Strojniški vestnik - Journal of Mechanical Engineering 54(2008)12, 855-865 UDC 658.562
Paper received: 28.01.2008 Paper accepted: 25.09.2008
Economic Design of Control Charts
Rok Zupančič1* - Alojzij Sluga2 1 Elan, d.o.o., Begunje, Slovenia 2 Faculty of Mechanical Engineering, University of Ljubljana, Slovenia
Control charts are widely used in industry for monitoring and controlling manufacturing processes. They should be designed economically in order to achieve minimum quality control costs. In this paper, an economic design of Shewhart control charts for process mean is proposed that takes into account various parameters. Standards for sample size within statistical process control do not exist due to high diversity of modern production. In the proposed economic model process-mean shift is assumed as random variable. This is a better approximation of the real world, than the models that assume process-mean shift as a constant value. Probability density function is used for description of process-mean shift. The optimum sample size is computed on base of loss function, regarding to constraints of particular production process. The comparison of optimum sample sizes assuming process-mean shift as a constant value versus random variable is presented. © 2008 Journal of Mechanical Engineering. All rights reserved.
Keywords: control charts, quality control, economic design, statistical process control
0 INTRODUCTION
A quality of a product is usually understood as a capability of fulfilling needs of a customer. A manufacturer desires to produce such products that match the desired specification and also fulfil customer needs. Hence, quality control is an essential part of the manufacturing process. In modern production the product's specifications are not controlled directly. The quality control is carried out by controlling the manufacturing process. A process quality is usually deducted from the manufactured products. In this case, not all of the products can be inspected directly, but only statistically. The quality control is therefore called Statistical Process Control (SPC). A control chart is a primary tool for SPC. A purpose of the control chart is to detect possible changes of the process, and inform the process operator about it. Many types of control charts have been developed in the past. Shewhart control charts, Cumulative sum (Cusum) control chart, Exponential weighted moving average (EWMA) control chart, and others are presented elsewhere for instance in [1]. Control chart is a quality control technique. Some quality improvement techniques are discussed in [2].
Shewhart control charts are widely used in industry. The design of the control charts has economic consequences since the cost of sampling and testing, the investigating out-of-control signals and possibly correcting probable
causes, as well as costs of allowing nonconforming products to reach the customer are all affected by choice of the control charts parameters [3]. The most important parameter that affects the cost of quality control and indirectly the cost of production is sample size. In literature, description of standards that deal with sample size for acceptance sampling can be found in [3]. Due to their generality of use, these standards are very robust. The standards for SPC, which deal with sample size, do not exist. The reason is a standard's uselessness due to its generality and high diversity of modern production. It is not the same if low- or high-priced products are manufactured. Due to diversity of quality control costs, optimum sample size should be calculated for every production process respectively. In industrial environment, no statistical expertise is available. Normally, a sample size is defined by a rule of thumb, lacking the optimality in economical sense. Due to fierce competition on the global market, an economic control charts design can be a critical issue for a manufacturing enterprise.
Some approaches to the economic control charts design are given in literature [4] to [7]. In all of these approaches, the process-mean shift (PMS) is assumed constant. In [4], Duncan's loss-cost function is used, which is originally proposed in [8]. In [5], simplified theoretical backgrounds and directions of economic design of control charts are proposed. In [6], economic design of
*Corr. Author's Address: Elan, d.o.o., Begunje 1, SI-4275 Begunje na Gorenjskem, Slovenia, rok.zupancic@elan.si
control charts of cumulative count of conforming products is presented. In [7], an economic design of control charts is presented for variable sampling size and sampling interval. In approaches [4] to [7], a PMS parameter is supposed to occupy an exact constant value. PMS is actually not known in advance. If constant value for PMS is assumed, the economic model is valid only for that PMS. PMS directly affects the optimum sample size calculation.
In this paper, an economic design of Shewhart control chart for the mean value of the process (x chart) is discussed. The approach does not assume an exact value for PMS. We show that by finding its probability density function of PMS, we can compute the optimum sample size (n*).
In Section 1, some theoretical backgrounds and parameters of the Shewhart control chart are discussed. In Section 2, the sampling policy and the economic model are proposed. In Section 3, the optimisation is presented in form of optimum sample size as a function of the PMS parameter. These curves are created for different economic parameters (costs). In Section 4, discussion and conclusions are presented.
1 BACKGROUNDS OF SHEWHART CONTROL CHART
Shewhart control chart method is about 80 years old. It is a method for statistical process control, in sense of process-mean and process variability supervision. The statistical parameters are deduced from a statistical sample. The accuracy depends on the sample size (n). In this paper, we are concentrating on x control chart for PMS detection. The control chart is a kind of hypothesis testing hence the process is in a state of statistical control. The hypothesis testing is done with sample statistics. The sample statistics is assumed to follow a normal distribution with mean (m) and sample standard deviation (o<x>). A sample standard deviation depends on standard deviation of population (a) and sample size (n). The null hypothesis states that the mean of a measured process variable has a desired value. The alternative hypothesis is that this mean value may be changed by ka. That means that the new mean value is n+ko.
The centre line of the control chart is kept at the mean of the process (m). The upper control limit (UCL) and the lower control limit (LCL) are:
UCL = u + La/Jn	(1)
LCL = U- La/4n	(2)
where L is the control limit coefficient (usually 3 is used).
The interval of normal distribution between m-3o and m+3o covers 99.73% of the entire population. Statistically, this means that 99.73% of sample values are supposed to fall between the control limits. In spite of no change in a process, 0.27% of the sample values are to exceed the control limits. This is treated as type I error (false-positive). The probability of type I error (a) is defined as:
a = 20(-L)	(3)
where <P(.) is a cumulative distribution function of a standard normal random variable. Type I error, which means a false alarm, has a geometric probability distribution. Its expected value is ARL0. A type I error, therefore, appears after every ARL0 samples on average. ARL0 is calculated as follows:
ARL0 =-!- •	(4)
a
When L = 3, a false alarm appears about every 370 samples on average.
The type II error denotes a false-negative detection (process change was not detected). The probability of type II error (fi) is defined as:
P(k, n, L) =
= 0(L - k/Jn) -0(-L - k/Jn) . (5)
The probability of type II error (fi) depends on the shift of the process mean value, denoted by ka. It also depends on L and n. When the process-mean changes, it is not detected immediately. The shift is detected on average after a number of samples, which is denoted by ARL1 (Average Run Length). ARL1 is the mean value of geometric distribution of shift detection,
when the process is out of control and it is calculated as follows:
ARL1 (k, n, L) =
1
1 -ß(k, n, L)
(6)
The chart of / and ARLi can be found elsewhere, for instance in [9]. From these charts one can conclude that sample size (n) has a significant impact on the efficiency of PMS detection. Greater n enables faster detection of PMS. Process capability ratio (Cp) is defined elsewhere [3] as:
C =
USL - LSL 6a
(7)
USL and LSL are specification limits. The control limits are allowed values for quality characteristic. Cp will be used in the following sections.
2 PRESENTATION OF SAMPLING POLICY AND ECONOMIC MODEL
2.1 Background of Economic Model
Economic model is a framework for the economic design of control charts. It is closely linked to a sampling policy, which determines a method of sampling. A sampling policy depends on the production type. Economic design of control charts is carried out on the basis of a loss-cost function optimisation. When the sampling policy changes, the results of optimisation are not valid anymore.
The issue of quality control prevents to produce nonconforming products. A fraction of nonconforming products, which are manufactured on a particular work system, has a very important impact on design. The average fraction of nonconforming products is described with probability of nonconforming product (Pnc), which depends on specification limits (USL, LSL), n, a and PMS. PMS is expressed by the multiplier of standard deviation (k). A normal probability density function for a process nature
is assumed. The average fraction of nonconforming products can be expressed as follows:
Pnc (USL, LSL, n,a, k ) = 2 +
+0
" LSL - + k) "	+1 -0	'USL - k)"
_ a _	2	_ a _
(8)
The value of Eq.(8) defines also the probability of product's nonconformity. The same value can be used for the fraction of nonconforming products in population. The value of this expression is always greater than 0. That follows from the assumed normal nature of the observed production process. The expression is minimum, when the process mean is in the middle of the tolerance interval. When the process mean changes, P increases.
2.2 Sampling Policy
For the proposed economic model, the following sampling policy is assumed.
Suppose we have a discrete production process, which is not self-correcting. The production process is assumed as a series of independent cycles over time. Each cycle begins with the production process in the in-control state and continues until the process changes and its change is detected and identified. A statistical process control is carried out periodically lot-by-lot, where Nx is a lot size. From every lot a sample of size n is taken, so that the last n products of the lot form a sample (Fig. 1). When a false alarm appears, the production is halted until the alarm identification is done. After a false alarm identification, the production continues. When an alarm is identified as a real alarm, which means that the process has been changed, a process adjustment is needed. Meanwhile, the process is out of control and an increased fraction of nonconforming products is manufactured. After the adjustment, during which the process is returned to the in-control state, the new cycle begins. One cycle of the production run in sense of the assumed sampling policy is presented in Fig. 1.
The sampling policy is determined by its lot size, sample size and inspection strategy. The lot size indirectly defines the sampling frequency (v^), which means how often a statistical sample is taken.
Fig. 1. Diagram ofproduction run in sense of the assumed sampling policy
A sampling frequency is calculated using expression:
v =■
J_
N
(9)
The economic model of control chart is related to the sampling policy. It represents a framework for the mathematical formulation of total quality costs. Using our economic model, an optimum sampling size and sampling frequency for a specific production can be found. When a process changes, this change is located after ARLj lots of manufactured products. Due to different batch sizes, variable processing times and appearance of different technical-organisational malfunctions in an industrial environment, economic design of control charts is not based on time, i.e., on the number of manufactured products.
From every lot, a sample is formed that is located at the end of the lot. In our economic model the most inconvenient case of a process change is assumed. We assume that the process changes immediately after the statistical sample has been formed. This would produce the maximum possible fraction of nonconforming products.
The economic designs reported in literature assume PMS to be a constant value. In our approach, PMS is assumed to be a random value, described by probability density function (PDFk). The shape of probability function PDFk
depends on the nature of PMS. PMS is expressed by k in units of a.
If we assume PDFk to be a symmetric bimodal probability density function with mode at (±Mk) and standard deviation (ak), PDFk is described with the following function:
PDFk = PDFk (k|Mk ,CTk) =
(k -Mk )2
1 1
2 42x1
nck
2at-
_ ( k + Mk )2
+
(10)
2 \[2nc
The states of the process, where k = 0 (process in statistical control), are not assumed as
PMSs.
2.3 Economic Model
The goal of economic design of control charts is to obtain the minimum costs of the production process. With quality control we want to achieve that only conforming products reach customers. Within the proposed model, the costs related to quality control should be minimized. In the proposed economic model, the following costs are included: (1) costs of quality inspection, (2) costs of false alarm, (3) costs of manufacturing nonconforming products and (4) costs of location and repairing an assignable cause of the process. The economic model is the
Fig. 2. Probability density function of PMSs in units ofprocess standard deviation
	Production		
XJN,		1	
Process under control	
1	MARL,,
L
Supposed state (L,„)
Process out of control	
ARLr 1	1
False alarm (L„J
Process change not dctcctcd (L,J
Process change identified (L,,)
	
Sub-total loss function (L,)	
	
Sub-total loss function (L„)	
			
	TOTAL loss function (LmUL)		
Fig. 3. Interpretation of the loss functions
framework for a loss-cost function derivation. Loss-cost function is a powerful tool, due to its simplicity of problem representation in the engineering area. Loss-cost functions are presented in [10]. For the case of simplicity of notation the loss-cost function will be named loss function.
In the statistical process control, the following states are possible (Fig. 3). These
states are linked with partial loss functions (L01, L02, L11, L12), which are grouped in 2 of separated sub-total loss functions (LI and LII). The sub-total loss functions define 2 states: process is under control and process is out of control.
For the loss function derivation, the following cost constants are needed:
Cc1 cost per unit (product) controlled, Cnc cost per nonconforming unit, Cfa cost per false alarm, Crac cost to locate and repair an assignable cause of the process malfunction.
Loss function design is presented in Fig. 3. Loss function is designed regarding 1 cycle of production run, presented in Fig. 1. In 1 cycle of production run the losses of two states are considered. The states are named Process under control and Process out of control. Their contributions are weighted with Xp/Ni and 1 respectively. The sub-total and partial loss functions are carried out analogously.
Partial loss functions are as follows: 1. If the process is under control (k=0) and a sample mean value is between the control limits, we have
L01 (n,k,USL,LSL, u,c, Nt,L) = nCcl +N,Pnc (USL, LSL, u,c, k)Cnc .	(11)
The partial loss function L01 assumes the state of the process under control and a sample mean between the control limits. That means that PMS is 0. L01 defines a loss of production and the cost of control per lot. The first summation term defines a loss due to control costs and the second one defines a loss due to the fraction of nonconforming parts. 2. If the process is under control (k=0) and a sample mean value exceeds the control limits (Type I error - false alarm), we have
L22 (n, k,USL, LSL,u,c, N,, L) = nCcl + +NP (USL, LSL, u, c, k)Cnc + CJb .
(12)
The partial loss function L02 assumes the state of the process that is under control and a sample mean falls beyond the control limits. In this case, a false alarm appears. Cfa defines costs of false alarm identification. 3. If the process is out of control (k£0) and a sample mean value falls between the control limits (Type II error) then
L11 (n, k, USL,LSL, u, c, N,, L) =
= nCci +
(13)
+NlPnc (USL, LSL, u,c, k )Cm
The partial loss function L11 assumes the state of the process, which is out of control and a sample mean lies between the control limits. PMS, which is greater than 0, has not been detected yet. In this state, the fraction of nonconforming parts is increased. This situation contributes the most to the overall loss.
4. If the process is out of control (k^0) and a sample mean value is beyond the control limits, we have
L12 (n,k, USL, LSL, u, c, Nl, L) =
= nCd + C,ac +
(14)
+N,Pnc (USL, LSL, u,c, k)Cnc .
The partial loss function L12 assumes the state of the process that is out of control and a sample mean beyond the control limits. A real alarm is detected. Crac defines costs due to PMS identification of the assignable cause of PMS and its removal.
From two aforementioned process states and belonging partial loss functions, two subtotal loss functions are formed. First of them is used for lots when a process is under control. The second is used for lots when a process is assumed out of control. From ARL1 it is possible to calculate how many samples have to be inspected until a change in the process is identified (if k^0).
The first sub-total loss function is:
L, (n, k, USL, LSL, u, c, Nl, L) = C f
= nC 1	+
c1 ARL0
(15)
+NP(USL,LSL,u,C,kC .
The second sub-total loss function is:
L„ (n, k,USL, LSL,u,c, N,, L) =
= ARL1 (k, n, L)[nCc1 +	(16)
+NP (USL, LSL, u, c, kC ] + Cm .
The sub-total loss functions correspond to the assumed process states, described in Fig. 1. The first sub-total loss function expresses costs per inspected lot. The second sub-total loss function expresses cost per out-of control process state, considering quality inspection and a fraction of nonconforming products that are
manufactured in this state. The total loss function can be deducted from Fig. 3. For PMS the probability density function PDFk is used. The total loss function is computed per manufactured product. This offers a possibility to compare losses of different lot sizes. There is additional information that is needed for the economic design of control charts. It is the process failure rate (Ap) that denotes after how many products, manufactured by the observed work system, the process changes on the average. Xp should be found by a long-term observation of the process.
A total loss function Eq.(17) consists of numerator and denominator. A numerator is a total loss during one cycle of the production run (Fig. 1). A denominator is a number of manufactured products in one cycle of the production run.
considered as PMS, there are two integrals needed. Due to symmetry, (17) can be simplified to
Lt(X ,n,USL,LSL,ju,a,Nl,L,/uk,ak) = X • L, (.) + 2 -J L, (.) • PDFk (k, Mk ) • dk
_ Ni_o_(18)
X + 2 -J ARLX) • PDF, (k^ ) • dk 1 N
Since the loss is expressed per manufactured product, it is possible to compare losses of different lot sizes and for each of them it is possible to find the optimum sample size (n*). By changing the lot-size, within the proposed sampling policy, the sampling frequency is changed as well.
3 SIMULATION STUDY
Lt (Xp, n, USL, LSL, fj,a, Nl, L,^t ,ak ) =
X 0 œ Nr ■ L, (■) +J Ln (.)• PDFk (■)• dk+J Ln (.)• PDFk (■)' dk = _-w_0_
[X- +)äRL(.> PDFk (■)• dk+J ARLÇy PDFk (■)• dk\N
(17)
Eq. (17) does not have k among its arguments. PMS-multiplier (k) is handled by PDFk. Since the zero case (k=0) is not
The optimum sample size is the sample size that minimizes LT. All of the discussed parameters have impact on the optimum sample size. In Fig. 4 we can see loss function (LT) as a function of the sample size for different modes (M). The following constants were assumed: CC1=10, CnC=100, Cfa=Crac=2 (process capability), Nl=100, o=0.2.
Äp=500, Cp=1
Fig. 4. Loss function (LT) vs. the sample size (n), where Cc1=10, Cnc=l00, Cfa=Cra Nl=l00, ak=0.2; the ordinate is in a logarithmic scale
=200, Xp=500, Cp=1,
From these curves the optimum sample size for a chosen mode Mk can be found. The curve (Mk=0.25) is monotonically increasing. In this case, we can conclude that control of the process is economically unacceptable since the optimum sample size is 0. The curve (Mk=0.5) has the minimum at approximately n = 1, but it is not explicitly significant. If greater sample size is chosen, error gets smaller. Curves, with larger Mk, have a significantly expressed optimum sample size. Using expression LT, the curves for the optimum sample size in dependency of Mk were plotted for various parameters. The optimum sample size and lot size can be calculated numerically, using LT.
The lot size defines the sampling frequency. In the presented case, the sampling frequency is 1 sample per 100 manufactured products, which means vs=0.01. The optimum sample size, where lot size is 100, can seen from diagrams, presented in Figs. 5a, b and 5c. By changing Cfa and Crac, the optimum sample size curves change. When Cfa and Crac increase, the optimum sample sizes lowers and peaks of the optimum sample size curves move to the right. Cfa and Crac have impact on the optimum sample size, where Mk is small. For large values of Mk the cost constants Cfa and Crac have no significant impact.
	i'								
	1 ■	\ %							
	.7								
	< / . <		£¡3						
	•' i 11			LV.					
t		■							
' f 1/ J	V-	A-							
----Co1 = 1
----Ccl =2
- -a- - Cc, = 5
Ccl = 10 Ccl = 50 Ccl = 100
Fig. 5a. Optimum sample size (n*) vs. modeMk for different Cc1, where, Cnc=100, Cfa=Crac=100, X„=500, Cv=l, N^lOO, ok=0.2
Fig. 5b. Optimum sample size (n*) vs. modeMk for different Ccl, where, Cnc=100,Cfa=Crac=200, Ap=500, Cp=1, Ni=100, ak=0.2
Fig. 5c. Optimum sample size (n*) vs. modeMkfor different Cc1, where, Cnc=100, Cfa=Crac=500, Ap=500, Cp=1, Ni=100, ak=0.2
In Figs. 5a, b and c the optimum sample size in dependency of the absolute value of mode (Mk) are presented. In some cases when the optimum sample size is 0, that means SPC is not recommended, because it is too expensive. In those cases, the total loss function LT is monotonically increasing, as we can see in Fig. 4 (Mk = 0.25).
4 DISCUSSION AND CONCLUSIONS
In this paper, a new approach for the economic design of x charts is given. Optimum intensity of quality inspection, that means sample size and sampling frequency, can be found by the optimisation of the proposed loss function. A sampling frequency is defined by lot size. The majority of the overall loses in quality control are contributed by the state, where the process is out of control and the process-mean shift is not detected. The duration of this state, which is estimated by ARL1, depends on the sample size, PMS and the control limits. Due to different PMS values appearing in practice, we describe PMS in terms of probability. In our approach PMS is not assumed to be a constant value, but it is described by the probability density function PDFk. The proposed case study is obtained with bimodal probability density function within the proposed economic model. The curves for the optimum sample size for different cost constants are presented. In other approaches, proposed in [4] to [7], a constant
value for PMS is used, which may be questionable in real production.
The advantage of our approach is that PMS, estimated with PDFk, is closer to reality. In real world, PMS varies with time. Our approach takes this fact into account and describes PMS as a random variable. For PDFk an arbitrary probability density function can be used. Zero case (k=0), that means PMS is 0, is excluded from the integral area in Expressions (17) and (18). If there are any constraints for sample size or lot size, given by the observed production process, the optimum sample size (n*) and the optimum lot size can be found iteratively. For the loss functions in Fig. 4 and optimum sample size in Figs. 5a, b and c the value of Cp is 1. If the process capability ratio is greater (Cp>1), the optimum sample sizes lowers and peaks of their curves move to the right. For greater Mk (Mk>4) no significant difference of optimum sample size for different Cc1 was observed.
In our case study, ak = 0.2, but normally it depends on the nature of the process shift. In Fig. 6, the curves of the optimum sample size for various ak values are presented. Proposed PDFk is taken into consideration. When a variance of PDFk approaches 0 (ak ^ 0), Mk approaches a constant value. Due to this fact, the curve of the optimum sample size taking into account PDFk differs from the curve of the optimum sample size, where PMS is
14 12 10
8 6 4 2
0
										
		t	A							
		/								
		1 / I								
		!								
y										
/ .r										
- ok =	0
----crk =	0.2
—■— <^k =	0.6
0 0.5 1
1.5
2.5 3 M„
3.5
4.5
Fig. 6. Comparison of optimum sample sizes (n*) for different (ak), where Cc1=2, Cnc=100, Cfa=Crac=200, Xp=500, Cp=1, Nl=100
supposed to be a constant value. The difference is significant for small values of Mk, (Mk<1.5). If the ak increases (ak=0.6), the optimum sample size approaches a constant value. In spite of that, the optimum sample size cannot be generalised. In Fig. 6 comparison of optimum sample size curves for different ak is presented. The optimum sample sizes, calculated with the proposed economic model (where ak=0.2 and ak=0.6) are compared with the optimum sample sizes, where PMS is assumed as a constant value.
In this paper, the economic model for Shewhart control charts for process mean design was proposed. An economic design of control charts is a complex operation. In industrial environment, a statistical and quality control expertise is not available. With intention to support this expertise, a web service based operation support will be available soon and proposed in [11]. The web service for this kind of operation support will help operators in the industrial environment not only to control the quality of processes, but also to design the optimum lot sizes and sample sizes. Economic design of control charts is a key operations support for the manufacturing enterprises in order to achieve a competitive position on the global market.
5 ACKNOWLEDGEMENT
This paper was partially supported by the Ministry of Higher Education, Science and Technology Republic of Slovenia, Grant No. 3211-06-000521 and inspired by 6 FP NoE VRL-KCiP - "Virtual Research Lab for Knowledge Community in Production", Contract No. FP6-507487/2.
6 GLOSSARY AND NOTATION
ARL0	Average Run Length of false
	alarms
ARL1	Average Run Length of process-
	out-of-control detection
Cc1	Cost per unit (product)
	controlled
Cfa	Cost per false alarm
C ^nc	Cost per nonconforming unit
Cp	Process capability ratio
C ^rac	Cost to locate and repair an
	assignable cause of the process
Cusum	Cumulative sum
EWMA	Exponential weighted moving
	average
k	Multiplier of a process standard
	deviation
ko	Shift of the process mean value
L
LCL
LSL
Lt
lxx
m
Mk
n
n*
N
PDFk PMS
P
nc
Control limits coefficient Lower control limit Lower specification limit Total loss function Loss function Sample statistic mean Mode of PDFk Sample size Optimum sample size Lot size
Probability density function of k
Process-mean shift
Fraction of nonconforming products
SPC Statistical Process Control
UCL Upper control limit
USL Upper specification limit
a	Probability of type I error
P	Probability of type II error
Process failure rate
i	Process mean
v,	Sampling frequency
a	Process standard deviation
ak Standard deviation of PDFk
a<x> Sample standard deviation
<P(.) Cumulative distribution function of a standard normal random variable
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X
p
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