Informatica 41 (2017) 71–86 71 
Aggregation Methods in Group Decision Making: A Decade Survey 
Wan Rosanisah Wan Mohd and Lazim Abdullah 
School of Informatics and Applied Mathematics 
Universiti Malaysia Terengganu, 21030, Kuala Terengganu, Terengganu, Malaysia 
E-mail: wanrosanisahwm@gmail.com, lazim_m@umt.edu.my 
 
Overview paper 
 
Keywords: aggregation phase, Choquet integral, MCDM, interacting, survey, interdependent 
 
Received: August 3, 2016 
A number of aggregation method has been proposed to solve various selected problems.  In this work, 
we make a survey of the existing aggregation method which were used in various fields from 2006 until 
2016. The information of the aggregation method retrieved from some academic databases and 
keywords that appeared through international journals from 2006 to 2016 are gathered and analyzed. It 
is observed that eighteen over ninety five of journal articles or nineteen percent applied the Choquet 
integral to the selection process. This survey shows this method most prominent compared to the other 
aggregation method and it is a good indication to the researchers to expand the appropriate 
aggregation method in MCDM. Besides that, this paper will give the useful information for other 
researches since the information given in this survey provides the latest evidence about the aggregation 
operator 
Povzetek: Predstavljena je analiza metod za združevanje odločitev skupine.  
1 Introduction 
Decision making is one of the most widely used 
management processes in dealing with real world 
problems which is typically characterized by complex 
and difficult task.  Multiple criteria decision making 
(MCDM) has been one of the fastest growing 
knowledge areas in decision sciences and has been 
used extensively in many disciplines. For example, 
Roy [1] developed a multi criteria decision analysis for 
renewable energy sources where it deals with the 
process of making decisions in the presence of multiple 
objective. Many researchers used other diverse 
methods of MCDM to solve the decision problem [2]. 
Basically, MCDM is a branch of operation research 
models and a well-known field of decision making. 
Both quantitative as well as qualitative criteria and 
attributes can be solved using the methods and it can 
analyze conflict in criteria and decision makers as well 
[3]. MCDM problems usually divided into two types: 
continuous and discrete. MCDM problems have two 
categories: multi-objective decision making (MODM) 
and multi-attribute decision making (MADM). By 
MODM methods, the decision variable values can be 
determined in a continuous or integer domain as it has 
a large number of alternative choices. Thus, MADM 
methods are generally discrete, with a limited number 
of specified alternatives. Each matrix has four main 
parts, namely: (a) alternatives, (b) attributes, (c) weight 
or relative importance of each attribute and (d) 
measures of performance of alternatives with respect to 
the attributes [4]. 
MCDM deals with the problem of helping the 
decision maker to choose the best alternative according 
to several criteria. In order to meet this objective, an 
MCDM method has basically four steps that support 
making the most efficient and rational decisions. 
According to Pohekar and Ramachandran [3], the first 
basic step is structuring the decision process, 
alternative selection and criteria formulation. The 
second step is displaying tradeoff among criteria and 
determine criteria weights. Applying value judgment 
concerning acceptable tradeoffs and evaluation is the 
third step in most MCDM. Methods. The fourth step is 
calculating the final aggregation prior to make a 
decision.  Some literature also suggests that MCDM 
can be divided into two phase process. The first phase 
is called as the rating phase where aggregation of the 
values of criteria for each alternative is made.  Ranking 
or ordering between the alternatives with respect to the 
global consensus degree of satisfaction is another 
phase in MCDM. It can be seen that aggregation is one 
of the fundamental phases in MCDM. Many MCDM 
methods such as ELECTRE I, II, PROMETHEE use 
criteria weights in their aggregation process. Weights 
of criteria play an important role for measuring overall 
preferences of alternatives. It is necessary to aggregate 
the available information in order to make decisions. In 
other words, a central problem in multi-criteria 
problems are an aggregation of the satisfactions to the 
individual criteria to obtain a measure of satisfaction to 
the overall collection of criteria [5].  The overall 
evaluation value is then used to help select alternatives. 
It can be seen that aggregation is the fundamental 
prerequisite of the decision making, in which 
descriptions on how to aggregate individual experts’ 
preference information on alternatives is succinctly 
made [6]. 
Aggregation is easily defined as the process of 
combining several numerical scores with respect to 
72 Informatica 41 (2017) 71–86 W.R.W. Mohd et al.  
 
each criterion by using an aggregation operator in 
order to produce a global score [7].  Detyniecki [8] 
defines an aggregation as ‘mathematical object that has 
the function of reducing a set of numbers into a unique 
representative value’.  Xu [9] defines aggregation is an 
essential process of gathering relevant information 
from multiple source. According to the [10], 
aggregation is made to summarize information in 
decision making. Omar and Fayek [11] defines 
aggregation in the multi-criteria decision making 
environments as a process of combining the values of a 
set of attributes into one representative value for the 
entire set of attributes. Aggregation is important in the 
decision making problem because it is used to derive a 
collective decision made by the decision makers by 
representing in the individual opinions. In addition, 
aggregation of individual judgments or preference is 
used to transform experts’ judgment knowledge and 
expertise in relative weight.  
The interest in the importance of aggregation is 
enhanced by judgments or preferences made by a 
group of decision makers. In group decision problem 
where number of decision makers are multiple, it is 
assumed that there exists a finite number of 
alternatives, as well as a finite set of experts. Each 
expert has their own opinions and may have a variety 
of ideas about the performance of each alternative and 
cannot estimate his/her preferences with crisp 
numerical value. Hence, a more realistic approach to 
be used to represent the situation of the human expert, 
instead of using the crisp numerical values. Thus, each 
variable involved in the problem may be represented in 
linguistic terms. Under those circumstances, 
aggregation methods are the key to tackle the 
mechanism to realize the comprehensive features of 
group decision making [12].  Several related research 
have been conducted to deal with multiplicity features 
in group decision making.  Many researchers have 
studied aggregation operation using different 
aggregation methods.  The way aggregation functions 
are used depends on the nature of the profile that is 
given to each user and the description of items [13].   
In the real world of decision making problems, 
decision makers like to pursue more than one 
aggregation methods to measure the aggregation 
information. In these kinds of problems, many 
aggregation methods have been developed in this area 
for the recent years for judging the alternatives. For 
example, Ogryczak [14] proposed reference point 
method and  implemented to the fair optimization 
method in  analyzing the efficient frontier. The method 
was proposed based on the augmented max-min 
aggregation.   
Aggregation operations on fuzzy sets are 
operations by which several fuzzy sets are combined 
together in some way to produce a single 
representative either fuzzy or crisp set. Therefore, 
aggregation operation needs aggregation operator to 
deal with the situation where the aggregation operator 
is commonly tools that can be used to combine the 
individual preference information into overall 
preference information and deriving collective 
preference values for each alternative. That is to say, 
the information aggregation is to combine individual 
experts’ preference coming from different sources into 
a unique representative value by using an appropriate 
aggregation technique [15]. Aggregation operations are 
used to rank the alternative decisions an expert or 
decision support system, which are established and 
applied in fuzzy logic systems. The information 
aggregation has received much attention from 
practitioners and researchers due to its practical and 
significance in academic [16][17][18][19][20][21]. 
The first overview of the aggregation operators in 
2003 by Xu and Da [22]. The study is reviewed of the 
existing main aggregation operators and proposed 
some new aggregation operators that is induced 
ordered weighted geometric averaging (IOWGA) 
operator, generalized induced ordered weighted 
averaging (GIOWA) operator and hybrid weighted 
averaging (HWA) operator. In 2008, a review of 
aggregation functions which focus on some special 
classes of averaging, conjunctive and disjunctive is 
reviewed by Mesiar et al. [23]. Furthermore, Martinez 
and Acosta in 2015 [24] have made a review an 
aggregation operators taking into account of 
mathematical properties and behavioral measures such 
as disjunctive degree (orness), dispersion, balance 
operator, divergence instead of general mathematical 
properties whose verification might be desirable in 
certain cases: boundary condition, continuity, 
increasing, monotonicity etc. Since then, it is important 
to make a review of the aggregation operator which 
provide the latest method which will be used to solve 
the aggregation in MCDM. Obviously, there is no 
review paper on aggregation operator from year to 
year.  
Our aim in this survey article is to provide an 
accessible overview of some key methods of 
aggregation in MCDM. We focus on development of 
type of aggregation methods that have attracted many 
researchers in this area without neglecting some 
technical details of the aggregation methods.  
Throughout this survey, the terms aggregation 
function, aggregation operator, surveys on aggregation 
in MCDM, an overview of aggregation operation will 
refer to find the journal articles as well. The collected 
journals which is regarding the aggregation is 
retrieving  from the various fields such as engineering, 
medical, operation research,  image processing, 
selection problems, project management selection and 
etc.  
This paper is structured as follows.  Section 2 
begins by laying out the various methods of 
aggregation since 2006 until now.   Section 3 presents 
an analysis out of the survey. Section 4 suggests some 
works that can be extended as future research direction.  
The last chapter concludes.  
Aggregation Methods in Group Decision... Informatica 41 (2017) 71–86 73 
 
2 Review of aggregation method 
This chapter reviews aggregation methods that have 
been developed to aggregate information in order to 
choose a desirable solution, decision makers have to 
aggregate their preference information by means of 
some proper approaches. This review is made by 
analyzing the method used based on the journals and 
conference proceedings that are collected from selected 
popular academic databases such as SpringerLink, 
Scopus, ScienceDirect, IEEE Xplore Digital Library, 
ACM Digital Library, and Wiley Online Library from 
the year 2006 until the year 2016.  
The class of aggregation is huge, making the 
problem of choosing the right method for a given 
application is a difficult one [25]. In this paper, we 
review the methods of aggregation and its various 
applications. Firstly, we segregate the aggregation 
methods by the basic ones which is the most often used 
aggregation operators. For example, the average 
(arithmetic mean), geometric mean and harmonic 
mean. Then, we proceed to the next aggregation 
operator by presenting a generalization of the classical 
one such as Bonferroni Mean (BM), power aggregation 
operator, fuzzy integral, hybrid aggregation operator, 
prioritized average operator and the linguistic 
aggregation operator [26]. 
2.1 Basic operators 
2.1.1 Arithmetic mean operator 
In real life decision situation, the aggregation problems 
in the MCDM are solved using the scoring techniques 
such as the weighted aggregation operator based on 
multi attribute theory. The classical weighted 
aggregation is usually known by the weighted average 
(WA) or simple additive weighting method. A very 
common aggregation operator is the ordered weighted 
averaging (OWA) operator which provides a 
parameterized family aggregation operator between the 
minimum, the maximum, the arithmetic average, and 
the median criteria whose originally introduced by 
Yager [27]. There are two features have been used to 
characterize the OWA operators. The first is the orness 
character and the second is the dispersion. The OWA 
has been widely used because of its ability to model 
linguistically expressed aggregation. Since the OWA 
operator is coming out, the approach has been 
employed by the most authors in a wide range of 
applications such as engineering, neural networks, data 
mining, decision making, and image processing as well 
as expert systems [28]. However, OWA operator was 
assumed that the available information includes crisp 
number or singletons. In the real decision making 
situation, it is found this may not be the crisp number. 
Sometimes, the available information is vague or 
imprecise and it is not possible to analyze it with the 
crisp numbers. Then, it is necessary to use another 
approach that is able to represent the uncertainty such 
as the use of fuzzy numbers (FNs). With the use of 
FNs, it is possible to analyze the real situation into the 
fuzzy value. Here is the definition of OWA.  
 
Definition 1: An OWA operator of dimension n  
is a mapping OWA: RRn   that has associated 
weighting vector W of dimension n  with  1,0jw  and 



n
j
jw
1
1 , such that [27]: 
OWA   


n
j
jjn bwaaa
1
,...,,
21
 
where jb  is the j th largest of the ia . 
 
There are some of the authors that being used 
OWA as aggregation operators such that Xu [29] 
developed the intuitionistic fuzzy ordered weighted 
averaging (IFOWA) operator, and the intuitionistic 
fuzzy hybrid averaging (IFHA) operator. Xu and Chen 
[30] investigated the interval-valued intuitionistic 
fuzzy multi-criteria group decision making based on 
arithmetic aggregation operators such as the interval-
valued intuitionistic fuzzy weighted arithmetic 
aggregation (IIFWA) operator, the interval-valued 
intuitionistic fuzzy ordered weighted aggregation 
(IIFOWA) operator and the interval-valued 
intuitionistic fuzzy hybrid aggregation (IIFHA) 
operator.   
Li and Li [31][32] developed the generalized 
OWA operators using IFSs to solve MADM in which 
weights and ratings of alternatives on attributes are 
expressed in IFS. Chang et al. and Merigo et al.  
[33][34]proposed the fuzzy generalized ordered 
weighted averaging (FGOWA) operator as it is an 
extension of the GOWA operator for the uncertain 
situation where the information given is in the form of 
fuzzy numbers.  Zhao et al. [35] developed some new 
generalized aggregation operators such as generalized 
intuitionistic fuzzy weighted averaging (GIFWA) 
operator, generalized intuitionistic fuzzy ordered 
weighted averaging (GIFOWA) operator, generalized 
intuitionistic fuzzy hybrid averaging (GIFHA) 
operator, generalized interval-valued intuitionistic 
fuzzy weighted averaging (GIIFWA) operator, 
generalized interval-valued intuitionistic fuzzy hybrid 
average (GIIFHA) operator where the proposed 
method is the extension of the GOWA operators taking 
into account of the characterization both of 
intuitionistic fuzzy sets by a membership function and 
non membership function and interval-valued 
intuitionistic fuzzy sets whose fundamental 
characteristic is the values of its membership function 
is represented by the interval numbers rather than exact 
numbers.  
Shen et al. [36] presented a new arithmetic 
aggregation operator which is induced intuitionistic 
trapezoidal fuzzy ordered weighting aggregation 
operator and applied to group decision making. 
Furthermore, in 2011,  Casanovas et al. [37] introduced 
the uncertain induced probabilistic ordered weighted 
74 Informatica 41 (2017) 71–86 W.R.W. Mohd et al.  
 
averaging weighted averaging (UIPOWAWA) operator 
where it provides a parameterized family of 
aggregation operators between minimum and 
maximum in a unified framework between probability, 
the weighted average and the induced ordered 
weighted averaging (IOWA) operator. Merigo [38] 
developed a new aggregation model that unifies the 
weighted average (WA) and the induced ordered 
weighted average (IOWA) operator that is called 
induced ordered weighted averaging-weighted average 
(IOWAWA) operator by considering the degree of 
importance that each concept has in the aggregation. 
Xu and Wang [39] proposed a new aggregation 
operator which calling induced generalized 
intuitionistic fuzzy ordered weighted averaging (I-
GIFOWA) operator by considering the characteristics 
of both the generalized IFOWA and the induced 
IFOWA operator. In order to deal with the 
intuitionistic fuzzy preference information in group 
decision making, the induced generalized intuitionistic 
fuzzy ordered weighted averaging (I-GIFOWA) 
operator based on GIFOWA and the I-IFOWA 
operator. Yu [40] introduced generalized intuitionistic 
trapezoidal fuzzy weighted averaging operator to 
aggregate the intuitionistic trapezoidal fuzzy 
information. Xia et al. [41] proposed several new 
hesitant fuzzy aggregation operators by extending 
quasi-arithmetic means to hesitant fuzzy sets under 
group decision making. Zhou et al. [42] introduced a 
new operator for aggregating the interval-valued 
intuitionistic fuzzy values which called the continuous 
interval-valued intuitionistic fuzzy ordered weighted 
averaging (C-IVIFOWA) operator. Both intuitionistic 
fuzzy ordered weighted averaging (IFOWA) operator 
and the continuous ordered weighted averaging (C-
OWA) operator has combined to control the parameter 
and employed to diminish the fuzziness and improve 
the preciseness of the decision making.  
2.1.2 Geometric mean operator 
The geometric mean operator is the traditional 
aggregation operator that proposed to aggregate 
information given on a ratio scale measurement in 
MCDM models. The main characteristics are its 
guaranties the reciprocity property of the multiplicative 
preference relations used to provide ratio preferences 
[43].  
 
Definition 2: An OWG operator of dimension n  
is a mapping OWG:   RR n  that has associated 
weighting vector  Tnwwww ,...,, 21  with 0jw  and 



n
j
jw
1
1 , such that [43]: 
OWGw   


n
j
w
jn
jbaaa
1
,...,,
21
 
where jb  is the j th largest of the  nja j ,...,2,1 . 
 
Some authors used geometric mean as aggregation 
operator. For example, Wu et al. [44] defined same 
families of geometric aggregation operators to 
aggregate trapezoidal IFNs (TrIFNs). Xu and Yager 
[45], Xu and Chen [46], Wei [47], Das et al. [48] 
developed some geometric aggregation operators based 
on IFS, such as intuitionistic fuzzy weighted geometric 
(IFWG) operator, intuitionistic fuzzy ordered weighted 
geometric (IFOWG) operator and intuitionistic fuzzy 
hybrid geometric (IFHG) operator. Tan [49] developed 
a generalized intuitionistic fuzzy geometric 
aggregation operator for multiple criteria decision 
making by considering the interdependent or 
interactive characteristics and preferences.  
Wei [50], Xu [51] and Xu and Chen [52] proposed 
approach based on interval-valued intuitionistic fuzzy 
weighted geometric (IIFWG) operator, the interval-
valued intuitionistic fuzzy ordered weighted geometric 
(IIFOWG) operator and the interval-valued 
intuitionistic fuzzy hybrid geometric (IIFHG) operator 
in different point of view. Verma and Sharma [53] 
proposed geometric Heronian mean (GHM) under 
hesitant fuzzy environment by developing some new 
GHM such that hesitant fuzzy generalized geometric 
Herinian mean (HFGGHM) operator and weighted 
hesitant fuzzy generalized geometric Herinian mean 
(WHFGGHM) operator.  
2.1.3 Harmonic mean operator 
Harmonic mean is the reciprocal of the arithmetic 
mean of reciprocal which is a conservative average to 
be used to provide for aggregation lying between the 
max and min operators and is widely used as a tool to 
aggregate central tendency data which is usually 
expressed in exact numerical values [54]. 
Definition 3: An ordered weighted harmonic mean 
operator of dimension n  is a mapping OWHM: 
RRn   that has associated weighting vector 
 Tnwwww ,...,, 21  with 0jw  and 


n
j
jw
1
1 , such that 
[54]: 
OWHMw  
 



n
j j
j
n
a
w
aaa
1
1
,...,,
21

 
where       n ,...2,1  is a permutation of  n,...,2,1 , 
such that    jj   1  for all nj ,...,2 . 
 
Some researchers proposed harmonic mean as a 
method to solve aggregation in the decision making 
problem. For example, Xu [55] developed some fuzzy 
harmonic mean operators, such as fuzzy weighted 
harmonic mean (FWHM) operator, fuzzy ordered 
weighted harmonic mean (FOWHM) operator, fuzzy 
hybrid harmonic mean (FHHM) operator. The aim of 
this paper is to extend the induced ordered weighted 
harmonic mean (IOWHM) operator to fuzzy 
environment and propose a new operator called the 
Aggregation Methods in Group Decision... Informatica 41 (2017) 71–86 75 
 
fuzzy induced ordered weighted harmonic mean 
(FIOWHM) operator.  
Wei and Yi [56] proposed an aggregation operator 
including trapezoidal fuzzy ordered weighted harmonic 
averaging (ITFOWHA) operator and applied to the 
decision making.  
Wei [57] proposed a new aggregation operator 
called fuzzy induced ordered weighted harmonic mean 
(FIOWHM) for fuzzy multi criteria group decision 
making. Zhou et al. [58] proposed the generalized 
hesitant fuzzy harmonic mean operators including the 
generalized hesitant fuzzy weighted harmonic mean 
operator (GHFWHM), the generalized hesitant fuzzy 
ordered weighted harmonic mean operator 
(GHFOWHM), the generalized hesitant fuzzy hybrid 
harmonic mean operator (GHFHHM) using the 
technique of obtaining values in the interval to the 
group decision making under hesitant fuzzy 
environment.  
Liu et al. [59] proposed a generalized interval-
valued trapezoidal fuzzy numbers (GIVFTN) is an 
extended of ordered weighted harmonic averaging 
operators to solve the problems in multiple attribute 
group decision making.  
 
2.2 Bonferroni mean (BM) 
The Bonferroni mean (BM) originally introduced by 
Bonferroni [60]. The classical Bonferroni mean is an 
extension of the arithmetic mean and its generalized by 
some researchers based on the idea of the geometric 
mean [61]. The BM is differ from the other classic 
means such as the arithmetic, the geometric and the 
harmonic because this mean reflect the interdependent 
of the individual criterion meanwhile on the classic 
means the individual criterion is independent [62]. The 
BM was originally introduced by Bonferroni [60], 
which was defined as follows: 
 
Definition 4: Let ,0, qp and  niai ,...,2,1 be a 
collection of nonnegative numbers. If  
 
 
qp
n
ji
ji
q
j
p
i
qp aa
nn
aaaB
n


















 
1
1,
,
1
1
,...,,
21
 
Then qpB , is called the Bonferroni mean (BM). 
 
The Bonferroni mean (BM) operator is suitable for 
aggregating crisp data and can capture the expressed 
interrelationships among criteria, which plays a crucial 
role in multi-criteria decision making problems [63]. 
Since the BM introduced, this aggregation operator has 
received much attention from researchers and 
practitioners. Among them are, Yager [64] generalized 
the BM for enhancing its model capability and further 
Xu and Yager [65] developed intuitionistic fuzzy BM 
(IFBM) and applied the weighted IFBM to multi 
criteria decision making. Beliakov et al. [66] gave a 
systematic investigation of a family of composing 
aggregation functions which generalize the Bonferroni 
Mean (BM).  
Zhu et al. [67] explored the geometric Bonferroni 
mean (GBM) by considering both BM and the 
geometric mean (GM) under hesitant fuzzy 
environment. Xia et al. [68] developed the Bonferroni 
geometric mean, which is a generalization of the 
Bonferroni mean and geometric mean and can reflect 
the interrelationships among the aggregated arguments. 
Wei et al. [69] developed two aggregation operators 
called the uncertain linguistic Bonferroni mean 
(ULBM) operator and the uncertain linguistic 
geometric Bonferroni mean (ULGBM) operator for 
aggregating the uncertain linguistic information in the 
multiple attribute decision making (MADM) problems. 
Park and Park [70] extend the works Sun and Sun [61] 
by considering the interactions of any three aggregated 
arguments instead of any two to develop generalized 
fuzzy weighted Bonferroni harmonic mean 
(GFWBHM) operator and generalized fuzzy ordered 
weighted Bonferroni harmonic mean (GFOWBHM) 
operator. Verma [71] proposed a new generalized 
Bonferroni mean operator called generalized fuzzy 
number intuitionistic fuzzy weighted Bonferroni mean 
(GFNIFWBM) operator which is able to aggregate the 
fuzzy number intuitionistic fuzzy correlated 
information.   
 
2.3 Power aggregation operators 
Yager [17] was first introduced a power average (PA) 
operator which uses a non-linear weighted average 
aggregation tool and a power ordered weighted average 
(POWA) operator to provide aggregation tools which 
allow exact arguments to support each other in the 
aggregation process. The weighting vectors of the PA 
operator and the POWA operator depend on the input 
arguments and allow arguments being aggregated to 
support and reinforce each other.   In contrast with 
most aggregation operators, the PA and POWA 
operators incorporate information regarding the 
relationship between the values being combined. 
Recently, these operators have received much attention 
in the literature.  
 
Definition 5:  The power average (PA) operator is 
mapping  PA: RRn   defined by the following 
formula [17]: 
 
  
  






n
i
i
n
i
ii
i
aT
aaT
niaPA
1
1
1
1
,...2,1  
where    



n
ij
j
jii aaSupaT
1
,  
76 Informatica 41 (2017) 71–86 W.R.W. Mohd et al.  
 
and  ji aaSup ,  is the support for ia  from ja . The 
support satisfies the following three properties: 
(1)    ;1,0, ji aaSup  
(2)    ;, , ijji aaSupaaSup   
(3)     tsjitsji aaaaifaaSupaaSup  ,,   
Motivated by the success of the PA and POWA, 
Xu and Yager [72] proposed a power geometric 
average (PG) operator and a power ordered weighted 
average (POWGA) operator. Besides that, power 
aggregation operators have been further extended to 
accommodate multi attribute group decision making 
(MAGDM) under different uncertain environments. 
For instance, Xu and Cai [73] developed the uncertain 
power ordered weighted average (UPOWA) operator 
on the basis of the PA operator and the UOWA 
operator, Xu [74] introduced the uncertain ordered 
weighted geometric average (UOWGA) operator based 
on the PG operator and the UOWA operator.  
Xu [75] under intuitionistic fuzzy and interval-
valued intuitionistic fuzzy decision making 
environments, the linguistic power aggregation 
operators by Zhou et al. [76], generalized argument-
dependent power operators by Zhou and Chen [15] to 
accommodate intuitionistic fuzzy preferences and 
power aggregation operators under interval-valued dual 
hesitant fuzzy linguistic environment and the power 
aggregation operators by Wan [77] under trapezoidal 
intuitionistic fuzzy decision making environments.  
Zhang [78] developed a wide range of hesitant 
fuzzy power aggregation operators for hesitant fuzzy 
information such as the hesitant fuzzy power average 
(HFPA) operators, the hesitant power geometric 
(HFPG) operators, the generalized hesitant fuzzy 
power average (GHFPA) operators, the generalized 
hesitant fuzzy power geometric (GHFPG) operators, 
the weighted generalized hesitant fuzzy power average 
(WGHFPA) operators, the generalized hesitant fuzzy 
power geometric (WGHFPG) operators, the hesitant 
fuzzy power ordered weighted average (HFPOWA) 
operators,  the hesitant fuzzy power ordered weighted 
geometric (HFPOWG) operators, the generalized 
hesitant fuzzy power ordered weighted average 
(GHPOWA) operators and the generalized hesitant 
fuzzy power ordered weighted geometric (GHPOWG) 
operators. 
 However, the arguments of these power 
aggregation operators are exact numbers. In practice, 
we often confront situations in which the input 
arguments cannot be expressed in the form of exact 
numerical values instead, they have to take in the form 
of interval numbers Qi et al. [79], intuitionistic fuzzy 
numbers (IFNs) [80][81][82], interval-valued 
intuitionistic fuzzy numbers (IVIFNs) [83], linguistic 
variables [84][85][86], uncertain linguistic variables 
[67][87], or 2-tuples [88], hesitant fuzzy sets (HFS) 
[81]. Gou et al. [81] developed a family of hesitant 
fuzzy power aggregation operators, for instance the 
hesitant fuzzy power weighted average (HFPWA), 
hesitant fuzzy power weighted geometric (HFPWG) 
generalized hesitant fuzzy power weighted average 
(GHFPWA), generalized hesitant fuzzy power 
weighted geometric (GHFPWG) operators for multi-
criteria group decision making problems.  
Wang et al. [88] proposed a dual hesitant fuzzy 
power aggregation operators based on Archimedean t-
conorm and t-norm for dual hesitant fuzzy information. 
Das and Guha [89] proposed some new aggregation 
operators such as trapezoidal intuitionistic fuzzy 
weighted power harmonic mean (TrIFWPHM) 
operator, trapezoidal intuitionistic fuzzy ordered 
weighted power harmonic mean (TrIFOWPHM) 
operator, trapezoidal intuitionistic fuzzy induced 
ordered weighted power harmonic mean 
(TrIFIOWPHM) operator and trapezoidal intuitionistic 
fuzzy hybrid power harmonic mean (TrIFhPHM) 
operator to aggregate the decision information.  
2.4 Fuzzy integral 
Another types of aggregation operators is fuzzy 
integrals (FI). There are many types of FI, and most of 
the well-known fuzzy integral are Choquet and Sugeno 
integral. 
2.4.1 Choquet integral 
One of the popular aggregation operator fuzzy integrals 
is the Choquet integral which is introduced by Choquet 
[90]. Choquet integral is defined as a subadditive or 
superadditive to integrate functions with respect to the 
fuzzy measures [91].  
Definition 6. Let f  be a real-valued function on 
X , the Choquet integral of f  with respect to a fuzzy 
measure g  on  X  is defined as [90]: 
          

 








n
i
iAgixfixffdgC
1
1   
     (1) 
or equally by  
             
n
i
ixfiAgiAgfdgC 1 1
 
      (2) 
 
where the parentheses used for indices represent a 
permutation on X  such that 
               ,,...,,00,1 nxixiAxfnxfxf 
and   1nA . 
The Choquet integral is a very useful way of 
measuring the expected utility of an uncertain event 
Aggregation Methods in Group Decision... Informatica 41 (2017) 71–86 77 
 
[92]. It is a tool to model the interdependence or 
correlation among different elements where a new 
aggregation operators can be defined. Choquet integral 
has been proposed by many authors as an adequate 
aggregation operator that extends the weighted 
arithmetic mean or OWA operator by taking into 
consideration the interactions among the criteria.  
Yager [93] extended the idea of order induced 
aggregation to the Choquet aggregation and introduced 
the Choquet ordered averaging (I-COA) operator.  
Mayer and Roubens [94] aggregated the fuzzy numbers 
through the Choquet integral. In the other field, 
Hlinena et al. [95] used Choquet integral with respect 
to Lukasiewicz filters to present a partial solution to 
look for an appropriate utility function in a given 
setting. Ming-Lang et al. [96] proposed analytic 
network process (ANP) technique to get the 
relationships of feedback of criteria and Choquet 
integral is used to eliminate the interactivity of the 
expert subjective judgment problem and apply in the 
case study of selection of optimal supplier in supply 
chain management strategy (SCMS).  
Buyukozkan and Ruan [97] proposed a two-
addative Choquet integral to the software development 
experts and managers to enable them to position their 
projects in terms of associated risks. Murofushi and 
Sugeno [91] used the Choquet integral to propose the 
interval-valued intuitionistic fuzzy correlated 
averaging operator and interval-valued intuitionistic 
fuzzy correlated geometric operator to aggregate 
interval-valued intuitionistic fuzzy information and 
applied them to a practical decision making problem. 
Angilella et al. [98] proposed a non-additive robust 
ordinal regression on a set of alternatives by evaluating 
the utility in terms of Choquet integral which represent 
the interaction among thecriteria modelled by the fuzzy 
measure. Huang et al. [99] applied a generalized 
Choquet integral with a signed fuzzy measure based on 
the complexity to evaluate the overall satisfaction of 
the patients.  
Demirel et al. [100] proposed generalization 
Choquet integral by taking consideration of 
information fusion between criteria and linguistic 
terms and fuzzy ANP as a fuzzy measure which can 
handle the dependent criteria and hierarchical problem 
structure and applied to the multi-criteria warehouse 
location.   
Tan and Chen [101] proposed intuitionistic fuzzy 
Choquet integral based on t-norms and t-conorms 
meanwhile Tan [102]  extended the TOPSIS method 
by combining the interval-valued intuitionistic fuzzy 
geometric aggregation operator with Choquet integral-
based Hamming distance to deal with multi-criteria 
interval-valued intuitionistic fuzzy group decision 
making problems.  
Bustince et al. [103] proposed a new MCDM 
method for interval-valued fuzzy preference relation 
which was based on the definition of interval-valued 
Choquet integrals. Yang and Chen [104] introduced 
some new aggregation operator including the 2-tuple 
correlated averaging operator, the 2-tuple correlated 
geometric operator and the generalized 2-tuple 
correlated averaging operator based on the Choquet 
integral. Belles-sampera [10] developed the extensions 
of the degree of balance, the divergence, the variance 
indicator and Renyi entropies to characterize the 
Choquet integral.  
Islam et al. [105] proposed Choquet integral using 
goal programming to multi-criteria based learning 
which combines both experts’ knowledge and data. 
In addition, Choquet integral is applied in the 
hesitant fuzzy environment. Some authors that used the 
Choquet integral in the hesitant fuzzy environment are 
Yu et al. [106], Xia et al. [40]. For example, Yu et al. 
[106] proposed Choquet integral aggregation operator 
for hesitant fuzzy elements (HFEs) and applied it to the 
MCDM problems, Xia et al. [40] applied the Choquet 
integral to get the weights of criteria for group decision 
making, Peng et al. [107] proposed Choquet integral 
methods which is an approach to multi-criteria group 
decision making (MCGDM) problem to rank the 
alternatives where the criteria are interdependent or 
interactive. Wang et al. [108] developed some Choquet 
integral aggregation operators with interval 2-tuple 
linguistic information and applied them to MCGDM 
problems.  
2.4.2 Sugeno integral 
Sugeno integral is one of the fuzzy integral which 
introduced by M. Sugeno in the year of 1974 [109]. 
Sugeno integral is proposed to compute an average 
value of some function with respect to a fuzzy 
measure. In particular, the Sugeno integral uses only 
weighted maximum and minimum functions [16]. The 
definition of Sugeno integral [109] as follows: 
 
Definition 7: The (discrete) Sugeno Integral of a 
function  1,0: Xf with respect to   is defined as 
     ii
ni
Axfdxf  ,minmax)(
1   
where         nxfxfxfxf ,....,,, 321  are the ranges and 
they are defined as        nxfxfxfxf  ....321 . 
 
In recent years, some authors and practitioners that 
used Sugeno integral are Mendoza and Melin [110], 
Liu et al. [111], Tabakov and Podhorska [112], Dubois 
et al. [113]. 
Mendoza and Melin [110] extended the Sugeno 
integral with the interval type-2 fuzzy logic. The 
generalization composed the modifying the original 
equations of the Sugeno measures and Sugeno integral. 
This method is used to combine the simulation vectors 
into only one vector and lastly the system will be 
decided the best choice of recognition in the same 
manner than made with only one monolithic neural 
network, but the problem of complexity resolved.  
Liu et al. [111] extended the componentwise 
decomposition theorem of lattice-valued Sugeno 
integral by introducing the concept of interval fuzzy-
valued, intuitionistic fuzzy-valued and interval 
intuitionistic fuzzy-valued Sugeno integral. As a result, 
78 Informatica 41 (2017) 71–86 W.R.W. Mohd et al.  
 
the intuitionistic fuzzy-valued Sugeno integrals and the 
interval fuzzy-valued Sugeno integrals are 
mathematically equivalent. It shows that the interval 
intuitionistic fuzzy-valued Sugeno integral can be 
decomposed into the interval fuzzy-valued and 
intuitionistic fuzzy-valued Sugeno integrals or the 
original Sugeno integrals.  
Tabakov and Podhorska [112] proposed fuzzy 
Sugeno integral as an aggregation operator of an 
ensemble of fuzzy decision trees in order to classify the 
corresponding HER-2/neu classes. They used three 
different fuzzy decision trees which are built over 
different image characteristics, colour values and 
structural factors and texture information. The fuzzy 
Sugeno integral has been used as an aggregation 
operator to design fuzzy trees and the final medical 
decision support information generated.  
Dubois et al. [113] proposed two new variants of 
weighted minimum and maximum where the criteria 
weights play an important role of tolerance. The 
Sugeno integral is proposed to the residuated 
counterparts, which means, the weight support on 
subsets of criteria. Then, the dual aggregation 
operations called disintegrals are evaluated in terms of 
its defects rather than in terms of its positive features 
proposed. The maximal disintegral is when no defects 
at all are present and maximal integral when all the 
merits are sufficiently present.  
2.5 Hybrid aggregation operators 
The hybrid aggregation operator has been proposed by 
several authors. It is important to propose more than 
one aggregation operator so that a wide range of fuzzy 
aggregation operators can be used in a wide range of 
application in decision making problems. For instance, 
Jianqiang and Zhong [114] developed the intuitionistic 
trapezoidal weighted average arithmetic average 
operator and the intuitionistic trapezoidal weighted 
geometric average operator.  
Zhang and Liu [115] proposed the weighted 
arithmetic averaging operator and the weighted 
geometric average operator to aggregate triangular 
fuzzy intuitionistic fuzzy information and applied it to 
the decision making problem.  
Then, Merigo and Casanovas [116] proposed fuzzy 
generalized hybrid aggregation operators where further 
generalize the fuzzy geometric hybrid averaging 
(FGHA) and the fuzzy induced geometric hybrid 
average (FIGHA) by using quasi-arithmetic means and 
the new result are Quasi-FHA and the Quasi-FIHA 
operator. 
 Xia and Xu [117] first proposed fuzzy weighted 
averaging (HFWA), hesitant fuzzy weighted geometric 
(HFWG) operators, generalized hesitant fuzzy 
weighted averaging (GHFWA), generalized hesitant 
fuzzy weighted geometric (GHFWG) operators in 
solving decision making problems.  
Yu et al. [118] proposed the interval-valued 
intuitionistic fuzzy prioritized weighted average 
(IVIFPWA) operator and interval-valued intuitionistic 
fuzzy prioritized weighted geometric (IVIFPWG) 
operator to aggregate the IVIFNs.  
Verma and Sharma [119] developed some 
prioritized weighted aggregation operators for 
aggregating trapezoid fuzzy linguistic information 
motivated by the idea of prioritized weighted average 
introduced by Yager [122] such that the trapezoid 
linguistic prioritized weighted average (TFLPWA) 
operator, the trapezoid linguistic prioritized weighted 
geometric (TFLWG), and the trapezoid linguistic 
prioritized weighted harmonic (TFLWH) operator.  
Liao and Xu [120] introduced some new 
aggregation operators including the generalized 
hesitant fuzzy hybrid weighted averaging operator, the 
generalized hesitant fuzzy hybrid weighted geometric 
operator, and the generalized quasi hesitant fuzzy 
hybrid weighted geometric operator and their induced 
forms.  
In 2016, Verma [121] proposed a new aggregation 
operator that based on the generalization of mean 
called generalized trapezoid fuzzy linguistic prioritized 
weighted average (GTFLPWA) operator for fusing the 
trapezoid fuzzy linguistic information. The prominent 
characteristics of the proposed operator does not only 
take into account the prioritization among the attributes 
and decision makers but also has a flexible parameter.  
2.6 Prioritized  operator 
Prioritized Average (PA) operator is one of the 
aggregation operators which has a great interest among 
scholars. In practical situations, decision-makers 
usually consider different criteria priorities. To deal 
with this issue, Yager [122] developed prioritized 
average (PA) operators by modeling the criteria 
priority on the weights associated with the criteria, 
which depend on the satisfaction of higher priority 
criteria. The PA operator has many advantages over 
other operators. For example, the PA operator does not 
need to provide weight vectors and, when using this 
operator, it is only necessary to know the priority 
among the criteria. 
Wei [123] extended the prioritized aggregation 
operator to hesitant fuzzy sets and developed some 
prioritized hesitant fuzzy operators in multicriteria 
decision making. As Yager [122] only discussed the 
criteria values and weights in the real number domain, 
thus Wang et al. [124] developed some prioritized 
aggregation operators for aggregating interval-valued 
hesitant fuzzy linguistic information.   
Recently, some researchers have focused on  fuzzy 
prioritized aggregation operator into intuitionistic 
fuzzy sets (IFSs) such as Yu et al.  [117], Chen [125] 
proposed some interval-valued intuitionistic fuzzy 
aggregation operators such as the interval-valued 
intuitionistic fuzzy prioritized weighted average 
(IVIFPWA) operator and  the interval-valued 
intuitionistic fuzzy prioritized weighted geometric 
(IVIFPWG) operator, Verma and Sharma [126] 
proposed two new aggregation operators such as 
intuitionistic fuzzy Einstein  prioritized weighted 
Aggregation Methods in Group Decision... Informatica 41 (2017) 71–86 79 
 
average (IFEPWA) operator and the intuitionistic 
fuzzy Einstein prioritized weighted geometric 
(IFEPWG) operator for aggregating intuitionistic fuzzy 
information  meanwhile Liang et al. [127], Dong et al. 
[128] developed some new aggregation operator called 
generalized intuitionistic trapezoidal fuzzy prioritized 
weighted average operator and generalized 
intuitionistic trapezoidal fuzzy prioritized weighted 
geometric operator and apply to the multi-criteria 
group decision making.  Verma and Sharma [129] also 
proposed two new prioritized aggregation operators for 
aggregate triangular fuzzy information called quasi 
fuzzy prioritized weighted average (QFPWA) operator 
and the quasi fuzzy prioritized weighted ordered 
weighted average (QFPWOWA) operator.  
2.7 Linguistic aggregation operator 
Often, human decision making is too complex or too 
weakly defined to be represented by the numerical 
analysis. It is always considered the available 
information is vague or imprecise and impossible to 
analyze it with numerical values. However, this may 
not represent the real situation found in the decision 
making problem. Therefore, the possible way to solve 
such situation, it is necessary to use a qualitative 
approach which is the linguistic variable to aggregate 
the fused information. The linguistic aggregation 
operators are offered when the situations of the 
information cannot be assessed with numerical values, 
but it is possible to use linguistic assessment [130]. 
There are some authors used the linguistic variables to 
aggregate the information in MCDM. For instance, 
Wang and Hao [131] presented a 2-tuple fuzzy 
linguistic evaluation model for selecting appropriate 
agile manufacturing system in relation to MC 
production. Herrera et al. [132] proposed a fuzzy 
linguistic methodology to deal with unbalanced 
linguistic term sets. Chang et al. [33] proposed a 
linguistic MCDM aggregation model to tackle to solve 
two problems which are the aggregation operators are 
usually independent of aggregation situation and there 
must be a feasible operator for dealing with the actual 
evaluation scores.  
Xu and Chen [52] extended the well-known 
harmonic mean to represent the information in the 
linguistic situation and developed some linguistic 
harmonic mean aggregation operators such as the 
linguistic weighted harmonic mean (LWHM) operator, 
the linguistic ordered weighted harmonic mean 
(LOWHM) operator, and the linguistic hybrid 
harmonic mean (LHHM) operator for aggregating 
linguistic information.  
Wei [133] proposed a method for multiple attribute 
group decision making based on the ET-WG and ET-
OWG operators with 2-tuple linguistic information. 
Shen et al. [36] developed the belief structure-
linguistic ordered weighted averaging (BS-LOWA), 
the BS linguistic hybrid averaging (BS-LHA) and a 
wide range of particular cases. Wei [130] extended the 
TOPSIS method for 2-tuple linguistic multiple attribute 
group decision making with incomplete weight 
information. Then, Merigo et al. [130] developed 
linguistic weighted generalized mean (LWGM) and the 
linguistic generalized OWA (LGOWA) operator and 
applied to the decision making problems.  
Besides that, there are some authors that used 2-
tuple linguistic variables such as Wei [134] proposed 
the GRA-based linear programming methodology for 
multiple attribute group decision making with 2-tuple 
linguistic assessment information. Wei [135] utilized 
the gray relational analysis method for 2-tuple 
linguistic multiple attribute group decision making 
with incomplete weight information. Xu and Wang 
[136] developed some 2-tuple linguistic power 
aggregation operators. Wei [137] proposed some new 
aggregation operators which is the 2-tuple linguistic 
weighted harmonic averaging (TWHA), 2-tuple 
linguistic ordered weighted harmonic averaging 
(TOWHA) and 2-tuple linguistic combined weighted 
harmonic averaging (TCWHA) operators for multiple 
attribute group decision making. Zadeh [83] developed 
some new linguistic aggregation operators such as 2-
tuple linguistic harmonic (2TLH) operator, 2-tuple 
linguistic weighted harmonic (2TLWH) operator, 2-
tuple linguistic ordered weighted harmonic (2TLOWH) 
operator and 2-tuple linguistic hybrid harmonic 
(2TLHH) operator to utilize to aggregate preference 
information considering linguistic variables in the 
decision making problem. Then, Li et al. [138] 
developed a new multiple attribute decision making 
approach for dealing with 2-tuple linguistic variable 
based on induced aggregation operators and distance 
measure by presenting 2-tuple linguistic induced 
generalized ordered weighted averaging distance 
(2LIGOWAD) operator which extension of the 
induced generalized ordered weighted distance 
(IGWOD) with 2-tuple linguistic variables. The 
2LIGOWAD basically uses the IOWA operator 
represented in the form of 2-tuple linguistic variables.  
Furthermore,  Liu and Jin [139] introduced 
operational laws, expected value definitions, score 
functions and accuracy functions of intuitionistic 
uncertain linguistic variables and proposed two 
approaches with intuitionistic uncertain linguistic 
information to the weighted geometric average 
(IULWGA) operator  and ordered weighted geometric 
(IULOWG) operator for multi attribute group decision 
making.  
3 Observation 
Throughout this study, hundred three journal articles 
have been reviewed with different aggregation 
methods within 2006 until 2016 searching via IEEE 
explore, Science Direct, Springer Link and Wiley 
online Library. In this paper, the methods and 
applications of aggregation are discussed in various 
fields. Based on the Table 1, the most popular 
aggregation operator is Choquet integral which is 
17.48%, then it follows by linguistic aggregation 
operator (15.53%), arithmetic average operator 
80 Informatica 41 (2017) 71–86 W.R.W. Mohd et al.  
 
(14.74%), power aggregation operator and geometric 
mean operator (9.71%) respectively, prioritized 
aggregation operator (8.74), Benferroni Mean and 
Hybrid aggregation operator  (7.77%), harmonic mean 
(4.85%) and Sugeno integral (3.88%). The details of 
the percentage are shown in the Figure 1.  
Table 1: Numbers and percentage of the 
aggregation methods. 
Aggregation 
Methods 
No. of 
Authors 
Percentage 
(%) 
Arithmetic 
Mean 
15 14.56 
Geometric 
Mean 
10 9.71 
Harmonic 
Mean 
5 4.85 
Bonferroni 
Mean (BM)  
8 7.77 
Power 10 9.71 
Choquet 
Integral 
18 17.48 
Sugeno 
Integral 
4 3.88 
Hybrid  8 7.77 
Prioritized 9 8.74 
Linguistic  16 15.53 
Total 103 100 
 
Based on the Table 1 and Figure 1, the Choquet 
integral have attracted more attention because it is 
usually known in the literature as a flexible 
aggregation operator and it is a generalization of the 
weighted average (WA) or simple additive weighting 
method, the ordered weighted average, and the max-
min operator [130].  
In addition, the Choquet integral is the appropriate 
tool to solve the interactions among the criteria in 
decision making problem  as the traditional multi-
criteria decision making (MCDM) methods are based 
on the additive concept along with the independence 
assumption where, in fact, each individual criterion is 
not completely independent [95]. 
Furthermore, the aggregation operator based on the 
linguistic variable has received considerable attention 
too. The linguistic information is used when the 
information available is vague or imprecise, but unable 
to analyze it using numerical values [131].  
 
Figure 1: Percentage of aggregation methods. 
Other than that, one of the conventional 
aggregation operators which is arithmetic mean also 
received attention from many scholars. They have been  
widely used because the first aggregation operator 
which introduced by Yager in 1988 is OWA which 
provide parameterized the arithmetic mean. Then, it is 
easy to compute. Since then, many researchers have 
developed aggregation operator that based on the 
arithmetic since because it is practical in the decision 
making problem.  
Besides that, there are many aggregation operators 
used in the decision making problem depending on 
various kinds of factors investigated. For example, 
most of these operators, however, can only be used in 
situations where the input arguments are the exact 
values, and few of them can be used to aggregate the 
linguistic preference information.  
4 Future work  
From the observations, there are many types of 
aggregation operator that have been used by the 
researchers. Recently, the most appealing and great 
attention of researchers is Choquet integral because 
this method can represent the interaction between 
criteria, ranging from negative interaction to positive 
interaction. Even the classical of aggregation operator, 
power operator and linguistic variables have attracted 
numbers of researchers to apply to this field. It is 
suggested that the Choquet integral in order to build a 
more robust method and can be improved or extended 
by taking into account a weighted combination of both 
experts’ knowledge and data. This suggestion is based 
on the only expert opinion can be overly subjective and 
may not result in desired performance.  
Aggregation Methods in Group Decision... Informatica 41 (2017) 71–86 81 
 
The Choquet integral derives from the large 
numbers of coefficients  associated with a 
fuzzy measure, however, this flexibility can drive to be 
a serious drawback, especially when assigning real 
values to the importance of all possible combinations.  
Further research could be further in selecting the 
fuzzy measure to the Choquet integral. It is because 
different fuzzy measure will be impacted to the 
Choquet integral.  
Since the linguistic aggregation operator shows 
more than fifty percent of overall, it is possible to 
further to the next phase. In the future, it may extend 
this approach to other situations that can be assessed 
with other linguistic approaches and introducing the 
new aspects in the formulation by integrating them 
with other types of aggregation operators.  
The arithmetic aggregation operator also shows the 
highest percentage among other aggregation operator. 
It is expected to expand in the future by using a 
generalized aggregation operator, distance measures 
and unified aggregation operators. Moreover, it can be 
represented in the uncertain environment using fuzzy 
numbers and linguistic variables.  
5 Conclusion 
The main purpose of this study is to find the 
appropriate aggregation operator that is able to present 
aggregation by taking account the importance of the 
data that being fused. 
In this paper, we have analyzed the method used 
based on the journals and conference proceedings that 
are collected from selected popular academic 
databases.  
From the collected journal, we have separated 
them according to the method that being used by the 
authors. Each of the aggregation method has been 
presented in the percentage. See Table 1 and Figure 1 
in section 3. 
From the observation, most of the criteria of the 
classical and linguistic aggregation operator in 
decision-making methods mentioned above are 
assumed to be independent of one another, but in 
reality, the criteria of the problems are often 
interdependent or interactive. For real decision making 
problems, it does not need the assumption that criteria 
or preferences are independent of one another and was 
used to show as a powerful tool for modeling 
interaction phenomena in decision making [100]. 
Usually, there is interaction among preference of 
decision makers.  This phenomenon is called correlated 
criteria.  
In the real world of decision making problems, 
most criteria have interdependent, interactive or 
correlative characteristics. The interaction phenomena 
among criteria or the preference of experts is 
considered which is making it more feasible and 
practical than other traditional aggregation operator. In 
addition, it is not suitable for us to aggregate them by 
classical weighted arithmetic mean or geometric mean 
method which based on the additive measure. On the 
contrary, to approximate the human subjective decision 
making process, it would be more suitable to apply 
fuzzy measures, where it is not assuming additivity and 
independence among decision making criteria 
To tackle this problem, Choquet integral is a 
powerful tool to solve the MCDM problems with 
correlated criteria. In the Choquet integral model, 
criteria can be interdependent, a fuzzy measure is used 
to define a weight on each combination of criteria, thus 
making it possible to model the interaction existing 
among the criteria. Besides that, Choquet integral 
which taking into account for correlated inputs may 
give a more accurate prediction of the users’ rating. 
Furthermore, it is a very useful tool to measure the 
expected utility of an uncertain environment. 
6 References 
[1]  Roy, B. and PH. Vincke, Multicriteria Analysis 
Survey and New Directions, European Journal of 
Operational Research, vol. 8, pp. 207-218, 1981.  
[2]  E. K. Zavadskas, Z. Turskis and S. Kildiene, 
State of Art Surveys of Overviews on 
MCDM/MADM Methods,  Technological and 
Economic Development of Economy, vol. 20, no. 
1, 165-179, 2014.   
[3]  S. D. Pohekar and M. Ramachandran, 
Application of Multi-Criteria Decision Making to 
Sustainable Energy Planning-A Review, Renew 
Sustain Energy Rev, vol. 8, pp. 365-381, 2004. 
[4]  C. Diakaki, E. Grigoroudis, N. Kabelis, D. 
Kolokotsa, K. Kalaitzakis, and G. Stavrakakis,  A 
Multi-Objective Decision Model for the 
Improvement of Energy Efficiency in Buildings, 
Energy, vol. 35, pp.  5483-5496, 2010.  
[5]  R. R. Yager. On Generalized Bonferroni Mean 
Operators for Multi-Criteria Aggregation. 
International Journal of Approximate Reasoning, 
vol. 50, no. 8, pp. 1279–1286, 2009a.   
[6]  A. Sengupta and T. K. Pal, Fuzzy Preference 
Ordering of Interval Numbers in Decision 
Problems. New York: Springer, Heidelberg, 
2009. 
[7]  J. L. Marichal, Aggregation Operator for 
Multicriteria Decision Aid. Ph.D. Dissertations, 
University of Liege, 1999.  
[8]  M. Detyniecki, Mathematical Aggregation 
Operators and Their Application to Video 
Querying. PhD thesis. University of Paris, 2000. 
[9]  Z. S. Xu, Fuzzy Harmonic Mean Operators,  
International Journal of Intelligent Systems, vol. 
24, no. 2, pp. 152-172, 2009.  
[10]  J. Belles-sampera, J. M. Merigó and M. 
Santolino, Some New Definitions of Indicators 
for the Choquet Integral, pp. 467–476, 2013. 
[11]  M. N. Omar and A. R. Fayek, A TOPSIS-based 
Approach for Prioritized Aggregation in Multi-
Criteria Decision-Making Problems, Journal of 
Multi-Criteria Decision Analysis, 2016.  
 22 n
82 Informatica 41 (2017) 71–86 W.R.W. Mohd et al.  
 
[12]  J. Vanicek, I. Vrana and S. Aly,  Fuzzy 
Aggregation And Averaging for Group Decision 
Making: A Generalization and Survey. 
Knowledge-Based Systems, vol. 22, pp. 79-84, 
2009 
[13]  A. K. Madan, M. S. Ranganath, Multiple Criteria 
Decision Making Techniques in Manufacturing 
Industries-A Review Study with Application of 
Fuzzy,  International Conference of Advance 
Research and Innovation (ICARI-2014), pp. 136-
144, 2014.  
[14]  Ogryczak, W. On Multicriteria Optimization with 
Fair Aggregation of Individual Achievements. In: 
CSM’06: 20th Workshop on Methodologies and 
Tools for Complex System Modeling and 
Integrated Policy Assessment, IIASA, Laxenburg, 
Austria, 2006.  
[15]  L. G. Zhou and H. Y. Chen, A Generalization of 
the Power Aggregation Operators for Linguistic 
Environment and Its Application In Group 
Decision Making,  Knowl Based Syst, vol. 26, pp. 
216-224, 2012.  
[16]  J. L. Marichal, On Sugeno Integral as an 
Aggregation Function, Fuzzy Sets and Systems, 
vol. 114, pp. 347-365, 2000.  
[17]  R. R. Yager, The Power Average Operator,  IEEE 
Transactions on Systems, Man and Cybernetics, 
vol. 31, pp. 724-731, 2001.  
[18]  V. Torra, The Weighted OWA Operator, 
International Journal Intelligent System, vol. 12, 
pp. 153-166, 1977.  
[19]  H. F. Wang and S. Y. Shen, Group Decision 
Support with MOLP Applications. IEEE Trans 
Syst Man Cybern, vol. 19, pp. 143-153, 1989.  
[20]  R. Narasimhan, A Geometric Averaging 
Procedure for Constructing Supertransitivity 
Approximation to Binary Comparison Matrices, 
Fuzzy Sets System, vol. 8, pp. 53-61, 1982. 
[21]  S. Ovchinnikov, An Analytic Characterization of 
Some Aggregation Operator,  International 
Journal Intelligent System, vol. 7, pp. 765-786, 
1998.  
[22]  Z. S. Xu and Q. L. Da, An Overview of Operators 
for Aggregating Information, International 
Journal of Intelligent Systems, vol. 18, pp. 953-
969, 2003. 
[23]  R. Mesiar, A. Kolesarova, T. Calvo and M. 
Komornikova, A Review of Aggregation 
Functions. Fuzzy Sets and Their Application 
Extensions: Representation, Aggregation, and 
Models, 2008.  
[24]  D. L. L. R. Martinez and J. C. Acosta, 
Aggregation Operators Review-Mathematical 
Properties and Behavioral Measures, International 
Journal Intelligent Systems and Applications, vol. 
10, pp. 63-76, 2015.     
[25]  M. Grabisch, J. L. Marichal R. Mesiar and E. 
Pap, Aggregation Functions: Means, Information 
Science, vol.181, pp. 1-22, 2011.  
[26]  M. Detyniecki, Fundamentals on Aggregation 
Operators. AGOP, Berkeley, 2001. 
[27]  R. R. Yager, On Ordered Weighted Averaging 
Aggregation Operators in Multicriteria Decision 
Making, IEEE Transactions on Systems, Man, 
and Cybernetics, vol. 18, no. 1, 1988. 
[28]  T. Calvo, G. Mayor and R. Mesiar,  Aggregation 
Operators: New Trends and Application. Physica-
Verlag, New York, 2002. 
[29]  Z. Xu, Multi-Person Multi-Attribute Decision 
Making Models Under Intuitionistic Fuzzy 
Environment, Fuzzy Optim Decis Mak, vol. 6, 
no. 3, pp. 221-236, 2007.  
[30]  Z. S. Xu and J. Chen, Approach to Group 
Decision Making Based on Interval-Valued 
Intuitionistic Judgment Matrices, Systems 
Engineering-Theory and Practice, vol. 27, pp. 
126-133, 2007.  
[31]  D. F. Li, Multiattribute Decision Making Method 
Based on Generalized OWA Operators with 
Intuitionistic Fuzzy Sets, Expert Systems with 
Applications, vol. 37, pp. 8673-8678, 2010.  
[32]  D. F. Li, The GOWA Operator Based Approach 
to Multiattribute Decision Making Using 
Intuitionistic Fuzzy Sets, Mathematical and 
Computer Modelling, vol. 53, pp. 1182-1196, 
2011. 
[33]  J. R. Chang, S. Y. Liao and C. H.  Cheng, 
Situational ME-LOWA Aggregation Model for 
Evaluating the Best Main Battle Tank, 
Proceedings of the Sixth International Conference 
on Machine Learning and Cybernetics, pp. 1866-
1870, 2007. 
[34]  J. M. Merigo and M. Casanovas, The Fuzzy 
Generalized OWA Operator And Its Application 
In Strategic Decision Making, Cybernetics and 
Systems, vol. 41, no. 5, pp. 359-370, 2010.  
[35]  H. Zhao, Z. S. Xu, M. F.  Ni and S. S. Liu, 
Generalized Aggregation Operators for 
Intuitionistic Fuzzy Sets. International Journal of 
Intelligent Systems, vol. 25, pp. 1-30, 2010.  
[36]  L. Shen, H. Wang and X. Feng,  Some Arithmetic 
Aggregation Operators Within Intuitionistic 
Trapezoidal Fuzzy Setting and Their Application 
to Group Decision Making. Management Science 
and Industrial Engineering (MSIE), pp. 1053-
1059, 2011. 
[37]  M. Casanovas and J. M. Merigo, A New Decision 
Making Method with Uncertain Induced 
Aggregation Operator, Computational 
Intelligence in Multicriteria Decision Making 
(MCDM), pp. 151-158, 2011. 
[38]  J. M. Merigo, A Unified Model Between The 
Weighted Average and The Induced OWA 
Operator, Expert Systems with Applications, vol. 
38, no. 9, pp. 11560-11572., 2011. 
[39]  Y. Xu and H. Wang, Approaches Based on 2-
Tuple Linguistic Power Aggregation Operators 
for Multiple Attribute Group Decision Making 
Under Linguistic Environment, Applied Soft 
Computing, vol. 11, pp. 3988-3997, 2011.  
[40]  D. Yu, Intuitionistic Trapezoidal Fuzzy 
Information Aggregation Methods and Their 
Aggregation Methods in Group Decision... Informatica 41 (2017) 71–86 83 
 
Applications to Teaching Quality Evaluation, 
Journal of Information & Computational Science, 
vol. 10, no. 6, pp. 1861-1869, 2013.   
[41]  M. Xia, Z. Xu, and B. Zhu, Geometric Bonferroni 
Means with Their Application in Multi-Criteria 
Decision Making,  Knowledge-Based Systems, 
vol. 40, pp. 88-100, 2013.   
[42]  L. Zhou, Z. Tao, H. Chen and J.  Liu, Continuous 
Interval-Valued Intuitionistic Fuzzy Aggregation 
Operators and Their Applications to Group 
Decision Making, Applied Mathematical 
Modelling,  vol. 38, pp. 2190-2205, 2014.  
[43]  F. Chiclana, F. Herrera and E. Herrera-Viedma, 
The Ordered Weighted Geometric Operator: 
Properties and Applications in MCDM Problems, 
Physica-Verlag, Heidelberg, pp. 173-183, 2002. 
[44]  J. Wu and Q. W. Cao, Same Families of 
Geometric Aggregation Operators with 
Intuitionistic Trapezoidal Fuzzy Numbers, 
Applied Mathematical Modelling, vol. 37, no. 1, 
pp. 318-327, 2013.  
[45]  Z. Xu and R. R. Yager, Some Geometric 
Aggregation Operators, IEEE Transact Fuzzy 
Syst, vol. 15, pp. 1179-1187, 2006.   
[46]  Z. Xu and J.  Chen, On Geometric Aggregation 
over Interval-Valued Intuitionistic Fuzzy 
Information. Fourth International Conference on 
Fuzzy Systems and Knowledge Discovery, IEEE 
Computer Society Press, pp. 466-471, 2007. 
[47]  G. U. Wei,  Some Geometric Aggregation 
Functions and Their Application to Dynamic 
Multiple Attribute Decision Making in the 
Intuitionistic Fuzzy Setting,  International Journal 
Uncertain Fuzzy Knowledge-Based System, vol. 
17, no. 2, pp. 179-196, 2009.  
[48]  S. Das, S. Karmakar, T. Pal and S. Kar, Decision 
Making with Geometric Aggregation Operators 
based on Intuitionistic Fuzzy Sets, 2014 2nd 
International Conference on Business and 
Information Management (ICBIM), pp. 86-91, 
2014.   
[49]  C. Tan, Generalized Intuitionistic Fuzzy 
Geometric Aggregation Operator And Its 
Application to Multi-Criteria Group Decision 
Making, Soft Computing, vol. 15, no. 5,  pp. 867–
876, 2011. 
[50]  G. W. Wei and X. R. Wang,  Some Geometric 
Aggregation Operators Based on Interval-Valued 
Intuitionistic Fuzzy Sets and Their Application 
To Group Decision Making, International 
Conference on Computational Intelligence and 
Security, IEEE Computer Society Press, pp. 495-
499, 2007.  
[51]  Z. S. Xu, Approaches to Multiple Attribute Group 
Decision Making Based on Intuitionistic Fuzzy 
Power Aggregation Operators, Knowledge-Based 
Systems, vol. 24, pp. 749-760, 2011.  
[52]  Z. S. Xu and J. Chen, An Approach to Group 
Decision Making Based on Interval-Valued 
Intuitionistic Judgment Matrices, System 
Engineering Theory & Practice, vol. 27, no. 4, pp. 
126-133, 2007.  
[53]  R. Verma and B.D. Sharma, Hesitant Fuzzy 
Geometric Heronian Mean Operators and Their 
Application to Multi-Criteria Decision Making, 
Mathematica Japonica, 2015. 
[54]  Z. S. Xu, Harmonic Mean Operator for 
Aggregating Linguistic Information,  Fourth 
International Conference on Natural Computing, 
IEEE Computer Society, pp. 204-208, 2008.  
[55]  Z. S. Xu, Fuzzy Harmonic Mean Operators, 
International Journal of Intelligent Systems, vol. 
24, pp. 152-172, 2009. 
[56]  G. Wei and W. Yi,  Induced Trapezoidal Fuzzy 
Ordered Weighted Harmonic Averaging 
Operator, Journal of Information & 
Computational Science, vol. 7, no. 3, pp. 625-
630, 2010.  
[57]  G. W. Wei, Some Harmonic Aggregation 
Operators with 2-Tuple Linguistic Assessment 
Information And Their Application to Multiple 
Attribute Group Decision Making, International 
Journal of Uncertainty, Fuzziness and 
Knowledge-Based System, vol. 19, no. 6, pp. 
977-998, 2011.   
[58]  H. Zhou, Z. Xu and F. Cui,  Generalized Hesitant 
Fuzzy Harmonic Mean Operators and Their 
Applications in Group Decision Making. 
International Journal of Fuzzy Systems, pp. 1-12, 
2015. 
[59]  P. Liu, X. Zhang and F. Jin, A Multi-Attribute 
Group Decision-Making Method Based on 
Interval-Valued Trapezoidal Fuzzy Numbers 
Hybrid Harmonic Averaging Operators, Journal 
of Intelligent & Fuzzy Systems, vol. 23, no. 5, pp. 
159-168, 2012. 
[60]  C. Bonferroni, Sulle Medie Multiple Di Potenze. 
Bolletino Matematica Italiana, vol. 5, no. 3, pp.  
267-270, 1950. 
[61]  H. Sun and M. Sun, Generalized Bonferroni 
Harmonic Mean Operators and Their Application 
to Multiple Attribute Decision Making, Journal of 
Computational Information Systems, vol. 8, no. 
14, pp. 5717-5724, 2012.  
[62]  M. Sun and J. Liu, Normalized Geometric 
Bonferroni Operators of Hesitant Fuzzy Sets and 
Their Application in Multiple Attribute Decision 
Making, Journal of Information & Computational 
Science, vol. 10, no. 9, pp. 2815-2822, 2013.  
[63]  P. D. Liu and F. Jin, The Trapezoidal Fuzzy 
Linguistic Bonferroni Mean Operators and Their 
Application to Multiple Attribute Decision 
Making, Scientia Iranica, vol. 19, no. 6, pp. 1947-
1959, 2012.  
[64]  R. R. Yager, Prioritized OWA Aggregation. 
Fuzzy Optimization Decision Making, vol. 8, pp. 
245-262, 2009.  
[65]  Z. Xu and R. R Yager, Intuitionistic Fuzzy 
Bonferroni Means, IEEE Transactions on 
Systems, Man, and Cybernetics, Part B 
(Cybernetics), vol. 41, no. 2, pp. 568-578, 2011.  
84 Informatica 41 (2017) 71–86 W.R.W. Mohd et al.  
 
[66]  G. Beliakov, S. James, J. Mordelova  and T. 
Ruckschlossova, and R. R.  Yager,  Generalized 
Bonferroni Mean Operators in Multi-Criteria 
Aggregation. Fuzzy Sets and Systems, vol. 161, 
pp. 2227-2242, 2011. 
[67]  B. Zhu, Z. S. Xu and M. Xia,  Hesitant Fuzzy 
Geometric Bonferroni Means,  Information 
Science, vol. 205, pp. 72-85, 2012 
[68]  M. Xia, Z. S. Xu, and N. Chen, Some hesitant 
fuzzy aggregation operators with their application 
in group decision making, Group Decis, Negotiat, 
vol. 22, pp. 259-279, 2013.  
[69]  G. Wei, X. Zhao, R. Lin and H. Wang, Uncertain 
Linguistic Bonferroni Mean Operators and Their 
Application to Multiple Attribute Decision 
Making, Applied Mathematical Modelling, vol. 
37, pp. 5277-5285, 2013.  
[70]  J. H. Park and E. J. Park, Generalized Fuzzy 
Bonferroni Harmonic Mean Operators and Their 
Applications in Group Decision Making, Journal 
of Applied Mathematics, vol. 2013, 1-14, 2013.  
[71]  R. Verma, Generalized Bonferroni Mean 
Operator For Fuzzy Number Intuitionistic Fuzzy 
Sets And Their Application to Multi Attribute 
Decision Making,” International Journal of 
Intelligent Systems, vol. 30, no. 5, pp. 499-519, 
2015. 
[72]  Z. S. Xu and R. R. Yager, Power-Geometric 
Operators and Their Use in Group Decision 
Making,  IEEE Transaction on Fuzzy Systems, 
vol. 18, pp. 94-105, 2010.  
[73]  Z. S. Xu and X. Q. Cai, Uncertain Power Average 
Operators for Aggregating Interval Fuzzy 
Preference Relations. Group Decision and 
Negotiation, 2010. 
[74]  Z. S. Xu, Approach to Multiple Attribute Group 
Decision Making Based on Intuitionistic Fuzzy 
Power Aggregation Operator, Knowledge-Based 
Systems, vol. 24, no. 6, pp. 746-760, 2011. 
[75]  L. Zhou, H. Chen and J. Liu, Generalized Power 
aggregation Operators and Their Applications in 
Group Decision Making, Computers and 
Industrials Engineering, vol. 62, pp. 989-999, 
2012.  
[76]  S. P. Wan, Power Average Operators of 
Trapezoidal Intuitionistic Fuzzy Numbers And 
Application to Multi-Attribute Group Decision 
Making, Applied Mathematical Modelling, vol. 
37, no. 6, pp. 4112-4126, 2013.  
[77]  Z. Zhang, Hesitant Fuzzy Power Aggregation 
Operators and Their Application to Multiple 
Attribute Group Decision Making. Information 
Science, vol. 234, pp. 150-181, 2013.  
[78]  X. Qi, C. Liang and J. Zhang, Multiple Attribute 
Group Decision Making Based on Generalized 
Power Aggregation Operators under Interval-
Valued Dual Hesitant Fuzzy Linguistic 
Environment, Int. J. Mach. & Cyber, 2015.  
[79]  T. Y. Chen, Bivariate Models of Optimism and 
Pessimism in Multi-Criteria Decision-Making 
Based on Intuitionistic Fuzzy Sets,  Information 
Sciences, vol. 181, pp. 2139-2165, 2011. 
[80]  K. H. Gou, and W. L. Li, An Attitudinal-Based 
Method for Constructing Intuitionistic Fuzzy 
Information in Hybrid MADM Under 
Uncertainty, Information Sciences, 189, 77-92, 
2012. 
[81]  J. Z. Wu, F. Chen, C. P. Nie and Q. Zhang, 
Intuitionistic Fuzzy-Valued Choquet Integral and 
Its Application in Multicriteria Decision Making,  
Information Sciences, vol. 222, pp. 509-527, 
2013.  
[82]  Z. S. Xu, Induced Uncertain Linguistic OWA 
Operators Applied to Group Decision Making, 
Information Fusion, vol. 7, pp. 231-238, 2006. 
[83]  L. A. Zadeh, The Concept of A Linguistic 
Variable and Its Application to Approximate 
Reasoning Part 1, 2 and 3, Information Sciences, 
vol. 8, pp. 199-249, 1975.  
[84]  L. A. Zadeh, A computational approach to fuzzy 
quantifiers in natural languages. Computers & 
Mathematics with Applications, vol. 9, pp. 149-
184, 1983.  
[85]  J. J. Zhu and K. W. Hipel,  Multiple Stages Grey 
Target Decision Making Method with Incomplete 
Weight Based on Multi-Granularity Linguistic 
Label. Information Sciences, vol. 212, pp. 15-32, 
2012.  
[86]  Z. S. Xu, Uncertain Linguistic Aggregation 
Operators Based Approach to Multiple Attribute 
Group Decision Making Under Uncertain 
Linguistic Environment. Information Sciences, 
vol. 168, pp. 171-184, 2004.  
[87]  L. Martinez and F. Herrera, An Overview on the 
2-Tuple Linguistic Model for Computing with 
Words in Decision Making: Extensions, 
Application and Challenges, Information 
Sciences, vol. 207, pp. 1-18, 2012. 
[88]  L. Wang, Q. Shen and L. Zhu, L.  Dual Hesitant 
Fuzzy Power Aggregation Operators Based on 
Archimedean T-Conorm and T-Norm And Their 
Application to Multiple Attribute Group Decision 
Making, Applied Soft Computing, vol. 38, pp. 
23-50, 2016.  
[89]  S. Das and D. Guha, Power Harmonic 
Aggregation Operator With Trapezoidal 
Intuitionistic Fuzzy Numbers For Solving 
MAGDM Problems, Iranian Journal of Fuzzy 
Systems, vol.12, no. 6,pp.  41-74, 2015. 
[90]  G. Choquet, Theory of capacities. Annales de 
I’Institut Fourier, vol. 5, pp. 131-295, 1953. 
[91]  T. Murofushi and M. Sugeno, An Interpretation 
of Fuzzy Measure and the Choquet Integral as an 
Integral with Respect to A Fuzzy Measure, Fuzzy 
Sets and Systems, vol. 29, no. 2, pp. 201-227, 
1989.  
[92]  Z. S. Xu, Choquet Integrals of Weighted 
Intuitionistic Fuzzy Information, Information 
Sciences, vol. 180, pp. 726-736, 2010.   
Aggregation Methods in Group Decision... Informatica 41 (2017) 71–86 85 
 
[93]  R. R. Yager. Induced Aggregation Operators, 
Fuzzy Sets and Systems, vol. 137, pp. 59-69, 
2003.   
[94]  P. Mayer and M. Roubens,  On The Use of the 
Choquet Integral with Fuzzy Numbers in 
Multiple Criteria Decision Support, Fuzzy Sets 
and Systems, vol. 157, pp. 927-938, 2006.  
[95]  D. Hlinena, M. Kalina, and P.  Kral, Choquet 
Integral with Respect to Lukasiewicz Filters, and 
Its Modifications, Information Sciences, vol. 179, 
pp. 2912-2922, 2009. 
[96]  T. Ming-Lang, J. H. Chiang and   L. W. Lan, 
Selection of Optimal Supplier in Supply Chain 
Management Strategy With Analytic Network 
Process And Choquet Integral, Computer & 
Industrial Engineering, vol. 57, pp. 330-340, 
2009.   
[97]  G. Buyukozkan, and D. Ruan,. Choquet Integral 
Based Aggregation Approach to Software 
Development Risk Assessment. Information 
Science, vol. 180, pp. 441-451, 2010. 
[98]  S. Angilella,, S. Greco and B. Matarazzo, Non-
Additive Robust Ordinal Regression : A Multiple 
Criteria Decision Model Based on the Choquet 
Integral, European Journal of Operational 
Research, vol. 201, no. 1, pp. 277–288, 2010.  
[99]  K. Huang, J. Shieh, K. Lee and  S. Wu, Applying 
A Generalized Choquet Integral with Signed 
Fuzzy Measure Based on the Complexity to 
Evaluate the Overall Satisfaction of the Patients. 
Proceeding of the Nineth International 
Conference on Machine Learning and 
Cybernetics, Qingdao, 11–14 July 2010. 
[100]  T. Demirel, N. C. Demiral and C. Kahraman, 
Multi-Criteria Warehouse Location Using 
Choquet Integral, Expert Systems with 
Application, vol. 37, pp. 3943-3952, 2010. 
[101]  C. Tan and X. Chen, Intuitionistic Fuzzy 
Choquet Integral Operator for Multi-Criteria 
Decision Making, Expert Systems with 
Application, vol. 37, pp. 149-157, 2010.  
[102]  C. Tan,  A Multi-Criteria Interval-Valued 
Intuitionistic Fuzzy Group Decision Making with 
Choquet Integral-Based TOPSIS, Expert Systems 
with Application, vol. 38, pp. 3023-3033, 2011. 
[103]  H. Bustince, J. Fernandez, J. Sanz, M. Galar, 
R. Mesiar and A. Kolesarova, Multicriteria 
Decision Making by Means of Interval-Valued 
Choquet Integrals. Advances in Intelligent and 
Soft Computing, vol. 107, pp. 269-278, 2012.  
[104]  W. Yang and Z. Chen, New Aggregation 
Operators Based on the Choquet Integral and 2-
Tuple Linguistic Information,  Expert Systems 
with Applications, vol. 39, no. 3, pp. 2662–2668, 
2012.   
[105]  M. A. Islam, D. T. Anderson and T. C. 
Havens, Multi-Criteria Based Learning of the 
Choquet Integral using Goal Programming. 
Conference on Soft Computing (WConSC), 2015 
Annual Conference of the North American, pp. 1-
6, 2015. 
[106]  D. Yu, Y. Wu and W. Zhou, Multi-Criteria 
Decision Making Based on Choquet Integral 
under Hesitant Fuzzy Environment, Journal of 
Computational Information Systems, vol. 7, no. 
12, pp. 4506-4513, 2011.  
[107]  J. J. Peng, J. Q. Wang, H. Zhou and X. H. 
Chen, A Multi-Criteria Decision-Making 
Approach Based on TODIM and Choquet 
Integral Within Multiset Hesitant Fuzzy 
Environment,  Applied Mathematics & 
Information Sciences, vol. 9, no. 4, pp. 2087-
2097, 2015. 
[108]  J. Q. Wang, D. D. Wang, Y. H. Zhang and X. 
H. Chen, Multi-Criteria Group Decision Making 
Method Based on Interval 2-Tuple Linguistic and 
Choquet Integral Aggregation Operators, Soft 
Computing, vol. 19, no. 2, pp. 389-405, 2015. 
[109]  M. Sugeno, Theory of Fuzzy Integral and Its 
Application. Doctoral Dissertation, Tokyo 
Institute of Technology, 1974. 
[110]  O. Mendoza and P. Melin, Extension of the 
Sugeno Integral with Interval Type-2 Fuzzy 
Logic, Fuzzy Information Processing Society, 
2008. NAFIPS 2008. Annual Meeting of the 
North American, pp. 1-6, 2008.  
[111]  Y. Liu, Z. Kong and Y. Liu, Interval 
Intuitionistic Fuzzy-Valued Sugeno Integral, 
2012 9th International Conference on Fuzzy 
Systems and Knowledge Discovery (FSKD 
2012), pp. 89-92, 2012.  
[112]  M. Tabakov and M. Podhorska-Okolow, 
Using Fuzzy Sugeno Integral as an Aggregation 
Operator of Ensemble of Fuzzy Decision Trees in 
the Recognition of HER2 Breast Cancer 
Histopathology Images, 2013 International 
Conference on Computer Medical Applications 
(ICCMA), pp. 1-6, 2013.  
[113]  D. Dubois, H. Prade and Agnes. Rico, 
Residuated Variants of Sugeno Integrals: 
Towards New Weighting Schemes for Qualitative 
Aggregation Methods, Information Sciences, vol. 
329, pp. 765-781, 2016.  
[114]  W. Jianqiang and Z. Zhong, Aggregation 
Operators on Intuitionistic Trapezoidal Fuzzy 
Number And Its Application to Multi-Criteria 
Decision Making Problem, Journal of Systems 
Engineering and Electronics, vol. 20, no. 2, pp. 
321-326, 2009. 
[115]  Z. Zhang and P. Liu, Method for Aggregating 
Triangular Fuzzy Intuitionistic Fuzzy Information 
and Its Application to Decision Making,  
Technological and Economic Development of 
Economy, vol. 16, no. 2, pp. 280-290, 2010.  
[116]  J. M. Merigo and M. Casanovas, Fuzzy 
Generalized Hybrid Aggregation Operators and 
Its Application In Fuzzy Decision Making, 
International Journal of Fuzzy Systems, vol. 1, 
no. 1, pp.  15-23, 2010. 
[117]  M. Xia and Z. S. Xu, Hesitant Fuzzy 
Information Aggregation in Decision Making, 
86 Informatica 41 (2017) 71–86 W.R.W. Mohd et al.  
 
International Journal of Approximate Reasoning, 
vol. 52, no. 3, pp. 395-407, 2011.  
[118]  W. Yu, Y. Wu and T. Lu, Interval-Valued 
Intuitionistic Fuzzy Prioritized Operators and 
Their Application in Group Decision Making,  
Knowledge-Based Systems, vol. 30, pp. 57-66, 
2012.  
[119]  R. Verma and B. D. Sharma, Trapezoid Fuzzy 
Linguistic Prioritized Weighted Average 
Operators and Their Application to Multiple 
Attribute Group Decision Making,” Journal of 
Uncertainty Analysis and Applications, vol. 2, pp. 
1-19, 2014. 
[120]  H. Liao and Z. Xu, Extended Hesitant Fuzzy 
Hybrid Weighted Aggregation Operators and 
Their Application in Decision Making. Soft 
Computing, vol. 19, pp. 2551–2564, 2015.  
[121]  R. Verma, Multiple Attribute Group Decision 
Making based on GeneralizedTrapezoid Fuzzy 
Linguistic Prioritized Weighted Average 
Operator, International Journal of Machine 
Learning and Cybernetics, 2016. DOI: 
10.1007/s13042-016-0579-y 
[122]  R. R. Yager. Prioritized Aggregation 
Operators,  International Journal Approximate 
Reasoning, vol. 48, pp. 263-274, 2008.  
[123]  G. W. Wei, Hesitant Fuzzy Prioritized 
Operators and Their Application to Multiple 
Attribute Group Decision Making, Knowledge-
Based System, vol. 31, pp.  176-182, 2012.   
[124]  J. Q. Wang, J. T. Wu, J. Wang, H. Y. Zhang 
and X. H. Chen, Interval-Valued Hesitant Fuzzy 
Linguistic Sets and Their Application in Multi-
Criteria Decision-Making Problems, Information 
Sciences, vol. 288, pp. 55-72, 2014.  
[125]  T. Y. Chen, A Prioritized Aggregation 
Operator-Based Approach to Multiple Criteria 
Decision Making Using Interval-Valued 
Intuitionistic Fuzzy Sets: A Comparative 
Perspective, Information Science, vol. 281, pp. 
97-112, 2014.  
[126]  R. Verma and B. D. Sharma, Intuitionistic 
Fuzzy Einstein Prioritized Weighted Operators 
and Their Application to Multiple Attribute 
Group Decision Making, Applied Mathematics 
and Information Sciences, vol. 9, no. 6, pp. 3095-
3107, 2015. 
[127]  C. Liang, S. Zhao and J. Zhang, Multi-Criteria 
Group Decision Making Method Based on 
Generalized Intuitionistic Trapezoidal Fuzzy 
Prioritized Aggregation Operators, Int. J. Mach. 
& Cyber, 2015. 
[128]  J. Dong, D. Y. Yang and S. P.  Wan, 
Trapezoidal Intuitionistic Fuzzy Prioritized 
Aggregation Operators and Application to Multi-
Attribute Decision Making, Iranian Journal of 
Fuzzy Systems, vol. 12, no. 4, pp. 1-32, 2015. 
[129]  R. Verma, Prioritized Information Fusion 
Method for Triangular Fuzzy Information and Its 
Application to Multiple Attribute Decision 
Making, International Journal of Uncertainty, 
Fuzziness and Knowledge Based Systems, vol. 
24, no. 2, pp. 265-289, 2016. 
[130]  J. M. Merigó, M. Casanovas and  L. Martinez, 
Linguistic Aggregation Operators for Linguistic 
Decision Making Based on the Dempster–Shafer 
Theory of Evidence, International Journal 
Uncertainty, Fuzziness and Knowledge-based 
Systems, vol. 18, no. 3, pp. 287–304, 2011.  
[131]  J. H. Wang and J. Hao, A New Version of 2-
Tuple Fuzzy Linguistic Representation Model for 
Computing with Words, IEEE Transaction on 
Fuzzy Systems, vol. 14, no. 3, pp. 435-445, 2006. 
[132]  F. Herrera, E. Herrera-Viedma and L. 
Martinez, A Fuzzy Linguistic Methodology to 
Deal with Unbalanced Linguistic Term Sets, 
IEEE Transaction on Fuzzy Systems, vol. 16, no. 
2, pp. 354-370, 2008. 
[133]  G. U. Wei, Extension of TOPSIS Method for 
2-Tuple Linguistic Multiple Attribute Group 
Decision Making with Incomplete Weight 
Information. Know. Inf. Syst., vol. 25, pp. 623-
634, 2010.  
[134]  G. U. Wei, GRA-based Linear-Programming 
Methodology for Multiple Attribute Group 
Decision Making with 2-Tuple Linguistic 
Assessment Information. International 
Interdiscipline Journal, vol. 14, no. 4, pp. 1105-
1110, 2011. 
[135]  G. W. Wei, Grey Relational Analysis Method 
for 2-Tuple Linguistic Multiple Attribute Group 
Decision Making with Incomplete Weight 
Information, Expert Systems with Applications, 
vol. 38, pp. 4824-4828, 2011.  
[136]  Y. Xu and H. Wang, Approaches based on 2-
Tuple Linguistic Power Aggregation Operators 
for Multiple Attribute Group Decision Making 
Under Linguistic Environment, Applied Soft 
Computing, vol. 11, no. 5, pp. 3988-3997, 2011.  
[137]  G. H. Wei,   FIOWHM Operator and Its 
Application to Multiple Attribute Group Decision 
Making, Expert Systems with Applications, vol. 
38, no.  4, pp. 2984–2989, 2011. 
[138]  C. Li, S. Zeng, T. Pan and L. Zheng, A 
Method based on Induced Aggregation Operators 
and Distance Measures to Multiple Attribute 
Decision Making Under 2-Tuple Linguistic 
Environment, Journal of Computer and System 
Sciences, vol. 80, pp.  1339-1349, 2014. 
[139]  P. D. Liu and F. Jin, Methods for aggregating 
intuitionistic uncertain linguistic variables and 
their application to group decision making, 
Information Sciences, vol. 205, pp. 58-71, 2012.