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The generally weighted moving average median (GWMA-eX) control chart is employed to monitoring the process sample mean/median. From the statistical point ...
Quality Engineering, 18:333–344, 2006 Copyright # Taylor & Francis Group, LLC ISSN: 0898-2112 print=1532-4222 online DOI: 10.1080/08982110600719431

The Generally Weighted Moving Average Control Chart for Monitoring the Process Median Shey-Huei Sheu Department of Industrial Management, National Taiwan University of Science and Technology, Taiwan ROC

Ling Yang Department of Industrial Engineering and Management, St. John’s University, Taiwan, ROC

e) The generally weighted moving average median (GWMA-X control chart is employed to monitoring the process sample mean=median. From the statistical point of view, the simulae chart outperforms tion result reveals that the GWMA-X e chart and the Shewhart-X e chart in both the EWMA-X detecting small shifts of the process sample mean=median. e chart is also In detecting the startup shifts, the GWMA-X e chart. An example more sensitive than the EWMA-FIR-X is given to illustrate this study. In general, the X charts are e charts are outliers-resistant. sensitive to outliers, and the X e In this paper, several X charts and X charts are used for e chart performs very comparison. Although the GWMA-X well in outliers-resistance, the GWMA-X chart is the best in fast detecting shifts. Therefore, the average quality cost is considered to be a criterion for choosing a control chart with outliers. The Lorenzen-Vance quality cost model is adopted herein. With various sfifts of the process sample mean=median, the average quality costs of control charts are evaluated under some contaminated normal distributions and cost parameters setting. We conclude that, from e control chart the economic point of view, the GWMA-X performs best with outliers. Keywords

EWMA charts; Generally weighted moving average (GWMA); Median charts; Outliers; Average quality cost

INTRODUCTION In monitoring the process mean, the traditional mean (X ) control charts, namely the Shewhart, cumulative sum (CUSUM) and exponentially weighted Address correspondence to Shey-Huei Sheu, Department of Industrial Management, National Taiwan University of Science and Technology, 43, Section 4, Keelung Road, Taipei 106, Taiwan ROC. E-mail: [email protected]

moving average (EWMA) charts have been investigated extensively. From past experience, there are some certain processes with outliers that happened occasionally. The outliers are the values of observations that are larger or smaller than the majority of the other observations. Suppose that the occurrence of outliers is due to the common causes and only the assignable causes will lead to permanent shifts, because the sample average, X , is sensitive to outliers. If the X chart is used to monitor the process mean, it will lead to a high level false alarms. In contrast to sample e , is a robust estimator average, the sample median, X of location for samples. Therefore, developing an e ) control chart for monioutliers-resistant median (X toring the process sample mean=median is a practical necessity. In general, the desirable properties expected from control charts are simplicity, fast detection of assignable causes, robustness to underlying assumptions, and economical usefulness. Castagliola (2001) proe control chart (EWMA-X e) posed an EWMA-based X with fast detection of assignable causes. Although the e chart was shown more efficient than the EWMA-X e chart in detecting small shifts of the proShewhart-X cess median, the discussion of the outliers-resistance was not included. Recently, Khoo (2005) proposed a e ) control chart. Shewhart-based median (Shewhart-X Under various contaminated normal distributions, the in-control average run lengths (ARL0s) of the e chart and the Shewhart-X chart were Shewhart-X evaluated by simulation. Their results showed that e charts are significantly the ARL0s of the Shewhart-X higher than that of the Shewhart-X charts. Khoo cone chart is more robust than cluded that the Shewhart-X the Shewhart-X chart in the presence of outliers. Howe chart and ever, the detection ability of the Shewhart-X the Shewhart-X chart for various shifts of the process

333

334

S. H. Sheu and L. Yang

sample mean=median in the presence of outliers was not discussed in Khoo (2005). Furthermore, the ecoe chart was considered by nomical usefulness of the X neither Castagliola (2001) nor Khoo (2005). Sheu and Lin (2003) proposed the generally weighted moving average (GWMA-X ) control chart, which is a generalization of the EWMA-X control chart. The GWMA-X control chart applied the method used in Sheu (1998) to the EWMA-X control chart and had been shown to outperform both the Shewhart-X chart and the EWMA-X chart in monitoring small shifts of the process mean. In this paper, e ) control the GWMA median (denoted as GWMA-X chart is employed for monitoring the process sample mean=median. We assume that the process characteristic follows the normal distribution. Through the evaluation of ARL0, the outliers-resistance of the several e charts and X charts is studied. Then, the LorenzenX Vance cost model (Lorenzen and Vance, 1986) is used for the evaluation of economical usefulness. The avere charts and X charts menage quality cost (AQC) of X tioned previously is studied and compared in the presence of outliers. The remainder of this paper is organized as fole lows: First, we describe the model of the GWMA-X chart. The simulation is used to evaluate the outof-control ARL (ARL1) of various process sample

mean=median shifts under various adjusted parae chart, meters. We compare the ARL1s of the GWMA-X e chart, and the EWMA-X e chart. In the Shewhart-X e chart is detecting the start-up shifts, the GWMA-X e chart. A sample compared with the EWMA-FIR-X for illustration is also given. To compare the economical e charts usefulness, the ARLs and the AQCs of various X and X charts mentioned above are evaluated under some contaminated normal distributions. Finally, some conclusions are included in the last section. e CONTROL CHART THE GWMA-X Suppose that the quality characteristic is a variable and the samples have been collected at each point in time (size of rational subgroups, denoted as n). Let ej be the sample median of subgroup j which is comX posed of n independent normal ðlj ; r2 Þ random variables Xj;1 ; . . . ; Xj;n , where lj is the nominal process mean=median and r2 is the common process variance. ej (derived by The distribution of sample median X e2 Þ normal Castagliola, 1998) is very close to the ðlj ; r 2 e e is the variance of Xj . If r e0;1 is distribution, where r the standard deviation of normal (0, 1) sample median, e¼rr e0;1 . For the values of r e0;1 , refer to we have r Castagliola (1998) for details.

Table 1 e control chart with time-varying control limits (ARL0 ffi 500) ARL of the GWMA-X q 0.0 0.5

0.7

0.8

0.9

a 1.0 0.5 0.7 0.8 0.9 1.0 0.5 0.7 0.8 0.9 1.0 0.5 0.7 0.8 0.9 1.0 0.5 0.7 0.8 0.9 1.0

(Shewhart)

(EWMA)

(EWMA)

(EWMA)

(EWMA)

L

d ¼ 0:1

0.2

0.3

0.4

0.5

0.6

0.7

1.0

1.5

2.0

3.128 3.099 3.095 3.095 3.093 3.088 3.087 3.074 3.054 3.054 3.035 3.070 3.027 3.001 2.983 2.970 3.012 2.909 2.865 2.841 2.827

439.2 307.9 327.1 334.5 342.8 347.0 223.5 242.5 247.2 261.1 277.2 171.3 184.2 198.6 210.6 222.5 132.8 133.4 137.5 151.6 165.4

285.1 131.8 140.9 146.7 152.7 154.4 79.6 85.9 87.5 97.0 100.4 61.8 61.5 65.2 69.4 76.0 49.8 45.8 44.9 47.8 51.9

175.3 63.6 64.9 69.3 72.0 72.1 38.5 37.2 40.6 41.2 44.6 32.1 29.4 29.4 31.1 32.8 27.0 23.9 23.0 23.8 24.4

110.4 35.0 33.5 36.1 36.7 37.9 22.6 21.3 20.8 22.6 23.4 19.9 17.6 17.5 17.9 18.1 17.2 15.1 14.5 14.6 14.7

69.6 21.3 20.8 20.7 21.3 22.3 15.2 14.1 13.7 14.3 14.3 13.9 12.0 11.8 11.7 12.2 12.3 10.7 10.2 10.2 10.3

43.1 14.1 13.5 13.7 13.7 14.2 11.1 10.1 9.8 9.8 9.8 10.3 9.0 8.9 8.7 8.8 9.4 8.2 7.8 7.7 7.8

28.3 10.8 10.1 10.2 10.2 10.5 8.8 7.9 7.7 7.8 7.8 8.3 7.4 7.2 7.0 7.1 7.7 6.8 6.4 6.4 6.4

9.7 5.5 5.1 5.0 5.0 4.9 5.0 4.5 4.5 4.4 4.4 4.8 4.5 4.3 4.2 4.2 4.5 4.6 4.0 4.0 4.0

2.7 3.0 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.8 2.8 2.9 2.8 2.8 2.8 2.8 2.9 2.7 2.7 2.7 2.7

1.4 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.2 2.2 2.2 2.2 2.3 2.2 2.2 2.2 2.2

335

Generally Weighted Moving Average Control Chart for Monitoring Process Median Table 2 e control charts from Table 1 The best performance X Best control chart

ARL1

e (q ¼ 0.9, a ¼ 0:5, L ¼ 3.012) GWMA-X e (q ¼ 0.9, a ¼ 0:8, L ¼ 2.865) GWMA-X e (q ¼ 0.9, a ¼ 0:8, L ¼ 2.865) GWMA-X e (q ¼ 0.9, a ¼ 0:8, L ¼ 2.865) GWMA-X e (q ¼ 0.9, a ¼ 0:9, L ¼ 2.841) GWMA-X e (q ¼ 0.9, a ¼ 0:9, L ¼ 2.841) GWMA-X e (q ¼ 0.9, a ¼ 0:9, L ¼ 2.841) or GWMA-X e (q ¼ 0.9, a ¼ 1, L ¼ 2.827) EWMA-X e (q ¼ 0.9, a ¼ 0.9, L ¼ 2.841) or GWMA-X e EWMA-X (q ¼ 0.9, a ¼ 1, L ¼ 2.827) e (q ¼ 0.9, a ¼ 0:9, L ¼ 2.841) or GWMA-X e (q ¼ 0.9, a ¼ 1, L ¼ 2.827) or EWMA-X e (L ¼ 3.120) Shewhart-X e Shewhart-X (L ¼ 3.120)

132.8 44.9 23.0 14.5 10.2 7.7 6.4

d 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1.0 1.5

2.0

4.0 2.7

1.4

e control From Sheu and Lin (2003), the GWMA-X statistic, Yj , can be represented as 0a

1a

ej þ ðq1a  q2a ÞX ej1 Yj ¼ ðq  q ÞX a a e1 þ qja l0 þ    þ ðqðj1Þ  qj ÞX

ð1Þ

where the design parameter q is constant (0  q < 1, a when q ¼ 0, let q0 ¼ 1), and the adjustment parameter a ða > 0Þ is determined by the practitioner. The expected value of Eq. (1) can then be computed by a

a

a

a

EðYj Þ ¼ ½ðq0  q1 Þ þ ðq1  q2 Þ a

a

a

e Þ þ qj l0 ¼ l0 ð2Þ þ    þ ðqðj1Þ  qj ÞEðX ej , j ¼ 1, 2, 3, . . . , are independent random variSince X e2 ¼ ðr  r e0;1 Þ2 , the variance of ables with variance r Eq. (1) is a

a

a

a

VarðYj Þ ¼ ½ðq0  q1 Þ2 þ ðq1  q2 Þ2 a

a

e2 ¼ Qj ðr  r e0;1 Þ2 ð3Þ þ   þ ðqðj1Þ  qj Þ2   r a

where Qj ¼ ðq0 q1 Þ2 þðq1 q2 Þ2 þ  þ ðqðj1Þ qj Þ2 : a

a

a

a

a

Then, the time-varying control limits of the e control chart can be written as GWMA-X pffiffiffiffiffi r0;1 ; UCL ¼ l0 þ L Qj re CL ¼ l0 ; ð4Þ pffiffiffiffiffi LCL ¼ l0  L Qj re r0;1 : When a ¼ 1 and q ¼ 1  k, Yj will reduce to e1 þ ð1  kÞj l0 ej þ kð1  kÞX ej1 þ  þ kð1  kÞj1 X Yj ¼ kX e2 = The variance of Yj will be r2Yj ¼ kð1  ð1  kÞ2j Þ r ð2  kÞ. The time-varying control limits will become qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e. These are the to l0  L kð1  ð1  kÞ2j Þ=ð2  kÞr e chart proposed by Castaequations of the EWMA-X gliola (2001). THE ARL PERFORMANCE e chart are The control limits of the GWMA-X varying with time. The simulation (see Ross, 1990) is e chart. used to estimate the ARL of the GWMA-X Let the in-control lj ¼ l0 ¼ 0. For simplicity, throughout this paper, we assume that the common variance is r2 ¼ 1 and the size of rational subgroup e0;1 ¼ 0:536 (from Castagliola, is n ¼ 5. We can get r 1998) for Eq. (4) to compute the control limits. We choose the design parameter q 2 f0; 0:5; 0:7; 0:8; 0:9g, and the adjustment parameters a 2 f0:5; 0:7; 0:8; 0:9; 1:0g only, and keep the ARL0 ffi 500 by changing e chart reduces to the L. When a ¼ 1:0, the GWMA-X e chart. When q ¼ 0, the EWMA-X e chart EWMA-X e chart. Let d denote the reduces to the Shewhart-X magnitude of the process mean=median shift (multiple e chart are obtained of r). The ARL1s of the GWMA-X by simulation too. Up to 20,000 runs of the process are simulated. The simulation programs are written in BASIC language. Table 1 shows the simulation results e chart. From the statistical of ARL1 of the GWMA-X

Table 3 e chart and the EWMA-FIR-X e chart (q ¼ 0.75, L ¼ 3) Comparison between the ARLs of the GWMA-X

d 0.00 0.15 0.25 0.50 0.75 1.00

FIR f ¼ 0.5 a ¼ 0.30

GWMA a ¼ 0:5

FIR f ¼ 0.6 a ¼ 0.21

GWMA a ¼ 0:75

FIR f ¼ 0.7 a ¼ 0.15

GWMA a ¼ 0:8

FIR f ¼ 0.8 a ¼ 0.10

GWMA a ¼ 0:9

FIR f ¼ 1.0

GWMA a ¼ 1:0

375.6 105.9 38.2 7.4 3.5 2.6

391.4 96.5 43.3 13.4 7.1 4.8

437.9 113.3 41.6 8.2 4.0 2.8

443.6 110.8 45.5 13.0 6.6 4.5

453.2 130.6 50.2 9.8 4.4 3.2

458.6 119.3 47.3 12.6 6.6 4.4

466.7 134.9 53.4 11.0 5.2 3.6

492.2 127.6 52.1 12.4 6.4 4.3

499.8 141.0 54.7 12.9 6.4 4.3

498.7 148.1 57.1 13.3 6.6 4.3

336

S. H. Sheu and L. Yang

Table 4 e control scheme and a GWMA-X e control scheme using data from a process initially in control Examples of an EWMA-X e e EWMA-X GWMA-X j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Xj1

Xj2

Xj3

Xj4

Xj5

ej X

Zj

UCL

LCL

Yj

UCL

LCL

1.588 1.261 1.499 0.994 1.132 0.189 2.222 1.174 0.608 0.835 0.448 1.123 0.002 1.492 0.059 1.285 0.603 1.129 1.832 0.236

0.162 0.831 0.132 1.424 0.885 2.530 2.101 0.430 0.202 0.359 1.335 0.981 1.853 0.404 0.439 0.100 0.447 0.218 0.810 1.195

0.415 0.778 1.063 2.636 0.189 0.089 0.924 0.956 0.363 0.716 0.465 0.011 0.166 0.706 0.504 0.800 0.761 1.340 1.773 0.145

0.921 2.056 0.898 2.587 0.378 1.227 0.565 0.505 0.524 0.982 0.826 1.037 0.473 0.426 1.358 2.124 0.447 2.047 2.506 0.092

0.617 0.445 1.407 0.202 0.195 0.768 0.494 0.018 0.159 0.086 0.693 0.091 1.167 0.294 0.770 0.522 1.046 0.292 2.016 0.804

0.415 0.445 0.132 1.424 0.189 0.768 0.924 0.505 0.202 0.359 0.448 0.981 0.166 0.404 0.504 0.800 0.603 1.129 1.832 0.145

0.104 0.189 0.175 0.487 0.318 0.431 0.554 0.542 0.457 0.253 0.078 0.303 0.269 0.303 0.353 0.464 0.499 0.657 0.951 0.677

0.402 0.503 0.552 0.577 0.591 0.599 0.603 0.605 0.607 0.608 0.608 0.608 0.608 0.608 0.608 0.608 0.608 0.609 0.609 0.609

0.402 0.503 0.552 0.577 0.591 0.599 0.603 0.605 0.607 0.608 0.608 0.608 0.608 0.608 0.608 0.608 0.608  0.609  0.609  0.609

0.042 0.070 0.062 0.192 0.113 0.176 0.228 0.226 0.205 0.139 0.090 0.195 0.170 0.188 0.210 0.256 0.268 0.336 0.454 0.341

0.155 0.182 0.199 0.210 0.219 0.225 0.231 0.235 0.239 0.242 0.245 0.247 0.250 0.251 0.253 0.254 0.256 0.257 0.258 0.259

0.155 0.182 0.199 0.210 0.219 0.225 0.231 0.235 0.239 0.242 0.245 0.247 0.250 0.251 0.253  0.254  0.256  0.257  0.258  0.259

point of view, the boldface numbers in Table 1 denote e chart outperforms the EWMA-X e that the GWMA-X chart (a ¼ 1) with the same q value. For instance, in Table 1, when q = 0.9 and the magnitude of shift e chart (with g = 0.2, the ARL, (44.9) of the GWMA-X a ¼ 0.8) is smaller than the ARL, ( ¼ 51.9) of the e chart with q ¼ 0.9 and a ¼ 1). When the EWMA-X e chart adjustment parameter 0 < a < 1, the GWMA-X e is more sensitive than both the Shewhart-X chart and e chart in detecting small shifts of the the EWMA-X process sample mean=median. From Table 1, the best performance control charts in terms of ARL1 are listed in Table 2. e COMPARING WITH THE EWMA-FIR-X CONTROL CHART Lucas and Crosier (1982) employed fast initial response feature (FIR) to monitor the process mean when a control scheme was initiated because of startup problems. Steiner (1999) used a single control chart but narrowed the time-varying limits even further for the first few sample points (named EWMA-FIR). He used an exponentially decreasing adjustment to further narrow the control limits to

e control scheme and the GWMAFigure 1. The EWMA-X e control scheme. X

Generally Weighted Moving Average Control Chart for Monitoring Process Median

( l  Lr

h

1  ð1  f Þ

1þaðj1Þ

) irffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2j ½1  ð1  kÞ  2k

Steiner suggested choosing a so that the FIR has little effect after about 20 observations. This leads to The choosing a ¼ ½ð2= log10 ð1  f ÞÞ  1=19. e chart with time-varying control limits is GWMA-X similar to that of the EWMA-FIR scheme. Simulation e is employed to compare the ARL of the GWMA-X e chart with the EWMA-FIR-X chart. As illustrated in Table 3, when q ¼ 0.75, L ¼ 3, and the shift of proe chart cess mean=median within 0:15r, the GWMA-X e performs much better than the EWMA-FIR-X chart by reducing the ARL1 for detecting an off-target process at startup.

337

EXAMPLE A set of simulation data is used herein to illustrate e control scheme. The values of the process a GWMA-X characteristic Xjk, where j ¼ 1, 2, 3, . . . , and k ¼ 1, 2, . . . ,5 are independent and have a common normal distribution with mean=median l0 ¼ 0, variance r2 ¼ 1. Let the target value be l0 , and the process be under control for the first 10 samples. Then, the process mean=median level shifts upward about 0:25r during the last 10 samples. These 20 simulation data, e control along with their corresponding EWMA-X e control statistics (Yj), statistics (Zj) and GWMA-X are listed in Table 4. Within this table, we set the parae chart meters k ¼ 0:25 and L ¼ 3.003 for the EWMA-X with time-varying control limits, ARL0 ffi 500. For a fair comparison, we set the parameters q ¼ 0.9 and

Table 5 ARL of different charts for data from a contaminated normal distribution with a desired ARL0 ffi 500 e charts X charts X (r0 ; h)

d

Shewhart

GWMA

EWMA

Shewhart

GWMA

EWMA

(0, 0)

0.0 0.1 0.3 0.5 0.7 1.0 1.5 2.0 0.0 0.1 0.3 0.5 0.7 1.0 1.5 2.0 0.0 0.1 0.3 0.5 0.7 1.0 1.5 2.0 0.0 0.1 0.3 0.5 0.7 1.0 1.5 2.0

500.1 439.2 175.3 69.6 28.3 9.7 2.7 1.4 264.5 236.8 126.6 48.6 24.1 9.8 4.0 2.4 218.8 204.5 119.8 56.3 23.8 10.0 3.9 2.4 158.9 160.5 95.5 37.2 19.8 8.9 3.6 2.4

500.2 151.6 23.8 10.2 6.4 4.0 2.7 2.2 352.0 116.2 22.4 9.9 6.2 4.0 2.7 2.2 317.0 131.5 21.6 10.2 6.4 4.2 2.6 2.2 256.4 102.3 20.0 9.9 6.1 4.0 2.7 2.2

499.2 165.4 24.4 10.3 6.4 4.0 2.7 2.2 343.9 127.4 23.3 10.0 6.2 4.0 2.7 2.2 314.3 137.2 22.4 10.1 6.4 4.1 2.6 2.2 250.2 107.2 21.3 10.0 6.2 3.9 2.7 2.2

500.2 405.3 128.1 41.5 16.3 5.0 1.7 1.1 87.1 78.0 48.2 24.5 13.8 5.7 2.7 2.1 106.9 106.5 61.8 27.4 12.4 6.0 2.7 2.1 34.1 48.9 24.9 15.7 9.9 5.2 2.6 2.1

499.4 128.5 19.9 8.5 5.3 3.3 2.4 2.1 179.2 70.7 16.4 7.9 5.1 3.4 2.4 2.1 192.75 94.3 15.8 8.1 5.2 3.4 2.4 2.1 74.4 45.6 14.5 8.1 5.1 3.4 2.4 2.1

501.0 136.3 19.2 8.5 5.2 3.3 2.4 2.1 186.1 85.2 17.1 8.0 5.0 3.4 2.4 2.1 198.3 104.0 16.6 8.0 5.1 3.3 2.4 2.0 78.9 48.9 14.9 8.0 5.1 3.3 2.4 2.0

(2.5, 6)

(2, 10)

(3, 10)

338

S. H. Sheu and L. Yang Table 6 AQC of different charts for data from a contaminated normal distribution with a desired ARL0 ffi 500 (Y ¼ 100) e charts X X charts

(r0 ; h)

d

Shewhart

GWMA

EWMA

Shewhart

GWMA

EWMA

(0, 0)

0.1 0.3 0.5 0.7 1.0 1.5 2.0 0.1 0.3 0.5 0.7 1.0 1.5 2.0 0.1 0.3 0.5 0.7 1.0 1.5 2.0 0.1 0.3 0.5 0.7 1.0 1.5 2.0

50.7 37.9 31.0 28.0 26.5 26.0 25.9 42.7 36.4 31.1 29.3 28.2 27.8 27.6 41.6 36.6 32.4 30.0 29.0 28.5 28.4 40.6 36.6 32.6 31.4 30.5 30.1 30.0

36.5 27.6 26.6 26.3 26.1 26.0 25.9 34.9 28.3 27.3 27.0 26.9 26.8 26.7 36.2 28.5 27.7 27.4 27.2 27.1 27.0 34.9 29.1 28.3 28.0 27.9 27.8 27.7

37.3 27.7 26.6 26.3 26.1 26.0 25.9 35.7 28.4 27.4 27.1 26.9 26.8 26.8 36.6 28.6 27.7 27.4 27.2 27.1 27.1 35.4 29.3 28.4 28.2 28.0 27.9 27.8

49.3 35.0 29.0 27.0 26.1 25.9 25.8 40.1 38.2 36.6 35.8 35.3 35.0 35.0 40.0 37.1 34.8 33.7 33.2 33.0 33.0 54.6 53.3 52.8 52.5 52.3 52.1 52.1

35.0 27.3 26.4 26.2 26.0 25.9 25.9 34.3 30.4 29.8 29.6 29.4 29.4 29.3 35.5 30.0 29.4 29.2 29.1 29.0 29.0 39.8 37.7 37.3 37.1 37.0 36.9 36.9

35.5 27.3 26.4 26.2 26.0 25.9 25.9 35.1 30.3 29.6 29.4 29.3 29.2 29.2 36.1 29.9 29.3 29.1 28.9 28.9 28.8 39.3 37.3 36.6 36.4 36.2 36.2 36.1

(2.5, 6)

(2, 10)

(3, 10)

e chart with time-varying L ¼ 2.884 for the GWMA-X e control control limits, ARL0 ffi 500. The EWMA-X statistics, Zj, display the out-of-control signal at the e control statistics, Yj, 18-th sample. The GWMA-X display the out-of-control signal at the 16-th sample. Under the assigned parameters as described above, it e takes 56.67 samples in average for the EWMA-X control scheme to detect an out-of-control signal, while e only 30.67 samples are needed for the GWMA-X e control chart can control scheme. The GWMA-X detect the small shifts more quickly than the e control chart. Figure 1(a) displays a plot EWMA-X e control statistics and Figure 1(b) of the EWMA-X e control statistics. displays a plot of the GWMA-X OUTLIERS-RESISTANCE AND FAST DETECTING SHIFTS In order to evaluate the ARL and AQC of different charts in the presence of outliers, the contami-

nated normal distribution used in Khoo (2005) is adopted. A contaminated normal distribution is that the observations ð100  hÞ% come from (N(0, 1)) normal distribution and h% come from (Nð0; r20 Þ) normal distribution, where h denotes the level of contamination and r20 denotes the variance of an outlier. We assumed that the outliers occur due to the common causes of variation and lead to a temporary shift. Only the assignable causes will make the process a permanent shift. We are interested in detecting a permanent shift. Three kinds of contaminated normal distributions, ðr0 ; hÞ 2 fð2:5; 6Þ; ð2; 10Þ; ð3; 10Þg, are used herein to evaluate the ARL and AQC of various e charts used for comparison are control charts. The X e chart (with L ¼ 3.128), the GWMAthe Shewhart-X e chart (with q ¼ 0.9, a ¼ 0.9, L ¼ 2.841), and the X e chart (with q ¼ 0.9, a ¼ 1, L ¼ 2.827). EWMA-X The X charts used for comparison are the Shewhart-X chart (with L ¼ 3.090), the GWMA-X chart (with q ¼ 0.9, a ¼ 0.9, L ¼ 2.840), and the EWMA-X

339

Generally Weighted Moving Average Control Chart for Monitoring Process Median Table 7 AQC of different charts for data from a contaminated normal distribution with a desired ARL0 ffi 500 (Y ¼ 500) e charts X X charts (r0 ; h)

d

Shewhart

GWMA

EWMA

Shewhart

GWMA

EWMA

(0, 0).

0.1 0.3 0.5 0.7 1.0 1.5 2.0 0.1 0.3 0.5 0.7 1.0 1.5 2.0 0.1 0.3 0.5 0.7 1.0 1.5 2.0 0.1 0.3 0.5 0.7 1.0 1.5 2.0

56.1 44.5 38.2 35.4 34.1 33.6 33.5 54.6 49.3 45.0 43.5 42.6 42.2 42.1 56.3 52.4 49.1 47.2 46.4 46.0 45.9 61.5 58.7 56.0 55.1 54.5 54.2 54.2

43.2 35.1 34.2 33.9 33.7 33.6 33.6 44.7 39.0 38.1 37.9 37.7 37.6 37.6 46.9 40.4 39.7 39.4 39.2 39.1 39.1 48.6 43.8 43.2 42.9 42.8 42.7 42.7

43.9 35.2 34.2 33.9 33.7 33.6 33.6 45.7 39.4 38.5 38.2 38.1 38.0 37.9 47.4 40.6 39.8 39.5 39.4 39.3 39.2 49.3 44.4 43.6 43.4 43.3 43.2 43.2

54.9 41.8 36.4 34.6 33.8 33.5 33.5 81.1 80.3 79.7 79.3 79.1 79.0 79.0 72.5 71.0 69.8 69.2 69.0 68.8 68.8 162.1 163.3 163.7 164.0 164.3 164.4 164.4

41.8 34.8 34.0 33.8 33.7 33.6 33.6 54.4 51.5 51.0 50.9 50.8 50.7 50.7 53.8 49.6 49.2 49.0 48.9 48.9 48.8 89.2 88.6 88.5 88.4 88.4 88.4 88.4

42.3 34.8 34.0 33.8 33.7 33.6 33.6 54.2 50.6 50.1 49.9 49.8 49.8 49.7 53.7 49.0 48.5 48.3 48.2 48.2 48.1 85.8 85.0 84.8 84.8 84.7 84.7 84.7

(2.5, 6)

(2, 10)

(3, 10)

chart (with q ¼ 0.9, a ¼ 1, L ¼ 2.835). The values of L for those control charts are based on the data which 100% come from the N(0, 1) normal distribution [i.e., ðr0 ; hÞ ¼ ð0; 0Þ], with a desired ARL0 ffi 500. Table 5 presents the simulation result. In Table 5, various combinations of ðr0 ; hÞ denote various contaminated normal distributions. For example, when ðr0 ; hÞ ¼ ð0; 0Þ, the data 100% come from the normal distribution N(0, 1). When ðr0 ; hÞ ¼ ð2:5; 6Þ, the data 94% come from the normal distribution N(0, 1) and 6% come from the normal distribution N(0, 2.52). When ðr0 ; hÞ 2 e charts are fð2:5; 6Þ; ð2; 10Þ; ð3; 10Þg, the ARL0s of X greater than that of the X charts. For instance, when e chart is ðr0 ; hÞ ¼ ð2:5; 6Þ, the ARL0 of the GWMA-X 352.2 greater than those of three X charts (87.1, 179.2, 186.1, respectively). It means that the X charts e charts are outliersare sensitive to outliers and the X e resistant. This robustness is the strong point of X chart. However, under various process mean=median

e charts are greater than shifts, all of the ARL1s of X that of the corresponding X charts. For instance, when ðr0 ; hÞ ¼ ð2:5; 6Þ and d ¼ 0.1, the ARL1 of e chart (¼ 116.2) is greater than that of the GWMA-X the GWMA-X charts (¼ 70.7). The shift-detecting abile charts is worse than that of X charts without ity of X respect to outliers. It becomes very difficult to make a e chart and the X chart. Therefore, choice between the X the quality cost model is employed herein to consider the economic usefulness of those charts mentioned previously.

THE AQC PERFORMANCE When we apply a control chart to monitor the process sample mean=median, if the plotted statistic is greater than UCL or less than LCL, the process is assumed to be out of control and a search for an assignable cause is initiated. Assume that the time

340

S. H. Sheu and L. Yang Table 8 AQC of different charts for data from a contaminated normal distribution with a desired ARL0 ffi 500 (Y ¼ 1000) e charts X X charts

(r0 ; h)

d

Shewhart

GWMA

EWMA

Shewhart

GWMA

EWMA

(0, 0).

0.1 0.3 0.5 0.7 1.0 1.5 2.0 0.1 0.3 0.5 0.7 1.0 1.5 2.0 0.1 0.3 0.5 0.7 1.0 1.5 2.0 0.1 0.3 0.5 0.7 1.0 1.5 2.0

62.9 52.7 47.2 44.8 43.6 43.2 43.1 69.3 65.5 62.3 61.2 60.6 60.3 60.2 74.6 72.1 69.9 68.6 68.1 67.8 67.8 87.7 86.4 85.1 84.7 84.4 84.3 84.3

51.5 44.5 43.7 43.4 43.3 43.2 43.2 57.0 52.3 51.6 51.4 51.3 51.2 51.2 60.4 55.2 54.6 54.4 54.3 54.2 54.2 65.6 62.2 61.7 61.5 61.4 61.4 61.4

52.2 44.6 43.7 43.4 43.3 43.2 43.2 58.1 53.1 52.3 52.1 52.0 51.9 51.9 60.8 55.5 54.9 54.7 54.6 54.5 54.5 66.6 63.1 62.6 62.5 62.4 62.3 62.3

61.8 50.4 45.6 44.1 43.3 43.1 43.1 132.4 133.0 133.5 133.7 133.9 134.0 134.0 113.3 113.4 113.5 113.6 113.6 113.6 113.6 296.5 300.7 302.4 303.4 304.3 304.8 304.9

50.4 44.3 43.6 43.4 43.2 43.2 43.2 79.4 77.9 77.6 77.5 77.5 77.4 77.4 76.6 74.1 73.9 73.8 73.7 73.7 73.7 151.0 152.2 152.5 152.6 152.7 152.7 152.7

50.8 44.2 43.6 43.4 43.2 43.2 43.2 78.0 76.0 75.7 75.6 75.5 75.5 75.5 75.7 72.8 72.5 72.4 72.3 72.3 72.3 143.9 145.0 145.2 145.3 145.4 145.4 145.4

(2.5, 6)

(2, 10)

(3, 10)

between occurrences of an assignable cause follows an exponential distribution with a mean of s occurrences per hour. The sustained shift occurs only once in each production cycle. In Lorenzen and Vance (1986), the time interval from the beginning of the in-control state to the adjustment of the out-ofcontrol state is called a production cycle. The expected production cycle time, PCT, is 1 ð1  c1 ÞsT0 PCT ¼ þ  u þ nE þ hðARL1 Þ þ T1 þ T2 s ARL0 ð5Þ The expected total cost per cycle, TC, is TC ¼

C0 þ C1  ðu þ nE þ hðARL1 Þ þ c1 T1 þ c2 T2 Þ s   sY a þ bn þ þW þ ARL0 h   1  u þ nE þ hðARL1 Þ þ c1 T1 þ c2 T2  ð6Þ s

where the cost parameters are n ¼ sample size h ¼ hours between samples C0 ¼ quality cost=hour while producing in control C1 ¼ quality cost=hour while producing out of control ( > C 0) h  u ¼ expected time between a shift (assignable cause) and the next sample   u ¼ 1  ð1 þ shÞesh =ðsð1  esh ÞÞ E ¼ time to sample and chart one item c1 ¼ 1 if production continues during searches, and 0 otherwise c2 ¼ 1 if production continues during the repair of the process, and 0 otherwise T0 ¼ expected search time when false alarm T1 ¼ expected time to discover the assignable cause T2 ¼ expected time to repair the assignable cause

Generally Weighted Moving Average Control Chart for Monitoring Process Median

Figure 2.

AQC of different charts for data from contaminated normal distributions (Y ¼ 100).

Figure 3.

AQC of different charts for data from contaminated normal distributions (Y ¼ 500).

341

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S. H. Sheu and L. Yang

Figure 4.

AQC of different charts for data from contaminated normal distributions (Y ¼ 1000).

Table 9 The best chart in terms of AQC with the contaminated normal distribution Y 100

500

1000

d

ðr0 ; hÞ ¼ ð0; 0Þ

(2.5, 6)

(2, 10)

(3, 10)

0.1 0.3 0.5 0.7 1.0 1.5 2.0 0.1 0.3 0.5 0.7 1.0 1.5 2.0 0.1 0.3 0.5 0.7 1.0 1.5 2.0

GWMA-X GWMA-X GWMA-X GWMA-X GWMA-X GWMA-X Shewhart-X GWMA-X GWMA-X GWMA-X GWMA-X GWMA-X Shewhart-X Shewhart-X GWMA-X GWMA-X GWMA-X GWMA-X GWMA-X GWMA-X GWMA-X

GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X

GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X

e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X e GWMA-X

Generally Weighted Moving Average Control Chart for Monitoring Process Median

343

s ¼ esh =ð1  esh Þ, the expected number of samples while in control Y ¼ cost to investigate false alarms W ¼ cost of locating and repairing an assignable cause a ¼ fixed cost per item b ¼ cost per unit sampled

e chart is the best presence of outliers, the GWMA-X e charts and the three X charts in among those three X terms of AQC.

From Eqs. (5) and (6), the expected average quality cost per hour, AQC, is

This work was partially supported by National Science Council of Taiwan under the contract number NSC94-2213-E-011-037.

AQC ¼

TC : PCT

ACKNOWLEDGEMENT

ð7Þ ABOUT THE AUTHORS

Those interested in more details are referred to Lorenzen and Vance (1986). In the quality cost model, let u ¼ 0.1, T1 ¼ 2, T2 ¼ 2, h ¼ 0.1, and Y ¼ 100, Y ¼ 500, Y ¼ 1000, respectively, and other cost parameters are adopted from Montgomery et al. (1995): 1=s ¼ 100, E ¼ 0:05, T0 ¼ 0, c1 ¼ c2 ¼ 1, C0 ¼ 10, C1 ¼ 100, W ¼ 25, a ¼ 0:5, b ¼ 0:1. By combining these cost parameters with Table 5, the AQCs of those control charts are calculated and compared in Tables 6 to 8, with Y ¼ 100, 500, and 1000, respectively. Figures 2 through 4 show e chart, the GWMA-X e the AQC of the Shewhart-X chart, the Shewhart-X chart, and the GWMA-X chart, with Y ¼ 100, 500, and 1000, respectively. Table 9 summarizes the best chart for detecting various shifts in terms of AQC. In Table 9, when ðr0 ; hÞ 2 fð2:5; 6Þ; ð2; 10Þ; ð3; 10Þg, most of the best charts are e charts. On the other hand, when the GWMA-X ðr0 ; hÞ ¼ ð0; 0Þ, most of the best charts are the GWMA-X charts. Therefore, from the economic point e chart is the best between those of view, the GWMA-X e charts and X charts when the process has the preX sence of outliers.

Shey-Huei Sheu received his MS degree in Applied Mathematics from National Tsing Hua University, and a Ph.D. degree in Statistics from the University of Kentucky. Currently, he is a professor in the Department of Industrial Management at National Taiwan University of Science and Technology. He has published in journals such as Naval Research Logistics, Journal of Applied Probability, RAIOR Operations Research, Microelectronics & Reliability, Reliability Engineering & System Safety, International Journal of System Science, International Journal of Reliability, Quality and Safety Engineering, Journal of the Operational Research Society, European Journal Operational Research, IEEE Transaction on Reliability, Production Planning & Control, Computers & Operations Research, Quality Engineering, and Annuals of Operations Research. Ling Yang is an Associate Professor in the Department of Industrial Engineering and Management at St. John’s University, Taiwan. She received her MS degree in Management Science from National Chiao Tung University, Taiwan, and a Ph.D. degree in Management Science from National Taiwan University of Science and Technology.

CONCLUSIONS e control chart is employed to moniThe GWMA-X tor the process sample mean=median. From the statistie chart outperforms cal point of view, the GWMA-X e e chart both the EWMA-X chart and the Shewhart-X in detecting small shifts of the process sample mean= e chart, median. Comparing with the EWMA-FIR-X e the GWMA-X chart still performs well in detecting startup problems. From the economic point of view, under some contaminated normal distributions and cost e chart outperforms parameters setting, the GWMA-X e e chart, the the Shewhart-X chart, the EWMA-X GWMA-X chart, the EWMA-X chart, and the Shewhart-X chart. Therefore, if the process has the

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