.:::O:

O,

'vr, s= 1, . .. n 2 •

We collect the inequalities (7) which are not yet covered by G(W RW7
N(W RW7 ) 2: O and arrive at the final rela,"'{ation, see [8, 10]

x = vec(X),.X E II}.

(4)

lf,)

We coUect all these eqnaiities iu the constraint G(YJ = O anc! arrive to the following relaxation,
see l8]
(QAPR2 J mi11{Lrace(B ® A 1 Diag(<JJY) G(Y; =\l.} EP}

Thus, the feasible set of the last forrnulation of QAP is

P := conv{xx 7

s=(i.k). 1Jr r=!j.i)s=(ki1jcf.k.

(QAPR 3 )

min{trace(B ® A

+ Diag(c))Y): G(Y)

(7)
)

= O in the constraint

= O, N(Y) 2: O, Y EP}

The resulting SDP has O(n 4 ) sign constraints and O(n 3 ) equality constraints. The relaxation
QAP Rs can not be solved straightforward by interior-point rnethods for interesting instances
(n .:::O: 15). In [8] the QAPR 3 relaxation is solved successfully (up ton= 32) by the Bundle
Method.

141

r
3

QAP Models Based on Copositive Relaxation

In [3j Anstreicher and Wolkowicz show I haL the quadratically ccmstrained quaclraLic programs
1
where Lhe quadratic constraints rnrrespond to Lhe malrix orthogonality co11dition X X = f
can equivalently be expressed I hrough semidefiuite programming. \\'e follow their i<lea arnl
find a uew Lagraugiau relaxation of the QAP. L'sing the following dmracterizatiun of the set
of permutation matrices 11 = {X 2_ O X I X = 1} we first rewrite the QAP as
min{ lracef AX BT

xr.xrx=xxr=r. x>o}.

+

Note that we ackled the (redumlant) coastraint XXT = l.
ort hogo nali 1\ co ast rai n ts ancl wri Le x = \·ec( X J. c = ved C).
OPT(,!\P

llllll

x~o

Lrace(AXD i CJX 1 t max (S.l

Ju lhe seqnel we dualize the

wit h some good l rac1 ahle consnaiut. t lieu we may ge: a good lower ur 11µper bound ;1Jr the
OPTQ.\P· H.ecent]\ Pari11u lias suggest.ed in :7] a hi.erardw ()f nmes. wlm h approximatcs the
copositirn nme from the insicle arbitrary close. The firs1 cone rn t !tis hierarch.1 , we will respect
t he usual aotat ion arnl denot,e it as Ko) is t he sum ,Vn +- s,; . and Lhe (r + i )-th cone (denotecl
is a sum
J!I:;
by K·,.) is the cone of those malrices }. for which the polynomial (~iJ Y11
of squares. The hierarchy of dna! cones approximates tlie nme C~ from t he outside. \Ne will
focus on these cones, For the first memher we have a cone of symmetric doubly nonnegative
rnatrices K0 = .1Vn ("S,;. while the K:f is already y_uite complicated and it 's description requires
11 matrices with II posilive semidefinite consnaints. see 7~
Om next model wiil he t herefore•

x7x7

mm

1
xxT\ t (Tl-X X1

(QAPT<w!

x 1(1J ® A

!.

+ Diag(c)

A + Diag(r

}'
~1

S,TESn.

+ trace(TJ

/jj 0

s. t

E

xf)"

i.}"

A~2 , S~ 2

I

yu

< ; < .J <

yiJ

r5; 1 •

L 1< i <J <

ri

1® S

T ® l)x

Y' 1 '.

max minLrace(S) i trace(T)+xT(D®A i Diag(c)-1008

T0f)x

\\'e have to emphasize that this is alread~ expensive model since the rnnstraim} E .\·11 2 impEcs
O(n 4 ) linear ineqnalities. We can gel a cheaper model hv demanding ouly } E S„2 but 1his

miu max trace(8)
Y:::oS.TESn
S,TESn

+

x:::o

max trace(S)

+ trace(T)

s. t. S, TE Sn

108

B@A+Diag(c)
min 1B ® A

T

Diag(c).

T®lECn2

n

lea<ls to a poor lower bou11d. whid1 is oul:, slightly be1 ter thau Hoffman- \,\'1elan<lt lmvur bouud
A good compromise seerns Lo be the Gangster constraim. see the zero pattern in ( 6 j 11 we
combine the sign constraim Y 2 O with second and the thir<l coustrai11L from QAPI<o• we get
1 hat the offdiaguual elemeuts in diago1tai hlocks an<l t he diagonal elemcnts in thc off<l,ag()ual
blocks of Y must be zero. This is exactly the Gaugster ,;oustraint Our fina! relaxatiou 1s

s. L. y E C~,

min

L1ii=l

s. t.

(QAPw)

(B ® A

Y

G(Y)

(QAPc:11-1 )

=/ Y,~

11. Y'';
ln the last formulation Y 11 is (1,J)th block of Y, i. e. }''J = [l~qj, p = (i l;n +l. ... m. q =
. )11 and ,i,J is the Kronecker delta. 'The last equalit.y above follows from strici
1 )11 + l.
U
feasibility of the last but one problem. i.e .. for T = S= -0:J the matrix 13 ® A + Diag( c) 1 ®
8 T@ l is positivc definite for (t sufficieut lv large. hence in the imerior of Cn" arn! therefore
strictly feasible.
There may exist arbiLrary large <luality gap between OPT(,!AP and solmion uf QAP u; Let
B = a(J - 1). A = f3J and C = O. where a. (3 > O. For every permutatiou matrix X is
trace(AXBT + C)XT = o.{3n(n - 1). hence OPTQAP = e.f3n(n - !). In Lhe relaxed problem
matrix B ® A and every feasible matrix Y have only nonnegative entries, hence OPTi,o 2:. O.
If we take Y = l/nin2, then Y is feasible for the relaxed problem and (D ® A. Y) = O. This
means that Y is optimal and OPTw = O. The <luality gap is o.f3n(n -1) and rnay be arbitrary
large.
In the sequel we add to the original problem redundant constraints trace(J X D, 1 XT) = l.
where D;j = ½(E;1 + E 1;). 1 Si <.} S n. By adding new constraints to the (QAPLn) we
obtain the following copositive program
min
s. t.

(QAP ep)

(B ® A + Diag(c), Y)
Y E C~2
I
I:;yii
8;1, 1 Si S j S n
trace(Yii)
1, lSi<jSn.
trace(JYii)

The new relaxation is tight. In [6] Povh shows that OPTQAP = OPTcp, This result leads to
severa! new approximation models for the QAP, If we replace the intractable constraint Y E C~2

142

ri

yiJ,

+ Diag(<'). }'
E

S~2

O
1
l. l -S 1 $ n
l. 1 S i < j S n.

In the following section we compare copositive relaxations wit.h SDP models.

4

N umerical results

In tliis section we compare the lower boumls for the pre\iousi.v derived relaxalions. Table l
reads as follows. The first column gives the pmblem instances and their sizes. In the second
colurnns we provide the optimal value for each instance, and the remaining columns are bounds
for the relaxations deri ved in this paper. The QAP R 2 and QAP R 3 bounds are computed by the
Bundle Method. see [9]. The QAPcM bounds for all listed instances. except Nug20 and Nug21.
we compute hy 8DPT3 sol ver (url: http•/ /www. math. nus. edu. sgrmattohkc/sdpt3. htmlJ.
Diffirulties arising solving problems with IPMs for n > 18 are rl.ue to the hardware limitations.
For computing approximate solutions for Nug20 and Nug21 we use a Boundary Point Method
(BPM) introduced in [5], which seems to be an eflicient method for solving large problems (see
Table 1). Our preliminary results in computing QAPKo• bounds with IPM are promising, i,e,,
the obtained bound for Nugl2 is 568, which is better bound than any computed SDP bound
for the same problem.

5

Conclusions

In this paper we recall SDP relaxations for QAP and derive new relaxations based on copositive
programming for QAP. Both approaches turn out to be strong tools for developing approximate

143

--

rf

i O PT
--+--~ qf\PR

problem

l\Jng12 ' 57P.
1014
Nngl4
l'i0
l\Jugl."i
l6
Nng16a
1240
:S:ugl6lJ
1930
l\Jugl8
2570
Nug20
24:38
:S:ng2

Talile

2

QAPR3

QAP c;"'.1

557
992
1 22
1570
l 188
1852
2451
2:52:3

529
959
1070
1528
11'.-38
1801
2380

528
958
069
1526

1136
1798
2380
2244

l3ouwis for SDP arnl

2244
relax.ations

co11s1_ raints and therefore
models for Lhe QAP. The resulLed relaxaLions conLain up
ll'Ms are not appropriate solving meihods for larger instances. Therefore. for SDP aud larger
coposiLive relaxations (n > 18) we use the Bundle Method ami the Bounc!arv Poiat Met.hod
that has been recently introduced b, Dnkanm·ic et aL [-'i]. The preliminary results show that
l he 8PM is a promising method for soh-ing ('Oposilive relaxations. Our results also show that
QAP r<o• provides strong lower bounds for QAP. Thus. our fntnre work will be so!ving QAP KO·
relaxation for large instances with the new Boundary Point Method.

S@R 'OS
Nova Gorica, SLOVENIA
September 28 · 30, 200S

Section

References
Anstreicher. K .. H.ecent Advances in lhe Solution
Program. jj_ 97:27 42. 2003.

Quadratic Assignmeut Problems. lvlath.

Solving Large Qnadratic AssigH!'. Linderot.h.
i:2] Anstreicher K Hrixius. N, Goux.
ment Problems on Computational Grids. Math Program. B. 91:563 -588. 2002.

i3] Anstreicher. K. Wolkowicz. H.

On Lagrangian Relaxation of Quadratic Matrix Constraints, SIAM J. Matrix Anai. Appl.. 22(1)41-55. 2000.

[4'

The 8th lnternational Symposium on
Operational Rmarch In Slovenio

Solving Lift-and-Project Relaxations of Binary lnteger
Burer. S Vandenbussche. D.
Programs. Technical report. June 2004.

,5] Dukanovic, L, Povh, .J., IlendL F .. Wiegele. A .. A Boundary Point Method for Semidefinite
Programs. University of Klagenfurt, Austria, Working paper, 2005.

l6] .J. Povh: Application of semidefinite and copositive programming in combinatorial optimization, PhD Thesis, University of Ljubljana, Faculty of mathematics and physics, Ljubljana,
2005 (in preparation).

[7] Parrilo, P. A.: Structured Semidefinite Programs and Semialgebraic Geometry Methods in
Ilobustness and Optimization, Dissertation, California Institute of Technology. 2000,

[8] Rendl, F., Sotirov, R: Bounds for the Quadratic Assignment Problem Using the Bundle
Met.hod, Technical report, University of Klagenfurt, Austria, 2003.

[9] Rendl, F,, Sotirov, R: Representations of Matrices with Sum of Row and/or Columns
Equal to One, Technical report, The University of Melbourne, Australia, 2005.

[10] Zhao, Q., Karisch, S, E., Rendl, F., and Wolkowicz, H.: Semidefinite Programming Relaxations for the Quadratic Assignment Problem. J, Comb, Optim., 2:71-109, 1998,

144

CROSS-TAB LATlON ANALYSES OF THE Sl RVEY RESEARCH
THE VIORAL VAU ES
Josip Arneri;.:. Elza .lunin
Spl1L Factdt, of Lcu110111ics
\lalicle hnatske 31
21000 Split Croat1a
Jarneric@efs1.hr, elza@eist hr

l nJ\'ersit\

Abstract

• gumg scientilic resean.:h on contemporarv family and moral
the e.\.tensi\·e
l'his paper is a pan
results the aurliurs hm e selected tlwse pertarn111g tu the slale of
, alues. All!\lilg a \\ 1de spectrnm
111g in the Split-Oaimatian
\ ie,, ,.1f indi, iduais 18 years and \,lder
!ll\1ra! \ a!ues from tile
lhis paper pa)S particuiar a11em1011 tuwards numewus cross-tabulat1011 analyses \,f tile
anal)sis was to detern1ine which
sune\ research hased un sample uf 1000 units. he purpc1 se
11fmorai \ai11es
and \\hich ones a slight mfluence
fac:turs hale a

Keywords sune\ research

moral 1alues. Ut>SS•tahulau,·n anahsis hribe. euthanas1a

L lNTRODlCTION
Motirnted h\ pre, ious research 011 values in the world and m Croatia at lhe beginning of the
third millennium. the authors have decided to carry out this study 011 the state of family and
going
values in the Split-Dalmatian County. On ,me hand. this paper is a part of the
extensi\e research. and rm the other hand is a continuation of the paper "Fairniy Values
Today - (ase Study of Split-Dalmatian Cnunty" with wh1ch it forms a coherent unity. The
Operation
paper as puhiished in "Proceedings of the 7th Internatitll1al Symposiurn
Research · and the cross-tahuiabm anal\ses \,·ere annuunced in the conclusion ofthe paper
Based on the imp01ian1 <lata nn the srnte of the contemporary family and moral \alues, the
scientific research, carried 0ut in the area of
authors ha\ e chosen the parts nf the
A sample 0f 000 interviewees 18 years and 0l der Vv ith currenl
Split-Daimatian
residence in the Split-Dalmatian County meets ali the criteria of the contemporan statist1cs
sex. age,
and makes a representati,e and random sample. Consequemly. accord,;ig
employment, profession, qualifications and marital status the structure of the mterviewees
gives a precise picture of the structure of the popula1ion of the Split-Dalmatian County
according to 200 census and ind1cates no significant statistical deviations
The given answers. which could at firsl be defined as easy-going, do not remain on the
leve! of description of forms of behaviour. but help us reveal the essence of interpersonal
relations within a maITiage and a family as well as relations of an individual to different
spheres ofhis social environment.

2. AN ABSTRACTION ON SOME MORAL VALDES IN SPLIT-DALMATIAN
COUNTY
For the purpose of this paper parts of the analysis of attitudes towards certain moral values as
a part of the complex output of the scientific research have been prepared in the adequate
form. Statistical analyses have been catTied out on the influence of sex and age of the
interviewees, their upbringing, place of the childhood residence until the age of 15, marital

147

r
:a; SlalllS
and hrihe.
gP l\l the \e:, heart
the autlwrs. these au1tudes are S\mpt,,mat1c hecause
the
the attitude
absuactl()n /\)cuses
of tile alliLudes 1<,warJs te. herel(,re.
mten 1ewees tuwards abortion. euthanasia. homosexuality and towards gn ing and taking a
the ,111e \\ hich
abstracl
the autlwrs anah se the atlituJe \\lm;h 1s
hribe.
the human bemg as the persun. l hese Jecisiuns ha\ e
dec1swns
manages
recognisable forms and far-reaching influence on the nmternporar) intimate.
religious. husmess and social fe and therefore create an atmosphere of the "cul ture f life"
ihis paper has been necessary to e:,;tract e,en a srnaller outpul
the limiteJ s1ze
Due
the cross-tahulati(m anah sis
h1ch wd ! be presented. lierefore. the analysis focuses
the attitucles of the inten iewees towards hribe anJ euthanasia as lU\\arJs the su ca!led
representati\es ofthe attitudes towards "husiness moral" and "life ethics".

sraius.

ahle 2.

o,

~--

NEVER

28.6%

4

11

O¾

63,6%

36 4%

100.CJ%

1.1%

1,1%

11¾

-·~1451
58 7%

102
413%

1

Asymp, S1g

100.0%

.O¾ 1 23.3% 1 27.3%

24,7%

~

UNDER SPECIAL

C:IRCUMSTANCES, lF I HAD

21
5%

1

IF THAT WOULO
CONTRIBUTE TO SOLVING

VITAL lSSUE {FLAT, JOB)
ALWAYS !F THAT S THE
SAFEST, EAS!EST WAY OF
ACHlEVlNG A AlM
TOTAL

281 [
63 3¾

1

50.0% 1 45.1%

1

SERIOUS HEAf. TH PROBLEM

166
2 1
69.5%
.8%
50,0¾

444

161

Pearson Chi-Square

21,967"!

8 '

.005

Likelihood Rabo

22.589 ;

8

!

004

N of /al1d Gases

:uu

8

'6.3% 1 100.0%

1

"Ii

29.7%

239

24

35

40.7%

59.3%

,O¾

3.9%

9.4%

4

623

!

444

9%

'%

5%

100.0%

49.5%

42.6%

44.4%

.4%

62.3%

100.0¾

100,0¼

50.2%

46.9%

23 8% [

26,6%
28.6%
34
"W,.iY:,IFTH;,~T~S~T~H~E---+-----,--

28.8%

-

239

5%

:_g%

100,0%

20.6%

~7 1%

14.9%

23.9%

~~T- cc-t----,-+---c-+- 59

3%

6%

<; "%

4%

:0, 1%

4,0%

4 0%

2.9%

4.3%

338

226

277

105

100.0%

"00,0%

100,0¾

100.0%

;oo.oo;.,

dl

20

A.symp. Sig
(2-sidedJ
,007

20

001

0%
.O¾

Ch,-Square

Tests

1/alue
Pear5on Chi-Squarn
Likehhood Ralio
.inear-by-Linear

1N of

✓ahd

000

14.205

-\ssocialIon

Cases

000

cells :36 %,' haveexpecledcountlessthan: The
min1mumexpecledcount1s.OB

00,0%

!

5,9%

~1000

ln this case general conclusion is the same as in
pre\wus cruss-analysis. As illustratiun of
significant influence ,Jf tlie age tu the
attitude lo\\'ards hribe can be underlined that 1 5
rhe age
times i 0.1%1 more ofthe mterviewees
to
,,ppcsed
as
alwa1s
bribe
gi\e
Ji(-;() v.ould
1hose who are in the age 51-60 ,2.9%J

RESIDENCE OF THE CHILDH00D UNTIL THE AG~ OF 15
1

ANSWERED

✓ ILLAGE

~owN WITH LESS
"HAN 10000
INHABITANTS

0%

455%

,O¾

--

·owNWITH

OVER 30.000
INHABITANTS

TOTAL

0%

9.'%

45.5%

100.0%

1.0%

,0%

.9%

',9¾

1?7

"'

26

63

247

0%

51.\%

-6%

10.5%

25,5%

1000%

.O¾

26.5%

20.7%

24.1%

..

o

~

236

'JNDER SPECIAL
JIRCLJr,15- :. NCES, IF ·W:
SERIOUS HEAL -H PROBLEM

JWNwr··

JP "O 30 000
INHABl"~ANTS

NEVER

1

51 I

2%

532%

2%

5%

100,0¾

49.3%

42.0%

47.2%

1

1.1%

93

24.7%
444

20.9%

100,0%

24.0%

1

1

01

IF THA T WOULD
CONTR!BUTE TO SOLVING
VITAL ISSUE (FLAT JOB)

-

44.4%

1

211

81

239

18.4%

8.8%

33.9%

100.0%

,O¾ 1

19.4%

29.3%

19.4% 1

30.9%

23.9%

1

100.0%

1
i

'2

91

1

203%

3.8% 1

8.0%

8.3%

150 1
15.0%

1081
10,8%

30.5%
479

1%

1

18

,0% 1

1000

1

35.5%

93 1
38.9%

.0%

5.9%

44

1

.0% 1

o

ALWAYS IF THAT S THE
SAFEST, EASIEST WA"' OF
ACHIEVING A A!M

59

100,0%

'00.0%

residence u,1 ihe chddhoocl unli! 1he uge of 15 un attztude r,n,.nds /wibe

TOTAL

3731

24.7%

20

2%

100.0%
1

36.2% '

52

37 7%

8%

1000%

26.6% 1 19.0% 1 23.9%

01
,0''/o

OD 0%

29.5%

!SSUE ; LA - • JB;

cells (40.J%, 1ave expected coL · Iess Ihan 5. TI-Je m1rnmum
expected count is .04

44.4%

43.2%

i

·-,9%

25.1%
'24

36.7% i

42.9%

(2-sidedl

df

247

·7 '

'""TH;.Twr.1" n

Chi-Square Tests

Value

1.1%

3'

1

%

YOU WOULO OFFER A BRIBE
HAVEN T ANSWERED

247

100.0%

St.Rl.'JS ·-IE-< ., JROBL·~M

-

TOTAL

MALE

1%
2.1%

1.0%
12.6% :

"-IAVEN

FEMALE
71

1.1%, 1

34.4%

1

.O%
.O%

.4%

-----+-----,~·

TOTAL

00 0%,

HAVEN T

ANSWER

"%

:,5%

;.•,::~:s,

Tahle 3.

HAVEN T ANSWERED

455%'

7 ·J~ND~E~R~S7PE=C~"~
..

6~ OR MORE

L 3%

)%

8%

Table l. l,1/luence oj the sex ofinterviewees on their af{i!udes towarcls bribe

YOU WOULD OFFER A BRIBE

5'. -60

50

4~

40

.O%

TOT~'

has been necessar;, to
,\mong many attitudes in the context of the worlJ of husiness
choose une wh1ch will act as a representati\e uf ali uthers. he auth,1rs ha\e clwsen the
all other
attitude ,1f the interviewee towards bribe belie, ing that
attitudes within the context ufhusiness nwral
at tbe anal\ sis
Hence. a 1 en Jelailed c:russ-lahulat1on anah sis has been carneJ
tm\ards
mlerviewee
tile
of
attitude
the
influence
of the factors wluch statisticallv s1gnifica11l
iewees.
ten
in
age
anJ
sex
of
analysis
bribe. We will give the results of the crosstabulation
employment
residence.
current
5.
of
age
place of childhood residence until the
qualifications, financial status. marital status anJ their attitudes towarJs hribe.

SEX

' 31

30

85

,SI· S
S1>: s:,IM
HE/INC

2.1. CROSST ABLLA TIOJ\/ Al\/ALYSES OF THE SlRVE\ RESEARCH 01\/ THE
BRIBE

'

•iAJEN /-,;·,JSWERED :

YOU WOULD OFFER A BRIBE
HA VEN T t.,NSWEREO

:IRC JMS

, h1•·he

!i ;_le,

47.9%
100.0%

l
i

1

100.0%

15.J%

1

100.0% 1

20

59

33,9%

\00.01/o

7,6¾

5.9%

262

1000

26.2%

100,0%

100.0%

100.0%

37 3% 1100.0%
100,0%

100.0%

As it can be seen in table 1. sex in general bas statistically significant influence on attitude
towards bribe ( Sig.x 2 = 0,05% ). For example 19% males would offer a bribe under special
circumstances (to solve a vita! issue like housing, job, education ... ), but even 26,6% females
would do the same under the above mentioned conditions. Just opposite, while 9,4% males
would offer a bribe always, if that is the safest and easiest way of achieving goal, only 3 ,9%
females would do the same.

148

Chi-Squ.:ire Tests

Asymp S19
Pe.:irsonChi-Square
Ukehhood Ra~o
Unear-by-Unear

Assodation

N ofVahd Cases

Value
33.257a

35257

12-slded)

df

16

,007

16

.004

8.433
1000

,004

In comparison to table 3. where residence of the
childhood until the age of 15 statistically significanl
influences on attitudes towards bribe , current
residence place in table 4. has statistically influence
at 6% leve! of significance.

a. 8 cens (32.0%) ha~e opected count less tllan 5. The minimum
e)pecledcounUs,01.

149

r
11rd1 hrihe

Tahle 4. /11/luence oj cw rent 1·e~1de11ce place 011 a/1//ucle

-HAt-. 10.000

JP ""O 30,000

INHAB!TANTS

INHABITANTS

-owNWITH
OVER 30000
INHABITANTS

7OTAL

1%

545%

100.0¾

7%

5%

11%

-;"QWNWITH

TOWN WITH LESS

HAVEN

YOU WOULD DFFER A BRIBE
HAVEN T ANSWEREO

ANSWERED

VILLAGE
273%

·.o%

6%

oa+

NEVER- - 2/,

32~

r--,,-7 1-

42.9%
7U~N~DE~R~S~P~EC~IA~L-------f--- --1
CIRCUr.1STANCES, IF HAD
SERIOUS HEAL TH PROBLEM

28.1%

CONTRIBUTE

2'~,

30.9%

14.3%

46.4%

8%

--:+

SAFEST, EASIES- WAY OF
ACHIEVING
AIM

23.3%

24.7%

67

165

444

372¾

100.0%

48.9%

40.5%

444%

27

120

31'
:o¼

!

'00.0%

50.2¾

'157%

20,1%

20.0%
22.0¾ 1

7¾

29.5%

23.9%

16

21

59

"%

356%

100,0¼

5.8%

5.2%

5.9%

( Sig.z

on
m,is hribe

attituJes

GOOD

o
0%

27,3%

36.4%

3%

:%

1,0%

3.4%

100

10"

24

O¾

12,5%

223%

5%

245%

%

4%

43,7%

37.5%

45.3%

44.2%

3%

26 1%

33

-9

444

4¾

0%

0001/o

37.5%

39.'%

44.4%

S8%

7%

\00,0¼

26.0%

194+

24

18

40.8%

IFTP!.,.WOl'~n
SOL /lNG

- ~L ISSUE (fl;,

JOB]

/<JAV"' IF TH/'-T ': Tl-[E
sr-cc=.;," tASIE·~ •
JF
~CHIE'/ING

AJM

1

46.5%

00,0°/2

000%

SERIOlJS HEI-- -H PROBLEM

cor,. "'RIBL -E -o

0%

.o~-~;

go~

2'>

'JNDER SPECIAI
CIRC Jf,1S-;, I.JCE,, IF HAD

,ow

SUFFICIEN-,

24.-%

·20

84

2%

50.2%

35,·%

,0%

18.9%

273%

4%

23.9%

7%

02%

3%

322%

9%

8%

100.0%

12.5%

11,3¾

'1.0%

4.9%

8.0%

"7.4%

5.9%

~

O¾

239

23.9%

,,-+-

6

'7

59

295

54

~01

·ooo

53

SJ9

389

88

23

1000

7¾

295¾

15.4%

137¾

40.7¼

100,0½

8¼

'i3%

439¼

3891/,

8%

_3%

·oo.03/o

100,0%

100.0%

100.0%

100,0%

100,0%

100,0¾

100.0%

·100.0¾

100.0%

100.0¾

100.0%

100.0%

100,0%

10.4%

Table 8.

emplnymenl on allitude tmvL!rds hrihe

I

HAVEN

PERMANENT

ANSWERED

I

HIPLOYEE

UNEMPLOYED

UNEPLOYED WHO

WHD RECENES

OOES NOT
RECEIVES

S/Li(US

YOU WOULD OFFER A BRIBE

TEMPORAR't

UNEPLOYMEW

UNEMPLOY/,1EW

EMPLOYEE

BENEFr:

BENEFI";

27,3%

.O¾

·:.2¾

36.4%

3.1%

7%

.0%

24%

',5%

101

IJ

20

75

3%

%

~

40,91/o
22.2%:

21.9%

20.6%

---jf-----+--~2~23+---2-,- ---HAD

HA VEN
RETI RED

;o: AL

~

1%

100.0%

"1%

i 1%

T

MARRIAGE STATUS

304%

12.6%

100.0%

~~:~

33.3%

24.7%

'¾

22.5%

:.7%

1000½

46.2%

44.4%

'JNDER SPE('I!'
::IRCJMST ;NCES, ~
SERL..:JS HE/· _ .. H PROBLEM

444

100

ANSWERELI ' SINGLE

ANSWEREO

247

27.4%

S,3%;

50.2%1

·---,
'

MARRIED ' Dr✓ORCEU W1UUW ER
o
?

;U~AL

J¾

8.7%

1QC 0%

~
6

'5,4%

1.1%

55,5%

4%

4.0%

27.1%

23.1%

28.6%

27,0%

'34

281

.2%

3C2%

63,3%

:: J%

3%

\00.0%

33.3%

38.6%

47.5%

42.9%

51.4%

44.4%

83

",t7

.4%

34,7%

6".5°/o

21%

"3%

100.0%

33.3%

23.9%

24.8%

23.8%

8.1%

23.9%

0%

36.4%

455%

,0%

"2%

.8%

94

137

0%

1 38.'%

.0%

·o

247
1

100 0%

24.7%
444

SERIOUS HEAL -~- PROBLEM

IFTHATWOULD
CONTRIBUTE TO SOLVING

44.4%

49.1%

43.8%

36,5%;

42.9%

----+---1~0~7-l---~15+------c27o+-----,,76+--~--239
44 8% 1

5%

VITAL ISSUE (FL• - JOB)

4%

8%

3'/.,
16,1%

23.8%

23.8%

27 7%

9%

),2¾

32.2½

;_ '¾

100.0%

4,4%

i1.l%

: 1%

.S.9%

3,2%

5,9%

20

ALWAYS lFTHAT S THE
SAFEST, EASIESTWAY OF
ACHIEVING AA//,1

TOTAL

J2

J2%

1

19

454

63

45.4%

6.3¾

1

23.9% j

31

,\'_w,:...vs

59

84

274

93

1000

4%

274%

.;!,J¾

100.0¼

100.0¾

100.0%

100.0%

~

CONTRIBL TE c.. SOL /ING
-~_ ISSLJEII=~:,~ ,)8)

100.0%

33 9%

23.6%

18.8%

6 3¼

TQT~L

1

100.0%,

100.0%

100.0%

lfTl-'!TST!.JE

Sf--.FES- 8,SIES"' WA- OF
;\!M
:, ::HIEVJNC\

100,0%

32 i

, 7% :

1rmr GRADUATED
YOU WOULO OFFER A BR\BE

At::~~~;D

I

T~o;~~iiL

I

FRo;~;~~tRY

r

I

QUALJFlCATIOIJS
GRAOUATED

I

FRm,,

GRADUATED

I FRO~~~~~tRY i

vosc::~~t

GRAOUATED
FRO!.,

!

GRADUATED
FROM'

':s::;:

1 ODOE~T:E~s

TOTN_

HAVEtl T Af1SWEREO

4%

100,0¼

,O'I,
IF THAT WOULD
cmHR!BUTE TO SOLVING
VfTAl ISSUE {FLAT, JOB)
AlWAYS !F THAT S THE
SAFEST, EASIEST WAY OF
ACHIEV\NG AAlM

.O¼
26

utmERSPECIAL
CIRCUMSTANCES, IF I HAD
SERIOUS HEALTH PROBLEM

5%

11%

59'/-.,

66.7¼

38.5¼

42.6¼

o·;,
.0'/2

.O¼

6¾

.O¼

15.4%

53.8¼

256
581%

!

62
14.0½

i

135%

44,93/,

49.2'/2

38.2¼

157

25

42

3)%

657¾

105%

176'/.

13,1¾

27.4°/2

19,8'/2

26.8½

36

9

•

i

7'/,

i

50.0'lo

01.,1

:1

444

01/, 11000¼
,O¼

44.4'/2

3
O'~ 110/0 :,

,O¼

.O¼

23.9%
100.0½

5.9¾

59

17%

17'/,

5!!½

51%

610¾

153%

85½

0%

O½

33.3%

7.7'/2

6.6"/2

5,8¼

6.3¾

7.1%

3.2¼

,O¼

.O¼

13½

61½

52½

574¼

126¾

15_7¾

6%

7'/2

1000¾

100.0¼

100,0¾

100,0¼

100.0¼

100.0¼

100,0¼

100.0½

100,0¼

100.0%

100.0¾

13

TOTAL
1¾

100,0¼

28
6)'1., 1

61

150

52

574

126

2

Sigx = 2,2% ).
ur example, thosc who
evaluate their financial
status as modest but
sufficient would offer a
hribe 1 " times less
!4.9%1
than
the
imerviewees
whose
financwl s1.m.1s 1s \ ery
luw ( 1'7,4%)
!

59

22

3-_3%

7%

3. 7%

4.8%

8,1%

592

21

37

100.0%

i

9.2% !

3

1

347

3% i

34 ..,%

59 2% i

21%

27%

100.0%

100.0%

100.0% '

100,0%

100,0%

100.0%

5.9%

!

1000

:VIarriage status ,s among
1he factors of strong
influence
vn
attitude
towards bn be
t S1g.x = lJ,07%)
ln
c1>mpansun
to
widow(erl who would
,1ffer bribe to salve vita!
problems in 8. l % cases,
married people would do
the same generally in 3
tirnes
more
cases
(24,8%).

2.2. CROSSTABULATION ANALYSES OF THE SURVEY RESEARCH ON THE
EUTHANASIA

1

~~~~~~; i F:~~~TUEL~

SESCCO~i;~y

239

JJ.J¾

100.0%

Table 6. Influence of quali(ica/inns un atlirude wwards hrihe

54,2%

hnanc1al
status
also
significantly influences
clll att1tude towards bribe

urritude 1owurds hrihe

HAVEN T

AN

)\%

:B¾

CIRCUl,1STANCES, IF

JER\ GOOD

4.4%

AN

-UNDER SPECIAL

ANSWER

\JE 1JER

Err1PLOYMENT

YOlJ WOULO OFFER A BRISE
HAVEN T ANSWER

MODES- BlJ"

-tAVEN

ucH~~~,\\~,~·v:sE~:OBRIBE

14.3%

TOTAL

Table 5. Influence

24.8%

15.1%:

167¾

24.7%,

28.6%

ALWAYS IF THA.7' S THE

24~
100.0%

the cross-tabulations
represented m table 5.
there are less signi ficant
mfluences
( Sig.i(, = 8.8%) than in
6 ..
where
table
qualifications
have
significan1
infl uences

--je----+----<

59

-o SOL JING

95
38.5¾

•

IFTHATWOULD
VITAL ISSUE (FLAT JOB)

34
8'1/,

130¾

336%

.
l

ab!e "

CURRENT RESIDENCE PLACE

1000

157

l

It has been difficult to choose the attitude which w11l in a ce1iain way urnte the overall
attitude of the interviewee towards "culture of life'· in general. Nevertheiess, the authors
have chosen the attitude towards euthanasia, because it reflects the respect to one's own life
as well as to the life of his beloved which has been primary given to us without our
contribution or our consent. Recently, we ali witnessed the reports in the media on the "top
matters" which refened to the situations when the closest relatives, or husbands and wives
needed to make an irrevocable decision not on their own lives but on lives of the persons
under their tutorage. Therefore, the authors have chosen euthanasia as a particular
representative of the complex of attitudes towards life and they have carried out the cross
tabulation analyses of the same size and structure as in the case of the attitudes towards
bribe.

151

COll\/TEGRATIOl\/ Al\/D Pl RCHASil\/G POWER PARITY IN
SELECTED l\iEW MEl\1BERS (W EL

Table 9 Jntiuenc·e oj re.1·1c/eJi<'e u; ihe, h1/dhood un 1he
RESIDENCE OF THE CHILDHOOD UNTIL THE AGE OF 15
TOWN WITH LESS
-HA!', 10.000

HAVEi\ -

EUTHANASIA IS

ANSWERED

VILLAGE

0%

- ·%

HAVEN T ANSWERED

INHABITANTS

TOWN WITH

-JWN WI- '"

UP T() 30 000
INHABITANTS

JVER 30.000
INHABITANTS

~OTAL

'4 3%

23.8%

100.0%

2,8¾

1.9%

2.1%

- 6%

00.0%

28.6%

42.6%

107

279

Daija Boršič. PhD
Lniversity ofMaribor. Slovenia
E-mail dai:Ja.bors1c(al,wli-mb.s1

12
48%

MURDER

426

46

0%

596%

.O¾

53.0%

ALLOWED IN SPECIAL
CASES, E G WHEN A

:8%

12.0%

100

01
0%

358%

0%

20.9%

:5.5%,

'1.3%

38.4%

24,1%

40.8%

27.9%

Abstract
The paper deals \\ 1th emp 1ncal testmg of purchasmg power pant, (PPP) theory Using Johansen
comtegralion test. the most re!axed PPP \ ersion 1s m 1·est1gated for Slovema, Czech Republic,
Austria_ Germam France and Itah in the period of Januarv
Hungar. and Poland m comparison
YY2 to December 200 l The results show no e, 1dence m fa, our of PPP theof\

PATIENT IS KEPT AUVE

----,+

~A~CT~Q~FM~E=RC~Y~T~Q~A---l--- PERSON IN GREA 1
PAlr-, WAITING TC _JIE

30.7%

33

•. O%

27.4%

·oo. •%

34.7%

30,6%

26.6%

27.4%

·so

08

262

1000

9%

0%

8%

262%,

000°/-

100.0%

100,0%

100.0%

100,0%

100.0%

23.6%

100.0%

~T~OT~A,-----------l-------1-~~
.::

100,0%

:_:n,;

52
'.10%

!

Ke, v,ords Johansen comtegration test stationant, pnces. exchange rates

Table 10. !n/!uence ofcurrenl reside11ce place on 1he opm1un 1uwurdJ euf/ii/nu.1,u

INTRODUCTION

CURRENT RESIDENCE PLACE
7AVEN

EUTHANASIA IS

ANSWERED

VILLAGE

-owN WITH LESS
-HAN· v.000

TOWNWITH
JP -o 30,000

TOWNWITI-·

INHABITANTS

INHABITANTS

INHABITANTS

TOTAL
21

28.6%

100.0%

1.5%

2.1%

HAVEN T ANSWERED
O¾

286%

. ,0'/,~,0%
MURDER

--- - - IN
SPECIAL

~

158

9%

371¾

57.1%

ALLOWED

CASES. E G WHEN A
PATIEN- IS '\EP- ALI /E

ACT OF MERCV TO A
PERSON IN GREAT
PAIN, WAI 'INC O JIE
TQTAL

-~

4

23.8%

c,,Q%

3.2%
53

1

124%

53.6% 1
55

34.4%

4%

197%

14.3%

18,6%

~,+

1

5

1

~-

1

2.9%

60

,.+

, JER 30.000

1

-~~-~+-

15,8%

'5%

28.6%

23.4%

15'

426

354%

00.0¾

37.1%

42.6%

'4'

279

52.7%

1000%

36,1%

27.9%

------~-274

52

4'

'%

27,7%

19.0%

15.0%

2 '6%

100.0%

28.6%

25.8%

33.8%

29.9%

25.3%

295

154

27.4%
'000

7¼

29 5%

154%

100,0%

100.0%

"00,0%

1

--~

407

137%

40,7%

100.0%

100.0%

100.0%

100.0%

3. FIN AL CONSIDERA TIONS

the
an ahstraction on the parts of extensive on-going research related
This paper is
defined frarne (1f the paper
cross-tabulmion anaiyses tuwards a few moral , alues.
limited presentation and analysis of the results, So, in the case of euthanasia only attitudes of
restdence ,,f the childhood until the age of 15and current residence place of interviewees
have been presented, They are the only factors of statistically significant influence. because
all others cross-tabulation analyses resulted at significant level more than 10%. Contrarv to
bribe which is under strong influence of numerous factors, the opinion on euthanasia is much
more stable and only surroundings during upbringing as well as current residence
environment could significantly destabilize "culture of life", This is only one of numerous
considerations which originates from the cross-tabulation analyses and the authors hope to
publish ali them in next research paper,

4. REFEREN CES
l. E. Jurun; L. Tomaševic (2003): "Family Values Today - Case Study of Split-Dalmatian
County", Proceedings ofthe 7th International SOR 2003, Podčetrtek, Slovenia
2. H,F, Weisberg; lA. Krosnick; B.D, Bowen: "Survey research, Polling and Data Analysis"
Third Edition, (2005) Ohio State University, Assessment Systems Corporation, USA

152

of purchasing power parity (PPP) is a rather sirnple theory, to which a lot of
The
th
1
was first invented · 16 century
research has been devoted in the last couple of decades
was invented and forgotten for a
in Spanish School of Salarnanca. During its history
couple oftimes This was rnainly due to authors not knowing previous research at that tirne.
th
Thanks to Gustav CasseL a Swedish econornist in the beginning of 20 century. the theory
th
became verv well known and has been a topic of rnany discussions during the 20 century
(Boršič 2004)
The purchasing power parity theory suggests that exchange rate system should provide a
rnechanism, which would enable a basket of goods being purchased in both analysed
countries to cost the same amount of money when recalculated in one currency
of purchasing power parity in Slovenia, Czech Republic and
This paper analyses the
companson with selected members of European Union: Austria, Germany,
Hungary
France and Italy, which are also main EU trading partners ofthe Central European countries
in question The observed period ranges from January 1992 (1993 for Czech Republic) to
December 2000 That is from the beginning of transition till the end of the individual
European currencies and the introduction ofEuro
The general model of testing for purchasing power parity (Cheung and Lai
following

993) is the

where e1 stands for nominal exchange rates, presented as the price of foreign currency in the
units of domestic currency, P are domestic prices and P* are foreign prices. Ali the variables
are in the logarithmic form. In the most restrictive form, there are the following restrictions:
ao = O, a1 = a2 =l. The symmetry restriction applies that a 1 and a 2 are equal, while the

1
Review articles in this field are: Officer (1976), Froot and Rogoff (1995), Rogoff (1996), Saino and Taylor
(2002) and Taylor and Taylor (2004).

153

r
limnation of a
Rogoff 1995)

1

and a 2 bemg equal to one 1s called the prnportionalitv restriction

(5),

The first empirical analysis started offwith the most restrictive version ofthe model (a1 =
= ). that is testing the real exchange rates. In the context of the relative PPP the movements
in exchange rates are expected to compensate for price leve! shifts Thus. real exchange rates
should be cnnstant m·er a long
and their time series should be stationarv Example of
such empirical analysis contain Boršič (2003. 2005), Holmes (2001). Parikh and Wakerly
(2000) and May (1999)
Relaxing the proportionality condition · equation ( ) allows us to test if nominal exchange
rates and relati\'e prices are cointegrated PPP holds the presence of long-run equilibrium
relation is confirmed The test used in this step is usually the Engle-Granger test of
cointegration Tavlor 1988), Kirn 990), Mark 1990) Pufnik (2002) and Boršič (2003
2005) are some of the examples of this approach

The maximum eigen
where 1" is 1-th maximum eigen \alue of matrix !l in equation
value statistic (LRmax) tests Ho number of cointegrating vectors is equal tor and H 1. number
of cointegrating vectors is equal
+ l LRmax can be calculated as follows.

Ll\n..~(i r+l)=-Tlog( -A, ,)=LR,,.(r k)·LR,,.(r+l k)

Results of Johansen c.:omtegration tests

Table

Number of
coint equations

1 Eigen
value

'
Slovenia
!

Czech
Republic

Austria
r=0

rs
'

--------

t-1

+ Bx1 + e1

where IT =

(4).

Matrix TI contains information about long run variation oftime series According to Granger
representation theorem (Engle in Granger 1987, Johansen 1991) matrix IT can be divided to
k x r matrices p in a with rank of r (r :S k-1 ), so that II= p a', if also IT bas reduced rank r <
k. Matrix a contains r linear cointegrating vectors, while matrix p presents adjustment
coefficients of error correction model.
Number of cointegrating vectors are tested by two statistics. Trace statistic (LR1,) tests H 0 :
number of cointegrating vectors is less or equal tor. The trace statistic can be written as:

154

r=0
r=
r=2

,LR1,

l

**42.244
**20.7690 •
2,7287
*21,4751
* 8.0402
2,7287 ,

23,9747
8,9509
0,4842
15,0238
8.4667
0,4842

**48,2695
*19,1047.
*4,5854
**29,1648
*14,5197
*4,5854

**56,5127
*17,9221
(J,8039
**38,5906
*17,1181
0,8039

i

-·t-···

1

1

fLRmax

'

r

-

25.971 !
6.7105
0,7913 ;
19.2606 7
5.9192 1
0,7913 !

**72,8139
**30,5526
0,7848
**42,2614
**29,76774
0,7848

*34,0048 1
**37,7476 i
*15,4926
12,3176
r<2
*4,4377
2,2730
1
18,5122
r=0
*25,4300
1
1
1
13,2196
r=l
jLRmax
7,87981
*4,4377
2,2730 1
r=2
i
1
Note: ** (*) presents 1% (5%) degrees offreedom.

**58,0352
*18,5722
2,0778 1
**39,4629
*16,4944
2,0778

**49,8470
**21, 1667
0,2165
**28,6803
**20,9502
0,2165

p

1=1

'

**37,2447
3,2543
O, 1168
*23.9904
]3 1375
O, 1168

*33.6738 :
4,6155
0,0275
**29,0583 !
4.5880 !
0,0275 i

(3 ).

LA; - I in f'; = - LA

LRmax

**44 0881 1
12.5636
0,8093
**3 .5245,
.7543
0,8093

29.5547
8,2806
0,7824 1
*21,2741 1
1
7.4982
0,7824 !

1=]

p

r::::Z

----------

r=0
rS l
r<2
r=0
r=1
r=2

p·

L f',LlY

i

r

23.3980 !
9.0'-'-9
0,7835 ,
4.342
8.2724
0,7835 :

France

where t ranges from 1 to T Y 1 is a vector of I variables. which are integrated of the first
order V AR in equation (2) can be also written as

Ll Y1 = TIY1.1 +

r<2
r=0
r=
r=2
r=0
rSl

1

(2).

LR1,

**6A312'
8'*4,7704
**47.8 27
8.848
*4,7704

Germany

JOHAN SEN COINTEGRATTON TEST
Since we are looking for a stationary linear combination of three variables. Johansen cointegration testis appropriate to use (Maddala in Kirn 998) This method is base ona V AR
and can be briefl v described as follows (Johansen 99

Poland

Hungary
1

'

When ali restnctions m equation ( i J are omitted. it becomes the least restrictive version of
PPP The nnl\ restriction that remains is the signs of the coefficients This i mplies that \Ve
are looking for any linear relation ship among the observed variables Taking into account
the unstable characteristics of non-stationan tirne series. the existence of stationary relation
ship among them are more 1mportant than deviations of coefficients from the strict PPP
theory (Liu 1992). lf a cointegration among nominal exchange rates. domestic consumer
prices and foreign consumer prices is found and it is presented by cointegrating vector of L
a1, -a2), the validity of PPP theorv is proven (Chen 995)

(6).

. LRi,
i

!

! LRmax

!

!

Italy
r=0
r:Sl

1

1LR1,

1

1

'

1

1

1

Critical values for Johansen cointegration test are stated in Johansen (1988) and Johansen
and Juselius ( 1990). However, this study applies improven critical values of Osterwald-

155

Lenurn ( 1992;, wh1ch are also used by EV1ews. To undertake the Johansen cointegration test
an appropriate lag had to be found Lags estirnated on he basi s \' AR's Akaike lnformation
Criteria (AlC) and Fina! Prediction Frror (FPE) are presented next to the individual countries
names in table 2

c,iiniegraung coeffic1ents are rung to c:onftrrn PPP theoP. ln comparison to Gerrnany,
there is
nimtegraiion e1ghter
the case of ·zech Repuhlic there is no evidence
of corntegration · cornpansun
Austria and Gerrnanv. while for ltalv and France there is a
conflict
results among the two test statistics Taking into account the different orders of
111tegration of individual tirne series. this is not surprising

In order to be cointegrated tirne series have to be integrated of the same order That's why.
ADF tests were conducted. in order to find the order of integration of mdividual tirne series
(norninal exchange rates, dornestic consumer prices and foreign consumer prices) The tests 2
showed that the tirne series are mostly 1(1 ). With the exception of Czech exchange rates and
consumer prices Thus, the results of Czech cointegration test are doubtful but are stili
presented as a matter of cornparison However. MacDonald 993 claims that even
the
event of different order of integration of tirne series, it is possible for their linear
cornbination to be cointegrated This is impossible
the case of three tirne series being
integrated ofthree different orders (Granger 1986)

Anahsmg Hungarian tirne series the results are the following ln comparison to Austria,
Italy and France, there is evidence of cointegration but the signs of cointegrating coefficients
are again not in favour of PPP validity In comparison to Germany. there is evidence ofthree
cointegrating equations. v,hich is accoi'ding to cointegration theorv impossible (arnong three
tirne series there can be at most two cointegrating equations) For Polanci the results are
s1m!lar In all four cases
crnntegration) 1s rejected but the coefficients in cointegrating
equations are ofthe opposite signs as proposed bv PPP theorv
The last three columns of table 2 present adjustment coefficients estimated from error
correction model. which is also a product of Johansen cointegration test. Adjustment
coefficients show the magnitude of correction of deviation from long term equilibrium in the
curent tirne period In e\erv case it can be seen that deviations are not diminishing, they are
rather getting larger. In analysing PPP theorv. variations of exchange rate deserve a special
attention Their adjustment coefficiems
are presented in the third coloumn oftable 2

Table 2. Cointegrat1on and adjustment coefficients
Slovenia

U1

Austria 3

-O 707'-

Germany6

-0,7090

France 5
Italy2

1

U2
1

- .8258

1

-1.1967

!,

-0,0424

-!

-0,0228

i

-0,07UJ

1

-U,3307
1

p1

1

-l .98E-05

i

Pe

2,23~0
-4,4198
0,9779

1

-0,0006

1

-

-

-0.0832

1

P2
-0.0020
-0.0050
-

1

,

2,24E-05

1

-0.0006

+----··-

CONCLl SION

0,0025
i

-0.0046

Czech Republic
Austria 1

-3,2421

Gerrnany2

-1 200 l

France 1

-2,8237

Italy2

-1,5897

2.3977

1

4.8959

1

-0.0872
1

-0.0746

-0,0030

0,01
t- ----·-

0.0406

-0.0033
~--·

12,993"-.
4, l 891

i

-0,0942 1
--+--0.0874

0,0313

o.o

'

-----------

-

-0.0039

-0.()099

5

Hungary
Austria6

-3,2617

9,8276

1

0,0092

-+-

0.0253

1

-0,0035

----~-- - - - -

Gerrnany6

-1,5443

4,8563

0.0316

i

0,0644

France6

-2,8604

15,3784

-O.O 178

!

0,0380

Italy6

-2, 1690

6,9434

-0. 17

i

0.0363

1
1

4,67E-05

0.0001

1

-7,92E-06

1

-0,005
-

-0.00 2

Poland
Austria 3

-116,0942

130,8193

0,0056

Gerrnany 2

-3,4839

10,9507

0,0457

France 2

-6,7725

36,3192

Italy 3

-6,3788

-2,9567

1

0,0130

-0,0009

0,0871

0,0137

-0,0007

0,0126

0,0027

7,76E-05

1

Tables 1 and 2 present the results of Johansen cointegration tests. Table 1 shows the
estirnated test statistics for different nuli hypothesis and table 2 presents cointegrating and
adjustrnent coefficients. In the case of Slovenia, the results show that there is cointegration
arnong the three tirne series in cornparison to Austria, Germany and Italy, but the signs of
2

Relaxing the assumption of proportionality and symmetrv in empirical testing of PPP theory
allows to test for cointegration of nominal exchange rates. domestic prices and foreign
prices. ln order to find evidence of PPP theory in Slovenia, Czech Republic. Hungary and
Poland, Johansen cointegration tests were conducted Though some cointegration among the
observed variables is proven. the coefficients of cointegrating vectors are not of an
appropriate sign to c:onfirm the validity of PPP The results show that none of the observed
currenc1es and prices exhibit such a long run equilibrium relationship that the purchasing
power parity would hold. These results are compatible with most ofthe rare studies ofthese
countries in the field of purchasing power parity
Further effort in searching for validity of PPP theory should concentrate on panel data.
Namely. panel unit roots test are well-known to be of a higher power that tirne series unit
root test. Thus, it would be interested to see the results of panel unit root tests in the case of
real exchange rates. However, further analysis is out ofthe scope ofthis paper.
REFERENCES
! .Boršič, Darja. 2003 Empirično preverjanje teorije paritete kupne moči v Sloveniji. NaJe
gospodarstvo 5/6 418-435
2.Boršič, Darja. 2004. Cassel in njegovi predhodniki: razvoj teorije paritete kupne moči.
Naše gospodarstvo 3/4: 68-79.
3.Boršič, Darja. 2005 Cointegration Among Prices and Exchange Rates in New Members of
EU. ERE Conference 2005. Luxembourg.
4.Chen, Baizhu. 1995. Long-Run Purchasing Power Parity: Evidence from some European
Monetary System Countries. Applied Economics 27: 377-383.
5.Cheung, Yin-Wong, and Kon S. Lai. 1993. Long-Run Purchasing Power Parity During the
Recent Float. Journal ojJnternational Economics 1/2: 181-192.

Due to limiled size of the paper, the results of ADF lesls are available upon request.

156

157

r
6 Cheung, Ym-Wong and Kon S La1 2000 On the Purchasmg Power Panty Puzzle. }ourna! of
lntcrnationa/ Eco110111ics "2{2) 321-130
7 lhnstev. Atanas and Abbas Noorbakhsh 2(J00 Long-Run Purchsmg Power Panty, Prices and
Exchange Rates m Transll1on lhe Case of Six Central and East European Countnes. (j/obal
87- 08
Finance Jo11rnal
987 Co-integration and Error Correction
W J Granger
8 Engle, Robert F m
Representation, Estimation and Test111g_ J,,co110111etnca 2 251-276
9,Froot Kenneth A and Kenneth Rogoff 1995, Perspectives on PPP and Long run Real
Exchange Rates, V: Grossman G, in K. Rogoff Handbook nl lnternational Economu.:s.
Vol. lil. Elsevier Science, 647-1688.
O Granger, C W.J. 1986 Developments 111 the Study of Cointegrated Economic Variables
Oxford B11/letin c!f E(:0/101111cs a11cl Sw11s11cs 2 3-228
l l Holmes, Mark J 200 l New Evidence on Real Exchange Rate Stat1onarity and Purchasmg
Power Panty 111 Less developed Countnes Jo11rnal Macroeco1101111cs 23(4) 60 -614
12. Johansen, S0ren, l 988 Statistical Analvsis of Cointegration \' ectors Journal
Eco1101111c Dynam,cs a11d Control 12 23 -254.
13 Johansen, S0ren 199 l Estimation and Hypothesis Testing of Cointegration Vectors in
Gaussian Vector Autoregressive Models Aconomet,·1ca 6 15" - 580
990 Maximum Likelihood Estimation and
14 Johansen, S0ren, m Katarina Juselius
lnference on Cointegration - with Applications to the Dernand for Money Oxford
B11/leti11 ojEco1101111cs a11d Sta11s11cs 2 69-2 O
5 Kun, Y oonbai l 990 Purchasmg Power Pantv m the Long Run A C omtegration Approach
.Journal of!vfoney ( 'reclir and Rankmg 22 49 -503
i 6. Liu, Peter C 1992. Pur.;hasmg Power Panty rn Latm Amenca. A Co-lntegration Analys1s
Weltwirtschafiliches Arclin 4 662-679
17. MacDonald, Ronald, l 993 Long-Run Purchasing Power Parity is lt for rea\'7 The
Review of Eco1101111cs a11d Staustics 690-695
18 Maddala, G S" and In-Moo Kirn. 1998 ['nu Roo/s. ( ·01111egratwn and S1111c111ral ( 'hange
Cambridge Lnivers,ty Press.
19 Mark, Nelson C 1990 Real and :\iomrnal Exchange Rates 111 the Long Run. An Empmcal
lnvestigation .Jo11rnal of111temational Ecrmu1111cs 28 l "- 36
20 May, Roger D ( 1999) An econometric Evaluation of Purchasmg Power Parity [on lme]
East Carohne Lrnvers1t\ Econom1cs Department Available h_U:pj/www ecu.eduJecon/ecer1
honors/may. p_gf [ 7 02 200 l J.
2 L Officer. Lawrence R 1976, The Purchasmg Power Panty Theory of Exchange Rates A
Review Article. IMF Stafl Paper 1-60
22. Osterwald-Lenum_ Michael.1992 A Note with Quantiles of Asymptotic Distribution of
the Maximum Likelihood Cointegration Rank Test Statistics Oxford Bulletin ~f
Eco110111ics and Statististics 3 · 461-4 72
23, Parikh, Ashok and Elizabeth Wakerly 2000 Real Exchange Rates and lJnit Root Tests.
Welt1Firtschqfi/iches Archiv 136(3) 478-490
24. Pufnik, Andreja. 2002. Purchasing Power Parity As a Long-Run Equilibrium Co-Integration
Test in the Case of Croatia ( 1991-1996). Croatian Economic Swwy /996-1999: 29-54.
25. Rogoff, Keneth. 1996. The Purchasing Power Parity Puzzle. Journal of Economic Literature
2: 647-668.
26. Samo, Luciana and Mark P. Taylor. 2002. Purchasing Power Parity and the Real Exchange
Rate. IMF Staff Papers 1: 65-105.
27. Taylor, Alan M., in Mark P. Taylor. 2004. The Purchasing Power Parity Debate. NBER
Worki11g Paper 10607,
28, Taylor, Mark P. 1988. An Empirical Examination of Long Run Purchasing Power Parity
using Cointegration Techniques. Applied Economics 20: 1369-1381.

158

SIMULATION OF THE BLSINESS PROCESS SALES_CJLAIM
NadJa Dami{ Janez Grad 2
1
Facult: Econom1cs
University of Ljubljana
Kardeljeva ploscad l '7
Sl-! 000 Ljubljana

'Faculty of Administration
University of Ljubljana
Gosarjeva ulica
Sl-l 000 Ljubljana
Abstract

The objective of this \,ork was to run the simulalion of the business process Sales Claim in order to
improve it to become a more effective and efficient process. To do that, a business process
Sales_Claim was first modeled using the flowchart technique and then the simulation process was
run using iGrafx software. The simulation results showed tlial the business process Sales_Clairn is
well modeled, effective and at the presen! time does not need further improvement.
Keywords: Business process, Process simulation, flowchart diagram

L Introduction
The majority of business processes were developed over tirne or badly modeled. Such
business processes became ineffective and need to be improved. For this reason, business
process modeling and simulation became an important way of ensuring changes in an
organization's functioning in order to create a better more competitive and successful
enterprise,
The aim of this work is to show that business process simulation could be used as an
essential tool to carry out business process improvements such as better resource utilization,
shorter transaction times. etc The process simulation is done using iGrafx software.
The paper consists of five sections. Sections 2 and J discuss the simulation of business
processes and introduce the flowchart technique as a tool used by different software
packages to run the simulation. Section 4 represents the results of simulation of a business
process called Sales_Claim. The fina! section contains some useful remarks and conclusions.

2. Simulation
Modeling, analysis, improvement and simulation of business processes are on the increase as
only a thorough comprehension of the business processes within an organization can lead to
effective, efficient and value-adding systems. According to Aguilar-Saven (2003), it is the
business processes that are the key element when integrating an enterprise. Furthermore,
conceptual modeling of business processes is deployed on a large scale to facilitate the
development of software that supports the business processes, and to permit the analysis and
re-engineering or improvement ofthem (Aguilar-Saven, Olhager, 2002). Business processes
are modeled with the aim of analyzing their current states within the organization, as well as
improving them through the execution ofpotential "what-if' simulation scenarios.

159

r
Currently, few organizations maintain a forma! model of their business process network
(Weyland, Engi!es, 2003 ). Even so. these forma! models were usually the product uf a series
of interviews, reviews, examinations, etc. conducted at a tirne \\ hen the urganization
encountered apparent and serious problems. Cunstruction uf a business process model makes
use of severa! accepted modeling constructs: delays associated \\ith perforrning \\ork.
statistical distribution of these delays to represent observed variabi!ity. dependence of
processes on completion of earlier processes. queuing of input emities waiting tu be
"processed", decision logic that directs entities into alternate flow paths depending on their
characteristics, application of resources to the work of a business process. and the costs
associated with these resources (Weyland. Engiles. 2003 J. Cunsequenlly. an "as-is" model
encompasses the above constructs wilh the aim of imitating the real business process under
inspection within the organization
Simulation modeling according to Pidd { i 998 l is based on very sirnple princip les the analyst
builds a model of the systern of interest. writes a computer program which embodies the
model and uses a computer to initiate the system's behaviour when subJect to a variety of
operating policies. Furthermore. it is thought that simulation modeling extends the potential
of business process modeling and analysis.
According to Weyland and Engiles (2003), modeling and simulation of a business process
network serves three immediate purposes: organizing the results of interviews and research.
identifying the cause of observed performance issues. and exploring alternate process
network configurations that improve performance. The aim 1s to efficiently image the
process network. Weyland and Engiles (2003) argue that hierarchical modeling tools are the
most useful, as they contain features like simultaneous understanding of high-level and
detailed views, as well as the manageability ofthe model.
Simulation is the imitation of the operation of a real-world process or system over tirne
(Banks et aL 2001 ). A simulation model enables the analyst to observe and study the
system's behaviour as it advances through tirne.
Simulation represents a powerful approach for analysis and quantitatiYe eYaluation of
business processes (Laguna, Marklund. 2005). Furthermore. they classify simulation models
into three groups depending on their attributes:
Static or dynamic. A static model is a model where tirne within the real business
•
process is insignificant, and a dynamic model incorporates changes over a tirne
period.
Deterministic or stochastic. A deterministic model is defined by a sequence of events
•
such as input recognition enabling the output definition, whereas a stochastic
simulation model has, according to Banks et al. (200 l ), one or more random
variables as inputs.
Discrete or continuous. A discrete model is a model that consists of discrete events
•
which are events that happen at particular times, and a continuous one consists of
variables changing continuously over tirne.

simulatiun descrihes hu\, a s,. stem \.\ iih disnete ll(\\\ unns , 1r 1,ibs evolves uver tirne
(Laguna. r-..fark!und. 2005) Therefore. ac,i1irding hl Banks et al (2001) discrete-event
simulation exarnines the mude!ing ,11' systems in \\hich the stale variable changes only at a
discrete set of points in tirne. \\'hat differentiaces a discrete-event model from a continuous
one is the fact that it deals with the attribute time only \\hen the event actually happens.
.

Consequently. according to J,aguna and Marklund (200'.'i ). such a perspective of the events
and tirne enables significant tirne compression because it makes it possible to skip through
ali tirne segments between events when the state of the system remains unchanged.
Simulation packages enable sirnulatioii runs of \ast and \arious numbers of events that may
in reality happen over a long period of tiine. A discrete-e\ent simulation model focuses on
the state of the business process at specific lime points \,hen the e\ents occur. Hence. when
executing the simulation run. the simulation clock jumps between the events and regards the
system as staying the same in the meanwhile.
A simulation model is normally based on a set of assumptions regarding the system's
operation. These assurnptions are expressed in mathematical. logicaL and symbo!ic
relationships between the entities. or objects of interest. of the system (Banks et al. 200 l ).
After a simulation model has gained form and been validated. it is deployed to examine
various "what-if" questions regarding the real-world system. so that any future alterations of
the system are first simulated and as a result it provides forecasts about the impact of the
alterations on systems effectiveness.
Laguna and Marklund (2005) summarized some of the main anrihutes that make simulation
po,verful:
Simulation, like analytical modeling. proYides a quantitative measure of
•
performance.
Simulation. unlike analytical and symbolic models. is able to take intu consideration
"
any kind of complex system variation and statistical interdependencies
Simulation is capable of uncovering inefficiencies that usually go undetected unti! the
•
system is in operation.
The availability of special-purpose simulation languages. massive computing capabilities at
a decreasing cost per operation, and advances in simulation methodologies have made
simulation one ofthe most widely used and accepted tools in operations research and system
analysis (Banks et aL 2001 ).

3. Flowchart
A flowchart is a simple diagram used by different software packages such as iGrafx to model
and run the simulation of a business process under discussion. iGrafx software was used in
this work to run the simulation ofthe business process "Sales_Claim".

Consequently, Laguna and Marklund (2005) state that business processes are in general
represented as computer-based dynamic, stochastic, and discrete simulation models which
are defined as abstractions of the actual business processes, represented in the computer as a
network of connected activities and buffers through which jobs or customers flow, and must
also capture the resources and various inputs needed to perform the activities. Discrete-event

A flowchart is defined as a formalized graphical representation of a program logic sequence,
work or manufacturing process, organization chart, or similar formalized structure (Lakin et
al., 1996). A flowchart is commonly used to show the flow of a process from its start to its
end. It usually consists of different symbols connected by lines, arranged in such a way to
lead us in correct sequence order through a series of steps. Process flow is traced by
following the connecting lines between the symbols drawn. These symbols include start and
end, activity, input and output, decision, and department.

160

161

A flowchart, according to Arlow and Neustadt (2002) begins with a starting point and
finishes with an ending point. The terminus symbo! is
used
flowcharting to
designate the beginning and the end.
An activity is represented by a rectangle and means an elementary task or a subprocess. The
path by which processes flow through the diagram consists of connecting lines between
activities. A set of activities could be contained by a container called a department. An input
is indicated by an arrow, which enters an activity. An output is shown by an mn1w, which
leaves an activity. An arrow connects one activity to another, showing the movement of the
diagram.

\\ e ran the
the process Sales_Claim sh\lwn
Figure l takmg into
cunsideration 600 claims. 80 claims
thern \\ere alread\
Jifferent phases of the process,
and we posrnlated that 30 claims were received by the urganization every day
To do that a standard calendar was used. that is, 8 hours/day. 5 days/week and 22
davs/month And the
resources were defined 6 Sales_Claim Clerk, 4
Sa,les-Clerk. Stock_Keeper and Warehouse_Cla imClerk

Register

Prm·

2.laim_Note

Glaim-.\Jote

~

If an activity is a decision, specifies alternative paths based on some Boolean expression
and is shown by a diamond. There can be only one input path to a decision, but there can be
many output paths (Arlow. Neustadt, 2002) A decisiun is a
a1 hich the process flo,,
can take one of severa! possible paths based un a defined criterion.

No

Sa!es

O\er-recei\ed
Products

...,._

•,

Require
Transport

Send

Explana\ion
Forward C!aim
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Products Back

Determina
:::.laim Solution
Pa\h

End

Send
Add11ional
lmoice

Yes

To model a task perfonned simultaneously by different departments or to model parallel
activities, we define different outputs from an activity as split outputs. A split is made by
defining multiple paths from a single activity to a set of activities. After parallel tasks have
been performed, outputs of those activities which performed the parallel tasks could be
modeled to enter a single activity; this is called ajoint input.

lssue
·:redil Note

-'es

Check

No

Appro'-01

i
Analyze

Create

According to Aguilar-Saven (2003) flowcharts are built to offer an enhanced comprehension
of the process, which is a requirement for process improvement. Dy grouping tasks into
logical areas of activity (processes) and drawing flowchmis of the events which occur it is
possible to get a concise picture of the way particular processes are completed within the
organization 1 . The flexibility of the flowchart technique is argued by some authors to be its
advantage as it allows each modeler to unite various pieces of the process together to obtain
the overall picture as he/she feels they fit best. On the other hand, other authors argue that
the technique is too flexible. describing !arge models without illustrating the hierarchy of
different layers.

•

Quantily of

Schedule of

Products

Transport

~

Warehouse

Retum :laim

'Slocumentatior

:t

No

Rejec! lhe
-II>{

Sand Schedu!e

C!aim

ofTransport

:t
1nform etx>ut
Shiptment

>i

End

Receplion

Figure . Flowchart of Sales_Claim business process

4. Results
Business process Sales_Claim represents a difficult problem in a large trading company.
Many customers are not satisfied with the solution obtained considering their claim
applications. The purpose of this paper was to model, simulate and improve the business
process of Sales_Claim. For this reason, in this section we are simulating the process
considered to find ifthere are possibilities for its improvement.
Figure 1 shows the flowchart of a reduced business proces s Sales _ Claim. As it is evident
from Figure 1, the process is performed in two departments; these are Sales and Warehouse.
In the framework of the Sales department 15 activities are performed, while 7 activities are
executed in the framework of the Warehouse department. The real Sales Claim business
process consists of more than 200 activities.

The results of running the simulation of the business process Sales _ Claim were as follows
- Average cycle tirne for one claim is 3 74 days:
- Elapsed tirne for considering 600 claims is 22.73 days.
For this reason, it seems that the process was improved earlier and therefore there is no
reason or possibility to carry out further improvement ofthe process.
5. Conclusions
The objective of this work was to run the simulation of the business proces s Sales_ Clai~ in
order to improve it to become a more effective and efficient process. The results of runmng
the simulation ofthe reduced business process Sales_Claim show that the process considered
is well mode led and effective. Nevertheless, these results also confirm the necessity of using
the simulation as a tool for improving business processes.

1http://w-ww.hci.com ..1u,~1cisitc2/toolk
it/llowchar .htm

162

163

References
l. Aguilar-Saven R. Business Process Modeling - Review and framework. International
Journal of Production Economics. 2003:90(2) 29-49.
2. Aguilar-Saven R, Olhager J. Integration of Product. Process and Functional
Orientations - Principles and a Case Study. Preprints of the lnternational C:onference
on Advanced Production Management Systems. APMS 2002. IFIP. The Netherlands.
2002,
3. Arlow, J., Neustadt, I. UML and the Unified Process - Practical Object-Oriented
Analysis & Design Pearson Education. Boston. 2002.
4. Banks J. Carson II JS. Nelson BL 1\icul D!Vl. Discrete-bent System Simulati.on
Edition. Prentice Hall International Series in lndustrial and S) stems Fngineer(ng.
Prentice HalL Inc„ Upper Saddle RiveL New Jersey. 200
5. Laguna M, Marklund l Business Process Modeling. Simulation. and Design Pearson
Education, Inc. New Jersey. 2005.
6. Lakin R, et aL BPR Enabling Software for the financial Service lndustry
Management Services. ISSN 0307-6768. 996.
7. Miller AR. Medica[ Diagnostic Decision Support Svstems-Past. Present, and Future.
J Am Med Inform Assoc. 1994: ( ) 8-25.
8. Pidd M. Computer Simulation in Management Science. Chichester: John Wiley.
1998;118-32.
9. Weyland JH, Engiles M. Towards Simulation-Based Business Process ManagemenL
Proceedings of the 2003 Winter Simulation Conference, iedsl Chick S. Sanchez PJ.
Ferrin D, Morrice DJ.. 2003;225-27.

Boom and bust phenornena in electnc1ty market*
Andreas Ehrenmann 1 . Marek Kočan 2 and Yves Smeers 3
1

CORK Gniversite catholique de Louvain, Louvain La Neuve, Belgium
eh renmann@core. uc L ac. be
2

Iustilute of :\lathernatics. P.J.Šafarik

Košice, Siovakia

kocan@science.upjs.sk
3 CORK

L'niversite catholique de Louvaill. Louvaiu La Ne11ve. Belgium
smeers@core. ucL ac. be

arnl bust cvcles in electricit_\ market.
Abstract: This paper studies a possibilit of
A single-no<le model is presented. where the lioom au<l bust ('Veles emerge. The model
the real <lata from electricity market represented hv Gerrnam France,
is simulated
Netherlands and Belgiurn.

Keywords: Electricity market. deregulation, boom and hust cvcles. Cournot game.

1

Introd uction

The <leregulation of elect.ricity rnarkets should help to decrease the prices and increase
2000 and 2001 conviuced, along with
the competition. But as the crisis iu California
prnblern was discussecl in Ford [8],
Tllis
Lhese benefits. instabilitv in prices can arise.
market.
electricit,v
in
where the author studies boorn and bust cycles
of underinvestment,
periods
general. the boom and bust cycles are characterized by
overinvestment and falling prices
rising prices an<l profitability (the boomJ followed
(the bust) [1]. One can observe the boom and bust cvcles for example in markets with
commodities (even if the products of those markets can be stored in inventory as a
buffer between production and consumption) and in real estate markets f9]. Because of
the special properties of electricity (impossibility to store significant amounts of it) the
booms and busts in electricity market can have even more severe impact on electricity
producing industry than on any other. This was the case of the crisis in California that
started two years after the deregulation of the electricity market in 1998 and continued
until the spring of 2001 [8]. During this tirne unprecedented outages appeared. On the
other hand, prices started to increase sharply along with marginal costs (the price of
gas for CCGT turbines increased), but since rnany end customers had no incentive to
reduce demand because of contractual protection from price increases, several electricity
producers went bankrupt.
The research was carried out during a placement in Electrabel, Be!gium within the project
.
SK/04/ A/F /PL-401 286 of the Leonadro da Vinci program.
*This is a preliminary version of paper appearing in Studies in Nonlinear Dynamics and Econometncs,
please do not quote. For quoting please write to kocan@science.upjs.sk.

164

165

r
AccorcliHg to Electrzc1ty d't/'ectwe 20()'//54/ 8(' _4]. eled ricd\ markets slmll bernme
open markets not la.ter thall Jul.\· 2007 in all El. cutmtries. Cmmtries in west,ern Europe.
Sranclinavia and also Great Britain have alread\· implemented this directive ami electricity markets in these countries are cleregulated ancl connectecl with each other. Also the
new EU countries like Slovakia. Czech republic etc are on their way to deregulation (the
process of privatization of transmission ancl proclnrtion capacities has alreadv started)
so a natural question arises: "Shal! Europe and its electricity markets expect similar
problem, like those in California at the ene! of the 20-th century?"
Thero are several articles clealiug with hehaYior of electricit y producers. These art icles
presellL an iclea of what quautities the producers shall offer to the market when Lhey kuow
the demand '10] or what prices the producers shall nffer to the market a11d cunsequeutl\
what quantities tu max1mize their profits ·1:11. But the mo<lels prese11tecl iu thal articles
are static and they do not focus on price evolution ancl the factors that affect the price
stability.
In this article we present a simple <lynamic model. where the boom arni lmst ('ydes iH
electricity market can be studied. The electricih' market ,vill be represent.ecl b\ one uude.
where the consumption and production will be concentrated. \Ve assume t,wo producers
of electricity. To increase the capacities. the proclucers have to make some iuvestmeut
decisions in each period (e.g. year) based on their expectations of future prices. The
model consists of two simultaneous processes. In the first oue we calculate the price of
l unit of electricity for each period and in the seconcl one the investments of each player
will be predicted. To model the investment behavior. the Cournot game [12] will be
used. Then real <lata from the market representecl hy 1-\·ance. GerrnaHy. 1\/etherlau<ls and
Belgium are applied and the price behmior is studiecl.

2

\\'e ass11me that rn eadi periud tLe pla.vers produce their full capacity. l sing this
full < apacit '. all(i quaHt itv derna11decl t Le part ic 11lar price wlil be detcrminccl for tlmt
period. \\·e furtlier suppose a linear dernami fuiwtion with a constant slope k > O, where
the increase of 4.ua11tity dernamled eorrespon<ls to the shift of the demancl function. The
demaml functiou for period 1 + l has the form:
p(t-~ lJ=n(l-~-1)

where Q 0 (t+ l) is the variable representiug quantit,· demandecl in period t+ 1 and a(t+ 1)
is a coust ant.
in period 1. the players ohserve l he total rnpacit v ( '( 1 i alJ(i t he price p(t) for which the
eledri('it.\· is soid in that period and dema11ded q11antli', IJ[i). if tlie price in the fqilowing
period t + l stayed at the same rnlne as in t he previoils period. 1.e. p(t -t l = p(f. then
iu periodi+ l tu l + r;,T)lt,
the im:rease in demarnl wuuld set the dernamlecl q,taJd
Ha\ing the puint JJ(f). ( l ; ijlJ(t): O!l the demaw-\ frnictiou i4) we get
u(!+l)=pU)+k!i t rJU(t)

After snbstituting this expressiou iuto '41. the demaud funl'ti11n iu period/+ l bec"n1es;
p(t + l)

p(t

C;(t + L) = C;(t + L - 1) + l;I,(t)
This implies that the total capacity of electricity production in period t

(1)

We further suppose that the demanded quantity of electricity increases linearly over
periods. Let r denote the rate of this increase (% per period) and D(t) the quantity
demanded in period t, we can therefore write

166

p{tJ + k(l + r)D(tJ

kQu(t + lJ

(5)

+

l)

=

p(t)

+ k(l

+ r)D(t:

kC(t ·

(6)

L'sing a similar reasoning for L periods aheacL we deri ve that players expect the price
in period t + [, tu be:
p(t + L)

= p(t) + k(( l + r) 1 1J(tJ

- C(l

+ L))

(7)

Iu this equation. expression (1 + r)L D(t) - C(t + L) has a direct impact on price
change. When the expression is positive (undercapacity). the price increases and when
the expression is negative (overcapacity) the price decreases.
Equation (7) creates a mistake in predicting prices. We can see that when L > 1 the
players do not take into consideration possible changes in price between periods t and
t + L. This is caused by their myopic behavior That mistake will adversely influence the
value of investments of both players.

+ L is:
(2)

D(t+L)=(l+rtD(t)

=

Ou the other han<l. t he pla.vers produce their full capacih in ead1 period. so tile llUant1ty
supplied in period t + l e4.uais the total capacit\. 1.e. Q 8 (t , 1 = ('(t + l J The
intersectiou of fuudion /5) wilh Qs(t + l) = C:t, l J will result rn the price fur w!uch
electricity in period t + 1 will he sold:

Modeling price expectations

Let us assume a single-node electricity market. Iu this market two electricity producers
(players. i = l. 2) are operating. These players are myopic (i.e. they are not able to
predict the future accurately). vVe suppose that the market evolves over periods (e.g.
months, years, ... ). a generic period denoted by t. Let t 0 he the initial period. Suppose.
that the power plants used to generate electricity do not retire and to build a new
power plant ta.kes L periods. Let Z; represent the new capacity obtained by investing
1 monetary unit. Denoting by Ci(t) the capacity of player i in period t, Ii(t) the new
investment of player i in period t, we can express the capacity of player i in period t + L
by:

(4)

kC,h,(t+l;

(3)

3

The Cournot game

The second step of the model studies the investment behavior of players and it will be
represented by a Cournot game [12],
When playing a Cournot game, the players try to maximize their profits by choosing
quantities to supply. In our case the quantities that will be chosen are the investments
in each period. Since it ta.kes L periods to build a power plant, to find the optimal
investments in period t, the players maximize their profits in period t + L. Denoting by

167

-----1

1

m the marginal costs for producing l unit uf electriciLy (we suppose tha! marginal eosts
also include the discounted past investment expenditures). t he profit funcrion uf pla_ver

·i = 1. 2 for period t

+L

is following:

TI;([, (t). J2 (t))

=

p(t + L )C,(t

Substituting (l) aud (7) we get from (8) for i

II;(Ji(t). l 2 (t)) = [p(t)

TI;(J 1 (t). J2 (t))

=

+ L- 1) + /J;(t)j

+ k ( ( 1 + r )1 D( t)

- C (t

rnC; (t

+ L)

(8)

l. 2:

+ k((l + r)L D(t) - C(t + [

-l2J2(t))] [C;(t
denoting A = p( t)

+ L) -

l) - l1Ii (t)-

- m[C,(t + L - l) + IJ,(t)]

+L

- i)) we have that for i = l. 2

= AC (t + L - 1) - kl 1li(t)(,.\(t + L - 1) -- kl2l2(t)C;(t + L - l)+
1

+AlJ,(t)

kitlJi(t)I,(t) - kl 2lJ2(t)l,(t) - rnC,(t +

l, ··

1) - ml),(t)

To calculate the investments, both players solve the following optimization problem:
(9)

i = l. 2

Being in period t both players know p(t), D(t) and C(s) for s= to ..... t + L - l. So the
only unknowns in the profit functions TI; are the iuvestrnents themselves. This implies
that the best response functions are:

člIIi (Ji (t), 12 (t)) = -kl 1 C 1 It+ L - 1) + Al 1 - 2kl~ It (t) - kl1l2I2(t)
a1i~)

'

·

= O ( 10)

ml1

Using (10) and (11) we get that if J1 (t) > O. l 2 (t) > Othe formulas for optimal investments
for period t are
p(t) + k(l + r)'D(t) - 3kC 1 (t + L- 1)
11 (t) =
3kli

p(t)

h(D=

+ k(l + r)' D(t) -

3kC2 (t + L

l)

3kG

If Ji(t) = O and J2 (t) > O, only the second player infiuences the profit of both players
and then the formulas are:
J1 (t) = O
J2 (t)

=

p(t)

+ k(l + rl D(t) -

2kC2 (t + L - 1)- kC1(t + L-1)
2kl

Because of symmetry, the formulas for J1 (t)

Ii(t) =

p(t)

+ k(l + rl D(t) -

.

2

> O,

J2 (t)

= O are

2kC1 (t + L - 1) - kC2(t + L - 1)
2kl

J2 (t)

1

4

Sirnulation

\,\'e performed simulations usiug real dala frnrn elenrwit._\ market represented by France
and Belgium (the first playerJ. arni 1\Jetherlall(ls arnl Ceruam (seconcl player).
The capacitx of the first player in initial period t 0 is the joinecl capacity available
in France aud Delgium at the end of 2004 aud its \alue is C\(t 0 J = 84725.18 1v!Wh.
siruilarh. the capacitv of secoud player is C 2
= 9515!. 87 iv!Wh ,3].
The initial price taken is the average week-day price frum the beginning of 2005,
a11nom1ced hv E11ropean Enery Exchange r5] and it is p(f 11 i = 07 1 Euro/1WvVh Quantity
dfmianded is taken frum 5i asa possible demand scenario. for electric1ty market presented,
for 20()5 The rnlue is lJi/. 0 ; = l68!H5 .vlUh.
\\'e assurne that hoth pla:,rers operate and lniild l he ( '( ·c [ t mhines V. 11,e expec
tati1ms for 200!'i are that rn = 36.4 Eur0,if'vfvVh when 1wrning ('('(;T h!l'11tnes. fhe
im·est went i osts fur \)liilding 1 !1!1Yh uf aew i apatit'. are 51900 Furus and tl,ev are f,xe<l
u\·er 1he nm uf the simulatiun. So / 1 = 12 = 1/51900 0.
fo general. the gruv,1L ra1e uf the demand for elecl
1s highly ,:urrelated w1tl1 ,lic
grnwth rate of GDP. 80 for r we take the average uf forecast of GDP of France. Hegil!lrn
Cermam and Netherlands for 2005 l7] r = 2%.
The long-term price elasticity of the electricity market in western Europe JS apprux u1ateh U,1 [11]. Using this elasticity and the data presented before we set k= O. 002.
Having all the values necessary we can start a simulation by calculating the tJptnnal
investments of both players using formulas derived in previous section and by ,alnilating
the µrices for each period using equation (6).
For the simulation a spreadsheet (Microsoft Excel) was used. Results of rm,ning the
simulation of the model over 100 periods for L = 1, L = 2, L = 3 are presented on Figures
l to :-l. On the horizontal axis are the periods, on the primary vertical axis we show pricc
Emos/lVTWh) and on the secondary vertical axis the values of total investments ! in
mill1o!ls of Euros). The !ight curve represents the price evolUI 1un. 1he dark one represents
the joined new investments of both players.
All three Figures shuw the same hehavior in the first four periods. Price decreases ancl
players do not invest. This is caused by the init,ial overcapacity. Then a sharp increase
uf price in all three Figures appears until it is optimal for players tu invest. The first
investments come in the 10-th period but with the difference that after that period for
l = l the players invest in all succeeding periods and the price increases in a steady way
until the end of the simulation. With L = 2 and L = 3 the evolution is different. After
the 10-th period in both cases the players are investing over next five periods. But with
L = 2 they invest one and a half more and with L = 3 almost twice as rnuch as with
L = l. This overinvestment forces the price in next periods to decrease in hoth cases, but
for L = 2 it decreases only for one period, and then the price increases in a steady way
until the end of simulation. With L = 3, the decrease of the price is sharper. but after
the price starts to increase, this increase is not permanent. The periods of decreasing
prices and overinvestment alternate with increase of prices over the run of the simulation.

= O.

Using the formulas derived, the players decide in each period how much to invest. According to their investment decisions, actual price, demand and capacity available, the
price for next period is set using equation (6). When repeating the process of finding
optimal investments and prices over periods, we get the price evolution scenario and we
can search for the triggers of boom and bust cycles.

168

5

Conclusion

As we have seen in the simulation, when the players decide about the investments, they
have to make some price expectations. The more they have to predict into future ( the

169

1400
1400
1200 - 1 - - - - - - - - - - - - - - - - ~ ~ 1200
1000
1000
800
800
600
000
400
400
200
200

~~~~~~~~~mm 'r,n',,mtrrmmm -mntt1-0

0

1

bigger is L lil t LP. rnudelsy t he b1gger 111ist.ake t he,\ wake i11 predi 1 tmg the priccs ancl so
t l1P. higger is the mistake wlieIJ zaknlat ing the im·estmems. \t\ heE we have appliecl the
rP.al data we l!arn seP.H that this mistake had a huge irnpact ,>11 pnce stability aiong with
appP.arance o! buom arn l bust cvdes.
ThP. model presentecl is a single-node model where two electri('it\ producers are concelltrated. :\s a topil' for fmther research it woulrl be intP.resting to see the evolution of
t he pri('es ill models with a greater lltllllher of pla,\'P.l'S. or a rnuil 1-nocle network.

References

J,_;,-1.;.........

1

7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97

'l) Boston Cunsult ing Group report Keepiug t he lights on. J,foy 200S.

,,\c';.r~;~.)8itf$~i!I•~:

,\ <omparison „f electri(ity markel desig11s :n
uetworks. Cambridge vl/ul'king Pupers w. bronomics CWI'E 0941. August 200.7

'2 A. Ehremna1m K

Figure l. Price evolution and the joined new im·estrnents for L

= l

1400

1400

1200

1200

1000

1000

BOO

800

600

. 600

400

400

200 .

200

o

o
1 6 11 16 21 26 31 36 41 46 51 56 61 66 11 76 81 86 91 96

:\euhuff:

Electrabel database: Elec:trabel S A .. Belginm, 2005
4] Electricity directive 2003/54/EC
http,//enropa. eu. int/comm/energy/electricity/legislatwn /legisiation_en.htm

'.5i Electricity markets: http//www. electricitymarkets info;
,6] European Energy Exchange: http://www.eea·.de/

:7] Eurostat: http//epp. eurostat. cec. eu.int/porial/page'2_
pagetd=l 090, '30070682, 109(L'ici076516f<J_dad=porialf1 srhema=PORTAL
Sinmlation Scenarios for the Western Electrieity Market. A Discussion Pnper far the Cnlifornia Energy Commission Workshop on Alternative Market
Strncfores for Califomia, November 2001.

:8] A. Ford:
Figure 2: Price evolution and the joined new investrnents for L

=2

[9] A. Ford: Waiting for the Boom:
struction in California, March 2001.

1000
1400
1200

4000

1000
800

3000

600

2000

400

1000

200 ·

o

o1

1 13 19 25 31 37 43 49 55 61 61 13 79 85 91 91

~ l l lhwlt$4'16
~:: ', Pr1M
.1..
.
,'!" ~, •• ' . ;

~

'

\>, ~. !-. '---

::

170

)O] D. Muitre: Nash equilibria in competitive electric energy rnarkets. Electric
Power Systems Research 60, 2000.

[llj F. Murphy, Y. Smeers; Forward Markets May not Decrease Market Power
when Capacities are Endogenous, GORE DISCUSSION PAPER 2005/28, Louvainla-Neuve University, Belgium, 2005.
[12] Hal A. Varian: Interrnediate Microeconornics, 3-rd Edition, W. W.Norton
Company, USA, 1993.

f3

[13] H. Xu, E. J. Anderson: Nash equilibria in electricity markets with discrete
prices, Mathematical methods oj operational research, 2004.

'

Figure 3: Price evolution and the joined new investments for L

A simulation study of power plant con-

=3

171

r
EVALUATION OF PARAMETERS WHICH JNFLLFNf'F THE ASSOCHATION OF
PRIV ATI•: FOREST OWNERS BY THE LSR OF AHP AND DEXi
Špela Malovrh
Ljubljana. Biotechnical Faculty.
!000 LJublJana. Slovema
spela.malovrh@hfuni-l,1.s1

University

Abstract
In this artide the two multi-criteria decision making methods AHP and DEXi are compared with
regard to estabhshing which factors were more important for private forest owners in making their
decision whether to become members
the pnvate forest owners association of the Mirna Yalley.
Both methods rely
hierarchical decomposition of criteria. The companson was carried out on a
real-life example of pnvate forest owners becommg associated.
Keywords: association of private forest owners. multi-cnteria dec1s1on makmg, Analytic herarchy
Process (AHP), DEXi

L INTRODUCTION
With social development, the demands of people towards nature have changed. For y_uite a
long while, the forest has not only been a producer of wood, but has had to assure a variety
ofnon-material functions which are becoming more and more important for society
Small and fragmented forest property is one of the main reasons for the low
efficiency and non-competitive position ofprivate forest management. Small and fragmented
forest property distributed among a large number of forest owners and co-owners is a typical
characteristic of Slovenian forestry, creating limitations for private forest owners. This is
why many people have decided to establish associations ofprivate forest owners. But before
a forest owner becornes a member of an association he/she always wishes to know which
factors influence his/her decision to join the association.
People have to make decisions in every day life over and over again. Some of these
decisions are easy and cause no damage if the decision is wrong; others can be complicated
and exert a strong influence ifthe decision is wrong.
When one must pick out of severa! alternatives (decisions) and when each alternative
consists of severa! criteria (attributes) we speak about multi-criteria decision making. One of
the problems of multi-criteria decision making is how to define the importance of the
alternatives and attributes f5]. lJndoubtedly, one alternative is more important than another
for pri vate forest owners, but it is difficult to ascertain how much important one alternative is
than another.
The goal of this paper is to establish which factors can influence a private forest
owner to become a member of the private forest owners association of the Mirna Valley.
With the help of the decision making methods AHP and DEXi we attempted to find out
which altematives and attributes were more important for the private forest owner and had a
decisive influence on his joining the association. The forest owner's property size was taken
into account.
2. MULTI-CRJTERJA DECISION MAKING
Multi-criteria decision making is based on the fact that the choice of a solution is affected by
numerous criteria, the importance of which varies. A decision making problem is thus
broken down into smaller subordinate problems (parameters, criteria, attributes), and these

172

173

r
are then assessed separately for each parameter f-<mal assessmem ,s obtained hv means of a
specific cornbining procedure.

the process assessment the problem of consistencv o!' assessments 1s encountered
assessment rnlues are inconsistent, they should be corrected, or the results
ai, akj = aij·
will be ofno use l

()
~

F (Xi, X 2,

A =

l\i

~1"

_a,.,

1
.... ,

function

Xn)

1

1V

IV

1

H1

W2

w,:

w.,

ll•

1V

w:

\\

1V

(1)

"

Utility vector w is obtained by searching for eigenvalues /\ of matrix A: Aw = \ 110 xw, where
A and w is the corresponding eigenvector. In
practical cases, this tends to be a complex calculation procedure. The eigenvector which
the maximum eigenvalues of positive reciprncal matrices can be obtained in
belongs
various ways:
method ofpowers - the matrix is put to a sufficiently high power, the values are then
e
added and normalized by lines
" normalization the matrix is nomialized so that the sum according to columns is 1
and the average element the line is calculated [6l
The eigenvalues which correspond to the eigenvector are obtained as:

!lmax is the maximum eigenvalue of matrix

(

x,

Parameters
, atmbutes. critena)

)

__/

n

Alternatives

~
LJ

( variants, decisionsJ

,

Figure

The 11111/ti-criteria decision making model

3. BRIEF DESCRIPTION OF DECISION SUPPORT METHODS
3.1 AHP METHOD
The analytic hierarchy process (AHP) is a well known method for a numerical evaluation
and analysis of options [l]. The AHP method was developed in the early seventies by
Thomas L. Saaty. The method is based on step by step mutual comparison oftwo parameters
at the same leve!. For comparison, a scale of 1 to 9 is used, where l means equal importance,
so two activities contribute equally to the objective, and where 9 means extreme importance
of one activity over another. The comparisons between elements are then entered into a pairwise comparison matrix, matrix A = au (i = 1 ... m, j = 1.. .m), if there are m parameters.
1

aij

=--.
aJ;

174

L-J

In

As evident from Figure 1, the model 1s fo1med hy means of parameters {attrihutes) X,.
These are variables which represent subordinate problems of a decision making process
!i..n, ;10n F is a rnie ac,,;ordmg
1attributes wluch define the qi1ali,y uf al:ernatl\eS) L
which the values ofindividual parameters are combmed to form rnriable Y, wh1ch represents
a fina! assessment of usefulness of alternatives [3]. Alternatives are described according
basic parameters by values ai, On the basis of these values, the utility function will provide a
fina! assessment of each alternative.
ln the case ofmulti-criteria decision making, two notions are encountered:
preferential relation S; x,S x 2 ; 1.e. x 1 is preferred to x 2
s
utility function - F(x,), which determines the degree of suitability (preference,
e
priority) ofparameter x,.
a
In actual cases we know what we prefer, but we are unable to assign a ce1iain value
which
procedure
solution (the utility function is not known). What is needed, then, is a
converts preference relation into utility function. One of the procedures which make this
possible is the AHP method [l], [6].

A is a square matrix with ones placed diagonally. Symmetrical values are inverse:

_ __!__ '\;' (Aw),

/.,max -

i~i

(2)

.

Wl

the difference ( ,\rn,

The measure of inconsistency is defined

- m).

It is expressed by the

= A-ma, - m) I (m consistency index CI:
A random index is then introduced, which is gi\en in tabular form [6]

jm

RI

/1

Io

..

ro

_i2

i3

„l2,,58

~--

[5

.

;6

i

7

8

(3)

i9

• !O

--,

...-___-1L-1-,3-~---+1-1-,4-l--+1-1-,4-5--l-1-1,-5-1~~
10,90 „Tu2_--11-1-,2-_~-

and inconsistency (CR quotient) is calculated as: CR = CI / RL lf CR < 1 the matrix is
sufficiently consistent In the opposite case, the matrix should be c01Tected; otherwise the
results will not be correct
For the purpose ofthis paper we used AHP as implemented in the program Expert Choice.
3.2 DEXi
DEXi is a multi-criteria decision method which is based on the construction of a decision
problem in a hierarchic structure of attributes. In contrast with AHP, DEXi uses qualitative
attributes. Each attribute in hierarchy is defined as a discrete variable which can take its
values from a set of symbols. These symbols need to be defined for each attribute separately
and typically consist of words like not important, very important ... In DEXi, the use of
qualitative attribute requires a different approach to aggregation than AHP. As opposed to
using weights, the aggregation is carried out by decision rules. These are simple "if-then"
rules defined by the designer, with which the utility function is expressed point-by-point for
all possible combinations of alternative values [5].

4. PRACTICAL CASE
The comparison and evaluation of AHP and DEXi was carried out on a real-life case of
private forest owners joining the association of pri vate forest owners of the Mirna Valley.
The problem was complex and simple enough to enable a sound comparison of the both
methodologies.

175

r
Because of the problems that forest owners meet, they decided to found the
association of private forest owners of the Mirna Valley, aiming to improve the
competitiveness of the private forestry sector. Dut every new member had to decide which
to the association. A poli was carried out on
altematives were important for his choice
With the information obtained from the interviewed members
the pattern of 40 members
of the association, mutual comparisons were assessed and a pair-wise cornparison matrix
was formed for each leve! of the decision tree.
We distributed private forest owners into four groups
conesponding to criteria
(attributes) with regard to the size of estate: private forest owners with 1-5 ha of forest,
private forest owners with 5-10 ha of forest, private forest owners with 0-25 ha of forest
and private forest owners with more then 25 ha offorest, because we expected the size ofthe
property would influence the reasons for
the owners' association. Each of these
criteria was then supported by alternatives - reasons why private forest owners became
elaboration of forest
members of the association: education, counselling, cooperation
management plans, conclusion of timber sale contracts, joint purchase of machinery,
equipment and literature (Figure 2).
FACTORS WHICH INFLUENCE JOININ(; ASSOCIATION

-5 ha
education
- counselling
cooperation in
elaboralion of
tbrest management
plans
- conclusion of
timbersale
eontracts
- joinl purchase of
cquipment

10 - 25 ha

5 10 ha

- education
counselling
- cooperation in
elaboration of
forest management
plans
- conclusion of
timber sale
contracts
- joint purchase of
equipment

- e<lucation
- counsetling
- cooperation in
elaboration of
forest management
plans
conclusion of
timber sale
contracts
- joint purchase of
equipment

wmni iNFU ENCE JOININ(;

ASSOUATION

l-5 ha

I0-25 ha

+25 ha

0.189

O 109

0,351

Education

Educatio.n

Education

E<lucation

0.251

0352

0,284

0,381

Counselling
IJ.251

( ounselling
~95

( ·ounselling

1'ounselling

O 324

0,264

( ·ooperation in
elaboration

( ·ooperation in
elaboration

\ 'ooperntion in
elaboralion

0.08?

IJ.092

0,224

( 'onclusion of
timber sale
contracts

( 'onCJusion of
timber sale
contracts

( ·onclusion of
timber sale
contracts

(-onclusion of
limber sale
contracts

0,165

0,205

0,216

0,084

Joint purchase of
equipment

Joint purchase of
equipment

Joint purchase of
equipment

Joint purchase of
equipment

0,082

0,061

0,083

0,046

( ·ooperation
elaboration

+25 ha
education
- counselling
cooperation in
elaboration of
forest management
plans
conclusion of
timber sale
contracts
- Joint purchase of
equipment.

Figure 2: Decision tree far private forest owners in the Mirna Valley
The analysis which we made with the AHP method in the program Expert Choice showed
that for the private forest owners the most rewarding alternative regarding the criteria was
education; counselling took second place, followed by cooperation in elaboration of forest
management plans, then came conclusion of timber sale contracts, while joint purchase of
machinery, equipment and literature were stated as least important for the owner's decision
to join the association (Figure 3).
When we compared the private forest owners by property size we discovered that
there were only minor differences between them. The most important factor why owners
became members of the association was education, since private forest owners wish to
improve their knowledge on safe work in the forest and on silviculture, and become
acquainted with new technologies. The second factor that was important for all ofthe owners
was receiving counselling about the timber market and being informing about the best
contractors.
The difference between owners with regard to the size of property was obvious with
respect to the following two factors: cooperation in elaboration of forest management plans
and conclusion oftimber sale contracts. For small (1-5 ha) and for big (+25ha) forest owners
cooperation in the elaboration of forest management plans was more important than for the
other owners.

176

FM I'ORS

Education
Counselling
Cooperation in elaboration
Conclusion of timber sale contracts
Joint purchase of equipment

l- 5 ha
0.25
0.25
0.25
0.165
0.082

5-10 ha
0.~52
0.295
0.087
0.205
0.06!

10-25 ha
0.284
0.324
0.092
0.216
0.083

+25 ha
0.381
0.264
0.224
0.084
0,046

Attributes
0.189
0.35
0.109
0,351

Result
0.336
0.279
0,167
0.156
0,062

Figure 3. Results oj evaluation with AHP
Concluding timber sale contracts was more important to ali the small proprietors that
to the biggest ones (+25 ha). Joint purchase ofmachinery, equipment and literature was least
important for all proprietors without regard of size.
The analysis that we made in DEXi shows that the most important alternatives for
private forest owners are education and counselling, less important is conclusion of timber
sale contracts and not important for the owners are cooperation in elaboration of forest
management plans andjoint purchase ofmachinery, equipment and literature (Figure 4).

i
Factors
1- 5 ha
5-10 ha
10-25 ha
+25 ha

Education

Counselling

verv important
less important
very important
less important
verv important

very important
very imoortant
very imoortant
verv imoortant
less important

Cooperation in
elaboration
not important
not important
not important
not important
not important

Figure 4: Results of evaluation with DEXi

177

Conclusion of
timber
sate
contracts
less important
less important
not important
not important
very important

Joint purchase
of equipment
not important
not important
not important
not important
not important

lf we compare pri vate forest owners with regard to property size we see that education and
counselling are important for ali owners. Conclusion of timber sale contracts is
important for owners with property sized -5 ha and for those with the largest property
25
ha). For small proprietors the conclusion of timber sale contracts is important because they
carry out tree felling in longer tirne intervals and thus have only scant information about the
timber market. But this factor is also important for big proprietors (+ 25 ha) because they seli
timber during the whole year and need to get a fair price during this whole period.
Cooperation in elaboration of forest management plans and joint purchase of machinery,
equipment and literature are not important for private forest owners.

5. CONCLUSION
Both AHP and DEXi use hierarchical decomposition for developing decision models. The
structure of attributes is identical, but the methodologies use different techniques for
describing the options and aggregating the values of input attributes into the final result. The
AHP uses comparison matrices, while DEXi is based on decision rules that aggregate
qualitative values. When comparing the results of evaluation, DEXi gives very
comprehensible symbolic results, while AHP produces a numeric evaluation. So the DEXi
method evaluates several factors with the same values, while AHP determines a unique value
for every project. The results of evaluation with both methodologies were almost the same,
excluding cooperation in elaboration of forest management plans. AHP evaluated it to be
important (in a third place of five), slightly better than conclusion of timber sale contracts
and joint purchase of machinery, equipment and literature. DEXi evaluated it to be not
important.
DEXi with its qualitative values is suitable for mode11ing problems that are by their
nature qualitative and would be difficult to describe with numbers. In our case all the
attributes were qualitative, so for this kind of problem DEXi is more suitable than the AHP
method [5].
Interesting future work could be implementing a combination of DEXi and AHP,
where DEXi would classify an option into a certain class and AHP would be used for
detailed evaluation within a certain class. The most important advantage of AHP is the
ability to distinguish between similar options, for which creating a comparison matrix is
easier than for complex problems. Such two-step models usually give better results than
models based ona single methodology [3].

6. REFERENCES
[ 1] LIPUŠČEK, I.: Assessment of ecologically most favourable manufacture of wooden chairs by
means of AHP method. Zbornik posvetovanja, SOR 2001, SDI, Ljubljana, pp.307-312
[2] MALOVRH, Š.: Prispevek društva lastnikov gozdov na razvoj podeželja (študija primera:
Društvo lastnikov gozdov mirenske doline), seminarska naloga, Ljubljana, 2005
[3] MANRAI, A.K.: Mathematical models of brand choice behaviour, European Joumal of
operational research 82, Elsevier Science B.V., 1-17, 1995
[4] SAATY, T.L.: Fundamentals of decision making and priority theory. RWS Publication,
Pittsburgh, 1994
[5] ŠPENDL, R., RAJKOVIČ, V., BOHANEC, M.: Primerjava kvalitativnih in kvantitativnih
odločitvenih metod: DEX in AHP pri ocenjevanju projektov. Zbornik posvetovanja, Organizacija
in management, XV. posvetovanje organizatorjev dela 1996, Univerza v Mariboru, Fakulteta za
organizacijske vede, Kranj, pp.190-199
[6] WINSTON, W.L: Operations research: Applications and algorithms. Duxbury Press, Bellmont,
CA, 1994

178

Sinmlation Analysis during Law Adoption
BoJan Peček, M.Se,
University of Ljubljana, Faculty of Admimstration
Gosarjeva 5, !000 Ljubljana. Slovenia
bojan.pecek@fu.uni-lj.si

Abstract

Simulations are commonly used m techrncal systems, Th1s paper demonsh·ates their applicability to
systems where the human factor is more 1mportant. lt d1scusses an example of a procedure in public
admm1strat1on where the effects reorgamzat1on are simulated. rhe mteniton of rhe reorgamzation
1s pos1tive - but easmg the load on chents means rhat the procedure takes longer. With simulations it
1s poss1ble momtor the procedure and obtain data which are otherwise 1mpossible to oblam.
Keywords: simulauons. discrete event simuiations, iGrafx
INTRODl:CTIO N

Public administration is almost entirely a senice
for which it is characteristic that
the quahty of the product (service) cannot be measured. It is probably the most expensive
human actlv1ty for wh1ch we have no, even approximate, data on the costs of the services,
Up to the present: simulatio.ns have been used mainly in technical systems. Now they are
becommg mcreasmgly apphed to processes which are unpredictable. "Manv services are
stoch.astic complex processes operating in resource-constrained environn1ents: Most of them
are d1screte processes" [4]. In this paper we demonstrate the utility uf the method of discrete
simulation in puhlic. administrati?n procedures. The hypothese which we wish to prove
through our research 1s the followmg: Simulations can be used during the adoption oj a new
law.
Tn the research we analyzed the procedure of granting social assistance benefits. With the
help of the discrete simulation method we compared the course of the existing manner of
performmg the procedure with the procedure prescribed by the new Law on General
Administrative Procedure.
2

METHODOLOG Y AND SOFTWARE

Software packages for the simulation of processes are as old as the first computers [8].
Today we can find several types of software for performing discrete simulations: software
wh1ch operates ~n one _of the standard computer languages (Fortran, Pascal, C, etc.), special
langua~es for s1mulahon, block-structured systems such as GPSS, and visual interactive
modellmg systems (VIMS) [5].
One of the interactive systems for modelling and simulation is the iGrafx Process 2000
program from _the Micrografx Company. Since the program does not have a strict drawing
method?logy hke some other programs (cf. [3], or [7]), one of the existing techniques must
be ~pphed. W~ used the technique of the swimlane diagram from the methodology of the
Umfied Modellmg Language (UML), which some people refer to as the "de facto industrial
standard object-oriented language for modelling" [2], [10]. The symbols which we use are
show_n in Table 1. Solutions for more complex situations are represented in Table 2. It is the
notat10n as suggested by Scott [9], and the methodology shown by Lunn [6].

179

Table} /)e11oli11g para/le/, and a/1ema/ive., in mi111/a11e dtagrams

Table l Swi111/a11e diagram .,ymbols

_____ Beh-"een{fr10pl BeM•een(1.J_5pL, __ _

Meam_'n-"'g_ __

Situation

T he orgamzat10nal unit ,s t he
[ performer of the activity. can
be any kind of unit, such as a
1
1
department, clienl, ministry,
1 secretary, department head, etc.
1

CI~

• C
o,

ci,~
C

s

o,

1

1

'

-- ~ ' " ' " ' " " " '"""'""' s<,p
""" - 1
whi,h '""" be <ak'" wHhi" 1h,
process

---------'

~-/ "--~lternativ-~>

-~

--1
Start
-------

1

'

1

--7

i Start/end
1

'

each diagram has at
least one start and one end

--

-

-~-

1
!

__ _ j

THE EXISTING PRACTICE OF THE
PROCEDURE

3

TT
-

,----------,
end-

_L__

pozitive

'___j,___-

+

drafiing
notification or
lemination of

L proc~ure

[

'[

.
1

-Ji,,(

clear
case?

%1

procedure' ) No 5
)ustified'',

Yes 9So/-:<

draflingof
decisionof
grJntingof
benefits

f-

1

- Yes~___J

Bew.~en(1;2)m

Belween(1;2)s

1m

issuing
certiftcate

--- ,---recording
-t,,j applkation and
forwardingto
cledc

Splitting:
Procedure
continues along
bolh branches
simultaneously

Be•veen{,;2)d

~

a:~:~~;';rk

-------------+--1

10s

l'isuing
--t,I
certificate
~
1regarding currentl
assels

forwarding

+1 Between{1;2)m'

----t,

revfo\~:fdata

/ / is ue ',,, Yes 99%

-><'. .

-~

~rb

1e;;/,, ;:,-

, y No 1%,

Figure 1

'Bew1een(t;1;rn'
issuing of
u"'rr,,loyn~nl
certincate

->i

L - -

1-

J

__J

Existi11g practice in performing the procedure oj granting social assistance benefits

The average cycle per transaction is 7.14 days. This is the tirne from the beginning of
collecting the documents to receiving the final decision. The tirne for waiting for
decisions to become binding as prescribed by law was not considered.
The average work tirne (recording, processing, deciding, etc.) was 40.2 l min. That is
0.41% (less than one percent!) of the total average tirne the transaction was in the
system. The rest ofthe tirne (99.59%) was spent waiting, in delivery, etc.

For calculating the efficiency of the process, the number of approved applications is
unimportant. But we should stili remember that 1693 applications were approved and
307 rejected. This data is important because the process will be compared with other
models. Through the number of accepted and rejected applications we will convince
ourselves that the new model is of equal value to the previous one.

Occasionally (estimated at one case in twenty), the client is called in for a discussion. When
this is completed, a positive or negative decision is made. In the case of an unjustified (i.e.
unreasonable) procedure, the applicant is notified that the procedure has been terminated. In
the case that the procedure is reasonable but the applicant is not entitled to receive social
assistance benefits, a decision is drafted. This is usually written more precisely than a
decision in favour of the applicant, since there is a greater chance of an appeal, and the legal
findings must be accurately defined. The percentage ofrejections and unfavourable decisions
is surprisingly small, around 15%. The model ofthe procedure is represented in Figure l.

The formal correctness of the model was confirmed, since the simulation ended normally.
For verification we repeated the questioning of the operative performers of the procedure.
The average transaction cycle of around 7 days seemed much too long to them. When we set
in the activity 'clear case?' the probability of continuing along the 'Yes' branch at 100%, the
average cycle was reduced to 6.5 days. Another change was made to a sub-procedure at the
Tax Office, as if they issued their confirmations on the spot when clients go there in person.
This reduced the average transaction cycle to 3.44 days. Taking into consideration the fact
that clerks are not aware ofthe tirne clients lose collecting their documents, we quickly come
to an acceptable approximation of the response we got from the operators to the question:
"How much tirne does it take for a client to receive a processed application?" "It's fast: in
two to three days, four at most." In short, we can be quite satisfied with the estimate of the
transaction cycle for an individual application.

The results of the simulation for an individual transaction show:
It takes 357 days to process 2000 applications. This is essentially in keeping with

expectations, as we received data for one year of operations.

180

lirrit

Bet'ween(10,30)m
interviewing

The procedure begins at a Center for Social
Work when a client files an application along
with the necessary certificates and documents which they have previously obtained from the
competent institutions - the Tax Office, Employment Service and Registry Office, or already
has in his or her possession. For the rest of the simulation it is important that the clients
themselves obtain the documentation. In the case of picking them up in person they are
issued on the spot and registered in the logbook. The issuing of certificates does not require
the opening of new files or the starting of new cases, and therefore no additional tirne is
required.
After the filing of the completed application at the Center for Social Work, the case is
entered into the records. When the matter reaches the clerk who will process it, it waits its
tum to be processed. From the point of view of work organization this phase is the most
controversial, since it cannot be excused.

•

5d
WJilingfortirre

---,

~I

~-~-~-e_~_t_h_fl_m_v_s~------~-!_______
Alternative
continuation:
Procedure
continues along
one of the
alternatives

waitingfor
responses and
fillingout
applkation

_., ~:::~~~
certificates

1-----------+-----1branch

Alt~rnative has two possible
1
meanings: a node where a
1
decis1on 1s rnade about thc
1
1 continuation ofthe procedure, or
i
\ a condition for the _beginn.in.g of \
the performance ot an acttvtty .

!

Start

1
1

Joining of partial
transactions:
Activity is
performed only

request for

__ liML notation following L1mn_

Alternative starts:
Activity is
performed when
transaction
arrives along any

;j

j

181

4

necessary forms and that it will be sufficient if he enters the information once and the
computer prints thern automatically. This is a very optimistic scenario . that the centers will
be well organized and equipped. Therefore we have predicted a very optimistic tirne for this
operation a total of 2 minutes with the supervisor's signature and dispatching.

COURSE OF THE PROCEDl RE Pl'RSLANT TOTHE NEW LAW ON GENERAL
ADMINISTRATIVE PROC'EDllRE

The Law on General Administrative Procedure LGAP) has brought major changes to lhe
operations of official bodies. In point three of Article 139 it prescribes that _bodies shall have
the obligation that "the clerk who is in charge of the procedure shall obtam mformation ex
officio on the facts about which the body competent for decision making keeps official
records". Similar is prescribed also in the article 75 ofthe same Law.

'drafting requests for reviewing and forwarding office' are followed by a gate
The
which indicates the splitting ofthe transaction into a new family ofpaiiial transactions. With
this \,e simulate that each document travels through the procedure independently ofthe other
two. The gate opens and the transactions are performed each day at the end of the day.

Obviously clients no longer have to obtain the documents themselves_, but can file the_ir
applications without enclosures. The necessary certificates must be obtamed by the clerk m
charge ofprocessing the matter. But what is apparently an elegant and user-fnendly solution
contains a number ofhidden traps.
A significant change which the new approach introduces has to be taken into consideration.
In the former case the client went to the body in person to obtain the desired docurnent and
the body only recorded the issuing of the document. In the case of an official inquiry by the
body the mail is recorded and filed pursuant to the rules on office operattons. In th1s case the
course of the procedure for obtaining missing documentation is as follows: The clerk m the
first body (the addressing body) drafts a request for the second body (the a_ddressed _body),
where they explain their request. This request is sent to the clerk's superv1sor for s1gmng.
The signed request is sent to the forwarding office, the matter is entered into a logbook. The
addressed body receives the request, and pursuant to the rules on office operations records
and classifies and partially marks it. The clerk in charge of the matter at the addressed body
puts the request into their stack of things to be done. When the matter _comes to the top_ of the
pile, it is processed and sent to the department head for signing. The s1gned document 1s then
sent to the forwarding office, which
sends it to the addressing body. The head
watingDr
Start
-J>j signatureof
41 retum to derk
office of the first body matches the
tirre mnit
head
document with the case. After the tirne
___L
recording
limit from the logbook expires the entire
produc!ng 1forwartllng
response
request
L
matter with an enclosed opinion is
retumed to the first clerk. The course of 1 - - - + - - - - - - - - - + - - - - - - - + - - - - the entire procedure of obtaining
____L__7
ves
/~
recording,
fimvardTlg
--t4 waimg fur
cbssifying, i---+-<,, nnrking ,
information from other organizational
~
parthl
~~
proci~mg
,,
units is shown graphically in Figure 2.
____i.No
~
_J

,___J

___J

1

The transactions, i.e. the opinions issued by the Registry Office or the Tax Office are joined
before the activity 'justified?' If the applicant is justified, the joined partial transaction is
fwiher joined by the certificate of active participation in the empioyment program. This is
followed by an entry into the data base and the drafting of a positive decision. Procedure is
presented in Figure 3. After execution ofthe simulation the following is concluded:
0

@

First we check whether the modelled process is sti!l comparable to the previous one: as
the previous process 2000 transactions were processed. 308 were rejected, the rest
were approved. Therefore we can conclude that the models are equal.
The average cycle time grew to 0.27 days. which is a 43.8% increase in the duratton of
the procedure 1 Interestingly, the effective amount of work per transaction is reduced by
over 5% - from 40.24 to 3:l.8 minutes. This decrease is most of ali due to the amount
of tlme the applicant is busy, since in this model he only fills out his applicahon once
and no longer has to wait to obtain his documents - from 16.41 to 8.55 minutes. The
amount oftime the clerk is busy is slightly increased from 19.40 to 20.43 minutes.

The conclusion of the experiment demonstrates that the tirne needed for processing a request
is significantly increased if Article 139 of the Law on General Administrative Procedure is
strictly adhered to.
We can therefore conclude that: ifwe take into account the recommendations ofthe Law on
General Administrative Procedure, the average tirne for processing an application is
increased from 7 to at least 1O days. We have to be aware that these applications are filed by
people who are in fact in serious trouble. For them the reaction tirne of the body is most
important. As the saying goes: "He who gives quickly gives twice"

1

1

processug -

t-further
sjgnmg
! preparing ~
In the case of the procedure in question,
nurldng
_J
~
response
'-------'
an application for social assistance
benefits, the addressing body is the
Center for Social Work. There are three
Figure 2: Course of officia/ request far issuing an
addressed bodies: the Registry Office,
opi11io11 between administrative bodies
the Tax Office and the Employment
Service.
The new procedure differs from the old one in that the applicant_ simply file~ his application
and his part of the procedure is then over. He is no longer reqmred to obtam the necessary
documents; this is now the responsibility ofthe clerk who is in charge ofthe procedure.

5

CONCLUSION

The research confirms the initial hypothese that using simulation we can monitor not just
technical systems but also difficult to predict systems which are almost entirely based on
people and their stochastic behaviour. The calculations showed and proved the exceptional
utility of simulation.

The Center for Social W ork stili first records classifies and marks the application. Then it is
put into the stack of matters which are waiting for an available clerk. The clerk first
determines whether the procedure is reasonable. If it is reasonable, he prepares requ~sts for
the issuing of opinions. We suppose that he will have a program prepared for prepanng the

At the end of the research we are justified in asking whether there is any other method of
obtaining similar data from which we could make calculations on the data which we
obtained from the described system of simulation. How could we calculate the 'cost of the
product', in this case the cost of processing one application? Definitely not with the existing
information system used in puhlic administration. There is simply no data on how many
minutes and seconds a person spends on an individual matter or an individual document, in
this case an application for the granting of social assistance benefits. We could (over the
course of a year) monitor the processing of the applications using dozens of stopwatches.
Even if this could be done, the clerks would perform their work completely differently if
they knew that they were being watched. From this we can conclude that simulation is more

182

183

r
DlSTRTBLTlOl\! CHAl\/1\!ELS lFOR ORC,Al\!H PRODUCED
AGRI(TLTL RAJL PRODlCTS

than Just experimenting with data - it represents the augmen!ation of an infonna1ion system.
Therefore the sentence written by Bosilj Vukšic and Kmačič, "the use of s1mulat1on
modelling will have to become standard business practice „ l l should be expanded to "the
use of simulation will have to become a component part of information systems''.
r
~

.g_
t

hlling oul
request

['"

Start

,----

1

½__J

, ._ ___ __./

1

__ .J

·1

i,,co.run.,g 1 ~' B1v";iJW,nl½,'jld
g
avaibble
<1pp0Caoon 1---cbk
1 ard assigning I

1

t

~
u

1

drafling I
request fur
rcview aixl
forwarding ,

h

'--,'--,,

I

1

1
L__

1

~

l->i

>-~

Reg

1 1

Q[[

7

1

Q[[

waitn,g ftlr
cbssifying
aOO irnrking l J availlble
request at Tax f'7 clerk at Tax
1
01[

1

infunning of
hproperly
aJTilnOPd
"'o~

stan~

T

T

I+
No

's...

5m

living in

household

;

,<,.certili:ate'

entJ:'. into
records

1

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-·

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,

,---105

of

1 1Be"'ll's'fui"\g7Jm 1

c___ _

~

Keywords:

organic farmmg. organic products, marketing. distr1but10n channels, benchmarking,
lmear programming

Introduction

IOs.

certifi:ate of H fb~•,-ardmg at
HLstatc:
, 0ft
oras.sets
1

<lX

L

___J

wa-itinglor
Bet.\-een(1;7)m ~
availabl, 1 revi::wing of ~ furward~ at
Erri>.Semce
file
clerk at 1
request al
Emp.Service 1 I Enq:,.Servi:e
classifyi,,g

and nurking

Figure.3.

The organic fanuers, who intend to appear on the market, have IO dec1de what strategy to choose to
benchmarking and qualitative analysis,
the optimal way. On bas1s
enter the organic market
for determmmg the optimal strategy
models
the
surveys and linear programmmg we have generatecl
distribution channels for Slovenian organic products. The goal of dete1mined strateg1es is to
reduce the recognizable. physical and econom1c d1stances between the producers and consumers of
the organic products. W1th linear programming we have established that organic markets, which are
at the emerging stage, do not explo1t ali given possib1hties.

Reg. ()ff
I

.____,___,._ __,.S~ffiSA~(-~,1~,i >--l><Be~,,.~se~A( ~12~~j------------

l

Abstract

f-->j forwanl,ng at

connuon
1 1

ves.

·
neg;:ilI\-e
decision

_ 1___, 90%1
~

1 - -

,,!~~
r'1

10%

>

'appocant
~tiliedJ/

l~~J - ~

1

tenninati:mor

~

95%
__J

waiting for
av.,ilabl,

of

~tiiled:' S% ~procedure

· 1 clcrk at Reg.

request at

1

1

f

Bel\\'een(1;2)j

and nnrking

~

-s,.( p.roced~

LYes 95%

l~b,-;;-~7

L

discussim

00 tifix: ation

0

Bet.veen(2,4)m

1

T

[se••aHmlf,il 5~

drafling

1

L---=-=-=----_-_-_-_-_-_

1

- T

/4, /s. ~ No

y.

===;

~~~~J

1

~-

o/.
,/"'-..
0
N~
clear
~-- case? /

end positive

end
negative

Bet:~~~)n~---2pl~---t----t--8,al""'")(3;2-----.~---7

dra3fil,g
invitat.bn to
'-

t

2m

. - --t,j

warting fur time
linlit

L

1---J-....Be""",iw,ip i__~---~-----,-~~-- -+----,

L_ - ~

5d

,--

1Bel"'1en(5;1Dirij

~

1

Petra Ulamec, ARSO. Vojkova l b, lOOO Ljubljana
Lidija Zadnik Stirn, Biotehmška fakulteta, Jamnikarjeva 101, 1000 Ljubljana
Katja Vadnal, Biotehniška fakulteta. JammkarJeva lOL iOOO Ljubljana
petra.ulamec@go,. si. lidija.zadnik@ht: uni~lj .si_ katJa.vadnal@bf,uni-lj.si

Diagram oj 1/ze procedure of obtaining social assistance benejits p11rs11a111 to Ar//cie 139 oj the ihAP

The simulations have shown that otherwise good intentions - the reduction of the burden on
can turn into an e~en more ttme~consummg
people requesting necessary documents
and effechve changes to mformahon
appropriate
by
followed
procedure if new laws are not
and communications equipment

.

REFERENCES

.

Forniation of new distribution channels and selection of the existent ones belong to the more
important strategic decisions of every organic faim that produces the products with intention
of marketing the products (Ulamec, 2005). Slovenian organic farms market their products
through one or two distribution channels. They use the distribution channels that were
typical for the start of organic farming in Western Europe. The prevailing distribution
channels in European countries today are of !esser importance for the Slovenian farmers; this
may threaten their competitive position on the organic market (Bratuša, 2000, Vadnal and
Bratuša, 2000). When programming distribution channels the producers have to decide
between what is ideal and possible and what is applicable (Kotler, 1998). When choosing
distribution channels for organic products produced in Slo\ enia the models for optimising
strategies can drasticall y reduce the recognisable 1 physical and economic distance between
the producers and real and potential consumers of those products (Zadnik Stim, 995 and
Ulamec, 2005).

Material and methods

[ 1] Bosilj Vukši6, V., Kovačič, A. (2004): Upravljanje poslovnim procesima, Smerg1Ja, .za~eb _
[2] Engels, G., Heckel, R., Sauer, S. (2000): UML - A Universal Modelmg Language!, ICATPN
.
.
.
2000, (Ed. Nielsen M., Simpson D), Springer Verlag, Berlm
1
McGraw-H1L
Arena,
w1th
S1mulation
(1998):
A.
D.
Sadowski,
P,,
R,
Sadowski,
W.,
D.
[3] Kelton,
of
[4] Laughery, R., Plott, B., Scott-Nash, S, (1998): Simulation of service systems, Handbook
.
.
Simulations (ed. Banks, J.), John Wiley & Sons
McGraw-Hill
analys1s,
and
modelling
Simulation
(2000):
D.
W.
Kelton,
A.,
M,
Law,
[5]
.
[6] Lunn, K. (2003): Software development with UML, Macmillan
[7] Pegden, D. C., Shannon, R. E., Sadowski, R, P. (1995): IntJ·oduchon to simulation using
.
.
Siman, McGraw Hill
[8] Pidd, M. (1998): Computer simulation in management sc1ence, John W1ley & Sons
[9] Scott, K. (2001): UML explained, Addison-Wesley
[10] Siau, K., Halpin, T. (2001): Unified Modeling Language: System analysis, design and
development issues, Idea Group Publishing

With benchmarking we have analysed and compared the stage and development of the
European organic market with the situation in Slovenia. According to Watson (1993) we
have with the purpose to identify the best solutions on the whole European organic market
determined the following spheres (Ulamec, 2005):
impact of agricultural political measure on the development of the European organic
markets
development ofEuropean organic markets
e
• evaluation of the importance of individual distribution channels for organic products
impact of the media
•
• constraints in the development of supply with organic products
• and constraints in the development of demand for organic products,

184

185

@

our research we paid a special attention to organic farms from Gorenjska region. The goa!
of research was to find out whether the used methods and v.ays of distribuling the organic
products frorn farms of Gorenjska region are different from the methods used when
distributing conventional products. On the basis of analysis we established wha1 part of
production the organic farmers from Gorenjska region use for:
„ direct sale on their own farms
„ sales through wholesalers
., their own use.
The questionnaire was distributed to 50 % uf organic farmers from Gorenjska region
2002 (Ulamec, 2005). We made the qualitative analysis of sales of organic products with
focus group in October 2004. We considered the princip le of homogeneity of the group. As
the goal of the survey was known to the group members beforehand, we used the method of
the direct interview (Vadnal, 2002). The goal of implementing the yuantitative analysis was
to find out tliat the marketing strategy of organic products based on price difference is
supported by strategy of distribution channels ( l;lamec, 2005 ).
With linear programming we have defined (Ulamec, 2005):
market share that can be achieved on ffereutly deveioped orgamc markets by organic
agricultural products whereby evalualion of importance of different <listribution channe!s
isto be considered.
"' market shares, that can be achieved on differently developed markets hy organic
proclucts whereby evaluation of importance of different distribution channels is to be
considered, as well as share of organic farms and share of land area under organic
management,
„ maximal income, that can be reached by organic producers by using different distribution
channels on differently developed organic markets and considering the interest of
consumers.
"' market share by using different disliibution channels on differently developed organic
markets whereby considering achieved market share on individual distribution channel
and the interesi ofthe consumers.

„

With linear programming we also have considered the difference development of organic
markets. We used the dala of evaluation ofthe importance of individual distribution channels
(Padel et al., 2003). We defined the quoted data of achieYed market shares on differently
developed organic markets as the average of the countries taken into consideration by their
share in the total food sales in the year 2002 (1nternational Trade Centre, 2003). We
summarized the dala on income share achieved by producers using different distribution
channels, by the results of survey distributed in Slovenia in the year 2000. The survey
included all Slovenian organic farms (Bratuša, 2000). Survey results gave us <lata on the
interests of consumers (Kvas, 1999). We obtained the <lata about the shares of organic fanns
and share of land area under organic management by doing sums of average value of
countries taken into consideration, which consist of individual groups with varied developed
markets (Willer and Yussefi, 2004). We also applied our own calculations with separate
linear programs on achieved market shares on particular distribution channels, on differently
developed organic markets, whereby considering the share of organic farms, share of land
area under organic management and evaluation of different distribution channels (Ulamec,
2005). For a solution of problems with more parameters where the restrictions are passed in
form of linear non-equations we used the method of benchmarking (Winston, 1994 and
Zadnik Stirn, 2001).

186

T
1

Results

We only present here the calculation of market share for different distribution channels in
urban regions; other calculations are in (Ulamec, 2005). Table 1 indicates the <lata of the
maxunal mar~et share of organic .products in total food trade, which can be achieved by
different d1stnbut1on channels
d1fferently de,eloped urban organic markets.
Here _we took into accou?t ~he value of different distribution channels applying the current
s1tuatlon. The only restnctlon 1s mqrk 5, which represents the highest possible value of
importance of separate distribution channel.
Table l · lmportance of d1fferent distribubon paths for reachmg maximal market share on differently
developed urban organic markets
IDistribution channels /Stage of organic Established market Growing market Emerging market Constraints
market development
3,8
3,9
2,9
Multiple retailers
5
l,9
1,9
[)irect marketmg
5
'

"'

Specialist organic shops

2,4

2,4

2,7

iOther shoos

1,7

l,6

2,1

leatering/public services

0.6

2

2,5

1,1

Market share

l

0,3
1

0,6

5
5
5
max

Mathematical fom1ulation of the problem in the form of LP:
X1 ~ 0,
2,5 X1
3,8 x,
1,9 X1 +
2,4 X1 +
l,7 X1 +
0,6 X1 +

X2 ~ 0. X3 ~
l,lx2 +
3,9 X2 +
2 x,
2,4 X2 +
1,6 X2 +
2 X2 +

(1)
0,6 X3
2,9 X3
.9x 2
2.7x1
2,1 X3
0,J X3

~

max
5

~

5

~

5
5
5

~
~

The solution ofLP (1):
Maximum value of objective function = 3,29
variable
value
X l
1,3
X2
.00
X3
,00
slack
value
S1
,00
S2
2,50
S3
1,84
S4
2,76
S5
4,21
constraint shadow price
Cl
,65
C2
,00
C3
,00
C4
,00
C5
,00

We introduced the unknowns x 1, x2, X3. The x 1 represents the maximal increase in market
share of organic products on developed organic market, the x 2 represents the maximal
growth of market share on organic market, which is in stage of progress and growth, and the

187

x3 represents maximal increase in market share on organic markets, \\ hich are based on
starting development level.

The solution of problem (1) tells that market share of organic products produced through
different distribution channels on urban organic markets is 3,29 % of total food tra<le in
European countries. The result given by linear program ( 1) also shows that it is possible to
reach the maximal market share on developed organic market and this is in accordance with
expectations. On the developed organic market we could reach the maximal market share by
growth of 1,31 (value x: ). The increase of the developed organic market depends on
development of organic markets that are currently in stage of progress and growth. In
optimal solution marketing the condition is fulfilled with multiple retailers. At direct
marketing we consider only half of the rnlue of importance of distribution channeL Optimal
solution of linear programme increases the value of impmiance on this distribution channel
in comparison with the value given by experts. To reach the maximal market share within
specialized organic shops the value considered is 3, 16 of that particular distribution channeL
Available is still 1,84 of the value. At other shops there is 2,76 avai!able value, at catering
the value is 4,21 of importance of distribution channeL
The dual programme indicates how the maximum value of objective function could change
if we change constrains - distribution channels: u 1 = 0,65. u2, U;. tL1. us= O. If the marketing
importance through multiple retailers increased for one more unit, then the market share
would increase for the additional 0,65 %. ln case of increase of importance of other
distribution channels there would not come to increase of market share.
In countries with developed organic market the market share of organic products represents
0,8 % - 2,5 % of the total food market We believe that the calculated market share,
considering predicted growth of 15 - 30 % per annum, will be reached within few years.
With linear programming we calculated that the market share that depends on marketing
through multiple retailers could reach 3,29 % on total food market in urban regions. The
increase of developed organic market is possible to reach with the development of other
lesser-developed organic markets. Within the optimal solution the condition for the
importance of marketing through multiple retailers 1s fulfilled. The optimal solution
considers only 4,6 % of the included organic farms. There is only 2,6 % of land under
organic management spare (Ulamec, 2005). On developed organic markets there are 26 % 90 % of all organic products marketed by multiple retailers (McDonald, 2004).
With linear programming which we do not present here due to the limited space we have
calculated that the market share within direct marketing can reach 4,3% of the total food
market. The only condition to reach the market share is that it increases for 1,72. In the
optimal solution we get partially unrealised condition of importance for direct marketing.
Included in the solution is 3,11 worth of value. Optimal solution does not include the total
value of organic farms shares. Included are 6,04 %, to be used there are still 1,96 % of
organic farms (Ulamec, 2005). Direct way of marketing is characteristic for the beginning of
development of organic markets. In the developed organic markets' countries the direct sales
is of the lesser spread distribution channels, as it has from 7 - 19 % share. In Slovenia there
are 85 % organic farms that use this distribution channel (McDonald, 2004).
According to the optimal solution the current market share of organic products within
marketing them in specialized organic shops could reach 4,3 %. If the developed organic
markets increased for 1,72, then maximal market share would be achieved. Optimal solution
of the linear programming implements the condition of the average share of land under
organic management in total. In the optimal solution there is partly unaccomplished

188

condition on 1mportance ofmarketing '-"ithu1 specialised organic shops. In the solution there
are 4. 14 worth of value, una,ailable are stil] 0,86 % of this distribution channel (Ulamec,
2005). At the revue of marketing data within the specialised organic shops we have found
out that in severa! countries with developed markets this particular distribution channel has
developed differently. ln some countries with developed organic markets it has not got a
major effect. The difference can be even up to 4 5% (McDonald, 2004).
With optimal solution the market share of organic products can reach 9,048 % with regard to
the total food market in urban regions, inclusive the use of different distribution channels on
differently developed organic markets, as well as considering the created income share on
separate distribution channels and buyers' interests. In linear programming we have
cousi<lered the maximal value of the individual distribution channels, which we calculated.
This is also the reason for reaching such a high market share, as ali three most expanded
dist1ibution channels are considered. The obtained results present us the final market share,
which the organic products could reach ( Llamec, 2005 ).
Wishing to consider the importance of distribution channels from consumers' point of view
we have used lhe models of the goal programming (Zadnik Stirn. 2001 ). In the prog:ramme
we included ali parameters, which we had used or calculated in the previous linear
programrning. In the optimal solutions we came \·ery close to the consumers' wishes on two
from three distribution channels. The results of goal programming reveal that in urban
regions we can consider 48,65 % of consumers' wishes, in rnral regions but up to 60,2 %.
This is the way to enable fulfilling the expectations of the consumers, who wish to buy the
products in specialised shops as well as through the direct marketing. In models we did not
wish to limit the achievement of the maximal market shares neither with shares of inclusive
organic farms nor with land under organic management The data demonstrate that both used
parameters are higher than the current situation in Slovenia. We have to emphasise that
random effects are not considered in calculations, although we are aware that their influence
can be very important (Ulamec, 2005).

Conclusion
With benchrnarking of European organic markets we have discovered that with forming the
measures for stimulation of organic farming in Slovenia we need to pay more attention to
organic fa1ms with market production. With benchrnarking we confirmed that for the
development of Slovenian organic market the insufficient supply ofproducts as well as their
production resources are very limited. The surveyed farmers use the same means of
marketing, which were used before tuming into organic farming. For the further market the
surveyed farms offer very small part of their production. Organic farmers will have to stand
on the market far more organised if they want to come closer to the potential consumers. The
consumers often do not distinguish between organic products from other products, which
appear on Slovenian market. In Slovenia the acknowledged selling channel are multiple
retailers. The selling channels are expanding within the development of the organic market.
However it depends on developing market which selling channel is the most important.
Organic markets at their emerging stage do not use all their given possibilities.
W e have also discovered that on developed organic markets new challenges are to be looked
for, as within the investigated parameters the further growth is not possible. The results
directed towards the future and which are based on development of all discussed organic
markets prove this statement. Consecutively the differences between them are becoming
smaller.

189

The models which are based on modified units of more linear programrnes prove that more
developed markets currently grow mainly with extension into the undeveloped regions. To
prepare the strategy for the adequate distribution channels we have to consider the potential
consumers as well as the current situation on the field of organic farm ing. The forming of the
strategy for the distribution channels is dependent on more parameters. The generated model
is based on consideration of optimal parameters of European organic markets with the
emphasis on S!ovenian consumers. Based on goal programming we have proved how
important it is to consider the interest of the consumers. They represent the most important
factor in development of organic market and herewith they help in promotion of organic
farming, which is directed to marketing production.

The gth lnternotional Symposium on
Operational Research In Slovenio

References
Bratuša A. 2000. Načim trženja ekoloških pridelkov. Diplomska naloga. LjublJana, Biotehniška
fakulteta, Oddelek za agronomijo: 129 str.
2.

International Trade center. 2003. Overview world markets for organic food & beverages
2004)
(estimates): str. http://www.intracen.org/mds/sectors/organicioveniew.pdf (27

3. Kotler P. 1998. Marketing, management, trženjsko upravljanje. anal12a,
kontrola. Ljubljana, Slovenska knjiga: 831 str.

načrtovanje,

R'OS
Novo Gorico, SLO\IENIA
September 28 30, 2005

izvajanjem

4. Kvas K. 1999. Modna muha ali preudarna izbira? Ona - tedenska priloga Dela, . 9: 8 - 9
5. McDonald P. 2004. The organic food market in Europe. Queens land govemmenl trade &
investment office Europe: 46 str.http://www.dpi.qld.go\.au/extra/pdf/organiceurope.pdf ( 8 maj
2004)
6. Padel S., Seymour, C., Foster C. 2003. Organic marketing initiatives and rural development
QLK5-2000-0l 124 SWP 5.1 Report of ali three rounds of the Delphi inquiry on the European
market for organic food*: 64 str. (23.maj 2004) http:/ 1WW\v.irs.aber.ac.uk/omiard/publicationspdf/DelphiFinal_Report.Oct_2003 D.pdf
7

Uamec P. 2005. Optimiranje tržmh poti ekološko pridelanih kmetiJskih pridelkov. Magistrsko
delo. Ljubljana, Biotehniška fakulteta, Oddelek za agronomijo: 12 str.

8.

Vadnal K., Bratuša A. 2000. Tržne poti slovenskih ekoloških kmetij V: Zbormk simpozija
(Moravske Toplice 14 in 15 december 2000) Novi izzivi v poljedelstvu 2000. Ljubljana,
Slovensko agronomsko društvo: 284- 288

9.

Vadnal K. 2002. Trženjski model oživitve etnične jedi na primeru ajdovega močnika. V: Zbornik
Biotehniške fakultete Univerze v Ljubljani. Ljubljana, Biotehniška fakulteta, Oddelek za
agronomijo: 47-52

10. Watson G.1993. Strategic Benchmarking. New York, John Wiley and Sons Inc„ 269 str.
11. Willer H.,Yussefi M. 2004. The world of organic agriculture statishcs and emerging trends.
Bonn, Germany, IFOAM -International Federation of Organic Agriculture Movements: 167 str.
http://www.ifoam.org (24 junij 2004)
12. Winston W.L., 1994. Operations research: applications and algorithms. Duxbury Press, Belmont,
CA: 1318 str.
13. Zadnik Stim, L. 1995. Razvoj tržnih potencialov slovenskih kmetijskih pridelkov (!.osnutek
kvantitativnega modela - poročilo raziskovalne naloge). Ljubljana, Univerza v Ljubljani,
Biotehniška fakulteta: 18 str.
14. Zadnik Stirn L. 2001. Metode operacijskih raziskav za poslovno
šola za upravljanje in poslovanje: 182 str.

190

odločanje.

Novo mesto, Visoka

SectionS

r
f

HIDDEN LOGISTIC POTENTIALS IN THE COMPANY
1

T Berlec l Kušar'.
Lenart2 • \1. Starbek 1
University of Ljubljana, Faculty ofMechanical Engineering, Ljubljana, Slovenia
2
Institut Josef Stefan, Ljubljana. Slovenia

1

Abstract
The companies operate in a turbulent env.ironment. b1 these circumstances only those companies are
successful which can respond to market changes as fast as possible. The art1cle explains how the
tools for modeling the flow of orders through workmg systems can be used in order to discover
hidden logistic potentials of workmg systems in a company and how these potentials can be
exploited. Presented are the results of finding hidden logisttc potentials of working systems m the
tool-production department company and the measures proposed for exploiting these log1stic
potentials.

Keywords:
working system, lead tirne. inventory, performance, logist1c potentials

1 INTRODUCTION
Analysis of research results regarding inventory of orders, performance, range and lead tirne
of working systems [l
[4]. [5] has shown that the tools presented in Figure l
should be used to model the flow of orders through the company working systems.
.1'1 tool: · ---

·-··

1Layout of the company
1

working systems and

~1200 tool:

· -·

Funnel models of the
company working
I

: lhe order~flow 1/0 card i systems

3~ tool:
Funnel model of the
particular company
worklng system

tool:
Order-fiow diagram of the
particular working system

i 5~ tool:
ProducUon characteristic
diagram of the particular
working system
Mean
RANGE(Wdj
Mean
.EAD TIME (Wd]

Mean

INVENTORY

(Nhj

Figure 1: Tools for modelling the flow of orders through working systems

2 FINDING THE LOGISTIC POTENTIALS OF WORKING SYSTEMS IN THE
COMPANY
The term "logistic potentials" of working systems in a company refers to hidden logistic
potentials of mean inventory (Mm), mean range (~Rm) and mean lead tirne (~TLm) of ali
working systems in a company.
Research has shown that hidden logistic potentials of working systems in a company can be
found in the following ways:

193

the
On the basis of the measurements of the order-flow tiuough workmg systerns
company the mterval P. an order-flow diagram is designed for each working system
5 [6] and actual operating point of a working system is defined.
2) Target operating point of a working system is defined [6
Logistic potentials of the j-th working system are shown
Mean
performance

Figure 2.

Mean range Rm {Wd]

PEfTI

___::.:.--~=---i
l(PEm1)act

[Nh/Wd]

OIT\)~,

Mean lead tirne Tlm , [Wdj

(lfT\)act
Hidden potential s of
mean inventory

" - - - ' " ' - ' - - • 11 the

2.2

Computer-aided finding of hidden logistic potentials of working systems in the
company

Manual calculation and drawing of flow diagrams and production characteristics and
detem1ining the hidden logistic potentials of working systems is rather tirne consuming, so
the "DIAGRAM" computer software was developed for this purpose.
The software was written Java programming language, runs ona PC and it can be added
to any ERP system (figure 3).
Software DIAGRAM

target
of
Definition
operating point of a
working system, i.e. the
target mean inventory of
orders (Itnj)tar target mean
(PEmj)tar,
performance
target mean range (Rmj)tar
and target mean lead tirne
(TLm)tar of the working
on
based
1s
system
equations. developed by P
P.
H.
and
Nyhuis
Wiendahl [6].

Figure 2: Logistic potentials of the j-th working system
Figure 3: Linking the ERP system with the DIAGRAM software [7]
2.1

Procedure for finding and exploiting the hidden logistic potentials of working
systems in the company

Procedure for finding the hidden logistic potentials of working systems can be described by
the following steps:

Step J · Order--jlow measurements through working systems
Step 2. Design of order-flow diagrams for working systems and definition oj actual
operating points of working systems
Step 3. Calculation of actual mean range and mean lead timefor processing ali orders in the
working system
Step 4: Calculation of actual mean invent01y, mean range and mean lead timefor processing
ali orders in ali working systems
Step 5: Designing production-characteristic diagrams of working systems and definition of
target operating points of working systems
Step 6: Calculation of target mean range and target mean lead !ime for processing ali
orders in working systems
Step 7: Calculation of target mean inventory. mean range and mean lead timefor processing
ali orders in ali working systems
Step 8: Calculation of logistic potentials in the working system
Step 9: Calculation of logistic potentials of ali working systems
Step JO: Selection of measures far exploiting logistic potentials of working systems

Using the Diagram software it is possible to draw or calculate for any working system of the
company: order-flow diagram of any working system, actual mean inventory performance,
utilization, range and lead tirne of the working system, production characteristic diagram of
the working systern, target mean inventory, performance and lead tirne of the working
system, logistic potentials of the working systern.
For ali working systems of the cornpany it is possible to draw or calculate logistic potentials
of all working systems in a cornpany.

3 CASESTUDY
Procedure for determining the logistic potentials of working systems was tested in a toolproduction department of ETI d.d. cornpany frorn Izlake, Slovenia.
It produces tools for transforming and cutting, jet rnachines for thermoplastic rnaterials, jet
and press rnachines for duroplastic rnaterials, and autornated assembly appliances.
There are 38 working systerns in the tool-production department.
The rnanagernent decided that hidden logistic potentials of working systems in this
department should be found out in the interval frorn Wd 291 till 350 and that these potentials
should be exploited.
Steps for finding and exploiting the hidden logistic potentials ofworking systerns in the toolproduction departrnent:

Step 1: Measurement of the flow of orders through working systems in the tool-production
department

194

195

r
l

ln each working system the company employees recorded:
., order numbers and dates of order anival at and departure from the working systems,

„ contents of work of orders which arrived at the working system or departed from

Mear,
performance
PEm

and

• number of orders which departed from the working systems.
There have been 2(J

[Nh/Wd]

Mean range
Rm,[Wd]

t
i

Mean lead Lime
Tln, [Wd]

Target operating
point for the
»Milling machine
ALG 200«

orders with 3883 working operations processed.

Step 2: Design o{ orcler-jlow diagram far working systems in the tool-production department
and clefinition ofact11al operating points o{ working systems
The Diagram software was used to draw lhe flow diagram and to calculate actual mean
inventory of orders, relative actual mean inventory of orders, actual mean perforrnance,
actual mean range and actual mean lead tirne for each working system ofthe tool-production
department.
Figure 4 presents results of designing the order-flow diagram for the "Milling machine ALG
200" and calculation of the actual operating point.

i

i

L

-:__

i

~

1

---~-!

E-~~-~---------------'- -~
2_91205 300 305 3 10 316 320 325 330 335 3'10 34<3 360

N

~"

~

~

~

~

00

.

ID

Mean inventory
lm 1 (Nhl

Actual operallng
point

-----1

4. (Rmj)act - 6.06 Wd
1
, 5. (TLmJ)ac, -- 6.06 W d
1

-:s
·~C:

~

Steps 6 and 7: Calculation of target mean range and target mean fead time far processing
ali orders in a particular working system, target mean inventory, target mean range and
target mean lead time far processing ali orders in ali working systems oj the tool-production
department.
These results are used in step 9 for calculation of hidden logistic potentials of aH working
systems in the tool-production department.

j 1 (lmj)act = 48.45 Nh
i 2. (Im,rj)act = 604.42 %
~(PEmj)act = 8 Nh/Wd
1

1~

Figure 5: Results of calculation of the target operating point and drawing the production
characteristic diagram for the "Milling machine ALG 200''

the »Milling machine ALG
• 200«

Histogram of
departures of
orders

10

Target operating
point

1

'100

(RmJ)tar = 1.90 W d

1

i Actual operating point for

4<'.50

(Im,rj)tar =180 %
(lmj)tar = 14.43 Nh

Ji~i

8bb
48

200

Figure 5 presents the results of calculation of the target operating point and drawing the
production charactenstic diagram for the '•Milling machine ALG 200"

Time
[Wd]

k i - - - - - _ _ _ __.:.P_ _ _ _ _ _ ~

Figure 4: Results of designing the order-flow diagram for the "Milling machine ALG 200"
and calculation of the actual operating point

Steps 3 and 4.· Calculation oj actual mean range and mean lead time far processing ali
orders in a particular working system oj the tool-production department, and definition of
actual mean inventory, mean range and mean lead time far processing ali orders in all
worki11g systems.
These results are used in step 8 for calculation of hidden logistic potentials in a particular
working system in the tool-production department.

Step 8: Calculation of hidden logistic potentials in u particulur working system in the toolproduction department
Hidden logistic potentials of the "Milling machine ALG 200" (working system No. 7)
reveal that by transition from actual to target operating points it would be possible to reduce
the mean inventory of orders by 70.22 %, to reduce mean range of the working systems by
68.65 %, and to reduce the mean lead tirne ofthe working systems by 79.04 %.
Step 9: Calculation oj hidden logistic potentials oj ali working systems in the toolproduction department
Hidden logistic potentials of working systems in the tool-production department reveal that
by transition from actual to target operating points of working systems it would be possible
to reduce the mean inventory of orders by 69.69 %, to reduce mean range of the working
systems by 68.55 %, and to reduce the mean lead tirne ofthe working systems by 75.36 %.

Step 5: Definition oj target points and drawing production-characteristic diagrams oj
working systems in the tool-production department
The Diagram software was used to draw the flow diagram and to calculate relative target
mean inventory of orders, target mean inventory of orders, target mean range and target mean
lead tirne for each working system ofthe tool-production department.

Step 10: Selection oj measures far exploiting the hidden logistic potentials oj working
systems in the tool-production department
If it was found in a particular working system that the relative actual mean inventory of
orders is larger than the target mean inventory, then it is possible to exploit the hidden
logistic potentials of the working system either by temporarily increasing capacities of the
working system, or by temporarily stopping arrival of orders, or by a tirne- or space transfer
of a part of a working-system load to a system of similar technology.

196

197

lf it was found m a particular workmg system that the relative actual mean inventory
orders is less than or equal to the target mean mventory then is necessary (by harmonizing
or reducing the order times or by reducing interoperation times) to find new !ogistic
potentials ofthe working system.

REVERSE LOGISTICS l N VALUE CHAIN
1

Ludvik Bogataj and Marija Bogataj
University of Ljubljana
F-mail Mari1a.Bogataj@guest.arnes.si

4 CONCLUSION
Because of ever fiercer domestic and international competition of companies, and transilion
from the market of sellers to the market of customers, the companies have to continuously
increase their economy which largely depends on inventory of onlers, range and lead times of
the companies' working systems [8],[9].
The article proposes a procedure for finding and measures for exploiting hidden logistic
potentials of working systems in a company: these can be found only by continuously
monitoting inventory of orders, range and lead times of working systems. Controlling allows
for a continuous monitoring of actual values ,)f i1n-enlory. range and lead limes of working
systems; if these values deviate from the target values, measures can be taken in order to
bring the actual values closer to the target ones. Only after logistic potentials of working
systems have been exhausted it is necessaiy to remove the barriers (by additional
harmonizatton of order times. or reduction of interoperation times. or manufacturing by
overlapping, or increasing flex,bility of working systems) that prevent further reduction of
mean inventory, range and lead tirne of working systems.
Case study of finding and exploiting hidden logistic potentials of working systems in the
tool-production department of the ETI d.d. company from Izlake has shown that there are
considerable logistic potentials which should be exploited future.

5 REFERENCES
[ l Wiendahl HP.: Load-Oriented Manufacturing Comrol. Springer, Verlag, Berlin, 995. pp.
41-199.
[21 Starbek M., Grum J. Operation lead time control. Robotics and Computer Integrated
Manufacturing 16, 2000. pp. 443-450.
[3] Starbek M., Grum J.. Control of the state of orders on machines. Computers & Industrial
Engineering 40, 2001, pp. 35-49.
[4] Kušar J., Brezovar A., Grum J., Starbek M"' Realistic lead tirne scheduling of operations
oforders. Machine Tools & manufacture, Vol. 44, 2004, pp. 1037-1046.

[5] Starbek M., Grum J., Kušar J.: Realistic lead times of Operations and Efficiency of the
PPC System. Journal ofMechanical Engineering, Vol.48, 2002, pp. 98-111.
[6] Nyhuis P., Wiendahl HP.: Logistische Kennlinien. Springer Verlag Berlin, 1999, pp. 8194.
[7] Berlec T.: Diagnostika pretočnega časa operacij, Magistrsko delo, Univerza v ljubljani,
Fakulteta za strojništvo, Ljubljana 2003
[8] Starbek M., Menart D.: The optimization of material flow in production. lnternational
Journal ofMachine Tools & Manufacture, Vol. 40, 2000, pp. 1299-1310.
[9] Lenart L., Kušar J.: Separable programming in optimal control. Proceeding of the 7th
International Symposium on Operational Research in Slovenia, Podčetttek, Slovenia,
Slovenian Society Inf01matika, Section for Operational Research, 2003, pp. 235-240

198

Abstract: To control the flows in total supply chain the analys1s in frequency dornain is more
compact and assures better visibility of_total chain. Location decision influence lead tirne in any
activity celi of chain and often simultaneously also the contribut10n of reverse logistics. It 1s the
reason that the approach to the analysis
locatwn dec1s10ns m frequenc\ domain is suggested here
indJrectly through changes of lead tirne. llow to choose the best location of activity cells in reverse
logistics pmiicularly'J l!ow the (simultaneous changes)
d1fferent locations along the whole supply
chain influence financial flow which reflects on the Net Present Value (NPV) can be studied using
extended Matena! Reqwrements Plannmg (MRP) model. It 1s extended, mcluding not only
distnbutJon but also recyclmg (reverse logistics) formaltzation. The recycling rate a is ernbedded in
the mode!. The suggesuon how to consider the perturbations of lead times in reverse logistics is
gn-en. The sensitivity analys1s approach to the lead tirne perturbat10ns 1s suggested m case when
recycling is introduced m a supply cham.

Key words: Reverse log1stics. Supply network. Material Reqmrements Planning, Laplace
Transforms, Input- Output modelling. Lead tirne.

1. lntroduction to tbe reverse logistics, studied in integrated supply cbain
Optimal decisions
where to produce, (ii) how to distribute the product and (iii) at what
tirne to order or deliver the items in integrated supply chain can be successfully discussed
and evaluated in transfom1ed environment, where lead times and other tirne delays can be
consider in linear form. The site and capacity selection, as for instance the problems where it
is best to locate a facility and what capacity is needed to achieve the most rapid response, are
discussed more easily in transformed environment. Lead times in the entire supply chain can
be analysed in compact form using MRP and 1-0 analysis 111 Laplace transformed space.
We can consider the activities · integrated supply chain as three distinct parts:
materials management,
physical distribution and
recycling part,
All three parts can be defined in tirne or in frequency space.
Various operating stages in the logistic chain (nodes of the chain) can be represented by a
simple model of a material - transformations or location - transformations processing celi. In
each processing celi a value is added and some costs are incurred. At each processing cell
there is a supply and a demand and often both are stochastic by nature.
Inventories are insurance against the risk of shortage of goods in each cel! of logistic chain.
They are limited by the capacity of each processing node of the chain and transportation
capability of input and output flows and influence all kind of inventory costs„ Ordering
goods (input flow) in distribution centres can be studied asa multi-petiod dynamic problem.
The demand (output flow) during each period has to be considered asa stochastic variable.
The distribution of this variable is often described with a certain probability function (see
Bogataj & Bogataj /6/ or Tang /18 /).
The flows of items in supply chains influence transportation costs and costs of activities
in logistic nodes of globa! economy and consequently the net present value (NPV) of ali
activities that have to be performed in such logistic networks. The costs of item flows

199

between two ceils depend on their location wh1ch can have advantages or disadvantages
considered in detail in /5;, 17/, 18/ and / 0/.
An integrated supply chain includes the purchasing of raw materials, manufacturing with
assembly and the distribution of produced goods to the final clients. A third part which is
more and more often included in the analysis of such integrated system, especially because
environmental restrictions, is the reverse logistics, having the same fom1al prope1ties in the
networks. In a supply chain the key variables that have be considered at each activity celi
are activity level and timing, inventor leve! and lead times between the order tirne and the
moment of the arrival of items in the required activity cell. The managers of a supply chain
have two main goals: (a) to keep the level of inventory in the supply chain as low as
possible, to reduce inventory costs; (b)To move the inventory in its continually changing
fmm from raw material to final product and its delivery through the physical distribution part
of the supply chain to the final consumer as fast as possible.
The final goal is mostly to achieve the maximal net present value NPV of ali activities in
the supply chain and not only to reduce the costs of operations.
Raw materials or parts of supply are delivered and stored in the raw material warehouses
at the production centres. From there, raw material and parts are withdrawn as much as they
are needed by the production centres. There the raw materials are transformed into
subassemblies. The subassemblies are transformed in semi- finished goods and so on to the
last production centre where final products are produced. From there finished goods are put
into final product storage. From storage the finished goods are delivered to the distribution
centres, or to their own warehouses or directly from there to retail outlets. There the goods
are available to the fina! consumer. The third part, which can be considered in the same way
as the first (material management) and the second (business logistics) part of the logistic
chain, is reverse logistics part. Here the question appear when to include activities of the
reverse logistics in a supply chain and how the increased percentage of recycling items in
production influence NPV of to tal integrated chain. This is particularly difficult to evaluate
when lead tirne is varying The compact analysis in frequency domain, where lead tirne
perturbations can be studied in linear form, can help to give also direct evaluations of EU
directives about environmental protection by recycling and remanufacturing activities.
A subject closely tied with production and inventory control is material requirements
planning (MRP). It consists of a set of logically related procedures, decision rules and
records designed to translate a master production schedule into tirne phased net
requirements. To fulfil the planned coverage of all these requirements, a schedule needs to
be implemented for each activity cell and its component of inventory items. A modem
approach using Input - Output analysis and Laplace transforms has been introduced by
Grubbstrom (1967) and his Linkoping School (see: 1989, 1990, 1994, 1998 ... ) The list of
papers considering this problem is available at http://www.ipe.liu.se/rwg/rwg.htm Such an
approach gives us good theoretical and practical results also for the extension of the analysis
to the distribution and especially to the reverse logistics part of supply network.
The concept of all three parts of an integrated logistic chain can be illustrated as in Fig.1.
From one stage to another, physical characteristics of the items and their locations are
changing step by step. In the process of globalization, the distances between the pairs of
production cells or between production and distribution cells are increasing, which
influences the transportation costs and a lead tirne for the delivery ofitems.

MatenaL f1uw

M aierials management (MRP require
Dusiness loghtics 1,DRP required/

-~
- - . ; ; ; ; ; ; - ~ irai1®
Fig. 1: The prese11tatio11 of tl,e itmer structure i11 logistic chai11, haviug prod11ctio11 pari, distrib11tio11 pari
a11d reverse /ogistics.

2. Numerical presentation of a supply chain containing reverse logistics
Fig.2 presents a part of dishibution and reverse logistic chain where 00 units of product A
are packed in one unit (let it be the palette) of A 1 and transported to the certain destination
0 1 At the destination n 1 the palette A 1 is opened and two different products are made A 21
and An. From one palette of products A po/o of pallet A 21 is filled with p good products
and (l 00 - p )% = q% pallet of defected products A 22 are collected.

A21

100

1

1

A22

X

A
Fig. 2: An example of a simple arboresce11t product structure, 100 1111its ofproduct A are packed in one tmite
of A 1. Unit A 1 is repacked into p% ofpallet containing good products A 21 and q% ofpallet containing bad
products col/ected in pa/let A22 •

Let us suppose that 3 components of C are needed for production of 1 component of B and 4
components of C and 2 components of B are needed for 1 product A. This is described by
Input - Output matrix H . In case when B in this production is covered

200

201

r
OOa % by recycling of An, the extended lnpu!
1s:
A
B C A1

in this supply chain

Omput mmrix

rhe total inventory in tirne domain can be written as the sum ofthe available inventory and
the allocated component stock:

1

o
H/O/

1

O; 00

o

H
=llR(a)

(l)

Here production and

(2)
distribution levels as control variables are collected into a vector

P'"' (s):
ptot(s) =

3. Supply chain lead times in fundamental equations of integrated supply chain
containing reverse logistics

(3)

For describing tirne developments and tirne lags of the relevant production, demand and
inventory variables in a compact way, the use of Laplace transforms for the production part
of the supply system and input- output approach has been suggested by Grubbstrorn /11-16/
and could be extended in to tal chain by introduction of l_O rnatrix described above. He
introduced the fundarnental equations which could be extended as follows. We have N items
01
10
(produced, distributed and recycled in N activity cells) in the system. Demand D \ stock S
01
and production P are presented by N dimensional column vectors. These vectors are
Curnulative values are
transformed to frequency space and are denoted by tildes or by

where x:

The lead tirnes

Ti,

j

= 1,2, ... N

Curnulative net
production

(I - H 101 (q,a)P / s lnitial inventory

Generalised input
matrix

jjtol

A vailable inventory
Curnulative gross
production
Cumulative interna!
dernand

=

(q,a) =

Hlol

(q,a)r

'ii_tot

(s)

]5101

(s)/ s

H/ol

Curnulative interna!
dernand together with
its required tirning
Allocated cornponent
stock
Backlogs

(q,a)P' 01 (s)/ s External dernand in the

stol {O)

H 101 (q,a)P(s)I s

101

(s)

jjtot

(s)

jjtot

(s)

pT

cP'"'(p)

K'"''i"'(p)v'"(p)

-

K/011101

(p )v/O( (p) + M(NPV(L'1 T' a ))

Let us note that in this case we can split the perturbed value of expected net present value
EP (NPV) into unperturbed part E(NPV) plus the effect of the perturbations M(NPV),
which is particularly useful in the sensitivity study to get a sirnple, quick and easily
achievable estirnate of the sirnultaneous effect of response tirne perturbation in the case of

frequency space

202

E(B'"'(T))e

101
distribution activities in the case when complex frequency equals to p, ?' (p) is the lead
tirne rnatrix, ? 101 (p) = diag( er,s •. erNs ), which depends on the tirne distance between two
activities in production or distribution cells and on input or output capacities which define
setup, production or distribution tirnes, which can be changed due ti changing location and
capacities by 3r 101 (p) = diag( er,s L'1 Ti,· ., erNs L'1 r N) . For details see analogous production
part / 16/. A sensitivity study of the influence of long term changing locations of activity cells
(the rnatter of strategic logistics and strategic alliances) on the lead times and consequently
on the expected net present value of the activities in the logistic chain can be carried out in
the presented cornpact form frorn 6/ directly:
EP(NPV) = r(E(J5tot (p))- pE(Btot (p))- E(Btot (T))e-pt -čJ>tot (p)(5)

ptot (s)/ s

A.

x0

different production levels and different locations which appear at different production and
101
is vector of set- up costs at production cells and
distribution cells in the supply chain, K
distribution cells for each activity (for exarnple, in transportation there are always some fixed
costs, which do not depend on the transportation volurne ). Here č is the row vector of unit
incrernental production costs, v'"' (p) is the vector of all completion tirnes of production or

r.

Cumulative deliveries

-rh production process.

(4)
at
products
ofthe
prices
sales
are
r
of
components
as
where E(.) denotes expected value, r1

create the

having e'' in the jof activity cells. They are represented by a diagonal lead tirne rnatrix
th diagonal position, where s is Laplace frequency. We denote the generalized input rnatrix
by if 101 = H 101 r (being disturbed by rp {s)= r(s) + lr(s)s ), which describes component
requirernents in each production cell together with their required timing. Grubbstrom 116/ introduced and developed the following notations for the production part of the logistic
chain, which are here extended for entire supply chain including reverse !ogistics and
therefore rnarked by ( )1°1•
stol

= 1,2, ... ml, denotes the activity level for

E(NPV)=r(E(D"'(p))-pE(B'"(p))

dynarnics of interna! dernands and are changing especiaily because of changing the location

Total inventory

,

i=m1+ 1111+2, .....m1+m2 , denotes the activity leve! for -th distribution process but the
levels ofrecyc1ing activities are detem1ined through lhe subrnatrix R(a) presented in
The expected net present value of the production part derived frorn 116/ has got the
extended form on the tirne horizon f as:

(O.

denoted by bars and Laplace inverse by ( '

1

j

203

non-robust perturbations ( l'l r. • o. 'dk. k= lL .. X . )and reverse logistic ratio a in R( a). In
this case the perturbation ofthe net present value is:
M(NPV(tw,a)

+

=

cl

1

/ "

~

kz)Ul

H (a)r(p)J

(a)r(p))* I ~i(p)P'(p))p +

k;ei

-1cli

-- . )
~- l<j (/ -II .(a)r(p)
H " ( a )/1 r (p )p

-1 -

P'(p)-

-Kf:.r(p)v(p)p + I'(o(Lir(p)p))
(6)
In (6) lhe amounts of end items and the timing oftheir production in unperturbed distribution

environrnent are ali captured in the vector P'(p). The original input matrix in (6) is denoted
by H

In the matrix H" the lower M rows of the extended matrix of H with capacity

requirements are included. In (6) the term

r(o(i\r(p)p))

is of lower order than

o(Lir(p)p).

4. Conclusion
Using (6) offers us the opportunity to study the influences of delay perturbations and reverse
logistics ratio simu!taneously. As we can see, a similar procedure can also be introduced to
study more robust perturbations. In this case the changes in (6), which are included in the
perturbation part of expression, have to be calculated more accurately, considering the
sufficient number ofterms in the appropriate expression ofTaylor series in lE(NPV)
The obtained sensitivity results for non-robust pe1turbations are paiticularly useful in
practice when it is necessary to make a quick estimate of the risk value owing to the
unexpected perturbations of lead times which appear in a certain activity celi at a certain
location and compensation of it with changing reverse logistics ratio a . This approach helps
to balance the costs of a certain level of inventories as the insurance of lean and/or smooth
production and supply.
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2001, pp. 25-29.
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chains ". Lipičnik, M. (Ed.). Transport, promet, logistika, Portorož, Slovenija, 2.
Annual proceedings, 2000, pp. 11-16.
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The Intemational Society of Logistics, 2001, pp 1-12.
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204

L."

6. Bogataj, M._. Bogataj.
Compound distribution of demand in Iocation - inventory
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11. Grubbstrom, R.W., Molinder, A., Further theoretical consideration on the
relationship between MRP, lnput-Output analysis and multiechelon inventorv
'
systems", Int.J.Prod.Econ., Vol. 35, 1994, pp 299-311
12. Grnbbstrom, RW. "On the Application ofthe Laplace Transform to Certain
Economic Problems", Management Science, Vol.13, No.7, 1967, pp. 558-567
13. Grubbstrom, R. W., "Stochastic Properties of a Production-lnventory Proces s W ith
Planned Production Using Transform Methodology", Int.JProd.Econ, Vol 45. 1996,
pp 407-419.
.

14. Grubbstrom, RW., "Material Requirements Planning and Manufacturing Resources
Planning", in Wamer, M., (Ed), International Encyclopedia ofBusiness and
Management, Routladge, London, 1996.
15. Grubbstrom, R.W., Bogataj, L. (Eds).Input-Output Analysis and Laplace Transjorms
in Material Requirements Planning. Storlien 1997. FPP Portorož 1998.
16. Grubbstrom, R W., "A Net Present Value Approach to Safety Stocks in Planned
Production ",IntJ.Prod.Econ, Vol. 56-57, 1998, pp 213-229.
17. Segerstedt, A., "A Capacity- Constrained Multi-leve! Inventorv and Production
Control Problem" Int.JProd.Econ, Vol. 45, 1996, pp 449-461 .·
18. Tang, Ou, "Application oftransforms in a compound demand process" Promet-

Traffic-Traffico, Vol. 13/5, 2001, pp. 355-364.
19. Waller, D.L,"Operations Management, A Supply Chain Approach, ITP, London, 1999.

205

lNTERMLNICIPAL GRA VITY MODEL OJF SLOVENIA
Samo Drobne· and MariJa Bogataj'"
University of Ljubljana. Faculty of Civil and Geodetic Engineering, Ljubljana, Slovenia,
sdrobne@fgg.uni-lj.si
university of Ljubljana, Faculty of Economics, Ljubljana. Slovenia, marija.bogataj@guest.ames.si

Abstract

In this paper. an extended Lowry-like intermunicipal model of daily commuting in Slovenia is
presented. The regression model has been already implemented as a web-based application, where
citizens could participate and give an opinion to policy-makers about the forecasting role and results
ofthe model.
Keywords: gravity modeL spatial model, commute, mumcipahly, Slovenia.

l INTRODUCTION
As a cause and a consequence of social change, human migration and periodical (daily or
weekly) commuting to job, school or for other activities are regarded as one of the most
important factors underlaying the demographic and socioeconomic composition of regions of
di fferent sizes ( varying from states and administrative regions to municipalities and/or
communities). Thus, for anyone attempting to analyze the general process of regional
change, an understanding ofinterregional migration is vital [l]. Cadwallader [9] has pointed
out, that policy-makers have become increasingly aware of the role of migrations as
migration of the human resources for any production or services and in the context of any
other socio-economic issues, especially as regional growth.
The growth of regions relates closely to population growth, which is mostly a result of
migrations and daily commuting. The migration between regions can be slowed down by
daily commuting, which is becoming a surrogate for migration, if the commuting is bringing
higher social well-being than any migration. If the contacts between regions, because of
improved transportation abilities and removed barriers, are becoming less expensive and
easier, the inhabitants often prefer dai ly commuting [ 13 J.
In our previous investigations [2,3,4,5,6], we proved that daily commuting also has an
important role in the context of a socio-economic issue in Slovenia. In [7], we summarized
all results: we investigated the main factors of interregional migratory and daily commuting
flows in Slovenia, previously predicted in above mentioned papers. For most regional
anaiysis, interregional extended gravity model should be precise enough: it can even predict
how regionalization and changes of regionalization influence gravity. However, more
precise forecasting results about daily commuting can be calculated only using smaller
spatial units than regions. For this purpose, we calculated an extended Lowry-like
intermunicipal model of daily commuting in Slovenia for 2005. The model has been already
implemented in web-based application, where citizens could participate and give an opinion
to policy-makers about the forecasting role and results ofthe model.
2 PROBLEM FORMULATION

The gravity models, which belong to the family of spatial interaction models, offer a
framework for building integrated models of land use compared to the econometric models.
The Lowry model, designed in 1964 for the Pittsburgh metropolitan region [12] and revised
severa! times later, is the basic in this group of models. However, Lowry-like models miss
many other aspects of integration; the impact of the transportation network on the land use,
which were particularly emphasized in [7], where we calculated an Lowry-like interregional

206

207

model of daily commuting. Howevec for more precise forecasting results, we studied
and irnplemented thern
mtennumcipal daily commuting flows for persons in
the gravity model.
Let us study 192 municipalities of Slovenia with a total of population
municipality and with
the
P = 998 079 inhabitants. We denote with i the
destination rnunicipality. The number and strncture of external daily commuters, persons in
employment,
were
obtained
from
Census
2002
data
and
was
DC;,J = DC = 440 299 (i -cf. j) , Data on Slovenian population as well as the number of

r
TDCs

.

MUNI'I'
= .k. DC ! f
p

Following to the principles of the gravity models, the distances between studied pairs of
municipalities should be included in the modeL In [4,7], we proved that a new number of
daily commuters DC'1,j 1s:
TDCs
1

LL

ne· = d''·-(Y'
!,}

where

O

(3)

,.;

j

external daily commuters were obtained from statistical data (Census 2002 and Statistical
news l
Figure shows population and
commuters persons
employment in
S!ovenian municipalities in 2005.

(2)

is a new, expected (improved) distance between municipalities after

investments
the network,
denotes that a distance can be treated in different ways
(Euclidian, road or time-spending distance), and K is an exponent - usually defined in the
regression analysis between actual distance d;,;0 and ratio of actual to hypothetical daily
commuters
DCs

hJ

DCi,j

(4)

= TDCs
l,J

The model
enables the forecasting of number of daily commuters between pairs of
analyzed spatial units. However, to extend gravity model some additional economic
coefficients were added to the model. To be included in the model, the following coefficients
was investigated:
K . . = GDP(i) K
!JDP.,
GDP(,W) ' GEAR.i

1

5500 Daily cornmuters • persons in employment

100 Kilome\e1s

50

25

J

Figure 1: Population and daily commuters persons in employment in Slovenian
municipalities in 2005
The simple gravity model helps us determine the expected number of daily commuters
which originate in municipality i and te1minate in municipality under the assumption that
no costs and no tirne are involved in undertaking a trip from one municipal to another (for
the staiiing analysis, the friction of distance is zero ), If there are no mentioned differences
among the municipal, we may expect for a particular person in employment, living in
municipality i, the probability to travel to municipality j is P / P, where P1. is the total
J'

•

population of the j -th municipality. By homogeneity assumption we can estimate tpe
number of daily commuters between pairs of municipalities.
1
Now, we can define the daily commuting coefficient between the municipalities k~ iN:
ewN =DC= 0.13
DC

=

GEAR(i)
K
GEAR(SI) ' EMP,i

EMP(i)
EMP(SI)

UEMP(i)
KUEMP,1

UEMP(SI)

where GDP(i) is Gross Domestic Product per capita in municipaiity 1, and in Slovenia
(SI) respectively, GEAR(i) is an average gross earning per person in municipality i and in
Slovenia SI) respectively, EMP(i) is the number of persons in employed in the
municipality and in Slovenia
) respectively, and UEMP(i) is the level of registered
unemployment in the municipality i divided by the leve! ofregistered unemployment in the
country, and UEMP(SI) is the leve! of registered unemployment in Slovenia, and
i=
92. The model was tested for Euclidian, the shortest road distances, as well as for
the quickest time-spending distances.
3 PROBLEM SOLUTION

The Census 2002 <lata enable the study of the volume of daily commuting to job and home
between pairs of municipalities in dependence of time-distances between the municipal
centres. Time-spending distances were calculated in transport planning software
OmniTRANS. It is impossible to show the intensity of all flows on a map; Figure 2
represents the intensity of daily commuting flows in 2002 for the workers employed out of
their municipality of living - only for Pomurje region.

(1)

p

Following the principles of simple gravity model and assumption mentioned above, the
hypothetical number of daily commuters - persons in employment between the pairs of
municipalities (originating in i and terminating in j, i -cf. j), is then:

208

1
The smallest spatial unit, that Gross Domestic Product was defined in Slovenia in 2005, is statistical region.
So, the presumption was, that ali municipalities in the same region have the same GDP.

209

Daily COrrn'IUleJS.
pe1sons in en-ployment
1 · 40

4 MODEL APPLICATION

41-140
-=- 141 300

lJsing the regression model of daily commuters - persons in employment, who trave!
between Slovene municipalities (5a) and data on new, improved time-spending distances
after highways will be finished in 20 3, new volumes of daily commuters were evaluated; a
model of calculation of time-spending distances in a raster-based GIS environment is
discussed in [10]. Figures 3 and 4 shows the growth of daily commuters - persons in
employment due to shmtening of time-spending distances (new highways) in municipalities
of origin in Slovenia from 2005 to 2013, and in municipalities of destination respectively.

301 · 580
581 1685
Municipal centre
DMunicipafity
i=ssa

@

Figure 2. Intermunicipal flows of daily commuters - persons in employment in Pomurje
region in 2005
In the regression analysis of daily intermunicipal commuters - persons in employmenl, who
trave! by car, only the time-spending distance and following coefficients gave the best results
(R 2 is the highest and P-values are the lowest):
•
DC ..
1,1

aPa, P.al

.Kr,
.Kr, .Kr,
' 1 Kr,GDP,1.Kr,
= d(t)/J.
UEMP.j
UEMP.1
GEAR,J
GDP,J

(5)

Go'.'it:h of daily corrrnuters
- persons in employment
in municipality of origin
0-250
D
D 251-500
D 501 -1250
li!II 1251- 2500
li!,ll 2501 5800

1,J

We got the regression parameters for the intermunicipal daily commuting flow equation,
where R-square is 58 % for 7099 observations and where d(t) is the road time-spending
distance needed for travelling by car; see Table 1

Table J .· Extended gravity model coefficients and summary output for intermunicipal daily
commuting ( DC;~ 1 ) - persons in employment in 2005.
Name o/ Coefj/cie11ts
Ln(a)

a,
al

Y,
Y,

r,

r,

r,

Multiple R = 0.76

Valueof
/
, Coefj/cie11ts in
(5)
-2.0567

tS/al

P-1 1a/11e

lo11,•er 95%

Upper 95%

-2.4012
0.3897
0.6907
-1.7674
-i.0699
-0.3923
0.4361
-0.0348
-0.2303

-1.7121
0.4395
07446
-1.6940
-0.7272
-0.0588
0.8607
0.1129
-0.0772

Vallle o/
Coefj/cie11ts
,vit/,0111

-11.7022

0.0000
0.0000
0.0000
0.0000
0.0000
0.0080
0.0000
0.2999
0.0001

32.6054
52.2147

0.4146
0.7177
-1.7307
-0.8986
-0.2255

-92.4236
-10.2788
-2.6514

0.6484
0.0391
-0.1538

5.9864
1.0367
-3.9385

25

"'

100~1orrelers

Figure 3. Growth of daily commuters - persons in employment due to shortening of timespending distances because of new highways in municipalities of origin in Slovenia from
2005 to 2013

r,

-2.0676
0.4 i 61
0.7164
-1.7303
-0.9526
-0.2229
0.6488
-0.1438

R Square = 0.58
Go'.'it:h of daily corrn1uters - persons in employment
in munidpa!rty of destination
D 0-200
D 201-300
301-900
D

The intermunicipal regression model of daily commuters - persons in employment is thep:
0.13P o.4t P. o.n
K-o.9o K-0.23 Ko.6s Ko.04 K-o.1s
1
1
DC~. =
(5a)
UEMP,j
UEMP,i
GEAR,j
GDP,j
GDP,i
d(t)~·'.3
l,J

D

901-6000

ii!11 6001 - 16 600

1,/

Using data from Census 2002, the results of regression analysis show that coefficient y4 is
not statistically significant (see also Table l ). It was expected that unemployment in the
municipality of origin was one of the most important factors pushing inhabitants to commute
to their work. However, excluding this coefficient from the analysis does not change a lot the
values of other coefficients (compare values in second and last column in Table 1).

210

25

.,

Figure 4: Growth of daily commuters - persons in employment due to shortening of timespending distances because of new highways in municipalities of destination in Slovenia
from 2005 to 2013

211

The model (Sa) is implemented also as a web-based application of interrnunicipal grav;ty
model of Slovenia l l - interested reader is invited to test at http:l/193.2.92.34/ where
decision-makers and citizens can work together to find constructive solutions · the decision
making about investments in the infrastructure, municipality growth and development.

WILL THE FUTLRE MOTORW AY NETWORK IMPROVE THE
ACCESSIBILITY TO ADMINISTRATIVE CENTRES IN SLOVENKA?
Samo Drobne'. MariJa Bogataj", Dejan Paliska '" and Daša Fabjan'"
University of Ljubljana, Faculty ofCivil and Geodetic Engineering, Ljubljana, Slovenia,
.. . .
.
sdrobne@fgg.uni-lj.si
Lmv~f~1ty of Ljubljana, Faculty of Economics, Ljubljana, Slovenia, marija.bogataj@guest.arnes.si
University of Ljubljana, Faculty of Maritime Studies and Transportation, Portorož,
Slovenia, dejan.pahska@fpp.edu, dasa.fabjan@fpp.edu

5 CONCLUSIONS
The results of intermunicipal gravity model of daily commuters - persons in employment
shows that the reduction of time-distances between Slovene municipalities increases the
daily commuting flow. Though it is expected that high tirne delays appear between the
spatial investment and the reaction of the population, this significance is already recognized
and proved in [7]. However, these results require a further study of the influence of the
newly built motorways in terms of employment and commuting between the spatial units of
different size in Slovenia. Finally, we need to ask ourselves how will the new investments in
roads, which will induce the transit traffic in Slovenia, influence the demand of land and
land use in the central places of Slovenia. This is one of the main parameters needed for
successful spatial planning.

Abstract
A two-stage raster-based GTS approach for evaluation of contmuous accessibility fields bas been
presented. Tbe proposed metbodology can be essential for a complete mtegration of accessib1ltty in
many d1fferent studies - fully integrated in GIS environment - at macro or mezzo leve!. In tbe model
applied. tbe improvement of time-spending distances to administrative centres due to new motorways
bas been studi.ed for tbe years 2005 and 2013.
Keywords: accessibility, trave! tirne, administrative centre. GIS, Slovenia.

References

1 INTRODUCTION

[ 1l Anjo mani, A., 2002: Regional growth and interstate migration, Social Eco110111ic Planning Sc1e11ce.
(36):239-265,
[2] Bogataj, M., Drobne S., Bogataj L., 1995a: System approach to urban struchue and growth. In: Rupnik
V. and M. Bogataj (ed.), SOR '95 proceedings, International Symposium on Operational Research,
Portorož, 37-46.
[31 Bogataj, M., Drobne S., Bogataj L., 1995b: The influence of investment and fiscal policy on growth of
spatial structure, Suvremeni promet, Zagreb, 15(5):239-245.
[4] Bogataj, M., Drobne S., 1997; The influence of investments in highways on gravity and interaction in
Slovenia. In: Rupnik, V., L. Zadnik Stirn and S. Drobne (ed.). SOR'97 proceedings, [nternational
Symposium on Operational Research, Preddvor, 55-60.
[5] Bogataj, L., Bogataj, M., Drobne, S., Vodopivec, R., 2003: Management of investments in roads and in
capacities of border regime to induce the flow of human resources in and out of region, In: Zadnik Stirn
L., M. Bastič and S. Drobne (ed.), SOR '03 proceedings. International Symposium on Operational
Research, Podčetrtek, 43-46.
[6] Bogataj, L. Bogataj, M., Drobne, S., Vodopivec, R., 2004: The Management oflnvestments m Roads and
in Capacities of Border Regime to Improve the Human Resources in Regional Development. Suvremeni
promet, Special issue, Mostar, Vol. 30, to appear.
[7] Bogataj, M., Drobne S., 2005: Does the improvement of roads increase the daily cornrnuting? Numerical
analysis of Slovenian interregional flows. In: Zadnik Stirn, L., Indihar Štemberger, M., Ferbar, L.,
Drobne, S., Selected Decision Support Models for Production and Puhlic Policy Problerns, Slovenian
Society Informatika, Ljubljana.
[8] Briassoulis, H., 2000: Analysis of Land Use Change: Theoretical and Modeling Approaches. Regional
Research Institute, West Virginia University.
[9] Cadwallader, M., 1992: Migration and residential mobility: rnacro and micro approaches. University of
Wisconsin Press, Wisconsin.
[10] Drobne, S., Bogataj, M., Paliska, D., Fabjan, D., 2005: Will the future motorway network improve the
accessibility to administrative centres in Slovenia?. In Zadnik Stirn, L., Drobne, S., Proceedings of
SOR'05, Nova Gorica, here.
[11] Gaiduchenko, O., 2005: Development ofWEB-based gravity model for forecasting the cornrnuters' flows
in Slovenia. Diploma work, Technical University ofRiga, Latvia.
[12] Lowry, I. S., 1966, Migration and metropolitan growth: lwo analytical models. Chandler Publishing
Company, San Francisco.
[13] Nijkamp, P., 1987: Handbook of Regional and Urban Economics. Vol. 1, Regional Economics, Norih Holland.
[ 14] SURS 2005: Statistical Office of the Republic of Slovenia, URL: http://www.stat.si/eng/index.asp, <late
accessed: 03-May-2005.

212

!he

concept of accessibility can be used in many contexts and in different ways; for
mstance, as a goal in transportation policy, as measures in rnral development policy, as an
indicator of rural deprivation, and as a variable in location analysis. Accessibility is one of
the main issues in development and physical planning policies, which are mostly concerned
with equity and a better distribution of people and activities in the te1Titory. So, accessibility
is considered, by planners and other technicians, to be a key variable for territorial
development and planning.
Accessibility is the ability to control the transportation facilities that are necessary to
reach the desired location at a suitable tirne [7]. However, the concept of accessibility
incorporates not only the transport link between origin and destination, and the ability for
travelling by the target group, but also characteristics of the destination and the objective of
the trip.
The concept of accessibility can be applied to many spatial problems. Thus, the analysis
based on this concept is therefore suitable to be integrated within geographic information
systems (GIS).
Accessibility can be measured in severa! different ways, already described in [8].
However, every measure of accessibility must consist of two elements. The first is distance
between the person or place and the destination; the second is the activity in the various
destinations. These two elements are combined by using comparative or composite
measures. In the context of GIS, both measures are easy to handle - although the composite
measures are more useful, than the comparative measures.
The strength of GIS lies in their flexibility; they can represent irregular spatial shapes and
pattems, as well as the spatial relationships between them. This characteristic gives the
possibilities for modelling, where proximity and coincidence of irregularly spaced
phenomena are very important. The relations between these phenomena can be modelled
with standard GIS functions - such as recode, buffer, and overlay - implemented in
cartographic modelling.
This paper expands our work in modelling accessibility fields [3, 4, 5 and 6], originated
from Donnay and Ledent [2] and tested by Juliao [9]. In the model applied we tested,
whether the future approved motorway network is going to improve the accessibility to
administrative centres in Slovenia or not.

213

2 PROBLEM FORMULATION
The assessment of accessibility can be sol ved using various algorithms, the so called shortest
path algorithms from graph theory. The research ofthe shortest path in a graph constitutes an
application frequently implemented GlS software. The issue fits in with geographical data
available in vector f01mat. However, Donnay and Ledent [2J stressed two major restrictions
that feature the vector solution of accessibility. First, accessibility is computed only at the
nodes of the network. The accessibility of intermediate points on one segment can only be
assessed with the interpolation between its two ending nodes, which is valid only for the
segments lying radially from the origin. And second, the accessibility of the points located
away from the network is completely ignored by the vector approach. Consequently, the
drawing of contour lines (isodistances or isochronous lines). which is often an important
goal ofthe application, is subjective and not reliable.
An overall solution would be to consider accessibility as a field; i.e. a spatial continuous
geographic variable. Then the accessibility can represent an actual value at every point in the
region of interest, a minimum value at the origin, and a maximum value inside the network
grid when the accessibility is expressed in terms of distance or tirne-distance. The handling
ofthe accessibility field presupposes a transcription ofthe problem· the raster format. This
can introduce some restrictions on <lata, but offers new processing possibilities concerning
surface modelling.
The raster-based GIS methodology for accessibility evaluation, proposed in this paper,
required a two-stage modelling. Layers describing the puhlic road network, administrative
centres, and dwelling houses with data on residents were used. The image resolution
obviously influences the accuracy of accessibility evaluation. In our application, the vector
layers were rasterized with the resolution of 100 m, which is accurate enough when working
at the administrative level.

3 PROBLEM SOLUTION
The key issue in modelling accessibility fields is the definition of cost surfaces. Cost
surfaces are defined by data on distances from origin (features) in terms of costs measures
(e.g. cost distances). Evaluation of cost surfaces requires a friction surface that indicates the
relative cost of moving through each celi. ln our application, costs of movement across the
region were expressed as trave! tirne. These represent the tirne necessary to move through
areas with certain attributes.
The friction surface was calculated using a simple model that fixes the value of the cell
crossing tirne (see also [3]):
CCT = PS. 60 · 9
TS ,1000

(1)

where CCT is Celi Crossing Time (in minutes), PS is Pixel Size (in meters), TS is Trave!
Speed (in kilometres per hour), and 9 is the correction coefficient explained below.
To use properly the raster-based approach, stereological sampling has to be considered
where the road distance between two points is determined by number of the intersections
between the road and the raster grid. In accessibility analysis, the category of road
determines the average trave! speed between two nodes in the vector-based approach, and
the average trave! speed between two intersections of that road with the raster grid. The
expected number of intersections of each road category has the same structure as is the
structure of road categories in reality. Using the solution of Buffoon's Needle Problem [l],
we can conclude the following: "In the structure of intersections of roads of different
categories in the analyzed area, the ratio of intersections of category X compared to the total

214

number of intersections is equal to the ratio between the length of ali roads of category X
and the total length of roads m the analyzed area" Thus, in [ 51 we calculated the correction
coefficient for CCT, which is 9 = lf/4.
While in [3] we proposed a one-stage raster-based GIS approach for modelling of
access1b1hty fields, cartographic modelling of accessibility fields at higher levels
(administrntive, regional, interregional, state or interstate leve!) makes it necessary to work
this case, two different cost surfaces are needed: one
out specific methodologies.
the whole road network, and another one excluding the
considering
tirne
indicating the trave!
moto1;-·ays. Two-stage mode!ling of accessibility fields as a spatial continuous geographic
vanabte 1s necessary because one can not get in and out of a motorway at any point - which
1s the case for ali other road categories.
Two-stage modelling of accessibility fields applied our cartographic model follows
the next major six steps:
Calculation of trave! tirne to the centres outside (without) the motorways and major
roads.
2. Calculation oftravel tirne from the centres using rnotorway or major road to mot01wav or
•
major road connections.
to
size)
pixel
by
defined
location.
3. Calculation of trave! tirne from hinterland (from each
connection
each
for
allocation
territorial
And,
the mot01way or major road connections.
according to the trave! tirne to the connections.
4. Adding up trave! tirne from hinterland to connections (step 3) with the trave! tirne from
the centres to the connections (step 2).
5. Determining the minimum trave! tirne from the administrative centre (comparing trave!
times from the centres without mot01ways or major roads (step 1) with trave! times using
motorways and major roads (step
shows the cartographic model designed and operating in GIS tool Idrisi where
Figure
violet rectangles denote input data and intermediate and final results, white re~tangles
attribute <lata for values assignment, green rectangles vector intermediate results, and red
rhombuses denote analytical modules in Idrisi.

4 MODEL APPLICATION
The calculations of travel tirne to administrative centres in Slovenia were performed for the
base year 2005 and for the year 2013, when the approved motorway network is to be
finished. To calculate the friction surface using the simple model ( 1) and to later add the
correction coefficient 9, the average speed was defined for every category of road network
as well as for the hinterland. The overall data and results are presented in Table 1.

Table 1: Average travel speed and celi crossing tirne according to the road category
Road ca/egory
Motorway (MW)
Major road (MR)
Main road 1 (MR 1)
Main road 2 (MR2)
Regional road 1 (RR 1)
Regional road 2 (RR2)
Regional road 3 (RR 3)
Regional road 3 - tourist road-/RT)
Local road (LR)
Other road (OR)

011tside the road network (OUT)

!

Average speed
(ill k111/I,)

1

CCT (Celi Crossi11g
Time,· in 111i1111tes)
0.0545
0.0667
0.0857
0.0923
0.0923
0.1200
0.1500
0.1714
0.1500
0.2000
0.4000

110
90
70
65
65
50
40
35
40
30
15

215

!

cer *9
(in 111i1111tes)
0.0428
0.0524
0.0673
0.0725
0.0725
0.0942
0.1178
0.1346
0.1178
0.1571
0.3142

Once the cost surfaces were determined. different accessibility analyses coulcl be
perfom1ed. [n our
we compare the location of dwelling houses in Slovenia with
the irnprovement of time-spending distances
administrative centres. In 2005, there were
about l 997 000 inhabitants living in almost 483 000 dwelling houses in Slovenia. Figure 3
shows dwelling houses of 2005 defined according to the improved time-spending distance to
administrative centres due to new motorways frorn the year 2005 to the year 2013.

,~-/
}

r-1,.,'-....

,,,/

,/

;

. >·/

J

Figure · Raster-based two-stage cartographic model for ca!culation of time-spending
distances to selected objects
In our application, we analyzed the accessibility of every location in Slovenia to the nearest
administrative centre. Therefore, the cost value (tirne distance) was calculated as the less
cumulative costs starting from the origins (administrative centres) and moving through a
friction surface. Note, that the time-spending distance was calculated for the ideal
circumstances without consideration of traffic flows by day tirne schedules, traffk
restrictions on road segments, etc.
Comparing time-spending distances to the selected centres in analyzed years 2005 and
2013, when approved motmway network will be finished according to the plans, the
improvement of time-spending distances due to new motorways was calculated. Figure 2
shows the improvement of time-spending distances due to new motorways from 2005 to
20 3.

6 10
5

D.ve!ling house ~vithout irnproverr.ent
0

LJ

Administrative regior
Yeff of construcUoo of htghway

1972 - 2005
-

Table 2. Number of dwelling houses and population in Slovenia in 2005 according to the
improved time-spending distance to administrative centres due to new motmways from 2005
to 2013

1 5
6 - lO
11 - 27

together:

0

Administrative centre

D

Administrative region
YeEI' of conslruction of highway

1972 -2005
2006 -2013
Higt"ruay connection

Figure 2: Improvement oftime-spending distances to administrative centres due to new
motorways from the year 2005 to the year 2013

216

2013

Figure 3: Dwelling houses in Slovenia in 2005 according to the improved time-spending
distance to administrative centres due to new motorways from 2005 to 2013

f------

C] 6-10
LJ 11-•5
C 16 20
O 21 and more

2006

'"t

lmproved time-spe11di11g distance
(in 111in11tes)

5

Administrative centre

··--,

1

N11111ber of dwelli11g
ho11ses
28 389
5602
1454
35 445

1

'

1

------

--~-

Perce11tage of
f!.O[!Ulatio11

Pop11/atio11
88 448
16 333
3717
108 498

1

4.4%
0.8%
0.2%
5.4 %

When talking about the improvement of accessibility to administrative centres in Slovenia
due to the new motorways from the year 2005 to the year 2013, we can realize that this will
affect only 5.4 % of ali population in Slovenia (app. 108 500 inhabitants), which can be seen
in Table 2 above. For most of them (4.4 %; app. 28 500 inhabitants), the time-spending
distance will be improved for only 5 minutes or less. However, there are only 2 %o of
Slovene population, whose accessibility to the administrative centres will be improved for
more than 10 minutes, when ali approved motorways will be completed in 2013.

5 CONCLUSIONS
In this paper, a two-stage raster-based GIS approach for evaluation of continuous
accessibility fields has been presented. The proposed methodology can be essential for a

217

complete integration of accessib1lity in many differenl studies. The application made here
order to obtain a
G IS environment
showed that accessibility analyses fully integrate
make the accessibility analysis more
dynamic and full territorial coverage. However,
the model regarding the
be considered
complex. some additional methodologies have
land surface are
regarding
information
other
barriers and traffic data. Physical barriers and
that considering
especially
accessibility,
territorial
of
results
important to get more accurate
especially
network,
road
outside
travelling
of
simulation
for
way,
this
In
off-road travelling.
surface
cost
simple
a
calculating
of
instead
And,
considered.
be
to
has
slope
terrain
on
data
based on a fixed travelling tirne, it is possible to integrate in the cartographic model an
accessibility behaviour that can often be expressed with traffic flows by day tirne schedules
and traffk restrictions on road segments. Ali discussed improvements of calculating the
accessibility fields should be incorporated as weighted layers in the model.
A raster-based cartographic model suggested here should be used only in accessibility
analysis at mezzo or macro leve! (at a country level, or statistical or administrative regions'
levels: conditionally also at municipal levels) where the high accuracy of the results is not
or town leve!) or
necessary. But, when calculating accessibility fields at micro level
use different
can
one
importance,
when the high accuracy of results is of a big
l
(
methodologies implemented in vector-based GIS approach
References

[1] Buffon G. L. L. 1777: Comte de. Essa; d'Arithmetiljue Morale. In: Supplemenl a l'Histoire
Naturelle, 4. Paris: Imprimerie Royale.
l2] Donnay J. P„ Ph. Ledent 1995. Modelling of Accessibility Fields. Proceedings JEC-GI '95,
I, pp. 489-494.
the
[31 Drobne S., 2003: Modelling Access1b1hty Fields in Slovene Municipalilies. Proceedings
7th Symposium on Operation Research in Slovenia (SOR'03), Ljubljana. Slovenia. p. 89-96.
Slovenia')
141 Drobne S., 2005: Do Administrative Boundaries Fit Accessibility Fields
Proceedings of the 6th international conference Environmental engineering, Vilnius, Lithuama,
pp. 537-542.
[5J Drobne S. and M. Bogataj. 2005: Korekcija (časorne) razdalje v postopku modeliranja
dostopnosti v rasterskem GIS-u = Correct10n of (time-spending) distance m accessibility
modells using raster-based. Proceedings DSI Dnevi slovenske informatike 2005 (DS! 2005),
pp. 484-490.
[6] Drobne S., D. Pahskan and D. FabJan 2004: Rastrski pristop dvostopenjskega modeliranJa
dostopnosti v GIS-u = Raster-based two-stage approach of modelling accessibility in GIS.
Geografski informacijski sistemi v Sloveniji 2003-2004. Ljubljana, ZRC. pp. 265-276.
[7] Geertman S.C and J. R. Ritseman van Eck, 1995: GIS and Models of Accessibility: An
Application in Planning. Jnt. J. Geographical I11for111atio11 Systems. 9(1 67-80.
[8] De Jong T., J. R. Ritseman van Eck, 1997: Location Profile-Based Measures as an lmprovement
on Accessibility Modelling in GIS. Computers, E11viro11ment and Urban Systems. 20(3): 181190.
[9] Juliao R. P., 1999: Measuring Accessibility Using GIS. GeoComputation Proceedings,
http://www.geovista.psu.edu/sites/geocomp99/Gc99/0l0/gc 010.htm, <late accessed: 18-042003.
[10] Paliska D., S. Drobne and D. Fabjan, 2004: Uporaba GIS-a za proučevanje prostorske
dostopnosti v analizi povpraševanja po storitvi JPP = Using GIS to study the transit accessibility
in puhlic transportation service demand analysis. Geografski informacijski sistemi v Sloveniji
2003-2004. Ljubljana, ZRC, pp. 79-88.

218

SOME CO:\JSIDERATIO NS ON OPTIMISATION IN lLOGISTIC
SYSTEMS
Peter Kochel
Chemnitz Umversity of Teclmology. Department oflnformatics, D-09107 Chemnitz
pko@mformatik.tu-chemnitz.de

Abstract

The paper mvestigates two decis10n probiems for multi-location logistic systems. The first one is the
fleet-s1zmg-and-allocauon problem, wh1ch•is deahng with questions about the size of a vehicle f1eet
and !he ernply veh1cle aliocauon. We briefl'; J1scuss the s1mulat1on optim1sation approach. Next we
,;ons,der so-cal!eci hub-and-spoke systems. Sc1ch a svsterr, consists a smgle ..:enh·al, the hub. and M
locations or spokes. The spokes generate a random demand for transportanon orders. For a defined
cost structure the dec,s1on problem ;s to find such a number of transp01iation units that minimizes for
the steady state the expected cost per tirne unit We formulate a solution algorithm and demonsh·ate it
by an example.
Keywords: fleet s;zing and allocation prohlem, swmlat10n opwrnsalion. hub and spoke systems

lntnuhllction
Optimal des1gn an<l contrul of Jynarnic stochastic svstems is a11 actual and mteres1ing field for
theorists as weli as practnioners Among ali apph,atwn <lomams logistics '.S one of the most
1mportant. W ilh respect to log1st1c systems exisl \ery Jifferent fonnulations and models for
optimisatiun purposes. Here we w!l, concentrate on some aspects of how to use transportation
resources in an optimal way More explicitly. we will discuss solution approaches to the
1.ransportation resources. In logistic literature we
problem or the optimal size and allocation
can find mullitudinous models and solution methods dealing with that problem. As a rule
optimisation in logistic systems leads to very complex decision problems. Thus it 1s not
astonishing that corresponding mathematical models suffer from various snnplification
assumptions. For instance. most models assume deterministic known demand for
transportation sen ices. Common approaches are then linear and non-linear network
programming models. More sophisticated mode!s and solution methods investigate the case of
stochastic demand. 1n that case inventory and queueing 1heory models are applied. However,
in general no closed-fonn solutions are available. One can say that the maJmity of papers
investigate algorithms that define an approximate solution for the discrete-time case with
lmown demand and infinite transportation capacity. To come to more realistic models and
solutions the broadest applicable approach is to build a s1mulation model. Simulation however
is not an optimisation tool. What can we do?
In our paper we are dealing with this question. We will demonstrate and discuss by means
of two decision problems the potential possibilities of both the simulation and the queueing
theory approaches. Simulation we combine with an appropriate optimisation tool. Through
such a simulation based optimisation we get an approach to find reasonable solutions for the
fleet-sizing-and-alloc ation problem. This will be discussed in the second section. In section
three we use queueing theory to investigate a multi-location system with hub-and-spoke
structure. A briefresume finishes the paper.

2. The Fleet-Sizing-and-All ocation problem
Practical needs of modem transportation networks lead to a control problem, which in the
literature is known as the fleet-sizing-and-allocation problem (FSAP). In fact the FSAP itself
combines two closely connected problems - the fleet sizing problem, which is dealing with
the question about the size of a vehicle fleet, and the empty vehicle allocation problem, i.e.,

219

r
1
l

how to redistribute empty vehicles among the locations of the network The term vehicle
transportation service. Such
applies to a single reusable unit, which realises a given kind
units may be cars, trucks, railcars, airplanes, containers, material handling equipments and so
on. A long tirne the two problems were investigated independently from each other (see e.g.
/2/, /3/). Investigations, which consider the FSAP as a whole, can be found for instance
1
4/ and /8/. Common approaches to solve the FSAP or at least one of its partial problems are
as well as inventory an<l
linear and non-linear network programming models (e.g. /
queueing theory models (ep. /4/, /6/). Dynamic programming models (see for instance 121. 171,
/9/) are also used. From the complexity of the problem follows that we cannot expect to get
closed-form solutions. Therefore most papers are dealing with algorithms for approximate
solutions in the discrete-time case with known demand and infinite transportation capacity.
Simulation optimisation is a promising approach to overcome these restrictions. is applied
in /8/, where the FSAP for M locations is investigated under the assumption of continuous
tirne, Poisson arrivals of clients, given queueing capacities for waiting clients, homogeneous
vehicles without failures and aging, infinite capacity to tranship vehicles between locations,
and random transhipment times. The decision problem is to choose a fleet size K E l. 2 ....
and a reallocation policy nEll such that for the steady state the expected cost per tirne unit
will be minimised, Le.
MIN '
•
C'(K.n)
(P)

to node
M

Figure

lut

Part of tile BCMP queueing network for the fleet sizing problem without reallocation

We notice that for closed queueing networks the steady state probabilities for being k vehicles
= .. M, exist With these probabilities the needed
in a node, p;(k) and PuJJ(k), k= 0„K,
A

steady state performance measures can be calculated as
of waiting vehicles in location i and R,

= l, · Pi (O) as the mean number of per time unit in

location rejected clients, = .. M, where A; denotes the arrival rate of clients to location i.
Thus goal function (2. ) reduces
M

CrK

·=C(K,n 0 )=2'.[wvi Wj+r; R;]
i=1

To get an impression on goal function C(K,n) we introduce the following notions:
- the waiting cost per tirne unit and vehicle available (not used) in location
wv;
- the waiting cost per tirne unit and client waiting for a vehicle in location
wc;
- the rejection cost per rejected (!ost) client in location
r;
- the mean number ofwaiting vehicles in location i;
W;
- the mean number ofwaiting clients in location
Q;
- the mean number ofper tirne unit rejected clients in location
R;
Then we can express the goal function as

and the initial optirnisation problem (P)
(P')

{Define K*

E

problem

N: C(K*) ~ C(K) for all K

= I[wv; · Wi + wc,

·Q; + ri R; ]+ T( K,nJ,

(2.1)

i=I

where T(K, 1C) denotes for fleet size K and reallocation policy n the expected transhipment
cost per tirne unit in steady state. However, we have no analytical expression for T(K, 1C). The
same holds for the performance measures W;, Q; and R;, which also depend on K and n. Thus
we can try either to simplify the problem and to get an approximate solution or to use a nonclassical approach like simulation optimisation. A variant ofthe second way is realised in /8/.
In a first step policy set ll is restricted to a single element Jlb, which realises no reallocations at
all. In this case term T(K, Jlb) can be omitted in (2.1) and we get a jleet sizing problem without
vehicle reallocation, which is analysed by queueing network theory. On the basis of that
analysis we restrict policy set llin a second step to policies with heuristic ingenious structure.
Thus let us consider the case ll ={ JllJ} for the 110-backorder case, i.e., demand which cannot
be satisfied by an available vehicle will be lost and all Q; vanish. We further assume that a
vehicle rented in location i will be retumed in location j with probability Pij after a random
using tirne, which has a Coxian distribution with expected value µij, i,j = l „M (see e.g. /10/).
These assumptions allow us to model the M-location system as a closed BCMP queuing
network with K jobs, where the vehicles are the jobs (ep. /10/). That part of the network,
which is related to location i, is shown in Figurel. With each node i are connected M artificial
nodes (i, j) representing that a vehicle is rented at location i and will be retumed in location
j = l..M. Figure 1 contains also the corresponding transition probabilities between nodes.

220

E

N.}

Similar as in /8/ to solve (P') we can use the following greedy algorithm:
M

Initialisation: Set K= O; compute C(0)=

L'z ·A;

i=l

REPEAT
K :=K+l;
Compute C(K)
lJNTIL C(K) > C(K-1).
K* :=K-l; C(K*)=C(K-1).
Output:
To compute for given K goal function C(K) we need the performance measures W; and R;. For
that we adapt the well-known Mean Va/ue Algorithm (see /8/ and /10/). Regardless of all
simplifications in (P') results on the fleet sizing problem without reallocations may be helpful
for the FSAP. First, solution K* of (P') is a certain indication for the search of the optimal
fleet size for the FSAP. Next, C(K*) is an upper bound of the minimal cost for the FSAP. And
third, realising the mean value algorithm we get the information which locations are surplus
and which are shortage locations. That knowledge we can exploit for simulation optimisation.
As already mentioned simulation optimisation combines simulation and optimisation. To
realise that very promising idea we have to go through the following three stages:
1 Parameterisation ofthe optimising problem.
2. Choice of one or more optimisation approaches.
3. Realisation of simulation optimisation, i.e., iterate proposal of a solution by the
optimiser with performance estimation of that solution by the simulator until a defined
stopping criteria is fulfilled.
Stage 1 we accomplish through a restriction to reallocation policies with heuristic ingenious
structure. Since we have a continuous-time model we concentrate on situation-dependent
policies (see /8/). Situation-dependence means that a reallocation order will be created as a
result of the occurrence of a defined event in the system, similar to an (s, Q)-policy in
lteration:

C( K,n)

(k) for the mean number

k-1

IK,rr)ENxFl

M

W = I;k

221

a shortage-event when a
mventory systems. Two groups of such events are of mterest
location has a shortage of vehicles. and a swplus-event when a location has too many
vehicles. Therefore we define three policy sets: IIi for shortage-based, f72 for surplus-based,
and fh for shortage/surplus-base d reallocation policies. A second attribute for a reallocation
policy is the number of locations, which are taking part in a given reallocation. lf ali random
variables in the system are assumed to be continuous then only following cases are relevant:
single-supplier-single -receiver, single-supplier-multi- receiver, and multi-supplier-singlereceiver case. As in /8/ we introduce now the notion of location balance B, as the sum of
vehicles waiting in location i or being in transport to location i minus the number of clients
waiting in location i. The parameterisation ofthe optimising problem is realised in /8/ through
the following parameterisation of the policies:
(1) Location i applies a (s;, Qi)-policy ifit observes the following rule:
IF B; < s; THEN place a demand order for Q 1 2 vehicles.
(2) Location i applies a (S;, q;)-policy if it observes the following rule:
IF B; > S; THEN place a delivery offer for q; 2 B, - S; vehicles.
(3) Location i applies a (s;, Q;, S;, qi)-policy if it observes the following rule:
IF B;<s;
THEN place a demand order for Q, 2 1 vehicles
ELSE IF B; > S; THEN place a delivery offer for Q 1 2 B1 - 81 vehicles.
In the currently simulator, available at Chemnitz University of Technology in a laboratory
version, these policies are implemented for ali three supplier-receiver cases.
As optimisation approaches corresponding to Stage 2 we prefer heuristic search methods like
Genetic Algorithms or Simulated Annealing. Especially for the FSAP we have used a Genetic
Algorithm (see /8/ for more details). In /8/ you will also find some results for a system with
five locations. From these empirical investigations follow at least two conclusions, i.e., wise
reallocation policies decrease the cost in orders of magnitude and the optimal solution for
policy 7lo can be a very bad approximation for the FSAP solution.
some main pros and
To finish our excursion to simulation optimisation we want to point
cons ofthat approach. Advantageous is that
problems respectively systems with practically arbitrary complexity can be handled;
2. the influence of a policy on the dynamics of the whole system as well as on particular
elements ofthe system are observab!e:
3. estimation ofindividual cost or gain components is enabled.
The main disadvantages are the need for an appropriate simulator, which increases cost and
tirne of the development process, and a considerable increase of computing tirne.
However, regardless of the great potential of simulation optimisation we believe generally
that introducing some strncture into an optimisation problem will lead to considerable
improvements in the optimisation process. This means for instance to move from the FSAP
for an arbitrary multi-location system to one with a defined and in reality existent structure.
An example will be given in the following section.

''s are available, i.e. the fleet includes K vehicles. Spoke i generates a demand for
identical
a single TtJ in accordance with a Poisson process with parameter Ai > O, i = l..M. TU's move
from the hub to a spoke and retum to the hub. The tirne for a trip from the hub to spoke i, for
unloading and the retum trip to the hub is assumed to be an exponentially distributed random
Let all random variables be independent. Further, let the hub
variable with parameter µi >
accordance with first-come-first-serve d (FCFS) policy. If all
serve transportation orders
TU's are on the trip aniving transportation orders will be queued. Following cost parts are
considered:
> O- fixed cost per TU and per tirne unit,
waiting cost per tirne unit and waiting transportation order.
w>
> O cost per tirne unit per TtJ in use.
i.e., a fleet size K*, which
Now lhe decision problem isto choose such a number K* of
unil.
tirne
per
cost
long-run
average
minimises for the steady state the

Remark 3,1
With the above stated assumptions we can model the HS-system as an M/M/K;oo queueing
I
system with arrival rate and service rate µ for a single server. where

In the previous section we have seen that the general FSAP is too complex for an analytical
solution. Now we want to investigate, what is possible if we assume some structure for the
multi-location system, i.e., we concentrate on so-called hub-and-spoke systems (HS-systems).
Let us assume that our HS-system consists of M+ 1 locations. Location O denotes a single hub
and locations l to M the spokes. For instance, we may think location O as a central storage for
a given product, whereas the locations l to Mhave to meet a random demand for that product.
At the hub is sto red an ample amount of product. Transports of product from the hub into the
spokes have to be realised by transportation units (TU's) with given capacity. Altogether K

222

1

M'
' µ=,~'·,
--'-X;~ 71, µi

Subject to Rernark 3 we consider now the IIS-system as an M/Ivl;K/cc queue. From
queueing theory we know that for the existence of the steady state regime the number K o[
servers must be greater than a = A / µ. For ali K > a we have following formulas for the
different performance measures {see e.g. 0/):
For the steady-state probabilities Pn(K) that n jobs resp. clients are in the system it holds
p O( K )

=

r

l / ~

l

K

I:

a

II

I n ! + a K +1

)

//

K

1(

K - a )] ~

II=

J

(3. )

and
,n5,K,
f lini
n=0,1,2,
K
1 11
lll(K K - ),n2K,

p 11 (K)=p 0 (K)·a" <

(ii) The average number Q(K) ofwaiting clients is given by
Q(K)

= p 0 (K)

x aK I KI x K x a / (K - a/

(3.2)

(iii) For the average waiting tirne W o[ a client holds

W=J

3. Multi-location systems with hub-and-spoke structure

. .
E(
, serv1ca1111e)=

and

1

aK

Q = µ·(K-l)!(K-a/ (K-a/-f_1 a" ln!+aK l(K-1)!.
n=O

(iv) The average number of clients in the system can be computed as L = Q + a .
(v) The average number ofbusy servers is equal to E(number ofbusy servers) = a

= AIµ.

1
M/M/K/oo queueing system means Poisson arrivals, exponentially distributed service limes, K identical servers,
and infinite waiting capacity.

223

To get a mathematical formulation of the decis1on problem above all we have to define the
the steadycriterion function. Therefore let CHs(K) denote the average cost per tirne
state regime ifthere are K servers. With the assumed cost structure it holds that

CHs(K) = K · c + Q(K) · w + a · c., .
Having in mind that for the existence of steady-state regime K should be greater than a (see
Remark 3.1) we formalize the decision problem as optimisation problem
MIN!
(' HS (K) •
(P-HS)

K>A-iµ

KEN

"s. Table 2 cont.ains the results of the numerical computations.
least 7 seners respectively
These results show the following:
. The probability pr;(K) that the system is empty is an increasing function of the server
number K From the formula for p 0 (KJ we see that lim p 0 (K)=e-a =e---{, = 0.00247875.
K----,oo

L The average number QrK) of waiting transportation orders in the steady-state regime is, as
stated Theorem 3. , a convex function ofthe number K ofTU's or servers.
3 The average cost per tirne unit in the steady-state regime if there are K servers, CHs(K), is a
convex function of the number K ofTU's or servers.
4. The optimal number ofTU's or servers is equal to K*= 11 For the given values ofthe cosl
parameters and rates underestimation of K* is more dangerous than overestimation.
5. With increasing K function CHs(K) becomes a linear function with grade c = 20 EUR/day.
This follows from the fact that Q(K) vanishes for K • ~-

415'1
L½ _ J~~\1.3_._,Je-·-=_-1_.-2_-_-+-+--_-o_.-6_-_-+_::-2-.4--+,-1~.5~~:

To define an optimisation algorithm prope1iies of C'1dK) with respect to K will be useful. lf
we could prave that CHs(K) is convex with respect to K, the design of an optimisation
algorithm is straightfo1ward. Since Kc is a linear function and a is a constant we have
consider Q(K) only. In /5/ is praven the following

3

2

~i

i

i _!:!l!. l_o.5

1 1o

2.0

o.3

2~_j

1

Table L Arrival and service rates for Example 3. .1

Theorem l.
For the M/M!Klco system the performance measure Q(·) is a decreasing convex function of
Kfor K~a.
With Theorem 1 the c1iterion function CHs(K) is convex with respect to the number K of
servers. Now the validity ofthe following optimisation algorithm lies on hand.

Algorithm optimal nwnber of servers
. Initialisation:

K := La J+ 1, C

K
7 .
8

1--=-

T

9

~
i

Hs(K); CO = MAX.

• 13
1 14
1

2. WHILE (Cl < CO) DO
BEGIN
K :=K+!,
CO:= Cl,
Cl := CHs(K)
END.

1

1

i

Po(K)
..
0.00157878
0.00214238
0.002352310.00243174
0.00246166
0.00247273
0.00247670
---0.00247808
- - - --

.

1
J_
1

Q(KL
3.682981
l .070945
0.391962
0.151949
0.059066
0.022474
U.008269
0.002924

CHs(K) [ELR/dayj
1 841.4904
l 295.4726
975.9810
-t
1
875.9744
. -849.53~
8512371
.
864.1346
881.4618

+

1

1

l

Table 2. Numerical results for Example 3.1

3. RETURN K* =K-l and CHS(K*) = CO.
Within the algorithm we have to calculate for different server numbers K (fleet size values)
function CHs(K) from (3.3). The most tirne consuming step is thereby the calculation of p 0 (K)
and Q(K) from (3.1) respectively (3.2). Its complexity is O(K). We remark that the notion LaJ
means the greatest integer not exceeding a.

Obviously the presented model can be extended in various directions.
A) General distributions.
For the M/MJK/co system the interarrival times as well as the service times are exponentially
distributed random variables. In case of arbitrary distribution functions we get a GIGIJK/co
system. In /12/ is praven that for the G/GIJK/co system the performance measure QO is a
decreasing convex function of K far K~ a. Thus the above formulated algorithm works again.
However, for G/GIJK/co systems we have for Q only approximate formulas. Now let Ai(,) and
Bi(,) denote the distribution function of the generation tirne for transpmiation orders in
location i and the service tirne by the centre, respectively. We assume only that the first
moments m 1(Ai) and m 1(B;) are finite for all locations. If A; = 1 / m1(A;) denotes the arrival
M

To demonstrate the algorithm we consider
Example 3.1:
Let lvf = 5, c = 20 EUR/day, w = 500 EUR/day, and cs = 100 EUR/day. The arrival and
1
service rates with measuring unit day- are given in Table l. From the <lata in Table 1 and the
1
formulas in Remark 3.1 we calculate A = 6 day" , µ = 1 dal. Since a = 1clµ = 6, we need at

intensity of transportation orders from i then we geta G/GIJK/co system by setting 1e = I: le; ,

224

225

i=l

M

M

i=l

i=I

A( t) = I: 1e; I 1e · A;( t ), t ?. O , and B ( t) =

I: 1e; / 1e · B ;( t ), t

?.

o.

B) Fini te capacity of the order queue.
We can assurne that no backorders or only a finite number ofbackorders is possible. Then we
get the lost-case models MIMIK/O or G/GIJK/0 respectively the finite models M/M/SIL or
G/GIJKIL.

C) Variarions ofthe decision problem.
Other criterion functions as well as other constraints are possible.
D) Differentj,-om FCFS service disciplines.
The hub can serve waiting transportation orders in accordance with other service disciplines,
e.g., in accordance with some priority rules.
E) Existence of extern transportation resources.
's we have to answer two questions If it is possible to rent from outside the HS-system
how much own TU's and how much rented TU's is optimal.

4. Resume
We have investigated two different formulations of the so-called fleet-sizing-and-allocation
problem. The main conclusion is that in general only a combination of severa! approaches will
lead to satisfying solutions. Thus for instance analytical approaches can give some
information on properties of the goal function as well as on the structure of optimal or at least
good solutions. On the other hand simulation optimisation can exploit that information for
is more promising to
searching optimal values of the parameters of those policies.
investigate problem classes with well-defined structure instead to deayl with general problem
formulations. Finally, we want to point to the fact that various tools from computer science in
connection with powerful computer technique open new horizons for solving realistic
problems.

TRAVELIJ\JG SALESMAN PROBLEM AT THE POST OF SLOVENIA
Andrej Lisec
Post of Slovenia, Ltd .. Business Unit Postal Logistics Center Ljubljana
Cesta Mestni log 8 l. SI - l 002 Ljubljana, Slovenia
E-mail: andrei.lisec@posta.si

Marija Bogataj
University of Ljubljana, Faculty ofEconomics
Kardeljeva ploščad 17, SI l 000 Ljubljana, Slovema
E-mail: mariia.bogataj@guest.arnes.si

ABSTRACT

the
In 1h1s paper the theory and methodoiogy of Roberto \\'olfler, 'alvo (2000J, descnbed
the fraveling Scdesman Problem wi1h Time Windows, are presented and
art,de A New Heuristic
their application to the Post of Slovenia is made. The reduction of log1stic costs using this approach 1s
,·ery h1gh. Suggestions for further improveme11ts are g1ven.
KEY WORDS: Postal system, Traveling Salesman Problem, Postal Log1st1cs Center. Reg1onal
Parcel Center. Parcei Post
, INTRODUCTION

Acknowledgements
The author wants to thank an anonymous referee for very helpful comments on a first version
of this paper.

References
Beaujon, G.L Turnquist, M.A. A Model for Fleet Sizing and Vehicle Allocation
Tra11sportalio11 Sci., 25, 19-45, i99l
'2/ Cheung, R.K.; Powell, W.B„ An Algorithrn for Multistage Dynarnic Networks with
Random Are Capacities, with an Apphcation to Dynarnic Fleet Management
Oper. Res., 44, 951-963, 1996
/3/ Crainic, T.G.: Service network design in freight transportation.
European Journal of Opera/iona/ Research, 122, 272-288. 2000
/4/ Du, Y .; Hall, R.: Fleet Sizing and Empty Equipment Redistribution for Center-Terminal
Transportation Networks. Man. Sci., 43, 145-157, 1997
/5/ Dyer, M.E.; Proll, L.G.: On the validity ofmarginal analysis for allocating servers in
M/Mic queues. Ma11age111e11t Science, v,23, No. 9, 1019-1022, 1977
/6/ Koenigsberg, E.; Lam, R.C.: Cyclic Queue Models ofFleet Operations.
Oper. Res., 24, 516-529, 1976
/7/ Ki:ichel,P.: An Approximate Solution for the Fleet Sizing and Allocation Problem.
Promet-Traffic-Trafjico, 9, 195-200, 1997
/8/ Ki:ichel,P.; Kunze, S., NieHinder, U.: Optimal control ofa distributed service system
with moving resources: Application to the fleet sizing and allocation problem.
Internatio11a/ Joimwl of Prod11ctio11 Economics, vol. 81-82, 443-459, 2003
/9/ Powell, W.B.; Carvalho, T.A.: Dynarnic Control ofLogistics Queueing Networks
for Large-Scale Fleet Management. Tra11sporlatio11 Sci., 32, 90-109, 1998
/10/ Trivedi, K.S.: Probability & Statistics with Reliability, Queueing and Computer
Science Applications. Prentice Hall, 1982
/12/ Weber, R.R.: On the marginal benefit ofadding servers to G/GI/m queues.
Management Science, v.26, No. 9, 946-951, 1980
/13/ Wu, P.; Hartman, J.C.; Wilson, G.R.: Fleet Sizing in the Trnek Renta! Industry.
Working Repar/ No. 98W-009, Lehigh University, 1999

226

Central European regions are improving their postal services. The Post of Slovenia,
also like to follow these efforts. This paper presents an approach to the spatial
would
Ltd.
optimization of postal services, particularly as applicable to the Post of Slovenia in respect to
the Traveling Salesman Problem.
The Traveling Salesman Problem with Time Windows - TSPTW developed by Wolfler
Calvo [ O] is described and applied to the Slovenian postal network by a set of fully
interconnected posts (nodes), each one characterized by a tirne interval or a tirne window.
One of these nodes represents a Parcel Post (depot), where the parcels from ali local post
services are collected and distributed. Each are has an associated trave! tirne and cost. The
problem consists of finding a minimum - cost Hamiltonian tour visiting each node during its
tirne window. The tour starts and ends on the lower level ofthe Parcel Post. The TSPTW 1s a
useful generalization of the classic Traveling Salesman Problem - TSP and it has
applications in many vital sequencing and distribution systems.
This problem is only a part of logistic problems of Slovenian postal network when we
wish to set up an optimal allocation of modem Postal Logistics Centers in given 4-level
hierarchical structure of regional central places in Slovenia. A verified model of spatial
optimization by Bruns [l], developed for the Post of Switzerland, has been previously
studied and has been partly applied and extended to evaluate Slovenia's second Postal
Logistics Center Maribor. The previous analysis is based on the paper of Bruns [1]:
Restructuring of Swiss Parcel Delivery Services, where O - 1 programming has been used.
So far two Postal Logistics Centers - PLCs have been opened up in Slovenia in the last
nine years. We have made the evaluation of logistic costs considering two instead of one
PLC and evaluated whether the two existing Postal Logistics Centers in Slovenia already
have optimal macro - locations of required capacity by using the model of Bruns [l].
Simulations demonstrated that the PLC Maribor in addition to the PLC Ljubljana is
acceptable if the variable costs of service from the PLC Maribor are lower or at least the

227

the costs of services of both do not
same as the variable cost of the PLC Ljubljana and
Posts {with short tirne depots) have
Parcel
leve!
iower
the
exceed a certain critical value. On
far,
so
exist
not
do
They
be established.
Not only in Switzerland but also in other Central European Countries, the problem of
postal hub location has been presented in some papers as being vita! for efficient postal
logistics. Wasner and Zapfel [9] have described the hub transportation network for parcel
delivery service in Austria in the paper An i11tegrated multi-depot huh-location vehicle
routi11g model far 11etwork planning oj parcel service. According to them the problem of
severa! parcel posts, their location and their coverage of area by the post connected in cycles
is the basic problem on the lowest leve!.

2. TRA VELING SALESMAN METHODOLOGY EMBEDDED IN HliB LOCATION
PROBLEMS
Postal hub is a special type of central facility, which can act as transshipment point in
postal transportation systems with many origins and destinations.
E-commerce influences the increase of parcel services and the number of parcels ali
over the world. How can this demand be addressed properJy? New technologies and modem
organization in postal services could improve the design of postal systems and especially the
logistics of these systems.
By using the basic approaches of graph theory we improve the approach to the hub
location problem with the model named Multi hierarchical hub location problem and also
combine in with the vehicle routing problem ona special way.
Given a collection of Posts and the cost of trave! between each pair of them, the
Traveling Salesman Problem is to find the cheapest way of visiting ali of the activity nodes
and returning to the starting point. ln the standard version, the trave! costs are symmetric in
the sense that traveling from Post i to Post j .The basic problems of companies that are
dealing with parcels are:
1. To determine the optimal number of Posts as hubs patronizing separate areas
(Regional Parcel Centers), and to determine the location and size ofhubs;
2. To determine optimal fixed locations of Postal Logistics Centers, patronizing
Regional Parcel Centers, location of Regional Parcel Centers, patronizing Parcel
Posts and Parcel Posts patronizing area of inhabitants. In these posts we need to
achieve:an effective collection and sorting ofparcels;
a reliable delivery;
an effective transport.
The criterion here is that the total sum of logistic costs in this service of parcels should
be minimal, often under certain capacity constraints.
The Posts on the leve! of local communities, patronizing a certain area, have to be
assigned to the proper Parcel Post.
The transport network has to be built, which connect Posts, Parcel Posts, Regional
Parcel Centers and Postal Logistics Centers, where the costs of daily transshipment of
parcels would be minimal.
These problems are well discussed and overviewed in the theory and practical
applications ofEbery [3], O'Kelly [7], Campbell [2], Wasner [9], Bruns [1], Marin [6], Ernst
[4], Skorin~Kapov [8] and Gendreau [5].
Successful parcel service application of postal services model was developed by the
Swiss post, described in Bruns [2], which deals with restructuring of the parcel service

228

network. with choosing transsh1pment points among the nodes in the network. They used a
discrete facility location modeL
Austrians W asner and Zapfel f91 describe why optimal design of depot and hub
necessary to develop a
transportation networks for parcel service providers makes
model
routing
vehicle
and
location
generalized hub
In this article the focus is on the Traveling Salesman Problem only on the lower leve!.
The results by Wolfler Calrn [ O] are being used. Let (i=(N, A be a graph, where N
and the are set is A = {(i,j): i,j EN, i j}.
l ,... ,11 is the set of n nodes plus the depot
The time window is represented by l ei, !, and, to each are J)E A. there is an associated

*

) and trave! time

cost (

positive entries ( lu

tij

The trave! time matrix is symmetric

tu

= rji and with strictly

f-lere e; is the earliest time for delivery (or collection) and

the

latest time for delivery (or collection).
A possible fommlation ofthe TSPTW is as follows
( )

min 2:C1t'ij
; )eA

subject

LX1;=
JEN

(3

Vi EN

if

xif

=l

e,~A~

Vi EN

E

v(t,J)E

xij

EA

then A + tij ~ pi

The variable xij

=l

(5)

(6)

A

means that are (i, j) is used, otherwise xij

=O

The variable P;

represents the departure tirne from node
Equations 2 and 3 define an assignment problem of dimension (11+ 1). Equations 4 and
5 represent the tirne window constraints.

3. APPLICATION TOTHE POST OF SLOVENJA
Our application is based on an analysis of Slovenia's postal service. The country is
divided according to NUTS system on NlJTS 2 and NUTS 3 regions, patronizing 192 local
areas.
Spatial hierarchy of postal services is more or less embedded in geographically and
politically determined regionalization.
We used the data from the Post ofSlovenia and optimized the hierarchical structure for
picking processing and delivery of parcels from Post to Post.
The Posts are allocated to Parcel Post on the criteria ofminimum number ofkilometers
and especially on the experience of daily transport from Post to Post to Postal Logistics
Center. We took the experience of postal workers in Business Unit Postal Logistics Center
Ljubljana into consideration and used the method ofthe Traveling Salesman Problem.

In Slovenia there are 555 posts in year 2005. The flows of parcels are directed from
Post to Post until the trnek is fully loaded and then sent to PLC Ljubljana or PLC Maribor
and back. Potential Regional Parcel Centers are not open yet, but could be opened in
Regional Business Units. They will patronize Parcel Post.

229

According to the results of our research, proper capacity and allocation of Regional
Parcel Centers and Parcel Posts should be carried out ln this case the hierarchy for Parcel
Posts would be as follows: the Parcel Posts should patronize twenty or less posts.

MARIBO:\l
• (MB)

KRANJ
• (KR)

Picmre

shows the structure of parcel flows in the Post of Slovenia.

:

l\IlTRSKA
SOBOTA

Ml'RSKA SOBOTA
• (MS)

CELJE
•(CE)

L,J{TBLUNA
!L,J\
NOVA GORICA
f(NGJ

1111-------------,Novo MEs:ro
-----------j •

(NM)' -

I<O:!)ER
• (KP)

Pic ture 3: The structure of parcel flows
Picture 1: The average daily number of collected parcels from regions

PLC Ljubljana

Picture presents the average daily number of collected parcels in PLC Ljubljana for
ali Business Units and Picture 2 the average daily number of delivered parcels from PLC
Ljubljana.

1. ,,

:

,.
-,
l\URJBO'(l.
•(l\IB)

tJ_FRSKA ŠQBOTA
• (1'-IS),:;

"/'\)
'-\

Iill~J '·
• (KR}'

CELJE
• (C'E)

LJllBLJANA
·;

NOVA GORICA
··'j(NG)

(LJ)

(

·---- _,_ --·-- --- - - 7

i

NOVO l\lE,~:ro.J
•
~l\Ij

\
KOPER
•• (KP)

',

\ .. j

(,,
·~ ,."...\

,
'

..._

.'

Picture 2: The average daily number of delivered parcels from PLC Ljubljana to regions

230

Posts stati to operate approximately two hours before the post offices open and close
two hours after the post offices do so.
Our application is made for the covering service area of the Postal Logistics Center
Ljubljana.
W e determined the optimal cycles for areas covered by the Postal Logistics Center
Ljubljana: 4 Parcel Posts patronized by the post of Kranj. 3 Parcel Posts patronized by the
post of Nova Gorica, 4 Parcel Posts in the Koper and 5 Parcel Posts patronized by the
regional post ofNovo mesto (Picture 4).
Our optimization is applied only to the area patronized by the Postal Logistics Center
Ljubljana because we have experience here and ali the necessary data also only from the
PLC Ljubljana. We propose the use of the same optimization methods for the Postal
Logistics Center Mai·ibor as well.
By using the Traveling Salesman methodology, the data on distances between Posts,
the statistics ofparcels from the years of2003 and 2004 and intensity offlows between posts
from the year 2005 we got an optimal new transportation network connecting the post and
Parcel Posts patronizing them with the criteria of24 hour tirne window delivery.
Optimization is focused on the minimal length of the sum of cycles by using the
shortest paths between Posts and Parcel Post.
The results are presented in the sums of lengths of cycles (in kilometers) for ali Posts,
which are patronized by the Parcel Post areas. This is the only possible way to compare the
difference of daily length of cycles between the new and the old transport structure.
Here each Post is visited more than two times per day from patronized Parcel Post.
Our results, presented in Table 1 show sums of the lengths of cycles in the previous
regime, but Table 2 presents new optimal sum ofthe lengths of daily cycles.

231

Table

rhe transportation costs are reduced (because of reducing the total number of
kilometers per day) from 15686 km (which is the situation at present) to 10105 km. This will
be achieved when Parcel Posts will patronize the sequence of Posts on the local area. The
difference is 5581 km. !fwe also consider fixed and variable cost, then the costs are reduced
by 30 percent in total.

kilometers

Previous sums of length of ali cycles

old Ljub!Jan-; [Ki·~n.fT Nova Gor~per! Novo mesto i sum
!688 T _- 20Q5 _j(}(},t~
2047
1711
lfuck 3129
3397 i
597
: 233 :
367
424
l 776
, van
122491
672
1 }:?~J
374
435 ·· 44{[_
i car
1156861
3274
: 2245
2788
2039
5340

!

-n

1

1

5. CONCLUSiONS AND FURTHER RESEARCH
Table 2: New sums of length of ali cycles - improved solutions in kilometers
,------

.

-

!

------:----------r,!Kranj INova Gonca )<:o_peri Novo mesto i sum i
!_lelBLjubljana

truc-4
i van,
car

807
658
l 406

:sum:

2871

1

i

i

_878 _
534
_J ~6~~ 665
1

602

1

03.ll_J

~14021

2077

: 769
j 135
j 888

626
672
665

;}68.l__j
2537]
1
!38861

1963

:!0105

1

1

1

.17921

The transport structure of the Business Unit Novo mesto is presented a bit closer here.
It encompasses the Parcel Post Sevnica, which has the following associated distance matrix:

Sevnica
Bostanj
Studenec

,1

Radece 1
ZidaniMo.1
Loka pri z.i

Sevnica

Bostanj

Studenec

/

2

15

2
5

13

Radece

Zidani Mo. Loka En

12
2

References

3
12

2
2
2

12

f.

2

4
4

Five iterations have been made . The value of the objective function is 58. The
sequence of arcs on this cycle is the following (constrains in tirne windows are based on the
Studenec (13) Radece (25) - Zidani
average speed of70 km/hour): Sevnica - Bostanj
total distance is 58 km.
The
l
(
Sevnica
4)
(
Mostu
Zidanem
pri
Most (2) - Loka

NOVO MESTO

Legend
Flow of parcels
-----· 1. 3
-4-5

20,000

• ,a:m

80,CIJO

-4

In this paper the use of the Traveling Salesman Problem method has been described
the Post of Slovenia, to the area patronized by the Postal Logistics Center
and applied
solving the real problem of the Postal Logistics Center
Ljubljana. As presented,
Ljubljana, the number of kilometers inside prescribed time window can be drastically
reduced.
Our further research will be based on the assumption that the areas, which are
patronized by the PLC Ljubljana, and the complementary areas, patronized by the PLC
advance. The same relaxation will be given for Regional Parcel
Maribor, are not given
Centers. By usmg this approach, the Traveling Salesman Problem in the area of the Postal
Logistics Center Maribor also has to be sol ved for ali belonging parcel posts.

hm

[l] Bruns, A .. et al. (2000). Restructuring of Swiss Parcel Delivery Services,
Operations Research - Spektrom, p. 285 - 302.
[2] Campbel, J. F. 1994). Integer programming formulations ofdiscrete hub location
problems, European Joumal of Operation Research, 72, p. 387 - 405.
[3] Ebery, J. et al. (2000). The capacitated multiple allocation hub location problem:
Formulations and algorithms, European Joumal of Operational Research, 120, p. 6 4 •· 631
[41 Ernst, A. T., Krishnamoorthy, M. 996). Efficient algorithms for the incapacitated
single allocation p hub median problem, Location Science. 4/3, p. 39 - 54.
Seguin, R. (1996): Stochastic vehicle routing. European
[5] Genreau, M., Laporte
Journal ofOperational Research, 88, p. 3 - 12.
[6] Marin, A. (2005): Formulating and solving splittable capacitated multiple
allocation hub location problems. Computers and Operations Research, 32/12, p. 3093 3109.
[7] O'Kelly, M. E. (1987): A quadratic integer program for the location of interacting
hub facilities, European Journal of Operational Research, 32, p. 393 - 404.
[8] Skorin - Kapov D., Skorin - Kapov J., O'Kelly M. (1996): Tight linear
programming relaxations of incapacitated p - hub median problems, European Journal of
Operational Research, 94, 1996, p. 582 - 593.
[9] Wasner, M., Zapfel, G. (2004). An integrated multi - depot hub - location vehicle
routing model for network planning of parcel service, Production Economics, 90/3, p. 403 419.
[10] Wolfler Calvo, R. (2000). A New Heuristic for the Traveling Salesman Problem
with Time windows, Transportation Science, 34/1, p. 113 - 124.

-6-8
-9-12
-13-15

Picture 4: Regional Parcel Center Novo mesto and their Parcel Posts

232

233

Dealing with the flu.ctuations of the travel times in real-world disfribu.fom
problems
Ana Osvald'. Raffaele Pesentt Walter Ukovichc
·' E-mail: ana.osvald@bayern-maiLde
b

DINFO- University of Palermo

Viale delle Scienze
1~90128 Palem10. !tal,
Phone: (+ 39) !J9 I 65981143 Fax ( · 39) 09! 659804-1 E-ma1l pesentt@unipa.it

c DEEI and
lJniversity of ·frieste
Via A. Valerio,
!-34127 Trieste, !tal:,
Phone: ( ·39) 040 558 7848 Fax (+39) 040 558 3460 E-maii ukovich@units.it

Abstract
h1 real-life routing probiems w1th llme constraints, it 1s ve1y important to consider the fluctuations of
the trave! times, in order to find good soiutions wlnch estimate the correct customers· delivery time.
h1 this paper, we present a method to solve the Vehicle Routing Problem with Time Windows
(VRPTW) where the trave] times between two locations depends not only on the distance, but also on
the tirne of the day. The paper proposes a solunon approach to th1s problem by first providing a
simplified estimation of the time-dependent link trave! times and then using an innovative heuristic
which sol ves the related routing problem.
Keywords: Routing, Distribution, Log1stics. VRP. Time-dependent trave! times.

L Introduction
Nowadays, intelligent planning offers potential savings in costs related to stock and
distribution of goods. In particular, the Vehicle Routing Problem with Time Windows
(VRPTW) [11] considers the classical distribution problem for the management of a fleet of
vehicles where a route for each vehicle of the fleet must be found in order to minimize the
total distribution cost, to satisfy each customer's demand and to consider all operational
constraints (including delivery tirne windows, capacity ofthe vehicle).
A lot of research effort has been devoted to various aspects of the VRPTW and many
techniques to solve this problem have been presented in the literature. Unfortunately, in
many cases these techniques do not provide useful and acceptable solutions to logistics
companies, because they do not take into account the fluctuations of the travel times due to
the vaiiable traffic congestion during the day. To achieve always useful solutions, it is
necessary to include the additional information on the trave! tirne variations into road
network.
Problems where a time-dependent modelisation of the trave! times is used are called Vehicle
Routing Problems with Time Windows and Time Dependent trave! times (VRPTWTD). So
far, they have not been much considered in the literature.
In this paper, we present a solution approach for the VRPTWTD suitable for logistic
companies; hence the solution shall be always feasible and shall be obtained in a short tirne
of about few minutes. We achieve it by first identifying a suitable model for representing the

234

235

time-dependent fluctuations of trave! times and then sohing the VRPT\\'TD w1th soft
windows
Section 2 we give a brief literature review ofthe VRPs with Time-Dependent lravel umes
represent the fluctuations of the travel limes,
and show differem approaches used
to estimate the Time-Dependent Originmethod
Section 3 we present the developed
ofthe developed heuristic for the solution
overview
an
give
and
matrix
Destination (TD-ODJ
real data. The
of the VRPTWTD, Section 4 reports the results of the developed apprnach
advantage of the presented approach and possible future research are presented in Section 5.

2. Literature review
Malandraki and Dial l6 intrnduced first in 1992 the time depemient Traveling Salesman
center with traffic
Problem (TSP). Bentner et al. l considered for the TSP a zone the
jams in the afternoon and showed how the simulatecl annealing aml threshold accepting
algorithms are able to handle such time-dependent problems.
Park et al. [8] presented the Time-dependenl Vehicle Scheduling Problem (TDVSP)
which the travel speed between two locations depends on the passing areas and the tirne
the day. They proposed a model for estimating the tirne \arying trave! speed.
lchoua a! al. [3] proposeed a time-dependent mode! for the VRPTW Their mode! is based
property. They
on time-dependent trave! speeds and satisfies the First-!n-First-Out
problem. They
the
ve
sol
to
O]
l
[
al.
et
Taillard
by
developed
extend the Tabu Search heuristic
completely
known
not
are
services
of
(number
dynamic
and
static
in
tested the algorithm
ahead oftime) environments and showed that the time-dependent model provides substantial
improvements over a model based on fixed travel times. In that work, there is no mention of
any experiment on real world problems.
Ali of the mentioned works use the Euclidean distances to estimate the shortest, timedependent trave! limes and so also the TD-OD matrix. In the real-worl<l problems. this
approximation is usually not acceptable, and real distances and real travel times must be
used.
In the literature. severa! representations of the link trave! times and related methods
estimate the shortest path are used to cope with fluctuations. The three most common
representations ofthe link trave! times fluctuations, usually used pre-trip planning, where
ali data are supposed to be known before the routing process begins, are as follows:
e Time-dependent trave! times; the trave! tirne d;;(t) from vertex to vertex 1s a
function of the departing tirne t from vertex i. An example is presented in Figure
.. Stochastic times; trave! times du are random variables.
• Stochastic and time-dependent times; the link travel times are considered a
continuous tirne stochastic processes, i.e., random variables and with probability
distributions which are functions ofthe tirne ofthe day.
~1-----d=''~-'-t)- - - - •
Arrival tirne: t+ d,,(t)
Start tirne: t
Figure 1: An example of Time-dependent representation of link trave! times

The time-dependent representation ofthe trave! tirne fluctuations are generally used to model
time-of-day congestion effects and are indicated for short term planning [4], to which usually
also the VRPTW belongs. Since we have to deal with this type of problems, we focus our
attention on the time-dependent representation ofthe trave! times.

236

Lsing this representation. is possible in a second tirne to improve the calculated routes in
acywrmg real-tirne the actual trave! times and partially recalculate the routes.
real•time

3. Estimation of tbe time-dependent link travel tirne and the heuristic approach for
the VRPTWTD
We split the solution of the real world VRPTWTD into two steps. lnitially we estimate the
TD-OD matrix and then we solve the VRPTWTD. An outline of the whole solution
approach is presented in Figure 2.
First step Estimation of TD-OD matrix
Extraction from the diqital map of the road network on which the customers. are defined
Estimate TD-OD matrix, represented by 8 constant matrixes:
o Classify the arcs of the street nelwork in 24 classes, according to their legal speed limit and
traffic congestion.
o Divid~ _a day into. fo_ur tirne periods and to each period and for each class assign a
coefflctent of vanatton of the speed (four different tirne periods are used, since during the
day there are usually two periods of rush hours). For each period, consider a constant
average speed (so the speed is a step function of tirne).
o ~alculate the shortest paths between ali customers defined on network for 8 given start
limes (one at the beginning and one in the middle of each of the four tirne periods) using an
'
algorithm based on the FSM algorithm and presented in [9].

Jj.
Second step: Solution of VRPTWTD
Solve the Vehicle Routing Problem with Time Windows and with Time Dependenl trave! limes with
a heuristic approach:
Repeat for different Start Customers and different insertion costs
Flnd a feasible tour:
Solve classical VRPTW with an average trave! tirne and trave! distance
Correction Tour
end Repeat
Selec! the three besi tours
For each tour
Tour Reduction (reduce the number of tours)
lmprovement of lhe solution with TabuSearch
end For

Figure 2: An overview of the solution approach

The estimation of the TD-OD matrix usually requires a lot of effort, since nowadays there
are very few specialized centers that manage a detailed traffic infonnation available to the
transportation companies. To represent the TD-OD matrix we use 8 constant matrixes. The
trave! tirne is in this way a step function of tirne. To assure that the fina! solution of the
VRPTWTD satisfies the FIFO property it is necessary to have continuous trave! times. To
this end, we use modified (tirne continuous) trave! times in the Second step of the solution
approach.
To solve the VRPTWTD we have developed a heuristic that initially generates an initial
solution using a constructive heuristic and then uses an improvement heuristic based on the

237

Tabu Search to furthermore improve the solution. The outline of this Second step is
represented in Figure 2.
lnitially a sequential constructive heuristic that solve a classical YRPTW is used. ln this part
the average trave! times and the average travel distances calculated on the TD-OD matrix are
used. Then, a correction procedure is used in order to take into account the actual tirne
dependencies of the trave! times. ln this procedure. the customers' delivery times are
recalculated using time-dependent trave! times. When necessary. some customers are
relocated in the tour, or removed from it in order to make the tour feasible. The removed
customers are, when possible, reinserted in other tours. Otherwise, new tour or tours are
initialized for the remmed and not reinserted customers.
To develop the improvement heuristic based on the Tabu Search we ha\·e used EasyLocal++,
an object-oriented framework for the design and the analysis of local-search algorithrns
The abstract classes that compose the framework specify and implement the invariant part of
the algorithm. The user can so focalize on the problem-dependent part
In the improvement heuristic we have used the Cross Exchange techni4ue for the
neighborhood generation.
A detailed description ofthe whole solution approach is given in

r

the tirne intiependent approach and have recalculated the delivery
start limes achie,ed
tirne at each cus1umer, using the times given hy the TD-OD matrix.
We observe that the recalculated tirne (Effective tirne) and the recalculated lateness
(Effective lateness) are higher than the Calculated tirne and the Calculated lateness. Above
ali, we notice also that some of the calculated tours are not feasible anymore, since some
depofs tirne windows turn out to be violated (in the problems with soft tirne windows, the
tirne window of the depot remains hard),
The second part of the table reports the results achieved with the method presented in
Section 3, that takes into account the time-dependency ofthe trave! times.

,',

1,

Figure 3: Time independent

4. Performance of the proposed approach and computational results
In this section we present the performance of the proposed solution approach for the
customers defined on a real road network. In the tests presented in following, three sets of
30, 50 and 100 customers, have been defined ona restricted map ofthe Friuli Venezia Giulia
region with 50.703 geo-referenced nodes. We calculate the TD-OD mahix for each instance
and so Ive the problem with two different approaches:
constant trave! times and travel distances, equal to the average travel times calculated
o
on TD-OD matrixes oftravel tirne and trave! distance,
time-dependent trave! times.
lil
50 customers

30 customers

100 customers

Time independent approach
Distance

245 km

376km

859km

Calcu lated tirne

438 min

726 min

1.326 min

Calculated lateness

42min

75 min

351 min

Effective tirne

508 min

867min

1.734 min

Effective lateness

239 min

806 min

894 min

lnfeasibility

20%

25%

12,5%

Time dependent approach
Distance

249 km

383 km

887km

Time

493 min

842min

l.693 min

Lateness

44min

89min

347 min

shortest pa th

.J

Figure 4: Time-depenclent shortesl Figure 5: Time-depenclent shortesl
path (Scenario 11)
palh (Scenario l)

We obserre that the time-dependent model provides significant improvements over the timeindependent model: we haYe a small improvement ofthe total traveled tirne and a significant
improvement of the total lateness. At last, using the time-dependent model, we achieve
always useful and feasible solutions.
The cumulati ve computational tirne for the instances of l 00 customers is about l 05 seconds
(45 for the estimation ofthe TD-OD matrix and 60 for the solution ofthe VRPTWTD) using
a Pentium IV, 1.6GHz, 256MB PC.
To show how the time-dependency influences the tra,e.l times and the trave! distances
between the customers, we present in Figures 3-5 three shortest paths (obtained minimizing
the trave] tirne) between the same pair of customers, for three different start times. To
minimize the total trave! tirne of the path on the tirne dependent network, it is sometimes
more useful to chose longer, but less congested route. Using the path achieved in the timeindependent environment with the start tirne in Figure 5, we have found that the trave! tirne
lasts 23 min 26 s, about 3 min more than path in Figure 5
' Trave! tirne
,~ Fig~re 4 ..
Figure 5
Figure 6

Trave! distance

9 min 5 s

9,979m

15 min 13 s

10.148 m

20min 43 s

12.474 m

~

Table 2: Trave! tirne and trave! distance between two points in different scenarios

Table 1: Results ofthe VRPTWTD

Table 1 shows the results achieved with both approaches. In the firs part, we show the total
distance of ali routes (Distance), the calculated total travel tirne (Calculated tirne) and the
calculated total lateness (Calculated lateness). The results obtained in the tirne independent
environment do not always give a useful and feasible solution for the real-world problems.
We have recalculated the routes, i.e. we have considered the sequences of customers and the

5. Conclusions and further researcb
The results of experiments defined on real road networks show that the chosen timedependent model provides significant improvements over the model with time-independent
trave! times. Above ali, the developed solution approach for the VRPTWTD always
produces a feasible solution, a crucial condition in the solution of the real-world distribution
problems.

238

239

The comput ational tirne for 1.he real-'.'. orlu distribut10n problern
s of up to 10() custom ers.
defined ona restricte d map as the one of the Friuli Venezia Cimlia
region, used · the tesrs
Section 4, is less than two minutes . Some weakne ss has been
noticed for some problem s
defined on extende d network s where the comput ational tirne, necessa
ry
estimat e the TDOD matrix for problem s of about 00 custom ers, is about 15
minutes. To speed up the
calcula tion of the fina! solution , it is necessa ry to use alternat ive
method s to calcula te the
TD-OD matrix for these problem s. A useful method that allows in
the standar d shortes t path
problem to reduce the comput ational tirne of about 40-60% , isto
use straight -line distanc e as
a lower bound on the distanc e betwee n any two nodes in the networ
k (5J. This method would
be useful to implem ent also for the time-de penden t network
, in order
reduce the
comput ational tirne for the estimat ion of the TD-OD matrix on large
network s.

Literature

2.

3.
4.
5.
6.

7.
8.
9.

l O.

11.

12.

Bentne r J., Bauer G., Oberma ir G. M., Morgen stern
"Optim ization of the timedepend ent travelin g salesma n problem with Mante Carlo method
s", Physica l review E,
Volume 64, 036701 ,2001
Di Gaspero L., Schaer f A. "Writin g local search algorith ms using
EASYL OCAL+ +".
In S. Vo/3 and D.L. Woodru ff, editors, Optimi zation Softwa re Class
Librari.es, OR/CS .
Kluwer Academ ic Publish ers, 2001.
lchoua S., Gendre au M., Potvin J. Y., "Vehicl e Dispatc hing With
Time-D epende nt
Trave! Times" , Europe an Joumal of Operati onal Researc h, pp. 379-39
6, 2003
Kalidas A., "Estima tion and Predict ion of Time-D epende nt OriginDestina tion Flow",
Ph.D. Thesis, Massac husetts Institut e ofTech nology , 996
Lysgaa rd J., "A two-ph ase shortes t path algorith m for
networ ks with node
coordin ates", Europe an Joumal ofOper ational Researc h, pp. 368-37
4, 1995
Maland raki C., Daskin M.S., "Time Depend ent Vehicle
Routing Prob-le ms:
Formul ations, Propert ies and Heurist ic Alg01ithms" Transpo rtation
Science 26
pp.
185-200, 1992.
Osvald A.. "The Vehicle Routing Problem with Time Window s
and Time Depend ent
trave! times: a heuristi c approac h", Ph.D. thesis, Univers ita <legli
studi di Trieste, 2005
Park, Song, "Vehic le Schedu ling Problem s with Time-V arying
Speed" , Compu ters &
Industri al Enginee1ing, pp. 853-85 6, 1997
Powell W.B., Jaillet P., Odoni A., "Stocha stic and Dynam ic Networ
ks and Routing ", In
Bali M., Magant i T., Monma C. and Nemha user G., editors, Networ
k Routing , NorthHolland , 1995
Sung K., Bell M. G.H., Seong M., Park S., "Shorte st paths in
a networ k with timedepend ent Flow speeds" , Europe an Joumal of Operati onal Researc
h, vol. 12 l, pp. 3239, 2000
Taillard E., Badeau P., Gendre au M., Guertin F,. Potvin J.
Y., "A Tabu Search
Heurist ic for the Vehicle Routing Problem with Soft Time Window
s", Transpo rtation
Science 31, pp. 170-186 , 1997.
Toth P., Vigo D., "The Vehicle Routing Problem ", SIAM, 2002.

240

Al..LO CATlO N OF SPE( !ALI Y EQUIP PED FIRE BRIGA DE
UNllTS: A CASE
STlDY FOR 11\JTERVENTION IN TUNN ELS ON SLOVE NIAN
ROAD NETW ORK
Dejan Paliska, Daša Fabjan
Faculty ofMariti me Studies and Transportation
Pot pomorščakm 4. 6320 Portorož
Samo Drobne
Faculty ofr,eode sy and Ci,il Engineering
Jamova 2, 1000 Ljubljana
Abstract
This paper presents the applicallon of the MCLP (:viaximai Covering
Location Problem) location-allocation
model in a ( ,IS (Geograph1c !nformation System) envlfonment used
to optimize the number and locations of
fire bngade units with trained personnel and special equipment for mtervent
ion in case of road tum1el accidents.
For s1mulation pul]loses a potential demancl for intervention was calculate
d considering the existing locations
and classification of hmnels on the Slovenian road network and
the potential spatial accessibility was
mvestigated. Stress was laid upon the hmnels rhat require specialized
emergency vehicle intervention in cases
ofincide nts. fhe paper describes the model and presents the results ofthe
simulations.
Keywords: GlS, Iocation-allocation problem, fire bngades, mnnel acc1dents
, emergency systems

L Introduction
There are almost 33 kilomet ers o f road timnels in Slovenia, around
30 percent of which are
longer than 500 meters. Compa red to other Europe an countrie
s, even relative to the
countiie s' area, this is not a lot; howeve r the possibi lities of tunnel
tire or similar inciden ts
exist. With regard to the better driving conditio ns due to newer infrastr
ncture and taking into
accoun t the freight traffic growth on the Sloveni an road network ,
especia lly after joining the
EU, the threat is even greater as most of the incidents happen
due to drivers who do not
respect the speed limits in tunnels. Thus it is necessa ry to se1ious
ly conside r the risk and to
foresee and take all due precaut ions, as well as prepare ali types and
means of intervention.
ln case oftunne l road accidents, speciall y equippe d fire brigade units
can intervene, thus
based on a demand and supply approac h our work focuses on
investig ating the potentia l
spatial accessi bility from tire brigade location s to tunnels , which
would involve as few fire
brigade s as possibl e to keep the costs lower, but at the same tirne
to be much more efficien t
in tirne and space. For this purpose a Maxim al Coverin g Locatio
n Problem (MCLP )
location -allocat ion model was used in a GIS environ ment.
2. A GIS approach and the model used
GIS technol ogy is increas ingly used and recogni zed as an impmia
nt plannin g tool for the
acquisit ion, organiz ation, manipu lation, analysis and display of
large volume s of spatiall y
referen ced <lata. It enables spatial data creation , task automa
tion and enhanc ed map
produc tion [3]. The integrat ion of model- based method s and
GIS technol ogy have
substan tial benefits for managi ng and analyzi ng data to produce
informa tion relevan t to
decisio n making and in simulat ing the effects of differen t plannin
g decisio ns [4].
For the simulat ions, the comme rcial GIS packag e ARC/IN FO 8.3
was used. There are six
location -allocat ion built-in models availab le in ARC/IN FO,
each designe d to salve a
differen t type ofprob lem. To effectiv ely use these models , it is importa
nt to underst and each
model thoroug hly - the types of problem s it is designe d to solve and
the types of solution s it
tends to produce . Locatio n-alloca tion models determi ne the location
s for centers and the

241

allocation of demand to centers according to a specific ubJective. Some models also unpose
constraints on the solution ( such as make sure no demand is further than O kilometers or
minutes from its closest center). Among these there are emergency service location rnodels.
where the objective isto serve as many calls as possible within a given tirne or distance.
In our case we used MAXCOVER (Maximal Covering Location Problem) to locate a
fixed number of facilities in order to maximize the demand that is covered within a specified
tirne or distance [2]. and MAXCOVER - constrained (Maximal Covering Localion Problem
with a Mandatory Closeness Constraint) to maximize the demand that is covered within a
specified tirne or distance, while ensuring that no demand is further than a second tirne or
distance threshold. Both models assign the demand to its closest facility and maximize the
coverage of fixed number of located facilities linearly [8]. however the first has only one
coverage constraint. the second has two tirne or distance constraints.
A demand point is considered covered if it is within the distance or time threshold of its
assigned facility. The facilities are located so that the demand covered is maximized. Ihe
actual distances traveled from demand points to centers, whether inside the distance
threshold or not, are not relevant. For an adequate emergency facility location a required
response tirne to demand points must be as short as possib!e. A tirne or distance threshold
defines the surrounding area that a facility can serve (demand within this area is so coveredJ.
When it is not possible to cover all demand with the existing facilities due to e.g. limited
funds, both models try to maximize the amount of demand that can be covered with a fixed
number of facilities within a given tirne or distance threshold. Running the MCLP model,
each time adding a facility, it is possible to analyze the amount of demand covered and find
out the trade off between the number of facilities and the coverage. There may be many
different location configurations of facilities that provide the same degree of coverage. The
Constrained MCLP model can then be used to analyze a variety of configurations.
The MCLP with Mandatory Closeness Constraints locates the facilities so that ali
demand is covered within a reasonable response tirne within a second distance threshold
when a certain demand is not covered within a first tirne or distance threshold. This means
that there is a second larger distance or tirne threshold beyond which there is no other
demand.
One crucial limitation of the simulation is a limitation of the implicated model (MCLP)
that does not consider the possible situation in which no road accident intervention squad is
available at the closest fire stations to respond to a call within the tirne or distance threshold.
The second important limitation is that potential spatial accessibility is calculated, which
does not consider other geographic factors, social---economic characteristic of target
populations, and organizational characteristics of emergency service. These factors and
characteristics may either increase or decrease accessibility.

3. Data used in tbe simulations and demand and supply formulation
Data analysis and graphic presentation of the study results were effective, fast and simple
due to the GIS approach, which enabled simulation and forecasting. Geographic and attribute
<lata from different databases were merged to make the analysis simpler and clearer, which
increased its efficiency in changing and testing the parameters and conditions. The <lata used
were mostly provided by the Administration for Civil Protection and Disaster Relief of the
Republic of Slovenia. Slovenia has a non-uniform population distribution, which also affects
the accident location distribution.
The simulation was run on the road network in the Republic of Slovenia. The road
network consists of local roads of relative importance. As the database did not include the

242

I hese had to be c t d
ls to . the highway tunn e1 pm•t as,
ent . access trai
data on mten
t
.
. .
•
ap ure onsite. Ass1g111ng the mtervention tra1ls a value o for drt' ve t·trne, the model algorith
I d
.
. .
. .
m la to
.
.
t'
instead of other roads The drivi ng trne on mtervenhon trail
mclude smrn!ation these
·
.
·
s was
a dde d to the total tlme ol the optimal access path Jater on.

3.1 Potential demand formulation
of demand were calculat ed fior eac h tunnel The
Allowing for different scenarios ' three types
"
:bi
three types of demand were needed to describe the facts of a case as realistically as
\
DEi~~D
intervention
The first type of demand for specialized emergency vehicles
accordi;;-g
classified
were
tunnels
the
case,
requJred a 5 rank class1ficahon of tunnels. 1n this
to tunnel length (less or more Ihan 500 m), tunnel type (single- or double-barreled) road
category and threat degree. The threat degree level was detennined considering ~raffic
volume on each road.
. According to the category, very short tunnels do not require specially equipped fire
b11gade mtervention as a rule, however 1t 1s 1mportant for all categories that the location ·
· tis
a foreseeable tirne in case of an accident in the tunnel. Tunnels that f:a II 111
access1ble
o
·
·h
·
categones wit_ a relatively high demand for specially equipped vehicles intervention are
usually access1ble by road or by special intervention trails for faster accessibility.
The Karnvanke tunnel_ (3336 m) falls into a categmy with the highest demand, which is
a result of 1ts charactenshcs and traffic vol_ume. Owing to the building of new highways
most t~nnels on these roads are more soph1shcated and better equipped. There are 28 tunnels
that fa1i mto th1s category,_ and they measure approximately 29,800 m all together.
Bes1des the d1fferent1ated demand based on the aforementioned factors, we used two
more types of demand. All the tunnels have the same value in DEMAND 2. On the other
hand, DEMAND_3 consisted of simulation conditions where two emerge~cy units had to
mtervene m case_ of an accident. In this case, very short tunnels with relatively small traffic
volume were om1tted (a value O was assigned).
3.2 Supply formulation
At present, 44 fire brigades are considerably well equipped and trained for technical
rescue on roads. T_hese are selected within a Decree on the Organization, Equipment and
Tra111mg of Protechon, Rescue and ReliefForces [13]. The fire brigades are contained in the
GESP_ layer and are well located in space. The optimal special equipment distribution was
111vest1gated among them.
Taking into c?nsideration the different scenarios, three types of supply were estimated.
SUPPL Y_ 1 took mto_ account the fire brigade units' qualification and equipment. All units
w1th at least two quahfied teams who can respond in Iess than 3 minutes were given a value
of 20. Th1s means that such a unit could intervene in cases of fire or tunnel accident
autonomously.
If the fire brigade unit has only one team that can respond in less than 3 minute s a value
1O was assigned; if one team can respond in more than 3 minutes but less than 5 :Uinutes
th~n a val~e 5 ~as as~igned. ~uch a fonnulation assured the intervention of at least two fir~
bngade umts w1th their teams 111 case oftunnels with a higher degree ofthreat.
SD_PPLY_ 2 assumed t?at all_ the fire ~rigade units are equally qualified and equipped
and so 111 such case each umt can mtervene m every tunnel autonomously. While 011 the other
hand, SUPPLY_ 3 presupposed that no fire brigade unit is qualified enough to intervene in
any tunnel autonomously.

243

With respective constraints thal no demand is outside 5 minutes dri ve tirne and that the
demand covered in O minutes drive tirne is maximized. more
sirnulalions on the
Slovenian road network were carried out to search for the optirnal numher and locations of
fire brigade units:
among the existing units,
among the existing units and subsequently added units.

f
100

4. Simulations results
With all types of demand (DEMAND_l, DEMAND_2 and DEMAND_3) the results of the
simulations showed that considering the tunnels' locations the existing fire brigade units are
not optimally located. In fact, the existing fire brigade units could not cover the demand
within a maximal drive tirne of 15 minutes. The average response tirne (7.71 min) remained
the same whether all 44 units or just 4 chosen units were taken into account. There was also
no difference in the demand area covered, which was in both cases 71 percent.
All together 12 tunnels were not covered, which implies that the potential demand with
a very high threat degree due to a large traffic volume cam1ot be satisfied by the supply
given. Further reduction to 8 fire brigade units slightly increased the average response tirne
while the demand area covered changed almost imperceptibly.
The simulations results suggested that the maximum response tirne should be increased
to 22 minutes to cover all demand with just 14 fire brigade units; the average tirne would
then be 12 minutes.
In the next simulation we considered also less qualified and equipped fire brigade units
in Slovenia. This simulation included only the more important tunnels with a higher threat
degree owing to the traffic volume. The maximal response tirne was estimated to be 11.92
minutes within which all demand was covered by 7 units. Five of these were extant
specialized fire brigade units. while two were extant but less qualified units. The average
response tirne this case was 5.88 minutes, which is shown in Table and Figure

Figure : Driving times to tunnels with a higher threat degree for 7 fire brigade units

Table l: Descriptive statistics of driving times for 7 fire brigade units
--------------------~
!

1
Descriotive statistics for the 11earest 1111its
5.88_
Average
0.42
Standard deviation
5.80
Median
0.27
Standard deviation
10.20
Range
1.79
Min
11.92
Max
1
270.82
Sum
46
N

By all criteria, this solution is rather satisfactory and practical, as a maximal coverage can be
achieved at lower costs, and it is relatively easy to put into effect. The two new units ought
to be tumed into qualified and properly equipped units to follow EU directives, and at the
same tirne such an action would probably be the most economically justified in the long
term.
Figure 2 shows that tunnels with a higher traffic volume and thus representing a higher
threat could be covered by 7 fire brigade units within a response tirne of 12 minutes' drive,
involving the five existing and two new qualified and specially equipped fire brigade units.

244

0

TUOflel portals

8

Seleded Rre Brlgades

~Tl..Hlnels

Figure_ 2:_ Optim~l coverage of all potential intervention demand for more important tunnels
(5 ex1stmg quahfied and equipped units and 2 existing, less qualified and equipped units)

5. Conclusion
The simulations ran in the GIS environment implementing the location-allocation model led
to _interesting fin~ings and _suggestions for the optimal allocation of qualified fire brigade
umts that could mtervene m case of road tunnel accidents in the Republic of Slovenia.

245

Testing a variety of scenarios the simulation generated more alternatives according
the
constraints and conditions given.
The first simulations did not provide satisfactory results. In fact, the 44 qualified and
properly equipped existing fire brigade units could not cover all dernand within a maximal
tirne of 15 minutes. Even reducing the number of units to 14 could not decrease the average
response tirne and increase the demand area covered.
After severa! scenarios were nm we had to consider less qualified and equipped fire
brigade units. This simulation estimated the maximal response time of 11.92 minutes within
which ali more important tunnels were covered by 7 units. Five of these were the existing
qualified fire brigade units, the others were two extant, less qualified, units. Since the less
important tunnels with relatively low traffic volume do not require special equipment is
possible to state that this proved to he the optimal solution, satisfactory in the practical sense
and economically justified.
The simulation results are a good basis for deciding on the number of the quahfied and
specially equipped fire brigade units as well as their allocation in Slovenia. However, even
the fact that the most qualified squad reaction tirne can be as sh01i as minute, the road
condition most of the times is not ideal, which in turn increases the calculated response times
for 5 minutes or more. Additionally, taking into account that tunnels represent a very specific
environment for fire to evolve rapidly, the tunnel road accidents with fire involved are
practically impossible to dominate by experience. The fire brigade intervention in such cases
can be too late. Thus, considering all these facts the education and reminders to the drivers to
observe the rules in tunnels, and the control of the speed limits play an important role in
preventing tunnel road accidents.
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[4]

[ 5J

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[8]
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246

LNIVERSAL ROUTING ALGORITHM FOR CITIES AND OTHER
BUILT-LP AREAS
Pavels Patlins
Riga Technical Cniversity. Faculty ofEngineering Economy. Logistic and Transport Economics
Department.

Abstract
50 years ago the main business problem was to develop a production: after that ~ to organize
different marketing processes, but the key topic today is to organize the logistic process by
optimal way. In other case it is impossible to achieve success in business. Auto transport is
the corner-stone if we plan goods delivery processes for cities, towns and other built-up
areas. Actually, we may use only this mode of transport to satisfy customers and organize
from-door-to-door delivery technique.
Keywords: Planning, routing, logistic methods. algorithm.

L INTRODUCTION

It is not so easy to create the optimal route within cities, towns and other built-up areas,
because there are many different ways how
complete
Sometimes logistic specialists
make a typical mistake during the routing process, they try to minimize only vehicle's way,
serving customers within the particular route.
The problem is really topical especially for multi-drop and multi-pick circular routes,
when each vehicle should serve many customers (for instance, 50) and return to the main
depot. It is necessary to optimize this process to achieve the best solution.
There are many special methods and also computer programs that may be used to solve
the particular problem, connected with routing. But every method or program has different
cons, talking about route planning within cities, towns and other built-up areas. In this case
is possible to achieve the optimal result, only using specials methods, programs and
specialists' experience combination.

2. ROUTE CLASSIFICATION
Routing is the top problem within the modem logistic environment.
There are two main types ofroutes:
11
circular route - connects more than 2 points. There are three types of the pendulum
route: multi-pick route, multi-drop route and combined route - (figure l);
" pendulum route - is a route, connecting only two points (for instance, it may connect the
particular depot with the customer) - (figure 2).

247

3. Ll'\11\ERSAl. ROLTlNG ALGORITHM FOR CITIES AND OTHER RUILT-UP
AREAS.

C

B

D

Trnnspon ~ompan;

How many suppliers does
the customcr have
depending on the particular
day of thc wcck 0

<

E
Figure

appreciate
staffs qualification
and sk11ls.

Example of a circular route (2,p.12

Figure l illustrates a very popular mode of circular route. which is multi-drop route.
Production is delivered from the one point (<lepot) lo many different customers.
Multi-pick route means. that production is collected from many different places; after
that it is necessary to delivery this production to the particular place. For instance, this route
is very significant for dustman companies.
Combined circular route associate both of the above-mentioned types.

A

L5: I'.~'.':~ ................ /CJ\
a)

How often does the
customer have the
suppliers row 0
6. step. Does the leve! of demand of the customer exceed
vehicle ·s capacity 0

B

A

b)

c)

Create the circular routc. Use the besi
combination ofthe own and commercial
transport. Combine multi-drop and multi pick
routes. lmprove the leve! of the consumer
service.

Create the pendulum route. Use the
best combination of the own and
commercial transport. Try to load
vehicle's run back. !mprove the level
of the consumer service

~>0,5
A

Does customer
have special
discharging
equipment°

B

Technique

Technique
Minimize
vehicle's
idle

Maximize

running.

utilization rate

vehicle's
cubic capacity

Traveling

Minimize the
lota! tirne of the
turn.

salesman
mcthod.

Minimal growth
mcthod [2,p.144148]

Using of
the spccial
soft.

B

Figure 2. Pendulum route's types
7. step. Check the result. It the optimal result achieved?

Figure 2 illustrates types ofthe pendulum route.
a) Pendulum route with the idle running back.
b) Pendulum route with when the rnnning back is partly-loaded.
c) Pendulum route with the full-loaded running back [2,p.121].
Actually, c) is the best type of the pendulum route; the running utilization rate (~) is
equal by 1 or 100%. Unfortunately, it is quite difficult to create this route in the real life.

Figure 3. Universal routing algorithm for cities and other built-up areas.

248

249

Accc1rding to the universal routing aigorithm for cities and other bu!lt-up areas, it is
e.,pedientl) to use the folkm ing steps. soking routing problem:

r

,'-,rep l. First of ali it is necessary to put serviceable territory into coordinate system. This slep
provides visuality and cleamess ofthe task.

lt is e:,pedienily to use specials methods and techniques to design the best route for
cities and other built-up areas. The most efficient way ,s to use these methods, teclmiques
and professional experience combination. Universal routing algorithm for cities and other
built-up areas (chapter 3) makes easy and more efficient route planning process.

Step 2. A~er that it is expediently to m\estigate ways between customers and lerritory's
infrastrncture in general. It is necessary to know everything about roads. its quality,
distance and other aspects.

5, REFERENCES

Step 3. Then to find the minimal distance between the start-point (depot where production is
located) and every customer. Minimal distance means way, which provides auto nm
tirne, going to the customer.

l. 13,iJ)KHHCI<HH A. JiorncnIKa: y qe6I1,HI( ,iJ,jHJ: Bb!CillHX H cpe,TJ,HHX CITel\HaJibHbIX yqe6Hb!X
3aBe,n:eHHH. 3-e H3,Ll,a!me. nepepa6. H ;(OIL- M.:vIHqiopMa!~HOHHO- 1me,n:peHqecKHH l_\eHTp
··Map1(ernm ... 2000.- 375c.

Step 4. Fourth step
imestigate customers' particularities as well as their discharging
opportunities and volume. There are 7 key-areas within this step (please. see figure
3, fornih step).

2. Hepyill 10. JlorttcTnKa: Yqe6mm ;i_;m B}'30B. - 2-e
,llAHA". 2000. - 389 c

Step 5. After that routing is started. LJsing special Svir's method [l,p.194], it is expediently

3. John J. Coyle, Edward J. Bardi. C John Langgley. Jr. The rvianagement uf Business
Logmics. (8-th Edition), West Publishing Companv, 200 l pp. 580.

to divide the serviceable territory in some sectors as weH as group objects and
connect them with particular company's vehicles.
Every vehicle serves the pmiicular sector and particular customers are connected with
appropriate vehicle.
Step 6. When logistic specialists start routing, they have to ans,\er to the following question:
does the leve! of demand of the customer exceed vehicle's capacity9 In order of the
answer as well as depending on some other factors, it is necessary to model a
circular route or a pendulum one. There some similar principles in the planning of
both types of the route. For instance, either circular, or pendulurn route requires to
use the best combination of the own and commercial transport. Of course, the main
task of every effective route is to improve the leve! of the consumer service. This
step is really very significant, because it may increase or decrease company's
transportation costs.
There are some additional tasks as we!L For example, to reduce company's
transportation costs, decreasing fuel costs, and other cost types. But this task is not
the most significant in the modem business. Depending on the route type, it is
expediently to use a particular technique, to achieve the best result. It is very
important to use special methods, creating circular routes or use special principles,
creating pendulum routes (increase the leve! of the cubic capacity utilization rate and
decrease the idle rnnning of the vehicle ).

uepepa6. n

}lOIT.

M.: "IOHI1TI1 -

4. S.EngewalL Cost a!locat10n in some routing prob!ems
a game theoret1c approach.
Dnision of oplim1zatiou. Department of mathematics. Lonkopmg msUute of
Technology, SE-582 83. Linkoping, Sweder1 -pp l 34.

Step 7. The last step is: to check the result lf the optimal result is achieved, logistic
specialists create the plan of delivery. If the result is not optimal, it is necessary to
improve it and check one more tirne.

4. CONCLUSION
Route planning process is a very significant logistic chain's operation. It is necessary to
draw routes very carefully, in other case delivery, transportation and logistic costs in general
will be increased.
Routing process helps to satisfy ali the customers in the best way, to supply them
production in tirne. It is not so easy to achieve it, especially within city's route when one
vehicle serves many clients.

250

H1.L

251

POLICE PATROL ROUTE OPTIMIZATION
Marko Urh, Anton Čižman
liniversity of Maribor. Faculty ofOrganization Sciences, Kidriceva 55a, 4000 Kranj, Slovenia
marko.urh@fov.uni-mb.si, anton.cizman@fov.uni-mb.si

Abstract
This paper treats the problem of police patrol route optimisation according to the locations with higher
incident frequency. In the course of our research we have analysed 9537 traffic accidents that occurred
between the years 1994 and 2002. In the case of these incidents the police patrol reaction tirne is
extrernely importanl. To perform the simulation of police patrol movement we have constructed a model
of streets and roads
the area covered by the police station in KranJ, using a geograplucal mformation
system. The model contains 1012 links and 69 nodes with a total length of 439,68 kilometers. The
developed and described model was prepared in the cooperation w1th the Operations and Commumcations
Centre of the General Police Directorate (OKC GPD) and can be use for solving other route problems.
Key Words: optimization, mathematical programming, geographical information systems. police

L Introduction
Continual changes in the world as well as in Slovenia have become constant and they dictate
different kind of work and life. With this change also the values of individuals and society are
being modified. The fact is, that security represents one of the key points which influences on
quality of life. Police has the power and legal authorization for ensuring safety and represents
public service for citizens. Patrolling as the part of a police work is a diverse and dynamic
process. For arrival on the place of a ce11ain event, police needs some tirne, which is called police
reaction tiine and represents fone, which elapses from a citizen call on intervention number to the
arrival of police patrol, criminal-police group or some other competence service to the place of
intervention event. Solving complex problems has become more efficient with improvement of
information technology with association with procedures and technique of optimization.
Geographical information systems (GIS) represent an impm1ant tool for making analyses.
Geographical information systems are computer supported information systems for capturing,
saving, searching, analyzing, displaying and distribution ofspatial <lata and information [l]. The
main advantage of this kind of analyses is a chance of seeing new connections between available
<lata. This gives us information, which we haven't had so far and can be used for better decisionmaking.

2. Some of the aspect of police work
Within every day work, police collects enormous number of <lata. Unfortunately, this <lata are
rarely used for strategic and tactical purposes. Police should be looking for problem area using
information systems, which are available according to the trends of events. Because of enormous
amount of <lata this kind of information systems can be one of the best systems for solving and
analyzing problems which are connected with some location. Results of this kind of analyses can
be used for making appropriate police strategies and tactics, which are used for solving ce11ain
problems. It's a known fact that the calls on intervention number 113 (before 92) increases every
year [2]. This confirms a demand for more rational and optimal work of Slovene police.

252

253

g
According to characteristics of individual events handled by the police and comparin
traffic
40.000
ately
approxim
about
statistical reports of the Slovenia police. every year happens
accidents. 35.000 violations ofpublic order and 90.000 crirninal offences.
the first
The most obvious example of negative events are spatiaL From the year 930, when
actions.
negative
of
kind
some
that
d,
discovere
been
has
it
done,
been
has
research in this field
they took
where
places,
the
according
differ
crime,
of
degree
and
rs
perpetrato
by
used
methods
with
place [3] This is also trne for traffic accidents. Reaction police time is critical when dealing
events
these
ali
However
cases.
criminal
traffic accidents, violation of public order and in some
are highly predictable considering the place where they occur.
Some facts, which confirm legitimacy ofmaking this kind ofresearc h, are:
• number of events increases every year (positive index growth),
and
• number of police patrols can't be increased, so we can only optimize their routing
work.
methods of their
the
• shorter reaction tirne of the police tirne can bring very positive consequem:es
lives.
rescuing
rs,
perpetrato
the
anesting
order,
lic
pub
of
violation
g
preventin
of
meaning
increasing satisfaction of citizens with services offered by police and other positive
things,
combination of information technology with procedures and technoiogies of
offers possibilities, which haven' been available before,
„ methods and techniques of operation research can be used for optimizing bigger systems
such as police GIS, regard to the nature ofthe problem and
for
„ adequate <lata, skilled people and suitable information systems are the basict and
unpleasan
other
some
or
crime
fight
us
help
can
which
e,
conquering new knowledg
unpredictable events.
task we have tried to answer the question, how to reduce police response tirne.
research
In this
on
We have combined three different areas such as operation research. geographic inf01mati
systems and police work.
•

this research we analyzed 9537 traffic accidents, which happened
cause. etc.
<late,
considers
between years 1994-2002 on area covered by police station in Kranj. Slovenian police
(3rd
injured
physically
badly
was
category),
(4th
d1ed
had
t
partic1pan
where
aH traffic accident
only
was
there
where
category). was hghtly physically H1Jured (2nd category) and traffic accident
European
malena! damage ( st category), ~articipants can 't agree about responsibility or to fulfill
on
accidents
traffic
analyzed
we
,
mentioned
report and wish the police to arnve [4J. As already
called
software
computer
used
we
task
this
h
accomplis
police station Kranj. Jo
area covered
with
ArcView Spatial Analyst. With this software we could <livide researched area on squares
accidents
traffic
analysing
when
resolution
te
appropria
is
which
meters.
00
x
Jnnensmn s of 00
Based on such distribution we counted the e\ents and detemiine the points with higher
a
critical points with higher frequenc) of traffic
frequency of events (hot spots). We recognized
s.
crossroad
on
p!ace
taken
accidenls and most of thern were
33. Rstablishing optimal route
route
Points_ wi1h higher frequencv of traffic accidents were a base ground for determine optimal
cvents
of
frequencv
higher
with
points
of police patroL ( ·mnbination of suitable network model.
for police
and with software catled Arc\iew l\Jetwork Analyst we could detennine optimal mute
order
optimal
in
points
11
all
visit
must
patra!
police
that
patrol. Assumpt10n m th1s mode! was
the
on
patrnl
ofpolice
route
optimal
represent
Figure
needed.
time
cons;denn g shortest poss1ble
traffic
of
frequencv
higher
with
points
the
to
according
Kranj,
station
police
by
area covered
le for
accidents. At simulation we assumed that this area is covered bv one police patrnl responsib
traffic accidents.

3. Optimization model under GIS (ArcView -ESRI)
3.1. Construction of network model of streets and roads

streets and
First step, necessary for doing the computer simulation was constructing the model of
police
five
top
the
of
one
is
which
Kranj,
in
roads on the area covered by the police station
streets
of
tion
Vectoriza
there.
live
who
people
of
number
the
to
stations in Slovenia according
of real
and roads was made by software ArcView-ESRI. Model was containing characteristic
and
limit
world, such as one-way street, closed street, restraint tuming left or to the right, speed
area
the
on
roads
and
length of individual streets or roads. On the network model made by streets
will
covered by police station Kranj, there was necessary to build a proper topology, which
The
O.
ARC/INF
software
used
we
phase
this
In
represents a basic for execute network analyses.
roads.
and
streets
between
relations
logical
fine
de
to
purpose of constructing topology is
3.2. Designat ing the points with higher density of events

on
Every traffic accident treated by Slovenian police is entered into the geographical infonnati
as
such
stics
characteri
tant
all-impor
with
map
ona
system of the police, where it represents point

254

Figure 1 Optimal routejor visiting alt 11 points with higherfrequency oftraffic accidents
area covered by police sla/ion Kranj.

011

3.4. Realization of simulation
d
Realization of simulation required a suitable network model of streets and roads establishe
have
we
~ase
first
In
ion.
optimizat
of
procedure
suitable
on
based
patrol
optimal route of police

255

r
assumed, that police patrol was located at a random spot somewhere on the area cm-ered by
police station Kranj, In second case we put police patrol random somewhere ona optimal route,
which we have dete1mined and described earlier, We have randomly chosen 50 traffic accidents
in this area, which took place between years 1994-2002.
In first simulation we assumed that police patrol was somewhere random on the area covered by
police stati on Kranj. Procedure of simulation was as followed:
<>
first it was necessary to choose one among 50 random points (traffic accidents) and it is
presented on figure 2 as point A,
„ then we have randomly chosen one among 1012 links (street or road), which composed
network model,
., programme (wtitten in Avenue) has random chosen length on ..;hosen link,
" after that programme puts a point on chosen length of chosen link, whi..;h is presented on
figure 2 as point B and represents starting point of police patroL
" then the programme establishes two points, which represen1s starting (B) and emling poin!
(A) or the beginning and the end ofthe route, which has to be traveied,
" tirne needed for trave! the route by the police patrol is written in special file as tirne
needed by this patrol. After 50 simulations we have calculated average tirne needed by
patrol to arrive on one of 50 randomly chosen points,
<>
simnlation was made for 50 randomly chosen points and for every one of them the
simulation was done 50 times, which gave us 2500 simulations.

"
"'
"
•
"'

"

then we have randomly chosen one among links, which represents optimal route (streets
and roads on figure 1
programme has random chosen length on chosen link,
after that programme puts a point on chosen length of a chosen link, which is on figure 3
presented as point H and represents starting point of the police patrol,
then the programme establishes two points, which represent starting (B) and ending point
(A) or the beginning and the end'ofthe route, which has to be traveled,
tirne needed for trave! the route by the police patrol is w1itten in special file as tirne
needed by this patrol. After 20 simulations we have calculated average tirne needed by
patrol to arrive on one of 50 randomly chosen points,
simulation was made for 50 randomly chosen points and for every one of them the
simulation was done 20 times, which gave us 1000 simulations,

A

Figure 3: Route oj the police patra! jrom ra11dom location on optimal route (B) to random chosen
trafjic accident (A).

4. Results

Figure 2: Route oj the police patrolfrom random location (B) to
random clwsen traffic accident (A).

In second simulation we assumed that police patrol was random somewhere on the optimal route
within the area covered by police station Kranj. Procedure of simulation was as followed:
" first it was necessary to choose one among 50 random points (traffic accidents), and it is
presented on figure 3 as point A.

256

For the purpose of simulation we have chosen 50 random traffic accidents, which happened
between years 1994-2002 on area covered by police station Kranj. In first scenario the police
patrol was placed somewhere random on network model (on the street or road) and had to come
to the place where traffic accident had happened. Second scenario assumed that the point of an
accident was the same as in first case but police patrol was located somewhere random on
optimal police route. In first case average tirne needed for arrival was 7'04" and in the second
case the tirne was 5'26". Comparing those times, we can clearly conclude that the second
alternative routing is 23,1 % shorter. In research we additionally established, that majority of
traffic accidents happen on Fridays and between 14:00 and 16:00 hour. This is the most probably
a consequence of tiredness, which is result according to finished working shift and other factors

257

r
connected with tiredness. According to the locations with higher frequency of traffic accidems
we established. that this locations in
represents higger crossroads in area.
S. Conclusions
For the successfully realization of the simulation it was necessary to build a suitable network
model of streets and roads on area covered by police station Kranj. Model contains 10 2 links
and 691 nodes, which measure 439.68 kilometers. Critical points with higher frequency oftraffic
accidents were established based on 95]7 traffic accidents. which happened in the researchecl
area between years 1994-2002. Reaction time of the police patrol is extremely important in these
cases. Based on these points we determine optimal route, which have to be traveled by the police
patrol under condition that every point has to be visit precisely once., Properly bui!d network
model in geographical information system was representing a base for realization of simulation.
From the research and results we can conclude that the police patrol reaction tirne is shorter ifwe
put patrol on optimal route, which was defined earlier, Researches from the pas! confirm the
theory that there is a strong connection between the locations where certain events are taking
place. Our research has shown that hy planning optimal route we can reduce the tirne that is
needed to patrol to arrive on the place where events took places. Improved reaction tirne of the
police patrol can have positive effects. Former arrival on the place where the accident happened
can save lives, prevents further accidents and has other positive effects. \Ve believe that police
offers a good service to citizens. From this point ofview we can conclude that by the reducing of
reaction tirne the satisfaction of citizens would be increased. Other positive thing about optimal
police routing is that we can reduce variable transportation costs, which are nowadays more
important than ever.
From logistical aspect we can say. that:
<>
we can reduce transport costs ifwe consider optimal routing for police patrol and
„ proper tirne and spatial supply represents added value, which means, that the subjects or
services are available, when and where is necessary [5].
The negative side of using an optimal route for police patrol is a chance that this kind of
patrolling could become boring and cause monotony. However in spite of bad things we can say
that advantage of this kind of work is essentially bigger. This kind of model can be used in other
areas in Slovenia with small modification. With this kind of work we can upgrade the police
planning and enable advantages of the police work and in the society.

References:
l. Korte G.B.: The GIS Book, On World Press, tretja izdaja, Santa Fe USA, 1994.
2. Internet: http://www.policija.si/si/preventiva/ 113/index.html
3. Trojanowicz, R.C., Kappeler, V.E., Gaines, L.K., Bucqueroux, B.: Community Policing. A
Contemporary Perspective. Anderson Publishing, Cincinnati, OH, 1998.
4. Zakon o varnosti cestnega prometa (ZVCP-1) Ur.!. RS, št. 83/2004, 35/2005, 67/2005
Odi.US: U-I-32/05-13, 69/2005.
5. Čižman A.: Logistični management v organizaciji, Moderna organizacija, Kranj, 2002.

258

The 8th lnternational Symposium on
Operational Research in Slovenia

S@R'OS
Nova Gorica, SLOVENIA
September 28 • 30, 2005

Section

MliLTI-CRITERION APPROACH VERSLS MARKOWITZ IN
SELECTION OF THE OPTIMAL PORTFOLIO
Zdravka Aljinovic, Branka Marasovic, Neli Tomic-Plazibat
Faculty ofEconomics, Cniversity ofSplit
Matice hrvatske 3 2 000 Split, Croatia
E-ma1l: zdra,ka.aljin0V1c@efst.hr: branka.marasonc@efst.hr: neh@efst.hr

Abstract: In the paper we select an optimal portfolio on the Croatian capi tal market. First, selection
the optlmal portfolio based
is carried out using classic Markowitz mo"deL which allows selection
upon the mean-variance (M-V) criterion. Second. we illustrate how the multi-criterion approach
makes poss1hle to integrate. within the portfolio select10n process. the conventional M-V criteria
with other market criteria.

Key words: pmifolio selectwn, Croatian cap1tal market, efficient portfoho. multi-criteria melhods

L INTRODUCTION
The Markowitz's theory of portfolio selection describes how to calculate a portfolio that
gives the lowest variance of return of ali portfolios having the same expected retum, or the
one that has the highest expected retum of all portfolios having the same variance. i.e. how
to calculate an efficient portfolio.
In accordance with the modem portfolio theory maximisation of retums at minimal risk
should be the investment goal of any successful investor. Nevertheless, contrary to the
expectations of the modem portfolio theory, the tests carried out on a number of financial
markets reveal the existence of other indicators important portfolio selection. Considering
the importance of variables other than retum and risk, selection of the optimal portfolio
becomes a multi-criterion problem.
In the paper we select an optimal portfolio on the Croatian capital market using
conventional Markowitz model and then, we illustrate how the multi-criterion approach
makes it possible integrate the conventional M-V criteria with other market criteria.
2. THE MARKOWITZ MODEL OF THE PORTFOLIO OPTIMIZATION
Now, we show how to calculate efficient portfolio and efficient frontier (the set of ali
efficient portfolios).
Throughout this chapter we use the following notation: There are N risky assets, each of
which bas expected retum E(r1 ). The variable R is the column vector of expected retums of
these assets, and S is the N x N variance-covariance matrix.
A portfo/io oj risky assets (when our intention is clear, we shall just use the word portfolio)
is a colurnn vector x whose coordinates sum to 1. Each coordinate x 1 represents the
proportion of the portfolio invested in risky asset i. The expected portfolio return E(r,) of a
portfolio x is given by the product of x and R:
N

E(r,)

= xr R = "I,x1E(r)
N

The variance ojportfolio x's return, a-: = a-.._, is given by the product xr Sx =

N

L "I,x xja-if .
1

i=l j=l

Mathematically, we may define an efficient portfolio as follows: For a given portfolio
variance cr~ (or standard deviation), an efficient portfolio x is one that solves

261

N

max E(r, ) = x T · R =

L x, · E(,;)

N

portfolio P has to be <D( PJ $ <D( PJ $ <D( P J (where P = XPS = L·\s,.

i=I
N

subjectto

X;

is proportion of

i=!

the share

xrSx=a_:,Ix, =L 0$x, $xM,'

s;

in portfolio P).

For criterion Ci to maximize we will have:

i=l

where xM, is the rnaximurn proportion to invest in share i.

f o if ci(P)= cjP)
P/P.PJ=[H;(dJ ifC/P)>Ci(P)' d1 =CJPJ C/P_J

3. THE MUL TI-CRITERION APPROACH

and
Considering the importance of variables other than retum and risk, selection of the
optimal portfolio becomes a multi-criterion problem.
Generally, the distinction is made in literature between two categories of criteiia: the
accounting criteria and those based on market value. The accounting criteria are indicators
of the financial situation of a company. There is a number of such indicators and their choice
depends on the manager's approach and objective.
Market criteria are indicators obtained by analysing the situation on the stock exchange.
They include: expected return, total risk (variance of expected returns), system risk (beta),
PER (price earnings ratio ), share liquidity, etc.
As there is a strong interdependence between some of the accounting criteria and market
criteria, in order to avoid their redundancy we limit this study to the following set of criteria:
l . Expected return E ( r, ) .
2. Beta coefficient: Beta coefficient is the measure of the systematic risk. For the whole
market, beta is by definition equal to 1.0; for a security with lower value of systematic risk,
i.e. with lower variability of expected rate of retum, beta is lower than 1.0, and higher than
1.0 for those with higher rate of risk. It is defined as:
COV (rm,rj)

p1 =

VAR(r,J
where r, is the expected rate of retum of the market portfolio and rj is the expected rate of
return ofthe share. It is the criterion to be minimized.
. 'd'
. . ) ri
total number oj shares treated
3 . T umover (l 1qm 1ty cntenon : H = - - - - - - - -----tata/ number oj shares outstanding
This criterion is to be maximized.
4. Price earnings ratio (PER): In many papers it is shown that shares with lower PER have,
on average, higher absolute and risk-adjusted retums, than randomly selected shares, so this
criteria is to be minimized.
5. Market-to-book value ratio: Shares of the companies with higher market-to-book value
ratio represent investments with higher risk, and so this criterion is to be minimized.
The method ofmulti-criterion programming, used in this paper, is based on PROMETHEE II
approach. With PROMETHEE II, each pair of actions is compared according to each
criterion. Since the overall portfolios that can be made up from a set of pre-selected shares
are infinite, it is impossible to compare all of the pairs of portfolios. In the paper [5] it is
proposed the method that allows an absolute evaluation of each possible portfolio by
comparing it with two fiction portfolios: the ideal ('p) and the anti-ideal (P). For criterion

-

,

1

pre-selected shares. In the same way C j ( P) = m[n C j (s;) . The set of possible solutions is

1

l H: (d ) if · c _ (J5).· > c_. (P)'

-

\..

J

}

.

f

'

!

'

a·=

C r P J C 1 { PJ

'

·-

·

where
.\

C/ P) = Ix;C'Js,),
i=l

i=1

H;

1r; are preference functions.

and

According to PROMETHEE II, net flow of a portfolio P is:
<D( P)

=

I,

wj

(Pj (P, I:_ )- P1 (P. I' ))

i=l

The preference functions requires thresholds q;. p;.

a:, i.e. q

1

,

p ,a

1

•

Finally, the optimal

portfolio is one that solves:
Ma.x<D( p)

subject to;
N

Ix,

11

Cj to maximize we'll have CJP)=m(JxCJs;), where S={si,s 2 , ... ,sN} is the set of N

f O if C,(P)= c.(P)

P1 (P PJ -

=l

i 0$x; $xu 1 ,

i=l

where
N

<D( PJ

= Iwi<DJ P),
i=1

P=X~S,
Xp =(x 1 ,X 2 , .. ,XN),
SP =(s 1 ,s 2 , ... ,sN),

= 1,2, ... , N) in portfolio P,
xM; maximum proportion to invest in share i in portfolio P,
N is the number ofpre-selected shares which can be included in portfolio P.
X;:

proportion invested in share i (i

4. AN APPLICATION
Using the above models we subsequently calculate the optimal portfolios on the Croatian
capital markets. From the total number of securities quoted on the Zagreb stock exchange in
2004 and 2005 a sample of fourteen shares has been separated. The shares sample contains
the most traded shares in the second part of 2004 and first quarter of 2005: CROS-R-A,

the set of portfolios which can be formed from pre-selected shares. For any possible

PLIVA-R-A, PODR-R-A, ATLS-R-A, ADRS-P--A, DLKV-R-A, ERNT-R-A, IGH-R-A,
KOEI-R-A, KRAS-R-A, PBZ-R-A, PLAG-R-A, ZABA--R-A, ZAPI-R-A. For each

262

263

-

1

security from 1.he sample we lake the
price at
each
First we
calculate the monthly return for each security. This is the percentage return that would be
earned by an investor who bought the securily at the end of a particular
1 - and sold
at the end of the following month. For month t and security A, monthly return r 1s

Table 2 Stock's values for the constructed criteria
r--·Mean return
! Turnover
1

; Stocks
.
(%)
'
1
I
CROS-R-A
· 6,104554647 !
1
!PLIVA-R-A _i_:3, 101549283
1 PODR-R-A
__j_ 4,299254057
I ATLS-R-A
16,56070642
i ·- 3,586385321_
1 ADRS-P--A
-r
1
1 DLKV-R-A9,52658376
i
1 ERNT-R-A
11,5267951
1
IGH-R-A
' ----15,45432582 T1
0,03302109:
W5OEI-R-A
4,809726606
1 KRAS-R-A
7,8134573604 •
PBZ-R--8.
5,861165236
: PLAG-R-A
1 ZABA--R-A
9,154158959 1
1
: ZAPI-R-A
f----

i

defined as:

!

Now, we calculate monthly mean, monthly va1iance and monthly standard deviation. By
programming in MATLAB, we calculated ten efficient portfolios (Table 1) and efficient
The range of solutions is
frontier for Croatian shares marketu observed period (Figure
given from the lowest to the highest possible monthly return, with a share ofshares in the
portfolio which lies on the efficienl frontier, and their monthly risk and return.

18----------~-~
l61
1

'l
l'l

i 10f- --

7/
·!· / ·

;;r

-1

sf ··· f ...

·······+·······i

·,l:/!-~~
-

O

6

10

l

1

I

1

1

I

i

_,

1

1

i

1
1

M/B
1,512619
0,943449
0,670162
1,131153
0,597637
1,595817
1/543534
1.043264
0,37464:
0,705089 1
2, 132389 1
O, 150894 1
1,469018 !
7.532247 ·

fn this case the linear preference criterion is used, which includes one threshold. For each
criterion J and for both of functions H; and H I we need thresholds p; and p ~ which

·t·

i "(·······l··

I

I

Mean-Variance-Effidentfron!ier

l

§
j
(%) ___ j
PER
i
0,3617)4,35275912 1 13,51974
Q,6437 7,91014276 ' 8,521907
1,0165 32,7235612 i 14,71509
1,0833 8,84472735 86,52308
1,0892 37,1017526 7,496381
__1_,7629 9,07602715 i 8,642446
1,0633 4,21995269 ; 8,784485
2,0765 40,8714844
:
--- ~~~--- '___ -3.08793
~-~---,
,7023 139,14772551 52,84407:
0,7125; 15,743644 j 15,08071 j
1,4761 0,82862545 i 12,31563 :
1.267~+~,91054294 1 15,4925 j
0,8201 0,28580844 8,772863
0,541 1_&1479571 15.50568 !

de.fine strict preference area. Then, function <D I for cnterion which isto be maximized. is

~

ofthe form:

____i__ __.....L___~__J
15
20
25
lJ

when C. (p)\ ~ C (P)' ~ , \
j

1

_,,

:

respect

the

Figure 1 Efficient frontier for Croatian share market

<!>j(P)=PJP,~)-PJP.P)=

-1=0

Table l Efficient portfolios for Croatian shares market
1

i Port

1

·------i------

1

i 4

!

when

p; -cj(P)+ci(P)

0,4477

0,1175

o

o

'o

o

p;

o
o

o

o

()

o

o

o

We propose these values for our application:

o

o

o

o

o

lo

0,0624

0,0891

0,1210

0,1497

0,2115

0,2774

0,3899

l

o
o
o
o
o

o
o
o
o
o

o
o

o
o

o

o

o

o

o

o
o

0,0641

0,1937

0,3048

0,4095

0,4763

0,2058

o
o

0,1993

0,4043

0,1790

0,1791

0,09620

o
o
o

o
o

o
o

o
o
o
o

o
o
o
o

ZABA--R-A

o
o
o

o
o
o
o
o

0,0665

0,1746

o
o
o
o
o

0,0610

0,1318

0,2377

0,4279

0,3126

0,0470

ZAPI-R-A

0,3477

0,2021

0,0695

o

o

o

o

o

o
o
o
o
o
o

i 1

CROS-~~~-io,4777

2

3

0,5832

0,6281

o
o

o
o

0,0356

KOEI-R-A

o
o
o
o
o
o
o
o

KRAS-R-A
PBZ-R-A

PLIVA-R-A
PODR-R-A
ATLS-R-A
ADRS-P--A
DLKV-R-A
ERNT-R-A
IGH-R-A

PLAG-R-A

!

'0,6188

5

6

7

8

9
!

'o

--

-··

p;

11,5607

1,0383

32,8715

7,5036

4,8491

o

P;

8,1015

0,6765

7.7142

71,5231

2,5322

o

This

o
o
o
o
o
o
o
o

p; + P7 =IC)P)-c)d.

Exn, Ret.%

3,2114

4,6946

6,1779

7,6611

9,1444

10,6277

12,1109

13,5942

15,0774

16,5607

Port. S:D:

4,6235

4,7931

5,2359

5,8280

6,7705

8,5713

11,1281

14,5149

18,5610

27,3281

To apply the proposed multi-criterion model, we had to calculate: values of beta, tumover
rate, market-to-book value ratio and PER for pre-selected set of shares. The values of all
criteria are shown in table 2.

264

.P)+ P1-

1

Risk(Standard Oll"tiation)

set

of

values

for

the

thresholds

p;

and

p~

relation

These values must also reflect the preferences of a decision-

maker, exactly as relative importance of criteria. In order to take into consideration the
behaviour of investors, we proceed to the change of weights. Table 3 shows weights of three
possible scenaiios.
Table 3 Weight of each criterion
Criterion
Mean return l .a
1 0.30
Scenario 1
0.30
Scenario 2
0.50
10.10
Scenario 3
0.50
0.20

Liquidity
0.15
0.20
0.10

PER
0.15
0.10
0.10

M/B
0.10
0.10
0.10

We note that this application is a simple illustration of the proposed approach. No real
decision-maker is implied. By programming in MATLAB we get solutions given in table 4.
Firstly, an optimal portfolio is calculated without maximum propmtion constraint, secondly

265

considering maximum proportion constraint xM

. and finally

THE CHOICE OF PRICE INDEX FORMULAE FOR ELEMENTARY
INDICES

considering

XM=02.

Table 4. Optimal proportions
to invest in shares
·-

r--1

Stocl<s

!
!

~

1

1

PODR-R-A

o

...

1

j ADRS-P::J\

-

1

U.1469

o
o
o
0.3000

1

1

1

DLKV-R·A

i ERNT-RA

: IGH-R·A
1

1
j

1)
1

!

i

i

PLAU-R-A

!

ZABA--R-A

o

!
i

1

o

o

o

o

o

u

I

o
0.2000

o

1

u

i
i

(/

+

o
o

i

I

o

1

1

1

0,2000

O
-

O
O

i

0.3000

'M

1

.. ···-t
1

o
o

u

!

;

Xj

=0,2

1

1

,;;--~
1

1

'M=

XM =iJ.21

o

0.1000

0.2000

1

l

0.3000

1

0.3000

1

lt-

i

u

1

o

()

o

o

o

o

o

1

0,3000

0.2000

{J

o

i

o

o

o

o

-

1

o

0.8886

0,3000

o

o

o

o

o

o

u

o

o

o

o

0.1 i 14

0.0445

o

0

U.2000

-

1

o
i

o

l

()

V.2000

1)

()

0.2000

IJ

o

o

- o
u

o
o

0.3000

0.2000

o

o

·-·

- -----

-- 1

i

0.2000
C----------

---

i),2000

-

o

1

IJ.0555
-----

i

0.2000
---·

o

o

1

i

___ _!__

1

0.2000

0.2000

o
i

-+----

!

-

1

0,2000

o

1

1

x;

= 0.3

!

o
1

:q
Xhf

o
o
o

1

o

1

1

i ZAPI-R-A

o

!

Scenario 3

1

o
o

0.3000

KRAS-R-A

!
1

Xj

= 2

Draženka Č'1znuc
Faculty ofEconomics. l;mversny ofZagreb
Ke1111edye, 1rg 6. l O 000 Zagreb. Croatrn
de 1z1rnc@efzg.hr

·1

!

Xj

XAf

+

0.8531

1

l PBZ-R-A

i

1

KOEI-R-A

!

!

0.1

o

1 ATLS-R-A

=0,.1

····- , , - - - - -

Scenario 2

1

o
o

PLIVA-R-A

!

Xj

XAf

CROS-R-A

r

Scenario 1

o

o

_ 0.3000

Abstract
Recently the mterest
the caiculatwn meth,Jds that are apprnpnate at the lowest le, e' of
aggregauon
a Consumer Pnce lndex (CP!) has mcreased.
has
been known that d1lr'erent
calculation methods for eiementar, mdices g1ve d1fferent results.
The geomelnc mean 1s bemg m1rodu.::ed hy more and more countries. The approach is to calculate
a geometric mean
prices m both penods and then deri ve the pnce relative or calculate a geometnc
average of the price re!atives. Both calculations w!ll yield 1he same results.
Keywords: Consumer Pnce lndex. pnce mdex formuia, e!ementary aggregate. elementary mdex,
elementary mdex bias

-

---·

l. Introduction

0.2000

u - - - -o- ~

1

5. CONCLUSION
The two different approaches have provided significantly different resu!ts. Cndoubtedly,
the optimal relation of retum rate and risk is what should be taken into account in selection
ofthe optimal portfolio, but the question is how to consider other, very imp01iant market and
accounting indicators for companies whose shares are traded. In this work we have shown
how that it can be done by a new multi-criterion programming approach.That approach
allows a stock market expert to build, in accordance with the decision-makers' preferences,
all the available information into the model.

Measuring prices aml their rale of change accuratel) is central to almost every economic
issue. Consumer Price Imlices (CPTs) published
national statistical agencies are mostly
described as measures of price change obtained
comparing the cost of a fixed set of
commodities through tirne.
The CP! is the hest measure currently a,ailahle., hut is nota true cost-of-livmg index It
suffers from a variety of conceptual and practical problems There are a number of issues in
the design of a CPI where there is no uni4ue best ,\ay of doing things.
Recently the interest into the calcularion methods that are appropriate at the lowest level of
aggregati~m of a CPI has mcreased. It has long been known that different calculation
methods for elementary indices gl\e different results.

BIBLIOGRAPHY
2. The nature of these indices
[l] Aljinovic Z., Marasovic B., Tomic-Plazibat N., (2005): The Selection Of The Optimal
Portfolio On The Croatian Capital Market, Proceedings of Sixth International
Conference on Enterprise in Transition (extended abstract, 299-301,fitll text on CD ROM)
[2] Anderson, P. L., (2004): Business Economics and Finance with MATLAB, GIS, and
Simulation Models, MA TLAB & Simulink Based Books, Chapman&Hall/CRC
[3] Benninga, S., (2000): Financial Modeling, The MIT Press, Cambridge
[4] Black, F., (1972): Capital Market Equilibrium with Restricted Borrowing, Journal of
Business, 45: 444 - 455
[5] Bouri A., Martel J. M., Chabchoub H, (2002).: A Multi-criterion Approach For
Selecting Attractive Portfolio, Journal OJ Multi-Criteria Decision Analysis 11 :269-277
[6] Prohaska, Z., (1996): Analiza vrijednosnih papira, lnfoinvest, Zagreb
[7]
The
Zagreb
Stock
Exchange
(2005):
History
Trading
2004-2005,
http://www.zse.hr/history. php? sessionld=62b23 Oda73 98c66c9 57 5bdb 8 62 5d416e

266

Elementary aggregates 1 play a strategic role in the index structure. They serve as strata in
price sampling and provide primary blocks to build up indices at higher aggregation levels.
The overall CPI is a weighted sum of ali the sub-indices at the lowest level of aggregation.
The choice of elementary index fonnulae is a very important decision, likely more
important than choosing an index fonnula at higher aggregation levels. Different countries
use a variety of techniques to calculate elementary indices. The Boskin Commission (1996)
1 Item, composite item and elementary aggregate are related
terms. The term "item" is used loosely to mean
any good or service included in the expenditure covered by the index. The tem1 "composite item" is used more
precisely to signify the most detailed leve! of aggregation of consumption for which weights can be estimated.
If a breakdown of the consumption of a composite item by region or type of outlet is not available and cannot
be estimated, then that composite item constitutes an elementary aggregate. But if such further breakdown can
be made, then the composite item is decomposed into a number of elementary aggregates.

267

rnaintamed that rhe use uf particular mdex number fornwla for e!ementar,, mdices 1mparted
an upv,anl bias ofapproxrn,a,el: ii 2'- percentage pomts annuall) m the merall L S CP!c
Research un elernenlan rndices has mostly been restricted to um\eighted formuiae
(arithmetic and geometric means) because detailed quantity dala are not a,ailabie \,ithin
elementary indices. Thus the problem is to compme elementa.[)' aggregate indices using only
sample price Jata and without any weights.
The choice of formula for elernentary aggregate indices needs to take account of three
features of such price data, which are of major practical importance:
!) There are sometimes missing obsenations.
2) Forced replacements occur the disappearance of one product frnm the outlet \\ here
its price has been obsef\ed or the permanent closure of the outlet requires the
selection of anuther product to replace it.
3) Optional many-to-many replacements occur \\hen the sample of products and,or
outlets is revised.
In response to an mcrease m the price, consumers could respond by adjusting their
spending to favour relati vely lower-priced goods or senices. Substitulion can take several
forms (substitution arnong brands of products, substitution arnong product sizes, substitution
among outlets, substitution across tirne, substitution among types of items within the
3
category. substitution among specific items in different index categories).
An elementary aggregate index can be computed:
„ as a ratio of mean prices or a mean of price ratios
„ using arithmetic or geometric or some other mean
„ as a chained index or as a fixed base index
There are certain tests or axioms, which an index should meet:
„ The identity test: whether or not the index has a \alue of unity if ali prices return to
their price reference-period leYeL
The permutation test: if price ''bounce", in the sense that the price of each of two
o
products becomes the same as the price of lhe other was in the preceding month,
while other prices remain unchanged, the index should show no change.
„ The unit of rneasuremeut test for example, if a price per kilo is used m al! periods for
any observed product, instead of the price per 500 grammes the elementary aggregate
index will not be affected,

3. Recent developments in elementary indices
3.1 Australia
It was recognised that a more effective index processing system might be developed if the
index compilers could select an elementary aggregate calculation formula from a suite of
such formulae, The specific fonnulae being considered are:
geometric mean of price relatives
I

anthmetic rnean uf µnce relatives
II
HI relatiYe of anthmetlc mean prices
The focus \\as placed on the use nf the unweighted (or equai weighted) variants of the above
fornmla
The logical starting point are the known properties of the alternative approaches from the
perspectiYe oftheir effecl on index outcomes:
„ the three approaches deber an identical outcome when ali price movernents are
equal
„ approaches II and 111 deliver id~ntical outcomes when ali base period prices are equal
" approaches I and lll deliYer 1dentical omcumes under conditions of price "bouncmg"
„ approaclles ! deliYers tile same result as an approach based on the relative of
geometric mean pnces under all conditions
The approach based on the relative of anthmetic mean prices is attract1ve due to 1ts
imrnediately dernonstrable equivalence to the standard Laspeyres fornrnla, its ability to deal
equitably with price houncing and the fact that ,t produces an average price as an automat1c
by-product of index construction. Arguments against this approach are: 1) in practice, outlet
weights are at best based on knowiedge ofrelative value shares rather than quantil!es, and 2)
the use of average prices a!one denies the index cornpiler ready access to individual
measures of long term price behaviour (price relat1ves;.
The strengths of the approach based 011 ihe arithmetic mean or price relatives are: l) the
better alignment of omlet weights with the underlying source data, 2J its ability to cope more
equitably wirh store standards, and 3) the additional mfornrntion embodied in price relatives,
The major weakness is seen as the counter intuitive way m which it deals witb price
bouncing.
The strengths of the geometric mean of price reiati ves approach are: l) it assigns base
period outlet weights in accordance with the source data. 2) it handles the problem of price
bouncing equitably. 3) it pro\ides the index compiler with price relatives, and 4) 1t deltvers
outcomes identical to those obtained by the reiau \·e uf geometric mean prices approach. Its
major dra\\ backs are 1) the d;fficuity statisl!cians would encounter explaining its use to the
larger comrnunity and. 2) the fact that it does not presene fixed underlying qua11tiry we1ghts.
In Australia (unweightedJ geometric mean fmmula is the preferred formula for calculating
the elementary indices primarily because it modeis, better then alternative fonnulae,
substitutions that consumers make in their consumption of goods and services m response to
changes in relative prices.
The geometric mean cannot be used to calculate the average price in ali elementary
aggregates, It cannot be used in cases where the price could become zero. lt is also not
appropriate to use the geometric mean in elementary aggregates covering items between
which consumers are unable to substitute, For these elementary aggregates the relative of
arithmetic mean prices formula is used,
3.2 Canada

MJ. Boskin, E,R, Dulberger, RJ, Gordon, Z. Griliches, D,W Jorgenson; The CPI Commission: Findings and
Recommendations; American Economic Review, May 1997 (Papers and Proceedings), 87(2), pp,78-83.
3
In response to an increase in the price charged by a store for a certain brand of ice cream, a consumer could
respond by redish·ibuting purchases along any of severa! dimensions represented by other priced items in the
category: to another brand of ice cream whose price had not risen, to a larger package of ice cream with a
smaller price per ounce, to ice cream at a different store where ice cream is on sale, or to a brand of frozen
yoghurt. The consumer also could respond by postponing purchase until a later dale. Finally, the consumer
could substitute from the ice cream brand to a specific alternative dessert item that is in another CPI category,

It was at beginning of the eighties when Prices Division encountered some anomalies in
the calculation of elementary indices. The anomalies were diagnosed as a result of linking
arithmetic means of price relatives, The purpose of the study, carried out in the Central
Research Section of Prices Division, was to assess the long-term effect of using alternative
elementary index formulae.
Six elementary index formulae have been taken into consideration, of which three are
ratios of mean prices:
• a ratio of equiweighted arithmetic mean prices

268

269

2

F
„ a rallo of equ1we1ghted geornetric mean rrices
" a ratio of eguiweighted harmornc mean prices
Three other formulae are means of pnce relatives:
" an equiweighted arithmetic mean of price relati,es
" an equiweighted geometric mean of price relali\·es
" an equiweighted harmonic rnean ofprice relati\·es
There are relatively small differences between index numbers derived from elernentarv
indices calculated as ratios of arithmetic, geometric and harmonic mean prices. For som~
commodities the highest index numbers were obtained with ratios of arithrnetic mean prices.
for some with ham1onic means prices, while index numbers obtained with ratios of
geometric mean prices fell between these two.
Differences between index numhers derived from elementarv mdices calculated as
arithmetic. geometric and hannonic means of price relatives are m~ch larger. Signs of these
differences are certain: an arithmetic mean of price relatives is never lower than their
ham10nic mean, while a geometric mean lies between the two. The amnlitude of the
'
differences grows with the frequency of linking.
mean prices
geometric
of
ratio
the
to
equal
always
is
relatives
price
of
A geometric mean
derived from the same sample. The index numbers obtained using these two elementaf\
index fomrnlae are also always equal to each other. This is not case with arithmetic o-r
ham1onic means.
Transitive formulae, such as ratios of mean prices or geometric means of price relatives,
will give the same index number whatever the frequency of linking.
The ratio of arithmetic mean prices was used in the Canadian CPl until 1994 as the main
elementary index formula to estimate month-to-rnonth price change for any given
4
commodity within a given spatial stratum. Geometric means started being used from
January 1995, to avoid problems with those categories of products that exhibit a broad
spectrum ofprices.

differs from the l 'PT-l T-XG only in the use of the Laspeyres fonnula far
(
aggregation ofprice quotations.
Fram December 990 to februar, 997. the CPI-U-XG rose 16.2 percent, compared with
18.6 percent for the CPI-L-XL The average annual rate of growth in the CPI-U-XG over
this period was 2.46 percent, 0.34 percentage points lower than the annual growth rate of the
CP!-U-XL
The geometric mean estimator has been introduced in both the CPl-U and the CPI-W 111
January 1999. Components retaining the arithmetic mean (Laspeyres) fommla are: 1)
selected shelter services. 2) selected utilities and govemment charges, and 3) selected
medica! care services.

3.4 European Lnion
European Lnion. Harmonised lndices of Consumer Prices HICPsl are consumer
the basis of a hannonised coverage and methodology.
price mdices compiled
CL,mmission Regulation (EC) No 749/96 defines eiementary aggregates referring to the
e:xpenditure or consumption cO\-ered by the most detailed leve! of stratification of the HICP
which reliable e:xpenditure information is not a,ailable for weighting purposes.
and
An "elementary aggregate index" is a price index for an elementary aggregate comprismg
price data.
The ratios of geometiic or anthmetic means of prices are the two farmulae, which should
be used in elementar, aggregations. The arithmetic mean of price relatives should not
normally be used. as it will · rnany circumstances result in failure to meet the comparab,
may be used exceptionally where it can be shown not to fail comparability
requirement.
requirement.
The price index for an elementary aggregate may be calculated as a chain index using one
the two preferred formulae. The arithmetic mean ofprice relatives must not be used where
chaining is more frequent than amrnal.

3.3 United States
The possibility of using the geometric mean formula to calculate elementary indices in the
5
CPI was first raised by BLS researchers in 1993. In December 996, the Advisory
Commission to Study the CPI (the Boskin Commission), recommended the use of th~
geometric mean famrnla for the aggregation of prices within ali item categories in the CPI.
In April 1997, BLS began issuing an experimental CPI that used the geometric mean
estimator in the calculation of ali elementary components of the index. This index was issued
to provide users with a quantitative estimate of the impact that use of a geometric mean
fomrnla at the lowest leve! of index construction would have on the perfarmance of the CPI.
The CPI-U-XG indices (the experimental CPI using geometric means) were a supplement
to the price infonnation available from BLS (CPI-lJ and CPI-W indices 6). Historical
differences between the CPI-U and CPI-U-XG arose not only because of the different
formulae used, but also because of methodological changes made in the CPI-U since 1990.
Far historical comparison, BLS alsoissued an experimental Test Laspeyres series called the
4

Each ratio was calculated taking into account only prices from "bilaterally matched samples", i.e. only prices
that have been collected during two consecutive montl1> in the same outlet and that relate to the same item.
Long-term estimates of price change are obtained indirectly, through linking of these direct month-to-month
elementary indices.
5
Bureau of Labour Statistics
6
The Consumer Price Index for ali Urban Consumers (CPI-U) and the Consumer Price Index for Urban Wage
Eamers and Clerical Workers (CPI-W)

270

4. Elementary indices in Croatia
The CPI bas become the official measure of inflation in Croatia in January 2004. It
replaced Retail Price Index and Cost of Living Index. The index covers more than 540
representative goods and services divided in 2 major consumption groups according to
COICOP. 8
Most prices are collected locally. Central price collection is applied in cases of state
monopolies and nationally homogeneous prices. Far products far which prices are collected
locally, the Central Bureau of Statistics supplies loose specifications to price collectors.
Collectors are free to initially select particular product far pricing in a particular outlet. In
order to make sure that the same products are then re-priced each month, the price collectors
also record a pre-defined set of additional product characteristics.
Elementary aggregate indices are calculated as a ratio of geometric means of the current
and reference period prices of the products within an elementary aggregate. The data
required for the estimation ofweights are not available within an elementary aggregate.
7

The CPI, which is calculated using a fixed-weight Laspeyres formula, does not reflect the fact that consumers
can and do change spending pattems as relative prices change. A measure of change in consumer prices that
uses geometric mean formula successfully accounts for this consumer behaviour. The index using geometric
means provides a closer approximation to a cost-of-living index.
8
Classification oflndividual Consumption by Purpose

271

5. Conclusion
The choice of elementary mdex fonnulae is a \ erv serious busmess. \vith strong potential
consequences on the resulting mdex numbers. · most practical cases, choosing an
elementary mdex formula 1s more 1rnportant than choosing a macro index fommla for higher
aggregat1on levels.
In ~10s_t discussion, the elernentary index number problem has been viewed as "lower leve!
subst1tut_1on" or "dementary index bias". An emerging body of research suggests that the
geometnc mean estimator for elementary indices adjusts for this bias fairly well.
The geometnc mean 1s bemg mtroduced by more and more countries. The approach is to
calculate a geometnc mean of prices · both periods and then derive the price relati\e or
calculate a geornetnc a\erage of the price relati \ es. Both calculations will vield the same
·
results.

References

2.

3.
4.
5.

6.

7.
8.
9.

l O.

11.
12.
13.

Australian Bureau of Statistics; Australian Consumer Price lndex: Concepts, Sources
and Methods; http:: www.abs.gov.au
Boskin M.J., Dulberger E.R., Gordon R.J., Griliches z., Jorgenson D.W.; The CPl
Commission: Findings and Recommendations; American Economic Review, Mav
·
1997 (Papers and Proceedings), 87(2), pp. 78-83
Formula.
Index
Piice
Consumer
The
In
Change
Planned
Statistics,
Bureau of Labour
1998., http:,/www.bls.gov
Bureau of Labour Statistics, The Experimental CPI using Geometric Means (CPI-UXG), 1998., http://www.bls.gov
Commission Regulation (EC) No 749/96 on initial implementing measures for
Council Regulation (ECJ l\Jo 2494/95 conceming ham10nised indices of consumer
prices, OJ No l 229/3
European Commission, Report from the Commission to the Council on
hannonization of consumer price indices in the European Union, COM/2000/0742
fina!
~MF; Croatia, Summary Methodology, Price lndex: Consumer prices; http:;/www.
unf.org
OECD, Main Economic bldicators, Comparative methodological analysis: Consumer
and producer pric.e indices,_ Vol_ume 2002, Supplement 2, http://www.oecd.org
Schultz B., Cho1ce of pnce mdex fonnulae at the micro-aggregation leve!: The
Canadian empirical evidence, First Meeting of the Ottawa Group on Price Indices,
Ottawa, 1994., http://www.ottawagro up.org/meet.shtml
Triplett J.E., Elementary indexes for a Consumer Price Index, Fourth Meeting of the
Ottawa Group on Price Indices, Washington, 1998., http://www.ottawagro up.org/
meet.shtml
Turvey R., Consumer price indices, An ILO manual, Intemational Labour Office.
Geneva, 1989.
Turvey R., Elementary aggregate indexes, 1999., http://turvey.demon.e o.uk/index.
htm
Woolford K., A Pragmatic Approach to the Selection of Appropriate Index Formulae
First Meeting of the Ottawa Group on Price Indices, Ottawa, 1994., http://www'
ottawagroup.org/mee t.shtml

272

SAMPLE SURVEY RESEARCH ON FINANCIAlL RISK
MANAGEMENT : A CASE OF CROATIAN COMPANHES
Ksenija Dwničic
University of Zagreb, Graduale School ofEconomics and Business, Depart:menl of Statistics
Kennedyjev trg 6. HR-10000 Zagreb, Croatia, kdwnicic@efzg.hr
Mirjana Čiž:mcšija
University of Zagreb, Graduate School ofEconomics and Business, Depart:menl of Statistics
Kennedyjev trg 6. HR-10000 Zagreb, Croatia, mciz:mesija@efzg.hr
Anita Pavkovic
Umversity ofZagreb. Graduate School ofEconornics and Business. Depart:menl ofFinance
Kenned,jev trg 6. HR-!0000 Zagreb. Croaua, apavkovic@efzg.hr

Abstract The paper presents the authors · sample survey research findings on statisticalh s1gni ficant
dependence of usage of financial risks managernent instrurnents on characteristics of Croatian firrns,
such as ownership type and application of controlling funct10n. The mam activity and managcrs · nced
for further educat10n about risk protect1on did not appear to be in s1gmficant dependence on managmg
nsks act1vity The liquiditv risk appeared to be more senous than currency risk and interest ratc risk.
Low practical efficiencv of mstruments was the main reason for non-usage of more protect10n
Keywords financial risk management mstrurnents. surve, research, random sample. X -test of

independence, Levene ·s test for equality of population variances. t-test of means difference

L Introduction
According to Jorion (2001 it is about a decade since financial industry emerged from prehistory of financial risk management, and now the circumstances are met for introducing
lntegrated Risk Management or Enterprise-wide Risk Management, which covers company ~
wide risks business and non-business risks, where financial risks are considered to be a set of
non-business risks including market risks, credit risks, liquidity risks and operational risks. In
the praxis the role of CFO (Chief Financial Officer) becomes more and more important and
more under stress, so they wish to share the weight of risk pressure using modified decisions
and modified managing strategies. Financial risks come out the financial transactions, but,
since, sooner or later, all the risks create financial consequences, as Peterlin (2003) said,
indirectly ali kinds of risks could be considered as to be financial. While economic risk is
connected with uncertainty ofproducing real goods and services in an economy, financial risk
is uncertainty connected with the price of financial contracts that perform the affirmation of
these goods and services (shares, bonds, and currency itself), and this risk can not be
eliminated, but could be only dispersed After recognizing the type of risk, financial managers
may use hedging, insurance or a kind of diversification strategy, prescribed instruments or
financial limits.
Companies in transition countries such as Croatia are particularly exposed to the problem
of financial risks, especially because they do not have enough experiences with protection
against them. So, the authors found financial management instruments usage to be quite
challenging topic to investigate.
The aim of this paper is to present the main results of an original empirical survey
research conducted using random sample of Croatian companies in October 2004 about the
extent to which large and medium-sized companies were using financial risks management

273

instruments to protect themseives against three kmds of nsks
nsk. currency nsk and
characteris11..:s
rnterest rate risks. The authors have mvestigated the 1mpac1
attitude towards financial management means apphcauon
The research hypothesis that financial managing in Croatian companies was independent
on their selected characteristics were tested as follows: the hypothesis H was that the usage
or non-usage of risk protection instruments was independent on Croatian companies · main
activity, the hypothesis H2 stated that the usage or non-usage of protection instruments was
independent on the type of ownership; the hypothesis H3 included the statement that the
usage or non-usage of financial risk protection means was independent on the application of
controlling function. and the last hypothesis H4 stated that the usage/non-usage of financial
risk protection was independent on the managers' need for some additional education about
financial protection services
Further purpose of the paper was
discover if managers knew or used certain protection
instruments for each kind of risks. why they did not use them more, or why they did not use
these instruments at ali.
In the analysis the appropriate statistical tests: Levene' s test for equality of population
variances, t-test for means difference and X -test of independence were applied (for tests
McClave et al. 2005, could be seen)

2. Survey Sample
Survey research based on a telephone interview as data collection mode with financial ( or
accounting) managers from a random sample of n= O Croatian companies was carried out in
October 2004. The methodology of stratified random sampling according to the number of
employees as a stratification criterion was adopted An approximately equal allocation of
sampling units was applied: n 1 =50 large (with more than 250 employees). with sampling
fraction

J;=0.12; and

n 2 =51 medium-sized companies (51-250 employees), sampling

fraction equals f 2 =O 03, from each of respective s trata was applied Sampling frame used was
a FINA's list of Companies from 2002. Later on, after the survey was finished, it was
recognised that distinguishing the companies by number of employees was not that important
for the main variable under study But, because this was an originally conducted pilot survey
for Croatia, as it has been described in more details in Dumičič et aL (2005a). it seemed to be
important to test if the size of a company would influence or not their usage or non-usage of
financial risks protection instruments.
Methods of estimation took into account the procedure of random sampling of units in
each of defined strata, so, margin of errors could be given, as well. Considering unbiased
point estimator ofthe proportion and with normal approximation of its sampling distribution,
in each strata an interval estimate for the proportion with n=50 and 95% confidence leve! (z =
1 96), was within maximum margin oferror of + 14.2%.
In the sample the majority of 56% (23 out of 41) of financial risk protection instruments
users were from large companies and 44% from those medium-sized. The users under study
had got an average number of employees of 565 with coefficient of variation of 107%.
Among non-users there was a majority of 55% of medium (33 out of 60) and 45% of large
firms. An average number of employees for non-users was 3 71 with coefficient of variation of
83%. Also, 56% (28 out of 50) of large companies; and 43% (22 out of 51) those mediumsized were users of financial risk protection instruments. But, the dependence of protection
instrument usage and company size was not shown to be statistically significant, because in
testing the hypothesis of independence for categories in Table 1 the empirical x 2 did not give

274

r
'l

enough evidence for rejection of the null-hypothesis ( x =1.200, p-value=0.273), compare to
the detailed research project description in Dumičič et al (2005a)
2

Table l. Usage offinancial risks protection instruments by company size
! Usage
I Medium-sized companies I Large companies
'Use
Don't use
! Total
1

!

8
33
51

1
1

23
27
50

For companies under study that were usrng financial risks protection instruments an
average annual revenue in the previous year (2003 was higher (35029725 €), than for the rest
of companies that were not using them
6923357 €) With the assumption of different
variances (Levene's empirical ratio F=l5 355 and p-value=0 000), the t-test of the means
difference for annual revenue for these two types of companies was applied, and, it showed
that this difference was significant only with significance level a higher than calculated pvalue= 0.066 (t=l888) Also. in firms that apply risk protection, the revenue was growing
more dynamically than in the rest of firms The appropriate ;r 2 -test of independence gave the
result which was highly significant, so it is possible that the revenue growth was statistically
dependent on usage of financial risk protection, and vice versa ( empirical x 2 =9. 711, pvalue=0. 008)

3. Research Results
Considering the main econom1c achv1ty of companies, for the purpose of testing the
research hypothesis HL the structure of the sample was as follows: 64% of companies were
from industry sector (manufacturing; electricity, gas and water supply; and construction); and
36% from services sector (retail trade: wholesale trade; hotels and restaurants; transport,
storage and communications, real estate, renting and business activities). Even though the
majority of instruments users were from industry sector. the dependence of risk protection
instruments usage/non-usage on the two roughly defined sectors of activity (industry sector
and services sector) did not appear to be statistically significant ( x =0.466: p-value= 0.495),
2

so, the research hypothesis H 1 could not be accepted
In testing the hypothesis H2, the survey results showed that the companies that were using
these instruments were more often (69%) registered as share holding companies than as
limited ones, and, dependence of usage/non-usage of protection instruments on the type of
ownership appeared to be significant only with significance level a higher than calculated pvalue=0.071 (x 2 =3.261).
Further, the hypothesis H3 came out from the idea that companies that apply a controlling
function in their business tend to apply some kind pf risk protection means, too. Among users
of risks protection instruments there were 87% of companies with developed financial
controlling and among non-users there was a majority of 65% of such companies. The x

2

-

tests of independence allowed the rejection of the null-hypothesis ( x 2 =5.097, p-value
=0.024), so, upon this test, with a =0.05, the categories of risk protection usage and
application of controlling were significantly dependant, so, the research hypothesis H 1 could
be accepted.
The hypothesis H4 stated that active financial risk management of a company was
influenced by the managers need for some additional education about financial protection

275

serv1ces, or vice versa. It was also found out that 40% of managers interviewed wanted to get
additional education about financial risk protection instruments services When testing the
mdependence of two categories mentioned in Table 2, the empirical X don't appear to be
significant, so, the null-hypothesis was accepted as it would be possible ( x' =O 684, pvalue=0.408), and the alternative H 1 rejected

Table 2. Dema nd for add1tional education about financial nsk protect1on b>v co mpany s1ze
Medium-sized companies

/ Education

Large companies

-----1
1

1

Need
Don't need
Total

1

l

24
27
51

agamst this kind of risk. Currency risks, as well as interest rale risks, were not considered as
to be so dangerous for Croatian managers The frequencies for financial risk protection
instruments "used" were in many cases smaller than for "known" category. This difference is
the most evident for liquidity risk protection method called "credit capability analysis".

Table 5. No. of managers that were acquainted by or were using certain types of financial risk
protection instruments (more than one answer was poss~l)_le) __ - - ~ - - - - - - ~
Against Liquidity Risk
Cash flow investments analysis
Analysis of assets, liabi!ities and sources
Credit ca~abilit}'. ana!ysis

20
30

so

Against Currency Risk
The reasons, mentioned by interviewed managers. for not using the protection mstruments
more often and more intensively are listed in Table 3, and this list indicates that the reasons
seem to be quite different across the surveyed companies "Protection instruments are not
enough efficient" was the most often mentioned reason for that

Table 3. The main reasons why companies did not use financial risk protection instruments
more often (one answer was possible)
Reasons
lrrotection instrumen~ are not well known - rotection instruments are not enough efficient
rotection instruments are too expensive
inancial market is not enough confident
Financial services have too many forma! requirements
Something else

Medium-sized
compames

!Large companies

2

3
2
2
2
2
10

6
5

2

Table 4. The main reasons why companies did not use financial risk protection instruments
,
(one answer was possible)

Medium-sized companies Large companies

tI>rotection instruments are not well known
Protection instruments are not enough efficient
IProtection instruments are too expensive
IFinancial market is not enough confident
Financial services have too many forma! requirements
Bad quality of financial services
Something else

14
12

o
1
3
1
2

6
2
3
1
2

o
12

Table 5 shows which instruments were known to financial managers interviewed and
which of them were actually used, compare to Dumičic et al. (20056 ). It is evident from the
highest frequencies for liquidity risk that most of users were concentrated on protection

276

Used

34
]O
38

29
29
20

Known

Used

9

5
10
12

14
10
2
15
9
8

-------

Against Interest Rate Risk

Also, the reasons for non-usage ofthe risk protection that were mentioned by interviewed
managers are listed in Table 4. It could be noticed that certain reasons for no-usage of
prolection by surveyed companies appeared very differently "Protection instruments are not
well known" was the most often mentioned one.

Reasons

Netting
Leading and lagging
i Selling prices policy
1 Assets and liabilities management
Currency forward
Currency futures
Currency swap
Currency options

Known

Interest rate management at the money market
Forward rate agreements
Interesi rates futures
Interesi rate swap
Interest rate option
Caps, floors, and collars

o
14
8
6

Known

Used

4
4
6
5
4
3

4
3
5
3
3
3

4. Conclusion

After the survey research for the purpose of this paper was carried out the enterprises in
Croatian transition economy seemed to be not enough aware of dangers that have arisen from
financial risks. The results from 2004 Croatian companies sample survey have shown that two
fifths of medium and large companies in Croatia did use instruments to protect themselves
from them.
There was a variety of reasons for non-usage mentioned by interviewed managers, but,
those that prevailed were: "Protection instruments are not well known" and "Protection
instruments are not enough efficient".
The main reasons for not using protection more often seemed to be quite different across
the surveyed companies, but the most significant reason was: "Protection instruments are not
enough efficient".
The research results indicated that Croatian companies were not sufficiently acquainted
with adequate protection instruments. Managers interviewed considered liquidity risk and
currency risk to be the most serious types of risks, and interest rale risk did not seem to be that
dangerous. These larger firms included in the sample were better informed and they used risk
protection approach, instruments or services, more often than medium-sized ones. Numbers of

277

companies that "used' these instruments were in many cases smaller than for the companies
that only "knew" about them.
_ Considering research hypothesis, sample survey research results led to some new
findmgs.
.
Afler t~sti~g the hypothesis
, the Croatian company' s usage/non-usage of financial
nsks prntect1on mstruments appeared not to be in statistically significant dependence on the
mam act1v1ty ofthese companies (H, accepted)
The test of independence ofusage/non-us age of financial risks protection instruments on
the ~ype of ownership (the hypothesis H2), showed that the two categories could be
cons1dered as dependent H, rejected) with significance level a higher than p-value=O 07
which is not highly significant
The test ofindependence ofusage/non-us age offinancial risks protection instruments on
applying the controlling function (the hypothesis
has shown that the ernpirical X-· -value
was significant with a =O 05. so the independence of categories under study could not be
accepted as possible ( H 0 rejected)
_
Afler testing the hypothesis H4, there was a surprise because nevertheless 40% of
mterv1ewed managers had got a desire for additional education about risk protection
mstruments, th1s characteristic was not m statistically significant dependence ( H, accepted)
on the fact that this financial managers used ( or did not use) risk protection means
Following the research findings,
should be recommended to banks and olher financial
mstJtutJons in Croatia,_ such as insurance companies, and consultants, to make stronger efforts
to mform corporate clients about financial risks protection services, methods and instruments.
and to suggest them what is available for application
The _limitati_on of this survey is that relatively a srna!! sample size was applied, so, in
perspechve a b1gger sample should be recommended !n a future samp!e survey the authors
plan t_o mclude not oni~ large and medium-sized, but also small Croatian companies to
mves_t1gate the financial nsks mfluence to this category of firms and their attitude towards this
real hfe phenomenon, as well

REFERENCES
. [~] Dumičic. K.. Č!žmešija, M._, ~avk?vic. A, Andabaka. A (2005a) Istraživanje
pnmJene metoda upravlJanJa financ1Jskim nz1c1ma u hrvatskim poduzecima - Anketa na
uzorku poduzeca. Projekt Ekonomskog fakulteta - Zagreb (Unpublished).
[2)_ Dumičic, K., D~mičic, M., Cukrov, R. (2005b). FinanciaJ Risk Management Instruments
Usage m ~~ge ~d Medmm-Sized Enterprises - Business Survey in a Transition Country. Bulletin
of the 55 Sess1on of the Intemational Statistical Institute, Sydney, Australia, April 2005
CD-rom.
'

[3_) Jor(on, ~h. (2001). ~alue,_risk and control: a dynamic process in need ofintegration.
In Fmancial Tm~es Mastenng R1sk. Volume 1: Concepts. (Edt. Pickford, J.). FT, Prentice
Hall, Pearson, Edmburgh, pp. 119-124.
[4] McClave, J. T., Benson, P. G., Sincich, T. (2004). Statistics for Business and
Economics. 9 th Edt., Prentice-Hall Intemational, London.
[5] Peterlin, J. (2003). Upravljanje financijskim rizicima - evidentiranje i izvješcivanje.
revizija i financije, RRIF, 10, pp. 46-54.

Računovodstvo,

278

MODELLIN G EXCHANGE RATE VOLATILIT Y IN CROATIA
Nataša Erjavec and Boris Cota
University of Zagreb. Faculty of Economics, Trg J.F.Kennedya 6, 10 000 Zagreb, Croatia
nerjavec@efzg.hr, bcota@efzg.hr

Abstract
There is no accepted model of firm behaviour subject to risk arising from fluctuations in exchange
rates and other variables. The consensus on the appropriate method for measuring exchange rate
volatility does not exist. A wide variety of methods that could be used to generate predicted values of
exchange rate uncertainty have been proposed. We employed GARCH type models and used data on
the percentage change of daily exchange rate between Croatian kuna and US dollar.

Keywords: volatility models, exchange rate volatility. GARCH

l. Introduction
The aim of this paper is to define a good volatility model that would capture the basic
characteristics of the percentage changes of daily exchange rate between Croatian kuna and
US dollar. Forecasting volatility of financial tirne series is the major task in many financial
applications. A model should capture the commonly held facts about financial tirne series;
such as volatility clustering, pronounced persistence, mean-reversion or asymmetry. Many
financial series display volatility clustering or autoregressive conditional heteroscedascity,
i.e. is commonly seen characteristic in financial application that large changes are followed
by large changes and smail changes are often followed by small changes. Volatility
clustering implies a strong autocorrelation in squared returns. The result of such a volatility
clustering is that external shocks influence the expectation of volatility many periods in
future, e. the volatility exhibits persistence.
Another characteristic of financial series volatility is that it evolves over tirne in a
continuous mam1er, i.e. the volatility jumps are rare. Volatility does not diverge to infinity,
which means that volatility varies over some fixed range. Mean-reversion of volatility is a
further implication of volatility clustering. It means that there is a normal level of volatility
to which volatility returns. As a consequence, there is a convergence in the term structure
forecasts to the long-term average volatility leve!. Furthermore, innovations may have an
asymmetric impact on volatility. The asymmetric structure of volatility generates skewed
distribution of forecasts.
In order to define an adequate volatility model we analysed ali mentioned characteristics
of financial series. We employed data on percentage change of daily exchange rate between
Croatian kuna and US dollar. The liberalization of capital flows and large increase in the
scale of cross-border financial transactions have increased exchange rate movements. The
magnitude of exchange rate movements increased in countries with underdeveloped markets
and without stable economic policies (Prasad et al., 2003 ). High exchange rate volatility is
consequence of currency crises in emerging market economies. Some other changes in world
economy may have reduced the impact of exchange rate volatility. For example, extensive
use of financial hedging instruments could reduce firm's vulnerability to risks arising from
volatile currency movements. Moreover, fluctuations in different exchange rates may have
offsetting effects on profitability of multinational firms.
There is no consensus on the appropriate method for measuring exchange rate
volatility. Commonly accepted model of firm behaviour subject to risk arising from
fluctuations in exchange rates and other variables does not exist. Therefore, theory cannot
provide definitive guidance as to which measure is most suitable. The scope of the analysis
will to some extent dictate the type of measure used (IMF, 2004). There is a wide variety of

279

m~tho?s that could be used to generate predicted values of exchange rate. We emploved the
um:vanate GARCH _type of model proposed by Bollerslev ( 986) in which the mean ~nd the
variance of the seri_es are s1multaneously modelled allowing the variance term to de end
upon the lagged :V~nances as well as lagged squared residuals 1. On the basis of such a m~del
part of the volahhty can be forecasted (based on the past values of the exchange rate) and
firms engaged m trade would make an effort to develop such forecasts2. Such a model can be
further used to forecast the absolute magnitude ofreturns and to predict the quantities.

2. Model
!he dependent variable, the input to the GARCH volatility model, is always a return
se~1e\ The returns are assumed _to be generated by a stochastic process with time-varying
vo at1_ ity. GARCH modd cons1~ts of t':o equations. The first is the conditional mean
equat~on ~n? th~ second 1s cond1t10nal variance equation. In this studv, \Ve defined the mean
•
equat10n m 1ts s1mplest possible form, i.e.

1i=µ+u1

l)

where µ is a constant and denotes the average o f re t urns over th e data period and
u1 ~N(O.a}).
In G~R~H model _variance term depends upon the lagged variances as weil as the lag ed
squared 1es1duals. 1:h1s allows for pers1stence in volatility with a relatively small numbe; of
· .
parameters. The vanance model for the standard GARCH(p ,q) mo d e11s.
q

p

2
er2t = a O + "a
,t_, jerl-j
L, ;111_2 1 + "fJ

~

.

(2)

H

2
0-,

2

2

2 _

er, -ao+a1u1-1+/J1er,_1•

•

.

=

a0

1-(a +/J,)
1

(4)

as long as a1 + /31 < • The model with such a property is a stationary GARCH mod 1 p
the mo~el, the conditio~al_ varian_ce forecasts converge upon the long-term average v=l~e :;
·
1 ·
the variance as the pred1ct10n honzon increases · For a 1 + fJ1 -> l the uncond't'
1 10na variance 1s
not d e fime d .
· kn own as a " unit root in variance" also termed Integrated
The case of a 1 + /J, ~ l 1s
?AR~H or IGARC~. Th1s phenomenon is commonly observed in practice and it leads to
im~osmg the constramt a, + /3, = 1 in a GARCH(l,l) model. Thus, IGARCH models are
umt-root GARCH models, with the key feature that the impact of the past squared shocks
2
'l, = u, -; - er,~; for i>O, on u,2 is persistent. Consequently, IGARCH volatility forecasts do
~ot r.nean-revert at ali. For such models, the conditional variance forecasts will tend to
mfimty as the forecast horizon increases. The IGARCH(l,1) model can be written as:

; In t~e CJ_ARCH model the conditional variance is modelled as an ARMA process
· · · k
Est1mat1on and forecasting volatility is important feature of financial risk modelli'ng , 1.e.
m ns management
,
·
d 11·
f tTty
vo a 1 1 mo e mg prov1des a s1mple approach to calculating value at risk of a financial position.

280

The GARCH(l l) model is the most common specification for GARCH volatility models,
being relatively easy to estimate and generally having robust coefficients that are interpreted
naturally in terms of long-term volatilities and short-run dynamics, (Alexander, 2001). The
sizes of the parameters a 1 and ~ 1 determine the short-run dynamics of the resulting volatility
tirne series. Large ~ 1 coefficient indicates that shocks to conditional variance take a long
tirne to dye out (so volatility is persistent) while large a 1 coefficient specifies that volatility
reacts quite intensely to market movements.
A feature of many financial series that is not captured by GARCH model is the
"asymmetry effecf' also known as the ··[everage effect" or ·'risk premium effect". In the
context of financial tirne series analysis the asymmetry effect refers to the characteristic of
the tirne series on asset prices that an unexpected drop tends to increase volatility more than
an unexpected increase of the same magnitude (or that bad "news" tends to increase
volatility more than the ·'good news" GARCH models do not capture this effect since the
lagged error terms are squared in the equations for the conditional variance, and therefore a
positive eJTor has the same impact on the conditional variance as a negative error. There are
a number of ways of paran1eterising this idea. We employed the so-called GJR model of
Glosten, Jagannathan and Runkle (1993) which is a GARCH(l 1) model where the variance
equation is defined:

a} = ao + a1u(_1 + /J1a/_1 + d1uf_Ju>o(u1-1)
an indicator function, in this case, for u > O Negative value

(6)

(3)

1s known as the conditional variance since it is a one-period estimate for the variance
calcul~ted based on any past information thought relevant. The conditional variance .
is
.
1) spe 'fi t·
changmg, but the unconditional variance of 111 under a GARCH(
c1 1ca 10n 1s constant
,
~
by:
·
an d g1ven
Var(u,)

(5)

of d 1 means that
with J being
negative residuals tend to increase the variance more than the positive ones.

The s1mplest case ofGARCH model is the generic or "vanilla" GARCH( .l) model:

.

where O < /31< 1 .

GARCH models are popular forecasting models when dealing with financial tirne series.
When evaluating the accuracy of a particular model for computing conditional variance
forecasts we calculated the -step ahead forecasts at the forecast origin h. W e used the
recursive formula where the forecasts ofthe early periods are used in the later periods.
When forecasting the conditional variance the observed values of the series are not
available for comparison. As a proxy, it is traditional to use the squared values of the data.
The forecasts can then be compared by with this proxy and the forecasts can be evaluated in
the same way as forecasts ofthe series itself. We employed standard measures offorecasting
accuracy derived from the forecast errors namely; mean absolute error MAE, mean absolute
percentage error MAPE, mean squared error MSE and root mean squared error RMSE.
3. Empirical results
We used data on the percentage changes of the log of daily exchange rate between
Croatian kuna and US dollar (K/$US) from 1 January 1997 to 15 January 2005. Excluding
days for which no prices are quoted (New Year's Day, etc.), this resulted in a total ofN =
2013 observations. Percentage changes ofthe log of daily exchange rate are calculated from
indices of daily nominal exchange rate HRK vs. USD (CNB midpoint exchange rate,
2001=100). Data are taken from the Croatian National Bank. Summary statistics ofthe data
show that the mean of the series is 0,000478. It is almost zero indicating that the average
change is zero. The skewness coefficient of -0.14237 indicates that the distribution is
negatively skewed which is a common feature of many financial series. The kurtosis
coefficient (excess) which is a measure of the thickness of the tails of the distribution equals
0.55969 and is significantly higher than zero. A Normal distribution has excess of zero

281

implying that the assumption of normality for the distribution of exchange rate changes is
dubious for the series.
As (log) exchange rates are to a rough approximation usually described asa random walk
(Tsay, 2002) we consider a model (1) where a conditional mean equation only includes an
intercept. The result of the OLS estimation with standard error is:
Var(u 1 ) = 0,0234
r1 = 0,0005+ u,.
(0,0034 J

The analysis of the correlogram of the data indicates no serial correlation. However, the
sample autocorrelation function (SACF) and sample partial autocorrelation function
(SPACF) of the squared series indicate the possibility that the changes in exchange rate are
not independent. Combining the two plots, it seems possible that changes in exchange rate
are senally uncorrelated but dependent However, the dependence is not so pronounced on
the graphs and it is important to perform the forma! testing for ARCH effects. The results
give Ljung-Box Q-Statistics for the residual series of Q(24)= 27.5951 (p-value=0.2775) and
for the squa_red_series Q(24)= 63 1765 (p-value 0.0000) which is highly significant resulting
m a clear reJect1on ofthe homoscedascity assumption.
In order to model a variance equation (3) the GARCH( l, ) showed to be the most
parsimonious. The parameters of the model are estimated by maximum likelihood 3 under the
commonly employed assumption that the conditional dist;ibution of the errors is nonnal 4
Thus, ajoint estimation ofthe mean and variance equations is:

1; = 0.0002+ u,,

a;'

= 0,0004+ 0,0270 u}_ 1 + 0,956 a,2_ 1
(0,0002)

t'0,0032)

(8)

(0,0110)

(0,0063)

From the volatility equation the implied unconditional variance of u1 is 0,0237 which is
very close to that of equation (7). However, t-ratio of the parameter in mean equation
suggests that the parameter is insignificant In the variance equation, ali coefficients are

v, = ~'

significant at 5% level. SACF and PACF of the standardised shocks

and squared

(ji

2

v,

fail to suggest any significam serial correlation in the two processes. More
series are Q( 2)= 16.6429. p-value= 1635 and
specifically, Ljung-Box Q-Statistics for
Q(24)= 27.6810, p-value = 0.2737. For the squared series Q(l2)= 13.2571, p-value = 0.3506
and Q(24)= 22.4309, p-value = 0.5536. It is evident that there is no autocorrelation left in
the squared standardised residuals and the model seems to be well specified. When
evaluating the accuracy of a model for computing conditional variance forecasts we
calculated the !-step ahead forecasts, at the forecast origin h= 1895. The measures of forecast
accuracy derived from the forecast errors are presented in Table 1
The characteristic of financial tirne series is that the distribution is leptokurtic. For
allowing for kurtosis it is possible to explicitly allow for non-normality in the conditional
distribution as well as the unconditional distribution when estimating GARCH models.
Assuming that !11 follows a standardized Student t-distribution we re-estimated the
GARCH(l,1) model jointly estimating the degrees of freedom of the Student t-distribution
and the parameters of the model. A joint estimation of the mean and variance equations for
the Student t-distribution GARCH(l,l) model is:
process

1;

= 0,0013+
(0,0033)

u,,

2

a, = 0,0004+ 0,0315
(0,0002)

(0,0080)

u;_

1

+ 0,9526
(0,0125)

a;_

1

•

(9)

u1 follows Student t-distribution with estimated degrees of freedom of 13.84 ~ 14 and

standard error of 3.5502. Ljung-Box Q-Statistics for the standardised residual series of the
672 and Q(24)= 27.5504, p-value = 0.2795. For the
model are Q(l2)= 16.553 p-value
squared series Q( 2)=13.4008. p-value =0.34060 and Q(24)= 22.1384, p-value = 0.5710.
Since there is no evidence of autocorrelation in the squared standardised residuals we can
conclude that the model is well specified. Consequently. we cannot reject the hypothesis of
using a standardized Student t-distribution with 14 degrees of freedom at 5% significance
leve!.
From the volatility equation, the implied unconditional variance or long-term variance of
u1 is 0,0239. Parameters estimates anči their standard errors are close to those of Gaussian
model. Large lag coefficient 01 of 0.9526 indicates that shocks to long-run (conditional)
variance take a long tirne to dye out. The value of error coefficient a1 of 0,0315 specifies
that the volatiiitv does not react quite intensely to market movements. For the model we
plotted estimated conditional variance. Figure l. The standard measures of forecast accuracy
are given in Table .
In both models. non-negatively constrains of the parameters are reached. However. the
In Gaussian model
fitted models show that the sum of the parameters is close to
a 1 + /J 1 = 0.983 and in the Student GARCH( 1, ) model: a 1 + /3 1 = 0,984 signifying that
the estimated model is c!ose being nonstationary. The results obtained from the estimation
ofthe IGARCH( ,1) model (5) are:
(10)
a,2 = 0,0002+ 0,0660u,2_ 1 + 0,9340a?_ 1
1; = 0,0029+ u,,
/0,0088)

!0,0001 i

[0,0034)

Ljung-Box Q-Statistics for the residual series is Q(24)= 22.8375, p-value=0.5294 and for
the squared series Q(24)= 18.2368, p-value 0.7914. The parnmeter _estin_iates are close to
those of the GARCH( , 1) rnodels estimated above but there 1s a maJor d1fference between
the models. The unconditional variance of u1, hence of the r 1, is not defined under above
IGARCH(l,1) model. This seems hard to justify for the percentage change series. From the
theoretical point of view, the IGARCH phenomenon might be caused by occasional leve[
shifts in volatility. The actual cause of persistence in volatility deserves a careful
investigation. For the model the measures ofthe forecast accuracy are given in Tablel.
Table 1: Forecast evaluation measures for forecasts from selected models
Model
! Measures of I
1
IGARCH(l,l)
Student tGausian
fo.-ecast
d'IS t n'b U f1011
GARCH(l 1)
accu.-acy
!
1
'
GARCHO,ll !
0.02033
0.02101
0.02110
MAE
37890.70633
32122.32491
O
32580.5811
MAPE
9.06070e-04
8.83355e-04
8.84332e-04
MSE
0.03010
0.02972
0.02974
RMSE
1

1

1

1

To test for the asymmetry in the Student t- distribution GARCH(l,l) model we employed
a GJR model (6). The joint estimation of mean and variance equations gives:

r, = 0,0011+ u"
(0,0034)

2

a, = 0,0004+ 0,0349u;_ 1 + 0,9555a;_ 1 -0,0l09u;-J11 > 0 (u,_1),
(0,0002)

(0,0119)

(0,0089)

(0,0115)

(11)

with estimated degrees offreedom for the Student t-distribution of 13,80 (3,533).

3

We used quasi-Newton Broyden-Fletcher-Goldfarb-Shanno (BFGS) optimisation method
Bollerslev and Wooldrige (1992) showed that the maximum likelihood estimates of the parameters of the
GARCH model assuming Gaussian errors are consistent even if the true distribution of the innovations is not
Gaussian.

The asymmetry coefficient equals -0,0109 and has an expected negative sign, although it~ tratio is only -0,9477. Thus, the asymmetry is not statistically significant in the model wh1ch

282

283

4

is not unusual finding for exchange rate, ( Verbeek, 2000 ). The asymmetry is also rejected by
the sign bias test (SBT) the negative sign bias test (NSBT) and the positive sign bias test
(PSBT) proposed Engle and Ng (1993).
To find the best model for forecasting volatility of exchange rate we compared the
standard measures of forecast accuracy derived from the forecast errors for the estimated
models, Table 1 On the basis of these measures, Student t-distribution GARCH( l, l) model
( l O) produces the most accurate forecasts of the conditional variance.
The graph of the conditional variance forecasts from the Student t- GARCH(l, 1) model.
Figure 2, shows that model approaches a constant value within a few steps.
Figure 1: Estimatecl conclitional variance from
Stuclent t-clist. GARCH(l,1)

Figure 2: Forecasts of conclitional variance
from Stuclent t-clist. GARCH(l,l)

Eslimated Volalillty

00425

Forecasted Volatllity

00232

00400

00375

j

O 0350

I

0032S

I

MACROECO NOMIC MODELLIN G RELAXING THEORETIC
ASSUMPTIO NS IN THE CROATIAN FINANCIAL SPHERE
Elza Jurun. Snježana Pivac
Universit} of Split. Facultv of Economics
Matice hrvatske 31
2 l 000 Split, Croatia
elza@efst.k spivac@efst.hr

Abstract

,

o:

This paper establishes an approach to macroeconomic modelling under conditions
relaxing the
assumptions of the classical model. Macroeconomic variables of the Croatia~ fin_ancial market a~e
integrated in the same multiple linear regression model. After parameter estnnatton by _the class1c
method the problem of multicolinearity appears. Based on the same database the same vanables were
modelled by mathematical programming. The comparative analysis of results confirms the
advantages of linear programming especiall) when the problem of multicollinearity appears.

00231

Keyworcls:

o rooo j

0,0229

0,0275

i

o.ona

0,025-0

i

parameter estimation. relaxing
multicollinearity. Croatian financial market

theoretic

assumptions.

linear

programrning,

1. INTRODUCTI ON

:0227

0.0225
0.0200

0.0225 j_

,-~

1920

'

1936

1952

196a

1984

~

2000

4. Conclusion

The aim of the paper is to find a good model for forecasting the conditional variance or
volatility of the percentage change of exchange rate. The model is chosen by its ability to
forecast volatility. There is no accepted model of firm behaviour subject to risk arising from
fluctuations in exchange rates and other variables. Therefore. theory cannot provide
definitive guidance as to which measure is most suitable. We employed GARCH(1.1) model.
Estimation of severa! models shoved that Student t-distribution GARCH(l, 1) produces the
most accurate forecasts of the conditional variance.
References:

Alexander, C. (2001). Market mode/s: A guide to financial data analysis, John Wiley &
Sons; Chichester, UK.
Bollerslev, T. (1986). Generalised autoregressive conditional heteroscedasity, Journal of
Econometrics 31, 307-327.
Bollerslev, T. and J.M. Wooldrige (1992). Quasy-maximum likelihood estimation and
inference in dynamic models with tirne varying covarainces, Econometric Reviews,
11(2), 143-172.
Engle, R. and V.Ng. (1993). Measuring and Testing the Impact of News on Volatility,
Journal of Finance, 48, 1749-1778.
Glos ten, L.R., Jagannathan, R. and Runkle, D.E. ( 1993). On the relation between expected
value and the volatility of excess retums on stocks, Journal of Finance, 48, 1779-1801.
IMF (2004). Exchange Rate Volatility and World Trade, Occasional Paper No. 28.
Prasad, E., K.Rogoff, S. Wei and M.A. Kose (2003). "Ejfects of Financial Globalization on
Developing Countries: Some Empirical Evidence", !MF
Tsay, R. S. (2001). Analysis of Financial Time Series, Wiley and Sons, Chichester, UK.
Verbeek, M. (2000). A Guide to Modem Econometrics, John Wiley & Sons, New York

284

A faster development and wider application of the statistic modelling in the financial
sphere started in the 1970s. Those were the years when various finan~ial inno_vations on tl~e
capital markets of the developed countries emerged and their modellmg reqmred stochashc
methods. Contrary to these trends, the Croatian financial market is still underdeveloped and
poor in financial instruments. Development of _the capital n_iark_et i~ Croati_a started at the e~d
of the 990s along with the process of trans1t10n and pnvatisat10n, wh1ch stili remam 1ts
important determinant. As a consequence the stock market, although itself ~nderdeveloped,
imposed itself as one of the more important segm~nts of_the overall _c~p1tal market. 1:he
situation on the developed world markets is oppos1te. Th1s charactenshc of the Croatian
capital market, along the processes of transition and privatisation_ is _significantly i~fluenced
by traditional financing through loans, lack ofknowledge offu_nch?nmg and potentials ofthe
capital market, monopolistic banking system, madequate leg1slahon as well as the overall
govemment attitude to the capital market.
2. NERVOUSNES S ON CROATIAN STOCK MARKET
After the period of dramatic war and post-war changes in the so~io - ec?~omic s_ys~em,
Croatia has plunged in an equally unstable period of economy m trans1hon. W1thm a
relatively short period of a decade the transition period has encouraged fundamental changes
in all aspects of Jife (sociological, cultural, moral, legal ...)._ Those ch~ng~s caused feed?ac~s,
which in the economic sphere are manifested through h1gher quahtahve and quanhtative
instability of movement in almost ali segments of economy.
. .
. .
For the purpose ofthis paper, this thesis will be illustrated remamm~ w1thm t~e framework
of the characteristics of movement of the basic indicators of the Croahan financial market. •
The estimates show that the first effects of globalisation started determining the situat~on on
the Croatian market after its "stabilisation" in the late 1990s. It follows t?e pen?d of
transition economy ofthe complex offinancial sector, and therefore we can claim that it was
then that Croatia became an integral part ofthe world financial market.

285

By choosing the variables the presence of both characteristic spheres of the financial
market is wanted to be achieved: the one connected with the cash flms as well as the one
connected with the securities flow. Apart from that after the statistical-mathematical
analysis only the variables with confirmed linear connection the period from 1999 to 2005
have been introduced in the modeL
and clearly.
In order to illustrate the movement on Croatian financial market more
fundamental indicators of changes in its particular segments were chosen. In the deYeloped
economies movements in interest rates for short-term governmen! securities are widely
accepted as an indicator of stability. Due to the underdevelopment of the Croatian economv
in relation to the developed countries, this indicator wi be considered as a qualitative on~.
rather than emphasising its quantitative dimension. Figure 2.1, besides the average dail)
mterest rates for each month on the Zagreb Muney Market in the period from 999 to 2004
also indicates weighed interest rates for treasury notes which are due in 3 5 davs. Therefore.
·
let the graphic illustrations replace further comments.

Instabili1v of interest rates is combined with considerable fluctuations in other segments of
financial a~tivities, especially those under powerful influence of the exchange rate of the
Croatian currency The kuna exchange rate depends significantly on the money supply and
interventions of the Croatian National Bank which aim at maintaining it within the desired
limits.
Money Supply

40000

r ~-

1,

'

1'

35000 i
30000

!z

25000 1

§

20000 1

,5

15000
10000

+
1

5000

Interesi Rates: Zagreb Money Market and Treasury Notes

18

1999 -2004

Mi - Croat1a - monthly

>

>

1:S

1:S

>

1:S

-

Years, Months

2

-Average Daily
Interesi Rates :far
Each Month),
Zagreb Money
Market i 1999 -

10

;;! af
~

~

r-

2

+1999

2000

2001

2002

2004

2003

Years, Months

Figure 2.1: Interest Rates: Zagreb Money Market and Treasur~ Notes

[ 11

As illustrated at Figure 2.1. the average daily interest rates on the Zagreb Money market in
the given period range from % to 16. 3 %. The lowest weighted interest rate for treasury
notes due in 35 days amounted to .88% whereas the highest was 0.5%.
Observing the movement of average daily interest rates for each month can be noticed
not only their significant quantitative fluctuation, but also significantly different globa! levels
of movement in paiiicular years ofthe observed period. as it is shown at Figure 2.2.
r----

-----

-

Average Daily lnterest Rates for Each Month Zagreb Money Market

- l 1==.-=-,.,, 71
1

II
1-,,,.JI

::::-::::
1--,0,2
1~2003

4
2

1

-

o +---,------------------------j
2

3

4

5

6

7

8

9

10

11

12

M1(1)
=---'-'--· 1000.
Kr M(O)

( 1)

where:

M 1 - stands for the market capitalisation ofthe CROBEX shares on day I and timej,
M (O) - stands for the market capitalisation on the basic da)
M(O) - 12.942,773,686.
- the factor of adaptation of the index basis on the auditing day T
Kr
- U 01848.
In order to be included in CROBEX. shares should meet the following conditio11s:
• liquidity - shares were traded on more than 75% of the t.otal number_ of trading days in the
observed period or are among the 25% of shares class1fied accordmg to the number of
trading days,
.
• quotation 011 the Stock Exchange for more than six months,. . .
• market capitalisation higher tha11 the median of market cap1tahsat1on of ordmary shares on
the last day ofthe observed period.
Kr

- - --1

--~-

Croatia

This paper models the dependence of CROI3EX on the above mentioned variables of the
Croatian financial market. CROBEX is the official index of the Zagreb Scock Exchange pic.
Its issuing started on September st 997. The basic <late is July 1st 1997, and the basic
value .000. It is weighted according to market capitalisatiDn while the weight of a certain
share is limited to 35%. lf one ofthe CROBEX shares was not traded 011 the particular day,
its last closing price is used. It is calculated co11tinuously during the trade by using the last
price the equation:

2004,)

-Weil)hed nterest
Rates far Treasuryi
Notes iw hich are
due in 35 days !
(:(000,-2003.,

6

4

Figure 2.3 Money Supply MI

1

3. PARAMETERS ESTIMATION IN MULTIPLE LINEAR REGRESSION MODELS

Months

Figure 2.2: Interest Rates - Zagreb Money Market.

The database consists oftime series of 72 monthly <lata on CROBEX movement, (CROB),
average daily interest rates on the Zagreb Money Market (IR), money supply (Ml) and

286

287

interventions ofthe Croatian National Bank (lI\T J. rhe data refer
1999 to December 2004.

the period from Januar1

Table .3.3. Parameters estimation without including interest rates as dependent variable

CROB =/Jo+ /J 1 IR + /J 2 Ml+ /J, INT

Variable
lnterce t

INT
., R-0,76287.

. -

1

, t-ra t·w

/3,·'

•

*significant at a =1%,
••significant at a =5%.

1

0,64329

9,6055
-2,7 l !8 I
O 18267* I 8,9107 1
1
0,00476** ! 1.7061 '

I -0,0 l 032*

IR

Ml

INT
R =0,7863,
*significant at a =1 %.
**significant at a =i0%.

As it was expected. this model is statistically significant according to ali the statistic
criteria. It explains the dominant and irnpmiant influence of the quantity of money supply
movement of the shmi-term interest rates and inter\'entions of the Croatian National Bank on
the CROBEX movement, as well as of the linearity of their interdependence. Testing the
model, not only according to statistic but also according to econometric criteria, the
multicolinearity was detected.

Ml

j

INT

i

l~~O
-0,69*
-0,24*

*significant at

1
1

i

-O~~*
l ,00
0,17

i
j

-~~!*
O, 17
i,00

norm/"'.
3.2. Parameters estimation by linear programming
The Jinear regression model provides the possibility of parameters estimation by use of the
linear program with / norm, i.e. criterion of maximal absolute deviation maximisation.
The function is introduced:
f(j]) =

Table 3.2. The correlation matrix ofindependent variables
I

For this model too statistic and common econometric tests have been carried out. It has
been established that neither the problem of heteroskedasticy nor autocorrelation of enors of
relation exist, and that the level of multicolinearity is below the one that would according to
common levels of significance present a problem. Namely, for this model, the calculated
2
value according to Farrrar-Glauber testis x' = l.393. and does not exceed x ,ah = 7.815
ln order to estimate the preciseness of the results from one more methodologically
different aspect the model has been additionally estimated hy linear programming using the

00

3.1.1. Testing the multicolinearity problem

IR

i

l

i

i 12,7618 1
0,22056* 1 14,0631 i
0,00612** , 2,1290 i

Ml

Table 3.1. Parameters estimation ofmodel (2) bY classic method
.' In depen den t

! t-ratio

Independent \
/31
Variable
0,49163*
Interceot

3.1. Parameters estimation by classic method

jjx/J- rj\

= maxi=l
00

(6)

,nl(xfJ- d!

To minimise the maximal absolute deviations the problem becomes:

!

Min(Pl max 1=1

1

.11

l(X/J-

(7)

The substitution is introduced:

a =5%,

As illustrated in the above table there is, indeed, a statistical!y significant linear conelation
between the interest rate and the money supply. and the correlation between the interest rate
and the interventions is statistically more significant than the one between money supply and
interventions on the money market. It leads us to the conclusion that the interest rate is a
variable which in this model represents a significant factor of multicolinearity. Therefore
Farrar - Glauber test was canied out. The value of the determinant of the matrix of the
correlation coefficient of independent variables is: det R = 0.4937 The calculated value of the
chi-square test:

x2 =-[n-1-¼(2k+5)}ndetR=48,114.

(3)

z=maxi=l

,j(xj]-r);j.

(8)

1

Accordingly. the regression solution in the form of linear programme, where the CROBEX
is the dependent variable, becomes:
Min z

(9)

Xj]-CROB s; ze
- X/J + CROB s; z e,
z~ o

,

where e is the unit vector.
Table 3.4 Parameters estimation by linear programming
z

When it is compared with the table value for df = -k(k + i) = 6 ·

0,23876

2

1

(4)
it can undoubtedly be concluded that the multicolinearity exists.
Therefore the multiple linear regression model was estimated without including interest rates
as independent variables.
X tab=l2,59,

3.1.2. Parameters estimation without including interest rates as dependent variable
(5)

288

As it is obvious in table 3 .4: /Jt + j]1- = o ; which means that practically variable IR has not
statistically significant influence on the dependent variable.
After parameters estimation by classic method and by linear programming using norm /"'
.
. .
.
.
their comparative analysis has been carried out.
Valuating the results of the model from the point of v1ew of their apphcat10? w1th purpose
of control and planning the significance of both approaches has been estabhshed. Namely,

289

a satisfying wa:: and therefore
actuaJ data
the _estimated values in both cases adapt
JUstlfy thel!' 1mplementation not onl 1 for the purpose of analysis.

4. FINAL CONSIDER ATIONS
Using the same multiple regression model this paper links the macroeconomic relations
whtch reflec~ t_he cash flow with the mm·ernent on the stock market as the dominant segment
m th_e_ secunttes fl?w on the Croatian financial market. Modelling is carried out under
con~1~1ons of relaxmg the assumptions of the classical model. Along with that after the
stahshcal-~a thematical analysis only those variables whose indisputable linear connection
for _the penod between 1999 and 2005 was established were introduced in the model. On
bas1s of the same database the same variables were first modelled b, the dassic method and
·
then the parameters were estimated by mathematical programming. ·
From the point of view of econometric criteria model estimation bv linear programming
·
has a number of advantages.
After spe~ifi_cation and ~he overall procedure of parameter estimation by the classic
method. stahstlc econometnc criteria require each particular parameter as well as the model
as a whole to ~e tested. On the other hand, model estimation by linear programrning offers
completely vahd results from the point of view of all statistic and econornetric criteria
alr~ady on the first step. Such mathematicaJ processes have been built in the procedure
wh1ch were able, 111 our case, to offer parameter estimation but were also able to identif, the
variable which is_the ~ominant factor ofmulticolin earity in the model. Namely, apart from
all parameter eshmation fulfilling both statistic and econometric criteria the model was
estimated without the interest rate vaiiable which is highJy correlated with' other regression
variables of the mu Iti ple model.
Therefme, wher~ver.it is possible it is suggested parameter estimation using mathematical
pr~gra~1mmg wh~ch 1s undoubtedly. according to statistic and econometric quality of
estunat10n, supenor to the classic estimation methods. especiaJly under conditions of
relaxmg the assumptions of the classical model.
REFERENC ES
Benninga, S .. Financial Modelling, The 'VIIT Press, Cambridge, 2000.
2. Jurun E., Pivac S., Zlobec S„ Influence of Fiscal Policy in Transitional Environment.
Proceedings of the 5th International Coference on Enterprise in transition Split'
Tučepi, 2003.
3. Prohaska, Z., Analiza vrijednosnih papira. lnfoinvest Zagreb, 1996.
4. Zlobec, S., Stable Parametric Programming, Kluwer Academic Publishers,
Dordrecht, The Netherlands, 2001.
5. Zlobec S., Authorized lessons.· Solving Regression Problems by Linear
Programming, Depaiiment of Mathematics and Statistics, McGill University,
Montreal, 2003.
6. The Croatian National Bank: http://www.h nb.hr
7. The Zagreb Money Market: http://www.t rzistenovca.h r
8. The Zagreb Stock Exchange: http://www.z se.hr

290

A NONLINE AR MONETA RY MODEL OF INFLATIO N
Alenka Kavkler. Bernhard Biihm
Faculty of Economics and Business, Razlagova 14, 2000 Maribor, Slovenia
Institute for .\1athematical Methods in Economics, Vienna Universtty of Technology, 1040 Vienna, Austria
e-mail: alenka.kavkler@uni-mb.si, bernhard.boehm@tuwien.ac.at

Abstract

This paper investigates a well known monetary model of the inflation process. The basic tool for
identification and estimation of the model equations is the smooth transition regression approach.
From the s1mulation of the estimated nonliilear system asymrnetric policy reactions can be deri ved.

Keywords: Smooth Transition Regression (STR). Quantity Equation. Phillips Curve. Okun s Law.
l. Introduction
This paper is devoted to an investigation of asymmetric policy reactions. lt focuses on the
monetary approach to inflation control and uses a simple three equation model that is
specified with the use of the smooth transition regression approach. The next section
presents the theoretical specification, followed by a linear estimation attempt used as a
starting point The nonlinear estimation approach is given in section 4 and the policy
simulations are presented in the final part.
2. Theoretical specification
W e start with a model of monetary inflation theory which can be shortly characterized by
an equation describing the monetary system augmented by a Phillips curve and the equation
ofOkun·s law. The simple elementary system is given by (see [3]):
m 1 = x, +rc,,
rc, =rc;-b(u, -u·),
u t - u t-1 = -a(x l - x ·)
These three equations determine three unknown variables: real growth rate (xt), inflation rate
(ret) and unemployme nt rate (ut) in terms of given monetary growth. The first equation
representing the quantity equation in growth rates (assuming constant velocity) will be
substituted by a demand for real money equation, which may not be homogenous of degree O
in its arguments. We assume the nominal money stock to be given by the monetary
authoritv. The equilibrium in the monetary sector can be described by a general equation
expressing the relationship between money stock, output and prices, respectively inter~st
rates. The Phillips curve relates inflation to the deviation of the unemployme nt rate from 1ts
natura! rate (u•) augmented by backward and forward inflationary expectations. Okun's law
provides a relationship between the change in unemployment rate and the deviation of the
actual from the trend rate of real output growth (x°). By considering an Okun type
relationship and additional supply effects in the Phillips curve the resulting model exhibits
essential features of the "triangle model" as defined by Gordon ([4]). Supply effects may
include tax changes, rates of change in import or oil prices, or the change in the
unemployme nt rate which may give rise to hysteresis. The (excess) demand side is typically
represented by the unemployme nt gap and may also incorporate lagged effects of the growth
of the stock of money.
This monetary approach to an explanation of inflation will be applied to quarterly <lata
from WestGerma nybetween 1970:1 and 1998:4.

291

3. Linear econometric model

rr'
rr?

0.000746

!

growlh rates. ln a
The econometric model employs transformations of \ariables
preliminary specification, all equations are modelled as hnear relationships. This simplifies
the search for an appropriate nonlinear specification. On the basis of these empirical results
further investigations will have to re,eal any remaining nonlinearity in these relations.
The first equation specifies real money demand growth (mt) as dependent variable
explained by adjustment terms representing the adjustment of real money growth to output
growth (xt) and prices. We obtained the following results by the least squares estimation.

Sample (adjusted); 1970:l 1998:4 (Included observations: 116)
1

Variable

Coefficient

Std. Error

t-Stalistlc

Prob.

const.
x,

0,023727

0.005669

4.185161

0.0001

0.183880

0.127073

1.447037

O 1508

0211157

0.098361

2 14675

0.0341

-0.582542

0.264546

-2.202045

0.0298

O 883156

18.45990

0.0000

-5.2684 l 7

0.0000

m :~l
m, 4

-0.267305

0.047842
0.050 7 3 7

rr:

-0.005449

0,001437

-3 792490

0.0002

dummyl

0.112028

0,016734

u.
m

6.694747

0.0000

Sam le (ad'usted): 1969:3 1998:4 (Included observations: 118)
Prob.

const.

0.007590

0,002537

2.991486

0,0035

~rrt-t

0.418053

0.087216

4.793325

0.0000

rr1.1

-0,247808

0.055656

-4.452521

0,0000

rrt-4

-0.166698

0,067031

-2.486884

0,0145

~rrt-B

-0,089575

0,056247

-1.592533

0.1144

~rr1-12

-0.110270

0.055852

-1.974331

0.0511

rr 1-s

0.165211

0.053915

3,064292

0,0028

292

0.0000
0.0000

0.098185

0,019734

4.975404

0.0000

-0.073331

0.026445

-2.772931

0.0066

-0.000693

0.000218

174645

0.0020

0.029236

0.015522

1.883461

0.0625

-0.074374

0.02 147

-3.516986

0.0007

rn1-4

0.064679

0.016840

3.840904

0.0002

durnmy2

0.010957

0.002014

~.440388

0,0000

Sam le (ad'usted 1970:2 2000 3

n: )

t-Statistic

12,38415
-8,341309

Dependent Variable: ~u 1

equals
backward and f01ward looking components. Forward looking price expectation (
the difference between the nominal and the real rate of interest according to Fisher's formula
ri" = r/ +
As nominal rate, we use the long term govemment bond yield while the real
interest rate is represented by a proxy variable, the ratio of the sum of real GDP over four
quarters to the corresponding sum of real investment expenditures. In view of the
modification of the Phillips curve by Samuelson and Solow ([7]), labour productivity growth
(A1) is also considered in the equation. The results of the linear Phillips curve are given in the
table 2.

Std. Error

0.014223
0.024364

Okun's law describes the short-run relationship between the GDP gap and the
unemployment rate. This empirical relationship, developed by A.M. Okun in the 1970s, links
the output gap
*) to the change the unemployment rate. x • denotes the expected rate of
real growth following the long-run trend. For an application, see Kavkler and Bi:ihm
We found the following reasonable linear estimates.

The Phillips curve is modelled according to the considerations in Bohru ([l
The
inflation rate (nt) depends on the unemployment gap (u1-u•) (with u• the natural
unemployment rate), energy price inflation ( n~ and expected inflation modelled by

Coefficient

0.176137
-0,203229

Table 2: OLS results for the Phillips curve

Table 1: OLS results for the surrogate quantity equation

Dependent Variable: ~rr 1

0,0678

2

R = 0.855, S.E. = 0.022, SSR = 0.054. AIC = -4.690

n'..

l .846239

R = 0.809. S.E. = 0.004. SSR = O.OOL AlC = -8.027

2

Variable

rr ~1~2

1n 1

Dependent Variable: m,

xt-4
~rr,_,

rr;\

0.000404

Included observations: 122)

Variable

Coefficient

Std. Error

t-Statistic

Prob.

x,-x

-4,953883

0.805964

-6.14653

0.0000

const.

0.1313 1

0.031709

4.141159

0.0001

ut-1

-0.017967

0.004797

-3.745722

0.0003

' 0.428198

0.068666

6.235993

0.0000

0.636079

0.102657

6.196183

0.0000

-0.433570

0.143116

-3 029498

0.0030

ilu1-1

dummy3
dummy4
2

1

i

R = 0.695, S.E. =

141. SSR = 2.298, AIC = -1.036

Table 3: OLS results for the Okun's law

The dummy variables were introduced to reduce the ARCH effects caused by the outliers
in the years 1991 and 1992, when the German reunion took place. Inspection of the
properties of the estimated linear equations indicates in ali cases problems of parameter
stability or misspecification. CUSUM an CUSUM square tests reveal structural breaks in the
surrogate quantity equation in the period ftom 1991:3 till 1993:1, whereas the Ramsey
RESET test with 3 fitted terms rejects the nuli hypothesis of NID errors for the Phillips
curve and the Okun's law.
In order to improve specification we need to investigate the influence of nonlinearities,
which we shall assume to be ofthe smooth transition kind.
4. Smooth transition regressions
A detailed description ofthis technique can be found in Terasvirta ([8]), to which we refer
the reader for further information.

293

Bach of the three equations is now subJected 10 linear
ity tests. Ali independent variables
may be cons1dered as smtable cand1dates for a transi
tion variable. From the table 4 with test
res~ltsh we obtam a set of suggested transition variab
les frorn which we first choose those
t e srnallest p-value. Due to occasional problems
with convergence of the nonlinear
op imizdatwn procedure some experimentation to
find appropriate starting rnlue s was
reqmre .

Wl!

7.87

FJ

7_

94

F2
7, 101
Philli s curve

df
F

49. 5l

F4

16. 5 l

F3

16,67

F2

16, 83

trend
Ll1t 1-1
F-stat.
F-stat.
(p-val.)
(p-val.)
1.8274
20394
(O.O 174) (0.0064)
2.12!7
1.4443
(0.0217) (0.1592)
l.1073
2.4285
(0.3667) 10.0060)
l.6001
l .4994
(0.0869) (0.1196)

nol
df
F

49. 51

F4

16, 5 l

F3

16,67

F2

16, 83

x,
F-stat.
(p-val.J
2.1617
(0.0062)
1.5166
(0172 2)
1.7743
(0101 5)
3.1389
(0.0048)

trend
F-stat.
(p-val.)
2.3383
(0.0028)
2.6015
(0.0174)
2.6158
(0.0164)
1.4620
(0.1894)

df

F4

•ii

m, = O.O! !2 + 0.2213 x -'- O. l 766 x _ -0.6599
14
1

(0.0039) (0.1 ]94)

t1

1

22.87

l

The fina! set of estimates obtained for the parameters
of the surrogate quantity equation
and the Phillips curve is found below.

1

Quantity eguation

F

j

o

7t l-1

F-stat.
F-stat.
(p-val.)
(p-val.)
1.6030
1.3350
(0.0488) (0.1545)
1.2918
1.2738
(0.2386) (0.2498)
1.2029
0.6912
(0.2893) (0.7926)
2.0969
2.0742
(0.0158) (0.0172)
Okun 'slaw

df
F

13, 103

F4

4,103

F3

4,107

F2

4, 111

Ll1t 1-1
F-stat.
(p-val.)
2.3838
(0.0023)
17722
(0.l03 l)
2.9608
(0.0075)
2.1552
(0.0445)

X1-4

F-stat.
(p-val.)
14783
(0.1034)
0.7533
(() 6277)
l 3475
10 2370)
2 5564
{0.0182)

rr?.::
F-stat.
(fl-Val.J
13113
(0.1699)
1.1995
(0.3006)
0.7013
(0.7827)
2.1086
(0.0152)

trend
F-stat.
( -val.)
1.3097
(0.2193)
2.4543
(0.0504)
0.9118
(0.4599)
0.8000
(0.5277)

ffil-1

m1-4

n'1

F-stat.
(p-,aL )
2 5023
(0.0014)
l.1519
(0.3389J
3.0606
(O 0060)
3.1299
(0.0049)

F-stat.
(p-val.)
2.5366
(0.0012)
3.0211
(0.0069)
0.8138
(0.5780)
3.5351
(0.0019)

F-stat.
(p-Yal.)
l.8201
(0,0266)
2.4554
(0024 0)
12178
(0.3008)
1.7021
(0.1169}

n,_4

~n1-B

F-stat.
(p-val.)
l.2605
(0.20 7 3)
0.7590
/0,7214)
1.4331
(0.1535)
l.7142
(0.0597)

F-stat.
(p-val.)
0.8453
(O 7219)
0.4812
(0.9453)
1.1130
(0.3617)
l.2651
(0.2393)

"-1-:
F-stat.
(p-val.J
1.0655
(0.4110)
0.9075
(0.5654)
1.2097
(0.2842)
1.1526
(0.3233)

F-stat.
(E-val.J
18317
(0.0170)
1.8731
(0.0461)
11664
(0.3173)
1.8654
(0.0357)

7t 1-1

F-stat.
(p-,al. )
1.6366
(0.0419)
1.5258
(0.1270)
0.9073
(0.5643)
2.2298
(0.0098)

r

X

1

-X

ul

.

F-stat.
( -val.
1.5387
(0.1161)
2.0633
(0.091 O)
1.9709
(0.1042)
0.8253
(0.5118)

Ll1t ,.12
F-stat.
(p-val.)
l.3384
(0.1524)
l.2311
(0.2781)
1.3955
(0.1712)
12288
(0.2644)

Jt l

F-stat.
(p-val.J
l.3137
/0. 1683)
0.9873
(0.4842)
1.9154
(0.0340)
0.9602
(0.5065)

F-stat.
(p-val.J
1.8682
(0.0144)
1.3288
(0216 8)
17354
(0.0609)
2.0472
(0.0189)

mr-.:

m:-J

F-stat.
(p-val.)
2.4265
(0.001 O)
3.0185
(0.0014)
09951
(0.4727)
1.9842
(0.0236)

F-stat.
(p-val.)
1.9963
(0.0079)
1.6067
(0100 9)
19518
(0.0301)
.6771
(0.0675)

0 1-1

LlUH

F-stat.
( -val.)
0.8491
(0.6079)
0.2426
(0.9135)
1.4665
(0.2175)
1.1052
(0.3578)

F-stat.
( -val.)
1.2270
(0.2712)
0.6275
(0.6440)
1.8459
(0.1254)
1.5155
(0.2025)

f

7[ l·5

m1.4
F-stat.
(p-val.)
l.9241
(0.0110)
16857
(0.0803)
2.2049
(0.0129)
.1442
(0.3302)

(0.()878)

+ (-0.2096 Illt-t -O.O !04
(0.0793)
(0.0022)

1
t

Ll7tt-l -

(0.2259)

n; )[l + exp {-4. l l 74 (Illtr 0.0572) /0.057l}T
(1.6579)

1

(0.0101)

'i

T = 116, R' = 0.870, S.E = 0.021. ani 1 6-lin = 0.723.
JB = 5.053(0.080)
FAR<ll = 1.351(0.248), F~(4) = 1.072( 0374), FAR(8) =
0.915(0.507)
LM~CH(l)= 1.725(0. !89), LMARCH(4)= 4.754(0.314)
, LMARCH(8)= 15.168(0.056)

i

Equation 1: STR estimates ofthe surrogate quantity
equauon

i

'1
l

l
!

l

= 0.1874

f.1t
1

n

(0012 2)

(O

t

,

0.0548 n _ -0.0545 ,.,_ -0.0582111t-:, + 0.0894 n?_
2 + 0.0607 Illt-4 + O.OJ 18 +
15
2
(0.0240)
(0.0233)
(0.0154)
(0.0133)
(0.0150)
(0002 3)

+0.01 29dum my2-0 .4132 ~n,_ -0328 6 n _, -0.!478
~n 1.,: -O.l86 71t~ -0.001 2u,+
1
1
(0.0018)
(0.0291)
(0.0478)
(00591 )
(0018 8)
(0.0002)
+ (0. l 768

11

1;

0863)

Ll7t _
1 12

1
-0.0228 n?_ + 0.0004 u1) [l + exp 1-42 8009 (111i. 2 0.0599) 10 0569l] '
1
(O.OJ 1lJ
(0.0002)
(0.0057)
(0.0027)

T = 116, R 2 = 0.817, S.E. = 0.004, ani , 6-lin = 0.944,
JB = 1.084(0.582)
FAR(!)= 0.0004(0.995 ), F AR(4) = 1.587(0. l 84 ), F AR(8)
= 1.561(0.148)
LMARCH(I)= 0.382(0.537), LMARCH(4)= 4.100(0.393)
, LMARCH(8)= 6.032(0.644)

Equation 2: STR estimates ofthe Phillips curve

The models passed the diagnostic checking satisfactorily
. The p-values of the J arque-Bera
test and the test of no remaining error autocorrelation
show that the nuli hypotheses of the
normal distribution of the error term and of no error
autocorrelation, respectively, cannot be
rejected. The test of parameter constancy detects no
problems and there are also no ARCH
effects present in our models.

5. System of equations
The estimated nonlinear system is inspected for
its dynamic properties. Contrary to
estimation, the money stock is considered a policy
instrument to influence the inflationary
process as well as the unemployment rate. Addit
ionally, the reaction of a change in oil
prices, in the forward !ooking expected inflation
rate and to a productivity shock is also
evaluated. In the latter cases, reactions tum out to
be rather symmetric, while shocks in the
money stock exhibit significant asymmetries. The graph
s in the figure 1 show the results of a
unit shock of one standard deviation of the money stock
on inflation and unemployment.
Maintained unit shock of one standard deviation of
the money stock creates a permanent
increase in inflation by 2.4 % points and a decrease
in unemployment rate by 8.2 % points.
Also in this case the asymmetric reaction to monetary
policy is obvious (figure 2). This again
shows that working with a nonlinear model generates
a result that is known in practice, but
not achieved by linear models.

Table 4: Linearity tests

294

0.9274 111i. 1 - 0.0920 dummyl +
(0.050 0)
(0.0166)

295

r~

.002T-----------

···--i

lilVE CROATIAN BANKS FOREIGN CURRENCY POSITION
ANAL YSIS USING VALUE AT RISK

1.2,--------------~

.001 l
.000
·.001
·.002

08

Dubravko Mojsinovic
0.4

Privredna banka Zagreb_ d.d, Račkoga 6. 10 000 Zagreb. Croatia

1

dubravko.rnojsinovic@pbz.hr

· 003
.004
.005
006 i,
00

i

1

1

05

10

2'0

;5

1

35

2's • 30

4'0

1

45 •

1

s'~

1

05

negative money stock shock to inflation rale :
positive money stock shock to inftation rate

1-

\

~

-.02

,::::::::, ,1
10

:s

1

20

2s

30

3'5

4°0

4's

-8

1

1

1

2

s'o

maintained negative money stock shock to inflation rate
--- maintained positive money stock shock to inflation rate

~

·'

j

!

1

60 ~15 /O t'5'
1

1

20

1

~s

1

30

3~

1

40

1

4s

"5•~

1-

ma!nta!ned neg,a,tive money stock shock to unemployrnent rale
; ---- ma1nta1ned pos1t1ve money stock shock to unemployrnent rate /

Figure 2: effect of a maintained unit shock of the money stock to inflation and llllemployment

References
B. Boh!,n: Structura! Change in the Austrian Injlation Process ' in G. Chalo upe k et . al
. . Th .
(ed s...) Ok onom1e
eone und Praxis. p. 31 - 46, Springer Verlag, 2001
m
B. Bohru: Smooth Transition Mode/s of Stnictural Change, in J. Leskow , M . p uc h et, L .
( d ) N
F p
ew Tools of Economic Dynamics, Lecture Notes in Economics and
• unzo . e s.
Mat~ematical Systems, Vol. 551, Springer Verlag, 2005.
[3 J H. Fnsch: Theories of Injlation, Cambridge University Press. 1983.
[4] R. Gordon: The Time - va,ying NAIRU and its Implications far the Economic Po!icy
·
Joumal ofEconomic Perspectives 11 (1), p. 11 - 32, 1997.
G:anger, T. Terasvirta: Modelling Nonlinear Economic Relationships Oxford
[5]
'
Umvers1ty Press, 1993.
Okun 's law far Germany with Smooth Transz·rzon
Model!ing
B. Bohru:
[6] A.
.
•
· S
R Kavkler,
egresszon, ymposmm m 0 perations Research '03 Proceedings, p. 289 _ 294, 2003.
[7] P.A. Sa~uelso~, R.M. Solow: Analytical Aspects of Anti-lnflation Policy, American
Econom1c Rev1ew (50), p. 177-194, 1960.
[8] T. Terasvirta: Mode_lling Economic Relationships with Smooth Transition Regression,
Handbook of Apphed Economic Statistics, p. 507-572 New York Marcel Dekker
'
'
'
1998.
[l]
.
[2]

c.w_.

Keywords: Croatian Banks. Foreign currency open position, Value at Risk
lntroduction

1
1
i
t
'l

T

\'· ·-----·------------------------------________ j

-03~

os

5

-1 4

·.01

00

45"

8,1------ ---~

···· ·· 1

1,

40

1
~

effect of an mitiai unit shock uf the money stock to inflation and unemployment

.02

041

35

1

!

.03TI- - - - - - - - - - - - -

-.05

1'0

1

negative money stock shock to unemployment rate
]/ --a•- positive money stock shock to unemployment rate /

j

Figure

15 20 25 30
1

Abstract: Foreign currency open position is the difference between assets and liabilities
denominated in a certain foreign currency. The Croatian Kuna counter value of that difference
changes as the exchange rate changes. The banks differ regarding the amounts of open positions in
different currencies. It can be said that riskier is the bank which has relatively high long open
positions in cu!Tencies whose exchange rates are volatile and positively correlated. Value at nsk
recognizes these conditions and therefore is used compare five Croatian banks m 2002 and 2003.

J;

f
1
1
f

1

Assets and liabilities are frequently denominated in different currencies. Croatian Kuna
(HRK) counter \·alue is subject to foreign exchange rate fluctuations. Open position is the
1
difference hetween assets and liabilities denominated in a certain foreign cunency. Big
open position suggests the existence of bigger foreign exchange rate risk. Long or positive
open position means that the bank has more assets denominated in a ce1tain foreign currency
than the liabilities. Short or negative open position means that the bank has more liabilities
than the assets denominated in a certain cunency. More concretely, if the bank has placed
100 ELR as loans and received 50 ELR as savings, it means that the bank is 50 EUR long or
that the bank has positive open position. lncrease of foreign exchange rate HRICEUR will
increase the net value of the bank denominated in HRK and the decrease will lower it.
Analysis of open position is conducted in a way to include open positions which are under
the influence of the management and foreign exchange rates which are external to banks.
This is done using VaR or Value at Risk indicator which takes into account the magnitude of
the open positions. volatility and conelations of exchange rates. This is done for five big
Croatian banks at the end of2002. and 2003.
Data and preliminary analysis
2
Five hig Croatian banks were selected. For comparison total Croatian banking system assets3
on l 2.2003 amounted to 201,8 bilhon HRK, while profit for 2003. was 2,5 billion HRK.
It can be seen from the figure that ali banks moved to the right from 2002. to 2003. This
movement suggests bank growth. Assets, capital and profit ofthe selected banks increased.

4

Figure 1 Data for five Croatian banks in million HRK
,-----Comparison ofFiw Croatian Big Banks-circle area is
proportional to profit for 2002,(left circles) and 2003,
(right circles)

-.s==;,~

s.ooo T - - - - - - - - -

6.000

•

+---

~

4.000 1

u

2.000

+.--:§D=o~~- -----1
i

O

1111'41111

O+---~--~--~---<
O

20.000

40.000

60.000

jeZABA
l@PBZ
l@RBA

O ERSTE
OHYPO

80.000

Assets

296

297

As can be seen from the figure bellow ali banks ha\ e a significant amount of assets and
liah1l1ties denominated in foreign cmTency or linked
the foreign currency This suggests
that there 1s a need for managing foreign currency risk. Managing foreign currenc:: risk is
the first place expressed through the absolute amount of open position. Foreign currency
1)pen position is the Jifference between assets and liabilities denominated
a ce1iain
cuITenc\
can be assumed that the lower the amount, the better is the safety against the
exchange rate change risk Banks are longer in ELR. This means that they ha\e more assets
in EUR than the liabilities. Therefore there exists the risk resulting in unfavorable
movements of EUR exchange rate. However, present Croatian National Bank policy is such
that it keeps this exchange rate fairly stable.

!'he tigure bellow shows the movement of end of month middle Croatian National Bank
exchange rates for HRK/EUR and HRK/USD 5 from 2000. until 2004.

Figure 3 l\1iddie end of month HRK/EL'R and HRK/LSD exchange rates
Foreign exchange rates from 2000. until 2004.

8~

-..___,,..~~::..·=.. ;;:.,___::;::_===;:;;;:;:.-__;;;;i
·····,, ....

6

---- .. I

HRKf~

4+

Figure 2 Assets and liabilities in foreign currency and HRK for five banks

2

--HRK/EUR

--

Strnctural Compaiison ofilanks on 3 Ll2.2003.

The figure suggests the stability of HRK/EUR. Since most of the open position 1s in ELR,
the open position in EUR is also stable. HRK/USD is not stable, but since the USD position
is not so significant. so is the influence on open position much lower.
VaR nr Value at Risk is the maximum amount of expected loss in a certain tirne interval with
a certain prohability 6 VaR indicates the expected loss in case when the unfavorable event
does not happen. There are different ways to calculate VaR. 7 In this work VaR is calculated
using assumption that the loss is normally distributed. 8 Profit in this sense is negative loss.
l\ionnal distribution is described by two parameters µ and 0. µ is the expected loss and CT is
the standard deviation of loss.
Open pos1tions existence does not mean that there is no foreign currency risk management in
place. Because of this VaR wiil be calculated to show how much could the bank lose with
such a position with respect to the exchange rate volatility. Moreover banks regularly have to
reprice their assets and liabilities according
the middle exchange rate of the Croatian
National bank to meet the regulatory standards. The impact of the open position and foreign
currency exchange rate movement on profit is shown in the table bellow.

Figure 4 V aR and normal distribution

Table I Impact of open position and foreign currency exchange rate movement on
profit for the selected currency
µ

j

1

Open position in the
selected currency

Change of the exchange rate in respect to
the previous value in tirne
i

1

Increase (more
HRK/EUR)

!

Decrease (less
HRK/EUR)

VaR=o·ac1+µ

1

where c1c 1 is the value of x for the standard normally distributed variable (µ=O. 0= 1)9 which
di vides the area bellow the function in two parts. On the left side the area is cl (or confidence
level) and on the right side is 1-cl.

1

If expected loss is zero and standard deviation depends on the exchange rates and open
..
pos1t10ns
t h en 10

Assets-Liabili ties>0

Profit

Loss

Assets-Liabilitie s =O

o

o

i

Assets-Liabilitie s <O

Loss

Profit

1

298

YaR

299

r

Banks decreased VaR
decreasing their open positions which is a favorable direction and
indicates the existence of the risk management function. PBZ has the greatest VaR and did
much less on lowering than ZABA although in 2002 they were much alike. ZABA by the
end of 2003. decreased its VaR on the leve! of smaller banks.

where p; 1s the exchange rate of currency Var is the rnriance and cov the covariance. A is
1
the amount of assets in cuITency P; is the amount of liabilities currency
is the number
of cu1rencies.
The expression under the root is the to tal \ miance of sum of dependent normally distributed
open positions in HRK or portfolio of open positions. Since the series of exchange rates are
monthly. the VaR refers to the monthly loss. For the purpose of having lower VaR
is
fa\ orable to hold positions of the same sign in negatively correlated currencies and of the
opposite sign in positevely correlated currencies.

Figure 6 Comparison ofVaR, profit and assets movements in millions HRK
\'aR as end of2002. (left circle) and 2003. (right cin:le)

.ooc .

Smcc amounts denominated in other cuITencies where insignificant they were added to EUR
resulting in rnble bellow.

ZADA

Date
LlR

l,SD

ASSl'IS

IIRK
EllR
; i• bilitks t:Sll
IIRK

f----,)pen

position

H:R
ltSIJ
!!RK

ZP.BA

POZ

PBZ

RBA

RBA

ERSTE

ERSTE

HYPO

HYPO

31.12.2003, Jl.:2.2002, 31.12.2003. J 1.12.2002. 31.12,2003. 31.12.2002. 3 l.12,2003. 3 l.l2.2002. 31.12.2003.
31.12.2002.
5.032
4.743
3,034
2.626
1.730
1.295
1.752
679
1.342
994
587
625
712
555
263
218
196
172
204
167
16.547
13.926
14,885
11.413
5.249
3.354
4.925
2.125
J.797
2,545
4.960
4.543
2.915
2.456
1.753
1.283
'i.659
608
1.338
959
578
626
697
540
271
224
198
166
207
168
17.156
15,412
15.887
12.784
5.o28
3.396
5,619
2.706
2.562
2.052
72
200
119
170
·23
12
92
71
4
35
9
o
.7
15
15
c2
-7
7
-2
-l
-609
-1.486
-1.002
-1.371
.ni
-43
-694
-580
-16
-255

Standard deviation and covariance ofthe exchange rate distribution is shown bellow.

Table 3 Exchange rate distribution parameters

{ 'urrency

!1 Expectntion

IIRK/EUR
IIRK/LSD

7,5225
,,4885

i

4C

ioPBZ

,o ERSTEI
ieRBA

i

1GHYPO;

1

--B,ink

60C

[

Table 2 Assets. liabilities and open position in millions ElJR and USD for five banks

f.-.-

+

80(

~

Covariance
Correlation coefftcient
\'ariance l-:-c::-:-:-:- :-:.-----1--- ~----l
· HRK/EUR HRK/USD HRK/EUR HRK/USD
0,0167
O 0167
-O 00i6
1 0000
-0,0122
0,9968
-0,0016
O 9968
-O O122
1 0000

Results

1
1
t!
'
t

l

1

1

20, JOC

6C UJ{

80 V(Yv

Although RBA and ERSTE are of the similar size, RBA has smaller VaR. RBA increased its
profit and assets. but did not decrease VaR. ERSTE grew, but did not decrease VaR. ZABA
and PBZ grew proportionate ly increasing their profit and assets and decreasing VaR.
It seems that at the end of2003. ZABA PBZ and HYPO are m the better position comparmg
them to ERSTE and RBA who were not able to decrease their VaR values.

Figure 7 VaR and the expected value of open position at the end of 2003. in millions
HRK
VaR and expected open position

1-------

lf all banks would have had the same combination of open positions, they would have had
the same VaR. So the differences in VaR are explained by the different amounts of open
positions. It is obvious that banks have significant differences in their VaR values.

4C 000

Assets

1-

-25--'

.~-w-~

--H)--j--f----

-5-~BYPU

Figure S Five banks VaR movements in millions HRK
Expectedopen j>JSition in HRK

VaR M ovcmcnt for cl""95% Calculated on the Basis or
Historical Open Position

ZABA

PBZ

RBA

ERSTE

HYPO

300

The figure shows that the greater open position means greater VaR. It shouldn't be forgotten
that VaR grows with the value of open position, but that its value also depends on the
volatilities and covariance of exchange rates. It can be seen that ERSTE (680, 20) has greater
expected value of the open position and at this point it could be said that it has greater VaR.
However this is not so because ZABA (613, 22) has lower open position in EUR, but
significantly higher in USD whose exchange rate is much more volatile and thus more
contributes to the increase of ZABA's VaR.

301

Al\JAUSYS AND RlSK MANAGEMENT OF lNVESTMENT
PROJECTS LSING METHOD MONTE CARLO SIMUILATION

References

an Šmid

1

Saunders. Anthony: Financial Institutions Management: a modem perspective Irwin.
Illinois. 1994, page 241.
'
~ Hrvatska bankarska industriia, Banka. specijalno izdanje, July 2004, pages 6-7.
~ Hrva~~k~ b~n~a:~ka industrija, Banka, specijalno izdanje, July 2004, pages 6-7_
0
disn1e 1zv1esce 2003. Zagrebačka banka d.d, pages 49, 51, Godišnie izviešče 2003.
P11Vredna banka Zagreb d.d, page 46. Godišnie izviešče 2003. Raiffeisenbank Austria d.d.
Zag1::b~ page ~~- Godišnie izvješče 2003. Erste & Steiermiirkische bank d.d, pages 8. 9.
?od1sn1e 1zv1esce 2003. Hypo Alpe-Adria-Bank d.d, pages 50, 51.
·
Tablica H l O: Srednji devizni tečajevi Hrvatske narodne banke na kraju razdoblia Bilten
HNB nurnber l 00. January 2004
'

( entar za poduzem1št,o Krapinsko-zagorske županije. Krapina
e-rnaii 1.s1111d,1j v1,1_1_.irr
\ alerija Bublic
Lorana J v o . Zagrebačka 26. Krapina

e-rnail. lornn,i,:a•i--r.htnd.hr

_?

6

Dowd, Kevin: An lntroduction to Market Risk Measurement John Wi!ev & Sons Ltd
2002.pagel9.
'
0

•

•

7

Dowd, Kevin: An Introduction to Market Risk Measurement, John Wiley & S
2002, pages 55-87.
ons. Ltd,
8

Dowd, Kevin: An Introduction to Market Risk Measurement John Wiley & Sons Ltd
2()02. pages 38-39.
'
'
·
9

Šošic, Ivan Vježbe iz statistike I. dio, mikrorad d.o.o. i Ekonomski fakultet Zagreb. 1994.
page 32.
·
10
Saunders, Anthony:- F1·nanc1·a1 Ins t·tut·
1
10ns ,Management: a mo d em perspective, lrwin.
Illinois. 1994. page 248.

Abstrac!:
this paper we investigate the poss1ble 111tlue11c:e pf d1fferem 111pul variablcs in the
and
lhicken I ar111 un the output d1stribulio11 of thc inlcrnai
estment pru_1ed !Jui
P\')
lhe
After idenli1' mg which mput
rate
relurn i RR l anJ 1he net prese11t \ alue
,ariablcs ha,e the slrungest influence 011 the 1nvest111e11t pro_iect. ,\e consider lhe feasibiiny ,)1 lhe
add1tillnal
estme11t thal 111tluences the distr;butiu11 ulthe identified critical rnpuls In th1s paper 11e
appl1cd ,l1e metl]\)d \ldille Carl() ';i111ulat11)Ji.
Keywonls
esunenl prnJect, 111ternal rale
retur1L net prcscnt va)ue, risk anah•s1s. risk
111a11agcmc11t. \11,mte ar!,) si1uulati()11

L

11'iTROlH(TION
tile
and practice ol managemenl various acti\ 1es tiwt help tu 111Jluencc risk
e1ents J)r risk pltases ur the business pr,)cess are well known. Entrepreneurs qu1te ollen dt)
not ha1 e enough financial resources 10 act on ali potential risks s1multaneoush Therel,,rc in
the risk management practice we trv to recognize potemial risk causes. which have the
strungest mlluences 011 the final business result We undertake activiues lu prevent risk in
accurdance with rewgnized priorities Rec11gnition and estimalion of priurities anwng
polential risk causes are one ufthe most important managerial tasks [11.
Calcu!atiuns
the net present I alue and the interna! rate of return are verv importanl
lacts when estunating the profitabilit\ of the investment Based on this inforrnali,,n wh1ch is
an obl1gatury part ur each 1m eslmenl prnjecL banks take Jecismns about loans.
1n acn)rdance
the instructions ()f the European Lummission, sensili 1
,rnaiysis and
risk anal:,sis. 11hich inciude prubahilit) anal:,sis. are a recomrncnded content ul :nvestmenl
prnJecls and s1udies 2
is necessan to define the critical variables of the prnJCCl and on
the bas1s z.f serics uf historical data define their probahil
distribution, frorn which follows
thc prubabi
distrihution of the net present I alue and the interna] rale ol' ret urn. However,
sometimes
is very difiicult to collect statistical dala about probability progression ul' the
critical project variables. ln such cases inforrnat1011 ean be colleeted frorn literature or
estimated hy a group of experts.
In this paper will presenl how the risk managcrnent strategy can be bascd ,rn thc
simulation rcsults.

2. RISK ANAL YSIS
Dala neecssary for the calculation of the parameters for estimating the protitability or
the investment reter to the future events. Due to that, those inputs are uneertain and each
calculation involving them contains a particular leve] of uncertainty. When taking a deeision
bascd on unreliable inputs, it has to be considered whether the ranges within whieh critcria
values can be expected are aceeptable or not. 1f there is a possibility that the investment
eould be unprofitable, we can talk about the investment risk. In ordcr to minimize the
invcstment risk it is first of ali necessary to rccognize the possible causes of the risk and
then, to aet on recognized potential eauses of undesirecl events.

302

303

i: 1h1s paper nsk anaiys1s Oi tile deter1111inslic linanc1al n1<,del 1s baseJ ,q: the
dpp!1cdliun ul 111c1hod \ilulllc tari,, s1mulation" Vve "til tn lu stale hu\, the ullnbined
L111certainlv ,,r thc input \ ariahles 111l1ucn,:cs the risk

J. PROBLEM DESCIUPTIOJ\J
l'hc lnvcslur plans "!O huild and to equip a chicken larm" \,huse capacit: is ~,UH)()
ch1ckens rn d C)clc llas1c dma are as ltll!nws:
,,,) "'"",9-;" J,ll lil \\Ur k"
·1ssets a11d" "'.J
·mg·
inveslmenl 4.J 15.295 kn • J • 976 •000 kn m t~1xed
- Tota!
<
,
•
captlal.
- l·"inancial sourees: 2 J 15.295 kn ol' mvn capi tal and 2.UU(UJUO ,li bank Juan pa::, hack penod
·
1O) ears and intcrest rale 5'),·o.
- Tota! 1ncome: 4 026. 71 J kn
- !herage weighl oJ'ehickens · grnwlh 2.05 kg/pc
- i\ verage mortalily rale: 3% per cvcle
- Planned nurnber or C) clcs/) car· ""
- Planned sales price or chiekens: 8.1 (1 kn/kg
- Planned cosls ,,r ()!le-da) chickcns 3,00 k11 pc.:
- i>lanned fceding eosls are slandardize d as follO\vs
a) [Vlixture PPTI. priee 2"70 kn/kg, a\erage need :si kg,chic.:ken
b) Mlxlure l'Pl2, price 2.60 kn/kg, average need is 1 kg,chicken
c) M1xlure PPTJ, price 2.50 kn/kg, average need is 1,66 kg,c.:hicken
- Planned labour and olher eosls are calculated on the hasis ur real costs"
Prices ~ir inpul anJ oulput ~re based on market pric.:es in february 200'. 1lcnce lorward lhe cc.:ono1111c.: llow is shmvn / Figure 1), which deri\es fr()m the J)I\lJC<.:!ed
1 bas1s
., 11,11 ed 011 ,,1e
tnc.:orne and cash llmv slatcment. Thc interna] rate or returp 8 ·1 2°;'.0 1·s c.:•11L,"i..
ol the econo1111c llow or lhe project, as well as the net presen! \ alue in the arnuuut pf
1.086.-116 kn by using 5% discounl rale"
11 , ; \I, '111'1(1\lll 1

'

....

,

•

,,

L

"-t.c"~~.~'~;\,~;;~,,,~;;;~,;·;c~:::.~fi.;;_··:, .;,;~;,.:,.j ;;;-.,~ -·-.:~.'-,:~/p ,;;·,: :Jp ,;'.'" '•Jf, '": ,.~

- M1xlurc l'l' l l 111;11" 2.50 kn kg" expected 2,'U kn/kg and max 2.90 kn/kg
- Mixlurc PP 12 mm" 2AU kn kg. cxpened 2.60 kn/kg and rnax 2.70 kn/kg
- M1'-.turc l'I' l ~ mm" 2.40 k111kg, expectcd 2.'-(l k111kg and max 2.60 kn/kg
- One-Jay c.:hickens" rn111" 2.9() kn pc.:. expedeJ _,.no k11ipc anJ ma'-. ;" lO kn/pc.
APPLl(AT IO!\i OF THE METHOD MONTE CARLO SIMllLAT ION

-L

\\ e \\ ill appl) the met hod Monte Carlcl simulation 011 th"· abo\ e Jescri bed crilicul inpuls.
First ·.\c will assign \lne ol the classic.:al probability distribution lo the critical inpuls and lhen
the s,,lt\\arc \\ill generale a series of random numbers. 1\h1ch correspond lo the chosen
distribution As a result there is an atlained histogram and cumulauve graph ,,1 the
probahilit\ \llthe distrihution ,it chosen critical inputs and their 1n!1uence ,>1> ~PV and !RR
1he ohjecti\e ut th1s paper isto del1ne the prubahiiit) distrihutiun ,if the lk.k. am: ",i'V
on the real e'-.ample ,d' the investment prn1ect ,,t a cinckcn larm. us111g the M( lillc ( .,ri,
si1nulali,,;1 111cth\>cL lurthermor e. ,,bjcctives are lo 1den1i1:, \1hid1 ,,1 thc c1bl '-C n1-.·11L, :>.. d
inputs ha\ e the strongesl influence on lhe IRR and '\i I'\ ,,! the pru1ec1 and tu dei11:-: tm 11sk
111111111111.rng strateg::- b, means of investing m the equ1pme111 \\ hich wd' 1;1;pr,",1 l. 1.he
pr,,bahil1t\ d1stributi,111 pararneters of lhe cntieal rnpuls l ,,r the s1mulau,,n purpl•Sl l'll
suitv,are uk.IS!( Palisade Corporatio n [JI wiil be applied hrst. \\e 11ude tile 111<1dei ,, t K
0
im c:strnent prPJecl ,n ex...:eL I he r(.)ik,wing values are sltpuL1ted as uncerlarn u1put, ar1.il 1"s
averagc 1\ eight , ,! d1ickens gro,1 th ,Yic:1 ure 2). m erai,!,e rnurtaiil\ rale. 1he sales pric..:
chickens. c.:osts ,,f,me-Jay chickens and cosls ofthe m1xtures PPTI, PPT2 and l'P r,
s,,n,,are 1.1,KISK enahles us the cho1ce uf a largc number "f JiiTerent pr,,bahtl,t)
distrihullons l'i.ir inpul \ ariahles" Bec.:ause of the insuffic.:ient historical dala wc ha1c appiicd
lhe s,mple tnangular distribution ll>r eac.:h rele\mll rnput \artahlc 1 'sing the 1na,,suiar
dislnhution . 1t is nec.:essary to define three , alues the assumed mimmai 1 aiue ul the randi.lrn
variablc. its assumed maximal value, aml the musl pn,hahle assumed \ aiuc ul the ra11do11
vanable l4J in Figure 2 we quole lhe dis1rihut111ns „f rnpuls wb:ch we dScd 111 111,s •-clsc
applymg .t{){!SK soltwarc.
1

1

71
D,sii·ihution
-;o
0'i
2
90
'
,111~
r
R,sk
2)
(Picture
growlh
/\ \'et~ we,ght of chickens
--R1sk:!_~·11.1_~ -_O 01 ()_.O u /)6
Average 1110rtality rale
0
,~,50,7.%.8.
R1skTric111g
1
Salesrnce l!i~hickens
-+ R1skTdang (L.,90, 3,00--='- -'~J1_ __
____
. Cost ot one-day ~1ckens
1
~,skl1~mg'2 ,'i0 -,~, .=9(,
+Costolrn,x turePPTi
I R1skTriang (2,40; 2,6U. 2 :'_i:2_ ____ ,
____
Cnsl ot rnixture PPT2
_JI
R1skTriang (2,40: 2,.50: 2.60)~--~J
L'iJst ,Jfmixlure PPT3
~~~~~~~~~;.;.;;;;~=~~~~~~
1,

1

l
ti 1/1)\( 1

~ .11~.!•J{;J '-7u 'l!l2:.l.'-71.r,S! J.S'J,1,'-1 .1 '7~ 1,'-21

1,.

,17

1 ..'-~' 1,,

~I
2

'""""!

"'i"
4

5
6

Figure 1· Economic !1ow of the project

...,

1u1 \W

----1

,able

b~~~~~~~~~

~igurc 2" Distribution of inpul variables for sotiware @RISK

_i:s. lh~. c'.·it_i~a~ in~ul_ wc ~elined the_ average weighl of chickens growth and average
11101 tc1l1l) .' c1lc. 1he du1 al1on ot one cycle 1s 42 days. Stalistical data of tirne series of c.:ritical
1_n1:ul \ anablcs wer~ not_ available. We knmv from experience that the average \veight of
ch1c~en grm_vlh \artes Jrom 1,90_ to 2,30 kg/piece and the most common value is 2,05
!,g/]J1eee._ while lhe av_crage morlaltly rale varies from 1% to 6% and the most common value
t~ -1 '!/i,. l he_ reasons lor such. oscillations are humidity, exlernal air temperatur e anJ other
laclors ..\\ h1cl_1 Cc'.nnol he prec1sely delined. Furthermor e, as critical variables we defined the
· •t
lhe inpul costs of one-day chickens and the input costs o f 1111x
salcs pnee ol ch1ekens,
ures
.
PPTl p 1r 1„1
1
hyper
the
to
Due
kn/kg.
O
8.1
now
is
chickens
of
price
~ am. PP D. Sales
• .
.
· 1
the price lll !'ali to 7, 90 ki-u1k g antl ti1e 111ax1111a
pt .oduc.:l1011 011 lhe loeal market wc expect
,
other
pncc could be only 7,50 kn/kg. We also predicted probabilitie s of prices of severa!
111puls:

304

1

masa u kg

Figure 3: A verage weight of chicken growth

305

1listograms , 1 1gure 3, and analogically histograms lor ,,ther
\anah!es
lh Figure 1) show relative frequencies of
accordanc,
l (11)(1(1
\ariahles based
gcncratcd random numbers.
5. RESULTS OF MONTE CAKLO SIM:lLATIO!'\/
RR anJ
Lxcel muJel are connected \\
S111cc thc oscillations uf the aitical inputs
successiun
m
en
gll
are
simu1atiuns
lhe
,,r
results
!\J l'V ol thc pruJccl II 1gures 4 and 5 1 the
herc ali crit:cal rnputs are \ aried simultaneousl).

merage
rale. c,ists ui 1111xlures PP"I I PPT:?. and PPT3 and
ch1ckens.
nsk managcment for
tlim \ aluahie informatiun. \\e can Jeline
the chicken farm.

\'lile
c,,sts

K:JRELACIJA Lr, N~

masilpr1rasta

SAJASNJL

RJE )NC S
1

,B04

42dana

1,

·,299

K JMULA 'IV'JA iJIS iRIBV. ,.A LA RR

igure 6.

'igun: 4. C'urnulative probabilit) distribution

or the

RR

rielo ,adaSnJa vn1ednost ·,unama
Values in Mlllions

1 igure 5. Cumuiati,e probability dis1rihut1on l1fthe NPV

6. RlSK IVIANAGE!VIENT
S111ce die probabilit) ol the proJeel foilure is .'3%, which 1s considered to be a relativelv
h1gh rale. 1t 1s ncccssary to de!ine lhe priorities of acting on particular input \ ariable;
l'hercby thc sull,,are ifljKISK helps us to show the information about the correlation ,>finput
and output variables or thc project. lnformation is gi ven in the form nf a so-called 1 ornado
graph ( Figure 6 ).
Thc highcr the absolutc valuc of the correlation coenicient is, the bigger is the influence
the input niriablc on the result. Figure 6 depicts the Tornado graph with correlation
cocrticients. which show how the NPV of the project depends on particular input variables.
hll'thermorc. 11c 11ill shmv only the Tornado graph for the NPV, as the graph for the IRR
looks quilc similar. Input variables 11 hich influence the project based on Figure 5, following
thc dcgrec of inllucnce from high to low are: the average weight of chicken growih, the sales

or

306

ornadu graph with correlations ol mputs

l\JP\

C\,uld be slated that the project is acccptable tl the 1'.._p\f 1s highcr Ihan Leru
(iencral
aml I f the RR is higher thall the interesi rate un the market {in our case '.'% L!\s the support
thc cumu!aU\e J1slnhu11on
this case. the most useful are graphs
for decision-mnking
rum these graphs we c.an
l.
6
( igures 4 al\J 5) and the 1ornado graph ( igure
K R anJ \.
that lhe prnbabi „ ul
depiu
l
"i
and
4
figures
{
graphs
·umulati\·e
(
draw se\ era! c.·unclusiPns.
1s 1 .~.U8%. 1gure 4
5%.
below
being
lRR
the
and
zero
below
being
pwject
the
l\.._
the
9\%
i~
%.
11
lhan
lower
being
project
the
of
oflRR
probahility
the
that
depicts
lrum the ornado graph (Figure 6) we see that there are 1w,, 1anab1cs ,,n vvhich thc
chicken growth anJ lhe average mortal I rale m:c
Jn1 estor cnulJ acl. The m erage weight
highh correlateJ ith NPV The correlation for the average_ weight of c)1ickc11 gruwth 1s
o 804 anJ fN the murtalit, rate is O. 290 Acting on these 1anables b, opt1mal temperature.
the farm b:, !'og dispersion. the risk uf thc econom_ic
ai;d C\Wling
air humiJit\
act un these varit~bles by m1esl111? 111
is possible
t!1c project can be reduced.
failure
the computer air C\,nJitioning S)Stem. (producer Big Dutchman 51;, wh1ch m the ,;ase ol the
chicken farm is 'i00.000 kn more expensive than lhe une wh1ch the lnvestur Pngmally
, . .
.
bu:,.
planneJ
case of uti!izing the precise computer contwlled a1r-con01t1011111g ad,1ustment m the
farm. using the special software for the SYstem MC 216. !~FOMATIC, . is rcalisllc to
expect changcs nf the inpul probabilit) distributions of the average mort'.11Ity rale and the
average weight of chickens growth (Figure 7). With these c_ha_nges wc w1ll 1_11akc the ne:v
model using &RISK software in order to see how probable it 1s that NPV ol thc proJect 1s
bellow zero i.e. that the proJect is unprofitable.
Nb.
2.

! Distribulion

lnnut variablc
_ _ _ __
Avcrnge mortality rate
Avera1ee weight of chickcns growth

\ RiskTriang (O.O l; 0.03; 0,04)
RiskTriang (2.00: 2.20: :?.JO)

Figure 7. Redefined distributions of input variables for @RISK soltware
As thc result on the rcdefined model. by means of Monte Carlo simulation method.
analogically to the previous procedure, a new cumulative probability distribution of the N PV
is created (Figure 8).

307

r
REDEFINIRA ~A KJMULJC rnJNA DISTRIBL:c JAZA ~pl,

..-'a)ues 1nM1lhons

t

ir

The 8th lnternational Symposium on
0perational Research in Slovenia

liguri: 8. Redelined i:umulali\ e probabilit: distribution ut the '.\ P\
7

,R'OS

COl\iCU SIO!\i

Rcdci111cd d1mulat1\e probabii1tv Jistribution of the l\JPV (hgure 8) shuws that the
pn,hab11 il\ thal 1\, I'\. cuuld be 11egati, e and the project unsuccessful is 9. 92" ,,. \\ hic:11 is
Iti\\ er than .~ .~ .118°/o.

! hc pncc ul reduc1ng this risk is 'i00.000 kn which should be imcsted additional!\ in
cu111putcriLcJ a1r cnndithJllillg S) stem. and it is a11 acceptable price tur rcducing the risk ut
lhL· prujccl foi!urc bclz;w I ll%. The mean of the '.'JI'\ \\ llboul 1he aJditiunal investrnenl
7
1 Figure 5) is 46 .908 kn and the mcan of the l\JP\. 1vith additional investment (Figure 6J 1s
i 1J4() ·734 kn

l'hc Ji ffcrcncc bctwcen means ur the net present I alue \\ it!i anJ ,, ithout 1he adJitiunal
1m cslmcnt 1s '.'%.826 anJ it is highcr than tbe estirnateJ custs o!' additi,mal im eslmenL lhus
this 1rncslmc111 is rcasunable 11'1he critical inputs \,ili be put in the frame P! figure 2
In th1s papcr it is shuwn huw we can use the method Monte Carlu simulation un the
clctcrmin1sllc linanCial rn,)Jcl like a business plan by risk management \\hen we calculate the
IRR and thc \l'V ul thc pn,1ect. rhe rnanager has to identify and to focus ()ll the nwst
1rnpwtant pt,ssiblc r1sk sources. Bascd on tlie example of the investment project, \\ liere \\e
iliustratcd thc pruccdurc tol the nsk management, it is shown ho½ the combinali(1n ,,1 the
common managcrial kllll\\ !edge and methods of quantitative analysis form e:-;cellent
J'oundations !'ur dclinmg the risk managcment strategy.
l I rJ,R/\TI 'RF

llI
121

[31
141
151

l lunpk. T„ Rcdep. tvl„ Upravljanje rizicima u planiranju trajnih obrtnih sredsta, a
pomocu tvlonte Carlo simulacije, časopis RRiF, br. 7 12003. str. 74
***Guide to Cost Bcnefit Analysis of Investment Project. Prepared !~Jr L,aluatiun
unit DG Rcgiunal Policy European Commission ~ Working Team Coordinated by
Professor tvlassimo Florio
\YWw.palisadc.com
llunjak. T„ Redep, tvl.: Upravljanje rizicima u planiranju trajnih obrtnih sredstava
pomucu tvlontc Carlo simulacije. časopis RRiF. br. 7 12003 .. str. 76
***Katalog tvIC 236 Der Computer fuer ein optimales Stallklima injedem
Gcllucgclstall

308

Nova Gorica, SLOVENIA
September 28 • 30, 2005

Section 7

Mt;LTIP LE PRICES AND DE NOVO PROGRA MMING
Zorati Babic', Tihomir H1111jak"
'Facully of Economics. 21000 Split, Matice hrvatske 31, Croatia, e-mail:babic@efst.hr
'Faculty of Organization and Informatics, 42000 Varaždin, Pavlinska 2, Croatia, e-mail:thunjak@foi.hr

Abstract: Optimization ofproduction program is one of the crucial problems of optimization.

For that purpose differem methods and procedures were developed in order to present the decision
maker with solutions which can provide the maximal realization of his aims. These various methods
and procedures were mainly !ooking for s'olunons in accordance with the previously defined resources
and constramts.
In real product10n plannmg situations linear programmmg models may be difficult to apply
because of its assumpt10n of proportionality. One common phenomenon is multiple pricing of raw
materials because companies may have more than one source of raw material at different costs, or
quantity discounts being offered for purchases.
Key words: Multiple prices, raw materials, De Novo Programming
1. Introduction
De Novo presents a special approach of optimizatio n. lnstead of "optimizin g a given
system" De Novo suggests a way of "designing an optimai system". De Novo approach does
not limit the resources as most of the necessary resource quantities can be obtained at certain
prices. Resources are actually limited because their maximum quantity is governed by the
budget, which is an important element ofDe Novo. Using De Novo most various cases can be
handled more effectively than by the standard programmi ng model. Changes in prices,
technologi cal coefficients, increasing costs of raw materials, quantity discounts and other
similar and real production situations can be easily incorporated in De Novo model and can
give very satisfactory solutions.
In this paper we will show the application of De Novo in the real production system
which produces various fenoalloys and in its production uses a set of raw materials with
different prices.

2. Problem Setting
This paper analyses the production planning problem in one ferroalloys factory. It
produces four types of ferroalloys: FeCr, FeMn, FeSiMn and FeSi. These alloys are produced
in three electric fumaces by special carbon-elec trochemica l treatments.
The ferroalloys furnaces work continuous ly through the whole year except during the
repairing and cleaning break in the second part of the year.
The Table 1 contains data relating to annual capacities of the fumaces expressed in
hours, tirne needed for one ton production of ferroalloys and net-income per ton of the
product.
Tab e l. Fumace capac1hes and net mcome fior fierroa11ovs pro duchon
Capacity (in
Fe Si Mn
FeMn
Fe Si
Fe Cr
hours)
7920
0.2084
Fumace 1
7160
0.48
0.2604
Fumace 2
7920
0.495
3
Fumace
35.663
18.409
54.104
42.284
Net-income

311

th fi For _the ferroallo~s production various raw materials are used. Availahle quantities of
e ive mam raw matenals as well as the use ofthese raw materials per ton offerroallo ·s are
presented m the Table 2. !n the last column there are unit prices ofthese ra,v materials. y
Table 2. Main raw materials
Raw materials

Fe Cr

E~~=o~e
~ __mass (t)

"

Quartz (t)

Fe Si

FeMn

Fe Si Mn

i 4.75
+--

!

Available
quantities

Unit price of

(bi)

raw materials
( i)

357056

8

42795.7

19.5

0.5 .. !0291'0A65i_Q .4L_. _.:313~
2953.8
O 035 ! 0.0 7 , 0.022 · 0.06

35.5

o.2lf4

2

-t

0.02

-f-

O~

0.58

86

42407.J

7.l

Ac~ording to t~ese <lata the production planning problem can be posted as th l
1
q uantftyi:;~~
o:joe~~:v: ::n:~:~r(:~:~
42
284
·
x1 + 54.104xz + 18.409x + 35.633x ).
lvfax (

i~~;;;:;~~~

l
1

1

'
Electric energy
4 / 9.6 ! 2.63
000 kwh
t-----:'M~a-'---n'-g.::.:an-'-'e-'-'sL.e_+-----L - - [ - / 204
O
O i
ore(t)
!

F

-r:mg

:e~~f~~~~:~:~:~~sh~::~ ~et:a~~~i~~~~:.
3

4

d.
Capa_city co~straints for the first and the third furnace can be rnodified b b
y oun mg
first two vanables, I.e. X1 :o; 38004, and X2 :o; 16000.
(x)
Furthermore, because of connected production of FeMn (x,) and FeSiMn
1
4 our
· ·'
..
d •
d f, :
problem has 1mposed
a_ nee or mtro ucmg the add1t10nal constraint: x 3 ;::: 3.4x4.
F S M M(arket constramts (purchased quantities) also require limitation onto production of
. et· I ni ~l 2 4000), so that our complete LP. model acquires the standard L p form and 1·ts
'
· ·
op ima so ut10n IS:
xi:= 14 l 28 t (Fe Si),
xi:: 3_8 004 t (Fe Cr).
X4 = 4 000 t (Fe Si Mn).
X3 - ! 7 940 (Fe Mn),
.
·
w1th maximum value ofobjective function·. z*= 2 844 269 currency umts.
Let. us note tha_t .this model does not consider prices of raw materials, as it assumes that h
t e
avallable quant1hes have been already purchased · From the Table 2 the c os ts o f pure h ase d raw
· 1
·
f
58
5430419
=
b-p·
~
to
amount
they
and
calculated
be
can
s
matena
o currency umts
.
L.., 1 1

3. De Novo Approacb
limited ~e Nt~v~ formulati~n of the problem assumes that raw materials quantities are not
, .e. a any qua~hty can be purchased depending on the means available for that
~~rpoose .. In other words, ms~ead of _the ?onstraint on the available quantities of raw materials
~ta m ?nly. on~ con.stramt, wh1ch 1s the constraint of the available budget (B) If raw
·
· ·
· d
ma tena 1 umt pnce 1s. p·'r,(1 = 1, ... ,m ) an d th e reqmre
quanhty 1s bi, then the production model
Can h ave th e f,oIIowmg orm:
Max Z= c, x, + C2 X2 + ..... + Cn Xn
(1)
s. t.
a, I XJ + a12 X2 + ..... + a,n Xn = b,
a2, x, + a22 X2 + ..... + a2n Xn = b2

·····················································

aml X1 + am2 X2 + ..... + amn Xn = bm
P1 b, + P2 b2 + ..... + Pm bm :o; B
Xj, b; 2 O , j = 1, 2, ... , n; i = 1, 2, ... , m

312

(2)

r

1

that model can be made s,mplec by the subslitulion of b; equations into the budget
equation, where
(5)
Pi alJ + P2 a2j ·, ..... + Pm amj = v]
represents the unit variabie cost ofproducing productj.
In that way we obtain a very simple linear programming model with only one budget
constraint, and possibly with additional constraints for market and technological reasons.

s.t.

1

Max z= C; X1 + C2 Xz + ..... +. Cn Xn
V1X1 +v2xz+ ..... +vnxn:S; B
X1 2 (J, j = 1, 2, ... , ll

Let us assume that the available budget in our problem is exactly the amount spent on the
purchase ofraw materials from the Table 2, i.e. 5430419.58 ofcurrency units.
Using that we can reformulate our ferroalloys production model and obtain the optimal
solution:
x2 •= 16 000 t (Fe Si),
x,•= 38 004 t (Fe Cr),
4 747 t (Fe Si Mn)
x/=
(FeMn),
16142t
x/=
with maximum value of objective function: z* = 2 939 l 05 currency units.
It can easily be seen that, in that case, maximum value of the obJective function is
greater than in the previous model, while the same amount of money is spent for raw
materials purchasing as in first model. In this model raw materials are to be purchased in
exactly determined quantities (there is no surplus and unused quantities ofraw materials), and
the necessary quantities b; (i = 1, ... ,5) can easily be determined by inserting variables xi in raw
materials constraints, i.e. in relations (2).
4. Increasing cost of raw materials
The linear prograrnming model is sometimes difficult to apply in real business
situations due to its assumption of proportionality. A frequent phenomenon arising in practice
is the varying price of the same resource. Namely, if a company needs additional quantities of
raw materials it is possible to buy then from another supplier but at a different (usually
higher) price. Let us assume that i-th raw material can be purchased at the price p;, but only
for the quantity lower than Q. To purchase i-th raw material above that quantity it is necessary
to take another supplier whose price is p;' > Pi· Then the relation for the i-th raw material is
transformed into:
(9)
a;1 X1 + a;2 X2 + ..... + a;n Xn = b; + d;,
raw material
i-th
the
of
quantity
with additional constraint b; :o; Q, where d; is the additional
with unit price pi',
Since the same raw material has different price variable, income from end product unit
anymore. Therefore, maximizing the sum of Cj Xj, would not be an accurate
constant
not
is
measure of net income. Net income equation (6) should be recalculated as the difference
between sales and total cost of materials, where the objective function will include materials
at both prices. Consequently, if Sj is the sales price of j-th product, the objective function has
the following form:
II

m

/

Maxz= Lsjxj-LPibi- LPk'dk
(3)
(4)

(6)
(7)
(8)

j=l

i=l

k=l

313

(10)

r

\\here dk (k= 1, .. ,l) are those mat.eria!s which in adJitiona l 4uantities can be bought
only at a
higher price (pk').
ln the budget equation (J) it is also necessary to introduce costs for additiona l quantities
of
raw materials, so that it now takes the following form:

Sinularlv , if bk = () then the relation should be dk ~Q (the quantity required for
the
discount is reacl1ed). Due to that, let us introduce two additiona l constraint s:
bk + M dk ~Q + N Yi
( 18)
dk + N Y2 ~ + M bk
( 19)

I

III

1

LPih1+ LPk dk<:;,B
i=I

(II)

k=I

There is no need to specify that b, should reach the maximum value of Q first, before
allowing
di greater than zero. The optimizat ion model ensures b, reaching the maximum
value of Q
because ofthe lower penalty, i.e. lower price p;

4. Quantity discounts
Let us now consider such productio n situation when there are 4uantity discounts
granted for ?ulk orders of raw mate1ials. Therefore , in addition to the increasing
cost effect
we have to mtroduce this possibilit y into the model. Let us assume that for the k-th
resource
(b~) the valid price is Pk as long as the purchased quantity is below Q, and the discounte
d price
Pk 1s va~td f?r the e~tlfe 4uant1ty the purchased quantity is higher than Q. Conseque
ntly the
assumpho n 1s oppos1te to the one m the previous model, i.e. Pk' < Pk•
The previous formulati on is not applicabl e since the optimizat ion model will prefer
using the
less expensive mate1ial without satisfying the quota (Q). A different model
has to be
formulate d with a slightly more complica ted procedure .
Let
bk, Pk - the amount and price of k-th raw material if it is purchased at less than
the
quantity discount volume;
dk, Pk _the amount and price of k-th raw material if it is purchased with the quantity
d1scount.

:f

1

-

The new model, in that case, instead of one equation for k-th raw material has two
relations,
and those are:
akl Xt + ak2 X2 +
+ akn Xn $ bk + M Y1
(12)
ak! x, +ak2X2+
+aknXn $ dk+My2
(13)
and, according to this, two relations as budget constrain ts:
PI b1 + P2 b2 + ..... + Pk bk + ,.. + Pm bm $ B + M Y1
PI b1 + P2 b2 + ..... + Pk 1 dk + ... + Pm bm $ B + M Y2

(14)
(15)

where M is a very great positive number (M >> O).
Besides that new variables y 1 and y2 are integer O - l variables, for which is:
Y1 + Yz = 1
Y1, Y2 = O or 1

(16)
(17)

In the above model there are two 0-1 variables y 1 and y , where due to the relation
2
(16) only one of them always equals 1, and the other equals zero. Naturally , if
the model
comprise s a number of resources that can be purchased at discounte d price then
there are
more 0-1 variables.
The problem of mutual exclusive ness of the variables bk and dk can be introduce
d in
the following way:
If dk = O (there is no quantity discount) then the relation bk < Q is valid i.e. the needed
quantity ofraw materials is below the quantity required to obtain the discount.

314

where N >>M>> O, i.e. N and M are very great positive numbers, but N is also much
greater
than M.
The model objective function remains the same, while Pk stands for raw material
prices,
which are either discounte d or rising.
Let us now consider such situation •for our ferroal!oy productio n model. The
third raw
material (coke) can be purchased at a discounte d price ifthe bought quantity is Q
> ~5 000,
and this reduced price is valid for the entire quantity supplied, i.e. p3' = 30. In
add1tton to
this, let us assume increasin g costs for electrical energy and quartz in this way:
Permanen t lack of electrical energy (b 1) and limited possibilit ies of quartz (bs) that
has to be
imported result in different (higher) purchasin g prices The limit of electrical
energy
purchased at a lower price is 250 000 (000 kWh), while this limit in quart~ ~s 25 000
t.
The purchasin g price of the add1t10nal 4uanttty of electncal energy 1s p1 - 12, and
of quartz
ps' = l o currency units. Assuming the same budget leve! B = 5 430 O_OO, and product
sale level
s = 257, s 2 = 242, s 3 = 135 and s 4 = 148, the product10n model now 1s:
1
1

i

1

Max (257xi + 242xz + 135x + 148x -8p - l2d -19.Sbz - 35.5b3 -30d3 - 86b4 - 7
1
lb5 - LOd5)
1
4
3
0.2604 X1 + 0.48 X4 $ 7160
X3 - 3.4 X4
~ 0
4x 1 + 9.6 x 2 + 2.63 X3 + 4.75 X4 -b1 - d1 = O
2.04 X3 + 0.5 X4
- b2 = o
O.S X1 + 0.29 X2 + 0.456 X3 + 0.48 X4 - b3 - M Y1 $ o
0.5 X: + 0.29 X2 + 0.456 X3 + 0.48 X4 - d3 - M Y2 ~ o
0.035 X1 + 0.07 X2 + 0.022 X3 + 0.064 X4 - b4 = o
0.28 x 1 + 2 x 2 + 0.02 X3 + 0.58 X4 - bs - ds = O
8b + 12d + 19.5b2 +35.5b 3 +86b 4 +7.I bs+l0ds -MY1 ~ 5430000
1
1
8b + 12d + 19.5b2+ 30d3+8 6b4+7.l bs+ l0ds-MY 2$ 5430000
1
1
b3 + M d, ~Q + N Y1
d3 + N Yz ~ + M b3
YI + Y2 = 1
y 1,y2=0il i l
0 $ Xt $ 38004, 0 $ X2 $ 16000, X3 ~ 0, X4 ~ 4000
b 1, b2, b3, b4, bs, d1, d3, ds ~ O

This is the problem of mixed integer programm ing which will be sol ved by branch
and
bound algorithm by Winqsb program. With the budget amountin g to B = 5 430
000 of
currency units, the optimal solution obtained is:
x/= 38 004 t (Fe Cr),
x/= 13 600 t (Fe Mn)

x/= 15 545.69 t (Fe Si)
x/= 4 000 t (Fe Si Mn), y,* = 1, Y2* = O

with maximum value of objective function:
z* = 1O 527 090 currency units,
while the optimal quantity of raw materials is:

315

=250000. d1*=l06022.6
= 29 744
d,* = 31.754.25
= O
= 2 957.54
bs* = 25 000, d5 * = 19 324.5.
h1*
h2*
b,*
b4*

COMPARISON OF THE APPLICABILIT Y OF COMPUTER
SUPPORTED MUL TI-CRITERIA DECISION-MAK ING METHODS
Vesna Čančer
Facu!tv ofEconomics and Business Maribor. Razlagova l4. 2000 Maribor, Slovenia
E-mail: vesna.cancer@uni-mb.si
,

It is obvious that this solution provides a higher value of ohjective function than in the
previous models for the same budget, which is only logical, as the entire quantity of the
required third raw material (d 3) is purchased at a discounted price.
S. Conclusion
De Novo presents a special approach to optimization. lnstead of "optimizing a given
system" it suggests a way of "designing an optimal system". The basis of the standard
approach is a production model with resources defined in advance, so that the constraints and
the feasible set are fixed.
However, De Novo approach does not limit the resources as most of the necessary
resource quantities can be obtained at certain prices. Resources are actually limited because
their maximum quantity is governed by the budget which is an important element of De
Novo.
using De Novo approach most various cases can be handled more effectively than by
the standard programming models. Changes in prices, technological coefficients, increasing
costs of raw materials, quantity discounts and other similar and real production situations can
be easily incorporated in De Novo model and provide very satisfactory solutions.

1

Keywords: decision support, methods. mult1-criteria decision-making (MCDM), scales

1. INTRODUCTION

6. References
l. Z.Babic, N.Plazibat ( 1985): Application of the Uoal Programming to the Production
Planning Problem, Seventh European Congress on Operations Research - ElJRO VU,
Bologna, Italy, June 16-19, 1985.
2. M Zeleny (1986): Optimal System Design wiih Multiple Criteria. De Novo Programming
Approach, Engineering Costs and Production Economics, No. 1O, p. 89-94.
3. M.T. Tabucanon ( 1988): Multiple Criteria Decision Making in Indust1y, E!sevier,
Amsterdam.
4. R.J. Li and E.S. Lee ( 1990): Multicriteria De Novo Programming with Fuzzy parameters,
Computers Math. Applic. Vol. 19, No. 5, pp. 13-20
5. Z.Babic, !.Pavic (1991): De Novo Programming in MCDM, 16th Symposium on
Operations Research, September 9-11. 1991, Physica-Verlag, Heidelberg, 1992, p- 579582.
6. Y. Shi (1995): Studies on Optimum Path Ratios in Multicriteria De Novo Programming
Problems, Computers Math. Applic. Vol 29, No. 5, pp.43-50.
7. Z.Babic, !.Pavic (1996): Multicriterial Production Programming by De Novo
Programming Approach, lntemational Joumal of Production Economics, Vol, 43, No. l,
1996. p-59-66.

Abstract
This paper introduces and discusses the applicability of severa! multi-criteria decision-making
(MCDM) methods. In addition to theoretical asp~cts of dif'.erent types of scales and the assumpt1ons
about decision makers' abilities and preferences. 1t summanzes some of the conclus10ns, made by the
MCDM methods' users in business practice. By utilizing the author's experience, it offers a critical
overview of MCDM methods, together with an evaluation of adequate computer programs.

1

Methods for supporting business decisions are inevitable tools in problem solving, especially
in the phase of problem definition (to identify. analyze and define problem) and 1~ ?ec1s10nmaking (to choose and verify possible solutions to the problem) [l, 2]. By ut1hzmg the
author's experience in using multi-criteria decision-making (MCDM) methods, the paper
offers a critical overview of these methods (emphasizing MAVT, SMART, SWING,
SMARTER, AHP, and discussing ANP, outranking approaches, interactive methods and
preference programming), together with an evaluation of adequate co~puter programs
(HIPRE 3+ and Web-HIPRE, Expert Choice, Logical ~ecisions® for _Wmdows). Put mto
groups, they are introduced and discussed regardmg their bas1c advantages and
disadvantages, convenience in problem solving, applicability for different types ofproblems,
the types of the obtained results, and even the participants' knowledge.
MCDM methods have already tumed out to be applicable in business practice (see [l, 6,
11]). The author presents some of the real-life applications in Slovenia that result from her
involvement in severa! research projects:
creditworthiness assessment [4], benchmarking of business processes, environmentally
oriented business decision-making, investments in new production technology [3],
selection of organizational structures, selection of information systems (to support
business decisions - on the micro leve!), and
measuring a country's position and potential in a period of contemporary globalization
®
[5] (to support national economic policymaking- on the macro le_v~l) ..
Some practical solutions that were found to solve the problems ansmg m these research
projects are introduced as well: regarding, for example, phases o~ the process for _the goal
fulfillment, model structuring, hierarchy (re)structming and we1ghts (re)calculahon, and
consistency improvement.
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2. SOME ASPECTS OF THE SELECTION OF APPROPRIATE MCDM METHODS
Multi-criteria decision-making describes the set of approaches that can help individuals or
groups in researching important complex decision-making problems (see [l, 2]). They
should be used when intuitive decision-making is not enough for severa! reasons: because of
the conflicts between criteria or because of disagreement between decision makers about

316

317

relevant criteria or their importance and about acceptab!e altematives and preferences.
their applicability in solving complex problems contribute the following facts:
They do not replace intuitive judgement or experience and they do not oppress creative
verify ideas and support problem
thinking; their role is to complement intuition,
solving.
fn MCDM we take into account multiple, more or less conflicting criteria.
• In this type of decision-making process we structure the problem.
• Users can compare different methods and assess their convenience. The most useful
approaches are conceptually simple, transparent and computer supported.
• The aim of MCDM is to help decision makers leam about the problem, express their
judgements about the criteria impo1iance and preferences to alternatives, confront the
judgements of other participants, understand the fina! alternatives' values and use them
in the problem solving ac!ivities.
When applying these methods, the assumptions about decision makers' abilities and
preferences are as follow:
A decision maker is able to decide between two possibilities and to express his/her
preferences.
• Relation 'is preferred to' is transitive:
(1
A P B /\ B P C • A P C.
Transitivity is a basis for measuring decision makers' inconsistency.
Considering (1). the relation 'A is preferred to C' is stronger than the relation 'B is
preferred to C'.
Solvability. When measuring the altematives' values with respect to an attribute by
®
using value functions, it is possible to assign scale values, e.g. 50, 25, 75.
Finite upper and lower bounds ofthe alternatives· values.
In MCDM, ranging has a principal role. An analyst has to find a relative importance of
each criterion or a relative preference to each alternative. Scales can be defined as nominal,
ordinal, interval and ratio [2]. Nomina! scaies are the least restricti ve and - consequently _
the least informative: numbers, symbols or descriptions are used for identification. Ordinal
scales enable ordinal ranging, where specific differences between the criteria importance or
preferences to altematives are not known. On the contrary, interval and ratio scales enable
adjudging exact numerical leve! of importance or preference. With interval scales it can be
expressed for how much the criteria importance or preferences to alternatives differ from
each other. With ratio scales it can be expressed how many times the criteria importance or
preferences to altematives differ from each other.
One of the most widely applied sets of multi-criteria methods is multi-attribute value (or
utility) theory (MAVT or MAUT) (see [l]). From the late 1960's, this set ofmethods bas
been developed not only by management scientists, mathematicians, psychologists, but also
by practitioners in management, economic, environmental and puhlic fields. The need to
include different scientific, professional fields in the development of these methods results
from the need to manage complexity. It has been improved to SMART (a simplified multiattribute rating approach) and other approaches (for example SWING, SMARTER). They
are supported by severa! computer programs, e.g. HIPRE 3+, Web-HIPRE [8] and Logical
Decisions® for Windows [9]. One decade later the Analytic Hierarchy Process (AHP)
method was developed [11], together with computer program Expert Choice [7]. It was
completed to the Analytic Network Process (ANP), which is supported by Super Decisions.
Thus it overcomes the traditional OR/MS approaches.
The use of the discussed methods would lead to over-complications when decision makers
do not need so detailed results as they are obtained with these methods. Namely, often it is
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318

good enough to find out which of the altematives is the most preferred. Therefore the socalled "outranking" approaches have been developed since l 970's. The most widely
applied are ELECTRE in more variants and PROMETHEE (see [12]). Further, interactive
methods are especially applicable when a complete preference model is not constructed a
priori and when altematives need improvements (see [12]).
Since traditional methods can not satisfactorily support many complex processes, methods
for the approximate specification of preferences are gaining force in enterprises. Decision
makers can express approximate preference statements through interval judgements.
Preference programming describes approaches (like P AIRS, PRIME and RICH) that can be
helpful in group decision-making, too [10]. Easy-to-use software has been developed to
supp01i the interval techniques: WINPRE for supporting the preference programming,
PAIRS and Interval SMART/SWTNG; PRIME Decisions as a software implementation of
the PRIME method; RICH Decisions for supporting the RICH method [ 1O].
When applying MCDM methods to severa! decision-making problems, we concluded that
they should be approached step-by-step. W e followed the phases of decision-making
processes that are commonly acknowledged in the literature f !]: from identification of a
problem, through problem structuring - model building, its use to inform and challenge
thinking, to the creation and analysis of a plan of activities that can solve a problem, but we
adapted and completed them for the problem's type [3, 4, 5].

3. COMPARISON OF THE MCDM SOFTW ARE APPLICABILITY
Table 1 introduces software products for MCDM that have been paid much attention among
experts in different business fields (because of user capabilities, availability of graphical
elicitation techniques, and the possibility to transform subjective judgements into objective
measures). It delineates our findings about their applicability according to our experiences.

Table 1. Some Findings About the Most Preferred Software Support for MCDM.
Applicability (according to our experienc<D_
Software
HIPRE 3+, Especially applicable for the methods based on ordinal and interval scales:
Web-HIPRE SMARTER, SMART, SWING, and for the measurement of altematives' values
with respect to each attribute by value functions, although it supports also the AHP
[8)
method in the sense of pair-wise comparisons, and direct measurement of
altematives' values
Especially applicable for the AHP method that is based on a ratio scale (pair-wise
Expe1i
comparisons), although it supports the measurement of altematives' values with
Choice [7]
resoect to each attribute by value functions and direct method
Especially applicable for problems where describing the altematives is of special
Logical
value (utility functions, AHP, adjusted AHP), and for weights' assessment with
Decisions®
for Windows tradeoffs, as well as by direct entry, the SMARTER and the SMART method,
weight ratios and the AHP
[9)
1

In the last year we are faced with the group decision-making upgrades of the computer
programs that have been most preferred for individual MCDM in the last two decades. They
are, for example, HIPRE 3+ Group Link, Expert Choice 11 with EC Decision Portal, and
Logical Decision for Groups.
Representations of available decision analysis packages contain the information provided
by the vendors and surveyed by the OR/MS researchers. The appropriate information can be
easily found on the world's web pages. Before buying such a decision support package,

319

experts in enterprises can use trial-free versions of computer programs to find out whether a
product offers enough possibilities for a convenient preparation of their decisions. However,
when using the results in answer and sensitivity reports, decision makers in enterprises must
be provided with appropriate knowledge about the basics ofthe applied methods.
4. REAL-LIFE APPLICATIONS: FINDINGS

Among MCDM methods that we applied in several research projects in Slovenian
enterprises, we found the AHP method specially applicable m connection with other
decision-making tools, e.g. in:
• Creditworthiness evaluation. We delineated the process of the assessment of an
enterprise's business partners' creditworthiness that is made by a film itself (so-called
interna! ratings) where the problem is approached step-by-step l4J. When verifying its
applicability for the creditworthiness assessment, and therefore for the selection of
business partners, we concluded that the technique should involve the following steps:
problem definition, elimination of unacceptable altematives, problem structuring
(building a model), establishing priorities (on importance and preferences), synthesis
and sensitivity analysis with verification. Since qualitative factors come into play in the
creditworthiness evaluation, the AHP technique was applied. Special attention was paid
to the assessment of the criteria importance: we presented the procedure for the
improvement of the decision makers' consistency. Studying the corrected intensity
levels, they improved their understanding of the relationships among the criteria, and of
the criteria meaning and importance as well.
Environmentally oriented business decision-making. We developed the method for
selecting among environmentally oriented business processes with respect to multiple
criteria. Since the particularities of the business processes in the sample enterprise,
decision makers' preferences, their judgements on importance, practical <lata about the
business processes in this enterprise, as well as the research results on environmental
management in the sample enterprises of the processing industry and those found in
eco-balances were taken into consideration, the AHP technique was found appropriate
in connection with other decision-making tools [3]. Pair-wise comparisons were
successfully applied in the criteria determination, the assessment of the criteria
importance and the <lata calculation for different business processes.
• Process benchmarking. The multi-criteria method for benchmarking of environmentally
oriented business processes was developed and presented with a real-life case from the
Slovene enterprise in the processing industry. We considered not only ecologically most
acceptable manufacturing, but also other, different and conflicting criteria that are
relevant to this complex goal. For this reason, the AHP technique was used; together
with other decision-making tools, it is suitable for benchmarking in order to help in
decision-making about business process re-engineering and selection ofnew production
processes. The activity of benchmarking can be decomposed in severa! steps [6]. The
approach used in this study involves the following steps: define the business process's
critical success factors, identify the business processes to be included in the analyses,
make analyses to determine strengths and weaknesses, and set performance goals for
improvements.
Furthermore, we compared some of the popular MCDM methods in other real-life business
applications, e.g.:
• Investments in new production technology. The methods for determination of criteria
weights which base on an ordinat (SMARTER), interval (SWING, SMART) and a ratio
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320

scaie (AHP), as well as the ways of measurement of local altematives' values (direct
input, use of value functions and pair-wise comparisons) were used in a real-life
problem of estimation of altematives for acquisition of pattem sets in a Slovenian
foundry. We were involved as outside experts (tutors) in this application. One expert,
employed in this foundry, has appropriate knowledge about the used quantitative
methods. By using interviews and questionnaires, the experts in this foundry determined
the criteria, their importance and the altematives for the model's building. The obtained
fina! results showed the most convenient alternative for the acquisition of pattem sets
and were used in decision-making. These experts evaluated SMART, SWING and
SMARTER as more demanding in comparison with the AHP in this application.
Similarly, pair-wise comparisons and the distributive and ideal modes of synthesis were
recognized as the most important advantages of Expert Choice.
Selection of organizational structures. Decision-making about the reorganization in a
Slovenian service company was based on the evaluation of the interna! and external
factors that influence the selection of an organizational structure. The enterprise's top
managers used the questionnaire (prepared by an employed expert with the appropriate
knowledge about MCDM) to evaluate the factors' importance. The decision model was
entered by computer programs Web-HIPRE and Expert Choice. Decision makers can
select the alternative with the highest fina! value as the most appropriate organizational
structure. Completed with sensitivity analysis, this practical business research can be
used for the verification of the current organizational structure and as a guidance for
further organizational aims.
111
Selection of information systems. The discussed methods were used also in the selection
of suitable approaches to further information systems' development in a Slovenian
building company. The method based on an ordinal scale SMARTER is evaluated
convenient in decision situations, in which it is possible to evaluate only the rank of the
criteria importance. The enterprise's experts evaluated SMART, SWING and
appropriate software as excellent tools in solving complex problems. However, they
emphasized that the quality of the decisions made on the basis of their results depends
on the responsibility in establishing priorities about the criteria importance and
preferences to alternatives.
On the macro leve!, we developed a multi-criteria method for measuring and analyzing the
globalization of national economies with the intention to establish and study thoroughly a
country's position and potential for international integration [5]. The method includes some
advantages of the AHP, emphasizing model structuring and establishing priorities on
indicators' importance. When applying this methodology to measuring the globalization of
national economies, we concluded that it should involve the following steps: problem
definition, model structuring, <lata collecting and measuring, establishing priorities, model
restructuring and weights (re)calculation (in some cases), synthesis, and sensitivity analysis.
@

5. CONCLUSIONS

Experts and managers in co-operative Slovenian enterprises found MCDM methods
applicable in business practice for solving severa! complex management problems. Since
common-practice methods cannot satisfactorily support many complex processes, including
national policymaking, the multi-criteria method for measuring and analyzing the
globalization ofnational economies can define their key success and failure spheres.
Let us briefly emphasize the most evident common conclusions that arise from the use of
the above-mentioned methods in the presented applications. Experts and managers in the cooperative enterprises accepted MCDM methods as a new approach in solving complex

321

problems since they considerably contribute to favorable business decisions. Because of the
real-life problems' complexity, a chosen method (e.g. the AHP) should be used
connection with other decision-making methods. The most important advantages ofthe AHP
that were mentioned by the analysts, experts and managers are: pair-wise comparisons,
measuring inconsistency, the distributive and ideal modes of synthesis and sensitivity
analysis. The methods based on an interval scale SMART and SWING were chosen as more
convenient for solving problems with sufficient information basis. With some adaptations,
the models structured and the methods developed in the described research projects can be
used in similar cases on the micro and macro leve!.
When using the results in answer and sensitivity reports, decision makers must be provided
with appropriate knowledge about the basics of the used methods in order to read, interpret
and use these results for problem solving. The ability to leam and to co-operate with experts
inside and outside an enterprise is, therefore, of high importance.

References
Belton, V., Stewart, T. J. (2002). Multiple Criteria Decision Analysis. An Integrated
Approach. Boston, Dordrecht, London: Kluwer Academic Publishers.
2. Čančer, V. (2003): Analiza odločanja (Decision-making analysis. In Sloveniml).
Maribor: University ofMaribor, Faculty ofEconomics and Business.
3. Čančer, V. (2004): The multicriteria method far environmentally oriented business
1ecision-making. Yugoslav Joumal ofOperations Research, 14, L 65-82.
4. Cančer, V., Knez-Riedl, J. (2005): Why and How to Evaluate the Creditworthiness of
~MEs' Business Partners. Intemational Small Business Joumal, 23, 2: 141-158.
5. Cančer, V., Bobek, V., Korez-Vide, R. (forthcoming in 2005): The multi-criteria method
far meas11ri11g economic globalization of national economies. In: R. Scitovski (ed.),
Proceedings of the 10th Conference of Operational Research K.01 '04. Osijek, Zagreb:
University ofOsijek, Croatian Operational Research Society.
6. Dey, P. K. (2002): Benchmarking project management practices of Caribbean
organizations using analytic hierarchy process. Benchmarking, An Intemational Journal,
9, 4: 326-356.
7. Forman, E. H., Saaty, T. L., Shvartsman, A., Forman, M. R., Korpics, M., Zottola, J.,
Selly, M. A. (2000): Expert Choice 2000. Pittsburgh: Expe1i Choice; Inc.
8. Helsinki University of Technology: Web-hipre help. http://www.hipr e.hut.fi, consulted
April 2005.
9. Logical Decisions: Logical Decisions® far Windows. www.logicaldec isions.com,
consulted April 2005.
10. Mustajoki, J., Hamalainen, R. P., Salo, A. (2005): Decision Support by Interval

SMARTISWING - Incorporating Imprecision in the SMART and SWING Methods.
Decision Sciences, 36, 2: 317-339.
11. Saaty, T. L. (1999): Decision Makingfor Leaders. Pittsburgh: RWS Publications.
12. Vincke, Ph. (1992): Multicriteria Decision-Aid. Chichester: John Wiley & Sons.

AN INTERVAL AHP MODEL FOR EFFICIENC Y EV ALUA 'BON OJF
PRODUCTIO N UNITS
Josef Jablonsky
Department of Econometrics, University of Economics
Praha, 130 67 Czech Republic
jablon@vse.cz. l;RL: http:/ 1nb.vse.czi ,jablon/

Abstract
The paper deals with models and methods for evaluation of efficiency of production units. The
standard modeling approach for evaluatioi1 of efficiency 1s data envelopment analysis (DEA) based on
the definition of effic1ency as the ratio of outputs produced by the umt and inputs speni in the
production process. Standard data envelopment analys1s models divide the units into ineffic1ent and
effic1ent ones. The efficient units receive the efficiency score !00% by standard DEA models and can
be further classified by super-efficiency DEA models. The paper d1scusses the possibility of using the
AHP model with interval pairwise comparisons for evaluation and classification of effic1ent umts and
compares given lresults with super-efficiency DEA scores. The proposed _approach is appli~d in
assessing the efficiency ofpens10n funds in the Czech Republic - the results g1ven hy super-effictency
DEA models and by the interval AHP model are compared and discussed.
Keywords: AHP, interval AHP. <lata envelopment analys1s, effic1ency

1. Introduction
There are severa! standard tools for evaluation of efficiency of production units. One of
the most popular modelling approaches in this field is data envelopment analysis (DEA).
DEA models measure relative efficiency of decision making units (DMU) that are usually
described by severa! inputs spent for production of severa! outputs. Let us consider the set E
of n decision making units E = {DMU 1, DMU 2 , . DMU 0 } . Each_ of the units produces r
outputs and for their production m inputs are spent. Let us denote x1 = {xij,_i=l,2, ... ,~} the
vector of inputs and y1 = {Yii, i=l,2, .. ,r} the vector of outputs of the DMUj, Then X 1s the
(m, n) matrix of inputs and Y the (r, n) matrix of outputs. A DEA model evaluates efficiency
ofthe DMlJq, qE{l,2.,,,.n} by looking for a virtual unit with inputs and outputs defined as
the weighted sum of inputs and outputs of the other units in the decision set - X')., a Y,\,, where
').,=(')., , ').,2 , . . , ').,0 ), ,\>O is the vector ofweights ofthe DMUs. The virtual _unit shoul~ be bett~r
1
(at least not worse) than the analysed unit DMlJq, The problem of lookmg for a v1rtual umt
can be generally formulated as a standard linear programming problem:
minimize
subject to

z = 0 - E( eT s+ + eT s-) ,
(1)

Y').,-s+=yq,
XA+ s-= 0xq,
').,, s+, s- ;c: O,

where eT = (1,1, ... ,1) and E is a infinitesimal constant. The variables s+, s- are just sl~ck
variables expressing the difference between virtual inputs/outputs and appr?pnate
inputs/outputs of the DMUq, Obviously, the virtual inputs/outputs can be computed usmg the
optimal values ofvariables ofthe model (1) as follows:

xq' = xqe• - s-'
yq'=yq+s+.

The DMU is to be considered as efficient if the virtual unit is identical with evaluated unit
(does not e:ist a virtual unit with better inputs and outputs). In this case Y')., = Yq, X').,= xq and

322

323

the minimum value of the objective function z = 1. Otherwise the
is not efficient and
minimum value of 0<1 can be interpreted as the need of proportiona l reduction of inputs in
order to reach the efficient frontier.
The number of efficient units identified by DEA modeis and reaching the maximum
efficiency score 100% can be relatively high and especially in problems with a small number
of decision units the efficient set can contain almost all the units. In such cases it is very
important to have a tool for a diversifica tion and classificati on of efficient units. That is why
severa! DEA models for classificati on of efficient units were formulated. In these models the
efficient scores of inefficient units remain lower than 100% but the efficient score for
efficient units can be higher than 100%. Thus the efficient score can be taken as a basis for a
complete ranking of efficient units. The DEA models that relax the condition for unit
efficiency are called super-effic iency models. The super-effic iency models are always based
on removing the evaluated efficient unit from the set of units. This removal leads to the
modificatio n of the efficient frontier and the super-effic iency is measured as a distance
between evaluated unit and a unit on the new efficient frontier. Of course severni distance
measures can be used - this leads to different super-effic iency definitions . The first superefficiency DEA model was formulated in (Andersen and Petersen, 1993).
The paper discusses possibility to evaluate the efficiency of production units by an
alternative approach based on the analytic hierarchy process (AHP) model with interval
judgement s (IAHP model). It is organized as follows. The next section contains a brief
introductio n to AHP models with interval judgement s. Section 3 formulates an IAHP model
for evaluation of efficiency of decision making units. The nume1ical experiment s are realised
on the real-world example that consists in comparison of 12 pension funds operating in the
Czech Republic. The results of the analysis by both the techniques are presented and
discussed. The last section contains summariza tion of the paper and discussion about the
future research.
2. The interval AHP model
The AHP is a powerful tool for analysis of complex decision problems. The AHP
organizes the decision problem as a hierarchica l structure containing always severa! levels.
The first (topmost) leve! defines a main goal of the decision problem and the last (lowest)
leve! describes usually the decision alternative s or scenaiios. The levels between the first and
the last leve! can contain secondary goals, criteria and subcriteria of the decision problem.
The number of the levels is not limited, but in the typical case it does not exceed four or five.
Let us consider a simple three-level hierarchy that can represent a standard decision problem
with fini te set of altematives - evaluation of n-alternativ es X 1, X 2 , . . , Xn, by k-criteria Y ,
1
Y 2, ••. , Y k, (Figure 1). The decision mak er expresses his preference s or compares importance
of the elements on the given leve! with respect to the element of the preceding leve!. The
informatio n resuHing from decision maker's judgement s in the given leve! of the hierarchy is
synthesised onto the local primities. They can express, e.g. relative importance of criteria
(weight coefficient s - in Figure 1 denoted by Vj, i= 1,2, ... ,k) or preference indices of the units
with respect to the given criterion (wij, i=l,2 ... ,n, j=l,2, ... ,k), In the standard AHP model the
decision maker judgement s are organised into paired comparison matrices at each leve! of the
hierarchy. The judgement s are point estimates of the preference between two elements of the
leve!. Let us denote the paired compariso n matrix A ={aul aii = 1/aii, aii>O, ij=l,2, ... ,k },
where k is the number of elements of the particular leve!. Saaty (1990) proposes to use for
preference expression au integers in the range 1 through 9, where 1 means that the i-th and
the j-th element are equally important and 9 that the i-th element is absolutely more important
than the j-th element. The local priorities are derived by solving the eigenvecto r problem (2)

324

A.v = "'-max v.
(2)

k

"'
L.,

1

=l 1

i=I

where
to

,max
1

is the largest eigenvalue of A and v is the normalised right eigenvecto r belonging

Amax·

!

Figure 1 Tbree-ievel hierarchy.
In the standard determinis tic AHP approach the decision maker_ always spec!fies p~int
fmates
that express his preference relations between two elements m the g1ven h1erarch1cal
1
~=v eJ. It can often be ve1y difficult to fulfil this condition for decision makers. T~ey feel
eh better and closer to have the possibility to express their preference s as mterval
:~imates. For instance, instead of giving that the i-th element is four times as preferab~e as
j-th element, he can assert that the i-th element is at least two but no more than five t1m~s
preferable as the J-th element. The AHP model with interval decision maker JUd~em~nts_ 1s
called interval AHP (IAHP) model. It is characteris ed by mtervai pa1rw1se
c01noan:son matrices given as follows:

r

A=

l

< P!k•qlk >

1

< P21,q21 >

<

P2k•:q2k

1

< Pkl•qkl >

l

>\

(T
. )

J

Pii is Jower bound and % upper bound for pref~re~ce relatio~ (au) betv.:een the i-th ~nd
element. Due to the reciprocal nature of the parrwrse companson matnces the relatron
= 1 holds for all ij= 1,2, ... ,k.
.
judgement s in the IAHP can be considered as random vanables defined over t~e
interval. In this way the IAHP changes from the determini~tic model
the model w1th
stochastic features. That is why it cannot be analysed m the trad1t10nal way - ~y
the eigenvecto r problem (2). It is necessary to look_ for n~w appr~aches that w1ll
stochastic features. The random variables for descript10n of mterva_l Judge~e~t s ~an
selected from the available probabilist ic distributions. We will use the umform d1stnbut10n
~v,,.,,.,~ over the interval <pij, %> and in our numerical e~periment s _below.
Th~ p:eference s
1
nts derived form from matrix A are random vanables. Therr charactens trcs can be
o f e eme
.
h t.
. 1
d
computed by several approaches - we used a Monte Carlo simu 1atron
t a rs very srmp e an
offers lower and upper bounds for the preference s in a very short tirne.

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325

3. The JAHP model: evaluation of the Czech pension funds

DEA/VRS
I efficieucy I super-eff 1
model
score
Iscore
1
Big funds
c.:A_::cl::..::li:.::.:a_nz____--+---__l_0_0_.0_0+infeasible
Credit Suisse
96.12
CSOB Stabilita
100.00
159.24
ING PF
100.00
119.87
1-------=---- --+------+-CP PF I
100.00
122.40
CP PF II
95,64
CS PF
82.14
f----------4,----1---------------1
KB
PF
_
i
_
l()0,00
_1_12.40 i
1--------1
Small funds _I __ ~ - _1__
_ ~
CSOB Progres
l 00.00t!_!l..fe~i!'le j
Generali
93.72: _____ ~
PF Ostrava
64.06 i
i
PF Zemsky __ 1 _ 100.001 infeas1ble
1

The discussion concerning the DEA and !AHP models in evaluation of efficiency and
especially super-efficiency will be demonstrated on a small numerical example with a real
economic background. It is the problem of evaluation of efficiency of available pension funds
in the Czech Republic. We worked with the data set for 12 pension funds, each ofthem was
characterized by the following seven criteria (the data are from year 2003 ):
" lNP - the number of customers [ thousands],
" INP 2 - total assets [mil. CZKJ,
" INP 3 - equity capital [mil. CZK],
" INP 4 - total costs [miL CZK],
" OUT - appreciation ofthe customer deposits for the last year (2003)
" OUT 2 - average appreciation of the customer deposits for the last three years (2001 2003) [%).
" OUT 3 net profit [mil. CZK].
For DEA analysis, first four criteria were taken as inputs and the remaining ones as
outputs ofthe modeL The criterion matrix is given Table .

1

i Allianz
Credit Suisse
CSOB Progres
CSOB Stabilita
Generali
·INGPF
CPPFI
CP PF II
CSPF
KBPF
PF Ostrava
i
PF Zemsky
1

#of cust.
106
611
18
304
23
346

~~~I
401
285
19
14

assets
4095
22592
452
8508
789
9767
6348
12441
10954
11776
935
468

eQuity tot. costs
77.0
49.5
549.0
454.l
56.01
15. l
298.6
203.3
74.0
15.5
289.1
221.7
290.7
184.7
522.5 --- 297.3
223.5
238.8
441.6
166.0
71.0,
18.2
87.9
23.2
1

appr. l appr.31
3.00
3.69'
3.36
3.67
4.30
4.15
2.30
2.83
3.00
3.90
4.00
4.27
3.34
3.651
3.10
3.37
2.64
3.31
3.40i
4.14
2.44
2.68
4.01
4.24 i

1

-~-----------,

profit
1.29
5.22
--

1

1

Table 2: Efficiency measures given by DEA models.

1

____:_]_

_

10.871
0.45.
0.26
6.83 /
6.90

The !AHP model for evaluation of efficiency is very simple and it is presented on Figure
2. In this modeL q 1 is the total weight of inputs and q2 is the total weight of outputs, (ji qz =
l. qij, j=l,2, ... ,m, are the weights of single inputs and qoj, j=l,2, .. .r, are the weights of sin_gle
outputs. Preference indices Uij, i=l ,2, ... ,n, j=l ,2, ... ,m+r, express the preference of the 1-th
alternative (pension fund) with respect to the j-th input/output
EVALUATION OF EFFICIENCY OF
PENSION FUNDS

·110

i

1

1

6.40
0.04
2.03

Table 1: Pension funds - criterion matrix.
It is clear that the funds listed in the previous table are of different nature. Four of them
are very small (CSOB Progres, Generali, PF Ostrava and PF Zemsky) and the remaining ones
are significantly bigger. That is why we decided to analyse them separately. In Table 2, there
are results of DEA analysis of both groups of funds (big and small). We used the basic
envelopment DEA model with variable retums to scale with output orientation and the superefficiency model under the same assumptions. First column of Table 2 contains efficiency
scores of the evaluated units - higher score corresponds to more efficient unit. The same
holds for super-efficiency scores presented in the last column ofTable 2. Of course the superefficiency scores are available for units indicated as efficient by the standard model only. The
word "infeasible" for super-efficiency score of Allianz fund means that the corresponding
variable retums to scale (VRS) super-efficiency model has no feasible solution. This situation
can occur in VRS super-efficiency models very often and in this case it disables the
possibility to classify the efficient units.

U,j

Utj

Figure 2: The IAHP model for evaluation of efficiency.
The global preference indices of altematives Pi, i=l,2, ... ,n, are synthesized from previous
hierarchical levels as follows:
m+r

Pi = Luii, i=l,2, ... ,n.
j=l

In the analysis, the AHP model does not deri ve the weights of the inputs and outputs but
they are either to set up directly as constants or optimised as variables of the proposed

326

327

simulation model. In the model the pairwise comparisons of alternatives (pension funds) with
respect to all inputs and outputs are given as random variables
uniform distribution
defined over the interval <a,b>. The cornparisons retlect given criterion values but by this
way it is possible to use different returns to scale for different inputs and outputs. It is one of
the advantages of this approach. Our numerical experiments were realized on the set of five
big pension funds identified as efficient by DEA model with variable returns
scale. The
pairwise comparisons of ali alternatives with respect to first input (number of employees) and
output (appreciation of customer deposits) are presented in Table 3 (free cells are the
reciprocal values). Similarly the comparison matrices for the remaining inputs and outputs
are given.
Number of customers

1

~ _ Allianz ! CSOB I ING \ CP PF; KBPFl
: Allianz:
CSOB

l

<6.8> <7,9> _j_ <35> • <5,7> !
l <1,2>

1

+-

ING
CP PF i
KB PF

t

1

,

l
<l,3> <3,5>
< ,2> <2,4>

1 <2 3> j
~
l

IAppreciation or customt!J"de osits
i Allianz j CSOB I ING ICP PF IKBPF
!

<3,5> ·
--'---'-·
l

1

+

!

-----+---

.
1

f - - - - + - - _...;-_,I- ~ -----'-----J

<4,6> <5,7>'
~,3::>_, <3,5> 1
<1,3> <3,5>

<3,5>
l-+---

__

1

1

Table 3: Pairwise comparisons of the IAHP model.

results are presented in Table 4. The first column of Table 4 contains super-efficiency
computed by the Andersen and Petersen DEA model with variable retums to scale, the
remaining three columns contain maximized efficiency score of pension funds given by the
presented lAHP model with different weight constraints (weight sets I, II and III).
The DEANRS model is not able to evaluate the Allianz fund, the best among the others is
the CSOB Stabilita fund and the worse is the KB pension fund. The results given by the
IAHP models are quite differenL If we consider the same weights for all the inputs and
outputs (AHP HI), the Allianz fund is classified on the top and the remaining funds are very
close each other. The results for the set of weights AHP I and AHP II are almost identical.
The best is the Allianz fund, the mean efficiency score of the CSOB and ING pension funds
is more or less the same and on the bottom of the ranking are the last two funds.
4. Conclusions

The aim of the paper was to verify how the AHP models can be used for efficiency
evaluation of production units and compare the results given by proposed interval AHP
model with efficiency scores computed by DEA models. In contrary to super-efficiency DEA
models the advantage ofthe IAHP approach consists in severa! points:
o the IAHP model can use different scales for differenl inputs and outputs according to the
decision maker preferences,
in super-efficiency DEA models with VRS not ali the units receive their super-efficiency
score (the problems need not be always feasible),
• the IAHP model can deal with categorical inputs and outputs without any transformation,
• the IAHP model offers a possibility of sensitivity analysis of results with respect to the
inputs and/or outputs.
The main disadvantage of the AHP (IAHP) model comparing to the DEA models consists
in preparing the data set for analysis (pairwise comparison matrices) and in tirne consuming
length of the optimisation analysis. Nevertheless, by using the AHP model the decision
maker can receive new information useful for the globa! analysis of the efficiency of the
evaluated set of units.
Future research in this field will be focused on comparison other super-efficiency DEA
models (slack based models) with IAHP models and other multiple criteria decision making
techniques. Severa! real-life economic applications will deal as background for numerical
experiments.
@

The DEA models maximize the individual efficiency of evaluated units by looking for
optimum weights ofinputs and outputs. That is why we did not work with weights deri ved by
the IAHP model but we tried to optimise the weights in order to reach the best efficiency
score of the evaluated alternative. We used the following requirements to the set of the
weights:
The sum ofweights equals to l, all the weights have to be greater than 0.05 (AHP
2. The sum of weights of inputs equals to 0.5, the same holds for the weights of outputs. Ali
the weights have to be greater than 0.05 (AHP
3. All the weights are fixed to value 1/(m+r), ie. /7 in our example (AHP

i Pension funds lDEA/VRSi
I super-eff.
Allianz
infeasible
CSOB Stabilita
159.24
INGPF
119.87
CPPF
122.40
KBPF
112.40

1

AHPI AHPII AHPiiil
1

1

1

0.4574
0.3608
0.3652
0.2123
0.2397

0.3262
0.2583
0.2764
0.2056
0.2138

0.3334
0.1449
0.1894
0.1699
0.1625

Table 4: Comparison of super-efficiency measures.

Acknowledgements
The research is partially supported by the GA CR - project no. 402/03/1360.
References

The optimisation run was realized by means of Crystal Ball software that is an MS Excel
add-in application for Monte Carlo experiments. Crystal Ball contains a special tool for
optimisation under stochastic conditions called OptQuest. This tool can find optimum values
ofvariables (weights ofinputs and outputs in our case) in stochastic environment that can be
modelled within MS Excel. The optimisation criterion is the efficiency score Pi, i=l,2, ...n, of
the evaluated alternative that is to be maximized. Because the efficiency score under our
stochastic conditions is a random variable we tried to maximize its mean value. We always
used a five minutes optimisation run for all the altematives with 100 trials per one simulation.

[1] Andersen, P. and Petersen, N.C. (1993) "A procedure for ranking efficient units in data
envelopment analysis", Management Science, 39, 1261-1264.
[2] Cooper, W.W., Seiford L.M. and Tone, K. (2000) Data Envelopment Analysis, Boston,
Kluwer Pubi.
[3] Jablonsky, J. (2004) "Models for Evaluation ofEfficiency of Production Units (in
Czech)" Politicka ekonomie, 2, pp.206-220.
[4] Saaty, T.L. (1990) The Analytic Hierarchy Process, Pittsburgh, RWS Publications.
[5] Zhu, J. (2003) Quantitative Models for Performance Evaluation and Benchmarking.
Boston, Kluwer Pubi.

328

329

DEVELOP MENT OF A MULTI-C RITERIA EVALUATION MODEL
FOR CLASSIFY ING WOOJD PROJDUCTS
Igor Lipušček, Lidija Zadnik Stim, Leon Oblak
University of Ljubljana, Biotechnical Faculty
1000 Ljubljana, Slovenia
igor.lipuscek@bf.uni-lj.si
lidija.zadnik@bf.uni-lj.si
leon.oblak@bf.uni-lj.si

Abstract: This paper presents the process of developing a multi-critetia decision model for
classifying wood products according to their environmental burden during their life cycle. The aim of
the model development was to include the most important criteria of burdening the environmenl
life cycle of wood products and to calculate the umfonn estimate of environmental
appearing
burden. For the generation ofthe model we used the Expert Chotce methodology. The model 1s based
on expert opinions gathered with the Delph1 method and mathematically and slatistically processed.
Keywords: wood industry, classification of wood products, multi-criteria decis1on support model,
Delphi method, analytic hierarchy process

l. INTRODUCTION

Environment preservation and protection belong to the greatest challenges the world has
set for the next decade. The consequence is an increasing number of environmental
protection organisations, environmental standards, legislative regulations and limitations
forcing companies in production which impacts the environment as little as possible.
Furthermore, the consumers are becoming more ecologically conscious and are deciding on
purchasing products whose raw materials, production, usage and removal do not represent
too much impact on the environment. They are realising that certain products and their
manufacturing have a different effect on burdening the environment they live in. If in the
past, environment protection with a specific product was considered as merely something of
no consequence and an additional unnecessary expense, it is an important marketable
argument today, and a fundamental reason for the products to be sold at all in the future
(Oblak 1999).
Dealing with these problems are also wood manufacturing companies that otherwise
manufacture natura! material, but the process of production is environmentally controversial.
Therefore, the manufacturers of these products are facing a dilemma - which products to
produce in order to impact the environment as little as possible.
From the viewpoint of the wood industry the environment is impacted by a number of
factors: releasing of solid wastes (wood wastes, wastes of wood based materials, artificial
substances wastes, wastes from lacquering, etc.); burdening of air with smoke gases from
fire places and with waste air appearing in production, where wood dust, dissolvent and
diluter vapours in gluing and lacquering are arising; appearance ofwaste waters with organic
and inorganic parameters, etc. These parameters have various effects on the environmental
burden. W e namely distinguish severa! categories of impact, such as global warming,
stratospheric ozone depletion, photochemical smog formation, eutrophication, human
carcinogenicity, atmospheric acidification, aquatic toxicity, terrestrial toxicity, habitat
destruction, depletion of non-renewable resources, waste heat, malodorous air and water,
noise, etc. (Guinee et al. 2002; Rosselot and Allen 1999). Establishment and estimation of
the influences on the environment, respectively a comparison of ali these influences among

330

331

lhemselves due to every enumeration, therefore, presents an enorrnous problem that needs
dealing with. With the methods known up to the present day (ISO 4000) is possible to
estimate environmental burden for every category of impact separately (Guinee et aL 2002).
However,
is impossible to give an overall estimate from the results of each category
because they are incompatible. In presenting an overall estimate of environmental burden we
use different models (Oblak and Zadnik Stim 2000) based on methods of operations
research, especial!y methods of multi-criteria decision-making (Bohanec and Rajkovič 1995,
Zadnik Stim 2001, Bell et aL 2003). Therefore, this crucial problem of calculating the
overall estimation of environmental burden is separated into severa! sub-problems (Lai et al.
2002). These are then estimated separately according to all impact criteria they have an
affect on. For the overall estimation of environmental burden we use the multi-criteria
methods of the anaiytic hierarchy process (Saaty 1994, Winston 1994).
2. MODEL FORMULATIO N

comparisons we were establishing which of the compared criteria represented a bigger
burden on the environment and what the difference between the compared criteria of the
burden is. In defining the differences between the compared criteria of the burden we
derived from the assumption that there was no absolute environment suitability and from the
fact that the impacts on the environment are so complicated that it is impossible for us to
evaluate them correctly with the present knowledge of the consequences and the present
level of information. The differences between the compared criteria were therefore presented
by the experts on the basis of their expert knowledge of the criteria and the most important
and well-known characteristics of the studied criteria. A sample of the questionnaire for the
first level ofthe decision tree is shown in Figure 2.
Environmental burden
l.

For this reason we generated a multi-criteria mathematical model for classifying products
according to their environmental burden in the entire life cycle (LC) of product which
enables us a critical judgement ofproducts inside a certain organisation, ranking ofproduct
concepts from the point ofview of environmental burden and a comparison of environmental
burden caused by wood and substitution products.
In developing of the decision-making model, we considered all most important
parameters of environment burdening appearing in wood industry. In structuring the
decision-making tree we considered the mutual dependencies, textual connections and the
principles of perfection, operation, dismantlement, unredundancy and minimalism. The
decision-making problem was divided into a larger number of sub-problems according to the
principles of multi-criteria decision-making, which were then later divided into more levels
and sub-levels. A decision tree is presented in Figure 1.

individual phase oflife cycle ofproduct

Preliminary production of
substance ,materials and energy
(raw materials, ]ubricants. energy. water) average ( 1kg]

987654

2.

32

Preliminary production of
substance ,materials and energy

Emissions (solid wastes f0,25dm

3
],

emissions to water [51], to air 1125 m3], energetic
emissions)

2

3

4

5

6

7

Working conditions

8

9

(injurious substances,

noise, lighting, climate, vibrations, dust)

(raw materials, lubricants, energy, water) avernge !1kg]

987654

3.

32

Emissions (solid wastes [0,25dm J,
emissions to water [5 IJ, to air [125 m3 ], energetic
emissions)
3

987654

32

2

3

4

5

6

7

8

9

Working conditions (injurious substances,
noise, lighting, climate, vil.Jrations, dust)
2

3

4

5

6

7

8

9

Environmental burden In individual phase of life cycle of wood producti
Pr~Jrod.xlaid~e,n">!tcritls Wmesflll

l~~t_J
~

~m.'lt~

lwixidtn::lc~j
Cod.nQ:11

1OOoer~trl.!:lsl

E\:tlrkencrw,

~~ ~

~
WtJod p-eservd:icm

Ef/1

t~uir

l.scldWMtMI

t-mtol'l~cnj

!Wllt,rfc.h:Wry\

,~

jlfai;n,:pa~ns\.r[n-i:5sa'ISfrtrnp- Ocinal'1

Cetrm..r,.d~n 1

Wasl:Mdf\<iOdbM-Mtw.emls
lir~:ov-4;c-d~~nl

I

Figure 2. Sample of questionnaire for expert findings ofpair-wise comparisons

~

---,

f_e?!ZdJ-1"4~

1

10r~P!~ '·o/:=~=l!
~dPll~l1

ktl"~ 5\.Mo!nee WMtas

w.mesfrtl'llb;q.lCITQ\l

Figure l. Decision tree for wood product classification regarding their impacts on environment
On the basis of the structured decision-making tree we developed a questionnaire for each
of the decision-making levels, in which we pair-wise compared ali the criteria at an
individual level. It was determined which of the compared criteria presents a bigger impact
on the environment and what the difference between them was.
The questionnaire was distributed to experts licensed to perform opinions on environment
impacts, experts from the field of wood science and technology, ecology, environment
protection and safety at work. The expert findings were handed over from 48 experts from
university, Ministry of environment, research institutes, institutions for ecological research
and other organisations for studying and protection of the environment. In pair-wise

332

The instructions for completing the questionnaire were as follows: "In continuation
different criteria are being compared pair-wise. We are interested in which of the enumerated
criteria presents a bigger impact on the environment and what _the differe~ce in intensity of
the impact of a specific criterion is (according to ali categones of env1ronmental burden
which the criterion appears in) in comparison with the compared one, ifmeasured ona scale
from 1 to 9."
In presenting the pair-wise comparison estimations three possible groups of answers were
possible:
a) the two criteria have the same impact on the environment. This means there is no
difference between the intensity of the environment impact, meaning the ratio between
the two criteria is 1: 1 or the estimation is 1, situated in the middle of the assessment scale
shown in Figure 2;
b) the criterion on the left hand side of the assessment scale has a bigger impact on the
environment than the criterion on the right hand side ofthe assessment scale. This means
that the ratio of impact intensity of the left criterion against the right criterion is from 2: 1
to 9: 1 or the estimation is from 2 to 9 on the left side of the scale shown in Figure 2. The
estimation depends on how much bigger the impact intensity of the left criterion is.

333

cJ the criterion on the right hand side of the assessment scale has a bigger impacl on the
environment than the criterion on the ieft hand side of the assessment scale, This means
that the ratio of impact intensity of the left criterion against the right criterion is from 1:2
to l :9 or the estimation is from 2 to 9 on the right side of the scale shown in Figure 2,
The estimation depends on how much bigger the impact intensity of the right criterion is.
The gathering of expert estimations was carried out with the Delphi method (Pečjak 2001,
Ronde 2003; Chang et al. 2002; Handfield et al. 2002). In the first round, we collected ten
expert opinions and processed them statistically-mathematically. We entered the statistical
values into the questionnaire and repeated the round of gathering information on a larger
group of forty-eight experts. The acquired data was processed again and we filtered
disturbances,
After the expert opinion galhering was finished we calculated the mean values of the pairwise comparisons, Individual estimation subjectivity was made unbiased by using mean
estimation values of the impact intensity ratio for further calculations. For the measure of the
variable mean we used the median, which is the most representative mean measure for data
measured with an ordinal unit scale (Košmelj 200 The consistency for the calculated mean
values was verified.
On the basis of the expert opinion analysis, we used the AHP method and the Expert
Choice software to calculate and define utility functions in Figure shown as priority
factors written in brackets by parameters (L means local priority; G means globa! priority),
which we detem1ined the rules of movement on the decision-making lree with from the
lowest leve! all the way to the final estimation of the variant, i.e. the product (Lai et aL 2002,
Saaty 2003). Part ofthe decision-making tree is presented in Figure 3.
Erle ~~ Asseument 2JTll,hesize ~ - O ' ~

D c;;i: lil <!J
~

J

311

1 AEI(

~
1

[]).

=

m ".& l' l'!!l
§?'

1

'f.M!W

Iool:s tidP

§o

~ aro,w ~
'ilM

A •
1

1m

1

Piaiffilk•liilii§iiffiiDMM®NtfflidiShd•lilliiiiliffii¼•i•lihtiffi®'i
(L: ,590 G: ,590)
'

O Preliminarv productlon of substance, materials and energy
O Raw materlals (L: ,522 G: ,308)
, O Wood and celulose (L; ,070 G: ,02!)
, 0Glues (L: ,170 G: ,052)
, 0 Coatlngs (L: ,306 G: ,094)
, O Wood preservatlOflS (L: ,346 G: ,107)
, O other materials (L: ,108 G: ,033)
- O Lubrlcants (L: ,261 G: ,154)
41 Olls (L: ,500 G: ,077)
11111 Lubrlcants (L: ,500 G: ,077)
- 0 Energy (L: ,152 G: ,090)
II Electric energy (L: ,333 G: ,030)
, 0 Fuels (L: ,667 G: ,060)
- O Preparatlve of water (L: ,065 G: ,038)
• Water for industry (L: ,600 G: ,023)
II Communal water (L: ,400 G: ,015)
- OEmlsslons (L: ,291 G: ,291)
- O Solld wastes (L: ,136 G: ,040)
II Wood wastes (L: ,022 G: ,ODI)
II Wastes of wood based materlals (L: ,063 G: ,002)
8 Impregnated wood wastes (L: ,146 G: ,006)
9 Artlficial substance wastes (L: ,101 G: ,004)
II Wa,;tes from lacquerlng (L: ,344 G: ,014)
9 Wastes from glulng (L: ,250 G: ,010)
8 Wastes of ash containlng dangerous substances (L: ,074 G: ,003)

!Unoleum
Vinil co11ting
Wood

lr#Ofmofun Oocumenl

• () WA<tP w;,tprs (1 , .4A'.H~, .141)

Figure 3. Decision tree for product classification with recorded priority factors

334

,312
,574
,114

After calculatmg the priority factors we formed the fina! model for classifying wood
their impact on the environment during the whole life cycle of
products according
it is evident that the decisionthe presented hierarchy structure of Figure
product
making problem of evaluating environmental burden is first divided into three sub-problems.
These are further on divided into even smaller sub-problems, etc., ali the way to the lowest
criteria of the decision tree. The functions of passing over to a higher level are shown in the
Figure 3 shown in brackets by parameters), which are ascribed on
form ofpriority factors
each criterion ofthe decision tree,
3. CONCLUSION
The represented mathematical model for classifying products according to their
environmental burden in the entire wood product life cycle contains ali the most important
ciiteria of environmental burden present in wood industry itself. It is shaped and made
the principles of the multi-criteria decision-making, which enables easy
according
deducting and adding of new critelia into the model if the model tums out to be set too
widely or too narrowly. In the case of adding new criteria these are set at a definite leve! of
the decision-making tTee and we can estimate pair-wise comparisons between the entered
and the existing criteria at this level and by using the AHP method we can re-calculate
priority factors and c01Tect utility functions of the changed decision-making tree. Utility
functions at ali other levels need not be changed.
The model is shaped for the entry of numerical data, which are measured for these
compared products in the stage of the state of the studied variant life cycle inventory. In as
far these values for a certain criteria cannot be measured directly, the AHP method enables
us to enter the model in the shape of indirect comparisons and estimations as welL The AHP
method can namely operate also with descliptive variables and estimations.
The mathematical model for classifying wood products according to their environmental
burden in the entire life cycle, which is briefly presented here, enables not only calculating
the final estimations of the differences in environmental burden among the studied products
but also comparisons at all sub-levels of the decision-making tree and thus establishing
distinct environmental advantages and disadvantages of a ce1iain product compared to other
products included in the research. The methodology of the multi-criteria decision-making
also enables the sensitivity analysis of the final estimations to the changes of priority factors
at any level of decision-making. The presented model thus enables us a clitical judgement of
products inside a certain organisation, ranking of product concepts and a comparison of
wood products with substitute products. The model of classifying products according to their
environmental burden can serve as support in choosing among the altematives and deciding
on a product, which in its entire life cycle, not only in an individual stage, represent the
smallest impact on the envirornnent. The model can also be used as help in judging the
ecological quality of products.

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Bohanec, M. and Rajkovič, V. (1995), Multi-criteria Decision-making Models (in Slovene,
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335

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criteria to supplier assessment A study in the Application of the Analytical Hierarchy
Process, European Journal of Operational Research 4 70-87.
Košmelj, K. (2001 Applied Statistics (in Slovene. lranslated hv the authors). Ljubljana,
Biotechnical Faculty.
Lai, V. S .. Wong, B. K. and Cheung, W (2002), Group decision making in a multiple
software selection. European Journal of
criteria environment: A case using the AHP
Operational Research 13 7. 134-144.
Oblak, L. and Zadnik Stirn, L. (2000), The Possibility of Solving Economic and
Environmental Protection Problems in Wood Industry Companies by the Application of
the Method of Fuzzy Goa[ Programming. Ek6logia Bratislava 19( 4 ). 409-419.
Oblak, L. (1999), Ecological Information System to Support the Decision-making Process in
Manufacturing Firms, Proceedings of Slovenian lnformatics Conference, DSI 1999.
794-800.
Pečjak, V. (2001), Ways to New Ideas: Techniques of Creative Thinking, Ljubljana, New
moment.
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Germany and France, Technological Forecasting & Social Change
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Engineering, http://www.epa.gov/opptintr/greenengineering/chl 3 summary.html
Saaty, T. L. (2003), Decision-making with the AHP: Why is the principal eigenvector
necessary, European Joumal ofOperational Research 145, 85-91.
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2001, 452-460.

336

T

MULTI-ATTRIBUTE UTILITY THEORY IN SUSTAINABLE RURAL
LAND MANAGEMENT
Anka Lisec
lJniversity ofLjubljana, Faculty ofCivil and Geodetic Engineering
Jamova 2, Sl-1000 Ljubljana, Slovenia
e-mail: anka.lisec@fgg.uni-lj.si
Lidija Zadnik Stirn
University of Ljubljana, Biotechnical Faculty
Večna pot 83. SI-! 000 Ljubljana, Slovenia
e-mail: lidija.zadnik@bf.uni-lj.si

Abstract
In this paper the multi-attribute utility theory is discussed in the context of complex realistic decision
problems in the rural land planning and rural land management. The multi-attribute utility theory is
presented at a general leve!, where the main stages of model development are described. At the end,
the multi-attribute utility methodology is exposed as a sample case for supporting the decision
process when implementing land consolidation in the rural landscape, with specrnl emphas1s on
sustainable rural land management.

Keywords: rural land management, sustainable development, decision theory, multi-criteria decision
making, multi-attribute utility theory.

1 INTRODUCTION
Decision making is of impmiance in many fields in social and natura! sciences, including
rural land management and rural land p!anning. The types of decision problems in the rural
landscape (as for example the implementation of land consolidation) involve a large set of
feasible altematives, and multiple conflicting and incommensurate evaluation criteria with
the main aim of providing the decisions which are economically, environmentally, and
socially effective.
Rural land development and rural land management need to be based on balanced
information about multiple objectives and require the integration of interdisciplinary
knowledge from specialized fields, such as agriculture, biology, hydrology, economics, soil
science and management science. Therefore, the use ofmulti-objective models has become a
necessity to support and to guide constructive thinking about management decisions in rural
areas [7]. Over the years, severa! models of how individuals should make multi-criteria
choices have been developed. In general, the main role of multi-criteria analysis techniques
is to deal with the difficulties that human decision-makers have been shown to have in
handling large amounts of complex information in a consistent way. All approaches make
the options and their contributions to the different criteria, and ali require special techniques
of judgement. They differ however in how they combine the data and usually provide an
explicit weighting system for the different criteria. The methods can be used to identify the
single most preferred option, to rank options, or to distinguish acceptable from unacceptable
possibilities.
One of the approaches to the complex decision making process is to develop a model that
evaluates altematives giving an estimate oftheir utility for the decision-maker. Based on this
estimate, the options are ranked and/or the best one is identified [l]. In the presented paper,
the methodological framework ofthe multi-attribute utility theory (MAUT) is presented, and
a sample problem is discussed.

337

2 MULTI-ATTRIBUTE UTILITY THEORY (MALT): AN OVERVlEW
One of the major analytical too!s associated with the field of decision analysis is multiattribute utility theory, which derives from the work of von Neumann and Morgenstern
( Y47, In: [2J), and ofSavage (1Y54, In: [2]). These authors aimed to derive a theory ofhow
rational individuals ought to choose between alternatives. Their work provided powerful
lheoretical basis that is needed in order to make a quantification ofpreferences the process
of making decisions under uncertainty. Progress in the sense of developing methodology for
the practical usage was made in the seventies, when Keeney and Raiffa ( Y76, In:
developed a set of procedures, consistent with earlier normative foundations, which would
allow decision-makers to evaluate multi-criteria options.
Multi-attribute utility analysis of alternatives explicitly identifies the attributes that are
used to evaluate altematives, and helps to determine those altematives that perform well on
majority of the selected attributes. The basis for the MAUT is a utility theory, which is a
systematic approach to quantifying an individual's preferences. It is used to rescale a
numerical value on several attributes of interest onto a 0-1 scale where represents the worst
preference and 1 the best one. This allows the comparison of many diverse measures, which
is of significant importance in the multitude of relevant attributes. The end result is a rank
ordered evaluation ofalternatives that reflects the decision-makers' preferences [5].
The multi-attribute utility methodology for the evaluation of a set of altematives typically
consists ofthe following steps [5]:
Identification of alternatives and attributes.
Estimation ofthe performance of the altematives with respect to the attributes.
Development of utilities and weights for the attributes.
Evaluation of the altematives and sensitivity analysis.

2.1

Structuring the alternatives and attributes

Let as assume that there aren attributes X 1, . .. ,Xn whose values are denoted by x 1, .. • ,xn,
The alternatives (d) and attributes (X) form a matrix in which each row corresponds to an
attribute and each column describes the performance of the options against each criterion
(see Table 1). The individual performance assessments are often numerical, but may also be
expressed as a descriptive or other non-numerical valuable. Therefore, the information in the
basic matrix has to be converted into consistent numerical values.
The next step of the procedure generates (I) a single attribute utility function over each
attribute that is scaled from O to 1, a weight for each attribute, and (II) a multiple attribute
utility function derived from the single attribute utility functions and the weights. The
assessment of the single attribute utility function obtains possible ranges for the measures Xi
of each attribute Xi, scaling the measures by assigning preference value of O to the worst
level and 1 to the best level. Common forms of this function include concave for risk averse
behaviour, convex for risk seeking behaviour, linear for risk neutral behaviour, and 'S'
shaped for hybrid of the convex and concave functions [ 5). According to Clemen ( 1991, In:
[5]) one popular form for single attribute utility function u(x) is presented by the following
equation (1) where the quantities A, B, and RT are parameters that must be set by the
decision-maker.
u(x) = A-Be-(x!RT)
(1)
There are severa! assessment techniques for eliciting utility functions from decisionmakers. Several methods also exist for assigning weights (w) to the performance measures.
One of those is the Trade-off method where decision maker provides the preference ordering

338

the attrihutes. This procedure in conjunction with the constraint that the weights must
sum to one uniquely determines weights [6]:
(2)
2.2

Evaluating the decisions

To quantify the preferences of the decision-maker the utility u is assigned to each of
possible consequences (x 1, .. ,,xn), In the case when u(xi", .. ,,xn") > u(x,', .. ,,xn'), ~he
decision-maker prefers the consequences (x 1", ••• ,xn") of the alternative d". The aggregatton
of the seemingly disparate measures is based on utility theory, that is presented by Keeney
and Raiffa ( 1976, In [5]), and von Winterfeldt and Edwards (1986, In [ 5]). The use of utility
theory ensures that any recommendation reflects [5]:
the interactions between measures, if any,
the relative attractiveness of a specific level on a measure,
the relative attractiveness ofperformance on different measures.
When the decision-maker's preferences are consistent with the independence conditions,
a multi-attribute utility model u(x 1, ... ,xn) can be decomposed into an additive, multiplicative,
or other well structured forms that simplify assessment. An additive function of multiattribute utility model couid be presented as follows [5]:
(3)
u(x;,, .. ,x,,) =
w;11;(X;),
11

L
1~1

where xi represents the level of performance on measure i; Ui(Xi) is a single attribute utility
function over attribute x„ scaled from O to 1; w; is the weight for attribute x" and the we1ghts
w sum to 1
An example of the multiplicative model, which is based on a weaker independence
condition, is given as [51:
II
(4)
1 + ku(xi, ... , x,, =
[1 + kk;11;(xJ]

IT
i=I

where ki is a positive scaling constant satisfying O ::::;ki :C::: 1, and k is an additional scaling
constant that characterizes the interaction effect of different measures on preference. When
the sum ofk1 amounts to one, the multiplicative model (4) reduces to the additive model (3).
The advantage of using more complex and nonlinear functions is that more co~plex
preferential behaviours can be represented, although the assessment of such nonhnear
functions may require more tirne and possibly more cognitive burden [4].
3 SAMPLE PROBLEM

The main aim of land consolidation is on one hand the contribution of the productivity,
efficiency, and competitiveness of the agricultural sector, and on the othe~· the sustainable
development of the rural areas. The objectives of land consolidation are !o 1mprove the land
holdings of farmers by concentrating their farms in as few plots as poss1ble, and to ~upport
the farms with roads and infrastructure, when needed. It leads to better land use plannmg and
land management and if it is carried out in a comprehensive way, it should support
environmental protection and natural resource management.
By the implementation of land consolidation, the multi-functionality of the rural
landscape should not be overlooked. To a is:eat degre_e, the ~ral landsc_ape serves as
agricultural production area and on the other s1de as habitat for ~1ffere~t ammal an_d _P_l~nt
species. Furthermore, it is part of the cultural landscape and prov1des d1fferent poss1b1httes

339

for development of tourism, recreation, relaxation, and sport activities in the countryside.
Rural landscape with intensive agricultural production might cause visual and psychological
side effects (Figure l) and could hinder the additional activities. In the last decades,
sustainable land management is becoming increasingly important in the process of the rural
land consolidation, when a special attention to the conservation of natura! and semi-natural
ecosystems is being given [3],

ln the second stage, the altematives are identified and described by the values of the
chosen attributes. Each athibute is measured in different dimension and scales. Ali the
values of the attributes have to be converted to the numerical values and could be collected
in the performance matrix (Table l ). For the illustrative purposes, we assume that the
decision maker has to choose between four altematives (d) with respect to the sustainable
land management, where plots' amalgamation and plots' re-allotment consider:
d'
Conservation of over 75 % ofnatural ecosystems in the original place;
d" Conservation of over 75 % of natura! ecosystems, with the possibility of removing
some ofthem to new location or replacing with new ones.
d"' Conservation of 50 % of natura! ecosystem, with the possibility of removing some of
them to new location or replacing with new ones.
d"" No special attention
the natura! ecosystem conservation less than 50 % of such
ecological elements remain in the consolidated area (along the river bank, roads etc.).
-----

Attribute

d'

X,

x.

'
X2 '

Xn

Xn '

Xn

X1

Figure 1: The rural landscape with variegated composition (left), and of intensive agriculture production (right),

3.1

X1

X2

Land consolidation objectives and attributes

To address the problem of estimating land consolidation altematives in the selected rural
area, we adopted the multi-attribute utility theory (MAUT) approach of Kenney and Raiffa
(1976, In: [2]). Let us take a simple example of the multi-attribute utility model, where a
linear, risk neutral, single attribute utility function is assumed. Further, we will assume that
the decision-maker's preference structure is independent. Therefore, we may suppose a
separable preference structure which is in the form of additive functions
In the first stage of the multi-criteria decision analysis, the structure of multi-attribute
model for the evaluation of altematives has to be developed. The decision-maker must have
information on different attributes that characterize the selected criteria, 1.e. economics,
environment, social etc. (Figure 2).
CRITERIA

Attl'ibute

Table

-

Decisions/ AHernatives
d"
d'"
"

,,

,.

d ,,,,

XJ

X;

"~
X2

X2

Xn

,,,

1 ,,~

,,,.

,,,,
__ x-'l_ - - -

Performance matrix or consequence table,

As some of the attributes like social acceptance might be inherently inaccurate or difficult
to obtain, some vadations of the descriptions could be formed, anticipating either an
optimistic or pessimistic development. Bohanec and Rajkovič [1) suggested the solution for
taking into account such variability of the descriptive attributes with the increased number of
altematives (for example optimistic and pessimistic social acceptance) that provide a
foundation for the subsequent what-if analysis.
In the next stage, a single attribute utility function has to be assigned to each of the
attribute {Table 2), It is a scoring function that maps a performance measure from its range
of possible values to [O, 1], We assume that the decision maker is risk neutral and the linear
utility function for single attribute is given with (Anandalingam, 987, In: [7]):

x, agricultural production

u(x) = x-xworst

x2 ease of modem agricultural production

(5)

Xbest - Xworst

x;-1 tourism

-ENVIRONMENTAL }

Xi

Otherwise, according to (1), non-linear single attribute function would be assigned, where
pararneters A, B and RT have to be set by the decision-maker.

biological diversity

x;+1 aesthetics
_x,-1 wild life conditions

Xworst

Xbest

u(x)

X1

XJworst

XJbest

(X-XI wors1)/(x lbest-X 1worst)

X2

X2 worst

X2best

( X -X2wors1)/(X2best-X2worst)

...
...

...

X(n-1\ worst

Xin-1\best

(x-Xrn-1 lwors1)/(X1n-l )besrX/n-1 )worst)

Xn worst

Xnbest

( X -Xnworst)/( Xnbest-Xnworst)

; Attribute

...
...

Xn-t
Xn
Xn,1

preservation of cultural heritage

...
...

.. .

Table 2: Utility function for single attribute.

employment
Figure 2: Criteria and attributes for policy choice model.
Xn

340

341

Once the performance of each alternative on each attribute has been obtained, the next
step in the analysis is the aggregation of the seemingly disparate attributes and determination
of the desirability of each alternative. Assuming that decision-maker' s preference structure is
independent, an additive multiple attribute utility model can be used
In that case, the
overall evaluation of attribute X is defined as a weighted sum of its evaluation with respect
to its relevant value dimensions. The weight (w;) determines the impact of the alternative
policy on the measurable attributes (Xi), which are estimated by the decision-maker. The
attribute weights are normalized using the fact that the weights should sum to one (2).
The evaluation of the alternatives rank the alternative policies according to the value of
the multiple attribute utility function u(x 1,. •,xn), In the case that the decision maker prefers
the alternatives like d" > d' > d'" > d'"', the values of the utility function are ranked as
follows:
U(X1 ",, .. ,Xn ")

>

U(Xt ',., .,Xn ')

> u(x1 '", .. .,Xn '") > u(X1 "'", .. . ,Xn "").

(6)

The 8th lnternational Symposium on
Operational Research in Slovenla

S@R 'OS

CON CL USI ON

Rural landscape management has been identified as providing important non-market
benefits to society, which includes landscape preservation, environmental protection,
preservation of cultural heritage etc. Decisions in the rural land management involve
comparing alternatives that have strengths or weaknesses with regard to the multiple
objectives of interest. It is difficult to deri ve a measure for non-agriculture uses of the rural
landscape. Progress in this direction may be achieved also with the proposed multi-attribute
utility approach. Special challenges to implementing MAUT in the rural landscape
management are the valuations of the various externalities, and the determination of the
interactions between different measures.

REFERENCES
[1] Bohanec, M., Rajkovič, V., 1999. Multi-Attribute Decision Modelling: lndustrial
Applications ofDEX. In: Informatica, Number 23, Ljubljana, pp. 487-491.
[2] Dodgson, J., Spackman, M., Pearman, A., Phillips, L., 2005. Multi-criteria analysis
manual. Office ofDeputy Prime Minister, London, Internet source:
http://www.odpm. go v. uk/stellent/ groups/odpm_ contro 1/documents/ contentservertemp 1
ate/odpm_index.hcst?n=7096&1=2
[3] Lisec, A., Pintar, M., 2005. Conservation ofnatural ecosystems by land consolidation in
the rural landscape. In: Acta agriculturae Slovenica, Number 1, Volume 85, Ljubljana,
pp. 73-82.
[4] Malakooti, B., Subramanian, S., 1999. Generalized polynomial decomposable multiple
attribute utility functions for ranking and rating multiple criteria discrete alternatives.
In: Applied Mathematics and Computation, Number 106, pp. 69-102.
[5] Morrice, D.J., Butler, J., Mullarkey, P.W., 1998. An approach to ranking and selection
for multiple performance measures. In: Proceedings of the 1998 Winter Simulation
Conference.
http://www.informs-s im.org/wsc98papers/ 096.pdf
[6] Zadnik Stirn, L., 2002. Operacijske raziskave - Interno gradivo na podiplomskem študiju.
Biotechnical Faculty, Ljubljana, pp. 134-146.
[7] Zadnik Stirn, L., 2001. Analysing rural development projects under uncertainty. In:
Proceedings of the 6th International Symposium on Operational Research SOR'0l,
Slovenian Society Informatika, Ljubljana, pp. 313-318

342

Nova Gorica, SLOVENIA
September 28 30, 200S

SectionB

ON DISTRIBlTED SOLVING OF THE CAPACITATED VEHICLE
ROUTING PROBLEM WITH BRANCH-AND-CUT ALGORITHMS
Matko

Botinčan,

Goranka Nogo

Deµartment of Ylathematics, liniversity of Zagreb
mabotinc©math.hr. nogo©math.hr

Abstract. In this paper. we describe theoretical and implementation issues concerning solving
the capacitated vehide routing problem with hranch-and-cut algorithms. A distributed solver
for the capacitated veh1cle routing problem has been implemented within the branch-and-cut
framework SYMPHONY We present results obtained <luring experimental evaluation of the
solver on a cluster of computers and analyze its scalability.

Keywords: vehicie ronting problem. hranch-and-cut distrihuted computing, experimental
evaluatioll

1 Introduction

Vehicle routing µrolJlem \ VRP) is the name for a class of problems in which a
minimum-rnst set of routes far a fieet of deliverv vehicles originating and terminating at a <lepot must be determined to serve known customer demands. In most practical
applications. the VRP is enriched with additional particularities such as tirne windows,
backhauls. pickup and deliverv. etc However. in this paper we shall concentrate on
solving its most basic variaut in which 1w additional constraints are presented and every
vehicle has a limited capacity. This variant of the VRP is commonly referred as the
capacitated VRP (CVRP)
The µurpose of this paper is to present basic theoretical and implementation issues
concerning a branch-and-cut CVRP solver and report some of the experimental results
obtained during evaluation of the distributed version of the solver on a duster of computers. Particular umtribut
the paper is that sequellt ial and parallel runuing times
far one concrete CVRP instance are thoroughly analyzed aud related tu maJor implementation bottlenecks of a branch-and-cut algorithm.
The rest of the paper is organized as follows. In Section 2, we give a precise mathematical definition of the CVRP and its integer programming formulation. A theoretical
overview of branch-and-cut algorithms is given in Section 3. In Section 4 we describe
our implementation of the CVRP solver within the branch-and-cut framework SYMPHONY In Section 5. we present obtained experimental results and analyze scalability
of the solver. The final Section 6 gives concluding remarks.

2. Problem definition and its integer programming formulation
Let G = (V, E) be a complete graph with the vertex set V= {O, ... , n} and the set
of edges E. We assume that vertices 1, ... , n correspond to the customers, while the
vertex O corresponds to the depot. A nonnegative cost cij is associated with each edge
(i, j) E E representing the travel cost between vertices i and j. The cost structure is
assumed to be symmetric, i.e. for all i and j holds cij = Cji and cii = O. Each customer
i E {1, ... , n} is associated with a quantity d; 2 O of a single commodity which needs
to be delivered. A set of K identical vehicles, each with capacity C, is available at the
<lepot (it is assumed that d; :c:; C for all i E {1, ... , n} ). The objective of the CVRP

345

r

1

1

is tu ±ind a collection of K simple circuits ill the graph G with a minimum cost (each
( in uit corresponding lO a vehicle route) such that each customer is served b,- a single
vehide ruute. ead1 ,·eliide route leaves from aud returus to the
aml tlie s1m1 of
the demands of the customers Yisited
each vehicle route does not exceed giveu ,·ehicle
capacity C
From this description it is dear tliat the C'VR P subsumes several difiicult combinaLorial optimizatiun
as its special cases. For instance. if we take C = x: we get
an instance of the multiple traveling salesman problem. lf we additioually take K = l,
a TSP instance is obtained. On the other hand, one can obtain an instance of a bin
packing problem by putting Cij = O. It is worth uotiug that this BPP aspect makes the
CVRP a considerably more difficult combinatorial optimization problem tha11 the TSP.
In onler tu solve a gi\·eu C\"H,P instance with a branch-aud-cut algoritlnu. we first
need to formulate t.he µrnulem as an iuteger programmiug problem. Heme. we associate
an integer variahle Xe with each eclge e E F ancl ohtaiu the
integer programming
formulation
t he C\ RP

(6)

value 1s larger t ha11 t he value uf the best upper bound previously found, the subproblem
S is nut considered further. If the solution of the relaxation is a member of S, then it
is optimal and the subµroblem is solved. Otherwise a partition {S 1 , . . . , Sp} of the set
S is identified. where each S, represents a uew candidate subproblem. The algorithm
continues with selecting one of the candidate subproblems and processes it in the same
wav
, The bounding operation
the uranch-and-h01 md algorithm is often accomplished by
using linear programming techniques. Tvpically. the integrality constraints of an integer
programming forrnulat.ion of the problem are relaxed and the obtained LP relaxation is
sol ved wit h a LP sol ver
globallv valid ( usuallv faced-defining; inequalities are used
in the LP relaxatiu11. the resulting algorithm is calied branch-and-rut algorithm
For a more precise description. let us assume that the subprohlem S is described
by a set
iuequalities Ls such that S = {x , x is integral and aT x :S, /3 'v(a, /3) E Ls}
According tu tile classical result of Farkas. Wevl and Minkowski 4, there exists a finite
set .Cof inequalities snch that conv(Sj = {x i a"x :S, B. 'v(o /3) E .C}. The inequalities
in .C represent cuttmg plrrnes which will be used for strengthening the LP relaxations.
As the problem of finding a member of .C which is violated hv a given point in space has
the same computational complexitv as the original optirnization problem [71 it is not
possible to enumerar.e all of the ineqnalities
.C
µractice. lnstead. the.1· are dehned
implicitl,,· and separation algorithms have to be used to attempt to find at least some of
these inecmalities when i heY are --:iolated
Howe;er. separation aigorithms need not always succeed to find a vioiated member
of the set .C. If 110 inequalities from .C can he fonnd. the branch-and-cut algorithm is
forced to branch The branching operation is accomplished by specifying a set of planes
which will divide the suhproblern such that the current soiution will not he feasible for
anv of t he new subproblems. One simple method is to select an inequality aT x :S, /3
ha~ing iuteger coefficients on the left side and /3 r/:. Z on the right side, and then branch
by generating two subprohlems defined by 0' 7 x ~ l,/3 J and aT x ~ i/3l

(7)

4. lmplementati on issues

( lj

s.t.

L
L

'c/'i E V

Xe

2,

Xe

2I<

{O}

e={i,j }EE

(3)

e={O,j)EE

e

=
E

Xe

> 2b(S).

O <

Xe

< l

'c/e={iJ}EE .1

<

Xe

< 2.

've= {O. J} E E

Xe

E

'\:/S~ V

(4)

} E E.
j f. S

Z,

Ve E

E.

=/-0

(5)

Tbe quantity b(S; represents a lower bound on ,he 1mmber of rnhicles needed to
serve customers in the set S. Iu this paper we take au obvious lower bound b(S) =
iI:iES d;/ whereas the resulti11g family of facet-defining inequalities (4) is Lhe11 referred as capacity constraints.

Ci,

3. Branch-and-c ut algorithms
In this section we give a brief overview of a general class of methods for solving hard
combinatorial optimization problems which are encompassed under the term branch-andcut. Intuitively, every branch-and-cut algorithm consists of a classical branch-and-bou nd
approach strengthened with an additional use of LP relaxations and problem specific
cutting planes at every node of the enumeration tree.
Branch-and-bou nd algorithms employ divide-and-conq uer strategy for partitioning
the space of feasible solutions into smaller subproblems and solving each of the subproblems recursively. Let S denote the subproblem which is analyzed in a step of the
branch-and algorithm. Initially, S represents the set of all feasible solutions of a given
minimization problem. In its bounding phase, S is relaxed and the solution of the relaxation gives a lower bound for the value of an optimal solution to the subproblem. If its

346

To implement a C'VRP solver we have decided to use one of the existing open source
branch-and-cut software frameworks. We have chosen SYMPHONY [8] developed by
T. K. Ralphs and L. Ladany. which is a part of open source COmputational Infrastructure
for Operations Research COIN-OR [3]. The design goal of SYMPHONY isto provide an
easv-to-use framework within which distributed solvers for various types of optimization
pr;blems can be implemented independently of the underlying hardware architecture.
The internal library. which contains majority of the subroutines needed for an implementation of a distributed branch-and-cut algorithm (tree management, cutting planes
pool, interface to the LP solver, inter-process communication , etc.), interfaces with a
user through a well-documente d application programming interface. In order to implement a distributed solver, the user only needs to provide problem-specifi c methods and
connect them to the SYMPHONY's internal library.
SYMPHONY's functions are grouped into five independent computational modules.
This modular implementation allows easy and configurable parallelization. The modules
can be compiled as a single sequential code as well as separate processes intended for
running within the PVM distributed environment, hence, with a little additional effort, a
user can develop a sequential and a distributed version of the solver at the same tirne. As

347

we are interested in the analysis of scalabiliL,, issues. it is impurtant that the sequential
and parallel version of the solver are developed uuder the same framework Distributed
version of our CYRP solver was cornpiled i11 such a way that SYl\lPHONY.s master
module, tree manager and cutting planes pool constituted one master process. while
the cut generator and the linear programming module were buiit as a stand-aloue slave
process (seve ral of which ma,v run concurrent ly)
As it is not possible to give here a detailed descriptio11 of the implemematim1 of our
CVRP solver within SYMPHUNY's framework, we shall stress out only the most important remarks, Our implementation is based on the VRP sample application publicly
available asa part of SYMPHONY. However. it differs from it iu a number of ways, most
notably in the use of separation procedures. Namely. we have pro\·ided own implernentation of procedures for separation of the capacity constraiuts. Procedures are based on
connected components and shrinking heuristics given in [6] .:VIoreover. we have emploved
severa! separation procedures for other classes of facet-<lefiniug inequalities available in
the CVRPSEP package developed by J. Lysgaard :2] as well as those available in t he
CGL (Cut Generator Library) which is an integral part of COll\'-OR
Nevertheless.
far ease of presentation and possibility to cornpare the obtained results with those iu
'5], experiments in this paper emploved procedures for separatio11 of capacity coustraints
only.

prrn ,essors

(

2
3
4

5
6
7

.,

1I

II

~

8
9
lU
11
12

u
14

15
16

728.829
:366.152
285.473
219.%5
172.705
143.431
119.860
106,169
95.631
82.285
74 5.52
68.4:17
65 980
50.97:l
47.41:l
45.606

0.757
1506
1932
2.514
3.193
3.845
4.601
5.194
5.767
6.702
7.:)97
8 058
8.358
108Hl
11.631
12 0Y2

Parallel efficiency
0.757
0.753
0,644
0,628
0,639
0,641
0.657
0.649
0,641
0.670
0.672
rJ 672
0.643
077'.1
0.775
O 756

Table l Experirnental results obtained for P-n50-k7 CVRP instance

5. Experimental results
Experimental evaluation of our branch-and-cut C.TRP solver has bee11 performed on
the cluster lsabella located at the l'niversity Cornputing Centre, l'niversity of Zagreb. In
our experiments we have used 16 Liual-processor nodes Pyramid GX28 (AMD Opteron
248). Unfortunately, due to limitations on the size of this paper we are able w present
only a srnall part of the obtained experimental results,
Our mai11 imerest was comparison of the running tirne of the sequeutial versiun of
the solver (denoted as T0 ) with the running tiiue ofthe distrilmted ,·ersiun of the solver.
The distribmed vers10n uf the sol ver consists of one mast.er process aud p slave processes
r incorporating cm generator and LP sol ver), each process nmning ou its ow11 processor.
Its nmning tirne we denote with Tp. For each p E {L ... , 16} we cakulate the parallel
speedup, defined as the ratio T 0 /Tp, which describes the speed advantage of the parallel
algorithrn against the sequential one. vVe also calculate the parallel efficiency which is
defined as the ratio T 0 /(pTp) of the parallel speedup to the nurnber od processors.
As a first example, we present results obtained for the P-n50-k7 CVRP instance.
Running tirne of the sequential version of the sol ver on that CVRP instance was 551.472
seconds, Running times, parallel speedups and efficiencies of the distributed version of
the solver for a number of processors ranging frorn 1 to 16 are presented in Table l.
As we can see frorn the table, parallel efficiency is strictly less than one due to parallel
overhead, Parallel overhead generally has tree basic cornponents: idle tirne, redundant
work and comrnunication overhead. As argued in [5] and evidenced by a detailed analysis of our experimental results not shown here, cornrnunication overhead (tirne spent
for packing and unpacking rnessages) and redundant work (tirne spent performing a
work that would not have been perforrned in the sequential algorithm) is not an issue.
However, id]e time, which main contributors are rnmp-up tirne (tirne occurring at the
beginning of the algorithrn when there are not enough jobs available to employ ali processors) and handshaking tirne (time spent by the node processing rnodules waiting for

348

requesLS !U be serYiced h:, a1wther mudule
1 he t ree manager or the cutting planes pool
module) preseHLS a problem which cannot be significantl,· improved and th1s type of ove.rhead presents a ven challenging Lask fur irnplementors of branch-and-cut frarneworks lil
general.
,
.
,
As a secuud example. we show a graphical representat10n of runmng t1rnes of the
distributed versimi of the solver agaiust the number of processors for three CVRP 1~1stances: A-n53-k7. B-n51-k7 ancl P-n50-k7 These CVRP instances are rnoderate lil
size a.nd are choseu so that corresponding nrnni11g times would be approximately of the
same order of magnitude and hence could be drawn on the same graph. The graph is
presented in Figure l. Description of the analyzed CVRP instances can be found at [l].

6. Conclusions and future work
Branch-and-cut is an algorithrnic approach for solving hard combinatorial optimization problems which has becorne very popular in the last fifteen years, In this pap,er, we
have presented its theoretical aspects and described a distributed CVRP solver 1mplemented within the branch-and-cut frarnework SYMPHONY.
Although the performance of our irnplernentation is not cornparable to that of the
best state-of-the-art solvers which can solve CVRP instances of more than 100 nodes, the
resu]ts we have obtained are stili interesting. Our goal was not to compete with the best
state-of-the-art solvers, but to analyze scalability of our CVRP solver on moderate-size
CVRP instances. As there is a ]ack of detailed experirnental evaluation and perforrnance
analysis of distributed CVRP solvers in the literature, our results though present a
worthy contribution.
, ,
Moreover, we hope that with additional experirnents we will be able to analyze hm1tations of our implerneutation more precisely and improve its bottlenecks, Also, _a rnor,e
clever utilization of additional separation procedures as well as adaptive branchmg cnteria could improve the perforrnance of our CVRP solver on larger CVRP instances.

349

Rumng ume !s]
800 ·
III

---+-- A-n53-k7

600

*
400 ·

'\

B-n51-k7

SENSITIVITY ANALYSIS OF THE MINIMAL COST SOLUTION BY
AND/OR GRAPHS
DFšAN HVALIC' A

!

-1111-

P-n5 ·-k7

['.\I\ERS!T\ OF LJllRLJANA. FAU LTY UF ECONOMIC'S

1111

schednling, 's.\·st,ern nenousness· cau be addressed bv
appl::ing AND • OR graphs. specifically. bv performing sensitivity analysis of the
optimal solution suhgraph in the corresponding AND, OR graph. A suitable
redefinition of the cost of solution subgraph allows for ronsidering the cost
dimension of alternative solutions: we show how sensitivity analysis can be
performed in this context.
ABSTRA( 1

200

*
"*•

2

3

4

*

•·

5

6

..

*

*

7

8

9

~

...

* .

*

·*

11

12 13

~
-1/"

.±

14 15

____1_

16

N umber of processors

Figure l Running times for t hree C'VRP instauces with respect tu t he Humber of processors

Keywords: systern nerwJusuess. alternative process scheduling, AND; on grapL
analYsis
l

References

ff Capacitated Vehide Routing Problem Data Sets.
http://branchandcut org/ VRP / data
L21 J. Lysgaard. GVRPSEP package. http/;www.hha .dk; lys 1 C RPSEP htru

[3] COIN-OR weh page. http//www. r(;in

nrg

[4] G. L. Nemhauser and L. A. vVolsey. Integer and Combinatorial Optirnization John
Wiley & Sons, 1999.
[5] T. K Ralphs. Parallel Branch and Cut for Capacitated Vehicle Routing. Parallel
Computiug 29(5): 607 - 629, 2003.
[6] T. K Ralphs, L. Kopman, W. R. Pulleyblank and L. E. Trotter Jr. On the Capacitated Vehicle Routing Problem. Mathematical Programming Series B 94: 343
- 360, 2003.

[7] A. Schrijver. Theory of Linear and Integer Programming. John Wiley & Sons, 1986.
[8] SYMPHONY web page. http://brancha ndcut.org/SYM PHONY
[9] P. Toth and D. Vigo. The Vehicle Routing Problem. SIAM monographs on discrete
mathematics and applications, 2002.

350

INTRODCCT[ON

Iu the last years. uilernative process scheduliny has been steadily gaining importance.
since, among other things, it allows for integration of planning and manufacturing
, 2] As opposed to cornmon scheduling, here the choice to be made is not onh when
to perform every activity. but also which of available sets of activities is to be used.
As shown in '.6 .si. this approach can be effectivel,v modelled by AND/OR graphs.
In [6] it is also shown that using AND /OR graphs to model scheduling provides
a wa)· l o address the issue uf · systern neruousness, ! the fact that a relatively small
so that
cha11ge of t he e1 p:ironment can result in substantial change of the system c4,
that
shown
is
it
specific,
more
be
To
a cornplete schedule regeneration is neededJ.
the
interest,
primary
of
is
activities
of
duration
in the case when the minirnal total
bounds
the
of
solution,
optimal
given
any
for
determination,
(the
sensitivity analysis
within which it remains optimal) can be performed and thus help in resolving the
question, whether to reschedule or not.
Sometimes, on the other hand, the costs of alternative solutions are the most
important. Here we show that AND/OR graphs can still be used to model such
situations and that sensitivity analysis can be perfarmed in this case as well.

1:

2.

AND /OR

GRAPHS

In any directed graph, far any node x, r(x) will denote the set of child nodes of x
and far any are (x, y) its cost will be denoted by c(x, y). It will be assumed that
c(x. y) E IR*, where IR* = IR U { oo }.
An AND/OR graph is a directed graph C far which a partition of the set of its
nodes G = OUAUNa is given, where Na denotes the set ofthe tip nodes (the nodes

351

without child nodes): the uodes in O aud A are called OF?-nodes ami AND-nodes,
respectiveh AND- and OR-nodes differ by the way iu which the:, are established
an OR-node is establishecl when an:· of its child uodes is establishecL an AND-node is
established when all of its chilcl nodes are established. while a tip--uode is established
if and only if it belongs to some given set 8 C Ne: the nodes in c',' are called solved
nodes.
AT\JD/ ( lR graphs were intrnducecl to model problem reduction ~ the situation

D ca11 be solved hY soh·ing D 1 ur by solviug both B 2 aml B 3
can be represented by the AND/ UR graph in Figure l: a uode is established when
the problem, corresponding to that node. is solved . .\odes in Xc; represeut problems
which either can not be reduced to subproblems or can be sol ved without reduction
/ hence the term solved nodesJ. Node C is an AND-node; it is estal>lished when both
n2 aud B 3 are. T\Jode R au OR-node can be established by establishing a11y of

h can be shown i6.! that uader these assumptions a new relation P' C P exists
for whid1
Rp,:: RE = 0.
i.e„ 110 activity is in the range of both P' and E. but which, by (3), nevertheless
implies complete original information about precedence.
Clearlv such a project can be represented by an ANDiOR graph ~ activities iri
the range. of P' aud E are represeuted by A'-JD-nodes aud OR-nodes. respectively:
a node is established when the correspondi11g actiYity is finished. For instance, the
situatiou
[J

is triggered h, the liuish of B 1 or by t he iinish of both H2 and B3

can be represented by the A;-; D / OR graph iu Figure l
A weighted AI\D/OR graph is then obtaiued h~ settiug
c(x. YJ

B,. <,

,~.

~
(!y

ie

4.

(l)

It is tacitly understood that everv activity starts as soon as possihle, which implies
that A2 starts as soon as every A. such that AP A2 . is finished.
More flexibility can be gained by introducing yet another binary relation
the finish of A 1 enables the start of A2

80Ll'TlON SGBGRAPHS AND IHEJR COST

A solution subgmph of x E G is such a subgraph U C G that

3. ALTERNATIVE SOLGTIONS
A project is traditionally described asa set of activities, equipped with the precedence
relation

{c}

(2)

thus allowing alternative ways of accomplishing a goal. For instance, if the conditions
for the start of activity A 2 can be attained by performing any of A 1, A 3 then we have
A 1 EA 2 and A3EA2,
It will be tacitly understood that when introducing alternatives one is thorough,
i.e., when A 2 starts at least one of A, for which AEA 2, is finished. Hence the following
will apply:
'v'A (AEA 2 ==> EPA) ==> BPA 2
(3)

e x E G' and for every t E G' there is a path x .. .. t in G'.
e for every OR-node t E G' it contains exactly one are leaving t,
e for every AND-node t E G' it contains all arcs leaYing t.
• everv tip node t E G' is solved,
0

it contains no cycles.

Clearly every node of a solution subgraph is established; to establish x it suffices
to find any of its solution subgraphs.
.
.
.
When an AND /OR graph represents a project with poss1ble alternative reahzations then every solution subgraph corresponds to a feasible realization of the project;
the same apples to AND/OR graphs associated with job-shop problems.
In any AND/OR graph the cost oj the minimal solution subgmph can be defined
in severa! ways.
If it is given by the function w . G -+ ffi.*, satisfying
w(u)=

352

= d{y)

where d(y) denotes the duration of the activity represented by y.
.
.
Thus, AND/OR graphs provide a suitable environment for modellmg alternative
solutions; the areas where such approach wouid be advantageous, are manufactureto-order. concurrent engineering. desigu management, etc.

/~(\

Figure l

A 1 EA 2

4

h(u),
u E Ne
minvEr(u) {w(v)+c(u,v)}, uEO
{ maxvEr(u){w(v)+c(
u,v)}, uEA

353

(5)

Procedure L(x.yJ.
begin

where
hi. i = {

'UES

u E Ne; \ S

CX,,

t-

then when w!x. y) is equal to the duration of activity y. the cost of the minimal
solution subgraph is equal to the minimal tirne of the completion of the project.
If AND/OR graph is associated to a job-shop problem then any minimal solutiun
subgraph (in the sense (5)) corresponds to the solution of the problem
Here. we shall Jefine the cost of a solntion s1tbgraph 1'
w!P)

=

L

c(x.yJ

(6)

·x,y;EP

If d x. y) is equal to the cost of activity y. then the cost of a solution suhgraph equals
to the total cost of the corresponding realizatioH of the pruject and any minimal
solution subgraph gives the least expensive realization.
It should be noted that for the cost definecl hv (6J the u,11a,w,a,nJJ1a, ""'"'''"'
is much greater than for (5) - the problem of finding the minimal solution subgraph
with respect to (6) is N'P-harcl '.::J]. Of conrse. this is yet another reason speaking for
performing sensitivity analysis. as then scenarios can be made in advance that allow
for quick response to sudden changes.
5.

8ENS[T1VITY ANALYSIS

In this section we shall see. how - for any given optimal solution subgraph

to
determine for each are the bounds within which its weight must lie
all other weights
being constant - if the optimal solution subgraph is to remain optimaL
Our solution is algorithmic, i.e., we present algorithms that for each are (x. y)
determine the bounds of the interval '. 0:. 6 1 such that for
y! E
J] the minimal
solution subgraph remains unchanged.
We shall use the following notation c;x,yJ will clenote the subgraph of G. spairned
by the nodes that lie on any path P in G which stai·ts at t such that
o

(x. y) EP or

o P splits from P' at an AND-node and ( x. y) E P'.
Clearly, every solution subgraph of t which contains (x, y) must lie in G'(x )·
ll
,Y
By G(x,y) we shall clenote the subgraph of G obtainecl by removing (x. y). Furthermore, Srnin, S:Uin ancl s;:,in will denote minimal solution subgraphs of t in G, G(x l
and G'(x,y), respectively.
,Y,
Consider the following procedures:

(x. y; (/:. Srnin then
if 8:Uin exists then
n +X. y; - (u·1S:Urn1 end:
Procedure [ (x y)
begin

if

1 fif

cx,:

X.

if

½'
1 'iJ,
'-m,n1

y) E S'rnin then

s;:,lTI
/J

+-

exists then
+ (wrs;:,in)

r(x.y)

wiSrninlJ

end;
It turns out that the followiug applies
Proposition l. If Srnin is a rninimal solutfo11 subgraph uf I then far an.v m·r Ii· y
prncedures L and determine the lower and upper bound. respectively. of the interval
6] such that far r,(x. y) E [a. /J; - ali other weights being constant -- the minirnal
solution s11bgraph rernains unchanged.
Proof. To prove that L determines correctly. observe that when (x. y i E Srnin·
lowering
y) will not spoil the
of Srnin· while when (x. y) (/:. Srnin· the
snbgraph Srnin wiil cease to be minimal only if lowering c( x y) results
w !S:Uin) <
w(Srninl The correctness of [ follows from the facts that when (x. y) (/:. Srnin raising
dx.y) will not spoil the minimality of S'min· while when fx.y) E Srnin· the subgraph
S'rnm will cease to be minimal onl:, ifraising dxyi results in w(S;~in) < u(Srnin! III
6.

CO\/CLl SlON

We have shown how to conduct sensitivity analysis when AND/OR graphs are used
to model alternative realizations of the project ancl when the minimization of the cost
is in forns. In the description of our algorithms we have made use of an algorithm
for searching for a minimal solution subgraph. Unfortunately. we are still short of
an effective algorithm for this problem (however. it is not difficult to construct an
algorithm with exponential tirne complexity) As already mentioned, by performing
sensitivity analysis in aclvance this drawback can be. to a certain extent, avoided.
REFERENC ES

fll J. C. Bede, M. S. Fox, Constraint-directed techniques for scheduling alternative
activities, Artificial Intelligence 121 (2000), 211-250.

[2] G Erdos, T. Kis, P. Xirouchalcis, Modelling and evaluating product end-of-life
options, Int'l Journal of Production Research 9 (2001), 1203-1220.

354

355

CHINESE POSTMAN PROBLEM WITH PRIORITIES
3 G. Gallo. G. Longo. S. Pallottino, S. ~guyen. Directed H,vpergraphs and applicaticms. Discrete Applied l\.lathematics 42 (1993). 177-20
4 R. \Y. Grnbbstrom. O. Tang. ,\fodelling Rescheduliug Activities in a .\lulti-Period

Production-Inventory System. International Journal of Production Economics 68.
No 2 (2000). 123-135.

i5] D. Hvalica, L. Bogataj. Sensitivity results considering rescheduling

AND /OH
graphs. International Journal uf Production Economics 9::3-94. / 2005 ). 455-464.
D. Hvalica. AND/OR graphs aud project mauagemeut. Joumal of Computing
and Information Technology I CIT; 5. No :-i 1 1997) 193 .. 197.

;11 O Tang, R. W. Grubbstrom. Rescheduling Considerations for Productio11 Planning Csing Control Theory, in M. T. IIillery and II. J. Lewis, (Eds.). Proceedings
of the Fifteenth Ccmference of the lnternational Foundation for Production Research, Limerick, Ireland. Vol. l (1999) 447-450.

Tomaž Kramberger Janez Žerovnik'· 3
l 'ni, ersil\ ni Maribor. Faculty of Logistics. Mariborska 2, 3000 Celje, Slovenija
Maribor. Faeult1 of Mechanical Engineering, Smetanova 17. 2000 Maribor, Slovenija
l 1111·ers1t1
· 1\JF".1/DTCS. Jadranska 19. 1000 LJubiJmrn. SloveniJa
Einails• tomaz.kramberger@uni-mb.s1. Janez.zerovnik@imfm.uni-lj.si

Abstract. A generalizat1on of the Chmese Post man Pwblem 1s considered. tn which a linear orcler on a
1mportant nodes 1s g1ven ancl the task 1s to traverse ali eclges at least once 111 such a wav that the
set
higher pnorit 1 nodes are vis1ted as soon as possible.

Keywords: graph. eulerian tour. snow plowing, salt gritting

l. lntroduction
) be an undirected weighted graph. where V is the set of nodes, E 1s the set of
edges and u E -• R is a weight function assigning a positive cost of traversing edges. The
well-knuwn Cllinese Postman problem isto find the shortest postman tour traversrng each edge
of a graµh at least once. Severa] real-world problems. such as street sweeping, mati deltvery,
solicl waste rnllection, salt gritting and snow plowing cmi be modelled as Chinese postman
problems with some additional constraints.

Let c;(\ , /:

This article deals with optimal organization of scattering icy roads. To be carried out properly,
consideration both security and economical effects. Regarding security, the
we slwuld take
road net spots have priority From the economical poinl of view, ali
most e\posed and first
these roads have to be scattered one after another using the cheapest route. Here we assume that
the shortest rnute is also the cheapest.
Winter plougiling and scattering uf the road is of most irnportance and of great expense. If the
roads are not ploughed or slipperv roads are not scattered, participants in traffic are exposed to
great danger Weather conditions often cause traffic jams and have negative economic effect, at
Jeast causi ng great dissatisfact1011 in people. In several papers i see, for exarnple article [2], [8],
[71 or l 6] J this problem is considered using different formulations. Here we give a forma!
definiuon that 1s not equivalent to those m [2] and [Sl While it is natura] to model the problem
with a variant of the Chinese postman problem. the most appropriate definition of the cost
funct1on is not obvious. Here we define a multi obJective cost function where the most important
goal ts to minimize the total cost (time, mileage), and second, among ali Chinese postman
solutions we are looking for walks that will visit the priority nodes as soon as possible.

2. Problem definition
The problem considered bere is formally defined as follows:

356

357

C,i i en

a11

umli rected we1ghted graph G(V, E,u), and a (short) sequence of priority nodes

,1, , the oh,1ecli i'e is to find a walk, which is
1 ; the shortest walk, which traverses each eclge at leasl once, and
among the solutions ½hich satisfy i 1), find the walk that visits \ as soon as possihle
131 among the solutions wh1ch satisfy (2), find the walk that visits v 2 as soon as possih!e
i

I 21

1k1

among the solut1ons which satisfy lk-i ), find the walk that visits

1,

as soon as possihle

3, The algorithm
We propose a solution wh1ch is a combination of the well-known algorithms for minimum
matching, Fleurv' s algorithm for constructing an Eule1ian walk and the Dijkstra' s algorithm for
cDmputmg shurtest paths lsee, for example [ l, p,269-271] or [5, p,225-227]), In short the
algonthm proceeds as follows, First, as for the basic Chinese postman problem, the set of odd
nocles IS' 1dent1fied and shortest paths between pair of nodes are added to the original graph,
mtroducmg clouble edges 1f necessary (see, for example [9, p, l 32 J, or anv textbook in
combmatorial optimization [3, 4, 5)), The new graph will be denoted by G, , A~y polynomial
algorithm for minimum matcl11ng of an auxiliary graph can be used, In the second phase, an
eulerian walk on the new graph is constructed, First. a walk P, from depot to the first priority
node 1 is chosen according to the following rule: H is one of the walks from v0 10 i I which are

= c;,, - f\, Then, a walk
that G 2 = G 1 - P, remains

shortest uncler c:ondition that G. without P, is connectecl, Dellote G
P, is chosen such that it 1s a shortest walk from v, to

con11ectecl, lf v

v1

such

Let us ums1der node I Veh1cle should drive ali the roads and scatter them in the fastest
possihle way v1siting priority spots/arcs within the shortest tirne. The priority nodes are, in this
orcier: crnssing 1 ,, • crossing 1 ,. and nossing 1 _,
According to the 1dea of the algorithm presented in Section 2, we have to construct an Eulerian
graph 11.e. a graph with ali nocies uf even degreeJ, Tothe graph with odel degree nodes we add
new Lom1ect1011s, so that ali ocld degree nodes will ha\'e even clegree. We choose all the odd
degree nodes and form an auxiliary graph, which is a complete graph and the edge we1ghts
correspond to the distances in the original graph, In our exarnple. we hal'e s1x odel degree nodes
and the edge we1ghts i or. distance) matrix:
V3

Vz

V7

1 '

Vz

2

V3

,,

V7

L

V3

2

Vu

4

4

3

4 ! 2

v,,

i

1

2

.,

Vs

VIJ

v,,

2

4

'
4

1

'

-~

2

4

1

1--t-::;-i
L
.
2

-'
2

2

.,

1

1

2

Tahle I Matrix of distances between nodes of odd degree

1

has already been visited, then P2 is empty, ln this way, ali priorily nodes are

\ 1sited and finaliy, a walk back to v0 is constructed which covers all the edges in Gk

A rnin1murn matching has to be found, for exarnple using the algorithm for milllmurn perfect
4
match1ng wnh tirne complexll\ {)(n ) (see, for example [3, p,2361 or [ IUj1, In our case. the
opt1rnal solutiun 1s 1 _, - \ . , 1, - 1, 1 1 , - 1 1, which can be found quicklv e\'en hv mspecting
all poss1hle cases, Thus, we geta new graph G, , in which ali nodes have even degree

Usmg the construction, it is easy to pro\'e

Theorem. The algorithrn is optimal.

4. Example

Fig, 2: Eulerian graph.

Note that the new edge v1, - v 15 is in fact a shortest path v13 -v14 -v1.1 •

Fig, 1: Graph with six nodes of odd degree

Once we have the eulerian graph, we can find the eulerian walk with Fleury's algorithm. Recall
that in Fleury's algorithm we have a lot of freedom for the construction of the walk as long as

358

359

we do not cross bndges isee. for example [9. p. 26] In other words. we must not disconnect
the rema1~mg graph. _In our case we have priority nodes, therefore want
vi sit the priority
11ocles dS somi .is poss1ble. Howe\er. not any shortest path can be taken. We ha\e
find the
f' u· th e
path that 1s the shortest among the paths such that G.. = Go - p, is a connected gi·apli . T o .m
shortest paths. we can apph Diikstra·s
d, from node

11.

to the nocle 1·

to calculate the \alue

the node distances

Dijkstra's algorithm runs in tirne O( 11 2 ). In our example. we

•

get valuc of the clistances d, and other nocles. as seen in Fig. J.
F,g. 4: Shonest path from

Finali\

10 1·

to v:,

we repeat the procedure frorn the last priont\ node ,

between nodes

Fig. ~: Shortest path frnm v 1

19

I

5

\\

e find 1he shortest path

and , . The resulting graph is gl\en in Fig. 5.

9

To fincl the shortest path from v. to v9 , we start frorn the node v0 aml cunstruct the shortest
path backwards. Let d be the value uf the distance from the node to the
hetween
d

=

nocle

-u

1 'J

and observed neighbouring nodes.

are on the shortest paths from

Clearly.

11 u„ and distance
neighhours with value
Fig. 5 Shortest path from

1

19 .

\Vhen there are more nodes each is label!ed

ancl for each one the same procedure is repeated. In our case nodes
consiclered hecause d,

= 5- 2 = .'

and c/ 10

= 5-2 =

d

= .i - 1 = 2.

Procedure should be repeatecl for nocles

shortest path P between v, ancl v,J As ( , 0 -

Fi

1

2

and

node v,,

\\'e move

whole procedure. From th1s node we obsene only one node.

I

1

1

11

shuuld be

and repeat the

for which it !Jolds

ln this \, a) we have found a

and

is connectecl. we set G 1 = (/,

which satisfies the connectivity condition.

After repeating above procedure we get the shortest path from

1'

9

15

to v5

After we ha\e \lSlted ali priority nodes. the graph G is connected w:th exactly two nodes of
odel de gree.

and v. Fleurv·s algorithm can be used tu trace all the eclges slartmg frorn v5 ancl

encling in v
Surnmarizing. we have constructecl the walk as showecl in Fig. 6.

~ and continue.

As the next priurity node has not been visi teci yet. we are looking for a shortest path from
11 15

1

to v1, as in Fig. 4.

1-9

to

5. Conclusion
As ilustrated by the example, the algorithm g1ven in Section 3 provides a solution to the Chinese
postman problem which has additional property that it visits the first priority node as soon as
possible 1sirnilarly, it also visits the other priority nocles as soon as possible subject to the
previous conditions) For the cost function defined in Section 2, the algorithm provicles an
optimal solution. The proof is straightforward, but is omittecl here due to page limit.
However, it is perhaps more realistic not to insist on traversing the minimal length (i.e. the
Chinese postman tour), but rather to assign some cost to the edges traversed. On the other hand,
a cost can be assignecl to any delay in visiting the priority nodes. For various ratios between the
costs assigned, we woulcl in general get clifferent optimal solutions to the optimization problem.

360

361

Por e\ample. the solutions found here are \ alid for the case. where the cost of additional edges
lraverscd IS much b1gger that the cost of delays and delay for the first priority node is much
\arious cost
b1gger lirn11 the cost of delay for the second node. and so on. Mure detailed
functions ma: he an interestrng ropic for future research.

AN IMPROVED EVOLUTIONARY ALGORITHM FOR
SOLVING THE VEHICLE ROUTING PROBLEM
Krunoslav Puljič. Robert Manger
Department of Mathematics. University of Zagreb
Bijenička cesta 30. 10000 Zagreb. Croatia
E-mail: nuno@math.hr. manger@math.hr

Abstract. An improvecl evolut ionary algorithrn for solving the capacitatecl vehicle routing

problem is presented. The algorithm employs repeated mutations in a manner similar tu local
search. Experiments are described, where the algorithm has been evaluated ona well known
farnih- of benchmark problem instances.
capacitated vehicle routing problem. evolutionary algoritluns, experimental
evaluation. repeated mutations. local search.

Keywords:
Fig. 6: Eulcnan walk

l. Introduction

Ref'erences

11 E. W DiJkstra, A note on two problems in connection with graphs. '\umerische Mathematik
1 i 1959) 269-271
2)

G. Gh1arn, C.lmprota, An algonthm for the Hierarchical Chinese Postman Problem.
Operations Research Letters 26 (2000) 27-32.

3 1 B. Korte. Vygen, Combinatorial optimization Theory and
Heidelberg. New York 2002.

Springer. Berlin,

4.1 C H. Papaclimitrou, K. Ste1glitz, Combinatorial Optimization, Algorithms and
Dover puhlications inc .. New York 998.

5 1 S Skiena, lmplementing Discrete Mathematics; Cornbinatorics and Graph Theory with
Mathematica, Addison-Wesley. Redwood Cit:,. CA. 990. 225-227.
61 Sirno, L L1ster P.L Hart G.D .. Pearson H.B .. Thermal mapping: reliabilil_\ and repeatahility,
Mcteorol. Appl. 4 (1997) 131-137.
7) Th~rnes, J.E., _Thermal Mapping and road- weather information systems far highway

cngllleers. In Hlghway Meteorology. ed. By A. H. Perry and L. J. Symons, ( 1991) 39-67.
8) H.-F. Wang, Y.-P. Wen, Time-Constrained Chinese Postman Problems. Cornputers and

Mathematics with Applications 44 (2002) 375-387.
9) Wilson, R.J., Watkins,_ J.J., Graphs, An introductionary approach, John Wiley, New York,

1990. (Slovene translat1on: Uvod v teorijo grafov, DMFA Ljubljana 1997.)

1O) P. E. Black, "Munkres' assignment algorithm", from Dictionary of Algorithms and Data
Structures, ed., NIST. http://www.nist.gov/dads/HTMUmunkresAssignment.html

362

The uehtcle muting pmblem (VRP) is an interestiug combinatorial optirnization task,
whid1 deals w,th schednling a fleet nf vehicles to distribute goods between depots and
customers 11] The basic versior1 uf the VRP is called the capacdated VRP (CVRP)
Tt may be described bv using a complete directed graph. whose vertices correspoml
to the customers. and one additional vertex corresponds to the <lepot. A nonnegative
cost is assigned to each are, and it represents the travel cost spent to move between
the incident vertices. Each customer vertex is associated with a nonnegative demancl
to be delivered. A set of identical vehicles with equal capacities are available at the
<lepot. The CVRP consists of finding a collertion of elementarv f'.ycles in the graph with
minimum total cost of the involved arcs. such that each f'.vcle visits the <lepot vertex,
each customer vertex is visited by exacth· one cyde. and the sum of the demands of the
vertices visited bv a cycle does not extend the vehicle capacity. Obviously. the solution
to a CVRP instance specifies an optimal schedule for the vehicles delivering goods from
the depot to the customers, so that the demand of any customer is sat1sfied ancl no
vehicle is overloaded. Each cycle in the solution corresponds to a vehicle route.
Evolutionary algorithms (EAs) are a popular metaheuristic which tries to salve optimization problems by mimicking natural phenomena, such as reproduction, mutation,
survival of the fittest, etc [7]. An EA maintains a population of chromosomes where
each of them encodes a potential solution to a particular problem instance. The population is iteratively changed, thus giving a series of population versions which are called
generations. During one generation lifetime, new chromosomes are createcl ancl existing ones are altered by means of randomized "genetic" operators callecl crossovers and
mutations, respectively. A new generation is formed by choosing more fit chromosomes
from the previous generation, where the fitness measure is related to the objective function of the original optimization problem. It is expected that the best chromosome in
the last generation represents a near-optimum solution.
The VRP is known to be an NP-hard problem [8]. Thus it makes sense to consider
applications of metaheuristics, such as EAs, to the VRP. In recent years, there have

363

been manv attempts to solvP the \"Rl' hy EAs, The oht ained performa11ce results are
reported in L!:l:, General impressio11 is that a pure Avolutio11ary approad1 is not yet
eompetitive on the VRP compared to other metaheuristics, particularly tabu search
;11], It seems that the presently used chrornosomes aud geuetic operators are not able
to capture the full essence of the problem itseff Therefore, many authors have proposed
hybrid algorithms :2,3], where the perfonnance of an EA has been improved by replacing
its mutation operator bv a traditional local-search procedure [8],
The aim of this paper is to present and evaluate yet another EA for solving the
CVRP The same algorithm has already been introduced in our previous paper )O].
but now it has been improved and tested on a larger set of problem instances. Our
approach is purely evolutionary in the sense that ouly "geuetic" operators are used for
produciug or alteriug chromosomes. The 11ovelty withiu our approach is somethi11g we
call repeated mutations. Namely, genetic mutation operators are evaluated many times
in orcler to produce similar effects as local search in hybrid algorithms.
The paper is orgauized as follows. Section 2 reviews some huildiug blocks of EAs for
solving the CVRP, which have been used hv other authors. Section :l explaius how those
huilding blocks have been rearranged in a novel way to form our algorithm. Sectiou 4
reports on experiments where the algorithm has been implemented and tested ona well
known library of benchmark problem iustances. The final Section 5 gives a conclusion.

2. Chromosomes, crossovers and mutations
!n our algorithm, a chromosome is built as proposed iu .6]. thus it is a list of integers representing a permutation of customer vertices. This permutation is interpreted asa large
elementary cycle, which is obtained from a CVRP-instauce solution by concateuating
the vehicle routes aud by omitting visits to the <lepot. :-Jote that the same chromosome
can in fact represent many different solutions. Still. we use a u11i4ue decoding, which
is based on the greedy approach. Thus it is assurned that the tirst vehide visits as
many customers from the initial part of the chromosome as it is possible acrnr<liug to
the vehicle capacity. the second vehicle serves as many custoruers as possible frorn the
following part of the chromosome, etc. For instance, let the vehicle capacity be 20, and
suppose that we have 9 customers whose demands are in turn;

r

chromosome (wrapping around when the end of the list is reachecl). Far iustance, let.
the two parents and the two cut points be as follows:
p1 =

1

(12:3[5467189).

P2 = (45211876193).

Theu we can use the parents in both orders. thus obtaining the following two chilclren:
C1

= (2 l 8 5 4 6 7 9 3)'
1

1

C2

= (3 5 4 l 8 7 6 9 2) .
1

1

Iu our algurit.hm. we use three different mutation operators, called remove-andremsert (RAR\1). swap (S.\l) and im-ert (IM). as proposed in 6!. f9] aml [5], respectivelv. Generalk a mutation makes a small randorn change in a single chromosome.
thus creating its new version (mutant). Our operators start in the same w~y ~y randomlv choosing two positions within the chromosome. and then they proceed m cl1fferent
ways., l\iamely. llARM removes the vertex (yustomer) from the first p~s~tion am.l remserts it to the second position, SM swaps the vertices at the two pos1t1011s, wh1le IM
inverts the sequence of vertices between the two positions. For instance, let the startiug
chromosome be
p = (1 2 3 4 ,5 6 7 8 9)

Suppose that the randomly d10seu positions are 3 awl 7. Then the tliree operators
produce i11 turn the following three mutants:
'PRARM =

(124567389).

'PsM =

(127456389).

'P11v1

=

(127654389)

Each of the three mutation operators is fnrther on used in two variants called globa[ and
tacal. respectively. thus making altogether six variants denoted by RAR!VIG, RARML,
SMG. SML IMG and IML. In the global variant. ali positions within the r::hromosome
are considered. In the local variant. both the old and the new positions of the moved
customers should belong to the route of the same vehicle.

3. Algorithm description

It is assumed, but not explicitly written, that each route starts and ends in the <lepot.
In our algorithm, we use the crossover operator called order-crossover (OX) proposed in [9]. The operator creates a new chromosome (child) by combining parts from
two existing chromosomes (parents). First, two cut points are randomly selected, and
the chromosome part located between those cut points on the first parent is assigned
to the child. The remaining positions are then filled one at a tirne, starting after the
second cut point, by considering the customer vertices in order found on the second

The overall structure of our evolutionary algorithm follows the outline from Section l.
The initial population is created bv producing random_ perrr~utations accordin~ to the
uniform probability distribution. A series of generat10ns 1s produced by usmg the
crossover operator OX and six mutation variants RARMG, RAR~IL, ~MG, S~IL, IMG,
IML. Evaluation of chromosomes is done directly by using the obJect1ve funct1on of the
VRP, i.e. the fitness of a chromosome is equal to the total transportation cost of .the
corresponding solution, The algorithm stops when no further improvement is poss1ble
.
or when a prescribed tirne limit is reached.
As already mentioned, our algorithm uses mutations in a spec1al way called repeated
mutations. It means that mutation operators are never applied directly or separately.
lnstead, they are grouped within special procedures. Such a proce.dure considers some
(or all) possible mutations of a given chromosome, and finally apphes only the best one
among considered, i.e. the one that maximally increases the fitness of that chromosome.
.
A separate procedure is built far each of the six variants of mutation~.
If the number of considered mutations within a repeated-mutat10ns proce~ure _1s
small then those mutations are chosen randomly, and the whole procedure 1s st1ll

364

365

2, 4, 7. 5, 3. 5, 8, 6, l.
Then the chromosome
p

= (2 5 8 9 3 7 6 1 4)

is decoded by the greedy approach to the following three vehicle routes:

(2 5 8 9) (3 7 6) (1 4).

'

randomiz ed. However, in the extreme case when all possible nmtation s are considere
cL
the procedur e hecomes determin istic and very similar to a local search procedur e.
Now follows a detailed descripti on of the way how our algorithm transform s one
generatio n of chromoso mes into the next generatio u.
• Two "good" chromoso mes are chosen from the current populatio n,
using the
so-called to'Urnament selection
The chosen dll'omoso mes serve as parents for
crossover. A prescribe d number of children is generate d from the same parents
with different randomly generate d cut points in the OX operator. Only the most
fit child is retained, while the others are discarded .
• The only remainin g child is improved by repeated mutation s. First. the child is improved by applying for instance the RARMG procedur e: then the improved child
(mutant) is further improved by the same procedur e: then the further improved
child (mutant' s mutant) is even further improved again uy the same procedur e.
etc. vVhen there is no further improvem ent. we switch to the next procedur e. e.g.
RARML. . and so on until all procedur es are tried.
• The final improved child is inserted into the curreut popnlatio n so that replaces
a "similar" chromoso me. Two chromosornes being sirnilar means that relative
difference of their fitness measures is helow a prescribecl threshold . If there is
no similar chromoso me, then the improved child replaces a "bad" chromoso me
chosen by a form of tournam ent selection with elitfam
As we see, two consecuti ve generatio os differ

one chrornosome.

4. Experim ents
To enable experime nting, we have develope d a C++ impleme ntatiou of om algorithm
.
The irnpleme ntation consists of three C-t+ dasses. whose obje<'ts rnrrespon d to CVRP
instances , chromoso mes and populatio ns. respectiv ely Various compone nts of the
algorithm have been realized as rnethods of those classes. for instance crossovers
and
mutation s are methods of the chromoso me class. while tournam ent selection s are methods of the populatio n class. Repeated -mutatio n proceclures haYe heen irnplerne
nted
in a flexible way. so that they can consider either all possible mutation s of a
given
chromoso me, or only a specified number of randomly chosen mutation s.
By a series of prelimina ry tests. we have first determin ed acceptab le values for
some of the free paramete rs in our algorithm . At the same tirne. we have designed
an
additiona l procedur e that dynamica lly adjusts the remainin g paramete rs dependin
g on
the algorithm performa nce. Thus the populatio n size has been fixed to 30. Similarly
.
the percent of populatio n to be considere d by tourname nt selection of good and
bad
chromoso mes has been set to 70% and 15%, respectively. Next, the execution
tirne
limit has been chosen as 10 minutes, 45 minutes or 3 hours, dependin g on the problem
instance size. On the other hand, the number of mutation s considere d by a repeatedmutation s procedur e, the number of children of the same parents and the similarity
threshold have been left over for dynamic adjustme nt.
The impleme nted algorithm has been experime ntally evaluated on seven benchma rk
CVRP instances from the so-called Christofi des-Ming ozzi-Toth library. The whole
library is available at the on-line repositor y [4]. Table 1 gives some basic propertie
s

366

r
!

1

, C\'RP
instance

umber
custorner s

vehicles

50
75
100
150
199
120
100

5

C.\IT0l
C.\1T02
C:\ITo:3
Cl\IT04
CIVIT05
CMTll
', C.\1Tl2 !
1

1
1
1

~ .

_

".\nmber

_j_.,

Table 1:

10
8

12
16
9
10

Costofthe' - Costs of our solutions:
best known
soiution
Avg
Min Max StdDev
.521
521.7
528
521
2.2
848.0
830
830
859
9.5
841.7
815
817
860
13.5
1015
i 1124.6
1028 1198
49.2
l 1390.7 1349 1422 20.5
1289
1140.7 1034 1181
1034
50.0
8694
838
820
16.0
893
of expenme ntal results.

the cousidere d test examples . including the rnsts of their optimal solutions or best
solutions lmown so far. All seve11 instances
fact belong to a more restricted problem
1:alled the Eucl1dean s·ymmetr ic CVRP. where the vertices are associate d with points m
the plane. and the cost
an are is defined as the Eudidean distance between the two
involved points.
The results of experime nts are summari zed again in Table l. Each row presents
our solutions for one particula r problem instance. The presented values can easily
be compare d with the results obtained by other authors with other metaheur istics.
Since our algorithrn relies on ran<lom numbers, different nms with the same data
and
paramete rs can produce different solutions. Table 1 therefore presents statistica l values
compute d over exactly 10 runs.
As we can see from Table 1, our algorithm solves srnaller problem instances to optimalitv. The obtained best solutions to medium test examples are up to 2% worse
than the best from literature . The algorithm is slightly less successful in solving larger
instances . where the relative errors range frorn 2% to 5%. Our best result for CMT03,
being only 0.25% above optimum , is visualized in Figure 1 where the custorner
coordinates and the vehicle routes are shown.
During experime nting, we have also gained some rough impressions about computin
g
times required by our algorithm . On a PC with a 2.4 GHz Pentium processor
, the
solution is usually reached in 5-10 minutes. Smaller problem instances are often solved
faster, i.e. in less than l minute. For larger instances , it can happen that the algorithm
converges only after 1-2 hours.

5. Conclus ion
The presented results clearly indicate that our evolution ary algorithm is competit
ive
compare d to the other metaheur istics, and that our approach based on repeated
mutations can assure similar performa nce results as the approach based on local search.
Indeed on the chosen set of benchma rk problem instances , we have obtained nearlyoptimal' solutions at tolerable computa tional costs.
.
.
Our future plan is to further improve the algorithm by developm g a more soplusticated procedur e for dynamic adjustme nt of paramete rs. Also, we plan to try
more
appropri ate chromoso mes or better ways to decode a solution from a chromosome.

367

y
64

The 811, lnternational Symposium on
Operational Research in Slovenia

S@R 'OS
10

20

30

40

50

Figure 1 Our hest solution to C\IT(l:1

60

70

Nova Gorica, SLOVENIA
September 28 • 30, '2005

8 7).

Sectio n 9

References
l. Baker B.M .. Ayechew M.A. "A genetic algorithm for the vehicle routing problem". Computers and Operatwns Researrh. Vol ::lo (200:3). 787-800.
2. Beullens P. et al. "A guided local search heuristic for the capacitated are routing problem".
European Journai of Operational Research, Vol 147 (2003). 629-643.
3. Braysy O. et aL "A multi-start local search algorithm for the vehicle routing problem
with tirne windows". European Journal vf Operational Researr;h. Vol 159 (2004). 586-605.
4. Diaz B.D .. The VRP Web. Languages and Computation Sciences Department. 1-Jniversity
of Malaga, 2004. http./ /neo. 1cc uma es/radi-aeb/W ebVRP
5. Hwang H-S" "An improved model for vehicle routing problem with tirne constraint based
on genetic algorithm". Computers and Industrial Engineering. Vol 42 (2002). 361-369.
6. Larranga P. et al.. "Genetic algorithms for the travelling salesman problem: a review of
representatio ns and operators". Artificial Intelligence Review. Vol 13 (1999). 129-170.
7. Michalewicz Z„ Genetic Algorithms + Data Structures = Evolution Programs, Third Edition, Springer, New York. 1995.
8. Papadimitrio u C.H., Steiglitz K .. Combinatorial Optimization Algorithms and Complexity, Dover, Mineola New York, 1998.
9. Prins C., "A simple and effective evolutionary algorithm for the vehicle routing problem".
Computers and Operations Research, Vol 31 (2004), 1985-2002.
10. Puljic K., Manger R., "An evolutionary algorithm with repeated mutations for solving the
vehicle routing problem", in: Lužar-Stiffler, V., Hljuz Dobric, V. (editors), Proceedings
oj the 21th International Conference on Information Technology Interfaces - ITI 2005,
Cavtat, Croatia, June 20-23, 2005, University Computing Centre, Zagreb, 2005.
11. Toth P., Vigo D. (editors), The Vehicle Routing Problem, SIAM Monographs on Discrete
Mathematics and Applications, SIAM, Philadelphia, 2002.

368

r

A lPC-BASED DECISION SUPPORT SYSTEM FOR TAILORI NG OF
LOGS IN VENEERS PRODUCT ION
Anton Čižman and Marko Urh
(;niversity of Maribor. Faculty of organizational sciences
Kranj. Kidričeva cesta 55a
E-mail: anton.cizman@fov.uni-mb.s1, marko.urh@fov.uni-mb.si

Abstract
We report on decis10n supp01i system (DSS) that recommends solutions of specific cutting stock
veneers. A prototype DSS 1s developed for use in wood-process ing
problems from the production
crealmg or 1mproving ex1stmg tailonng of logs using
a1d them
enterprises a11d 1s des1gned
commercially available Operatmns Research software. The results of testing a typ1cal cutting-stock
out how such user-friendly DSS. wh1ch utilizes lmear and m1xed-mteger
problem are shown
mput matenal
programming, can reduce mventory costs and irnproves the explo1tat1on

lntrndudkm
Decision Support Systems (DSS) have emerged as the computer-based systern to assist
decision makers address semi struc!ured problems by allowing them to access and use <lata
and analytic models. Such systems have the following characteristics: thev are interact1ve
computer-based systems: they are aimed al semi-structured problems. They utilize models
with interna! and extemal databases. and they emphasize flexibility. effectiveness. and
adaptability. These characteristics have guided rnuch of the research in the DSS area, but the
potential benefits of DSS in the business environment have not been fully realized.
Although the definition of the DSS concept has been elusive L1, 2, 3 the field has
tlourished with the development of computer technology PC technologies are becoming
accepted and incorporated into organizalions and our personal lives. PC-based systems have
the potential improve both individual and organizational performance. As decision makers
recognize the potential henefits, many companies are investing in information technology.
PC-based systems have been generally hailed as a revolution that will change the nature of
professional work and transfo1m the way people work. It is expected that almost ali
knowledgeable workers are likely to have their own PC to perform both stand-alone tasks
and network serv1ces.
Despite the proliferation of microprocessor-based systems, the potential benefits of these
systems, as aids to managerial decision making, may not be fully realized due to poor design
and low acceptance by users. It is recognized that individuals are sometimes unwilling to use
these systems, even if the system may increase their productivity. While some of these
systems may have an impact on individuals and organizations, the adoption and acceptance
of these systems among decision-makers has been limited. This may be due to the
inflexibility in the systems as well as their narrow design. Therefore, it is important to
understand the environment of the decision makers and the type of support they need in
order to make effective decisions, and to examine the models appropriate for addressing their
problems [1, 5].
In this paper we describe a PC-based DSS which addresses the optimal cutting of logs in
the manufacturing of veneers. The development of DSS is proposed on pragmatic approach
[2, 3] that leads to more efficient and simpler use of such systems. The effectiveness of the
DSS is shown by relatively simple example which utilizes the pattem-oriented LP-based
methods for solving one-dimensional cutting stock problem. The results of testing show that

371

the application of such managerial DSS enhances proble
m solving capability for achieving
greater competiti veness of an organization.
In the first part of the paper, we shortly present a
description
major operations
management problems in the production of veneers focusi
ng on the issue of efficient log-cutting. The second part of the paper describes the formu
lation and development of a DSS
for this situation. Finally, we illustrate how the DSS
can be used to improve existing
decision making process using scientific approach.
2. Analyzing the decision-makiug process

Piima ry analysis of the decision-making process applie
d to the management of the
veneers production operations described in the paper
[3] reveals a number of decision-making situations. One of the relevant problems in this
regard, such as determination the
optimal log-cutting strategy and the quantity of logs to
be purchased, is related to optimal
exploitation ofinp ut material (logs). This situation is shown
Fig.
1

i

•·

-- 1

!.ogs kon

,

i

'

~

()pen customer
orders

JJ
,r

1'

li

r

i

J

· · - ·----,

11 for newlogs

~----,
: New venecr 1
' sheets to be
produced

logs to be purchased
logs needed to bc cut

S

cutting ar logs

Compose the shipment
to customer

P

p='. i=l

subject to
(2)

where
and integer,

Xjp ;c:

'

For cutting pattern

=

I, np; p

=

s

lip, P =

s

be valid:

·1
------ !__~

CaicuJate the optimal

11

min L,L,L pxjp

Purchase order ;

~-____,rD-e-ci-de_o_n:_,___ ___, ___ _ ]
•
•

be cut. A combination of required sub-logs lengths in
the log of len_gth Lp is ca_lled cutting
pattem. For each log length Lp, the numb~r lip (p = 1, 2,
... ,s) of all d1fferent cuttmg-pat~~ms
can be determined with regard to the part1cular sub-log
length !;. The element GiJp ofm~t11x 1!'
that describes each cutting pattem, represents the numbe
r of sub-logs of length l; obtame~ m
cutting pattem j related to log length ~P·. The n~mber oflog
s of length Lp to be cut accordmg
to cutting pattem j is denoted by a dec1s10n_vanable_X
jp.
In this problem situation, the objectlve 1s to mm1m1ze the
number of logs (Xjp) of l~ngth
Lp, that have to be cut according to thej-t h pattem. Thus,
our ld-C_SP represents a pme
integer programming (fP) problem that can be modeled
as follows.

~--- -~1
Veneer sheets
i

on stock

i

i=l

aiJp

c

O and integer,

j

= J,

i

"----r--~
-__j

Fig. 1. The decision-making process in the production of
veneers
The discussion is focused to the indicated initial decisi
on-making problem from Fig. l,
i.e. the optimal cutting of logs. The current decision-ma
king model in the production of
veneers is based on obtainable <lata samples about logs
and related Yeneer sheets that have
been produced in severa! pe1iods. lJp to the presen
t, the longitudiual tailoriug was
perfon ned by an intuitive "rule of thumb". Thereby, the
goal of the operations manager isto
establish a "reasonable minim um" of the edge-w
aste remainders of logs which,
unfortunately, does not take into account the end-waste
remainders of logs. This simple
approach to log cutting causes gradual accumulation
of an undesired inventory of veneer
sheets in the warehouse which significantly increases
inventory costs. Thus the idea is to
improve the existing decision-making process using scient
ific approach to minimize at the
same tirne both, the edge waste as well as the end waste
from cutting the logs.

The IP problem described above can be sol ved using .differ
ent LP-based method~ [\ 4] In
. r·
ost IP problems can be solved by the techmque ofBra nch-a
nd-Bo und 1.BBy or by
pt ac 1ce, m
f " [3 6]
using the LP-based Gilmore and Gomory's "delayed co I
umn genera 10n
, .

4. The system struct ure
B ed on work done with a case study from a Sloven
ian veneers factory, a simple
proto :~e DSS was developed for solving the specific cutt_in
g sto.c~ problem. The purpose of
the study was to check whether a simple-to-use manag
en~l dec1s1on support system can be
developed efficiently for users lacking detailed Operat
10ns Research (OR) knowledg~,
starting from available commercial Mathemahcal Progra
mmmg (MP) software. The DSS is
divided into three components: the user-interface, the model
mg base and the _database. T~e
integration ofthe three components ofthe DSS, i.e., the
structure ofour DSS, is presented m
Fig.2 .
OR praclioner / Planner

3. The problem-solving paradigm

The problem of optimal cutting of logs in veneer produ
ction is called one-dimensional
cutting stock proble m (ld-CS P). This problem consists
of determining the smallest number
oflog s ofleng th Lp. (P = 1, 2, ... , s) that are available in
sufficiently large quantity and have
to be cut in order to satisfy the required number b; sub-lo
gs ofleng ths !;, (i =], 2, ... , m).
The lengths of sub-lo gs l; are determined by the required
length of ordered veneer sheets. On
the other hand, the required numb er of sub-logs (b;,) can
be calculated from the required area
and thickness of ordered veneer sheets taking into accoun
t the diameter and quality of logs to

372

Solution
Technique
Base
(Solvers)

User-System interface
Windows-Based
Grafical Inlerface
Menu-driven

Model Base
(MP Models)

Database

Fig. 2: The structure of a decision support system

373

For the purpose of selection locally appropriate commercial MP software. the process of
setting up requirements for a proper SW tool was considered from the broad viewpoint ofthe
enterprise, the viewpoint of the OR expert and the pragmatic viewpoint of the user. If
compared on the basis of given requirements, a couple of MP software tools that are
available in Slovenia today have favorable features as candidates to be selected for further
development of a manager-friendly DSS. They appear to have an almost equal ranlc from the
viewpoint of the OR expert. Therefore, the choice of the basic DSS development software
may depend largely on the particular needs and circumstances of the enterprise where the
new DSS will be used.
A general argument for SAS/OR software selection was the fact that the SAS System
offers to an user a wider selection of software tools and technology, not only for MP [7, 8,
9]. Another important argument for having selected the SAS Systern is the possibility of
effective exploitation ofmodem technologies for effective developrnent and implementation
of end-user oriented DSS such as: Data Warehousing and On-Line Analytical Processing
(OLAP), object-oriented programming concepts (the SAS/AF software) with powerful
Screen Control Language (SCL), and interfaces that provide easily and timely access to data
from SAS-format and outside databases (the SAS/ ACCESS software ).

Table · Possible cutting patterns for a log oflength L, = 446 cm,j = 1, 2, ... , 12,p = 1
Cutting pattems U)
Pat. 1- xl
Pat. 2 x2
Pat. 3 x3
Pat. 4 - x4
Pat. 5 - x5
Pat. 6 - x6
Pat. 7 - x?
Pat. 8 - X8
Pat. 9 - x9
Pat. 1O - xl O
Pat. 11 - x11
Pat. 12 x12
Req.nb. of sublo~s

J 82-cm sublogs
1
2

15 l-cm sublogs

112-cm sublogs

o
o

o
o
o

o
o

1

o
o
o
o
o
o

o
o
o

350

250

2

o
o

1
2
2

2
3
400
r•:,,
};i/L~Y.

{;;\,., Gbbd, ()ptms -

0

He\>

.:] .ili!J~

~~

Edge waste (cm)
264
82
113
1
152
40
295
144
32
334
222
110

i,~,~,.,

LIHEM A IHTEGEII PflGRAHIIING Pll08lEJffl

S. DSS design, development and testing

J:'ITTH-

be used by the operations
The goal was to build an user-friendly screen-based DSS
manager for cutting of logs in the production of veneers, such that does not require any
special MP knowledge for operating, can be used through simple point-and-click commands,
has simple data-input and data-output procedures, can be operated from a standard personal
computer running the Microsoft Windows operating system, exploits as much as possible the
features and functions ofthe selected SAS/OR software and is flexible and easy to modify as
well as to maintain.
The standard SAS/OR graphical user interface provides a specific programming
language that was not considered appropriate for an operations manger. Therefore, further
development was focused around building a special window-based graphical user interface
(GUI) allowing easier use of typical MP functions. The graphical user interface was
developed specifically for this application by means of the SAS/AF software. With this
software, the developer was able quickly to prototype the basic functions and to verify the
functionality of implemented functions.

5.1. Test example
Let suppose that a customer orders three different quantities of veneers of equal
different sheets lengths: !1 = 182 cm and total area A 1 m 2 ; 12 = 151 cm
thickness and of three
2
2
2
and total area A m ; /3 = 112 cm and total area A 3 m • In this situation it is calculated that
1
length !2 and 400 sub-logs (b ) of
of
)
(b
sub-logs
the 350 sub-logs (b ) of length 11, 250
2
3
length of 446 cm is used. All
with
3
type
tree
of
logs
original
the
length [3 are needed if
possible cutting pattems that can be created from selected log of length 446 cm are
determined. Results ofthis step are given in Table l.
The operations manager has to input data into the DSS, according to the Table 1. Input
data relate to the computation objective (min or max), to the number of constraints and
decision variables and to the file name (existing or new). Fig. 3 and the upper-left/central
part of Fig. 4 illustrate how the test data can be fed into the new DSS by means of the two
data-input GUI screens.

OOJFIRI

l'_,;illlo,,:X,___ _

'-'I

cONSm

1.

3:

DECYM

j

12:

FILE 1W'E jCSPI

Fig. 3. Basic data-definition screen
After perfmming data input the presented test case brings the operations manager_to the
lt f th SAS "PROC LP"/IP optimization procedure. To get them, he has t_o chck on
;::u.. ;OoLVE~' button shown in Fig. 4. The results of the optimal solution. appear m the left~
bottom part of the same (main data inpuUoutput) screen. In th1s soluhon, the values o
of 446-cm logs that
· bles (Xl , x2, · ·, X12) practically represents the• number
· ·
d ecis10n
the
· T b 1 In th'
vana
is cas~,
, •.
a
m
given
pattems
cutting
12)
=
(11
have to be cut according to twelve 1
· · 11 r uired number of original logs to fulfill this customer order is 338. Opt10nal
::::~it~ :Jalysis is available by clicking either to the »R!'N<:_iEIPR~?E\ or to t~he
»RANGE/RHS« button. Recalling Fig. 4, the user can combme mput ata rom o er
. d
databases by clicking the ACCESS button.
The results of the optimal solution appear in the left-b~tt_om pa~ of the same (mam ata
in ut/out ut) screen. In this solution, the values of decision vanables. (Xl, X2, ... , x122
r~ctical re resent the number of 446-cm logs that ha':'e_to be cut a~cordmg to twelve_ (~11 · Tab . 1. In this case ' the mimmally reqmred
·
p
· y pattems
p12) cuttmg
r k'
·1 bl bof ongmal
. . number
m
given
a . e Y c 1c mg
a':'ai
IS
analys1s
sensitivity
lo s to fulfill this customer order is 338. Optional
Recallmg Fig. 4, the user
· gh t th RANGE/PRICE« or to the »RANGE/RHS« button.
h ACCESS b tton
. .
e1t er o e »
..
• h .
_t1
.
e
t
chckmg
by
can combine input data from other databases
mtmtive
e
t
to.
companson
m
DSS
new
the
using
of
benefits
the
discuss
us
Let
decision-makin used in the veneers manufacturing firm to date. The oper~tions manager
· t ·t· 1 gdecide to cut one original 446 cm log directly into the reqmred lengths of
· es to cut 400
Y 151 and 112 cm leaving an edge-waste o f 1 cm . Th'is reqmr
ive
mm
may
f 182
b 1
-i;;t :46-c~ logs and gives ; total edge waste of 4 m only and _a t~tal end w_aste of
m. The optimal solution result in total 338 original logs (OBJF m Fig. 4) reqmred for
31

~~

{s

374

375

cutting, where total edge waste is 44 98
m th at is much greater !hen 4 m, but total end
'
waste is zero.

r

CONJOINT ANALYSIS IN WOOD PRODlUCTS' DESIGN

i

Matjaž Feltrin, Zakraj la, 5283 Slap ob Idrijci, Slovenia, e-mail: matjaz feltrin@email.si
Lidija Zadnik Stirn, Jasna Hrovatin, University ofLjubljana, Biotechnical Faculty, 1000 Ljubljana,
Slovenia, e-mail: lidija.zadnik@bf.uni-lj.si; jasna.hrovatin@bf.uni-lj.si

~

1

INTEGffi

fNTEGcR

Operations research tools are widely used in wood products· market research which highly affects the
products' design. For successful wood products' design different tools can be applied, In this paper
Conjoint Analysis (CA) is used for detennining the benefits of the attributes and their levels which
best correspond to anaiyzed wood products or services. CA application is also employed in
evaluating already market placed wood products where redesigning or repositioning of products
should be established.

INTECER

scm.us
st.m.us
st.m.us

Key words: operations research, wood products' design, market research, conjoint analysis (CA)

J
j _TYPE
~X11VAA

fNTEGER

II

II

~~:

.

Abstract

INTEGER
INTEGER

72

X7

INTEGffi

SU\CI<

X8

JIHEGER

OOJECT

X9

INTECER

X10

INTEGER

VALl.lf

mEOER

338

J

Fig. 4. Main data lJO screen: input test data and the results of optimal solution

6. Conclusion
. l
The significance of this paper is primaril
~ prlactica. The proposed approach represents
t
an affordable alternative for a firm to
h.
rt'
ge a s1mp e-to-use DSS fu
pa icu 1ar, t is approach
. .
n_iay represent a convenient opportunit for small
. . . anhd medrnm-s1ze enterpnses. Another
s1gnificant contribution of this case st dy
f DSS d
u Y is m nsmg t e awar
evelopers and
,
eness O
f
·
f
need
real
the
about
OR researchers
OR users. These managers can con:r~bu~:~r:/o: ~anagers as a special class of DSS and
·1 bl ITY uch . mipro\ mg the competit1veness of
their finns when they have read1'ly
f · l
support m the fo
avai a e
• • .
d Th h nn o s1mp e-to-use DSSs.
of F Id
findings
recent
Tlus IS m accordance with
. ac enkary (2001) who urge the
OR community to empower front-lin d .. ou s :n
e eciswn ma ers m fiims by means of end-user
focused DSS.

Introduction
The problem we are dealing with is to analyze the main attributes and their levels of three
different furniture elements (living roorn table, dining room table and a chair), and to
evaluate, by presented attributes, three different chairs, one dining room table and one living
room table, which participated at an exhibition (only dinning room chair is displayed in this
paper to show the evaluation method). The method used for the analysis and evaluation is
conjoint analysis (CA). CA is widely used as market research tool for developing and
designing ofnew or redesigned products or services (Green and Srinivasan, 1978; Dahan and
Hauser, 2000; Wang et al., 2004). Using CA, we can answer questions such as: which
product's attributes are important or unimportant to the consumer, what levels of product's
attributes are the most or the least desirable in the consumer's mind, how large is the market
share of leading competitors· products versus our product, and many other market focused
issues (Green and Srinivasan, 1978; Gustafson et aL, 1999; Zadnik, 200 ). Its main
advantage is that it asks the respondem to make choices in the same way as the consumer is
when consuming (Green and Srinivasan, 1978). Attributes and levels to evaluate wood
products, used in the presented research, are given in Table
Table

References
l. Basnet, C. Foulds L Igba · M 1996
• Fleet Manager: a microcomputer-based
na, .. ,
' ·,
'
d . .
195-207
2 ~~:~~~up~ort?~tembfor vehicle routing, Decision Support Systems 16
··
·
1'
Y, ., a 1en erger, L. and Beheshstian M 1996 A
·
srna I busmess mventory DSS: design,
•.
~evelopment and implementation issues Com t ' & o·,
63-72
(!)
23
Research
perahons
ers
pu
·
'
.
.,
3 Cižman A Č
.
sim le-;o- ., emet1c, J., 2004. Improvmg competitiveness in veneers production b' a
y
use DSS, European Joumal ofOperational Research 156 241 -260
D
.
'
bi
4. yc ioff, H., 1990. A typology of cutting and ackin
g pro ems, European Journal of
P
Operational Research 44, 145 _ 159 _
5. Foulds L. and Thachenkary c 2001 E
systems and IT can aid the OR analy;t ~=~ ;o ~he P;ople - how decision support
0
ay, une 2 001, p. 3. (Downloadable
/
from website http·//www lionhrtp b 'm1
u .co orms orms-6-01/foulds html)
·
·
6 G'll
.
.
.
I more, P.C. and Gomory, R.E., 1961. A linear ro rammin
.
g approach to the cuttmg stock problem,
p g
Operations research, 9, pp. 849 _ 859 _
User's Guide: Mathematical Programming, Version 8, SAS Institute
7.

t

::~,~::,:;s/oR

A J' ·
8. SAS, 1995. SAS/AF Software F
1cat10n Development Concepts, Version 6,
First Edition, SAS Institute Inc r~me
.
., ary,
9 SAS 1994
. The SAS System for Information Delivery-Scientific and Technical Applications, SAS Institute
. Inc.'

J~

376

Analyzed attributes and their levels
1 not innovative
Innovativeness L..:::cmc.:e..:cd:c.iu.c:m.:.1cc·nccn_ov....:acctic:.v.:.e_ __
high innovative
limited functionality
Functionality
functional

Aesthetics
Safety

The present research includes personal interview with four respondents of different working
areas. The results demonst:rate which attributes and which levels must be of highest
consideration when designing the treated wood products. Further, analyzing different
products, the objective isto focus on repositioning ofwood products, demonstrating which
attributes and which their levels were not considered sufficiently when designing.

Methods and data
The first part of the paper includes three furniture elements (living room table, dining room
table and a chair) analyzed by CA, regarding respondents' preferences to attributes presented
in Table 1. Further, the second part of the research is focused on five furniture elements

377

which competed on Biennale ofin dust rial
design (BIO) in Ljubljana in the year 2004
, where
redesigning considering analyzed attrib
utes was our interest As noted, only the
analysis of
dinning room chair is shown in this pape
r.
Method used to analyze the data collected
from respondents included in this resea
rch is
CA, performed by SPSS program. The
name "Conjoint Analysis" implies the
study of the
joint effects and as a multivariate techn
ique has its applications in wood prod
ucts' design
where the decisions are studied (Green
and Srinivasan, 978; Gustafson et aL,
200 l ). Benefit
of using CA rather than any other exist
ing method lies in the methods ability
to incorporate
ali relevant aspects of the decision mak
ing process, what means that the decis
ion maker is
better able to asses the importance of attrib
utes as well as the levels of attributes (Gre
en and
Srinivasan, 1978). Marketing managers,
as well as decision makers, are faced with
numerous
difficult tasks when developing new wood
products or modifying existing products.
These
tasks include prediction of:
- the profitability and/or market share
for proposed new product,
the impact of new competitors' products
on profit,
- customer·s acceptability ofa new prod
uct (Green et al.. 986),
- competitive reaction to sellers' strat
egies when introducing a new product,
- the impact of situational variables on
customer preference,
response to alternative advertising strat
egies,
the customers' response to alternative prici
ng strategies (Green and Srinivasan, 978)
.
CA used for evaluation of competitive
strategy of a new wood product may be
implemented
by modifying the marketing mix, new
product identification, pricing, advertisin
g and
distribution (Green and Srinivasan, 1978
; Green et aL, 1986; Gustafson et ai.,
999; Haaijer,
999; Dahan and Hauser, 2000; Gustafson
et al., 2001; Wang et al., 2004). Such
strategies
may focus on new segments or products'
re-positioning, what is considered in this
paper.
Further, CA involves the measurement
of psychological judgments, such as
consumer
preferences, or perceived similarities or
differences between choice alternatives.
The main
methodologies which model CA are:
Stimulus Construction: Two Factors
at a Time; Full Factorial and Fractiona
l Factorial
Design,
- Data Collection:
Two Factors at a Time, Tradeoff Analysis;
Full Profile Concept
- Model Type:
Compensatory and Non-Compensatory
Models, Part Worth
Function; Vector Model; Mixed Model;
Idea
l Point Model,
Measurement Scale: Rating Scale; Paire
d Comparisons; Constant Sum; Rank Orde
r,
- Estimation
Procedure: Metric and Non-Metric Regr
ession; MONANOVA;
PREFMAP; LINMAP; Non-metric Trad
eoff; Multiple Regression;
LOGIT; PROBIT; Hybrid; TOBIT; Disc
rete Choice,
- Simulation Analysis: Maximum Utili
ty; Average Utility; LOGIT; PRO BIT
(Green and
Srinivasan, 1978; Gustafson et al., 1999
; Gustafson et aL, 200
Severa! different implementations of
conjoint measurement are manifested
in CA
algorithms and computer programs.
The most noticeable are: categorica
l conjoint
measurement, monotone ANOV A mod
els, OLS regression methods and linear
programming
methods (Green and Srinivasan, 1978;
Green et al., 1986; Haaijer, 1999; Gust
afson et al.,
2001). For purposes ofth is paper, only
OLS regression methods are discussed.
The Ordinary
Least Squares (OLS) regression appr
oach to CA offers a simple method
of deriving
alternative forms of respondent utilities
(part-worth, vector, or ideal point mod
els). The
attractiveness of the OLS model is partl
y a result of the ability to scale responde
nt choices
using rating scales, rather than rankings.
Based on this rated input, the CA meas
urement
procedures are applied to identify a
mathematical function of attributes.
Further, the
objective of OLS CA is to produce a
set of additive part-worth utilities that
identify each

378

enfs preference for each leve! of a set
of product' s attributes (Green and Srini
respon d
vasan,
d
l 978; Gustafson et al., 200 This mod 1 .
e IS expresse as.•
r.
J

= bo +bix/,J +b2X2.1 +b3x3.J +

(1)

+bmxm,J + 6)

where:
,
. )
r - observed evaluation of profile J. (resp
ondent s ratmg ,
)

.. ,p; (p-th attribute)

b0 - constant term
• •
g the effect of attribute leve! vana
b,
b _ model parameters quan t'fy'
tion on profile
I m
1
4
evaluations
.
f
fi1 ·
- variables indicating the combination of
attnbute level o pro ' e J
X1.;, ... ,xm,J
6 _ exogenous stochastic distributions
.
"
,,
al
l~ CA contribution ?f attribute' s leve!
0
to t;e ~ : ~ ut~~y
t~=
~~~ :~~: n!~~ :ttth e
utility of a profile is equal to the s~m
1
o a . p
function s, is defined as (Haa1Jer, 1999
).

~~~~'.'~~

s,

(2)

= "f,fp Y;p
p=i

w~r e:

.
s, - preference for the stimulus object at
leve 1l,
.
h f different levels of the stimulus obJe
f P - function representing the part worth
cts
of eac o
for the p-th attribute,
- leve] ofth e p-th attribute for the stimulus .
obJect at level
.
'
;~e estimation of utilities for each attrib
ute p;~m.it~ the e~~ ~t~: a~~ ra~~ ;~;:
attnb
ute
s
importance AP which is computed as (Gre
en an nmvasan
,
'
. )
(max,P-mm;p
(3)
A= "\'
P
,L,(max;p-mm. ,P )
where:
A _ importance of attribute P
P,·n m lp Iowest utility of analyzed attribute p
max - highest utility of analyzed attrib
ute p
,p

In our research 4 respo ndents were identified to answer questions about three produ~ts
.
nts The CA process started w1th
(designed as profiles) and five s~lected
1
furm ~t:: s e;~u s~d on the related prob
lems of
stimulus construction as the bas1s for
ChA.
d t (Gustafson et al. 2001). In the first
.
.
h'
h
tt
'b
t
s
to
prese
nt
to
t
determmmg w ,c a n u e
e respon en
'
. . "
uct cons1'd ere d, was described by profiles contamm
g iour
part of the ~esearc h ' e ach prod
d . 1,
.
level
s
(Tab
le
1).
The
num
ber of selecte st1m
attributes w1th three, respectlvely two
u'
d
d on the type of the model and must be
(profiles), to be presented to the re~pon~e
nts, rt:e ::al Regarding the fact that,
if using the
carefuily constructed, so that the stimuli
are o
t t~ evaluate all 36 (3x2x3x2) profiles,
fuil profile, it is an unmanageable for
the resp~~ tnus e fractional factorial statis
tical design
we have chosen th~ additive rule. So w~%
~~;a b~e ~). It is from this reduced ~et
of pr~files
to reduce 36 poss1ble profiles to. 9 pr~
..
. t d with each of the indiv1dual attnb
that we estimated the set of cho1ce utiht
ute
ies asst ct: e uestionnaire which was hand
ed to the
and their associated levels. Table 2 als?
presen s e \ rmed by personal inter
view with
respon dents . Further, the CA expenment was per o

J

379

respondents. They were also presented with a fold-out form that had 9 response categories,
The rated profile numbers were
where means the best and 9 the worst category (Table
program.
SPSS
in
utility
of
used for the evaluation
In the second part of the research five products, which competed at an exhibition, were
analyzed. The objective was to evaluate the products by presented attributes and their !evels
their
(Table l). First, respondents were asked to evaluate five furniture elements using
with
elements
fumiture
of
descriptions
the
for
used
further
were
results
These
pictures.
attributes. The aim was to illustrate how well the already made products correspond to the
consumers' preferences derived from the previous (first) part of the research. The
questionnaire designed in the second part ofthe research is given in Table 3.

Table 4: Respondems· evaluation and derived utility for dining room chair
~····-~----

Evaluation

i Profile
1

i

#1
#2
#3
#4

1

!
1

i

:
l

#5
#6
#7
#8
#9

Innovativeness
High innovative
High innovative
Medium
innovative
Medium
innovative
Medium
innovative
Low innovative
Low innovative
High innovative
Low innovative

1

1

Limited functionality i

Aesthetics
High level
Low level
High leve]

1

1

i

Functional

i

Low leve!

High leve!
Limited functionality
Low leve!
Limited functionality [
Limited functionalit:y l Medium ievel
Medium leve!
Functional

!nnovativeness

( rate l to 9)

1

l

1
1

i

Leve!
Low innovative (l)
Medium innovative (2
Hi h innovative 3)

1

1

i

l

l

lnappropriate

1

I nappropriate
Inappropriate
lnappropriate
Appropriate

1

:
i

Evaluation- sign

(x)

6,3
7,5
9,3
6,2

1 8,0
1 5,2
2,5
7,2
1

(r1 )

#5
5
4
6
5

i

2,7
2,7
3,8
2,7

7,3
7,3
7,5
7,7

1

l lnnovativeness , Functionality

1
1

lnappropriate

1

#4
7
8
7
8

#6
8
6
4
6

4,7

3
3

Part worth
max
min

l

5
5
2

1

l
1

1

1
1
1

Aesthetics
0,5
0,5
0,2

#8
6
7
5
7

#9
1
1
3
1

3,7
1,5
1,3 1 2,8
0,8 ! 4,2
l,7 i 3,2

8,7
9,5
7,0
9,2

#7
9
9
9
9

{s;)

1,8
3,7
6,2
2,7

5,0
5,0
3,7

Table 5: Demonstration of evaluation of respondent A for profile

Appropriate

Table 3: Evaluation ofproducts
Attribute

1

!

!

Limited functionality [ Medium leve!
1

Safet)
lnappropriate
Appropriate

#3
3
2
2
3

Deri ved utility for profile

A
B
C
D

1
1

l
4

#2
2
1
1
5
1
8
i 2
1

Respondent

Table 2: Stimulus combinations offractional factorial design
1
Functionality
Functional
!
Limited functionality

#1
4
3

Respondent
A
B
C
D

1

#l
Safety 1 Constant
4,5
1
-6,7
9,0 -i
1
4,5

Tota!

~

~
l,O

1

Table 6: Relative importance for dinning room chair
Relative importance (%)
Respondent
A

B
C
D

7

Tota!

Innovativeness
21
4
7

Function
27

ll

36
26
28

ll

29

1

4
24
56
ll

Safety
48
36
II
50

24

36

Aesthetics j
1
1

Sum
1

100

1

100
100
100

i

100

1

1

Function

(ll)
Aesthetics

(lil)
Safety
(IV)

Results

From the relative importance (Table 6) it is evident that safety (36 %), functionality (29
%) and aesthetics (24%) must be of great significance when designing new products,
however taking into account consumers' preferences (Table 5).
In the second part ofthe research products that already participated at an exhibition were
analyzed. The results collected from a questionnaire (Table 3) were compared with the
maximum and minimum utilities collected in the first part of the research. In Table 7 we can
see that among different chairs when analyzed by respondents, different acceptance levels
are displayed. This is due to different working areas of respondents. To get more reliable
results research should include more respondents.
Table 7: Evaluation and derived utility ofthree dinning room chairs

Utilities s; (2) and the relative importance AP (3) of each attribute were calculated for each
respondent separately with SPSS CA program. The total worth for the selected respondent
was calculated using the additive rule (1). The results for dining room chair when evaluating
the 9 selected stimuli are shown in Table 4.
These utility values were further used to obtain a total utility for each of the profile from
Table 2. For example, to find the utility of the first combination in Table 2, considering first
respondent we simply add the part worth ofthe respective levels, using equation (2).
The derived utilities calculated for each respondent separately were then aggregated to
obtain an overall result using equation (3), and are presented in Table 6.

380

1
i

Resp.
A

!

B

1

C
D

l

Evaluation (see Table 2)
chair 3
chair 2
chair 1
Il-III-III2-IV 1
12-112-IIII-IV2
12-112-III2-IV 1
Il-112-III3-IV2
Il-III-III2-IV2
Il-III-III2-IV 1
12-III-III3-IV2
11-112-III2-IV2
12-112-III3-IV2
12-III-III2-IV2
12-112-Ill3-IV2
12-112-1113-IV 1

381

Derived utility
ch.2
ch.3
eh.!
1,7
4,4
9,5
10,7
6,0
6,0
7,5
7,0
10,0
7,2
10,1
5,7

Comparison
max
min
10,8
1,5
11,0
2,3
0,8
10,3
1,7
10,7

Conclusion

We used conjoint analysis to find out which product's attributes are of highest importance to
consumer and must be of great concern when designing wood products - furniture. We also
analyzed five products that participated at an exhibition, by the same attributes. The results
showed that safety, which is at the highest leve!, must be of highest importance when
designing furniture. Second most important attributes are functionality and aesthetics (at the
highest leve!) and the last is innovativeness. Results are reasonable as it is known that
everyone pays the attention first at product's safety and functionality, and then how the
product looks out and how innovative it is. Regarding innovativeness, we can conclude that
is considered only after safety, functionality and aesthetics are established.
The second part ofthe research revealed which attributes and which their levels were, or
were not considered enough when designing the analyzed products. These results could be
used with real wood commodities when repositioning or redesigning the considered
products. Further, from this research we can conclude that among the participated products
only one nearly responded to the attributes at the highest leve!, which were of the highest
importance to respondents. This means, that when redesigning the analyzed products,
attention has to be paid on attributes by which the highest high !evel was revealed by the
research results.
Tools of operation research such as CA can be used in designing processes in wood
production companies. With CA we demonstrated how operations research tools can be and
must be used for solving the problems when designing furniture. In the first phase the
importance of deri ved attributes should be determined, in order to improve them in the redesigning phase.

HenrrnticR
Caµacitated Lot Sizmg
Problem
Sequence Dependeut St:tups and
Overtimes
Zrinka Lukač. Kristina Šoric. Višnja
Rosenzwcig
Facnlty of Ernnomics Zagrch
Kennedyev trg 6
10000 Zagreb, Croatia
e-mail: zlukac@efzg.hr ksoric@efzg.hr vvojrndic@efzg.hr
June 15. 2005
Abstract

The wellknown CLSP problem is generalized including sequence dependent set up times and overtimes and modeling as a quadratic mixed
D-1 integer programming problem called Capacitated Lot Sizing Problem
with Sequence Dependent Setups and Overtimes. We develop a heuristics
based on Lagrangean relaxation and tabu search for solving the problem
At the end. some computational results are presented.

References

2.
3.
4.
5.
6.

7.

8.

Dahan, E. and Hauser, J.R., 2000. Managing a Dispersed Product Development Process.
Paper 103. The Handbook of Marketing. Cambridge. Center for Innovation in Product
Development at M.I.T.. 62 pages.
Green, P. E. and Srinivasan, V„ 1978. Conjoint Analysis in Consumer Research. Issues
and Outlook. Journal ofConsumer Research, 5: pp 103-123.
Green, P. E., Carroll, J.D., Goldberg, S.M., 1986. A general approach to product design
optimization via conjoint analysis. Journal of Marketing Research, 45, 3 · pp 17-37.
Gustafson, A., Ekdahl, F., Bergman, B., 1999. Conjoint analysis: a useful tool in the
design process. Tota! Quality Management, 10, 3: pp 327-343.
Gustafson, A., Herrmann, A., Huber, F., 2001. Conjoint measurement. Methods and
applications. Second edition. Springer Verlag, Berlin., 542 pages.
Haaijer, M.E., 1999. Modeling conjoint choice experiments with the probit model.
Faculty ofEconomic Sciences. Groningen. University Library Groningen, 20 pages
http://www.rug.nl/bi bliotheek/collecties/d issertaties/index, 20.07.2005.
Wang, Q., Shi, G., Halbrendt, C.C., 2004. Market potential for fine furniture
manufactured from low-grade hardwood. Evidence from a conjoint analysis in the northeastern United States. Forest Products Journal. 54, 5: pp 19-25
Zadnik Stirn, L., 200 l. Operacijske raziskave (podiplomski študij biotehnike), for
interna! use, Ljubljana, Biotechnical Faculty, pp 149-153.

382

Key words: capacitated lot sizing problem, two machines, cap~city Hmitations, overtimes, sequcnce dependent sctup t,imes. quadratK mixca U-l mtcgcr
programming problem. heuristics, tabu search, Lagrangcan relaxat10n

1

Introduction

ln this paper we are generalizing the wellknown deterministk. singlc-lcvcl <lvnamic lot sizing the capacitated lot sizing problem (CLSPJ m the sensc 1,hat
the setup times are scquence depcndent and the overtimes are introduccd. Also.
the availabilitv of two machines is prescnted. The objects is to minimizc thc
sum of the costs of production, storage, sctup and overtime, A setup may im ply
two kinds of machinc consumption. One is setup cost. expresscd in monetary
tcrms· the other is setup tirne, consuming a certain amount of machinc-hours.
Havii:g the machine capacities expressed in tirne units, it is ~mporta~t to ha:c
the setup tirne as small as possiblc because in this way therc 1s more tn~e a:a1lable for the production. This problem can bc formulatcd as a quadrat1c m1xcd
0-1 integer programming problem called Capacitated Lot Sizing Problem with
Sequence Dependent Setups and Overtimes.

383

Capacitatcd lot sizing problem has bccn shown LO be NP-hard (Bitran and
Yanassc, 1982), Also, whcn sctup tirnes are includcd in the rnodcL finding a feasible solution to thc capacitatcd lol sizing problem also bccornes an J\P-,·umplct.e
problem (Garcy and Johnson. l!J7!J), Therefore, rescarch un devcloping cffect ive
heuristics bas becn a profi Lab[e rescarch area for a long t ime.
The paper is organizcd as follows, In thc sccond section wc are prcsenting
thc mathernatical model describing the considcred problem, Thc third scction is
devotcd to thc rclaxations, Cornbining thcsc rclaxations we are constructing thc
LTO hcuristics in Ll1c fourth scction and ,he cornputaticmal residts are prescmed.
We pcrformcd Lhrcc kinds of nurnerical sirnulations. First, we linearized the
problem for the small dirnensions in urder to get thc optirnal solmion. The
same problems wcre sol ved using onlv Lagrangean relaxat ion and after t hat,
applying the LTO hcuristics, For thc problems of !arger dimensions we appliecl
the Lagrangean relaxation and after that, the LTO heuristics. The cornparison
results are presented in the table considering the quaiity of thc solutions and
computational tirnes.

2

Formulati on of the problem

ln order to formula te the abovc problem. le! ns int roduce the followiug not,at ions:
l Index: i item tvpe. i = l 2, , . n.) machine 1,vpe . .1 = 1 2 aml 1
T.
planning period. 1 = l. 2.
2. Pararneters: d; 1 the <lemand for pro<luct i in period l. PitJ the unit
production cost of product i in period I on machine J. hit the unit storage
cost for pro<luct i in period t, k;tJ - the fixed setup cost for itern ·t in period t
itcm I in period / on machine
machi ne J, u;uJ thc set up tirne frorn item I
J. QtJ - t,hc overtirne cost of rnachine} in period /. Ct 1 the capacity of rnachine
the consumption of rnachine .f per unit nf itern i and O,J
J in period t, a;J
thc maxirnurn overtirne of rnachine j in period t.
3. Variables: XifJ - the arnount of item I produced in period ton machine J.
S;t - the inventory (stock) of itcrn i in period t, OtJ - thc ovenirne uf machine .7
in period t
and

l.

Yitj

if machinc :i is set up for i tem i iu period t
otherwise

= { O,

n

T

2

2

L I):r>itjXitj +

hasa

T

+ LkitjYitj) + L
j=l

i=l t=l j=l

2

LQtjOtJ

t=l j=l

2

Bi,t-1

+ L Xjtj = d;t + Sjt
j=l

384

Vi, t

(1)

n

n

1

1 )J YltJ

+

L a;

1 Xit 1 '.':'.

Ctj

+ OtJ \/t, j

(2)

i=1 l=l

(3)
n
LYitJ

s

\/t, J

(4)

i=l
Oty

<

\/t. J

!5)

(61
Ctj+Oq
a,;

whcre A1uJ is the upper bound on the production capacH.ies, ,/\;/,, 1 =
The constraints ( l) represent the flow conservation constraints for each item
in each period. Also. the constraints (2) describe thc capacity limitations and
ovcrtirnc decision for each rnachine in each period. Thc constraints [4) refcr to
a singlc mode of production.
For srnall dimensions wc linearized the model and applicd the branch and
bound method offered by Cplex. But in this way the nurnber of variables and
constraints increased a lot. In order to solve the problem for iarger dimensions
we introduce the heuristics based on tabu search and the Lagrangean relaxation.

3

Relaxations

In order to solve thc NP-hard problem defined in the previous section, we propose three relaxations.

The first relaxation: decomposition (Rf'l)
The rnotivation for thc generation of the heuristics starting points is as
follows. We introduced the 0-1 variables
Zij

The quadratic mixed 0-1 intcger programming forrnulation called (P) is

minf(x, Y, s, o)=

n

L L um Y;u -

={

l.
O,

if itern i is produced on machinc J
otherwise

\/i,t.j. If z; 1 = L \/i,:i (every item can be
and constraints YitJ '.':'. Z;j,
produccd on every machine). the dirnension of the problem is very large. The
feasibility set is as large as possible. But, by fixing some Z;j to O, we decrcase
n} We assign thc
the number of variables and constraints. Let N = {1.
iterns to the machines in order to allow their productions on them. Let N1 bo
the set of ali items that are going to be produced on machine 1, let N2 be the set
of ali items that are going to be produced on machine 2, and let No bo auxiliary
set. We allow an item to be produced on both machines as well. Because of
machine deterioration, without a loss of generality wc can assurne that setup

385

i.e.
timcs Uiltj are proporLional by some coefficienL of propor1ional ity b >
ua,t+l,J = bu;uJ, Thercfore we fix tirne to be t = l. Then we fix the rnachine.
If for _the sun_1 of the sctup tirnes on the first rnachine for a pair of iterns i and
/ th e mcquahty 11 ,11 l i U[i l; "o k holds for a certain value of k. the11 we allow
both items I ancl I to be produced on the first rnachine. i.e. 1 . / E 1 \\'e
u,112 . u,,:2 'S k. r.hen we aJlow items,
lile same for t.he sccond rnachine. i.e.
ihere exists an i1.em
and I to ~e produced on the second rnachinc. i. l E ,\·2 .
rf. /1[1
= {i E
z not ass1gncd to any machine, we put it to set
I
be produced
After wc ve done this, the items from the set Jv;, are allowed
on the machincs, randomly. Now the dimension of I he problem p 1 is reducec!
significantly.
z.=
= {1 E
nl
·sing thc notations /1/ = /l.
1 2. 1he
1.1
}
.
. d .
111 exmg 1 = 1
n:J = 1, 2 in every constraint is substil uted
1 E
2
.J=
resulting in the new problem :RPl:.
„

The second relaxation: fixing F1J and n 11

(

.·.'

max w
w <'. f(x' .y' s' .o")+ LAt 1 9tj(x' y".u'),r
t=

J ... R

18)

. .. T. J

=

2

..• T. J = l. 2 of the (15) represents
]
The optimal solution ), = (,\ 11 ).
the initial Lagrangean multiphers for (LR). Let ns note the optimal solution of
(LR) by (x(,\), y(,\), s 1,\), o(>.)). Now. we have the lower bonnd for the optimai
objective fonction value of the problem (P; as follows:

y(A). s(,\). o(,\))+

1

=

f,J

R P2,

eliminate the
Let us consider the quadratic constraints (2). !n orcler
·
·
· •1
Jy\ - ,- "
'llladratic terni F.tJ,
-L..J,L....Jftltlt_ ;Yil~. }YttJ1n~wca pacit-:constr anHsg, 1 !x.y,
~
uniform
the
usiug
Ot; < O. we w1l1 fix F,/y; randornly:
_'1,Y, + ':""'' a,1X,t1 - Ct1
d1stnhnt10~ offcred by Cplcx, tn some vai11c F[; from t he interval (O. max F,J ..
. T. Also. we will fix
where maxFtj = max{uat 1 1. l E Nj } . .1 = L 2: 1 = l
is I he maxifrom the interval [O. 0 1). where
the term o,_, to some val ne
., 1CJT1
pr(ll)
"1e
,-onstruc·
we
period l. [n 'his wav
J
machine
of
nvert1me
mnm
<,
~
~
•
IRP '
1 f'J but the •
·
t ra1nts
as the prohlen,
has Lhe .same constraints
2' wluch
cuns
,
\
.
,
.
.
.
h,
(2\ . . ,
, aie su st1<utcd w,U, the const1·aims obtained m :he following way·

F ()

(3), (4). (5) and (6) of the problem (P). The problem
subject to (
max{v(,\) ,\ ::> O} is the Lagrangean dnal of 1P) rclative to the complicating constraints 9t 1 (x. y. o) <'. O, 'eft. j
!low to select an appropriate value for the ,\? Having t he seqnence of optirnal
solntions for (RP2). (x". y". s". o'). r = L ... R. let us solve the problem (D)

L Atjg, (x(A) y(,\) o(,\)) 'S
1

fmin

t,J
1
Let /x(>.J,y(,\),s (,\),o(,\)J = (xR+l yR+ .sR+l aR+t) and solve the probconstraint generation method also
the
lem (D) for r = L .... R + l. We applied
called cutLing plane method (CP). One substantial advantage of iCP) nver
subgradient algorithms for (LR) is the existence of a uue termination criterion
the algorithm.
incorporated

•,i

L

a,1XitJ

'S

"ti

F,~

+ o;] ./ = l. 2:

t

= l.

T

Now we can substilute the constraints (:Ji
Fixing F and '1, ,
Vi, l.J and rcduce the constraints /3).
r;
. .
h
an so vmg t e resulting capacitated lot sizing problem without setup Lime;
and overt1mes, RP2). R times will give us the seqnence of optimal solutions
.
yr · s" ,0 ,r'\ r = l , R "'!
'x"
solutions may satis(v all : he nm, iese opl1rnal
.
·;
stramts ol -.P1. exccpt maybe the constraints i2). If some of the solutions
iX" .. y' s" 0 "_) is fcasible for (P) we have an upper bounc! on the optimal obJC~ti_ve funct10n valnc !min· Having more than one upper bound. we take the
m1mrnal one.
Let

Ct1

aijXit1 'S
1.
<l

F[. + o[1 = b[i

1

b[jYitJ

The third relaxation: Lagrangean relaxation (LR)
In order_ to obtain the Lagrangean relaxation, we do as follows. Let us dualize
the constramts (2) of the problem (P) obtaining the problem i_LR):
v(,\)= {minf(x,y,s ,o)

+ LAtj9ti(x,y .o;}

4

LTO Heuristi cs and Comput ational Results

In order to define finaly the tabu search heuristics, we have to construct the
neighborhood . For each point of the tabu search heuristics we construct exactly
four neighborhoo d points. First we find the pair of items i and l allowed to be
produced on the first machine for which the setup tirne uan is maximum. Thcn
we construct the first neighborhood point by leaving item l to be produced on
the first machine and sending item i to be produced on the second machine,
and second neighborhood point by leaving item i to be produced on the first
machine and sending item t to be produced on the second machine. The other
two neighborhood points are obtained by finding the pair of items allowed to
be produced on the second machine for which the setup tirne 1li!t2 is maximum,
and by sending either item i or I to be produced on the first machine.
The heuristics is implemented in the AMPL programming language and uses
CPLEX. Ali computation s were performed ona PC having Pentinm IV 2.4 GHz
processor and 1Gb RAM.
The next table shows the comparison of the results in terms of computational tirne, wherc Time-Imp LL denotes the average improvement (in%) of the
computation al tirne of Lagrange rclaxations compared to the linearized problem, and Time-Imp TL denotes the average improvement of the computation al
tirne (in %) of the tabu search heuristics compared to the linearized problem

t,J

386

387

:VIARKOV CHAINS IN MRP MODEL OF A MULTI-LEVEL
PRODLCTION SYSTEM
Aleš Silič
Goriške opekarne d,d„ Merljaki 7, 5292 Renče, Slovenia
e-mail: ales.silic@go-opekame.si

li

-- ;
-Awag~-tf;;"e-(in sec(}nds'i
Tabu, search ; Time-Impi Time-Impl
Lmeanzed [ Lagrange
model
1 relaxa tion I
1
heuristics I LL (in %) 1 TL /in o/c)
0
2.8
1
2. 4
1
,
/
; 14 29
u.9
/
1
4.8
2. 8--1.7 .. 16.8
4167
1
4301.9
265 .7 1
--·--1
9382
354.1
f- ·/205-t
·
-- -~
' _ 4-2- +-- 21~c-~---- 5()-----+
8.1
) -----+-)
,
,L
'
1
·.' 110425.l 1 >~130000
(20, 10)
. 9956-]
483.fi -t-' ...=....
=.,..+
'
1
'i- >DB 7
_
1523.4
130000> t3oooo
• .J.2.0 15) ~
1
- !
terms of rnr:1putmional LJme
Table 1 Comparison of rcsults
-=-cc~---

Problem Class
(T,n)
(10, 5)
(15, 5)
'.15, 8)

I

i

-+

t

L

·-·1

····-····+----

solution. the average differ·
~omparing the results in the terms of the qualit)
uicc bctwccn thc obJcct1vc function valuc obtaincd hv Lagratige i·elaxa,·. wn aml
.
l .
1t . d I
d. cr
. .
thc liucarizcd model is O·42°101 a 11 d •J
o l amc iy so Vlllg
lllercucc
" ie avcrnge
.
, .
,
b
ctwccu thc obJcctivc fnnct10n value obtained hv t.abu search heuristlcs and
obtamcd by soh mg thc lincarizcd model is 0.89%
the problem Lahu
In tcrnrn of computational tirne, for smaller dimensions
s.earcl'. heunst1cs is not as effective as Lag,ange relaxaticm method. or even
lmcanzcd modd. Howcvcr. for highcr dimensions of l ho problem it is by far
tl'.e t.nost effect1ve m~th~d :n terms of cornputational tirne, producing soiutions
w1thlll 1% gap and s1g111ficantly rcducing computational t.ime.

References
1

·1

the capacit.aced
G.~: Bit~an, lLH. Yanassc. ( 'ornpm.ational cornp!exit:;
lot stze problem, Management Science 28 ( l.982) 1174- !186

[2] M. Garcy. D. Johnson. Computcrs and lntractability A Guide
of NP-Complctcncss, Freeman, San Francisco, CA \ l979 j

the Theor~

[3] B. Karimi, S M_T Fatemi Ghomi, J.IVL Wilson. The capac:itated lot sizing
problem; a rcv1cw of models and algorithms, Omega, The Jnternational
Journal oj Management Science 31 (2003; .365.:'J78
f4l ~.' _Macs, J.O ..McClain. L.N. Van Wasscnhove, Multilcvel capacitated Jots1zmg complcx1ty and LP-bascd hcuristics, European Journal oj Operational
Research 53 11991) 131-148

Lidija Zadmk Stirn
University of Ljubljana. Biotechnical Faculty, Večna pot 83, 1000 Ljubljana, Slovenia
e-mail lidija.zadnik@bf.uni-lj.si
Janez Kušar. Marko Starbek
University of Ljubljana, Faculty ofMechanical Engineering, Aškerčeva 6. 1000 Ljubljana, Slovenia
e-ma1!: 1anez.kusar@fs.uni-il.si, marko.starbek@fs.um-h.si

Abstract: Material requirements planning (MRP) deals with relationship between the demand for
the final product and the components used to make it. Thus in manufacturing systems MRP are used
to schedule production_ The paper presents a MRP model which is based on the concept of inputoutput analysis. Further. the MRP model incorporates stochastJc external demand usmg Markov
analyze the dynam1c behavior of the multi-level production system the Laplace
chains. In order
h'ansfo1m in continuos tlme model developed by Grubbstroem and Molinder was used. A s1mplified
version of the presented model is illustrated ona fina! product - radia! ventilator
Keywords: material requirements planning, stochastic external demand, Laplace transfonn, Markov
chams, radia! ventilators

L INTRODUCTION
Material requirements planning (MRP) is a management information system to dete1mine
schedules in multi-level manufacturing systems. Normal!y a multi-level system consists of
raw materials, components, subassemblies, and fina! products, which are also known as
items. MRP reduces a master schedule of final products into a time-phased schedule of
requirements of intermediate items. In other words, it suggests decisions to purchase
manufacture and assemble items based on the extemal demand for fina! products [4]. There
are mainly three ways for representing a product and its quantity relationship with its
components. These three methods are the product structure tree [6], the Gozinto graph [8]
and the input matrix ofinput-output analysis [4].
Impmtant concept is application of input-output analysis in the MRP models, which has
been investigated by Grubbstrom and Ovrin [2l Whereas input-output analysis uses matrices
for the representation of the requirements of subcomponents for the production of items at
later stages, the MRP approach makes use of tableaus for describing the way in which the
demand for final products are broken down into requirernents (interna! demand) of
subcomponents, taking lead tirne, initial inventories and order quantities into account.
In order to analyze the dynamic behavior of the multi-level production system, Grubbstrom
and Molinder [3] used the Laplace transform theory in continuous tirne models. MRP
models have been upgraded in severa! directions to come closer to real situations. In this
paper we consider stochastic extemal demand in the MRP model including Markov chains.

388

389

2. MATERIAL REQUIREMEN TS PLANNING
For the problem considered here the following bas1c notations are introduced:
N = number of different items as raw materials. intermediates, as well as fina! products
Q =orderquantity ofitem
= 1,2, ... ,
Ki = number of setups of item at some point in tirne (production frequency)
r; = lead tirne for the production of item from the tiine its production is ordered until the
tirne the order is completed
S;= total inventory of item i at some point in tirne
R1 = available inventory of item i at some point in tirne
d, = external demand rate of item i at some point in tirne (the master production schedule)
fI iJ = number of units of i tem i required for the production of one unit of
/ = identity matrix
s= comp!ex frequency ofthe Laplace transform.
For convenience. the above quantities are arranged into vectors and matrices according lo Q
being (NxlV)-dimensional diagonal matrix with Qi in diagonal;
an (NxN)-dimensional
input rnatrix with Hi} in position (ij). H'=(J - J-/) 1 is Leontief inverse of /l. K.
S. R ali
being N-dimensional column vector with ith components K;,
R" respectively.
ln order to analyze the dynamic behavior ofMRP system we make use of Laplace transfom1
ofa function ofcontinuous tirne [1], which is defined as:

+ d, - R, (O)

Q l

i=

1l

(4)

1

Where means item and

itemj with lower index (/ <

3. MARKOV CHAINS
According to [9], a discrete-time stochastic process is a Markov chain, independent oftime t
for ali states andj
,2. ,s), if:
(5)
=pu
P(Xi- 1=i1 , 11v(1=i1,X11=i1-1, ".,X1=i1,Xo=io) = P(X,
Where Pu is the probability that given the system is in state i at tirne t, it will be in astat~~ at
tirne t + 1. Pu are referred as the transition probabilities and d1splayed as an sxs trans1t1on
probability matrix P:
P=

00

L{f(t)}=l(s)=

fe- 'f(t)dt
s

P.

P·>

P,:

pil

(6)
C

(1

Given that the state at time is

f

The function /{t) is inverse transfom1 of (s). Important property of Laplace transform is
transiation property: Ll}(t-r)}=e
. The lead times r 1. . rs form interna! demands
for components of these times ahead of the completion of the products they enter. Hence,
they correspond to translations backwards in tirne. For this purpose we introduce a diagonal
matrix. the lead tirne rnatrix r. having e ', in its diagonal position. The producl fi = II?
is defined as the generalized input matrix. When production takes place instantaneously in
lots according to Q. the gross production of all products at any pa1iicular point
tirne is
given byQic. This production requires cornponents in the amounts ofHQic Taking all lead
times into consideration, the interna! demand (when components are reservedJ is given
by H'iQ'ič which translates the requirements backwards · tirne compared to HQK Dne to
the lead times, we make a distinction between total inventory S, available inventory R and
allocated component stock S - R. Making use of a tirne integration operator s-:, total
inventory accounts for ali items in the system and is initial stock plus cumulate net
production less external demand [5 J:

i=S~+~-H~ K-J.

Pi:
p„

the process must be somewhere at tirne t + . This means

:t P(Xt+l =

s.

P(X! =

j i

(7)

j:==l

A concept of steady-state probabilities is used to describe the long-run behavior of the
Markov chain. There exists a vector ,r = [1r1 1r 2 • tr\ ] such that
7
r
TC,
rc,
i TCI
'
TC,
rc, rc,
(8)
iimPy(n)= 1r.
limP" =
' 11• 00
•
11

':1)

Lrc,

TC,

1CSJ'

Where Pu(n) is called the n-step probability of a transit~on ~rom stat~ i to state and is th~
ijth element of P'' For large n, P'' approaches to a matnx with 1dent1cal rows, and for ali 1,
Pu(n+l) zPu(n) zffJ:
k=s

P.(n + 1) = ~ p (n )p . , if n is large trj = L., ffkAi
U
f:i ,k

~)

kj

k=l

(9)

s

As seen from the point of view of total inventory, the interna! demand is instantaneous.
Available inventory is given by [3, 5]:

R = R(o)+(I-H{)Qx=-J.
s

Where

(3)

4.

ffJ

+

ff2

+ .. +

ffs

= 1.

AN ILLUSTRATIO N - RADIAL VENTILATOR S AS FINAL PRODUCTS

Suppose a company assembles radia! ventilators which are also the final products of the
company. The final product denoted A is composed by an e!ectromotor denoted B, screws
denoted c, a flange denoted D, an impeller denoted E, a frame denoted F_ and le_gs of the
frame denoted G. Figure 1 illustrates the product structure tree [6] of the radia! ventilator.

Nonnegativity of available inventory is written r' {'ff.(s )}~ O and constitutes the major set of
constrains in related optimization problems. Given fixed order quantities the decision
variable is the cumulate number of setups K;(t) of item i in tirne t, which is generated as the
nonnegative integer providing a nonnegative available inventory [3]:

f
390

391

:m . ~ -.[']

<,

[i] h GJ

R (d )

Order quantities Q; for cornponents (A, B, F, E, D, c;, and C) are 600, 1100, 1650, 1300,
2250, 4100 and 23000 pieces. The lead times 7; are l, 2, 3, 2, 2, 1 and 1 week and the mitial
available inventories Ri(0) are 450, 1000, 1300, 1200, 2000, 5000 and 2l500 pieces. The
company orders cornponents at the end of every week for the next week based on their
consumption in the cmrent week. The company has sales contract with one minor aud two
maJor purchasers. lf demand for final products in separate week comes only from the minor
purchaser, then the order is 100 pieces of radial ventilators. lf demand of one major
purchaser is inc!uded in the same week, then the order is 400 pieces. if hoth major
purchasers are included, then the order is 700 pieces. All transition prohahilities are ohtained
based on past sales experiences with these three purchasers and illustrate transition
probability matrix Pas defined by equation (6) aml (7):

P=400! 0.4
7ooL o,7

* 100

+ OA

* 400

+ O,17

* 700

322 Pieces

generalized input matrix. Order quantity diagonal matrix Q and initial available inventory
\·ector R(O) are based on order quantities Q, and initial arnilable inventories Ri.
A B F E /J c; (
AiO
B',
'j

0.2 !

o o

()

o

()

()

o o o
() o o
DIO ] o o
c;io () 4 o
F l

ll=E l

cLo

4 8 6

e··

T

o
o
o

o

el'

o
o
o

o
o
o
o

()

4e3s

o

8e 3 '

2
6e '

·'

r''° 1

o
o

!

R(0) =

o
o
o
o

01
o o'
o o
o o.

o

{'
,./

o

2

1300

o

o

o

2250

o

o

()

o

o
o

o
o

o

o

o

o

()

o

! o

i
Q=

()

1

4e'

'

o

o

o
o

l 100

!

o o
o

o
le

1650

o

L

o

1200

!21500 j

o
o
o

o

r-600

1000
1300

1 2000 1
I 5000 1

4

e'

4e

7

o o·
o O'

o

o o
o () () o·
o e·' o o o
o o e'' o o
o o o e·· o
o o o o e· J

o o
o o
()

o

2

==

o

o

e"

o o

o

o e'' o,.
()
o e·

o

e

11 =

o
o o
() o
o o
o o
o o

()

o

o
o
o

700
OJ5;

0,4

OAJ

r

Figure l: Product structure tree of the product A (radial ventilator)

1

=

n: X

The input matrix H is based on the product structure tree of the radia! ventilator and the lead
tirne rnatrix
is based on the lead times r, . The product if = Hr is defined as the

4X~

100 400
l 00[ 0,35 u,5

t

.=i

~ di ~ ctJ~, ~

1xd:J

=

o
o

o
o
o
o
o

o
o
o

o
o

4100

o

o

23000

OJ 5 OJ 5,

Considering only request of final products which happen once a week, the extemal dernand
Company wants to know the expected extemal dernand ofradial ventilators after a long tirne,
number of setups and cornponent stocks per week. The dernand of the final products in each
week with transition probabilities shown in transition probability matrix I' is described as a
Markov chain with states X 1 = 100, X2 = 400, X, = 700 [piece j. To find out the expected
external dernand E(d) after a long tirne we first solve the equations (9) for steady-state:
[n

n

n,J=[n.

n

:0,35

0,5

n,]; 0,4

0A

L0,7

OJ5

vector

d is according to [7] given as:

-l

1
322*es -- i '

1322*(e '+e Zs +e

o
o
o
o
o
o

1

OJ5;
1
0.2
OJ5j

n, = 0,35n, + 0A0n, + 0,70n,

o
o
o
o
o
o

n, =0,50n, +0,40n, +0,15n,

n 3 =0,15n, +0,20n, +0,15n,

Replacing the third equation by the condition 1 = ff1 + ff 2 + ff,, we obtain the steady-state
probabilities

1r 1 =

0,43, ff 2 = 0,4 and 1r 3 = 0,17 .

Long-run behaviour ofthe Markov chain is given by rnatrix frorn equation (8):
0,43

0,40

0,17!

limP" = 0,43 0,40 0,17
n->n

[

0,43

0,40

0,17

Cornputing severa! n-step transition of the P" rnatrix, steady state 1s reached (to three
decirnal places - 0,1 % rnistake) after only 5 transitions (weeks).
Now we obtain the expected extemal dernand E(d):

392

Because of limited space we give here according to (4) only a nurnerical exarnple for
curnulate nurnber of setups K;(t) for cornponent G, i.e. the sixth cornponent - legs of frarne,
to fifth week K 6(5). It is generated as the nonnegative integer providing a nonnegative
inventory according to r

1

{ii(s)}~ O:

r •-1
1I_H 6 /r.Q)čj

K.(5)= Q/i-'~j,I

r

1

s

\

1

,

j

,

s

,

L 4100

l

s

s

1

J

l

K (5) =[-l-L_,{13200 e•
6
4100
s

\1

1 -

+ld.-R.(O))r =1-l-L.j]4e''l650K,(8)e_,,+0-5000}]=

5s

_

50001] =[8200] = 2
s J
4100

Curnulate nurnber of setups and cornponent stocks for eight weeks shows Appendix 1.

393

5.

Appendi x . Numeric al example based on structure A in Figure 1

01\ICLl iSIONS

the MRP model using the
this paper we mcorpor ated stochast ic external demand
the MRP
analysis
Markov chains theory. lmporta nt concept is applicati on of input-ou tput
the
analyze
to
order
models. which has been imestiga ted by Grubbst rom and Ovrin. In
transform
Laplace
the
dynamic behavior of the multi-le vel producti on system. we used
theory in continuo us tirne models treated by Grubbst rom and Molinde r.
which represen ts a
A numeric al example is shown based on structure A ~ a final product
The tradition al
radia! ventilato r (Figure ) illustrati ng the simplifie d model propose d above.
week \Ve
each
MRP model with dete1111inistic extemal demand with one known state
n
transitio
rs)
purchase
upgraded with stochast ic external demand with three states (three
which
theory
chains
Markov
the
probabil ities shown in transitio n probabil ity matrix P using
after a
demanJ
is normally more close to a real situation . To find the expected external
of
1ivity
nonnega
The
ion.
distribut
tate
long tirne we sohed the equation s (9) for steacly-s
{R(s )} :C: O constitutecl the major set of constrai ns in a related
a, ailable inventor y
was the cumulat e
optimiza tion problem Given fixed order quantitie s lhe decision variable
tive integer
nonnega
the
as
d
generate
was
which
number of setups K;(t) of item i m time t,
compon ent
and
setups
of
number
e
Cumulat
y.
inventor
providin g a nonnega li\e mailable
stocks for eight weeks are represen ted in Appendi x

WeekiLJ - ~

/ ltem A _ ______ - - - - - - demand E(d)
I Expected e~ternal
· Available inventory R_ _ _ ________

[2J

l3J
[4]

[5J
[6]
[7]
[8]
[9]

394

,_
1

600
0
~ - --- . -!-- ---- -+-itl9 4J~~~ -o~;-~23~8'...\~--~o+---;;-28;;;2;-t---=-;4;-t--~onl
: Delivery Q _______ ___
...'.0'....1-_ __,_I+----'':-+ ---0;--i
1,
o:
-1 :
1
-t--0
_ __
._
1Net reguiremen t
:-=3+---,4;+--;c;:;4;,1
-=c2+----,-:
--'2:..J__....
I
l
1
____
_ _ _ __
0
1 Setups K
_ _Oc..L._6_00
:::.:0::....L
.::.0_1____:c60
-~o'.Ji_....'.6'.::'0
~
~
J
+-_ _ _ _ ___
K
6001
i Cumulate selups
Orderslar tQ · - - _______ _____ .L_,_ _L_
1

-+- -

+--

!

-

~Item B (lxltem ~ _ --~

----H'
~ ___tl___1:
1

= "___

- - - -j _~
__ _
400
1000 .
_Interna! demand _
--'f;+~
r
___
__
JL
invenloll
_Available
_____ ___ - - r
Delivery_Q
_ _ _ _ _ ---fLNetrequu ement
---+--______ __ . --+
Selups K
1
O,
'
K
---__
I
L
ales~
LCumu
1100
______
LOrder start Q

'.::'0.::.0+---'::-6~00+--=0;-t---;;6"'00n1
_..':o'--!--....'.6
_.8="=,0:.::0+---=8=-=0-:';-0+-_20-;0,1
O
O
o 1 l ,OO
o·

--;;,60~0
o
900 ,
400
IOO.
1

90(l..i-_..:č30~0+-

t+--

+---Q.'.=_3fOl_l<lQ.~
---t ~ _01 200
___I_~
(_ o _
--r ot-'_o , _ I.J----0
2
1
1:
I ,

-;tt-

-=t=' ·

_-t--_

0
--0!
01
1

:

~
llOOi

O/

O,

o1
lOO

600 j
l 150 i
1650

600'
550'

GOO,
,
.Item F (lx ltem A) _ _ _ _ _ - ----r- 0,
600
- 100
· - - - - +-- - - + Interna! demand
00
?O~;-~ ~
·Available invenlory R ___ _____ - +--110_()_ 1

~ - ~ '.
----tt
--t ----tt
--------+-------:
Selups
I
·-if~
Cumulate setups K_____ ____--t--1650
____.__
1Order start Q

,~::i;:~~i? ement _____ __ - _ _

+---r

K

1

--"-.L.--~

-----L-

.. _____ ----, ----.----

, Item E (lx ltem AJ
Interna! demand

E.. An intrnduc tion to Laplace transform ation and applicat ions,
Anderss on,
: symposi um,
output analysis and Laplace transfo1 ms and material requirem ents planning
pp 16-31
1998_
,
Pmiorož
ology.
ofTechn
Institute
ng
Storlien, FPP Portorož & Linkopi
between
hip
relations
the
of
zation
generali
poral
11.tertem
P.:
Urubbst rom, R. W., Ovrin,
of
Journal
onal
lnternati
ysis,
anal
tput
input-ou
and
planning
ents
mate1ial requirem
8
1-3
3
pp
Producti on Econom ics, Vol 26, 992,
Mo linder, A.: Further Theoreti cal Conside rations on the
Grubbst rom, R. W
Relation ship between MRP_ lnput-Ou tput Analysis and Multi-E chelon
pp 299-3
Systems . lnternati onal Jomnal of Producti on Econom ics, Vol 35, 994,
and inputMRP
of
topic
the
Grubbst rom, R. W Tang, O.. Early writings re!ated
planning :
ents
requirem
material
output analysis , and Laplace transform s and
Portorož ,
ogy.
Technol
of
Institute
ng
symposi um, Storlien, FPP Portorož & Linkopi
1998, pp 33-44
nje in zalog,
Horvat, L.: MRP, input-ou tput analiza in večnivojski sistemi proizvod
336-349
1994,
4,
št.
45,
letnik
a,
Ljubljan
revija,
ska
ekonom
Slovens ka
975
Or!iky, J. A.: Material Require ments Planning , McGraw -Hill, New York,
nih potreb,
Peterle, P.: Vključitev teorije Markov skih verig v model planiran ja material
2002
a,
magistrs ko delo, Ekonom ska fakulteta , Ljubljan
New York,
Vazsony i, A.: Scientifi c program ming in business and industry , John Wiley,
1958
, Belmont ,
Winston , W. L.: Operatio ns research : applicat ions and algorithm s, Duxbury
1994.

_,,_3~22~__:3ii2fi2i'---'3f,2~2;'t-~327'2!1
_L_I32~2W-i--=3~24,2 +,I _}__32~2~1!-,, --:3~2~2+--_
_,,3~6;:_2.J_____:4.'..:0'../----=3c:.l.:::.8+-----=5c:c9-::-6i-_2_7-;;-14
84
128 !
406 , - ot-'
O
600
600
o
600

,-- 1
---t- 450 !

-+-

1

6. REFER ENCES

-2~,--3~, ---------- c4c-r---- ;5~--:6;,- --;;7:-r1- -o13

o,--iT

-,--'--

__

_

~----t

!

·-~

r Available inventon R
J Delivery Q
- Nel reguiremen t

·-- -

!

0
,
O
1
0

-t-- l._

---_ ----

1

\ Delivery Q
requiremen t
1

I
:

900 1
0

-+
1

i

Setups
Cumulate selups K
Order start O

'

Item G <4xltem F)
Inlemal demand
Available inventorv R

1,

K

1

It

1

,

Setups

K

Cumulate setups K
Order start O

l
1650 :

2
11
O' ~

1'
Oi

600 i
600 •
o,
o 600
1 _i_O-;:o-r--=o'.-f!-,.c7.:::.oo:+800
lOO
1300
_lJ0~---~0-1---~'.:_'.'6300~00'...\ll_ _-;';-0°+•o---,;+-~
500
o
___,o~,
Oi
1
o
o
1
1
o
1!
O!
o
2
1
..--,
_--=-.L_J]QQ_;__
I
O
I
O
1300 i
o
o

I

o,

0

1

1

~

i

900 ,
0

oI
0]

1

oi
o

1
l
O

1
O

225

o

o

o

l

1
o

o

o

1_ ~o:~ o+l--;;~o~'f--~66~0~0-+
Q.0l,', ~ 6~60~0!..ji_=-=~
O
2500
2500
00
o 1
25
4100

5000
0
0

4100

4100
600
1 1
1
o

~

,,
+ lxltem D + 6xltem E + 8xltem F + 4xltem
15450
21500

O

o
o -t--n1
-i+~--.=:f~.cc:2~0:..::0:c~~o~~O:~---:oO, č-t---;;--ft-o
o
2250

o

___

O
o

x
em
Interna! demand
A vailable inventorv R
Deliverv 6
Net requiremen t

0

~

Setups K
Cumulate setups K

C (4 ltem B

O

O1

Delivery O
Nel requiremen t

~cr~O

o:
_ oi

-· o

5~

!_i!

2000 _

I

_

JLj[ ~: ~:
--ot _
~'

600
1600
1650

1100
01
wsot~95
o
-1:_™tto
~O*Ot'---;:;oio\
o~I-~o~l--;l',;l
_l'__-j__l__l o~o~l--:c::'
2050 1
1 900 1 2050
l
950
2050
2050

, Available inventory R
1 Nel

O:
550
O:

--ar!:_ _o! -~0;-t'-....'.'.:'.ft-~~-riT-Jl

--t-Jo4 -·
i -~o4
--i-

/ Setups K
___ - - / Cumulate selups K
i Order start Q _ _ _ _ -··---- _1.._._

o (lx ltem B)
i, Item
Interna! demand

i

o,

_0-'--+----::-10
-'2=-+----, -J
2
O 1100

~ 60~

600
600
1200
-· 1 - ~

,

__ -

6~

l
o

_o::.+__oc--1

20800
700

O
o
O
o

23000

395

8250
3000
2
14750
1
1
o

o

O

o

O

4100

o

o

1

1

1
4100

2

7800 20800
2650
450
O 23000
O 20350
1
O

o

1

23000

2

23000

o

otto
o

1

o
o
o

o

o

o

o
2

o
2

o
2

o

o

o

o

13200
12450
23000
10550
1
3

7800
4650

o

4400
250
o
o

o

o
o
o
3
o

4650
o
o

o

o

3
o

3
o

IMPROVEMENT DESIGN AND MANAGEMENT OF WOOD
PROCESSING MANUFACTURE SYSTEMS
Lyudmyla Zahvoyska
Department ofEcological Economics, National University ofForestry and Wood Technology,
Gen Chuprynky Str.,
79057,
Ukraine.
E-mail: zld@forest.lviv.ua

Abstract
Product10n and consumption reshaping according to the needs of sustainable development force us
towards stepwise and rational use of ali kind of production resources, mcluding material and energy
flows. In th1s sense production systems' operation tirne is a unique and non-renewable resource and its
losses become an aggregated indicator of produchon system effic1ency. Approaches, methods and
models of production logistics which lead to diminished tirne losses in wood processing manufacturing
systems are highlighted in the paper.
Key words:

eco-efficiency. manufacturing systems. transfer lines management, operat10n tirne
losses. stochastic s1mulation models, analytical models, optimal performance.

Introduction
Time is unique and non-renewable resource not only for some persons but also to ali
mankind. So that, its rational use in ali fields of human activity is of great importance,
especially in industrial production. To estimate efficiency of foms' industrial activity, namely
manufacturing systems (MS), we use such indicators as system capacity. prime cost of
production, income etc. These indicators strongly depend on leve! of MS' tirne use. Rapid
development of modem facility productivity induces simultaneous price increasing. Losses of
tirne that occur in MS involve losses of capacity, which increase together with cost of MS.
Hence, the important task of production logistics optimisation, especially in case of pulsation
production (De Cristofaro, 2003), we face now, is reduction or even elimination losses of
tirne, which arise in multistage transfer line systems because of mutual equipment blocking,
mutability of inputs, failures of operators and equipment. Ali these perturbations particularly
acute emerge in wood processing MS.
Stochastic simulation models are commonly used for MS investigation but this multipurpose instrument of operations research needs sound soft, relevant computer capacity and is
tirne consuming. Fram this point of view analytical models seem to be faster and more
convenient for practical application. But the way of their generation, verification and
validation is not very straightforward and comprehensive.
In this contribution we highlight the results of our exploration of analytical models,
which in proper way reflect dependence MS productivity on their structure, stability,
reliability, composite elements' characteristics, and linking way. Models of queuing theory,
which were used as a benchmark in our research, were improved and developed by means of
stochastic simulation and compared with relevant results, obtained by other scientists, and real
industrial MS.

396

397

r- -

1

2. Approache s to MS Parameter s/Perform ance Modelling

sophisticat ed analytica! methods (Budnick, McLeavey , Mojena, 1988). It is more easily to
understand the modeling process and obtained results.
Hence. following the idea of lean production and high profit, modem en~repre?eur
needs quicker analytical models, which deals properl~ with types of MS, operat10ns tn~e,
buffers capacity, and machines reliability (Gershwin, 1994). In _t~ble 1 we prov1d~
information about availability of models (both analyt1cal (A) and empmcal (E)) for transfer
hnes' performan ce optimisatio n.

Performanc e of MS strongly depends on a set of perturbing effects. such as equipment
interna! correspond ence of
reliability. adequacy of operators· knowledge and
frorn prescribed leve!
divergence
Minor
y.
productivit
ts·
departmen
parallel
successive and
causes thern, is an
ce.
circumstan
Common
losses.
shutdown
significant
to
lead
could
equipment failure. How to run MS to minimise the tiine losses is question of critical
modem
to
importance
rs.
entrepreneu
_
3 0 i - - - - - - . _ _
Some types of MS, for
!
processmg
wood
instance
are
manufactur ing systems,
201
especially exposed to such
i
stochastic perturbatio ns due to
biological origin of processed
10 1
inputs, which are wooden details.
1
Ontogenesis and phylogenie s of
wooden details being processed.
o~
changeabil ity of their physical
')
1)
and chemical characteris tics
bring strong variability mto
l-----~This
operation duration (Fig.
"
Fig I3ox-Wh1sker Plot ofwood processing operations
differs
noticeably
peculiarity
Veneer cuttmg: 2.
wood
of
logistics
production
3. Detai ls calibration
processing enterprises , makes it
more complicate d, production
recourse use less efficient and lask of such systems parameters optimisatio n more evident.
The first significant cont1ibution to MS probiems investigatio n made K. Palm and P
Finch in 947. They proposed to describe duration of technologic al operation by exponentia l
probability distribution. Smce then investigatio ns of MS, especially in engineerin g industry.
deal with Markov chain process models. R.Jackson ( 1954), G.Gantt ( 956), E.Koenigsberg
(Koenigsberg, 1956) proposed a new model of transfer line employing queuing theory. But
these approaches did not find an appropriate application because of two reasons: they were
not enough powerful and gave only approxima te results. They fail to generate meaningfu l
analytic solutions for non-Poisso n tandem systems like Ek/Ek/M, which real wood processing
MS are. That is why we use the models of queuing theory as a benchmark in our investigatio n
of MS performanc e.
Stochastic simulation is commonly used in MS problems investigatio n due to its ability
to incorporate in models different sources of perturbanc es. This kind of modelling enables
investigato r to make direct experimen ts with model of real object and let make the model as
precise as investigato r would like. But the process of simulation model generation,
verification and validation tend to be quite sophisticat ed. It takes a lot of tirne even for
experience d in simulation (Sargent, 2001).
It is known that simulation is good for didactic (teaching) purposes. Simulation not only
helps acquire a good understand ing of overall systems, it also gives good reason for that. It is
able to provide us with missing informatio n for different situations and case studies. It may
illustrate the model developme nt process, or allow one to better comprehen d process in
question, or better understand relationshi ps within decision-m aking setting and to develop
students' decision-m aking skills. Simulation tends to be relatively straightforw ard. Nontechnical decision-m akers and students prefer them especially when compare them with
1

1

398

Table · Availabilit v of Models for Transfer Lines' Performan ce Optimisat ion

1
1

Nnrnber of rnachines (n) and lheir
pararneters:

.
.

Er\angian parameter (K)
Productiviti es ratio ( µ

Transfer line machinery composition aiul rhythm

Without buffer
1

Number
of
machines,
II

Erlangian
parameter.
K

Prodnetivityratio

Parallel Composition

Serial Composition
With
buffer

1

lnd1v1dual
rhythm

Smgle
rhythm

lnd1ndual
rhythm

Smgle
rhythm

l

TuAE
L/A

L. Tl!E

?. l

Tl,B1A

Equal
-No equal

TL,
--·

TJA
Equal

B!E
--

---

-----

-

TL, µ, BIE
-

TJA

~3

------

-·--

TL, f..[BIE

No equal
l

----.

TL, µ'BI A

Equal
--

---

--

--

~-

2

?..l

5

Stages~

l

l

W1th
buffer

W1thout buffer

TLI E
-

for n ~ 500

-

---

---

-·

-

--

Tli E

Virtual Pairs
Method

2.
No equal

for n ~ 30
· · / T ype o fM o d e \s.
h
Notes:
Shape of the records in the table: Transfer Line Performanc e C aractenstJcs
L = expected number of units in system.
Transfer Line Pe,formanc e Characteristics:
TL = additional tirne losses in transfer line.

µ = machines in line productivity ratio.
B

= buffer capacity.

Type of Mode/s: A = analytical; E = EmpiricaL
Models in the shadowed sells are ours.

As one can see from table 1, we consider three cases of lines length, when numbe~ of
machines in transfer line (11 ) equals one, two or _greater t?an ~o. Ra~io (µ) of machi?es
productivity and stability of technologic al operations, wh1ch 1s 1escnb_ed by Erlang~an
2
parameter K (K=r/S , where t - means duration of operation and S - vanance of operation

399

duration) are analyzed in our investigations, too. For cases. when [( is approaching unity
( K ~ ) duration of operations tencls to be a stochastic variable. We can see
such wood
processing operations as veneer cutting, scarf joint, details calibration, etc. If Erlangian
parameter K is approaching infinity ( K ~ oc cluration of operations tends to be a
cleterministic variable. This situation is typical for transfer lines, especially engineering. As
one can easily see from table 1, the most of inclicated types of transfer lines there are
analytical or empirical models.
Time losses unambiguously determine transfer line capacity that depends on its
stmcture, stability of operations, machines reliability, composite elements' characteristics,
facility layout (Zahvoyska, 1999, Zahvoyska, 2003). Examining transfer lines performance
we marked out two types of transfer lines: lines with fini te buffers between them for details
storing and with direct connections between working machines
buffers). Each of
these two types of transfer lines can work
two clifferent conditions. The first of thern we
call single rhythm line. In this case ali rnachines of transfer line begin their technological
operations simultaneously. Finishing their own operations ali machines are waiting for the
longest operation completion and only after that all of them begin the next operation. The
second type of conditions is free individual rhythm of transfer line. In this case each machine
can start its operation independently at any tirne whenever the buffer before it is not empty
(Okamura and Yamashina, 1983.
Transfer lines as the main elements of MS belong to the real tirne systems for which
tirne is the most valuable and non-renewable resource. Additional operation tirne losses
induce chain ofwasted resources: first of all energy, material ancl financial flows, overnsed of
machinery system resource, and last but not least losses of labour. From this perspective
operation tirne losses become an aggregated indicator of procluction system efficiency. Their
climinishing is generally viewed as one of the most important tasks of MS management.

resources (machinery, energy. and labor flows). Such models let engineers design efficient
transfer lines with minimal losses of energy and material flows.
From other point of view they enable managers to use or to redesign MS in a proper
way. Such 1rnprovements make systems in question leaner and nearer to eco-efficient,
aclequate to the needs ofnew concept oftechnological progress.
Table 2: Assessment of Additional Time Losses in Transfer Lines (Serial Composition)

11

K

With buffer

Without buffer

-1

\
!nd1ndual rhythm

Single rhythm

=T1

\

--

-

+T1

-

1

;:;1

LqK =Lq1 +K-1
µ

-

-

..,

2

T1

=
~-

,,:,

).

B1 •

,

LqK = Lq)K ),, +

= 1,

00
---

T12 =(1-µ)

t1,

(KB + ✓ nK + 1

---

-

--

---

B=B 2
-

x;

1

-

BI =B2

1

--

1 2,,:,Lq K
1

--

(1 - µ8+3

1-1 = µ/

µ,

K= 2 !(K7 1 + K:;- 1 )
-

---

-

----

-~--~

--

-µ
TLI=µ TL-ITL2 T12 = (11
2
K = (1 + µ ) / 2(K 1 +K2 µ )

i •Tl

*

-1

TL =

1-1 2)

-------------- - - -

1---

=
~1

R:,

✓
K

,,:,

(1,900 - 1,800 In ).1(B + 3 ), n ~500

1

--l-

rr.,;-1 + i ,
2,11

1T

1

i

'- l,11

;:;3

K

L,'

TLl = µ B+2T12,

*

2· Ti = 1- Jc/ j..l 1
'
-

-~-

~

! -

1

= TI =(B+3t1; B=B 2

3, Results of Investigations

400

---

µ

;:; 1 -

For all shadowed cells from table l, which represent specific types of transfer lines, we
developed event oriented stochastic simulation models (Zahvoyska, 2003 ), which provide us
with missing information about tirne losses MS in questions.
Further statistical elaboration of generated results gives us empirical models aimed to
optimise design and management of transfer lines. ln particular one can estimate value of
operations fone losses, the optimal productivities ratio and the optimal buffer capacity (table
2). Besides we propose to analyze transfer line performance with different machines
productivity (when number of machines is less then 30) using virtual pairs method (Dudyuk,
Zahvoyska, 2003). Following our investigation this method gives the most reliable
information for such complex cases.
Using models, introduced in table 2, it is possible to estimate value of tirne losses
coefficients for more than 20 types of MS, to find dependencies for determination values of
the optimal machinery loading, the optimal productivities ratio, and the optimal buffer
capacity for the different types oftransfer lines.
Engineers, designers or MS managers can predict or assess tirne losses value for
clifferent types of transfer lines and make proper decision conceming their design or
management. For instance, using optimising techniques one can find the optimal lines'
parameters in question, like buffer availability and capacity, way of machinery composing or
other relevant matters which production logistics deals with. In case of transfer lines the time
losses elimination means the production of some given output leve! using a minimum amount

Transfer line machinery composition and rhythm

Number of
machi.nes
and their
parameters

-·-

L900 - ,800 n

T

,=:,-

l

---

-------

---

KB+-/iK+l

r

_;•

II
"\'121

K-iDLJ\!ri K,
L ;~1

'
1

*

Virtual Pairs Method, n ~ 30,
-

p=

Vn I 2 (rL _, - 1)

Notes:
Shape ofthe records in the table: Transfer Line Performance Characteristics / T)_'Pe_ofModels.
Transfer Line Performance Characteristics:
Lq = expected number ofumts m queue.
./4 = arrival rate.
µi = service rate.

401

----

)

J

J,

4. Conclusions
Quoting P.Hennicke, it is worthwhile to stress, that modem environmental policy
becomes aware that "the very quantities alone of energy used and substances moved as well
as increased land use create a problem. Every product and every service is linked throughout
its entire life cycle with energy and material throughputs" (Hennicke, 2003 Reshaping
physical exchange of anthroposphere and the environment in eco-efficient way society will
encourage environmentally benign technological progress. Sustainable enterprises wil! be one
of the numerous steps in this way. And we hope the proposed models will be useful
instrument to get this new strategy of development implemented.

References

The 8th lnternational Symposium on
Operational Research in Slovenia

S@R 'OS

1. Budnick. F.. McLeavey. D„ MoJena. R..
Management. Irwin, Illinois.

988. Principles of Operations Research for

2. De Cristofaro. E, 2003. The Efficiency Revolution. [n: Proceedings of the 5th Int-l Conference on
Ethics and Environmental Polices "Business Styles and Sustainable Development"
Dudyuk, D., Zahvoyska, L„ 2003 Estimation and Eliminat1on of Time Losses in Wood
Processing Manufacturing Systems. IZMN. Kyiv. (Ukr.).

4. Gershwin. S.B. Manufacturing Systems Engineering. 1994. PTR Prentice Hall. New Jersey.
5. Koemgsberg, E. Production Lmes and Interna! Storage
p.

:1

Management Sc,ence. - 1956 .... 5

4

6. Okamura, K., Yamashina, H., 1983. Analys1s ofin-Process Buffers for Multi-Stage Transfer Line
Systems /; Int. J. Prod. Res. Vol. 21 No 2, p. 83-195.
7. Hennicke, P. 2001. Decouphng Well-bemg from GDP. Towards a New Kind of Technological
Progress. In: Proc. of 5-th Intemational Conference on Ethics and Environmental Polices
"Business Styles and Sustainable Development. Kyiv. April 2nd-6th, 2003.

8. Sargent, R., 2001 Some Approaches and Paradigms for Verifying and Validating Simulation
Models Intelligence // Proc. of the 2001 Winter Simulation Conference / B.A. Peters, J.S. Smith,
D.J. Medeiros, and M.W. Rohrer (Eds.). - P. 106-114,

9. Zahvoyska, L., 1999. Analysis of Multi-Stage Transfer Line Systems using Simulation. In: From
Theory to Practice. Gaps and Solutions in Managerial Economics and Accounting in Forestry.
Sisak L., Joebstl H., Merlo M. (Eds). Proc. of the Symposium in Prague. May 3-15, !999. - P.
239-243.

10. Zahvoyska, L., 2003. Towards Eco-Efficiency: Operations research in wood-processing
enterprises management. In: Proc. of the 7th Intemational Symposium on Operationai Research in
Slovenia. L.Zadnik-Stim, M.Bastic and S.Drobne (Eds.). - Podcetrtek, S!ovenia, September 2426, MIGRAF, Ljubljana. - P. 209-214.

402

Novo Gorica, SLO\/ENIA
September 28 • 30, 200S

Section

TESTS OF NO ASSOCIA TION BETWEEN VARIABL ES _
COMPON ENTS OF CONSUM ER CONFIDE NCE INDICATOR AND
STRATIF ICATION VARIABL ES FOR CROATIA N CONSUM ER
SURVEY
Vlas ta Bahovec
Mirjana Cižmešija
l n iversitv ot Zagreb - Graduate School of Economics & Business
Trg F. Kennedya 6
10000 Zagreb, Croatia
bahovec@efzg. hr
mc1zmesiia@efzg.hr

Abstraci

Consumer sune:, is an important tool for tracking and predicting cha11ges in private consumpt1011 and
surve~ Such
111 a whole countr:, ecomJm\ Like a business surve:, this is a qualitative economic
in the
changes
predicting
for
especiallv
sis.
anal)
economic
short-term
for
survevs are intended
economic c,cles. The) are important because sur\'e) results uf economic \ ariables are a, ailable before
lhe same or the similar data from the official statistics is published. Time series of variable balances
Consumer onfidence !ndicator for Croatia are roo short for these pred1ct1ons.
and tirne series
find out whether there is no association between
However. the surve~ results can be used
coinponent , ariables uf CC! and variables: region, settlement size. household size and children
(households wirh clr without children under 14 ).

Keywords: CU - Consumer Confidence lndicator, Pearson 'hi square l'est, ML Chi Square Test,
Phi Coefficient, Contingency Coefficient, Cramer·s V

Introduction
1
Consumer surveys are conducted in ali EL countries lt can be seen that business and
consumer surveys are also caiTied out in the future member states like Croatia. This is
important for the integration of Croatian business and consumer surveys in The Joint
Harmonised bL programme of these surveys, and for the integration of Croatia into the EU.
The results of consumer survey are of the best quality if consumer opinions in the whole
country about variables in consumer survey are independent of stratification variables. The
aim of this paper is to conduct tests of no association consumer survey variables and
stratification variables for Croatia.

Consumer survey and Consumer Confidence Indicator - CCI
Consumer surveys are conducted on a monthly or quarterly basis, by means of questionnaire s
sent by post or as a telephone interview. The questionnaire has fifteen questions2 The most
used survey sampling method is applying stratified sample with a random choice. The survey
data are presented as "balance". Balance is defined as a difference between weighted

1
The first harmonised consumer survey was started in May 1972 (The Commission of the European
Communities), on the basis ofa decision by the Permanent Representatives Committee, 1970.
2
Jf consumer survey is conducted monthly, the number of questions is twelve, and three questions are asked
quarterly.

405

percentages of positive and negative answers of ..:onsumers on corresponding \ariables.·;
Balance is calculated far all 4uestions variables). Questions are grouped as: an assessmeut of
the present situation. an assessment of the year-on-year change and an expectation_
With the aim to summarise consumers' subjective assessments of the economic and social
trends in a countr). Consumer Confidence indicator
can be calcu!ated, Consumer
confidence indicator
is a composite indicatoL This is a simple average of seasonallv
adjusted balances 4 offour variables:
•
( 1 Financial position of households
(2) The general economic situation
3 l 1: nemployment expectations (with inYerted sign)
(4 Saving over the next 12 months 5

C CI = B,,~ + Bu,,.1 + (- B, i)+ B_,
4
Where are:
('(' l - Consumer Confidence Indicator
B,,s - Seasonally adjusted balances ofjinancial silualion ofhouseholds.

consumption expenditure at constant prices). It means that the series of variable balances and
CCI series can be used to predict changes in private consumption. This is especially useful for
countries which do not have quaiierly national accounts.
C onducting consumer survey is important for the calculating ESI Economic Sentiment
Jndicator \\hich is a composite indicator (more detailed: Nikic, G., Šošic, I., Čižrnešija, M.,
2004) and which is important for predicting changes in GDP and in a country's economy asa
whole. 6 our country the series
is too short to make this calculation.

Consumer survey in Croatia
In addition to the business surveys in Croatia, canied out since 1995, Centre for market
research I'rivredni ;;jesnik started
the first 4uarter uf 2003 conducting consumer surveys
too. Like business surveys, consumer surveys are carried out on a quarterly basis. The results
of the tests have been regularly published in «Pri vredni vjesnik». Survey is ?ased o~ a p~nel
of 620 households. The panel reflects regional structure uf populat1on \ 5 reg1ons), s1ze of the
settlements. size of the househoids. and households with and w~thout children belo:w the a_ge
of 14. The test was carried out by the GfK branch m Zagreb The sam~le s1ze ,n th_e last
Consumer test conducted for the 2004..
quarter is 600 households and 1s representatlve of
the entire Republic of Croatia.

B,a,.i· - Seasonal!y adjusted balances of the general economic situation.
B, 11:" - Seasonally adjusted balances of unempioymenl expectutions.

B.1. - Seasonally adjusted balances of expected savings.
Qualitative results collected through the consumer SUl'\'e, are doseh correlated with the
quantitative statistical data. "The role of this data is to pro~ide an initi;l advance indicator of
development in ce1iain areas of the economy. They are nota replacement far the quantitative
data, which describe the actual situation." ( .. The joint harmonised .., 2004. p. 387 Survey
results are used as such or summarised in synthetic. confidence indicators.
Each variable in business and consumer survey. such as expectation and confidence
indicator. must be a short-term forecasting indicator for the corresponding variables. lt is of
interest here to track the direction of changes in a variable and in a referent series. This is
important because the business and consumer survev data are usuallv available before the
cmresponding official statistics is published (Bahovec: V .. Čižmešija, M., 2003).
Referent series in official statistics for the series CCI can be private consumption (growth
percentage compared with that in the same quarter of the preceding year). Selected questions
(variables) can be compared with inflation, unemployment, and so on. The highest correlation
coefficients for most EU countries are between the CCI and private consumption (private final
3

More detailed in ... The joint harmonised EU programme of business and consumer surveys, user guide.
(updated: 26/05/2004). European Economy. European Commission, Directorate-Genera] for Economic and
financial affairs., p. 8-9
4
Because series of variable-balances in Croatian Consumer Survey are short, balances are not seasonally
adjusted.
5
( 1) «How do you think the financial position of your household will change over the next 12 months? (geta Jot
better, geta little better, stay the same, geta little worse, geta lot worse, don't know)», (2) «How do you think
the general economic situation in this country will develop over the next 12 months? (geta Jot better, geta Jittle
better, stay the same, gel a little worse, geta lot worse, don't know)», (3) «How do you think the leve! of
unemployment in the c?untry will change over the next 12 months? (Will it: increase sharply, increase slightly,
remam the same, fall shghtly, fall sharply, don't know)», (4) «Over the next 12 months, how likely are you to be
able to save any money? (very likely, fairly likely, fairly unlikely, very unlikely, don't know)».

406

No association test results
The aim of this paper was to test a hypothesis of no association between CCI componentvariables and four stratification variables:
1) Regi on ( 1=Zagreb. 2= Middle Croatia, 3= Nmih Croatia, 4= Slavonia, 5=Dalmatia)

2

The settlements size l="less then 000". 2="1000-4999". 3="5000-9999''
4="10000-4999 9" 5=''50000-99999 " 6="more then 99999" residents)
3) The households size: l =one member, 2=two members, 3=three members, 4=four
members 5= five members, 6= six and more members)
4) Children' l = households with children under 14. 2=households without children under
14)8

One of the most widely applied statistic tools for testing the presence of assotiation between
two categorical variables in the contingency table is Pearson ~hi-square test. a~ well as othe1:
types of chi-square tests (hkelihood-ratio chi square (ML- ch1 squar_e;, _Mantet~Haenszel c/11
square) and Fisher exact test (McClave. J. T., Benson,. G. P., Smc1ch, T. ,2005)). Th e
measures of association evaluate the strength ofthe assoc1at10n between the vanables.
The analysis of sample data was made using SAS corputer package. In the SAS outp~t ~he
Pearson and ML chi square test results are selected , as well as measures of assocrat1on
between two categorical variables (Phi Coefficient, Contingency Coefficient and Cramer's V.)

Referent series in official statistics for the series ESI is GDP -Gross Domestic Products.
There are many European Countries in which GfK is the organization for conducting consumer surveys.
8 On the basis of consumer survey results for 2004, IV quarter.
. .
9 fisher exact testis a more appropriate test of no association than chi square test when the sampl~ s1ze IS s mali.
This test was not selected, because the computational tirne can be prohibitive, depending on the s1ze ofthe table

6

7

n
and the sample size.This testis not practical (in terms ofCPU tirne) for (r -

407

l)(c -l) > 5 ·

In the Pearson Chi-square test the nuli hypothesis (Ho)
no association between two row
and column classification variables, and alternatiYe hypothesis fl 1 ) which contradicts the null
hypothesis are:
Hn P. =p ·p Vi.j.
(2)

where p,1 p . p

i = II

r

j = L2.

(3
where 111,, e
i = .2, · r
j = 1,2.
c are empirical and expected frequencies for celi ij
respectively. The expected frequencies are the frequencies that would be expected under nuli
hypothesis that there was no relationship between the variables.
For preceding statistics to be valid. the expected number of observations
each celi should
be at least 5. Tfthis assumption is met sampling distribution ofthe test \alue is approximately
a chi-square distribution with (r- )( c- l degrees of freedom.
The null hypothesis will be rejected if the test \alue is greater than critical \alue at given
significance leve! a or, equivalently if the empirical significance leve] (p-rnlue) is less than
theoretical significance leve! u.
Likelihood ratio (JvIL ratioJ chi square is formulated in the same way as Peurson chisquare, but the computation of the test rnriable is based on the maximum likel ihood method.
So computed test value is usually very close in magnitude to I'earson chi square.
Coe.fficient Phi is a measure ofthe association between two categorical rnriables, defined as
a following function of chi square statistics:

(4)
and takes nonnegative values. If the value of coefficient Phi is zero. the variables are not
associated.
The contingency coefficient is a chi square measure oj association between two variables
measured on the nominal scale. Its range is [0,1] (where O means complete independence).

C= ✓

Table

-

%2
'
.
r+n

(5)

L =min(r,c)

408

VE [0,1] .

(6)

··--·-

'Stat1st1cs for Table of region by financial position
Statist
·---··-------Ch1,Square
Likelihood Ratio Ch1 Square
Phi Coeff1c1ent
cont1ngency Coefficient

are probabilities (for cells

i=L ..r: j=
of two dimensional discrete random variable. marginal column and marginal row
probabilities. respectively. If the components of two dimensional discrete random variables
are independent, each two-dimensiona l probability equals product of marginal column and
row probabilities.
The expected frequencies are the frequencies that wou!d be expected under nuli hypothesis
that there was no relationship between the variables. The Pearson chi-•square statistic is
computed as:
i.:

FN the contingency table of «region» by «financial position» the results of tests are
summarized the SAS printout (Table 1):

Cramer s

( ·111·

Value

Prob

21 .7352
22.0999
O .1903
1870
O 0952

0.1519
0.1400

• test ··alue
leve! (p-value1
,
.is ~1 ·73'i2
· · · and the related ernpirical significance
.
. . is.
The null hypotheses is accepted, since the p-value 1s greater than the the01et1c~l
: - ·fi1cance
·
1e,.e J 0 -- o.O'il'he sample data lead to the conclusion that the households m
s1gm
-.
.
.
..
Jifferent regions have similar statements of thetr financial pos1t1on.
. .
In this example the ML chi square is 22.0999 and has p-value
400. Th1s 1esult leads t?
the same conclusion as ( ·11; square /esr. Ali measures of association have small valu.~s, and 1t
can be concluded that there is no association between two cons1dered. vanables. 1 he sam~
"
1
•
50;o significance leve! is made for all other contmgence tables unde1
1.,0llC,USJOn a t the ..
investigation. The statistics are summarized in Table 2 and in Table 3
SljUClle

o s9

Table 2

j

.,, .

'Statistics

Value _ _Prob _ _
Stat1st1cs
~Value
. Prob_ .
Table of regi on,~ financial position
+ Table of settlement,s by financial pos1t10n
. -S - - 21 7352'
0.1519 Chi-Square
17.2012
0.3727 1
eh 1- quare
•
o
756
LRChi-Square
22.0999
0.1400 LRChi-Square
17.1559
,3,
Phi Coefficient
O.1903
Phi Coefficient
0.1693
Contingencv Coeff
O. 870
Contingency Coeff. 1 0.1669
, v·
o 0952
Cramer's V
0.0847
~C ramer s
___,_ _ ~
-+
.
fi
•1
• ·
1
Table uf h~ehold size by financial position
Table of ch1ldren ~y mancia pos1t101~ ~
I
~
h
s
.
•
12
0228
0
7424
Chi-Square
2.3510
0.6 l.1
C 1- quaie
• ·
LRChi-Square
12.1409
7342 LRChi-Square
2.3478
0.6 72 j
· t
O. l 4 1, 6
Phi Coefficient
0.6721
Ph 1. c oe ffi1c1en
Contingency Coeff
0.1402
1 Contingency Coeff.
0.6721
.,
, y
0.0708
Cramer's V
0.0626 . .
C ramers
.
Table ofregion by economic situation
Table ofsettlements by econom1c s1tuat10n
I
1 Chi-Square
, 18.9353 1
0.2720 1 Chi-Square
124.2268
0.0847
L R Chi-Square
19.6319
0.2373 L R Chi-Square
24.1620
0.0860
· t
0.1776
1 Phi Coefficient
0.2009 1
Ph 1. C oe ffi1c1en
C ff
1970
Contingency Coeff. 1 0.1749 1
Contin~ency oe . 1 O
1
1
,
y
O
0888
Cramers
V
.
0.1005
.
C ramer s
---+
.
. . .
.
Table of household size bv economic situation
Table of children by econom1c s1tuat1on
1 Chi-Square
1 19.8366 1
0.2277 Chi-Square
3.9787
0.4089
\ L R Chi-Square
19.32381
0.2522 L R Chi-Square
4.0672
0.3970
1 Phi Coefficient
0.1818
Phi Coefficient
0.0814
Contingency Coeff.
0, 1789
Contingency Coeff.
0.0812
1 C
, y
o
0909
Cramer's V
0.0814
I
ram
er s
·
1
1
t
1
Table ofregion by unemployment
Table ofsettlements by unetnD ovmen
Chi-Square
14.9567
0.5278 Chi-Square
19.9108
0.2243
L R Chi-Square
15.0362
0.5220 L R Chi-Square
17.1559
0.2477
Phi Coefftcient
0.1579
Phi Coefficient
0.1822
Contingency Coeff.
0.1560
Contingency Coeff.
0.1792
, y
o.0789
Cramer's V
0.0911
C ramers
1,

1

1

409

lable 3

___________ _
- - - -- p -b- - - -Statistics
Val
· Statisdcs v 1
------_ -- ---- - ~ - - - _ro
Pr'?_b_ _ _
a ue
:able ot household size b) unemp lo}rn~ nt--r--~ - __ _
1
rnent
unemploy
chi~~
ot
ie
_
-----'
i
0.091
e
Ch1-Squar
O 92180.9190
e
Ch1-S_quar
0.4840
0.0911
R Chi-Square
092!4
0.922
LRLh1-Square
0514
01610
Phi Coefficiem
0.0391
Ph1 ( "oefficient
O 1590
, Contingenc) Coeff
0_039
Contingenc) Coeff
0.0805
~ Crarner's__y_
J_ o. 0391
Cramer's V _
---f- 1
-'-T bl
bi
-- -.- ___<l ~~~ saving rnone_y
- _Ta e of settl~ ~ saving mone_y
O 3684
13.0088 'r'
Ch1-Sq~are
th1 ·Square
O 2359
18.5294 '
12 7383
R Ch1-Square
020!
19.277
quare
LRChi-S
03883
0.1472
PhiCoefficient
0.1757
Phi Coefficient
,
Continge1ic, Coeff
14 7
0.173
Coeff
)
ontingenc
\
v
__Cra111er's
o
O· 0850
1 Cramer's V
I
-------~
- - --'- _-_l__Q_ll
mone
~- - _Ia\J_le ot'_househ_<Jld size by salv inrig _
_ ~ -1- , _Table of children b) savmg mone)
,-- 8 588 ,
Ch1-Square
176
TI
-,
Ch1 Square
O,
·
i L R_ Chi-Square
O 1020
· - -'JI
O 7180 L R-C _
8.8225
0.0788
8 3738
_ , hi Sq~are
I
0.1196
Ph1 Coefficient
O. 35
;,h1 Coeffic1en_t
0.1188
1 Contmgency Coeff.
0.1 128
'-ontmgency Coeff.
Cramer's v
--J_

1

- 0.0691

---

r

_L

-

-

Crame ~

_j__

-

l 135

Conclusion
'fi
The sample data lead to the conclus1·on that at •01.
'10 s1g111 1cance le- ·el th
ere 1s no association
'
.
d t ~_fi .
between compon ent-varia bles of ('CI
l
·
·
vanables
1cat10n
ratt
s
an
,
h
. r~g10n. sett ement size,
ousehold size and children (househo lds with or with
out childr_en unde1 4 ). The same results
b t
are conduct ed for the tests of no associati
ot~er vanables (no compon ems of CCI
ween
~t
n
variabi°
tion
in consume r test and stratifica
t at ali consume rs in Croatia have the
lmeands
.
des.
·c
econom1
the
of
similar assessments
b •
an soc1a tren s Theref,
- me. we e 1ieve the consume r
.
.
survey has to be continue d and the tim
(especially CC!) should
be used for observin g following e selnes of the sune) results
h
·
n
eh
ng
forecasti
and
ammg
exp
,
'
•
,
a ges m t e pri vate
. .
. C
consump hon (as a referent series) and for c·hanges m
.
economy
s
roatia

References

11

Bahovec. V., Čižmešija M. (200 3) S'1

_

•

.

c~n

·es~onde~ce of factors
;
se!ected byfacto r analysis a/busine ss sun'.e~ :.~::t!~c~:~~;ce_
7
111
1
p, oduchon 111 Croatia.
ia
ush
e_
.
. t ·
SOR Proceedings (6 p ) The ?ti1 m
•
ernat10na1 svmposm
.

· ·

Slovema. Podčetliek, 24-26. Septemb er 2003.

m on operat1onal research in

.

. .
[2] McClave, J. T., Benson G p Si · h T (200
5). Stallst,cs far business and Economics.
9th edition. Prentice Hall. New. Y~rk nc1c ' .
[3] Nikic,_G., Šošic, I.,

.
Čižmešija, M. (2004). Estimate oifES1111

. .
C, oalla - a case study oj an

economy 111 transition. 27th CIRET C onf,erence, Warsaw Septemb er 200 4
.
//
·

h

'

ttp: www.cir et.org/us er/warsa w2004

EU progr

ifb

.

[4] •••The Joint hannonised
amme o. z~smes~ and consumer surveys, User guide
(2004). Europea n Econom Euro
Comm1ss10n, Directorate-General for Econom ic and
pean
y.
affairs.
financial
[5J. ................ ,Program support SAS

410

MODE L Oli BASI.C TRAIN IN(; FOR INFOR MATIO N SYSTIEM
DEVE LOPER S
Mile Pavlic. Marija Marinovic, Sanja Čandrlic
Faculty of Philosophy. University of Rijeka, Rijeka, Croatia
n1pavlic;ivffri.hr, 111arinm@ffri.hr, sanjac@ffri.hr

Abslract :

training on basic
The paper presents the training model for information system designers, providing
participation in
and
models
reading
,
modeling
data
and
analyses
knowledge in business processes
nals actively
professio
y
project teams. The training model is used with information technolog
included in information system development
Keywords: information system. design. methodic. training

1. lntrodu ction
n process for
.earning methods for business system analysis and data modeling is an educatio
which
model.
system
business
a
produce
to
aiming
s,
training informat ion system designer
have
models
g
Teachin
(IS).
system
ion
informat
an
can be used as a prototype for building
reasons:
g
followin
the
due
authors
the
hy
years
been improve d during past lwenty
approach es and
improvin g of methodi c and methods, emphasi zing particular didactic
s.
designer
system
ion
informat
practical work in education of
in the one-year
Over tirne, severa! Yarious training models have been developed resulting
e of one-year
objectiv
The
model.
training
daily
and
model
training model. weekly training
e is to make
objectiv
the
model
training
daily
the
in
while
skiHs,
training is to acquire design
n.
educatio
further
for
es
guidelin
provide
and
methods
students familiar with
isto, during
The objectiv e of the basic training for designers (or so called weekly training)
with
design,
systems
ion
informat
of
es
techniqu
and
methods
six working days, present
g
modelin
relevant
omitting
not
yet
,
contents
abridged
an
with
.
enough details, however
take
and
models
concepts , so that students. after complet ing their lessons, are able to read
theories, with
didactic
various
on
based
is
Training
teams.
ent
developm
IS
part in design and
J.
promine nt role of didactic asa curriculum theory, i.e., objective approach l3
various working
of
ntation
impleme
the
includes
s
designer
system
ion
informat
Training of
has been gained
models. The training was introduced in 1984 and extensive experience
studies for
since. A total of 6 one days seminars, 24 weekly courses and 14 one-year
included
were
designers has been performed_ Eleven generations of full tirne students
(approxi mately 30 students per year), totaling to over 1000 students [9].
technology (IT)
The paper presents experiences gained in education of information
n in design
professi onals working in IT centers in various compani es (andragogy educatio
standard
by
ted
school through seminars ) and IT science students. The content is represen
process modelin g methods defined in the MIRIS methodic.

2. lnforma tion system design methodology
) for recording
The basic problem in modelin g is to find a precise enough system (method
methods, basic
various
relevant knowledge. In drawing up various types of diagrams using
aggregation,
ation,
principle s of organization are implemented, such as: classific
generali zation and hierarchy [7].
r phase up
The methodi c prescribes phases of developm ent and activities within a particula
sequence
and
s
activitie
r
to the required extent of details, defines relations between particula
used by
be
to
methods
of executin g activities. The basic task of methodic is to define
the
defines
c
methodi
The
informat ion system designers in building a system model.

411

uinnect1v1ty of the models at the system design !evel and. on the base of these. the
procedures for producing software as a part of a future IS.
defining phases of
Various methodic solve the problem ofdesigning and building an IS
of information
schedule
The
these.
in
methods
certain
the processes and implementing
prototype.
radicaL
(cascade.
methodic
particular
for
differs
system life cycle phases mostly
have
these
of
All
others).
and
approach
radical
and
«V» approach asa version of the cascade
swiftness,
design
tools,
methods.
foatures:
some
lile cycle phases, but put forward
implemented measurement , improvement s and adaptations.
The basic issues included in life cycle phases. according to the cascade model, are as
follows: design. realization and maintenance 12 or analysis, design and implementati on l l
with overlapping activities of some phases and evaluation of the system during maintenance
phase
In more details. the breakdown of life cycle into phases is represented by: strategy - wh~
/problem definition), analysis - what (analysis of a problem). design - how (working out a
solution), building, i e .. programming - production (making an IS), testing - quality
assessment (check). introducing - the first implementati on with testing and maintenance
I4j, [12j, [6j.
A methodic is said to be well defined if it: decomposes life cycle phases into activities.
prescribes method and output models for each activity, describes relations among particular
aclivities, i.e„ input models and information required for modeling. determines conditions
for slarting and finishing the modeling, prescribes quality of output documentatio n and it is
proved enicient in real life projects.

3. The MIRIS methodic life cycle
The MIRlS methodic (acronym from Croatian MetodJka za ,Bazvoj Jnformacijsk og §ustava
= Methodic for Developmen t of Information System) is a set of methods and activities in
tirne. where objectives are designing and building an information system (IS). It has been
developed since 1984 and it was published in 1995 1O]. The MIRIS methodic was
developed through expansion of and Iearning from other methodologie s and testing methods
in research projects f8] It was tested in approximatel y hundred projects of various sizes and
it was improved on the tly.
The fundamental hypothesis on which the methodic has been deve!oped is the "system
decompositio n" and it is as follows:
• Dividing life cycle in three phases.
•
•

In the first phase, producing an abstract model of the entire system, and dividing
system into subsystems.
In the second phase, producing a model of the subsystem and setting appropriate
method for process modeling.

In the third phase, modeling data for relevant processes and defining application
architecture depending on the process model and data model.
Unlike this approach, it is possible to assume «decompositi on ofthe design process».
It is quite natura! to answer what to do first and what to do Iater, so that the hypothesis
that life cycle is to be ordered according to the activities execution sequence has become the
«actual» assumption in most of the methodic [5], [4].
The life cycle suggested by the MIRIS methodic is quite original. It resembles to the «V»
model, so that on the left side there are phases in which a business system is decomposed,
while on the right side, linear development and introduction of application is proposed
regardless ofrespective complexity.
•

412

lhe basic concepts of the methodic are as follow: methods, phases, proj~c.ts, forms and
tructure model shows their mutual relations and connect1v1ty.
.
tea~hrn:~~l ~ :~;h~dic uses three basic methods, namely: data modeling method, process
lication modeling method. These methods are adapted and
e
nd oals set and have original concepts and original
modelmg metho_d and app .
"Arhitekture
expanded followmg the requ1remenhtso da (Jng c· ro atian·· method for . modeling
.
.
d ling The APP met
tu " -- S 0 ft w are Product Architectures ) is an ongmal method leanmg on
_process mo ePru1zvol
Programskug
DTP and ER methods., l . the MI.RIS methodic are divided in two groups of phase_s:
modeiing (building IS)
. T a bl e and phvsical
l:y_c e m
the
. shown. 111
,
shown m
IS des,an)
g hfe
dof !"111
• l
l Phases
0
· p hases
. s1x
e group has three phases, so t here aie
ogica 2.mo
.. (in prev1ous vers1011
. in. total
Each
Table
. .
.
there were seven Ph ase s defined •l· Phases are further d1v1ded m actlv1t1es.

Table 2: Physical modelmg - bmldrng

.
.
.
Table 1: Log1cal modelmg - des1gn SP)
1
1 Phase 1: STRATEGI C PLANNING O_F _IS (
A 1 · . Defining and tratntng a team. 1
list of documentation and
na_ ys1s. f
1.1.
,
o a process,
decompos1t1on
--1
.
navigation through the system
relat1011s
and
subs.yst.e'.11s
Defining
Subsystems:
1 1.2.
· 1,3, Priorities: Determining pnont1es
1 1.4. Resources: Defining complete infras~r~~ture
1
1. 5. Plan: Planning main project and actlv1t1es
Phase 2: MAIN PROJECT (MP). _ _ _ _ ··---1
p
.
2. 1• PT: Drawing up proJec_t task
/ 2.2. DTP: lnterview, analys1s, process modeltng (DT ~
an
1 2.3. MP Processes: Analysis of process, problem
.
for improvemen~s .
proposal
1
--1
.
2.4. MP Dala: Dala descnpt1011
! 25. MP Plan: Planning project realizat1ons -----::-:--1
Mp Resources: Defining resource models ot the :

7

'z"];.

Phase 4· . SOFTWARE REALIZATION (SR)
~si~ning physical data base
d" .
.
_
~
, 4.2. Registering DB scheme 111 data 1ct1onary . 1
; _ , Producing prototype and generattng 1
43
.
application tree
,
4.4. 4GL or 3GL programm111g
4.5. Writing instructions an d exp Iana t .ion llf I
.
.
program solutions
4.6. Entering test data tn BP and test111g _
ANO
IMPLEMENTATION
5:
Phase
APPLICA TION (IAA)

1

j

~~~~~~~~~~=~==-=--,

----i

1

~:-=--::=...-'---: ---:-:-:-=-main project
~~~~~~ ~~~=;:- :::;:----- -/
.
.
(PR)
REALIZATION
~ e 3: PROJECT
1
l 3,1. DEV lnterview, abstraction, data modeltn. {EV)
. 3.2. Translation: Translation of dala model tnto DB
order to:
--l
1 scheme (RMJ
Defining architecture of
1 3.3. PR Architecture:

program product (A PP)

-------= ~/

~~:!!!l::~~~~ ~=~::::...::.::: .:..:.-=-=-:.-'- -'---~

3.4. PR Operations: Design of operation on DB

scheme

::!:

th
~;~~:sinform atization, where a series of model~ ~re
4. Descrip!ion of t~e ~IRIS me
. d t ·1 The important outcome is the defi111t1on
IS the first p b
Strateg1c plannmg
defined, describing a system clearly _ut _n_ot m e a1 .
ther good and detailed methodic. This
of subsystems and informatizati on pnontles.
h se there are numerous o
• 1 ·
me other methodic can be used here, for
h d'
p anmng p a '
In strateg1c
is not a specialty of the MIRI S met o ic, so so
phase
[l].is a phase of process mo de J'mg. This phase is performed for each particular
example
design
Main SPIS
. d fi d 'th a scope to prepare users before the
subsystem within a strategic plan.
In this phase, a set of docum_ents. I~ e_ n;ingw1process modeling as well as drawing up
'
interview. Following the preparat10n, m erv1e
facilitated.
are
diagrams
of data flow

413

Detailed design is the phase of d~ta m~delin g and operatio ns on dala using
a software
p~oduct On~ subs~stem from the mam des1gn can be divided into severa]
compon ents and
atter t~e matn des1gn p_h~se. detailed design phase can be perform ed for
severa] times. The
res_ul~ 1s a model ~es~nbm g a ~art of a system in detail. which represen
ts a solid basis for
build1~g the app_hcat1on~. Bas1c_ met~ods in this phase are: method
for semantic data
~odehn~1 (ER fiom entlty~rel~t1onsh1p). method for presentation of
database relational
scheme and method for apphcatl on modeling.
Softwar e product ion is the stage where various developm ent tools are used
(languag es of
the lil and IV generati on, Sl TBP • RAD tools • C ASE tools etc ) and
.
th e o b'Jec t'1ve 1s
.
· to
produce complet e software.
·
The _methodic is a~d it aims to be independ ent in terms of producti on tool
and process of
the software product1on process. lf CASE tools are used. process and
data models are
ad~pted to the. tool sel~cte~, that is, it is necessary to set translation of
the model from the
mam
. and detaded des1gn mto
. .a. model. acceptab le for the CASF. to ol . M o d e J p1.o d uce d m
·
~es1gn phases represents the 1n1t1al vers1on of the project documen tation,
which is translated
111tof the model readable by the selected tooL while the further documentati'on
d · h'
m a 1•n tenance ·
1s
per orme w1t m a CASF tool The power of concepts establish ed in the
MIRIS methodi c is
then !ost, but the power of the generato r and C ASE tool are gained.
I_?troduction and _implemen~ation is the phase where produce d software
becomes tool in
~s~1 s hands for c_arry111g out busmess processe s and it can take as much as
twice Jonger than
~ottware. p~oduct_1on phase. ~he managem ent should move an organiza tion
towards changes
111 behav101 and 1mplementat1on of complet ed solution. The
responsi bilitv• for introd uc111g
·1
· 1s
·
managem ent, w h I e IT professio nals should actively cooperat e in training
users and
1mprovements of the software.
Mainten an~e is th~ phase consistin g of executin g activities from previous
required
pl~ases due to 111troduc111g new or changin g the existing business processe
s and correctin g
failures.

?n

S. Model of basic informa tion system design classes
The objectiv e of the basic training for designer s is to teach methods of
busi·ness
1 · ·
. • •
. .
opera t'10n
ana ys1s, 111te1\1e':111g user~, orgamza tion of databases. building applications.
program design
and document111g 111format1on systems.
On ~h~ ba~is of th~ develop ment of informat ion system phases and methods
therein used
for trammg 111format1on system designer s; a training model for the MIRIS
d f d
h
·
•
me th o d'1c 1s
e 111e , as s own_ 111 Ta_ble ~- _Nu'.11ber of seminars is not complia nt with
phases or design
s~quence; rather, it has 1~ ong1_ns m_ ~ear's long research of the contents
and sequenc e that
p10duced the best results 111 bas1c tra111mg about IS design.
A more complet e model of one-year training differs from the basic model
The basic tr~ining of desi_gners consists of logical units (seminar s) show~
in Table 3 The
sequenc e, durat10n and class1fication into develop ment phases are shown in
Table 3.
Table 3: Model of lessons in basic trainin for IS desi ners
Class

Duration in 60 minute
eriods

Conce tual modelin, and database desi n
Hands on desi n and roduction of own model
Exam ~ roduction ofa re uired model

414

2
8
10
8

4

Phase acc.
toMIRIS
1
2

3
2,3

fhe basic training model does not include building and software introduction
phases. but
these are mention ed the introduction.
5.L A brief descript ion of particul ar seminar
Iopics from a particula r seminar are stated further on in this chapter, without
describin g the
importan ce
a specific topic. which is described in Chapter 4.
J11troductio11 in IS design
So!l. ware engineer ing, informat ion system, methods, methodic, methodo
logies, model s,
design. modelin g. IS design !ife cycle phases and software production, project
manage ment
commun ication and relations with users,
2. Analysis of business and process modelin g (ANA)
stem analysis. master project, project task, team analysis, data flow chart:
data tlow.
process. data storage, external systems, decompo sition. context
chart, hierarchical
descripti on. process types, balance. recomm endation s for drawing, constrain
ts, interv1ew,
practical represen tation of structural research, recognition of process and
data tlow. how to
develop the !S in a company . analysis and modelin g using students' example
s.
3. Concept ual modelin g and deve/opment of dala base (DDB)
.
.
.
Abstract ion, method entity-relations (ER), structure of bR: ent1ty. relat1on.
attnbute ,
aggrega1ion. . Constrai nts in ER: relations hip's and attribute ·s cardinali
ty,. Data analys1s
documen ts and respective modeling, independent and team modehn g, database
. 4(1L, data
111
dictiona ry. relationa l model relation, attribute, domain, candidate for
key. relation key,
external ke). constrain ts, relational operators, operator, normaliz ation, translati
on of ER into
a relationa l model. Object oriented approach, uML/OM L, executiv e proJect,
exerc1se 111
modelin g.
4. Design practice (DES) and realization of own project
.
Solving example s, team modeling, case analysis, realization of own proJect,
interview
exercise . review and suggesti ons for models in our work.

5.2. Charact eristics of basic training model
Example s of answers given during seminars: Requirem ents changed during
the realiz~tion of
a project, how to documen t a project, standard ization of projects and methods
, plannmg and
managem ent of developm ent, distribut ion oftasks, etc.
Scope, depth and sequenc e of curriculu m are defined for eac~ seminar.
These are three
basic didactic dimensi ons of teaching, aiming to a successful teach111g.
Scope of curriculu m is the issue of extent of knowled ge and skills that designer
s should
acquire and develop during particula r seminar.
The depth of curriculu m changes over tirne. Some topics are presente
d as a mere
informat ion such as normaliz ation, while others become more important.
The seq~enc e ofpresen ting the cuniculu m is based on the following principle
s:
e
from easier to more difficult
„ from known to unknown
„ from simple to more complex
„ from existing to desired
.. from program ming practice to abstract modelin g of a system.
. .
The sequenc e of seminars follows the IS developm ent life cycle phases, unhke
m the oneyear training model.
.
.
Success rate in training design could be measured more accurately apply111g
the followm g
criteria:
• quick familiari zation with business technolo gy of organiza tion system

415

"'
"'

quality of the project
quality of the devel oped JS program support.
is consid ered that the lessons ha-:e been impro ved
over tirne, which was suppo rted bJ
the fact that lesson s have been contin uousl y perfor
med as of 986 as well as suppo rting
iterature, textbo oks and presentations.
Durin g teaching, nume rous misco ncept ions were
noted. At early stage. results were
poorer. Half of the studen ts would not sit for exam
or they would not pass the exam . A fe\\
studen ts actually becam e IS designers.
There were quite a great numb er of studen ts
attend ing first seminars. At the tirne. a
numb cr of variou s metho ds for the same task were
theoretically studied. The teaching was
oriented towar ds the conce pts of metho d structure.
Mode ling process was disreg arded .
It was teachers who defined and achiev ed aimed teachi
ng goals. Stude nts showe d quite a
high degre e of know ledge and enthu siasm . But
somet hing failed. The numb er of studen ts
that becam e design ers was not as high as desired.
Particularly efficient study ing did not help
so new metho ds, which would foster creativity. were
search ed 3J.
Tcach ing can be well elabo rated, mater ials cover
ing topics could be prepared. studen ts
could learn every thing as planned, so we could
speak of a good teaching. Yet, studen ts are
not able to indep enden tly design IS, and the excell
ent teaching produced poor results.
Over time, teaching included know ledge from
critica l-com munic ation didactic. It has
been perma nently aimed to impro ve the teachi
ng. In teaching, demo cratiz ation and
huma nizati on is the potential future horizo n and
any failure in teachi ng is seen in that light
and used to acqui re new useful know ledge . Teach
ing is a comm unica tion proce ss. There is
intera ction between teachers and students. Teach
ing starts, lasts and ends through an
intens ive comm unica tion. Teach er is a moderator.
who is, during teaching. perma nently
availa ble for comm unica tion with studen ts.
Any quest ion to any direction establ ishes a
relation of new quality and provid es
oppor tunity for other stude nts to learn. Such flashe
s of digression are the sourc e of new ideas
and they increase the powe r of design er's thinki
ng - design ers are bom. In comm unica tion,
roles are chang ed, at times teacher assum es a design
er's role. and at times he is a user. The
same roles are assign ed to studen ts in their comm
unica tion.
Teach er is a comm unica tion mode rator and he does
not allow ineffe ctive comm unica tion
(wast ing tiine). This is students' role, too. and
studen ts are also entitle d to join the
comm unica tion. Perso n and his/he r attitud e are respec
ted in comm unica tion.
From the comm unica tion process, teache r measu
res and reads expec ted teachi ng results
(acqu isition of desig ning skill), as shown in
stude nts' statements, action s and decisi on
referring to simul ated reality. There is an active
comm unica tion in order to presen t skill
acqui red durin g teaching, while if objec tives are
not achiev ed, teachi ng is intens ified in order
to reach necessary know ledge and obtain result
s, i.e. teach a designer. The objec tive of
teaching is to teach stude nts how to design , there
is a confid ence in stude nts and, at last,
differ ence betwe en the expec ted and the mode led
by studen ts is analyzed, which results in
assess ment of the quality of a project soluti on and
design speed.
The succe ss of training is contro lled imme diatel y
during the training. The control can be
obvio us or hidden, throug h feed back instru ctions
, advic es, desire s, opinio ns, and answers.
The contro l provid es and enabl es the progr ess
of teaching. Other wise, the progr ess in not
effective.

416

6. Conc lusion
l
d I
basic training for information system designers
and the
l he paper presen ts t 1e mo e l th ., of teachi
ng comm unica tion. The training model has
d d ·l . k•1owledge as a cnt1ca em,
. 1 ac ics .' .
. h d
,.,
d mode ling of business systems before addre ssmg
an s-on
sernm ars ctelmed_ tor an_al: s1s a1~e uired tirne is six
days, 1.e .. 24 hours of lessons.
soft,\ are produ ctwn. Mimm um q
.
f
rticular topic can be redesigned as one-day
laUon o a pa
Acce 1erate d anll abridg.ed. presen
.
.
,J re students are not intended to acqu1.rc
or briefe r cours e
prond mg mform atwn
\\ 1e
.
the design skiHs
.
the
basis of need for makin g the teaching
rhis paper resulted from expen ence, . on
, ., :
.
. .. f d . ers is a creat!ve commu,m:a
tion process.
enll\ e. .ctucauu,1 o es1gn
.
11·
.
ld
b
rade
•hrou,il1 establ ishing a metho d tor
effects
teac mg cuu
e
'
•
.
"°
o
urtlter researc h
.
, d d· . ,· 11 \ an ing ihe teaching by sequence.
. .
1ent·a1s after comp lete
e uca hl . ·•
measu ring d es1gnmg
po· 1 1
•
ns and emot1onal moo ds
d
, l effects effects m group. result re!atio
·
exam ples. m I\ •u.1ua
•
..
b r
easuring changes in attitudes and explo nng
(confiu ence anu_ rnethods. att1tudes, el1evfesJ, m
.
tion
lt
is
quite
clear
to
the autho rs that
.
.h .
o comm urnca
.
teach mg w1L ou, an d w1. th vanou s leve
. II
. t d and that initial states can cons1. derab lv
measu remen ts in this field are espec ia
y aggia va e
influence the resuits and ..:onc!usions.
References
.
ic planm ng ,f mform auon
D ,v k
l\, Vrcek .. t rameNOI·k r,,1 r s·,·ateg
L,
.
,
Hrumec.
ecsa . ,·
.. f the 7th An1ericas c,,nfe rence nn lnformaL1•11
Svsten1s.
s•,stems. In Proce eomg s u,
·
~
'7''
Boston. LSA, 2001 P· .u - 17n7
;u, .. .
.
H!Z. Sa\_Jetcnanje ASf 8. Opatq a,
Cippi co.
Faza
iz\'ed
be
infonnac1,1skng susla\ a.
2
v

.

•

996 .. P 9-4U

·
.
994
278
Pedagc,gija. temel.1na znan1,a, l~duca. Zagi~
b.
.. p. .
µ. ' , . - !ali.
,udjon s,
1 .. lntroduction
Syste ms Ana1vs1s and Des1gn, ier.t,c
4. Hawn szkie w,.:z.
e
l\iew York. 988 p 373 .
A 1 't t
i
Devel opers Prirner,
at1on
Svste ms rc 11 ec ur e - a Svstem
5. ,nmon
.. Vv' . H ··. Inform
.
··
·
·
235
·entic
e-l la!L l\iJ. 986 .. p,
Pl
.
· k 11J sistem
·
6. Kovačic A .. Vintar. M. Nacrtovan1e in gradnja informaciJs
ov DZS,
,juhljana, 994. p. 6 J H An Overw iev ofKno
wledg e Representat1on
7 Mvlo poulo s.
Levesque, · ·
.
C „
NY 1984 p 3-17
·
.
·
l
s
smm
on1er
ence..
•
·· ·
Proceedrngs oflnte mat10 na ' yrn~o . . ,.
..
kih
sustav a IV saviet ovanj.e o
., M
Metod
ika
pro1e
kt1ran
p
m1orm
ac1js
8, Pav l1c. • ··
,
.
.
· ..
5- do 5-16.
informatičkoj dj~latnost1 u ZO R1Jeka, Opati•·ja, 987
E5
ati'a, 993., p. 8 l do 8-8
9. Ppavl i~ MM ~eh~~~~og~:n~~~:~~:~;~faa~:;:;~a~:skih
s!ta~ a, CASE 7, Opati ja, 1995., p.
O. av 11c. .
v

·i

1- lt3
L Advanced Structured Analysis and Design, Prenti
ce Hall, Londo n, 1998., p.
P e ers, .
272
.
J h Wiley & Sons NY, 200L p. 327
12 Simon . J.C. Introd uction to Inform a~on Syste
~~' o n b 1992
104
·
•
•
•
v'
Skols ka knjiga Zagre ,
,, P
13. Srica, V.: Uprav!Janje ~r~at1vnosc
~, k
tava
I~fotr
end,
vol.
43,
br.
2., Zagre b, 1996.,
14. Varga. M„ Faze razvo ja mformacijS og sus
'
p. 54-57
l

417

APPLI CATIO N OF THE QUEU EING THEO RY ON CALC ULAT ION
OF THE OPTIM AL NUMB ER OF EMPL OYEE S OF THE CALL CENT RE
1
Marko Potokar .Gregor Miklavčič2
3
and Mirjana Rakamaric Šegic
1
Bankart (lo.o .. Celovška 150, !000 Ljubljana. Slovenia,
email: marko.potokar@bankart.si
2
Bank of Slovenia, Slovenska 35, 1000 Ljubljana, Slovenia
email: gregor.miklavcic@bsi.si
1
Politechnic ofRijeka, Vukovarska 58. 51000 Rijeka, Croatia
email: mrakams@veleri.hr

ABSTRACT
s) ofthe call-centre.
The paper deals with the calculat10n ofthe optimal number ofemplo yees (operator
the first part ofthe
In
theory.
queueing
on
based
e,
call-centr
ofthe
model
We have derived a stochastic
second part of
the
In
.
optimum
its
d
calculate
and
1)
(model
function
objective
the
defined
article we
y analys1s. All
sensitivit
d
performe
and
2)
(model
the artide we developed advanced obJective function
ofthe
statistics
working
the
from
gathered
we
which
data
the calculations are based on real
e.
cail-centr
the
of
operators

KEYWO RDS
sens1tivity
Call-cenlre, stochastic model, queueing system, Poisson process, stationary regime,
analysis.

L INTRO DUCTI ON
terminals, to
The Bankart compan y offers card and ATM processing services, as well as POS
in
tions
organisa
king
non-ban
d
intereste
to
and
all banks and other financial institutions,
card
debit
and
t
paymen
effecting
for
data
with
users
its
Slovenia. The compan y also provides
s.
machine
ice
self-serv
services
and
ons
payments, authorises transacti
ATM and
One ofthe most important services provided by Bankart compan y isto manage
ng
POS network and to provide conect information for its users regarding the processi
which
re
services of their cards. This services are provided within the so called call-cent
operates in non-stop regime.
ofthe operators
The mathematical models ofthe call-centre developed to optimize the number
theory.
queueing
of
being on duty shown in this paper is based on methods

2. THE MODEL OF THE CALL-C ENTRE
CTI server.
In the call-centre incomin g telephone calls are distributed by special server called
g calls are
The entrance in the system is represented with unique waiting line (queue). Incomin
callthe
delivered to operators by the method FIFS (First In First Served). We can interpret
centre as the M/M/m/oo queueing system where:
)..>0
• telephone calls (jobs) arrive in accordance to a Poisson process with intensity
• times of the services are distributed by exponential law

418

419

@

there are m ;::,_

Parameter h shows the opportunity costs ofwaiting of each individual customer per minute. We
estimated the value b = 3*a (b = 63 SIT/mi11.). Because it is difficult to estimate the value ofthis
parameter nght, we dec1ded to check the quality of models with sensitivity analyses.

identical servers operators)

• one server (operator) serves one job with intensity µ >
• an arrivingjob goes directly into service ifany ofthe servers are free; otherwise the
job joins the waiting line (the queue) with infinite waiting places

For calculation of parameters and µ of the objeclive function we have used the slatistical dala of the
applicalion of the CTI server for the period of one monlh (March 2005) that has 230 working hours.

• jobs leave the system after service completion

•
•
•

111 H 1°1°1

;i, =

o~---)
•

"'

1

'."_~~

(rp)2

(rp) 3

11 = ...!.~--

(rp)'

+ --

r:

- p + so. ; p

2!

3!

+ (rpY __
I_

i

(r-1)!

r!

n, = the number of calls in one hour U

=

0.427/111i11.)

J

t 1 =the i11divid11al serving /ime i11 111in11tes Iµ

= 0,29 l/111i11.)

The objective function is compound of two pa1is: the cost of serving the customers (a•s) and
the cost ofcustomers waiting in line(t•b•Wq)- So we define the objective function as
where:

= Ji/µs

a = the cost ofindividual server (operator)
s = the number of servers
b = the cost of individual call
'A, = the rate of incoming calls: the rate is calculated as one month average:
µ = service intensity
Wq = the average waiting tirne
minutes)

From conditions for stationary regime the condition p < 1 (s> Ji/µ) must hold:

S=l+rp+ (rp/ + (rp)i + ... + (rpY

;

3.2 The objective fi111ctio11

has the following form (Potokar et.al.,

(rp)'- 1 (rp)'
(rp)'
+---+--+-(r r'
r1

.

·l

II

2004):

S = l + rp + - - - + ---- - +
2!
3!

i-i

Z:ti 1

We have defined the model for tirne interval from Monday to Friday for the hours between 7
a.m. and 5 p.m. Values ofparameters ofthe objective function (see chapter 3.).are also
calculated for this interval. Within this interval the stationary regime exists.

L, ~

230

230-60mrn

Our statemenl that the waiting line has infinite waiting places is based on statistical Jata
gathered from the CTI server. It has never happened that waiting line was full.

In such a condition the sum S= +

l
.

1

2.,ni

;p=Ji/µs

(m

1-p

The average waiting tirne is:
w

nq

P(i;::,_s)
(s-p)'•p 0
"
d
1
=- - . w h ere p/·
1 t ~s)= -~-~-;p=AJµs an p 0=-- ----;-

s.µ - Ji,

s!•(l -

I

p)

;~o

(s·

4. ANALYTICAI. METHOD OF CALCULATlNG THE OPTIMAL NUMBER OF
THE OPERATORS

11
. ;p= N/JS

p) + isJ} _
i!
s!-(I - p)

We derive the optimal number ofthe operators by calculating the minimum ofthe objective
function:
d(a •s+ b ·Ji,· W (s))
miu J = - - - - - - '9- - =O; p = Ji/µs < 1 ~s> Ji/µ
ds
d(a·s+b J.,-W/s))_ d(a·s)+ d(b-J.,-W9 (s)) =a+b-J.,, dW/s);

3. THE OBJECTIVE FUNCTION AND ITS PARAMETERS (Model 1)
In our case we are interested in costs ofindividual costumer being served and in opportunity costs that
rise if costumer leaves the system without being served.
3.1 Data gat/1eri11g

ds
We can write Wq(s) as:

Ali the calculations are based on real <lata which we gathered from the working statistics ofthe
operators of the call-centre.

ds

17

2.,BP;

Wq(s)

We calculated the parameter a a s a = ~ where BPi is the operator gross salary in Slovenian
17
tolars, converted in tolars on minute of working (BPi = (basic gross salary * tariff coefficient +
bonus)/nurnber ofminuts in one month; a = 21 SIT/mi11.).

ds

ds

(S·~)'
l
(s·~} (s·~l
s•µ

= -------~__.:~-~:------:---::--=,
Ji, )

(s . µ -

Ji,)· s!·( 1 -

-s,-µ .

,_ 1

~

+ s!•(l - ~ )

i!

s•µ

420

..
1

421

program Excell to
The derived funcl1on 1s very d1fficult to resohe. We rnther used the
rs
operato
of
number
optimal
the
dete1mine

WITH EXCEL L
5. DETER MININ G THE OPTIM AL NUMB ER OF OPERA TORS
parame ter s and
We calculated values ofthe objective function for different values ofthe
function are shown
e
objectiv
ofthe
values
ted
given values ofparam eters a, b, A inµ. Calcula
in the Graph bellow.

g parameter a in the
In Tahle we presented the results of the sensitivity analysis. By changin
< a < 51,4. In relation to
model 2 we have calculated that the inte1val, where s= 3, equals 1,6
g a than model l.The
model 1 the interval has widened, so model 2 is less sensitive to changin
in relative terms.
a,
expected costs are increasing very similarly to increment ofparam eter
Tube/ 1 1IIrjl uence o,fCJIGI1ging a 011 s, C and C1/C*
( (in SIT/min)
C/C* (in%)
s
1
a/a* (in%)
~(~~ SIT/min)
-73,2
17,33
3
-75 i
5,25
-48,8
33,08
3
-50 1
10,50
f--24,4
4_8~
!
__
-25
15,75
.l - - - - - · - ~ ---+
- - ----o
i
3
+i
64,58
0'
21,00
-f-24,4
---ti- +- ___j9, 33
2 t t -_ _ --26,25
SO!
___
f48,8
3,
96,0§ [_
{
Jl,50 j -·--- ---+ --- · ~
.
'
1
i
100
..
42,00
t---148
52 00
t-- -ions
ca/culat
own
Source:

r--

1
1

1

ter s
Graph l. val11es ofthe ob;eclive fi111clio11 far dijJel'e/11 val11es of the parame
SIT
400

--

+--

----7
1

300

1

t

~---,
, __________

200

1

j

100

o
00
-200

4

5

6

7

8

9

10

11

12

13

1

~

14 1

----1i

l

j

-J

1

_ _ _ _______,
-300 ~ - - - - - - - - - - - - _ _ _ _ _ _ _ _

---

So11rce: own ca/c11/atio11s

----s

We can see that the minimu m ofthe objective function is when s equals
number of operators is 3.

So the optimal

6. ADV ANCE D OBJEC TIVE FUNC TION (Model 2)
costs and average
In ~~e fir~t m?d~I we presum ed that the relation between opportunity
this relation is more
that
wa1tmg tirne 1s lmear. But from practical experience we can assume
ied and so
unsatisf
more
and
more
is
exponential because the custom er waiting in the line
tirne spent in
with
rise
costs
nity
opportu
more likely to get another provider. The custom er's
form:
the
in
n
functio
e
objectiv
ed
advanc
write
waiting line as well. We

7. SENSI TIVITY ANAL YSIS

_J__

. _
2: Jnfluence o'chang ing b 011 s, C and C/C*
11
Tabel
fin%)--1
C/C*'--'"-~-:
SI'T'/
·
'
~ C (in -::·. ::.'. .:m.:::1:::n"-,l -f-.----'r-:h(in SIT/min) i 'L b/b* (in%)
-14,4
5653,~96 i
~5-- i-15,151
'
- 1,2 i
,, 1
~ --50 •
1---· ·--31,5 0
-0,6 •
I
64,!9
3 ,
-25 i
47,25
o~
1
64,58
o
63,00
i--··
1'2 1
64,98 :
- · · - - - · 25 i - - - 3
78,75 i
~
,
65,38
_
+
}
50
94,50
~
.
___66,17 • - - ~
3'
100 1
~····-- 12600
31 4
"'---!--+- --··· +--_,1
_,
'
'
·
•
84,86
_
_
4
l~ __
- -870
:L ___----~----Source: own ca/c11/atio11s

t- -

~=-

+-s

-¼

1

! __

i

.l.i. ---

+--

rate). Firstly, there ar:
In Table 3 we presented the results from varying p_arameter ,l (mTi~al
ter /4 must be 0,353 again no significant changes comparing both modeis. Secondly, patame
,1,, < 0,796, ifwe would like to have s= 3.
Tabel 3: Influence of changillf! ,l. 011 s, C an d C/C*
s
'}.,/'}.,*(in%)
'}., (number of calls/min)
-75
0,10675
-50
0,2135
·
-25
0,32025
o
0,427
25
0,53375
50
0,6405
100
0,874
Source: own ca/culat1011s

1

1

1

422

z+-

al cust~mer per minute
Parame ter h shows the opportunity costs of waiting of each individu
rs bemg o~ ~uty. The
operato
ofthree
case
in
and should be inside the interval 25,8 < b < 869,
me 1elat1vely
changes
that
means
wh1ch
one_
s
previou
results ofthe analysis are similar to the
ant.
s1gmfic
not
agam
are
2
and
1
model
n
betwee
ces
small (see Table 2). The differen

1

is sensitive to changes in
In this section we analyse how the optimal number of servers
presented here in a way
parameters a, b, ,l and µ. Only results of the second model will be
Sensitivity analysis is
that one parame ter will change and the others will remain fixed.
and then we looked
ter
perform ed in Excel, where we entered different values for each parame
model.
ofthe
solution
for the optimal

- -

-

1i
2
2

3
3
3
4

C (in SIT/min)
35,92
46,62
57,60
64,58
65,74
67,95
84,33

C/C* (in%)
-44,4
-27,8
-10,8

o

1,8
5,2
30,6

s = 3, if parameter µ is
In the last table we deal with parameter µ (service inten_sity), w~er~
ant change.
s1gn1fic
no
o, 156 < µ <0,352. Comparing both models we again reahsed

423

Tabel 4: Influence of changing µ 011 s, C mul C1/C '*
1 µ (numb~~f calls/min) j J:11µ* (in_o/c_o)..... i
s

"~;~;

f--

0,2 825
0,291
0,36375
0,4365
0,582
Source: ow11 ca/c11/atio11s

~,~

-25
O
25
50
100

!H. '~: ~~
j (

_3_
3
2
2

1

2

(1r1 Sll/min)

.;~:

C/C* (in%)

]
1

66~,3_0--1---_
2,7
64,58
O
61,65
-4,5
__5_2~,5_2-+---~-----18_,7___,
46,26
-28,4

Hence, from the present analysis we can sum up:

2.

From the sensitivity point of view models and 2 do not differ significantly from each
other.
When we are changing parameters u and h, the influence on the optimal number of
operators is only slight, while changing ,1 and µ, \Ve have more significant effect

8. CONCLUSION
In this paper the problem of determining the optimal number of the employees of the callcentre was solved by application ofthe model from the queuing theory. Tfwe compare the
results for both models we see they match. This is one indicator that confoms our
presumption oftheir validity.

7. lntemational Abstracts in Opeations Research: Volume 55 Number 2 March 2004;
Serial Numbers 85946-86395; A complete and simple solution for a discrete-time
multi-server queue with bulk arrirnls and detem1inistic service times
8 Karlin S. and Taylor H .. A First Course in Stochastic Processes. 2d ed. Orlando, Fla.:
Academic Press, 975.
9. Koechel P„ Stohastic Modelling; Lecture held at the University Ljubljana 1999
Legiša P.: Analiza!; zapiski predavanj, FNT 988
Potokar M. Miklavčič G.: Teoretična analiza in empirična preverbal modela za
določitev o~timalnega števila strežnih mest v klicnem centru. EF Ljubljana, April
2004. 37.
12. QTS software; John Wiley & Sons, Inc; http://www.wile y.com;
fftp://ftp.wiley.com/public/sci tech med!queuerng theory:
http://www.geoc ities.com/gtsplu s
.
.
.
SOR'0J Proceedings: The ?1h Intemahonal Symposmm on Operahonal Research
Slovenia; Podčetrtek, Slovenia, September 24-26. 2003
. .
. .
14. Whitt, Ward: lmproving Service by lnforming Customers About Antic1pated Deiays
Management Science, Vol ume 45, Issue 2, 1999, str. 92-207
_
15. Winston Wayne
Operations Research Applications and Algontl1111s. Helmont
Duxhury Press, 99 l 025
6. http://www.sigo v.si/zrs/publikac ije,skd!skd02s .doc

But we have to be aware that accuracy ofthe results we get with such models strongly
depends on the used data. In our case the most important factor for the validity of the results is
the rate of incoming calls ( ;i,) and service intensity ( µ) that we calculated. If the data are not
observed during a sufficiently long period oftime, serious problems might appear regarding
their accuracy.
The sensitivity analysis shows that models and 2 do not differ significantly. The influence
on the optimal number of operators is only slight when changing parameters u and h, while
changing ;i, and µ, we have more significant effect.

REFERENCES
l. Doob J.L.: Stochastic Processes. New York: Wiley, 1953.
2. European Joumal OfOperational Research: Volume 156, Issue 2, 16 July 2004; An
analyzing of queuing systems with multi-task servers; Z. G. Zhang and N. Tian
3. Gihman I.I. and Skorohod A.V.: The Theory of Stochastic Processes I-III. Berlin:
Springer, 1974.
4. Hudoklin-Božič A.: Stohastični procesi. Kranj: Univerza v Mariboru, 1986. 180 str.
5. Hudoklin A.: Stohastični model, Organizacija, revija za management, informatiko in
kadre, Založba Modema organizacija, letnik 32, št. 10, December 1999, str. 535 - 547.
6. Intemational Abstracts in Opeations Research: Volume 55 Number 1
January/Februar y 2004; Serial Numbers 85496-85945; Analysis of a finite-buffer
queue with discrete-Markov ian arrival process: D-MAP/G",b/1/N; 85911

424

425

The 8th lnternational Svmposium on
Operational Reseerch in Slovenia

S@R
Nova Gorica, SLO\/ENIA
September 28 30, 2005

,

426

Addreaes of SOR'OS

n

th
(The 8 lnternational Symposium on OR in Slovenia, Nova Gorica, SLOVENIA, September 28 · 30, 2005)
1

II)

First name

Sumame

1.

Zdravka

Aljino vic
-

2.

Josip

Arnerič

Street and
Number

Instituti on

Malice Hrvatske

University of Split, Faculty of
Economics
-+-------University of Split, Faculty of
Economics

+

3.

4.

Zoran

6.

7.

8.

Vlasta

Majda

Tomaž

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Jožef Štefan Institute

Marija

Croatia

21000
1

Babič

University of Split, Faculty of
Economics

i i

Sloc~oia

Matice Hrvatske
31

21000

Split

Croatia

Bastič

University of Maribor, Faculty of
Economics and Business Maribor

Berlec

University of Ljubljana, Faculty of
Mechanical Engineering

Bogataj

University of Ljubljana, Faculty of
Economics

Bogataj

University of Ljubljana, Faculty of
Economics

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tomaz.berlec@fs.uni-lj.si

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guest.arnes.si

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Street and
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1

Bernhard

Bohm

Vienna University of
Technology,lnstitute for
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Economics

11.

1

Matej

Borovinšek

University of Maribor, F aculty of
Mechanical Engineering

12.

1

Darja

Boršič

University of Maribor, Faculty of
Economics and Business Maribor

13.

1

Marko

Sotinčan

University of Zagreb, Department
of Mathematics

Post code

Austria

bernhard.boehm@tuwie
n.ac.at

Smetanova 17

2000

Maribor

Slovenia

matej.borovinsek@unimb.si

Razlagova 14

2000

Maribor

Slovenia

darja.borsic@uni-mb.si

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mabotinc@math.hr

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cesta

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Lorana d.o.o.

15.

1

Lorenza

Castelli

DEEI - University of Trieste

Via A Valerio, 10

16.

1

Katarina

Cechlarova

P. J. Šafarik University, Faculty of
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Jasenna 5

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University of Zagreb, Fa~:;; of
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18.

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University of Rijeka, Faculty of
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20.

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University of Maribor, Faculty of
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1

Uni ve rsity of Maribor, Faculty of
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anton.cIzman @fov.ummb.si
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Number

Institution

Draženka

čižmic

University of Zagreb, Faculty of
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23.

David

čreslovnik

25.

Nadja

Samo

University of Ljubljana, Faculty of
Economics

Drobne

Uolsecsily of Ljcbljaoa, f aoclfy of
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Dumičic

University of Zagreb, Graduate
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26.

Ksenija

0ukanovic

University of Maribor, EPF

27.

Igor

28.

Andreas

29.

Ehrenmann

CORE, Universite Catholic de
Louvain

Erjavec

University of Zagreb, Faculty of
Economics

Daša

Fabjan

University of Ljubljana, Faculty of
Maritime and Transport

Matjaž

Feltrin

--30.
31.

1001

Croatia

1

Kardeljeva
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samo.drobne@fgg.unilj.si

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ehrenmann@core.ucl.ac
.be

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University of Zagreb, Graduate
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liljana.ferbar@ef.uni-lj.si

Department of Computer Science
and lnformation Theory,
Budapest Univ. of Tehnology and
Economics

Muegyetem rkp.
3-9

H-1111

Budapest

Hungary

fleiner@cs.bme.hu

Department of lnformation
Technologies, Faculty of
lnformatics and Management,
University Hradec Kralove

Rokitanskeh6 62

50003

Hradec
Kralove

Czech
Republic

martin.gavalec@uhk.cz

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ID

First name

I Surname

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32.

Liljana

I

Ferbar

University of Ljubljana, Faculty of
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33.

Tamas

I

Fleiner

34.

Martin

35.

Janez

36.

I Gavalec

37.

Tihomir

38.

University of Ljubljana, Faculty of
Administration

I Gosarjeva ulica 5

Hrovatin

University of Ljubljana,
Biotechnical Faculty

I Večna pot 83

1

1111

Ljubljana

Hunjak

Faculty of Organization and
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Pavlinska 2

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40.

Hubertus
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RWTH Aachen University (D),
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41.

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Ju run

University of Split, Facu-lty ofEconomics

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49.

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Lipušček

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51.

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Mechanical Engineering

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1000

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I ~,;;,~,ity of W,~ljaoa, Fao,lfy o'J
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University of Ljubljana,
Biotechnical Faculty

Manger

University of Zagreb, Department
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Post code

Town

Country

E-mail

1000

Ljubljana

Slovenia

spela.malovrh@bf.unilj.si

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Bijenička

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David

Department of Computer
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GlasgowG12
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University of Split, Faculty of
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Glasgow

Matice Hrvatske
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57.

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58.

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Omladinska 14
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Privredna banka Zagreb d.d.

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University of Ljubljana, Faculty of
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Kardeljeva
ploščad 17

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61.

Luka

Trg J. F.
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Goranka

University of Zagreb ,
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63.

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University of Zagreb, Graduate
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Univer~ity ofR;eka, Faculty of
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Marko

Pivac

University of Split, Faculty of
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Matice Hrvatske
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Department of Mathematics,
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--1

The 8th lnternational Symposium on
Opernlional Research in Slovenia

S@R
Nova GoricQ, SLOVENI/.\
September 28 • 30, 2005

Appendix:

"

T

1

Austrian Science and Research Liaison
Office Ljubljana

ASO LJUBLJANA
AUSTRIAN SCIENCE ANO RESEARCH
LIA180N OfFICE LJUBLJAHA

Main tasks of the Austrian Science and Research Liaison Office Ljubljana

(ASO Ljubljana)
ASO-Ljubljana airns to facilitate and intensify scientific contacts and cooperation between Austria,
Slovenia and South-Eastern European countries. In close thematic cooperation with the new science policy of the
Austrian Federal Ministry of Education, Science and Culture (bm:bwk) in South-Eastern Europe, the ASO
Ljubljana serves together with ASO Sofia/Bulgaria as focal point in this reg1on.
By organizing and (co-)financing of international seminars, workshops, conferences, as well as training
seminars for researchers from SEE region the ASO Ljubljana prov1des and promotes opportunities for
collaborative work, scientific networking and cooperation with institutions of higher education and research
facilities in Austria, Slovenia and the SEE Countries.
Special attention is paid to joint research projects in Et:-prograrns, where ASO Ljubljana acts as project
partner as well as mediators between potential project partners.

Main fields of activities of ASO Ljubljana
11!!1 Contribution of science to EU-integration of the Western Balkan Countries 11!!1 Governance 11!!1 Civil Society Knowledge Society l!!I Science and Technology Policy l!I Technological and Regional development in SEE l1l'I
Entrepreneurship development in SEE li!!I Information and Communication Teclmologies (ICT) li!!I Ethnic
Minorities in Science

Austrian Science and Research Liaison Office Ljubljana
director:
office employees:
address:

dr. Miroslav Friedrich Polzer (economist and environmental engineer)
Gorazd Weiss, mag. Marija Malle
Avstrijski znanstveni institut v Ljubljani
Dunjaska 104; SI-1000 Ljubljana
tel.: +386 1 5684168; fax: +386 1 5684169
e.mail: polzer@zsi.at
homepage: www.aso.zsi.at

Head office/Coordination in Vienna:
Centre for Social Innovation - Zentrnm fiir Soziale Innovation
Linke Wienzeile 246; A-1150 Vienna
tel. +43-1-4950442; fax: +43-1-495044240
e-mail: institut@zsi.at

KNJI ZNICA FGG

Proceed ings SOR
Rupnik V. and L. Bogataj (Editors): The 1s
Research , SOR'93 . Proceedings . Ljubljana :
Section for Operational Research, 1993,

-

C

z

J 519.8

C
f"7

::o

INTERNATIONAL.8
SOR

N
::D
C

r

L.

C

OJ

r

L.

::D

llllllllllllllllllllll 111111111111111111

z

020050947
- ~ .,,~,f'=""'"'..-' ,- - Rupnik V. and M. Bogataj (Edi t ors): The 2
Operational Research in Slovenia, SOR'94 . Proceedings . Ljublj ana:
Slovenian Society Informatika, Section for Operational Resea rch, 1994,
COBISS I]

275 pp .
Rupnik V. and M. Bogataj (Editors): The 3rd lnternational Symposium on
Operational Research in Slovenia, SOR'95. Proceedings . Ljubljan a:
Slovenian Society Informatika, Section for Operational Research, 1995,
175 pp.
Rupnik V., L. Zadnik Stirn and S. Drobne (Editors .): The 4th lnternational
Symposium on Operational Research in Slovenia, SOR'97 . Proceedings .
Ljubljana: Slovenian Society Informatika, Section for Operational Research,
1997, 366 pp . ISBN 961-6165-05-4.
Rupnik V., L. Zadnik Stirn and S. Drobne (Editors .): The 5th lnternational
Symposium on Operational Research SOR '99, Proceedings . Ljubljana :
Slovenian Society Informatika, Section for Operational Research, 1999,
300 pp. ISBN 961-6165-08-9.
Lenart L., L. Zadnik Stirn and S. Drobne (Editors.): The 6th lnternational
Symposium on Operational Research SOR '01, Proceedings . Ljubljana :
Slovenian Society Informatika, Section for Operational Research, 2001,
403 pp. ISBN 961-6165-12-7.
Zadnik Stirn L., M. Bastič and S. Drobne (Editors): The 7th lnternational
Symposium on Operational Research SOR'03, Proceedings. Ljubljana:
Slovenian Society Informatika, Section for Operational Research, 2003,
424 pp. ISBN 961-6165-15-1.

ISBN 961616520-8