Adaptive sampling enhancement for Hotelling’s T2 charts

European Journal of Operational Research 178 (2007) 841–857 www.elsevier.com/locate/ejor

Stochastics and Statistics

Adaptive sampling enhancement for Hotelling’s T2 charts Yan-Kwang Chen

*

Department of Logistics Engineering and Management, National Taichung Institute of Technology, 129 Sanmin Road, Sec. 3, Taichung, Taiwan, ROC Received 3 November 2005; accepted 2 March 2006 Available online 2 May 2006

Abstract This paper makes a study of an adaptive sampling scheme useful to increase the power of the fixed sampling rate (FSR) T2 control chart. In our study, the three parameters of T2 control chart: the sample size, the sampling interval, and the upper percentage factor that is used for determining the action limit, vary between two values for a relaxed or tightened control, depending on the most recent T2 value. The average time to signal (ATS) and adjusted average time to signal (AATS) a shift in the process mean vector for the new chart are derived and regarded as an objective function respectively to optimize its design parameters. With some minor changes, the new chart can be reduced to the variable sampling interval (VSI) T2 chart, the sample size (VSS) T2 chart, the variable sample size and sampling interval (VSSI) T2 chart, or the FSR T2 chart. Numerical comparisons among them are made and discussed. Furthermore, the effects of the initial sample number (use for estimating the in-control process parameters) upon the chart’s performance and adaptive design parameters are presented.  2006 Elsevier B.V. All rights reserved. Keywords: Variable design parameters; Multivariate control chart; (Adjusted) average time to signal; Genetic algorithms

1. Introduction Recent advances in industrial technology such as modern data-acquisition equipment and on-line computer use during production have made it common to monitor several correlated quality characteristics simultaneously. As a result, various types of multivariate control charts have been proposed for statistical process control works. There were two review papers summarizing the advances in multivariate quality control up through mid-1980s by Alt (1984) and Jackson (1985), while Lowry and Montgomery (1995) updated the reviews, focusing subsequent developments. Assume that the (p · 1) random vectors X1 ; X2 ; X3 ; . . ., each representing the p quality characteristics of interest, are observed over time, where these vectors represent individual observations or sample mean vectors. A natural multivariate extension to the univariate Shewhart chart is the Hotelling’s multivariate control chart *

Tel.: +886 4 22196759. E-mail address: [email protected]

0377-2217/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.03.001

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Y.-K. Chen / European Journal of Operational Research 178 (2007) 841–857

(1947). This procedure assumes the p quality characteristics are jointly distributed as p-variate normal with mean vector l and covariance matrix R. When Xi ’s represent the independent sample mean vectors with sample size of n, the Hotelling’s multivariate control chart signals a change in the mean vector of p correlated quality characteristics as soon as 0

v2i ¼ nðXi  l0 Þ R1 0 ðXi  l0 Þ > k;

ð1Þ

where l0 and R0 denote the in-control process mean vector and covariance matrix, respectively, and k is a specified action limit given by the upper a percentage point of chi-square distribution with p degrees of freedom, denoting v2p;a . When Xi ’s represent the individual observations, the sample size in (1) is equal to one. If l0 and R0 are unknown, then they are estimated by the averaged sample mean vector X and sample covariance matrix S from m initial (p · 1) random vectors prior to on-line process monitoring, and a T2 statistic defined by 0

T 2i ¼ nðXi  XÞ S1 ðXi  XÞ

ð2Þ

is the approximate statistic for the Hotelling’s multivariate control chart. The action limit k for Hotelling’s T2 control chart to monitor future random vectors is given by Alt (1984) as k ¼ Cðm; n; pÞF p;m;a ;

ð3Þ

pðmþ1Þðn1Þ where Cðm; n; pÞ ¼ ðmnmpþ1Þ , m = mn  m  p + 1, and Fp,m,a is the upper a percentage point of F distribution

with p and m degrees of freedom if sample size n > 1. Moreover, Cðm; n; pÞ ¼ pðmþ1Þðm1Þ and m = m  p if sample m2 mp size n = 1. Hotelling’s T2 control chart has the advantage of simplicity but, similar to the univariate Shewhart X chart, it is slow to detect a small change in the mean vector of p correlated quality characteristics. In a univariate case, the idea of the Shewhart chart with variable sampling interval (VSI) was studied by Reynolds et al. (1988), Runger and Pignatiello (1991), Reynolds and Arnold (1989), and Runger and Montgomery (1993), where it was shown that the X chart using variable sampling intervals significantly improves the efficiency of the Shewhart X chart. Subsequently, the idea of the X chart with varying the sample size (VSS) (see Prabhu et al., 1993, 1994; Costa, 1994) and the idea of the X chart with varying both the sample size and sampling interval (VSSI) (see Prabhu et al., 1994; Costa, 1997) were proposed for detecting small changes in the process mean. In addition to the idea of the VSSI scheme, Costa (1999a) also considered the variable action limits. This new chart is called the VP X chart and it makes the X charts more powerful for detecting small process mean shifts. The idea of a control chart with variable parameters has been extended to the multivariate case to enhance the efficiency of Hotelling’s T2 chart in detecting small shifts. Three modified T2 charts with VSS, VSI, and VSSI features were studied by Aparisi (1996), Aparisi and Haro (2001), and Aparisi and Haro (2003) respectively under the assumption that the shift in the process mean occurs at the beginning when the process is just starting. It is shown that the modified T2 charts are faster in detecting a change in the process mean. Among previous studies, a fixed action limit was commonly used for monitoring the multivariate process, given the incontrol process parameters l0 and R0. He and Grigoryan (2005) proposed the multivariate multiple sampling (MMS) control chart scheme, which is a multivariate extension of a double sampling X chart (Daudin, 1992). The results of the investigation shown that MMS T2 chart outperforms the multivariate EWMA and CUSUM charts, but the time required to collect and measure the samples should be considered negligible. This paper proposes an adaptive sampling plan to increase the efficiency of Hotelling’s T2 chart for detecting small process changes, and we assume that l0 and R0 are unknown, conforming to a real case. With this assumption the width of action limit will fluctuate slightly according to the sample size at each sampling epoch (refer to Eq. (3)) if the VSSI (or VSS) control plan is employed. Due to the trait of the fluctuated action limit in the VSSI (or VSS) T2 chart, the VSSI feature can be extended with little additional effort to the VP feature to use as an alternative other than the adaptive T2 charts. In the next section, a description and an example of the VP T2 chart are presented. The average time to signal a change in the mean vector of p correlated quality characteristics at the beginning when the process is starting or at some random time in the future is given in the third section. On the basis of genetic algorithms (GAs), the VP T2 chart is statistically designed in the fourth

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and fifth sections to compare with the Hotelling’s T2 chart and other types of T2 charts in terms of their speed in detecting off-target conditions. Finally, concluding remarks are presented in the last section. 2. The VP T2 control chart Costa (1999a) proposed a control procedure for varying all X chart parameters (sample size, sampling interval, and action limits) simultaneously to shorten the X chart’s response time to process mean shifts. This procedure is called a VP X control chart. In this section we extend the control procedure to a multivariate process and explain its design principle. 2.1. Principle of VP control scheme When Hotelling’s T2 chart with fixed sampling rate (FSR) is used to monitor a multivariate process, a sample of size n0 is drawn every h0 hours, and the value of the T2 statistic (i.e., sample point) is plotted on a control chart with k 0 ¼ Cðm; n0 ; pÞF p;m;a0 as the action limit. The VP T2 chart is a modification of the FSR T2 chart. Let (n1, h1) be a pair of minimum sample size and longest sampling interval, and (n2, h2) be a pair of maximum sample size and shortest sampling interval. These pairs are chosen such that n1 < n0 < n2 and h2 < h0 < h1. The decision to switch between the maximum and minimum parameters depends on the prior sample point on the control chart. That is, the position of the prior sample point on the chart determines the size of the current sample and the time of its sampling (see Fig. 1). If the prior sample point (i  1) falls in the tightening region, the pair (n2, h2) should be used for the current sample point (i). On the other hand, if the prior sample point (i  1) falls in the relaxing region, the pair (n1, h1) should be used for the current sample point (i). Here the tightening and relaxing region are given by the warning limit wj and the action limit k j ¼ Cðm; nj ; pÞF p;mj ;aj (tightening region is given by (wj, kj] and relaxing region is given by [0, wj]) respectively, where j = 1 if the prior sample point comes from the small sample, and j = 2 if the prior sample point comes from the large sample. It is assumed that w1 > w2 and k1 > k0 > k2. Moreover, for the sake of simplicity we set wj as follows: p0 ¼ PrfT 2i < w1 j T 2i < k 1 g ¼ PrfT 2i < w2 j T 2i < k 2 g.

ð4Þ

The following function defines the adaptive principle of the VP T2 control scheme: ( ðn2 ; h2 ; w2 ; k 2 Þ; if wði  1Þ < T 2i1 6 kði  1Þ; ðnðiÞ; hðiÞ; wðiÞ; kðiÞÞ ¼ ðn1 ; h1 ; w1 ; k 1 Þ; if 0 6 T 2i1 6 wði  1Þ:

ð5Þ

When the process is starting or after a false alarm the sizes of samples are chosen at random between two values. Small size is selected with probability of p0, whereas large size is selected with probability of (1  p0). Not only the size of the sample is chosen at random, but also the length of the sampling interval. If the next sample was chosen to be small a long time interval is considered before taking the next sample, if the next sample was chosen to be large a short time interval is considered before taking the next sample. T2

Action region R

k1

k2 L

L

Tightening region R

w2

w1 R

0

h1 h1+h2

Relaxing region

2h1+h21 2h1+3h2 2h1+2h2

Fig. 1. The VP T2 control chart.

Time

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As the warning limit and action limit are varied, depending on the sample size, it is possible for the practitioner to employ one chart for a small sample and another chart for a large sample. However, this is a tedious process. To avoid it, one may construct a control chart with two scales, one on left hand side and the other on the right hand side. The observation from small sample can be plotted according to the left scale, and the one from large sample can be plotted according to the right scale. However, it is still difficult for a practitioner to plot the point because the left scale is not proportional to the right scale. Costa (1999a) recommended breaking the left scale and plotting these sample points anywhere inside the right region, regardless of the right position. In this way, the effort to monitor a process with the VP control chart or with the FSR control chart is almost the same. 2.2. Example Consider a soft drink fabrication process involving two quality characteristics of interest (p = 2). One of the quality characteristics is the pressure inside the soft drink bottle, and the other is the gas volume present in the drink. These directly affect the product quality and can cause taste problems in the soft drink. In order to maintain the two quality characteristics of interest in a statistically stable status, an FSR T2 chart with estimated in-control mean vector X ¼ ð6:93; 3:86Þ and covariance matrix  S¼

0:60

0:04

0:04

0:01



based on 300 initial samples (i.e., m = 300) is employed for on-line monitoring. At present, the sample mean vector Xi of size three is drawn every one hour (n0 = 3 and h0 = 1.00), and the T 2i values calculated by (2) are plotted on the chart with action limit placed at 10.74 (a0 = 0.005). The average time the control chart needs to signal an assignable cause is 55.29 minutes. Assume the process engineer is considering monitoring the process by using a VP T2 chart instead of the FSR T2 chart as shown in Table 1. Accordingly, if the current T 2i value falls into the relaxing region, the next sample with small size (n1 = 1) will be drawn after a long time interval (about 1.12 hour or 67 minutes). Otherwise, if the current T 2i value falls into the tightening region, the next sample with large size (n2 = 20) will be drawn immediately (about 0.01 hour, 0.6 minutes, or 36 seconds). The process is stopped if the T 2i value falls into the action region. The left scale of the VP chart with the warning limit of 4.55 and action limit of 13.91 is used when the T 2i values from small samples are plotted. On the other hand, the right scale with the warning limit of 3.95 and action limit of 6.51 is used for T 2i values from large samples. For the sake of simplicity we assume the chart is started at time 0, and the first sample is obtained immediately (36 seconds) or latter (67 minutes) at random. Without loss of generality, we suppose that the first sample with size 1 is taken at the time of 67 minutes. If the first sample mean vector X1 = (6.00, 4.00), then T 21 ¼6.81 falls into the tightening region according to the left scale. The second sample will be taken immediately (at 67 + 0.6 = 67.6 minutes) with size 20. If the second sample mean vector X2 = (7.00, 3.77), then T 22 ¼3.00 falls into the relaxing region according to the right scale. The third sample will be taken at a time of 134.6 minutes (67.6 + 67 = 134.6 minutes) with size 1 and so forth until some T 2i value falls into the action region (see Fig. 2). The average time the VP scheme needs only 13.02 minutes to signal the same assignable cause.

Table 1 Design parameter values for the example Design parameters

FSR T2 chart

VP T2 chart

Sample size Sampling interval Action limit Warning limit

n0 = 3 h0 = 1 k0 = 10.74 (a0 = 0.005)

(n1, n2) = (1, 20) (h1, h2) = (1.12, 0.01) (k1, k2) = (13.91, 6.51) (w1, w2) = (4.55, 3.95)

Y.-K. Chen / European Journal of Operational Research 178 (2007) 841–857 T

845

2 R

Action region 6.51

13.91 L

Tightening region L

4.55

3.95 R

Relaxing region

L

67

0

134.6 67.6

202.2 201.6

Minutes

Fig. 2. The VP T2 control chart for the example.

3. Measurements for statistical efficiency The length of time it takes a control chart to produce a signal is used to measure its statistical efficiency. If the process is in control, then this time should be long so the rate of false alarms is low, but once the process mean shift occurs, the time from shift to signal should be short so the detection is speed quick. When the interval between samples is not fixed, the length of time is measured by the average time to signal (ATS) or adjusted average time to signal (AATS). The measurement of ATS is used when a mean shift takes place from the start of the process until it is detected, whereas AATS is used when the process starts out as an in-control situation (l = l0) and then encounters a mean shift (l = l1) at some time in the future. The Markov chain approach has been used widely to evaluate the properties of the adaptive X control charts, including AATS (see, e.g., Costa, 1994; Aparisi, 1996). However, the drawback of using Markov chains is that a subroutine is required to find the transition probabilities (Costa, 1999b). Another approach, one that applies Wald’s identity, can be used to compute the value of ATS or AATS (see, e.g., Reynolds et al., 1988; Bai and Lee, 1998) and to enable simplification of the computer program (Costa, 1999b). Accordingly, we follow this approach in this paper to evaluate the properties of the VP T2 chart. In the following sections, the expressions (6), (9)–(22) using for the value of ATS or AATS are presented. All these expressions were obtained by Costa (1999b). 3.1. Average time to signal ATS is the time interval from the start of the process where a mean shift takes place until it is detected, so the ATS for the VP T2 chart depends on the first sample. When the first sample is small, the ATS is obtained based on the number of sample point, N1, in the relaxing region taken from the start of the process to the time the chart signals. Thus, N1 is a geometric random variable with parameter (1  q1), where q1 is the conditional probability of obtaining another point in the relaxing region, given that the current sample point belongs to the relaxing region. Thus, 1 X pi1 ð6Þ q1 ¼ p11 þ p12 22 p 21 ; i¼1

where p11 ¼ PrfT 2i < w1 j T 2i  Cðm; n1 ; pÞF p;m1 ;k1 g; p12 ¼ Prfw1 < T 2i < k 1 j T 2i  Cðm; n1 ; pÞF p;m1 ;k1 g; p21 ¼ PrfT 2i < w2 j T 2i  Cðm; n2 ; pÞF p;m2 ;k2 g; p22 ¼ Prfw2 < T 2i < k 2 j T 2i  Cðm; n2 ; pÞF p;m2 ;k2 g; where notation F p;mj ;kj represents the non-central F distribution with p and mj degrees of freedom and non-centrality parameter kj for j = 1 and 2. The non-centrality parameter kj is given by 0

kj ¼ nj ðl1  l0 Þ R1 0 ðl1  l0 Þ.

ð7Þ

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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 If we let d ¼ ðl1  l0 Þ R1 0 ðl1  l0 Þ, then kj can be rewritten by kj ¼ nj d2 ;

ð8Þ

where d is the Mahalanobis distance used to measure a change in the process mean vector. Let L1k be the length of time from the current sample point to another sample point outside the tightening region, given that the current sample point belongs to the relaxing region. The L1k ’s are independent and identically distributed with PrfL1k ¼ h1 g ¼ 1  p12 and PrfL1k ¼ h1 þ jh2 g ¼ p12 pk1 22 ð1  p 22 Þ for j = 1,2,3, . . . , 1. Moreover, the expected value and variance of L1k are EðL1k Þ ¼ h1 þ

h2 p12 1  p22

ð9Þ

and V ðL1k Þ ¼

h22 ½p12 ð1 þ p22 Þ  p212  ð1  p22 Þ2

;

ð10Þ

respectively. As a result, the time from the start of the process to the time the chart signals is TS1 ¼

N1 X

L1k .

ð11Þ

k¼1

Then, using the Wald’s identity, the ATS can be written as EðTS1 Þ ¼ EðN 1 ÞEðL1k Þ ¼

h1 ð1  p22 Þ þ h2 p12 ; D

ð12Þ

where D = 1  p11  p22 + p11p22  p12p21. Similarly, when the first sample is large, let N2 be the number of sample points in the tightening region taken from the start of the process to the time the chart signals. Also let L2k be the length of time from the current sample point to another sample point outside the relaxing region, given that the current sample point belongs to the tightening region. Then, the ATS is determined as the expected value of TS2-the time from the start of the process to the time the chart signals, and it is expressed as EðTS2 Þ ¼ EðN 2 ÞEðL2k Þ ¼

h2 ð1  p11 Þ þ h1 p21 . D

ð13Þ

Since the first sample is chosen at random with probability of p0 for being small and (1  p0) for being large, the ATS is given by ATS ¼ p0 EðTS1 Þ þ ð1  p0 ÞEðTS2 Þ.

ð14Þ

3.2. Adjusted average time to signal To develop an AATS that allows a process mean shift to occur between samples, let TS = the time from process mean shift until a signal, U = the length of the interval in which the process mean shift occurs, Y = the time from process mean shift to next sample, Z = the time from the next sample after the process mean shift until a signal. The relationship between TS, U, Y, and Z is shown in Fig. 3. It is clear that TS = Y + Z, and, therefore, the AATS is EðTSÞ ¼ EðY Þ þ EðZÞ.

ð15Þ

To determine E(Y), two assumptions following Reynolds et al. (1988) are made. The first is that when the shift falls in a particular sampling interval hj the time from mean shift to next sample is uniformly distributed over the interval, that is, EðY j U ¼ hj Þ ¼

hj ; 2

j ¼ 1; 2.

ð16Þ

Y.-K. Chen / European Journal of Operational Research 178 (2007) 841–857 Process starts

Process mean shift Last sample before process mean shift

847

Out-of-control detected

First sample after process mean shift

Time U Y

Z TS

In-control period

Out-of- control period

Fig. 3. The time to detect a process mean shift.

The second assumption is the probability of the shift falling in an interval of hj is proportional to the product of this length, and the probability of the occurrence of this length. That is, p 0 h1 ; p0 h1 þ ð1  p0 Þh2 ð1  p0 Þh2 PrfU ¼ h2 g ¼ . p0 h1 þ ð1  p0 Þh2 PrfU ¼ h1 g ¼

ð17Þ ð18Þ

According to (16)–(18), we obtain EðY Þ ¼

p0 h21 þ ð1  p0 Þh22 . 2½p0 h1 þ ð1  p0 Þh2 

ð19Þ

To determine the expected value of Z, E(Z), it should be noted that E(Z) depends on the region, R, where the first sample after shift falls. If this sample point falls in the relaxing region, then the expected value of Z is expressed as E(Z j R = R1); if it falls in the tightening region, then the expected value of Z is expressed as E(Z j R = R2); otherwise, the expected value of Z is zero. Thus E(Z) can be written as EðZÞ ¼ PrfR ¼ R1 gEðZ j R ¼ R1 Þ þ PrfR ¼ R2 gEðZ j R ¼ R2 Þ ¼ PrfR ¼ R1 gEðTS1 Þ þ PrfR ¼ R2 gEðTS2 Þ. ð20Þ Since the probability of this sample point falling in the relaxing region or the tightening region depends on the length of the interval in which the mean shift occurs, we have PrfR ¼ R1 g ¼ PrfU ¼ h1 \ R ¼ R1 g þ PrfU ¼ h2 \ R ¼ R1 g ¼ PrfR ¼ R1 j U ¼ h1 gPrfU ¼ h1 g þ PrfR ¼ R1 j U ¼ h2 gPrfU ¼ h2 g ¼ p11 PrfU ¼ h1 g þ p21 PrfU ¼ h2 g

ð21Þ

and PrfR ¼ R2 g ¼ PrfU ¼ h1 \ R ¼ R2 g þ PrfU ¼ h2 \ R ¼ R2 g ¼ PrfR ¼ R2 j U ¼ h1 gPrfU ¼ h1 g þ PrfR ¼ R2 j U ¼ h2 gPrfU ¼ h2 g ¼ p12 PrfU ¼ h1 g þ p22 PrfU ¼ h2 g.

ð22Þ

4. Statistical design of the VP T2 chart The approach taken to investigate the usefulness of the VP T2 chart is presented in this section. 4.1. Matching the in-control performances In evaluating the usefulness of the VP T2 chart, it seems natural to compare the performance of the VP T2 chart to the FSR or other types of T2 charts under equal conditions. As a result, before a comparison between the VP T2 chart and the FSR T2 chart (or other type’s T2 charts) their in-control performances should be ‘‘matched’’ in the sense that they have the same false alarm rate and that, on average, the two charts require

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sampling of the same number of items per unit time during the in-control period. Therefore, the eight design parameters of the VP T2 chart—(n1, n2), (h1, h2), (w1, w2), and (a1, a2) will be specified in accordance with the constraints in Eqs. (23)–(25): n1 p0 þ n2 ð1  p0 Þ ¼ n0 ; h1 p0 þ h2 ð1  p0 Þ ¼ h0 ; a1 p0 þ a2 ð1  p0 Þ ¼ a0 ;

ð23Þ ð24Þ ð25Þ

where n0, h0, and a0 are the design parameters of traditional T2 chart. The three constraints in Eqs. (23)–(25) allow one to choose one pair of (n1, n2), (h1, h2) or (a1, a2) as well as one element from each remaining pair, then determine the remainders by the above three constraints. Many options could be consequently chosen, for example, we can choose (n1, n2) and the elements h2, and a1. Then, following (23)–(25), it is given that n0  n2 ; n1  n2 h0 ðn1  n2 Þ  h2 ðn1  n0 Þ h1 ¼ ; n0  n2 a0 ðn1  n2 Þ  a1 ðn0  n2 Þ a2 ¼ . n1  n0 p0 ¼

ð26Þ ð27Þ ð28Þ

Table 2 Comparison between ATS for the schemes VP and FSR (p = 2, h0 = 1, a0 = 0.005) n0

m

d

n1/n2

h1/h2

a1/a2

k1/k2

w1/w2

ATSVP

ATSFSR

%

2

600

0.25 0.50 0.75 1.00 1.25 1.50

1/43 1/19 1/10 1/6 1/3 1/3

1.02/0.01 1.06/0.01 1.12/0.03 1.24/0.03 2.00/0.01 2.00/0.01

0.000/0.205 0.000/0.083 0.000/0.045 0.000/0.025 0.000/0.010 0.000/0.010

15.82/3.17 20.50/4.99 20.09/6.24 16. 91/7.42 17.87/9.30 18.36/9.29

7.54/3.00 5.81/4.03 4.42/3.79 3.24/3.04 1.39/1.37 1.39/1.37

65.80 21.62 8.65 4.10 2.17 1.42

145.15 76.20 37.35 19.13 10.48 6.16

54.67 71.63 76.84 78.57 78.29 76.95

3

300

0.25 0.50 0.75 1.00 1.25 1.50

1/58 1/21 1/10 1/5 1/4 2/4

1.04/0.01 1.11/0.01 1.29/0.01 2.00/0.01 3.00/0.01 2.00/0.01

0.000/0.140 0.000/0.046 0.000/0.022 0.000/0.010 0.000/0.007 0.000/0.009

19.12/3.95 15.72/6.20 17.98/7.68 19.41/9.30 19.41/9.89 14.79/9.46

6.28/3.56 4.66/3.93 3.04/2.87 1.40/1.37 0.82/0.81 1.40/1.38

44.41 12.82 4.79 2.22 1.33 1.12

127.92 55.21 23.62 11.10 5.79 3.37

65.28 76.78 79.72 80.00 77.03 66.77

4

200

0.25 0.50 0.75 1.00 1.25 1.50

1/66 1/21 1/10 1/5 2/5 3/5

1.05/0.01 1.17/0.01 1.50/0.01 3.97/0.01 2.99/0.01 1.99/0.01

0.000/0.107 0.000/0.032 0.003/0.008 0.000/0.007 0.000/0.007 0.002/0.008

23.28/4.50 18.09/6.91 11.89/9.63 16.16/10.19 17.61/9.94 12.54/9.82

6.31/3.84 3.87/3.48 2.22/2.18 0.58/0.57 0.82/0.81 1.39/1.38

34.04 8.93 3.49 1.53 1.12 1.04

113.75 42.13 16.43 7.33 3.77 2.25

70.07 78.80 78.76 79.13 70.29 53.78

5

150

0.25 0.50 0.75 1.00 1.25 1.50

1/71 1/21 1/9 2/6 3/6 4/6

1.06/0.04 1.25/0.01 2.00/0.01 4.00/0.01 3.00/0.01 2.00/0.01

0.000/0.087 0.000/0.024 0.001/0.009 0.000/0.007 0.001/0.007 0.005/0.005

25.25/4.91 17.99/7.51 15.22/9.48 17.00/10.21 13.69/10.12 10.94/10.65

5.91/3.97 3.30/3.06 1.41/1.38 0.58/0.58 0.82/0.81 1.39/1.39

27.67 6.79 2.43 1.21 1.04 1.01

101.94 33.38 12.15 2.73 6.26 1.71

72.86 79.66 80.67 80.00 61.90 40.94

10

80

0.25 0.50 0.75 1.00 1.25 1.50

1/80 2/20 5/11 7/11 8/11 9/11

1.13/0.01 1.79/0.01 5.93/0.01 3.99/0.01 2.99/0.01 2.00/0.01

0.000/0.043 0.000/0.011 0.000/0.006 0.002/0.006 0.005/0.005 0.005/0.005

20.31/6.39 18.52/9.19 16.28/10.47 12.74/10.44 10.69/10.90 10.84/10.81

4.58/3.82 1.68/1.62 0.37/0.37 0.58/0.58 0.82/0.82 1.40/1.40

14.18 2.86 1.14 1.01 1.00 1.00

64.28 14.25 4.40 1.96 1.25 1.05

77.94 79.93 74.09 48.47 20.00 4.76

a1 = 0.000 means that the optimal value of k1 make the risk of false alarm nearly zero.

Y.-K. Chen / European Journal of Operational Research 178 (2007) 841–857

849

Table 3 Comparison between ATS for the schemes VP and FSR (p = 4, h0 = 1, a0 = 0.005) n0

m

d

h1/h2

(a) a1/a2

k1/k2

w1/w2

ATSVP

ATSFSR

%

2

1400

0.25 0.50 0.75 1.00 1.25 1.50

n1/n2 1/49 1/23 1/12 1/8 1/5 1/3

1.02/0.02 1.05/0.01 1.10/0.01 1.17/0.01 1.33/0.01 2.00/0.01

0.000/0.234 0.000/0.106 0.000/0.053 0.001/0.030 0.000/0.020 0.000/0.010

23.12/5.57 22.34/7.64 22.40/9.35 19.01/10.73 22.93/11.74 22.93/13.36

11.63/5.39 9.76/6.81 8.05/6.94 6.89/6.45 5.41/5.24 3.37/3.33

78.42 28.31 12.13 5.95 3.19 1.92

160.30 99.31 53.89 28.68 15.76 9.12

51.08 71.49 77.49 79.25 79.76 78.95

3

700

0.25 0.50 0.75 1.00 1.25 1.50

1/69 1/26 1/13 1/7 1/4 1/4

1.03/0.01 1.09/0.01 1.18/0.12 1.49/0.01 2.97/0.01 2.97/0.01

0.000/0.168 0.000/0.061 0.000/0.029 0.000/0.015 0.000/0.007 0.000/0.007

25.73/6.45 23.24/9.03 22.55/10.81 22.45/12.44 22.45/14.05 22.45/14.05

10.89/6.09 8.42/7.01 6.53/6.13 4.61/4.51 2.39/2.37 2.29/2.37

53.85 16.96 6.89 3.25 1.79 1.29

146.82 76.05 35.16 16.71 8.56 4.79

63.32 77.70 80.40 80.55 79.09 73.07

4

500

0.25 0.50 0.75 1.00 1.25 1.50

1/82 1/28 1/14 1/7 2/5 3/5

1.04/0.01 1.12/0.04 1.29/0.03 1.98/0.02 2.97/0.01 2.99/0.01

0.000/0.129 0.000/0.044 0.003/0.012 0.001/0.009 0.000/0.007 0.000/0.007

22.19/7.15 24.67/9.80 16.49/12.87 19.43/13.54 20.69/14.10 20.70/14.10

10.36/6.58 7.61/6.75 5.64/5.51 3.39/3.34 2.40/2.37 2.40/2.37

41.89 12.05 4.93 2.23 1.32 1.10

134.93 60.05 24.69 10.93 5.41 3.05

68.95 79.93 80.03 79.60 75.60 63.93

5

400

0.25 0.50 0.75 1.00 1.25 1.50

1/90 1/28 1/13 1/6 3/6 3/6

1.05/0.01 1.17/0.01 2.49/0.02 4.92/0.02 2.98/0.01 2.98/0.01

0.000/0.103 0.000/0.032 0.000/0.015 0.001/0.006 0.000/0.007 0.002/0.007

21.22/7.73 21.85/10.59 28.52/12.40 19.06/14.55 21.12/14.09 17.53/14.32

9.92/6.88 6.88/6.35 4.64/4.51 1.66/1.65 2.39/2.37 2.39/2.38

34.53 9.27 3.49 1.65 1.14 1.04

124.40 48.58 18.28 7.72 3.79 2.20

72.24 80.92 80.91 78.63 69.92 52.73

10

150

0.25 0.50 0.75 1.00 1.25 1.50

1/106 1/28 4/15 6/11 7/11 8/11

1.09/0.01 1.50/0.01 2.04/0.14 5.00/0.01 4.00/0.01 3.00/0.01

0.000/0.054 0.005/0.015 0.000/0.009 0.001/0.006 0.005/0.005 0.005/0.005

22.47/9.36 23.79/12.52 21.14/13.71 20.12/14.59 15.12/15.05 15.09/15.05

8.56/7.06 4.74/4.53 3.12/3.08 1.67/1.66 1.93/1.93 2.39/2.39

18.25 4.05 1.74 1.03 1.00 1.00

86.53 21.51 6.41 2.60 1.47 1.11

78.91 81.17 72.85 60.38 31.97 9.91

a1 = 0.000 means that the optimal value of k1 make the risk of false alarm nearly zero.

Furthermore, using Eq. (4) and (26) we have wj ¼ Cðm; nj ; pÞF 1 p;mnj mpþ1 ðð1  aj Þp 0 Þ;

j ¼ 1; 2;

ð29Þ

where F 1 p;mnj mpþ1 ðÞ is the inverse of the F distribution function with p and (mnj  m  p + 1) degrees of freedom. 4.2. Minimizing the ATS or AATS Once the VP T2 chart is matched to the FSR T2chart or other charts, their statistical efficiencies in terms of ATS or AATS can be compared for various l1 to determine which chart will do a better job of detecting a change in the mean vector l. However, there are several options for the values of (n1, n2) and the elements h2 and a1, and all of them lead to different values of ATS or AATS when a process is out-of-control. Thus, it is necessary to find out the optimal VP T2 chart among the matched ones. In this section, we propose a procedure to select the optimal chart, in which the optimal design parameters (n1 ; n2 ; h1 ; h2 ; w1 ; w2 ; a1 ; a2 Þ are derived by minimizing the value of ATS or AATS for given process parameters (p, m, d), and the contrastive design parameters (n0, h0, a0) used for the FSR T2 chart. In doing so, we apply the genetic algorithms (GAs) because the minimi-

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Table 4 Comparison between AATS for the schemes VP and FSR (p = 2, h0 = 1, a0 = 0.005) n0

m

d

n1/n2

h1/h2

(a) a1/a2

k1/k2

w1/w2

ATSVP

ATSFSR

%

2

600

0.25 0.50 0.75 1.00 1.25 1.50

1/43 1/19 1/10 1/6 1/4 1/4

1.00/1.00 1.00/1.00 1.12/0.03 1.25/0.01 1.49/0.01 1.50/0.05

0.000/0.208 0.000/0.087 0.000/0.045 0.000/0.025 0.000/0.014 0.001/0.013

20.12/3.15 17.59/4.89 20.18/6.24 20.34/7.42 15.54/8.57 13.88/8.75

7.54/2.97 5.82/3.97 4.42/3.79 3.24/3.04 2.21/2.15 2.20/2.15

65.46 21.52 8.79 4.15 2.34 1.61

144.65 75.71 36.85 18.63 9.98 5.66

54.75 71.58 76.15 77.72 76.55 71.55

3

300

0.25 0.50 0.75 1.00 1.25 1.50

1/59 1/22 1/10 2/7 2/6 2/6

1.00/1.00 1.00/1.00 1.28/0.01 1.24/0.03 1.31/0.06 1.32/0.04

0.000/0.138 0.000/0.051 0.000/0.021 0.001/0.022 0.003/0.011 0.004/0.008

17.30/3.97 17.92/5.98 16.13/7.76 15.08/7.65 11.91/9.10 11.40/9.63

6.84/3.58 4.77/3.93 3.04/2.88 3.25/3.06 2.78/2.72 2.78/2.74

44.20 12.87 5.09 2.53 1.48 1.06

127.42 54.71 23.12 10.60 5.29 2.87

65.31 76.48 77.98 76.13 72.02 63.07

4

200

0.25 0.50 0.75 1.00 1.25 1.50

1/67 1/24 2/11 3/7 3/7 3/8

1.00/1.00 1.00/1.00 1.28/0.02 1.33/0.01 1.33/0.02 1.25/0.01

0.000/0.108 0.000/0.038 0.000/0.022 0.001/0.018 0.003/0.012 0.005/0.005

19.50/4.48 19.71/6.59 22.05/7.65 15.59/8.06 12.05/8.98 10.82/10.70

6.34/3.83 4.16/3.65 3.06/2.88 2.80/2.68 2.79/2.72 3.22/3.20

33.75 9.20 3.57 1.71 1.05 0.81

113.25 41.63 15.93 6.83 3.27 1.75

70.20 77.90 77.59 74.96 67.89 53.71

5

150

0.25 0.50 0.75 1.00 1.25 1.50

1/73 1/26 3/11 4/8 4/8 4/10

1.00/1.00 1.00/1.00 1.33/0.01 1.33/0.01 1.33/0.01 1.20/0.02

0.000/0.085 0.001/0.027 0.000/0.019 0.001/0.016 0.004/0.008 0.005/0.005

17.56/4.96 15.03/7.30 16.78/8.00 13.62/8.35 11.23/9.84 10.82/10.72

5.96/4.02 3.75/3.43 2.81/2.68 2.80/2.70 2.78/2.75 3.58/3.56

27.59 7.32 2.68 1.30 0.86 0.70

101.44 32.87 11.65 4.76 2.23 1.21

72.80 77.73 77.00 72.69 61.43 42.15

10

80

0.25 0.50 0.75 1.00 1.25 1.50

1/87 5/25 8/16 9/16 9/22 9/44

1.00/1.00 1.33/0.01 1.33/0.02 1.17/0.01 1.08/0.01 1.03/0.02

0.000/0.044 0.000/0.019 0.002/0.015 0.005/0.006 0.005/0.005 0.005/0.005

17.63/6.32 16.56/8.05 13.31/8.48 10.88/10.53 10.84/10.77 10.83/10.75

4.76/3.92 2.83/2.70 2.81/2.72 3.90/3.88 5.10/5.09 6.93/6.89

14.27 3.09 1.15 0.70 0.58 0.53

63.78 13.75 3.90 1.46 0.75 0.55

77.63 77.53 70.51 52.05 22.67 3.64

a1 = 0.000 means that the optimal value of k1 make the risk of false alarm nearly zero.

zation problem is characterized by a non-linear objective function, mixed continuous-discrete decision variables (design parameters), and a discontinuous and non-convex solution space. If typical non-linear programming techniques are used to solve this optimization problem, they will be inefficient and time-consuming. GAs are global search and optimization techniques motivated by the process of natural selection in biological system (Davis, 1991; Goldberg, 1989). They have been used successfully in the optimization field of the parameters of quality control charts (e.g., Aparisi and Garcı´a-Daı´z, 2003; He et al., 2002; He and Grigoryan, 2002; Chen, 2004; Chou et al., 2006). GAs are different from other search procedures in the following ways (Karr and Gentry, 1993): (1) GAs consider many points in the search space simultaneously, rather than a single point; (2) GAs work directly with strings of characters representing the parameter set, not the parameters themselves; and (3) GAs use probabilistic rules to guide their search, not deterministic rules. Because GAs consider many points in the search space simultaneously there is a less chance of converging to local optima. Furthermore, in a conventional search, based on a decision rule, a single point is considered, and that is unreliable in multimodal space. The primary distinguishing features of GAs are an encoding, a fitness function, a selection mechanism, a crossover mechanism, a mutation mechanism, and a culling mechanism. The algorithm for GAs can be formulated as the following steps: (1) Randomly generate an initial solution set (population) of N individuals and evaluate each solution (individual) by fitness function. Usually an individual is represented as a numerical string.

Y.-K. Chen / European Journal of Operational Research 178 (2007) 841–857

851

Table 5 Comparison between AATS for the schemes VP and FSR (p = 4, h0 = 1, a0 = 0.005) n0

m

d

h1/h2

a1/a2

k1/k2

w1/w2

ATSVP

ATSFSR

%

2

1400

0.25 0.50 0.75 1.00 1.25 1.50

n1/n2 1/49 1/23 1/13 1/8 1/5 1/4

1.00/1.00 1.00/1.00 1.04/0.55 1.17/0.01 1.33/0.01 1.50/0.01

0.000/0.235 0.000/0.108 0.000/0.059 0.000/0.034 0.000/0.020 0.000/0.015

23.78/5.55 23.76/7.59 24.23/9.09 23.76/10.40 24.23/11.72 23.64/12.40

11.64/5.37 9.77/6.78 8.28/6.98 6.90/6.39 5.41/5.23 4.59/4.50

78.03 28.15 12.31 6.01 3.31 2.10

159.80 98.91 53.39 28.18 15.26 8.62

51.17 71.54 76.94 78.67 78.31 75.64

3

700

0.25 0.50 0.75 1.00 1.25 1.50

1/69 1/28 1/13 1/8 2/6 2/5

1.00/1.00 1.00/1.00 1.20/0.01 1.40/0.01 1.33/0.01 1.50/0.01

0.000/0.168 0.000/0.065 0.00/0.027 0.000/0.017 0.001/0.018 0.001/0.012

24.87/6.46 22.18/8.88 19.61/11.03 22.00/12.03 19.95/11.93 18.10/12.86

10.88/6.10 8.61/7.05 6.52/6.15 5.05/4.91 5.42/5.25 4.61/4.52

53.50 16.97 7.13 3.53 1.95 1.29

146.32 75.55 34.66 16.21 8.06 4.29

63.44 77.54 79.43 78.22 75.81 69.93

4

500

0.25 0.50 0.75 1.00 1.25 1.50

1/81 1/30 1/13 2/9 3/7 3/7

1.00/1.00 1.00/1.00 1.33/0.01 1.40/0.05 1.33/0.01 1.33/0.01

0.000/0.133 0.000/0.047 0.000/0.020 0.000/0.017 0.001/0.016 0.005/0.006

28.28/7.07 24.21/9.64 26.05/11.72 25.05/12.05 18.21/12.21 15.27/14.34

10.35/6.50 7.80/6.83 5.45/5.24 5.07/4.92 5.41/5.27 5.38/5.35

41.34 12.11 5.00 2.42 1.37 0.96

134.43 59.54 24.19 10.43 4.91 2.55

69.25 79.66 79.33 76.80 72.10 62.35

5

400

0.25 0.50 0.75 1.00 1.25 1.50

1/93 1/31 2/14 3/9 4/8 4/9

1.00/1.00 1.00/1.00 1.33/0.01 1.50/0.01 1.33/0.01 1.25/0.01

0.001/0.100 0.000/0.037 0.000/0.019 0.000/0.014 0.002/0.013 0.005/0.005

19.98/7.79 31.04/10.21 22.61/11.79 21.72/12.49 16.79/12.73 15.02/14.94

9.99/6.95 7.17/6.50 5.46/5.25 4.61/4.51 5.40/5.30 5.98/5.96

34.45 9.41 3.79 1.81 1.06 0.80

123.90 48.08 17.78 7.22 3.29 1.70

72.20 80.43 78.68 74.93 67.78 52.94

10

150

0.25 0.50 0.75 1.00 1.25 1.50

1/110 4/31 7/17 9/14 9/18 9/29

1.00/1.00 1.28/0.00 1.43/0.01 1.25/0.01 1.12/0.06 1.05/0.01

0.000/0.056 0.000/0.022 0.001/0.015 0.003/0.012 0.005/0.005 0.005/0.005

22.56/9.26 23.26/11.55 20.09/12.41 16.11/12.96 15.08/15.01 15.08/14.99

8.66/7.08 5.80/5.55 4.93/4.82 6.02/5.92 7.50/7.48 9.38/9.34

18.07 4.39 1.54 0.83 0.65 0.55

86.04 22.01 5.91 2.10 0.97 0.61

79.00 80.05 73.94 60.48 32.99 9.84

a1 = 0.000 means that the optimal value of k1 make the risk of false alarm nearly zero.

(2) If the termination condition is not met, repeatedly do {Select parents from population for crossover. Generate offspring. Mutate some of the numbers. Merge mutants and offspring into population. Cull some members of the population.} (3) Stop and return the best fitted solution. When applying GAs to the minimization process, a decimal encoding of individuals is adopted so that each individual in the form of decimal string represents a possible solution (n1, n2, h2, a1). The fitness value of each individual is evaluated by the ATS or AATS. Based on the ‘‘elitist’’ strategy of above algorithm, that is, the survival of the fittest, the evolution of a population of N individuals has been pursued. The termination condition is achieved when the number of generations is large enough or a satisfied fitness value is obtained. 5. Numerical comparisons The genetic algorithms (GAs) described in the last section is considered to evaluate the usefulness of the VP T2 chart.

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5.1. Comparing the VP and FSR T2 charts The values of n1, n2, h1, h2, w1, w2, a1 (or k1), and a2 (or k2) that minimize ATS or AATS have been obtained by fixing h0 at 1.00 and a0 at 0.005 for the following cases: p = 2 or 4, d = 0.25, 0.50, 0.75, 1.00, 1.25, or 1.50, n0 = 2, 3, 4, 5, or 10. As for the value of m, Quesenberry (1993) suggested it should be large enough so that the charts based on estimated parameters perform in a similar manner to the charts based on true parameters during the on-line process monitoring stage. Lowry and Montgomery (1995) recommended the minimum m values necessary for several combinations of p and n. Nedumaran and Pignatiello (1999) followed these studies and considered the run length distribution performance of the respective control charts to speculate a range from 800p/3(n  1) to 400p/(n  1), which is necessary for minimum m. In this paper, the value of m corresponding to a combination of n0 and p is first chosen from above range, and a sensitivity analysis is then conducted to investigate the effects of m values on the chart’s performance and adaptive design parameters. EVOLVER 4.0.2 is a genetic optimization package that can function as an add-in to Microsoft Excel. In the statistical design of the VP T2 chart, we include Severo and Zelen’s (1960) approximation for non-central F distribution along with the program of EVOLVER 4.0.2 to minimize the ATS or AATS. The following settings of control parameters for the package manipulation have been used: population size N = 50; crossover probability = 0.5; mutation rate = 0.25; the number of generation = 10,000 for ATS and 100,000 for AATS due to different converging speeds. Since the minimum time-period between samples has to take into consideration to generate the required sample size, it requires h2 P 0.01 in the following numerical comparisons.

Table 6 Effect of m values on k1 built to minimize ATS (h0 = 1.00; a0 = 0.005) (n, p)

m

d 0.25 k1

0.50

1.00 k1

1.50

ATS

k1

ATS

(2, 2)

10 20 50 100 300 600 1200

59.39 27.42 19.14 17.21 16.09 15.82 15.69

67.26 67.18 66.57 66.31 66.12 65.80 65.75

111.79 40.90 25.94 22.74 20.92 20.50 20.29

23.06 22.26 21.78 21.62 21.62 21.62 21.62

69.15 30.27 20.68 18.48 17.21 16.91 16.76

4.64 4.37 4.22 4.17 4.15 4.10 4.10

ATS

k1 84.28 34.35 22.77 20.19 18.71 18.36 18.19

1.56 1.48 1.45 1.45 1.44 1.42 1.42

ATS

(2, 4)

10 20 50 100 700 1400 2800

234.15 51.16 28.92 24.73 23.86 23.12 22.55

80.70 80.00 79.27 78.89 78.53 78.42 77.48

302.73 58.35 31.69 26.83 23.53 22.34 22.17

30.94 29.73 28.84 28.54 28.28 28.31 28.26

192.28 46.18 26.90 23.17 20.60 19.01 19.00

7.36 6.57 6.19 6.07 5.95 5.95 5.95

224.26 50.04 28.48 24.39 23.21 22.93 21.28

2.52 2.22 2.02 1.97 1.93 1.92 1.92

(4, 2)

10 20 50 100 200 400 800

141.76 46.97 28.73 24.94 23.28 22.52 22.17

34.26 34.18 34.07 34.04 34.03 34.00 34.00

107.18 39.87 25.46 22.33 18.09 18.05 17.57

9.27 9.06 8.96 8.93 8.93 8.90 8.89

150.98 49.03 29.57 29.53 16.16 16.04 15.64

1.59 1.56 1.55 1.54 1.53 1.51 1.50

19.74 15.38 13.34 12.73 12.54 12.31 12.23

1.05 1.04 1.04 1.04 1.04 1.04 1.04

(4, 4)

10 20 50 100 250 500 1000

202.74 47.48 27.44 23.60 22.68 22.19 21.82

42.51 42.49 42.24 42.14 42.06 41.89 41.03

253.17 53.27 31.75 27.36 25.21 24.67 24.24

12.91 12.46 12.21 12.14 12.09 12.05 12.04

283.13 53.39 27.94 23.27 20.98 19.43 19.25

2.46 2.31 2.26 2.26 2.24 2.23 2.21

138.58 43.97 26.70 23.11 21.29 20.70 20.45

1.17 1.14 1.13 1.12 1.12 1.10 1.10

Y.-K. Chen / European Journal of Operational Research 178 (2007) 841–857

853

Tables 2 and 3 show the results of ATS for two and four quality characteristics of interest for the same VP and FSR charts, while Tables 4 and 5 show the results of AATS. Several findings as illustrated in Tables 2–5 are spelled out as follows: (1) The VP charts consistently have shorter ATS or AATS values than the FSR charts. Percent reductions in ATS or AATS are about 70–80% for most cases, but they are dramatically reduced if the averaged sample size used for moderate or large shifts is large. This is because the values of ATS or AATS are already small by themselves. In addition, the correlation between percentage reductions and average sample size is positive for a small shift, but negative for a moderate or large shift. (2) The optimal design parameters that minimize ATS or AATS have some properties in common, e.g., the maximum and minimum sampling intervals should be spaced far apart to detect a moderate or large shift in the process mean; as the process change becomes evident, the minimal sample size tends to be large and the space between maximum and minimum action limits tends to be narrow. However, there are still some properties that are different, e.g., the sampling intervals used to detect a small shift had better fixed to minimize AATS but changeable to minimize ATS. (3) The optimal warning/action limits and maximal sample size increase when the number of quality characteristics is augmented. The effect of the m value on the optimal design parameters has been studied for the following cases: p = 2 or 4, n0 = 2 or 4, and d = 0.25, 0.50, 1.00, or 1.50. From the results, we find that the chart’s performance and all Table 7 Effect of m values on k1 built to minimize AATS (h0 = 1.00; a0 = 0.005) (n, p)

m

d 0.25 k1

0.50 AATS

k1

1.00

1.50

AATS

k1

AATS

(2, 2)

10 20 50 100 300 600 1200

84.42 34.39 30.79 26.49 21.12 20.12 20.06

66.47 66.32 65.83 65.64 65.50 65.46 65.45

84.42 34.39 25.61 22.45 19.13 17.59 17.19

23.01 22.24 21.78 21.62 21.52 21.52 21.50

111.79 40.90 29.82 25.78 21.68 20.34 20.13

4.71 4.42 4.27 4.22 4.18 4.15 4.15

k1 44.32 22.53 17.76 15.42 13.93 13.88 13.78

1.77 1.68 1.63 1.61 1.61 1.61 1.60

AATS

(2, 4)

10 20 50 100 700 1400 2800

326.16 60.52 32.49 27.43 24.02 23.78 23.64

80.36 79.44 78.74 78.39 78.06 78.03 78.02

307.72 59.78 32.49 27.43 24.02 23.76 23.64

30.73 29.57 28.71 28.42 28.18 28.15 28.13

307.72 58.79 31.85 26.96 24.02 23.76 23.64

7.51 6.65 6.24 6.12 6.02 6.01 6.01

307.72 58.78 34.21 28.59 24.13 23.64 23.55

2.69 2.34 2.19 2.15 2.11 2.10 2.10

(4, 2)

10 20 50 100 200 400 800

86.08 34.81 23.38 20.68 19.50 18.95 18.69

34.01 33.94 33.83 33.78 33.75 33.75 33.75

86.08 34.81 22.41 19.90 19.71 19.14 18.88

9.55 9.36 9.25 9.22 9.20 9.19 9.19

26.25 19.58 17.43 16.42 15.59 15.39 15.29

1.84 1.77 1.73 1.72 1.71 1.71 1.71

16.43 13.11 11.52 11.05 10.82 10.71 10.65

0.87 0.84 0.83 0.82 0.81 0.81 0.81

(4, 4)

10 20 50 100 250 500 1000

639.26 83.61 39.59 32.62 29.34 28.28 27.82

41.41 41.74 41.51 41.42 41.36 41.34 41.33

328.08 60.75 32.49 27.43 24.96 24.21 23.86

12.88 12.47 12.23 12.17 12.12 12.11 12.10

252.45 61.55 33.13 28.43 25.83 25.05 24.66

2.89 2.63 2.50 2.46 2.44 2.42 2.42

29.61 20.79 17.02 15.98 15.39 15.21 15.11

1.08 1.02 0.99 0.98 0.97 0.96 0.96

854

Y.-K. Chen / European Journal of Operational Research 178 (2007) 841–857

adaptive design parameters except the maximum action limit are insensitive to the m value. Tables 6 and 7 depict the effect of m value on the optimal maximum action limit, k1, to minimize ATS or AATS. The maximum action limit is decreasing as the m value increases. But it remains almost constant as long as m value goes beyond Nedumaran and Pignatiello’s lower bound: 800p/3(n0  1). 5.2. Comparisons between the VSI, VSS, VSSI and VP T2 charts As mentioned in the introduction, there are alternative adaptive control schemes (VSI, VSS, and VSSI) available to enhance T2 control charts, and we have make comparisons between them. Before the comparisons, the optimal design parameters in the VSSI, VSS, and VSI T2 charts were separately selected to minimize their ATS or AATS for given p, n0, and d values. In the optimal design of VSSI T2 chart, with fixed aj (i.e., a1 = a2 = a0) we adjust the three design parameters: n1, n2, and h2 to search for the minimal ATS or AATS, and then determine the design parameters h1 and wj (j = 1 and 2) by Eqs. (27) and (29), respectively. In the optimal VSS T2 chart, we adjust (n1, n2) to minimize ATS or AATS with all other parameters fixed (i.e., h1 = h2 = h0, a1 = a2 = a0) except wj (j = 1 and 2). Finally, optimal sampling intervals h1 and h2 are adjusted to find the optimal VSI T2 chart, and p0 in Eq. (26) should be modified as p0 ¼

h0  h2 . h1  h2

ð30Þ

Tables 8–11 show the results of the comparisons among VP, VSSI, VSS, VSI, and FSR T2 charts for the cases of two quality characteristics, n0 = 2 or 4, and the values of d from 0.25 to 1.50 increased by 0.25. Tables 8 and 9 give the optimal design parameters corresponding to the shortest ATS while Tables 10 and 11 give that corresponding to the shortest AATS. From these tables, we find that the ATS or AATS values of the VP T2 charts are always shorter than other types of T2 charts over the range of d that we have studied. Secondly, as Table 8 Comparison between ATS for the schemes VP, VSSI, VSS, VSI and FSR (n0 = 2, m = 600, p = 2, h0 = 1, a0 = 0.005) d

n1/n2

h1/h2

a1/a2

k1/k2

w1/w2

ATS

ATSFSR

VP

0.25 0.50 0.75 1.00 1.25 1.50

1/43 1/19 1/10 1/6 1/3 1/3

1.02/0.01 1.06/0.01 1.12/0.03 1.24/0.03 2.00/0.01 2.00/0.01

0.000/0.205 0.000/0.083 0.000/0.045 0.000/0.025 0.000/0.010 0.000/0.010

18.50/3.17 15.86/4.99 20.09/6.24 20.09/7.42 17.87/9.30 18.51/9.29

7.54/3.00 5.81/4.03 4.42/3.79 3.24/3.04 1.39/1.37 1.39/1.37

65.80 21.62 8.65 4.10 2.17 1.42

145.15 76.20 37.35 19.13 10.48 6.16

VSSI

0.25 0.50 0.75 1.00 1.25 1.50

1/182 1/40 1/16 1/8 1/3 1/3

1.01/0.01 1.03/0.02 1.07/0.01 1.17/0.01 2.00/0.01 2.00/0.01

0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005

10.73/10.61 10.73/10.62 10.73/10.62 10.73/10.63 10.73/10.67 10.73/10.67

9.21/9.13 7.04/6.99 5.32/5.29 3.86/3.84 1.38/1.38 1.38/1.38

100.47 31.28 11.08 4.79 2.41 1.46

145.15 76.20 37.35 19.13 10.48 6.16

VSS

0.25 0.50 0.75 1.00 1.25 1.50

1/185 1/41 1/18 1/10 1/8 1/6

1.00/1.00 1.00/1.00 1.00/1.00 1.00/1.00 1.00/1.00 1.00/1.00

0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005

10.73/10.62 10.73/10.62 10.73/10.63 10.73/10.63 10.73/10.64 10.73/10.64

9.23/9.15 7.09/7.03 5.56/5.52 4.35/4.33 3.86/3.84 3.20/3.19

100.58 31.88 12.16 6.16 3.91 2.84

145.15 76.20 37.35 19.13 10.48 6.16

VSI

0.25 0.50 0.75 1.00 1.25 1.50

2/2 2/2 2/2 2/2 2/2 2/2

3.51/0.01 4.62/0.01 7.02/0.01 11.52/0.01 19.12/0.01 31.49/0.01

0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005

10.73/10.73 10.73/10.73 10.73/10.73 10.73/10.73 10.73/10.73 10.73/10.73

0.67/0.67 0.47/0.47 0.31/0.31 0.18/0.18 0.11/0.11 0.06/0.06

135.63 60.11 21.29 6.75 2.32 1.24

145.15 76.20 37.35 19.13 10.48 6.16

a1 = 0.000 means that the optimal value of k1 make the risk of false alarm nearly zero.

Y.-K. Chen / European Journal of Operational Research 178 (2007) 841–857

855

Table 9 Comparison between ATS for the schemes VP, VSSI, VSS, VSI and FSR (n0=4, m=200, p = 2, h0 = 1, a0 = 0.005) n1/n2

a1/a2

k1/k2

w1/w2

VP

0.25 0.50 0.75 1.00 1.25 1.50

1/66 1/21 1/10 1/5 2/5 3/5

1.05/0.01 1.17/0.01 1.50/0.01 3.97/0.01 2.99/0.01 1.99/0.01

0.000/0.107 0.000/0.032 0.003/0.008 0.000/0.007 0.000/0.007 0.002/0.008

19.89/4.50 18.09/6.91 11.89/9.63 16.16/10.19 17.61/9.94 12.54/9.82

6.31/3.84 3.87/3.48 2.22/2.18 0.58/0.57 0.82/0.81 1.39/1.38

34.04 8.93 3.49 1.53 1.12 1.04

113.75 42.13 16.43 7.33 3.77 2.25

VSSI

0.25 0.50 0.75 1.00 1.25 1.50

1/164 1/33 1/11 1/5 2/5 3/5

1.02/0.02 1.10/0.01 1.42/0.02 3.98/0.01 2.99/0.01 2.00/0.00

0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005

10.99/10.65 10.99/10.66 10.99/10.68 10.99/10.73 10.82/10.73 10.81/10.73

7.74/7.56 4.74/4.67 2.42/2.40 0.58/0.58 0.82/0.82 1.39/1.39

52.80 11.43 3.77 1.55 1.13 1.03

113.75 42.13 16.43 7.33 3.77 2.25

VSS

0.25 0.50 0.75 1.00 1.25 1.50

1/165 1/37 1/18 2/13 2/10 3/8

1.00/1.00 1.00/1.00 1.00/1.00 1.00/1.00 1.00/1.00 1.00/1.00

0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005

10.99/10.65 10.99/10.66 10.99/10.67 10.99/10.68 10.99/10.69 10.82/10.70

7.57/7.57 4.97/4.89 3.49/3.44 3.43/3.39 2.79/2.76 3.22/3.20

53.12 12.53 5.29 3.14 2.24 1.75

113.75 42.13 16.43 7.33 3.77 2.25

VSI

0.25 0.50 0.75 1.00 1.25 1.50

4/4 4/4 4/4 4/4 4/4 4/4

3.84/0.01 6.48/0.01 12.98/0.01 26.65/0.01 49.99/0.01 49.99/0.01

0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005

10.76/10.76 10.76/10.76 10.76/10.76 10.76/10.76 10.76/10.76 10.76/10.76

0.60/0.60 0.34/0.34 0.16/0.16 0.08/0.08 0.04/0.04 0.04/0.04

100.72 25.71 5.11 1.44 1.03 1.00

113.75 42.13 16.43 7.33 3.77 2.25

d

h1/h2

ATS

ATSFSR

a1 = 0.000 means that the optimal value of k1 make the risk of false alarm nearly zero. Table 10 Comparison between ATTS for the schemes VP, VSSI, VSS, VSI and FSR (n0 = 2, m = 600, p = 2, h0 = 1, a0 = 0.005) n1/n2

h1/h2

a1/a2

k1/k2

w1/w2

VP

0.25 0.50 0.75 1.00 1.25 1.50

1/43 1/19 1/10 1/6 1/4 1/4

1.00/1.00 1.00/1.00 1.12/0.03 1.25/0.01 1.49/0.01 1.50/0.05

0.000/0.208 0.000/0.087 0.000/0.045 0.000/0.025 0.000/0.014 0.001/0.013

20.12/3.15 17.59/4.89 20.18/6.24 20.34/7.42 15.54/8.57 13.88/8.75

7.54/2.97 5.82/3.97 4.42/3.79 3.24/3.04 2.21/2.15 2.20/2.15

65.46 21.52 8.79 4.15 2.34 1.61

144.65 75.71 36.85 18.63 9.98 5.66

VSSI

0.25 0.50 0.75 1.00 1.25 1.50

1/185 1/40 1/16 1/8 1/5 1/4

1.00/1.00 1.03/0.01 1.07/0.01 1.17/0.01 1.30/0.01 1.50/0.01

0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005

10.73/10.61 10.73/10.62 10.73/10.62 10.73/10.63 10.73/10.64 10.73/10.65

9.23/9.14 7.04/6.99 5.32/5.29 3.86/3.84 2.76/2.75 2.19/2.18

100.08 31.32 11.10 4.79 2.48 1.62

144.65 75.71 36.85 18.63 9.98 5.66

VSS

0.25 0.50 0.75 1.00 1.25 1.50

1/185 1/41 1/18 1/10 1/8 1/6

1.00/1.00 1.00/1.00 1.00/1.00 1.00/1.00 1.00/1.00 1.00/1.00

0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005

10.73/10.61 10.73/10.62 10.73/10.63 10.73/10.63 10.73/10.63 10.73/10.64

9.23/9.15 7.09/7.03 5.56/5.52 4.35/4.33 3.86/3.84 3.20/3.19

100.08 31.38 11.66 5.66 3.41 2.34

144.65 75.71 36.85 18.63 9.98 5.66

VSI

0.25 0.50 0.75 1.00 1.25 1.50

2/2 2/2 2/2 2/2 2/2 2/2

3.13/0.01 3.81/0.01 4.31/0.01 3.84/0.01 2.86/0.01 2.09/0.01

0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005

10.73/10.73 10.73/10.73 10.73/10.73 10.73/10.73 10.73/10.73 10.73/10.73

0.77/0.77 0.61/0.61 0.53/0.53 0.60/0.60 0.86/0.86 1.30/1.30

136.29 61.20 22.96 8.61 3.69 1.92

144.65 75.71 36.85 18.63 9.98 5.66

d

a1 = 0.000 means that the optimal value of k1 make the risk of false alarm nearly zero.

AATS

AATSFSR

856

Y.-K. Chen / European Journal of Operational Research 178 (2007) 841–857

Table 11 Comparison between ATTS for the schemes VP, VSSI, VSS, VSI and FSR (n0 = 4, m = 200, p = 2, h0 = 1, a0 = 0.005) n1/n2

h1/h2

a1/a2

k1/k2

w1/w2

VP

0.25 0.50 0.75 1.00 1.25 1.50

1/67 1/24 2/11 3/7 3/7 3/8

1.00/1.00 1.00/1.00 1.28/0.02 1.33/0.01 1.33/0.02 1.25/0.01

0.000/0.108 0.000/0.038 0.000/0.022 0.001/0.018 0.003/0.012 0.005/0.005

19.50/4.48 19.71/6.59 22.05/7.65 15.59/8.06 12.05/8.98 10.82/10.70

6.34/3.83 4.16/3.65 3.06/2.88 2.80/2.68 2.79/2.72 3.22/3.20

33.75 9.20 3.57 1.71 1.05 0.81

113.25 41.63 15.93 6.83 3.27 1.75

VSSI

0.25 0.50 0.75 1.00 1.25 1.50

1/165 1/33 2/14 2/8 3/7 3/8

1.00/1.00 1.10/0.00 1.20/0.00 1.48/0.03 1.33/0.01 1.25/0.01

0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005

10.99/10.65 10.99/10.66 10.99/10.68 10.99/10.70 10.82/10.71 10.82/10.70

7.75/7.75 4.74/4.67 3.60/3.56 2.21/2.19 2.77/2.76 3.22/3.20

52.61 11.68 4.02 1.86 1.05 0.82

113.25 41.63 15.93 6.83 3.27 1.75

VSS

0.25 0.50 0.75 1.00 1.25 1.50

1/165 1/37 1/18 2/13 2/10 3/8

1.00/1.00 1.00/1.00 1.00/1.00 1.00/1.00 1.00/1.00 1.00/1.00

0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005

10.99/10.65 10.99/10.66 10.99/10.67 10.99/10.68 10.99/10.69 10.99/10.70

7.75/7.75 4.97/4.89 3.49/3.44 3.43/3.39 2.79/2.76 3.22/3.20

52.62 12.03 4.79 2.64 1.74 1.25

113.25 41.63 15.93 6.83 3.27 1.75

VSI

0.25 0.50 0.75 1.00 1.25 1.50

4/4 4/4 4/4 4/4 4/4 4/4

3.39/0.00 4.27/0.00 3.62/0.00 2.32/0.00 1.62/0.00 1.29/0.00

0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005 0.005/0.005

10.76/10.76 10.76/10.76 10.76/10.76 10.76/10.76 10.76/10.76 10.76/10.76

0.70/0.70 0.53/0.53 0.65/0.65 1.13/1.13 1.93/1.93 2.99/2.99

101.52 27.28 6.90 2.35 1.18 0.79

113.25 41.63 15.93 6.83 3.27 1.75

d

AATS

AATSFSR

a1 = 0.000 means that the optimal value of k1 make the risk of false alarm nearly zero.

compared with the VSS T2 chart, the VSI T2 chart shows a significant improvement in detecting a moderate or large shift, but it has only a trivial improvement in detecting a small shift. Thirdly, similar to the VP T2 charts, the sampling intervals used for the VSSI T2 charts to detect a small shift should be fixed to minimize AATS but should be changeable to minimize ATS. Even so, the VSSI T2 chart and the VSS T2 chart have a similar performance with either ATS or AATS. This finding implies once more that the sampling plan using the VSI can not conspicuously help the FSR T2 charts in detecting a small shift. Finally, in the case of the VP versus the VSSI T2 charts, we find that adopting variable action limits provides a great improvement for VSSI T2 charts in detecting a process shift, especially a small process mean shift. 6. Concluding remarks In this paper the VP T2 chart has been developed to increase the power of Hotelling’s T2 chart, assuming the in-control process mean vector and covariance matrix are unknown. Also, an optimization procedure that minimizes the ATS or AATS has been proposed to conduct the statistical design of the changeable sample sizes, sampling intervals, warning/action limits in the VP T2 chart. The length of time it takes to minimize the AATS is longer than that for ATS. In our experience, it is recommended at least 100,000 trials (the number of generation for termination condition) for AATS while 10,000 trials for ATS. It has been shown that the VP T2 chart significantly outperforms Hotelling’s T2 chart by increasing the speed to respond to a change in the mean vector of p correlated quality characteristics. Moreover, as compared with the VSS, VSI, and VSSI T2 chart, the results indicate that the T2 chart with variable design parameters also outperforms the VSS, VSI, and VSSI T2 charts, especially in detecting a small change in the mean vector of p correlated quality characteristics. When a small change occurs, the policy of adapting sample size and action limit seems more helpful than the policy of adapting sampling intervals to enhance the power of the T2 chart. However, the usefulness of adapting sampling intervals becomes evident when the process change in the mean vector is moderate or

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large. Furthermore, the number of initial samples used to estimate the unknown in-control process mean vector and covariance matrix refers to the range proposed by Nedumaran and Pignatiello (1999) in our study. It would be worthwhile conducting further research on whether or not the range should be modified when a Hotelling’s T2 chart with adaptive sampling plan is considered. References Alt, F.B., 1984. Multivariate quality control. In: Kotz, S., Johnson, N.L., Read, C.R. (Eds.), The Encyclopedia of Statistical Sciences. John Wiley, New York, pp. 110–122. Aparisi, F., 1996. Hotelling’s T2 control chart with adaptive sample sizes. International Journal of Production Research 34 (10), 2853– 2862. Aparisi, F., Garcı´a-Daı´z, J.C., 2003. Optimization of EWMA and MEWMA control charts using genetic algorithms. Computers and Operations research 31, 1437–1454. Aparisi, F., Haro, C.L., 2001. Hotelling’s T2 control chart with variable sampling intervals. International Journal of Production Research 39 (14), 3127–3140. Aparisi, F., Haro, C.L., 2003. A comparison of T2 control charts with variable sampling schemes as opposed to MEWMA chart. International Journal of Production Research 41 (10), 2169–2182. Bai, D.S., Lee, K.T., 1998. An economic design of variable sampling interval X control charts. International Journal of Production Economics 54, 57–64. Chen, Y.K., 2004. Economic design of X control charts for non-normal data using variable sampling policy. International Journal of Production Economics 92, 61–74. Chou, C.Y., Chen, C.H., Chen, C.H., 2006. Economic design of variable sampling intervals T2 control charts using genetic algorithms. Expert Systems with Applications 30 (2), 233–242. Costa, A.F.B., 1994. X control chart with variable sample size. Journal of Quality Technology 26, 155–163. Costa, A.F.B., 1997. X charts with variable sample sizes and sampling intervals. Journal of Quality Technology 29, 197–204. Costa, A.F.B., 1999a. X charts with variable parameters. Journal of Quality Technology 31, 408–416. Costa, A.F.B., 1999b. AATS for the X charts with variable parameters. Journal of Quality Technology 31, 455–458. Daudin, J.J., 1992. Double sampling X-bar charts. Journal of Quality Technology 24 (2), 78–87. Davis, L., 1991. Handbook of Genetic Algorithms. Van Nostrand Reinhold, New York. Goldberg, D.E., 1989. Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Reading, MA. He, D., Grigoryan, A., 2002. Construction of double sampling S-control charts for agile manufacturing. Quality and Reliability Engineering International 18, 343–355. He, D., Grigoryan, A., 2005. Multivariate multiple sampling charts. IIE Transactions 37, 509–521. He, D., Grigoryan, A., Sigh, M., 2002. Design of double and triple-sampling X control charts using genetic algorithms. International Journal of Production Research 40, 1387–1404. Hotelling, H., 1947. Multivariate quality control-illustrated by air testing of sample bombsights. In: Eisenhart, C., Hastay, M.W., Wallis, W.A. (Eds.), Techniques of Statistical Analysis. Mcgraw-Hill, New York, pp. 111–184. Jackson, J.E., 1985. Multivariate quality control. Communications in Statistics—Theory and Methods 14, 2657–2688. Karr, C.L., Gentry, E.J., 1993. Fuzzy control of pH using genetic algorithms. IEEE Transactions on Fuzzy Systems 1 (1), 46–53. Lowry, C.A., Montgomery, D.C., 1995. A review of multivariate control charts. IIE Transactions 27, 810–880. Nedumaran, G., Pignatiello Jr., J.J., 1999. On constructing T2 control charts for on-line process monitoring. IIE Transactions 31, 529–536. Prabhu, S.S., Runger, G.C., Keats, J.B., 1993. An adaptive sample size X chart. International Journal of Production Research 31, 2895– 2909. Prabhu, S.S., Montgomery, D.C., Runger, G.C., 1994. A combined adaptive sample size and sampling interval X control chart. Journal of Quality Technology 26, 164–176. Quesenberry, C.P., 1993. The effect of sample size on estimated limits for X and X control charts. Journal of Quality Technology 25, 237– 247. Reynolds Jr., M.R., Arnold, J.C., 1989. Optimal one-side Shewhart control chart with variable sampling intervals. Sequential Analysis 8 (1), 51–77. Reynolds Jr., M.R., Amin, R.W., Arnold, J.C., Nachlas, J.A., 1988. X -charts with variable sampling interval. Technometrics 30, 181–192. Runger, G.C., Montgomery, D.C., 1993. Adaptive sampling enhancements for Shewhart control charts. IIE Transactions 25, 41–51. Runger, G.C., Pignatiello Jr., J.J., 1991. Adaptive sampling for process control. Journal of Quality Technology 23 (2), 133–155. Severo, N.C., Zelen, M.C., 1960. Normal approximation to chi-square and non-central F probability functions. Biometrica 47 (3–4), 411– 416.