2003 economic-statistical design of multivariate control charts for

Journal of Loss Prevention in the Process Industries 16 (2003) 9–18www.elsevier.com/locate/jlp

Economic-statistical design of multivariate control charts formonitoring the mean vector and covariance matrix

C.-Y. Choua,∗, C.-H. Chenb, H.-R. Liu c, X.-R. Huanga

a Department of Industrial Engineering and Management, National Yunlin University of Science and Technology, Touliu 640, Taiwan, ROCb Department of Industrial Management, Southern Taiwan University of Technology, Yung-Kang 710, Taiwan, ROC

c Department of Food and Nutrition, Hung-Kuang Institute of Technology, Shalu 433, Taiwan, ROC

Abstract

When a control chart is used to monitor a process, three test parameters should be determined: the sample size, the samplinginterval between successive samples, and the control limits or critical region of the chart. In this paper, we present the procedureto conduct the economic-statistical design of multivariate control charts for monitoring the process mean vector and covariancematrix simultaneously; i.e. to economically determine the optimum values of the three test parameters such that the statisticalconstraints (including the type I error probability and power) of the control chart can be satisfied. The test statistic�2�nL is appliedto develop this procedure and the cost model is established based on the function given by Montgomery and Klatt. A numericalexample is provided to illustrate the solution procedure of the design and then the effects of cost parameters on the optimal designare examined. 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Quality control; Control chart; Economic-statistical design; Mean vector; Covariance matrix

1. Introduction

Statistical process control is an effective approach forimproving product quality and saving production costsfor a firm. Since 1924, when Dr Shewhart presented thefirst control chart, various control chart techniques havebeen developed and widely applied as a primary tool instatistical process control. The major function of controlcharting is to detect the occurrence of assignable causesso that the necessary corrective action can be takenbefore a large quantity of nonconforming product ismanufactured. The control charting technique may beconsidered as the graphical expression and operation ofstatistical hypothesis testing. When a control chart isused to monitor a process, three test parameters shouldbe determined: the sample size, the sampling intervalbetween successive samples, and the control limits orcritical region of the chart.

Duncan (1956) proposed the first economic model fordetermining the three test parameters for theX-bar con-

∗ Corresponding author.E-mail address: [email protected] (C.-Y. Chou).

0950-4230/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0950-4230(02)00089-X

trol charts that minimizes the average cost when a singleout-of-control state (assignable cause) exists. Duncan’scost model includes the cost of sampling and inspection,the cost of defective products, the cost of false alarm,the cost of searching assignable cause, and the cost ofprocess correction. Since then, considerable attention hasbeen devoted to the optimal economic determination ofthe three parameters ofX-bar charts (Duncan, 1971;Gibra, 1971; Goel, Jain, & Wu, 1968; Knappenberger &Grandage, 1969). Montgomery (1980) gave a thoroughreview of the literature in economic designs of variouscontrol charts. A bibliography of related papers is alsoavailable in works by Ho and Case (1994) and Vance(1983). Since the solution from an economic design ofcontrol charts may result in poor statistical properties,Saniga (1989) presented the economic-statistical designfor a control chart in which the cost function is minim-ized subject to the constrained minimum value of powerand maximum value of the type I error probability.

Although many efforts have been put on economicdesigns of control charts that measure single character-istic, some industrial products and processes are charac-terized by two or more measurable characteristics, andtheir joint effect describes product quality. For example,

10 C.-Y. Chou et al. / Journal of Loss Prevention in the Process Industries 16 (2003) 9–18

in the production of synthetic fiber, the tensile strengthand diameter may be equally important quality charac-teristics. These characteristics are jointly distributed ran-dom variables and cannot appropriately be controlled byindependently applying a control chart to each variable.Some authors (Ghare & Torgersen, 1968; Hotelling,1947; Jackson, 1956) have developed quality controlprocedures for several related random variables. Amongthese procedures, Hotelling T2 control chart is probablythe most widely known. Montgomery and Klatt (1972)presented a cost model to economically design the T2

control charts. Chen (1995) conducted the economic-statistical design of the T2 control charts by adding thestatistical constraints to the design procedure.

Since the T2 control charts monitor only the processmean vector and the process covariance matrix may alsohave an impact on product quality, in this paper we applythe test statistic �2�nL (which will be reviewed in Sec-tion 3) to develop the economic-statistical design of themultivariate control charts for monitoring the processmean vector and covariance matrix simultaneously. Thecost function used in this paper is based on the modelgiven by Saniga (1977) and Montgomery and Klatt(1972). An example is provided to illustrate the solutionprocedure of the design and then some sensitivity analy-ses are conducted to investigate the effects of cost para-meters on the solution of the design.

2. Model assumptions

To simplify the mathematical manipulation and analy-sis of the control chart, the following assumptions aremade:

1. The quality of the process can be described by themean vector and covariance matrix of p character-istics and is monitored by a multivariate control chartusing the test statistic �2�nL.

2. The p quality characteristics monitored by the multi-variate control chart follow a multivariate normal dis-tribution with mean vector m and covariance matrix�.

3. In the beginning of the process, the process isassumed to be in control; that is, m � m0 and � ��0, where m0 is a given vector and �0 is a given

positive definite matrix.4. There are two out-of-control states caused by two

assignable causes, respectively. After the first assign-able cause occurs, the process mean vector shifts tom1 � m0 � �m and the process covariance matrixremains unchanged, where the p × 1 vector �m isknown. After the second assignable cause occurs, theprocess covariance matrix shifts to �1 � �0 � ��

and the process mean vector remains unchanged,where the p × p matrix �� is known.

5. The time the process remains in the in-control statebefore going out of control is assumed to follow anexponential distribution with mean l�1 hours.

6. When the process goes out of control, it stays out ofcontrol until detected and corrected.

7. During each sampling interval, there exists at mostone assignable cause which makes the process out ofcontrol. The assignable cause will not occur at sam-pling time.

8. When the multivariate control chart indicates that theprocess is out of control, the process is stopped forinvestigating the assignable cause.

9. The cost for investigating real and false alarms isthe same.

10.Considering the statistical properties of the multivari-ate control chart, the upper bound of the type I errorprobability is set to be 0.1 and the lower bounds ofthe powers for the two out-of-control states are all setto be 0.9.

3. The test statistic

Suppose that the output of a process can be describedby p quality characteristics, and Y is a p × 1 randomvector whose jth element is the jth quality characteristicand is multivariate normally distributed. Let E(Y) � mbe the p × 1 mean vector of the characteristics andCov(Y) � � be the p × p covariance matrix of Y. Gen-erally, m and � are unknown. For a random sample ofsize n from Y, say Y1, Y2, …, Yn, the sample meanvector and sample covariance matrix may be com-puted by

Y �1n�

n

i � 1

Yi, (1)

S �1

n�1�n

i � 1

(Yi�Y)(Yi�Y)T, (2)

where superscript T denotes the transpose operation. Thelikelihood ratio criterion of testing the hypothesis H0:m � m0 and � � �0 against alternatives H1: m � m0 or� � �0 may be expressed as (see Anderson, 1958)

L � �en�np/2

|(n�1)S��10 |n/2 exp��

12Tr��1

0 [(n�1)S (3)

� n(Y�m0)(Y�m0)T]�,

where Tr denotes the trace operation of a matrix. Thevalue of L is between 0 and 1. If L � ca, the null hypoth-esis H0 is rejected, where ca is the lower 100ath percen-tile of the distribution of L. However, under the nullhypothesis, the exact distribution of L is unknown.Therefore, statisticians (Davis, 1971; Nagarsenker & Pil-

11C.-Y. Chou et al. / Journal of Loss Prevention in the Process Industries 16 (2003) 9–18

lai, 1974; Sugiura, 1969) transform L to the statistic�2�nL, which may be expressed as a chi-square series,as the test criterion by obtaining its asymptotic distri-bution. Thus, the null hypothesis H0 should be rejectedwhen �2�nL � �2�nca � UCL, where UCL is theupper 100 ath percentile of the distribution of �2�nL,and is the upper control limit (UCL) of the multivariatecontrol chart in this paper, which indicates that theassignable cause may exist in the process. To considerthe statistical constraints (i.e. the type I error probabilityand the power) of the chart, the distribution functions of�2�nL under the null and alternative hypotheses shouldbe evaluated.

3.1. The distribution function of �2�nL under H0

According to Sugiura (1969), the distribution functionof �2�nL under the null hypothesis, which is related tothe type I error probability of the multivariate controlchart, may be obtained through the following five steps:

Step 1 The characteristic function of �2�nL under thenull hypothesis is

f(t) � E[eit(�2�nL)] (4)

�

(2e /n)�npit�pg=1��n(1�2it)�g

2 ��(1�2it)np(1�2it)/2�p

g=1��n�g2 �� .

Step 2 The approximation formula for the gamma func-tion in Eq. (4) is

�n�(x � h) � �n�2π � (x � h�1/2)�nx (5)

�x� �wr � 1

(�1)rBr+1(h)r(r � 1)xr � O(x�w�1),

where Br(h) is the Bernoulli polynomial ofdegree r. Taking the logarithmic operation onEq. (4) and substituting Eq. (5) into (4) results in

�nf(t) � ��p(p � 1) � 2p

4 ��n(1�2it)

� �wr � 1

(�2)rBr+1

r(r � 1)nr[(1�2it)�r�1] (6)

� O(n�w�1),

where

Br+1 � �p

g � 1

Br+1(�g /2).

.Step 3 Applying the exponential operation on Eq. (6)

yields the characteristic function of �2�nL, i.e.f(t) � e�nf(t), which can be expressed as thesummation of a chi-square series(Nagarsenker & Pillai, 1974).

Step 4 According to Theorem 2.6.3 in Anderson (1984),if Z � �2�nL, then the density function of Z is

f(z) �1

2π �

�

e�itzf(t) dt. (7)

Step 5 The distribution function of �2�nL may beobtained using its definition, i.e.

F(z) � P(�2�nL � z) � �z

0

f(z) dz. (8)

As the value of w in Eq. (6) is equal to 3, the distri-bution function of �2�nL under null hypothesis may beexpressed as (see Sugiura, 1969)

F(z) � P(c2f � z) � B2n�1[P(c2

f+2 � z)�P(c2f � z)]

�16

n�2[(3B22�4B3)P(c2

f+4 � z)�6B22P(c2

f+2 � z) � (3B22 � 4B3)P(c2

f � z)]

�16

n�3[(4B4�4B2B3 � B32)P(c2

f+6 � z) � B2(4B3�3B22)P(c2

f+4 � z)

� B2(4B3 � 3B22)P(c2

f+2 � z)�(4B4 � 4B2B3 � B32)P(c2

f � z)] � O(n�4),

(9)

where

f � p �p(p � 1)

2,

B2 �p(2p2 � 9p � 11)

24,

B3 ��p(p � 1)(p � 2)(p � 3)

32

and

B4 �p(6p4 � 45p3 � 110p2 � 90p � 3)

480.

The distribution function in Eq. (9) is sufficientlyaccurate only for large sample size (say, n50). In prac-tical operation of a control chart, the sample size is usu-ally small. Therefore, Eq. (9) cannot be used to obtainthe type I error probability for the chart. In this paper,to find the type I error probability for the multivariatecontrol chart, we apply the abovementioned five stepsdirectly by increasing the value of w in Eq. (6) and runthese mathematical operations in the software mathem-atica 4.0. (2000). In most of the cases, as w30, thechi-square series would converge and consequently thetype I error probability for the chart can be numerically


obtained. The values obtained from mathematica 4.0are consistent with those given in the study by Nagar-senker and Pillai (1974).

3.2. The distribution function of �2�nL under H1

In order to evaluate the powers of the multivariatecontrol chart, the distribution function of �2�nL underalternative hypotheses should be studied.

Sugiura (1969) developed an asymptotical distributionfunction of �2�nL under H1. However, this function is,again, appropriate only for large sample size. When thisfunction is expanded as what we do in Section 3.1, itcannot converge to a certain probability value. There-fore, in this paper, we apply the simulation andregression approaches to evaluate the distribution func-tion of �2�nL under H1.

The simulation and regression procedures for the firstout-of-control state (caused by the shift of mean vector)are described as follows:

Step 1 Select a value for UCL (called z value) as thecritical value of the chart. The selection of zvalue must meet the statistical requirements. Forexample, in the case of n � 4 and p � 2, afterthe five steps in Section 3.1 are run, we canobtain the upper 10th percentile of the distri-bution of �2�nL under H0 which is 14.386.Since the upper bound of the type I error prob-ability of the chart is set 0.1 in this paper, theUCL would be greater than or equal to 14.386.When we select a z value, the possible range isz14.386 and, say, 14.40 can be a starting value.

Step 2 Generate n p × 1 random vectors from the multi-variate normal distribution with mean m1 andcovariance matrix �0. This step may be doneusing the software mathematica 4.0.

Step 3 From the output of Step 1, calculate the value of�2�nL using Eqs. (1), (2) and (3)

Step 4 Let mj � 1 if the calculated value of �2�nL isgreater than the selected z value. Otherwise, letmj � 0. The subscript j denotes the run of simul-ation.

Step 5 Repeat Steps 2–4 by 10,000 times.Step 6 Compute the simulated power for the selected z

value as follows:

Power1 �

�10000

j � 1

mj

10,000. (10)

Step 7 Go back to Step 1, select another z value byincreasing a step of 0.5 from the last selectedz value, and obtain its corresponding simulatedpower. If the simulated power is greater than 0.9,which is the lower bound of the power for the

chart, repeat Step 7; otherwise, stop the simul-ation procedure.

Step 8 For a set of selected z values and their corre-sponding simulated powers, treat the z value asan independent variable and the simulated poweras the dependent variable, and obtain a poly-nomial regression equation by using forwardselection (Montgomery & Peck, 1992) for a cer-tain combination of n and p. This equation isused as a function to estimate the power for thecorresponding UCL.

For the second out-of-control state caused by the shiftof covariance matrix, the same eight steps are appliedto estimate and evaluate the power of the chart exceptthat in Step 2, we generate n p × 1 random vectors fromthe multivariate normal distribution with mean m0 andcovariance matrix �1. The simulated power based on thesecond out-of-control state is denoted by Power2 inthis paper.

4. The cost function

The expected total cost per unit of product associatedwith the test procedure may be written as

E(C) � E(C1) � E(C2) � E(C3), (11)

where E(C1) is the expected cost per unit of samplingand carrying out the test procedure, E(C2) the expectedcost per unit associated with investigating and correctingthe process when the test procedure indicates that theprocess is out of control, and E(C3) is the expected costper unit associated with producing defective products.

The cost of sampling and testing is assumed to consistof a fixed cost (denoted by a1) independent of the samplesize n and a cost per unit sampled (denoted by a2), i.e.

E(C1) � (a1 � na2) /k, (12)

where k is the number of units produced between suc-cessive samples. To simplify the analysis procedure, weassume that there are only three states for the process,as what Saniga (1977) did. State 0 denotes the processis in control. State 1 denotes the process mean vectorshifts such that the process is out of control. State 2denotes the process covariance matrix shifts such thatthe process is out of control. The state that the processmean vector and covariance matrix shift simultaneouslyis not considered in this paper. Let a3 be the averagecost of investigating and correcting an out-of-controlprocess. Then

E(C2) � (a3 �2

i � 0

riai) /k, (13)

where ri is the conditional probability that the test pro-


cedure indicates the process is out of control given thatthe process is in State i, for i � 0, 1 and 2, and ai is theprobability that the process is in State i at the time thetest is performed. Let a4 represent the penalty cost ofproducing a defective unit of product. Thus

E(C3) � a4 �2

i � 0

digi, (14)

where di is the conditional probability of producing adefective unit given that the process is in State i, and giis the probability that the process is in State i at anypoint of time. Therefore, by substituting Eqs. (12), (13)and (14) into (11), the expected total cost per unit ofproduct can be expressed as

E(C) � (a1 � na2) /k � (a3 �2

i � 0

riai) /k (15)

� a4 �2

i � 0

digi.

The ais are cost coefficients independent of the testprocedure. The probabilities ri, di, gi and ai in Eq. (15)will be examined in the following paragraphs. Furtherdiscussion of the general form of the cost function maybe found in works by Knappenberger and Grandage(1969) and Montgomery and Klatt (1972). Economic-statistical design of multivariate control charts is todetermine the three test parameters (i.e. n, k and UCL)such that E(C) in Eq. (15) is minimized and the statisticalconstraints (i.e. r0�0.1, r10.9 and r20.9) are satis-fied.

4.1. Determination of ri

The definition of ri is the conditional probability thatthe test procedure indicates the process is out of controlgiven that the process is in State i, for i � 0, 1 and 2.Note that r0 is the type I error probability of the chart,r1 the power due to the shift of process mean vector,and r2 is the power due to the shift of process covariancematrix. Therefore, if UCL � z, then

r0 � 1�P(�2�nL�z), (16)

where P(�2�nL�z) may be obtained through the fivesteps in Section 3.1. Also, r1 and r2 can be estimatedby Power1 (in Eq. (10)) and Power2, respectively, usingthe simulation and regression approaches given in Sec-tion 3.2.

4.2. Determination of di

The definition of di is the conditional probability ofproducing a defective unit given that the process is inState i, for i � 0, 1 and 2, whose values depend on the

specification limits for each quality characteristic. Definel and u as the lower and upper specification-limit vec-tors, respectively, whose elements �j and uj (for j �1, 2, …, p) represent, respectively, the lower and upper

specification limits of the jth characteristic. Therefore,according to the definition, di may be determined by

d0 � 1��u1

�1

%�up

�p

[(2p)p/2|�0|1/2]�1 exp��12

(Y (17)

�m0)T��10 (Y�m0) dyp%dy1,

d1 � 1��u1

�1

%�up

�p

[(2p)p/2|�0|1/2]�1 exp��12

(Y (18)

�m1)T�-10 (Y�m1) dyp%dy1,

d2 � 1��u1

�1

%�up

�p

[(2p)p/2|�1|1/2]�1 exp��12

(Y (19)

�m0)T�-11 (Y�m0) dyp%dy1.

4.3. Determination of ai

The element ai is defined as the steady-state prob-ability that the process is in State i at the time the testis performed, for i � 0, 1 and 2. To obtain ai, a tran-sition probability matrix B is required. The elements inB, denoted by bij, are the probability of the process shift-ing from State i to State j during the production of kunits, for i, j � 0, 1 and 2. Suppose Q units are pro-duced per hour and fractional units can be produced. Theprobability of remaining in State 0 (in-control state)while k units are produced is P0 � exp(�lk /Q). Theprobability assigned to States 1 and 2 (out-of-controlstates) while k units are produced is P1 � P2 � 1�P0 � 1�exp(�lk /Q). Knappenberger and Grandage(1969) developed a formula to determine the values ofP1 and P2 as follows:

Pi �2![1�exp(�lk /Q)]qi(1�q)2�i

i!(2�i)![1�(1�q)2],

for i � 1 and 2,

(20)

where 0 � q � 1. Proper selection of the value of q canprecisely describe the distribution of P1 and P2. Parti-cularly, P1 � P2 as q � 0.667, P1 � P2 as q � 0.667,and P1 � P2 as q � 0.667. In practice, the values of P1

and P2 may also be determined by past experience.The elements of B may now be defined. The prob-


ability of being in control at the mth sample and still incontrol at the (m � 1)st sample is the probability ofremaining in control during the production of k units,i.e. b00 � P0. The probability of the process being incontrol at the mth sample and being in the ith out-of-control state at the (m � 1)st sample is the probabilityof shifting to the ith out-of-control state during the pro-duction of k units, i.e. b0i � Pi, for i � 1 and 2. Theprobability of the process being in the ith out-of-controlstate (for i � 1 and 2) at the mth sample and still beingin the same out-of-control state at the (m � 1)st sampleis the probability of detecting the out-of-control state atthe mth sample times the probability of going the sameout-of-control state again for the production of k unitsplus the probability of not detecting this out-of-controlstate at the mth sample, i.e. bii � riPi � (1�ri), fori � 1 and 2. The probability that the process is in theith out-of-control state (for i � 1 and 2) at the mth sam-ple and is in control (or in another out-of-control state)at the (m � 1)st sample is the probability that the ithout-of-control state is detected at the mth sample timesthe probability of remaining in control (or going anotherout-of-control state) for the production of k units, i.e.bij � riPj, for i � 1 and 2, j � 0, 1 and 2, and i � j.Therefore, the transition probability matrix B may bewritten as

B � P0 P1 P2

r1P0 r1P1 � (1�r1) r1P2

r2P0 r2P1 r2P2 � (1�r2)�.

It is easily shown that B is the transition matrix ofan irreducible aperiodic positive recurrent Markov chain.Therefore, there exists a vector a such that

aTB � aT, (21)

where aT � [a0,a1,a2], a0 � a1 � a2 � 1, and ai isthe steady-state probability that the process is in State iat the time the test is performed, for i � 0, 1 and 2. Itcan be shown that the solution to Eq. (21) is

a0 � r1r2P0 / (r1P2 � r2P1 � r1r2P0), (22)

a1 � r2P1 / (r1P2 � r2P1 � r1r2P0), (23)

a2 � r1P2 / (r1P2 � r2P1 � r1r2P0). (24)

4.4. Determination of gi

The element gi is defined as the steady-state prob-ability that the process is in State i at any point of time,for i � 0, 1 and 2. Duncan (1956) has shown that givena shift between the mth and (m � 1)st samples, the aver-age fraction of time that elapses before the shift occurs is

t �1�(1 � lk /Q) exp(�lk /Q)

[1�exp(�lk /Q)](lk /Q).

Note that t is the conditional expectation of the occur-rence of the assignable cause within an interval of sam-pling. In this paper we assume that during each samplinginterval, there exists at most one assignable cause thatmakes the process out of control. Then, the probabilitygi (for i � 1 and 2) depends on the probability that theprocess is in the ith out-of-control state when a sampleis taken, and the probability that the process is in controlwhen a sample is taken and shifts to this out-of-controlstate during the production of k units. That is

gi � ai � a0Pi(1�t), for i � 1 and 2. (25)

Consequently, we have

g0 � 1�g1�g2 � a0P0 � a0P1t � a0P2t. (26)

5. An example and solution procedure

From examination of Eq. (15) and the probabilities inthe preceding section, it can be seen that determining theoptimal three test parameters is not straightforward. Toillustrate the nature of the solutions obtained from econ-omic-statistical design of multivariate control charts, aparticular numerical example is presented. We assumethat only two quality characteristics are of interest (i.e.p � 2), that the in-control state is

m0 � �0

0 and �0 � �10 0

0 15,

and that the other necessary parameters are

m1 � �2�10

2�15, �1 � �90 0

0 135,

l � ��3�10

�3�15, u � �3�10

3�15,

q � 0.667, l /Q � 0.001 (i.e. on an average, the processshifts out of control after every 1000 units areproduced), a1 � US$20 per sample, a2 � US$0.2 perunit sampled, a3 � US$10 per investigation, and a4 �US$10 per defective unit discovered.

The solution procedure is of two stages. In the firststage, the feasible solution area of the upper control limitof the chart for a particular sample size is determinedsuch that the search area can be reduced. In the secondstage, a grid search is applied to find the values of n, k,and UCL that minimize E(C).

5.1. The first-stage solution procedure

When the process is in control (i.e. State 0), the qual-ity characteristics, Y, follow a multivariate normal distri-bution with mean m0 and covariance matrix �0. The type


I error probability, r0, can be obtained through Eq. (16).For a particular sample size (n), to meet the statisticalconstraint of r0�0.1, the feasible solution area for theUCL of the chart may be found. Table 1 lists the feasiblesolution areas of UCL for some sample sizes underState 0.

When the process mean vector shifts to m1 (i.e. State1), the quality characteristics, Y, follow a multivariatenormal distribution with mean m1 and covariance matrix�0. The power estimation function can be obtainedthrough the simulation and regression approaches men-tioned in Section 3.2. For a particular sample size (n),to satisfy the statistical constraint of r10.9, a powerestimation function can be obtained and then the feasiblesolution area for the UCL of the chart may be determ-ined. For example, as n � 5, the power estimation func-tion is

Power1 � 1.04038�0.00887448z (27)

� 0.000662788z2�0.0000171429z3,

where z is a specific value of �2�nL. Fig. 1 shows thesimulated power value and the fitted power estimationfunction in Eq. (27). The determination coefficient forthis fitted function is 0.999151. From this function, thefeasible solution area for the UCL of the chart is z�30.492. Table 1 lists the feasible solution areas of UCLfor some sample sizes under State 1.

When the process covariance matrix shifts to �1 (i.e.State 2), the quality characteristics, Y, follow a multi-variate normal distribution with mean m0 and covariancematrix �1. The power estimation function can beobtained through the simulation and regressionapproaches mentioned in Section 3.2. For a particularsample size (n), to meet the statistical constraint ofr20.9, a power estimation function can be obtained andthen the feasible solution area for the UCL of the chartmay be determined. For example, as n � 5, the powerestimation function is

Power2 � 0.996079 � 0.00137878z (28)

�0.000216846z2,

where z is a specific value of �2�nL. Fig. 2 shows thesimulated power value and the fitted power estimationfunction in Eq. (28). The determination coefficient forthis fitted function is 0.998880. From this function, thefeasible solution area for the UCL of the chart is z�24.467. Table 1 lists the feasible solution areas of UCLfor some sample sizes under State 2. In Table 1, the lastcolumn summarizes the feasible solution area of UCLof the chart for a particular sample size by simply exam-ining the intersection of the feasible solution areas underStates 0, 1 and 2.

5.2. The second-stage solution procedure

The grid-search approach is used to find the values ofn, k, and UCL that minimize E(C). A computer programis coded for this purpose. This program is able to interactwith the software mathematica 4.0 and computes theprobability elements in the cost function. The result fromthe first-stage solution procedure greatly reduces therange of the search. From the output of the computerprogram, the optimum solution is n � 6, k � 62,UCL � 19.918, r0 � 0.009129, Power1 � 0.998334,Power2 � 0.968012, and E(C) � 0.77116. That is, theoptimal control procedure is to take a random sample ofsize six every 62 units and conclude the process is outof control if �2�nL � 19.918. The expected cost perunit of the control procedure is US$0.77116.

6. Effect of cost parameters

In this section, sensitivity analyses are conductedbased on the preceding illustrative example to study theeffect of cost parameters on the economic-statisticaldesign of the multivariate control charts.

The cost parameter a1 is the fixed cost of taking asample. Table 2 lists the optimal designs for differentvalues of a1. It can be seen that sampling intervalbetween samples increases as a1 increases. This is con-sistent with our reasoning.

The cost parameter a2 is the inspection cost per unit.Table 3 lists the optimal designs for different values ofa2. As a2 increases, the sample size decreases accord-ingly. This result is to be expected. In addition, increas-ing a2 leads to the decrease of the upper control limit.This tends to stabilize the power with the test.

The cost parameter a3 is the cost of investigating andcorrecting the process. Table 4 lists the optimal designsfor different values of a3. As a3 increases, both the sam-ple size and upper control limit tend to increase. This isprobably due to the expectation that increasing a3 maycorrespond to a decrease in type I error probability andan increase in power.

The cost parameter a4 is the penalty cost of producinga defective unit of product. Table 5 lists the optimaldesigns for different values of a4. It is noted that as a4

increases, the sampling interval decreases, which meansthat test/sampling should be conducted more frequentlysuch that a high penalty cost can be avoided.

7. Conclusions

Although many authors have discussed the economicdesign of control charts for the last two decades, thedesign of multivariate control charts still receives rela-tively little attention in the literature. In this paper, we


Tab

le1

The

feas

ible

solu

tion

area

sof

the

uppe

rco

ntro

llim

itfo

rso

me

sam

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size

s(N

ote

that

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spec

ific

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ple

size

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nar

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4z

14.3

86z�

24.6

11z�

18.6

3414

.386

�z�

18.6

345

z12

.754

z�30

.492

z�24

.467

12.7

54�

z�24

.467

6z

11.9

14z�

36.6

82z�

31.0

7611

.914

�z�

31.0

767

z11

.400

z�42

.950

z�38

.268

11.4

00�

z�38

.268

8z

11.0

53z�

49.6

43z�

46.1

9311

.053

�z�

46.1

939

z10

.802

z�56

.367

z�54

.164

10.8

02�

z�54

.164

10z

10.6

12z�

63.0

38z�

62.0

4110

.612

�z�

62.0

41


Fig. 1. The simulated values of Power1 and its fitted power esti-mation function.

Fig. 2. The simulated values of Power2 and its fitted power esti-mation function.

Table 2Effect of the fixed cost of taking a sample (a1) on the optimal design

a1 n k UCL E(C) r0 Power1 Power2

10 5 44 20.175 0.58154 0.013062 0.990333 0.93563320 6 62 19.918 0.77116 0.009129 0.998334 0.968012100 7 144 17.358 1.55910 0.015628 0.999955 0.989324

Table 3Effect of the inspection cost per unit (a2) on the optimal design


0.1 7 61 21.190 0.76112 0.004431 0.999420 0.9808000.2 6 62 19.918 0.77116 0.009129 0.998334 0.9680121.0 4 66 17.111 0.82831 0.053275 0.983580 0.915114

Table 4Effect of investigating and correcting the process (a3) on the optimal design


5 5 61 17.065 0.76522 0.031249 0.996753 0.95645910 6 62 19.918 0.77116 0.009129 0.998334 0.968012100 7 62 26.219 0.86182 0.000810 0.997110 0.964893

present the procedure to conduct the economic-statisticaldesign of multivariate control charts for monitoring theprocess mean vector and covariance matrix simul-taneously. The test statistic �2�nL is applied to developthis procedure and the cost model is established basedon the function given in the study by Montgomery andKlatt (1972). A numerical example is provided to illus-trate the design procedure and the effects of cost para-meters on the design are studied. From the results of thestudy, the following observations are obtained:

1. As the fixed cost of taking a sample increases, thesampling interval between samples also increases.

2. As the inspection cost per unit increases, both thesample size and upper control limit lead to decrease.

3. As the cost of investigating and correcting the processincreases, both the sample size and upper control limittend to increase.

4. As the penalty cost of producing a defective unit ofproduct increases, the sampling interval decreases.

Acknowledgements

This research was supported by the National ScienceCouncil of Taiwan, ROC under the grant NSC90-2218-E-224-004.


Table 5Effect of the penalty cost of producing a defective unit of product (a4) on the optimal design


10 5 41 20.021 0.60567 0.011537 0.998873 0.95583420 6 62 19.918 0.77116 0.009129 0.998334 0.968012100 7 153 17.276 1.42832 0.019836 0.999538 0.986430

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