Progressive Mean Control Chart for Monitoring ... - Wiley Online Library

Research Article (wileyonlinelibrary.com) DOI: 10.1002/qre.1386

Published online 28 February 2012 in Wiley Online Library

Progressive Mean Control Chart for Monitoring Process Location Parameter Nasir Abbas,a*† Raja Fawad Zafar,a Muhammad Riaza,b and Zawar Hussaina Control charts are widely used for process monitoring. They show whether the variation is due to common causes or whether some of the variation is due to special causes. To detect large shifts in the process, Shewhart-type control charts are preferred. Cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts are generally used to detect small and moderate shifts. Shewhart-type control charts (without additional tests) use only current information to detect special causes, whereas CUSUM and EWMA control charts also use past information. In this article, we proposed a control chart called progressive mean (PM) control chart, in which a PM is used as a plotting statistic. The proposed chart is designed such that it uses not only the current information but also the past information. Therefore, the proposed chart is a natural competitor for the classical CUSUM, the classical EWMA and some recent modifications of these two charts. The conclusion of this article is that the performance of the proposed PM chart is superior to the compared ones for small and moderate shifts, and its performance for large shifts is better (in terms of the average run length). Copyright © 2012 John Wiley & Sons, Ltd. Keywords: average run length (ARL); memory control charts; EWMA; CUSUM; progressive mean (PM); statistical process control

1. Introduction rocesses are subject to variation. Control charts are used to signal the occurrence of special causes. The power of a control chart partly lies in its simplicity: it consists of a graph of a process characteristic plotted through time. The control limits in the graph provide easy checks on the stability of the process (cf. Shewhart1). A process is said to be statistically in control if there are no signals in the chart (i.e. the variation is due to common cause variations). When the chart signals an out of control situation (e.g. an observation outside the control limit), then the process is said to be statistically out of control (cf. Montgomery2). Control charts are classified into two categories: memoryless control charts and memory control charts. Shewhart-type control charts are termed as memoryless because they only use the current information to detect special causes. They are less sensitive to small and moderate shifts in the parameter of interest. Runs rules can be used to enhance the ability of control charts to detect small and moderate shifts. However, their application results in an increment of the false alarm rate in general, which is not desirable. This problem was successfully tackled by resetting the control limits coefficients for different rules (cf. Riaz et al.3 and Abbas et al.4). Another approach to address the detection of small shifts is to use memory control charts. The most popular ones are the cumulative sum (CUSUM) control charts (cf. Page5) and the exponentially weighted moving average (EWMA) control charts (cf. Roberts6). These charts are designed such that they use past information along with current information, which make them sensitive to small and moderate shifts in the process parameters of interest. The idea behind the CUSUM control chart is that the sum of deviations from the target (usually the process mean) is zero if the process is in control. In case the process has deviated from its target value, the sum of deviations would increase or decrease substantially depending on the direction of shift. An alternative to the CUSUM chart is the EWMA control chart, which assigns exponentially decreasing weights to the observations (i.e. the weights decrease for the previous observations). If the value of weighting constant is close to zero, the EWMA control chart becomes more sensitive for small shifts; if the weighting constant approaches to 1, the chart performance becomes better for the large shifts (cf. Crowder7). A commonly used performance measure for control charts is the average run length (ARL), which is the expected number of samples before a shift is detected. The in-control ARL is denoted by ARL0, and the out-of-control ARL is denoted by ARL1. After the introduction of the CUSUM and EWMA charts in the 1950s, many modifications of their control structures have been proposed by different authors to enhance the detection of shifts of these charts. Lucas and Crosier8 introduced the idea of using the fast initial response (FIR) feature with CUSUM charts. The core of this idea is to use some positive value for the initial CUSUM statistics rather than setting them equal to zero. Jiang et al.9 proposed the adaptive CUSUM (ACUSUM) procedure with an

P

a

Department of Statistics, Quaid-i-Azam University Islamabad, Islamabad, Pakistan Department of Mathematics and Statistics, King Fahad University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia *Correspondence to: Nasir Abbas, Department of Statistics, Quaid-i-Azam University Islamabad, Islamabad, Pakistan. † E-mail: [email protected]

b

Qual. Reliab. Engng. Int. 2013, 29 357–367

357

Copyright © 2012 John Wiley & Sons, Ltd.

N. ABBAS ET AL. EWMA-based shift estimator in which the reference value k for the CUSUM chart is adaptively updated by the EWMA estimate. Riaz et al.3 proposed the use of two run rules schemes with the CUSUM charts and proved the superiority of their proposed schemes over the classic CUSUM charts. Similarly, on the EWMA side, Lucas and Saccucci10 proposed the use of FIR feature with EWMA charts. Capizzi and Masarotto11 proposed an adaptive EWMA control chart where the weights are assigned to past information using a suitable function of current error. Abbas et al.4 applied two runs rules schemes on the time-varying EWMA control chart and showed the superiority of their schemes over the classical EWMA charts. Many other modifications of CUSUM and EWMA charts are also available in the literature. In this article, we proposed a new control chart for monitoring the mean of the process called progressive mean (PM) control chart by considering the case of individual observations. Section 2 provides details regarding the design structure of the proposed PM control chart and also investigates its performance in terms of ARL. Section 3 gives comparisons of the performance of the proposal with those of some existing counterparts. Section 4 illustrates the applications of the proposed chart using numerical example. Section 5 concludes the findings of this article.

2. The proposed PM control chart Let X be a quality characteristic of which the mean will be monitored. We used individual observations from a normal distribution. If Xi, i = 1, 2, 3. . ., is the sequence of independent and identically distributed observations from the process under investigation, then PM is defined as the cumulative average over time. Mathematically, we may define the PM statistic as Pi j¼1 Xj PMi ¼ (1) i Hence, PM is a cumulative average. The difference between PM and moving average is that in moving average we have a fixed moving interval such that we excluded the most previous observation before adding next, but in PM we do not exclude previous but keep on including the next observation. We can say that moving average is a special case of progressive average. For more details on moving average control charts (and geometric moving average control charts), see Roberts12 and Sheu and Lin.13 s2

PMi is an unbiased estimator of population mean m0, and its variance is given by io , where m0 and s2o are the mean and variance, respectively, of an in-control process. According to the typical three sigma control limits, the control charting structure based on the PM statistic and its variance may be defined as s0 LCLi ¼ m0 3 pffi ; i

CL ¼ m0 ;

s0 UCLi ¼ m0 þ 3 pffi i

(2)

where all the parameters used in Equation (2) are as explained earlier. It is obvious that the control limits given in Equation (2) are time varying and exploit the past and present information using equal weights. Note that the design structure of the proposed chart is relatively simple and easy to execute compared with the CUSUM and EWMA control charts. A problem with the control structure in Equation (2) is that the control limits remain too wide for the larger values of i (wide relative to the plotting statistic), which result into a very small probability of getting an out-of-control signal for larger values of i (like i > 1000). We have solved this issue by imposing a penalty on the control limits such that the control limits are a bit narrower for the larger values of i. The penalized limits for the proposed PM chart are given by s0 c s0 c LCLi ¼ m0 3 pffi ; CL ¼ m0 ; UCLi ¼ m0 þ 3 pffi (3) f ð i Þ f ð iÞ i i where f(i) is an arbitrary function of i and c is a constant that is used to control the run lengths (RLs). To evaluate the performance of the proposed control chart, we have used ARL as performance measure. RL properties of a control structure can be obtained through Monte Carlo simulations. We used d to represent the state of control for the process (i.e. d = 0 refers to an in-control situation and d 6¼ 0 to an out-of-control process situation). Wherever needed in this study, we have performed 10,000 simulation runs to evaluate different RL properties. In Equation (3), we have used different possibilities of f(i) and have searched for a suitable constant c for each possibility to fix the in-control process properties in terms of ARL0 and optimum out-of-control RL properties in terms of ARL1. We identified i0.20 as the most suitable choice of f(i) in terms of optimizing the RL properties. For this optimum choice of f(i) (i.e. i0.20), we have worked out the values of constants c by fixing ARL0. These constants for some commonly used ARL0 values are provided in Table I. For these constants and ARL0 values given in Table I , we have carried out an RL study, and the resulting properties in terms of ARL0 and ARL1 are provided in Table II. Following Palm,14 Shmueli and Cohen,15 Antzoulakos and Rakitzis,16 Riaz et al.3 and Abbas et al.,4 we have also reported the standard deviations and percentile points of the RLs to have a clear idea about distribution of RL. The standard

Table I. Values of the control limit constant c for different choices of ARL0 for the proposed PM control chart ARL0 168 200 370 400

358

c

1.000


1.043

1.187

1.213

500 1.267


N. ABBAS ET AL. deviation of RL (SDRL) and the percentiles of RLs (Qi, for i = 10, 25, 50, 75 and 90) are provided in Tables III and IV, respectively (for the same constants and ARL0 values as given in Table I). Similar results for other values of ARL0 and d can also be obtained easily. The standard errors for the results of Tables 1–4 remain less than 1.3%. We conclude from Tables 1–4 the following: 1. The proposed PM control chart is really good in detecting small and moderate shifts and is still good in detecting large shifts (cf. Table II). 2. For a fixed value of d, the SDRL values for the proposed chart increase with an increase in ARL0, whereas for a fixed value of ARL0, it decreases with an increase in d (cf. Table III). 3. The distribution of RL of the proposed chart is positively skewed (cf. Table IV). 4. ARL1 decreases rapidly with an increase in d (cf. Tables I and II). Table II. ARL values for the proposed chart for different shifts d

Prefixed ARL0 500 400 370 200 168

0

0.25

0.5

0.75

1

1.5

2

3

4

5

498.14 400.938 369 200.82 170.33

47.2353 44.474 17.452 34.7366 33.0695

19.032 17.872 10.166 14.6914 12.1479

11.1564 10.313 10.166 8.6191 8.1779

7.5504 7.200 7.0941 5.9814 5.659

4.5204 4.261 4.184 3.5996 3.4086

3.1455 2.985 2.931 2.5415 2.4437

1.9803 1.877 1.721 1.6124 1.5555

1.4248 1.360 1.236 1.1981 1.1598

1.1186 1.0847 1.043 1.0338 1.0263

Table III. SDRL values for the proposed chart for different shifts d ARL0

0

0.25

0.5

0.75

1

1.5

2

3

4

5

500 400 200 168

987.244 738.2676 394.5643 245.41

37.913 35.877 30.743 30.14

11.958 11.543 9.955 9.782

6.0564 5.8134 5.257 4.987

3.631 3.592 3.227 3.095

1.929 1.841 1.659 1.636

1.187 1.149 1.068 1.0379

0.671 0.649 0.612 0.594

0.5169 0.490 0.399 0.366

0.3197 0.2817 0.174 0.150

Table IV. Percentiles (Qj ) values for the proposed chart for different shifts d Percentile

0

0.25

0.5

0.75

1

1.5

2

3

4

5

500

P10 P25 P50 P75 P90 P10 P25 P50 P75 P90 P10 P25 P50 P75 P90 P10 P25 P50 P75 P90

22 59.75 179 517 1264.1 19.9 50 150 429.25 1028.1 11 27 73 215 504 9 22 64 182 424

11 20 38 64 97 10 19 34 60 92 7 13 26 45 77 6 12 24 44 71

7 10 16 25 35 6 9 15 23 33 4 7 12 19 28 4 7 11 18 27

5 7 10 14 19 4 6 9 13 18 3 5 7 11 16 3 4 7 11 15

4 5 7 10 12 3 5 7 9 12 2 4 5 8 10 2 3 5 7 10

2 3 4 6 7 2 3 4 5 7 2 2 3 4 6 2 2 3 4 6

2 2 3 4 5 2 2 3 4 4 1 2 2 3 4 1 2 2 3 4

1 2 2 2 3 1 1 2 2 3 1 1 2 2 2 1 1 2 2 2

1 1 1 2 2 1 1 1 2 2 1 1 1 1 2 1 1 1 1 2

1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

400

200

168



359

ARL0

N. ABBAS ET AL. Note that if we apply the same set up with samples sizes n > 1 instead of n = 1, the results will be the same with the obvious adjustments in the control limits in Equation (3).

3. Comparisons To detect small and moderate shifts, EWMA and CUSUM charts and some of their modifications are available. We have introduced in this article a rather simple alternative to these charts, namely, the PM control chart, and in this section, we compare the performance of our proposed chart with some of its counterparts in terms of ARL. We compared the performance of the proposed chart with the classical EWMA, the classical CUSUM, the FIR CUSUM, the FIR EWMA, the runs rules–based CUSUM, the runs rules–based EWMA and the adaptive EWMA. We used for the ARL0 the values 168, 200, 370, 400 and 500 so that valid comparisons with each chart can be made. In the following sections, we present one by one the comparison of the proposed chart with its competitor. 3.1. Proposed versus classical CUSUM ARL values for the classical CUSUM are given by Page [5] and can be found in Table V. Comparison of the classical CUSUM with the proposed PM chart clearly shows that the proposed chart almost outperforms the classical CUSUM for all the values of d (cf. Table II versus Table V). 3.2. Proposed versus the classical EWMA Lucas and Saccucci10 did a complete ARL study of the classical EWMA chart in their article, and we have reproduced some of their results in Table VI. Comparison of Table II with Table VI shows the uniform superiority of the proposed PM chart over the classical EWMA chart. 3.3. Proposed versus FIR CUSUM and FIR EWMA The idea behind the FIR feature is to give a head start to the initial value of the plotting statistic. ARL values for the FIR CUSUM presented by Lucas and Crosier8 are given in Table VII with 25% and 50% head start (where Co represents the quantity of head start). The ARLs for the FIR EWMA from Lucas and Saccucci10 are also given in Table VIII with 25% and 50% head start values. The comparison of FIR CUSUM and the proposed PM chart indicate that the performance of proposed control chart is better for small and moderate shifts even when the ARL0 for the FIR CUSUM is not fixed at 168 (cf. Table II versus Table VII). We may conclude that the proposed chart performs really well for small shifts, but for the larger shifts the performance of the FIR CUSUM with larger values of Co becomes better than the proposed chart. The same is the case for the FIR EWMA, that is, for 25% head start, the performance of the proposed chart is superior to the FIR EWMA for every choice of l, but as

Table V. ARL values for the classical CUSUM scheme with k = 0.5 d 0 0.25 0.5 0.75 h=4 h=5

168 465

74.2 139

26.6 38.0

13.3 17.0

1

1.5

2

2.5

3

8.38 10.4

4.75 5.75

3.34 4.01

2.62 3.11

2.19 2.57

Table VI. ARL values for the classical EWMA scheme at ARL0 = 500 d 0 0.25 0.5 0.75 1 1.5 2

l ¼ 0:1 L ¼ 2:814 500 106 31.3 15.9 10.3 6.09 4.36

l ¼ 0:25 L ¼ 2:998

l ¼ 0:5 L ¼ 3:071

l ¼ 0:75 L ¼ 3:087

500 170 48.2 20.1 11.1 5.46 3.61

500 255 88.8 35.9 17.5 6.53 3.63

500 321 140 62.5 30.6 9.90 4.54

360

Table VII. ARLs for the FIR CUSUM scheme with k = 0.5 d 0 0.25 0.5

0.75

1

1.5

2

h = 4, Co = 1 h = 4, Co = 2

11.6 8.97

7.04 5.29

3.85 2.86

2.7 2.01

163 149


71.1 62.7

24.4 20.1


N. ABBAS ET AL. we increase the head start to 50%, the performance of the FIR EWMA becomes better than the proposed chart for the larger shifts with l = 0.1 (cf. Table II versus Table VIII).

3.4. Proposed versus runs rules–based CUSUM and EWMA Riaz et al.3 and Abbas et al.4 introduced the runs rules–based CUSUM and the runs rules–based EWMA, respectively. The ARLs for the runs rules–based CUSUM and EWMA are provided in Tables IX and X, respectively, where in Table IX WL and AL represent the warning and action limits, respectively. The comparison of the runs rules–based CUSUM with the proposed chart shows that proposed chart performs better than the runs rules–based CUSUM uniformly for both the schemes (cf. Table II versus Table IX). Similarly, the comparison of the runs rules–based EWMA with the proposed chart also reveals that the proposed chart is superior to the runs rules–based EWMA for most of the values of d (cf. Table II versus Table X).

Table VIII. ARL values for the FIR EWMA scheme d

% Head start

l ¼ 0:1 L ¼ 2:814

l ¼ 0:25 L ¼ 2:998

l ¼ 0:5 L ¼ 3:071

l ¼ 0:75 L ¼ 3:087

0

25 50 25 50 25 50 25 50

487 468 28.3 24.2 8.75 6.87 3.57 2.72

491 483 46.5 43.6 10.1 8.79 3.11 2.5

497 487 87.8 86.1 16.9 15.9 3.29 2.87

498 496 140 139 30.2 29.7 4.33 4.09

0.5 1 2

Table IX. ARL values for the runs rules–based CUSUM at ARL0 = 500 d Limits

WL

AL

Scheme 1

4.8 4.7 4.6 4.49 4.8 4.7 4.6 4.49

5.12 5.2 5.39 1 5.11 5.19 5.5 1

Scheme 2

0.25 141.111 150.372 145.1886 146.564 139.7048 142.1588 145.7868 149.0352

0.5

0.75

1

1.5

2

38.5986 38.5942 38.1954 38.4918 38.8562 37.9752 38.3342 39.9042

17.3916 17.5288 17.486 17.7254 17.4586 17.2674 17.3938 17.5682

10.5176 10.5994 10.5584 10.8566 10.5056 10.5826 10.734 10.9658

5.9052 5.8984 6.0066 6.3326 5.8222 5.8716 6.0526 6.4506

4.0574 4.14 4.2374 4.6894 4.0776 4.1036 4.2726 4.873

Table X. ARL values for the runs rules–based EWMA at ARL0 = 500 2/2 scheme

d

l ¼ 0:1 Ls ¼ 2:556

l ¼ 0:25 Ls ¼ 2:554

l ¼ 0:1 Ls ¼ 2:3

l ¼ 0:25 Ls ¼ 2:345

501.7558 103.3109 29.5748 14.3216 8.9561 4.9197 3.4498

505.5284 169.1349 47.0105 19.2776 10.5964 5.2578 3.5527

502.883 66.6864 21.4251 11.7427 7.5539 4.4676 3.4534

499.6153 97.0108 31.2023 14.4295 8.6761 4.7066 3.549



361

0 0.25 0.5 0.75 1 1.5 2

2/3 scheme

N. ABBAS ET AL. 3.5. Proposed versus adaptive EWMA The adaptive EWMA of Capizzi and Masarotto11 is designed so that it performs better for small and large shifts at the same time by giving weights to past information using a suitable function of the current error. Three functions of error represented by fhu(.), fbs(.) and fcub(.) are used in their article. ARL values for the adaptive EWMA with these three functions of errors are given in Table XI, where the value of d is targeted between 0.25 and 4. The comparison of the proposed chart with adaptive EWMA shows that the proposed chart also outperforms the adaptive EWMA for all the values of d (cf. Table II versus Table XI). 3.6. Proposed versus ACUSUM Jiang et al.11 proposed the use of an ACUSUM with EWMA-based shift estimator. They used the concept of updating the reference value of CUSUM using an EWMA estimator where the parameters for the ACUSUM are dþmin, l, g and h. ARLs for the ACUSUM with dþmin ¼ 0:5 and l = 0.2 are given in Table XII. Comparison of Tables II and XII shows that the proposed PM chart is uniformly superior compared with the ACUSUM with EWMA-based shift estimator. To summarize the results, we have made some ARL curves of the proposed PM chart against its existing counterparts. These are given in Figures 1–3. ARL curves for the proposed chart, the classical EWMA (with l = 0.1), the adaptive EWMA (with error function fhu(.)),

Table XI. ARL values for the adaptive EWMA at ARL0 = 500 Error function fhu(.) fbs(.) fcub(.)

d 0.25

0.5

0.75

1

1.5

2

3

4

98.51 135.01 97.03

40.94 42.72 41.54

25.04 21.99 25.68

17.59 13.91 18.16

10.11 7.12 10.52

6.08 4.25 6.36

2.29 2.01 2.35

1.26 1.28 1.27

Table XII. ARL values for adaptive CUSUM with dþmin ¼ 0:5 and l = 0.2 at ARL0 = 400 d 0 0.25 0.5 0.75 1 1.5 2 3 4 5

g ¼ 1:5 h ¼ 6:056

g¼2 h ¼ 5:105

g ¼ 2:5 h ¼ 4:633

g¼3 h ¼ 4:43

g¼4 h ¼ 4:348

g¼1 h ¼ 4:327

399.68 67.19 26.73 15.5 10.47 5.9 3.86 2.1 1.4 1.1

400.22 67.34 25.66 14.66 9.91 5.68 3.76 2.05 1.36 1.08

399.2 65.51 24.72 14.13 9.63 5.65 3.84 2.13 1.39 1.09

399.33 64.08 24.17 13.85 9.48 5.66 3.93 2.29 1.49 1.12

399.97 63.34 23.91 13.72 9.42 5.67 4.01 2.47 1.71 1.26

399.9 63.33 23.87 13.68 9.39 5.66 4.01 2.54 1.88 1.45

Proposed

EWMA

Adaptive EWMA

RR CUSUM II

RR EWMA II

150

120

ARL

90

60

30

0

0.25

0.5

0.75

1

1.5

2

δ

362

Figure 1. ARL curves for proposed chart, classical EWMA, adaptive EWMA, runs rules–based CUSUM and runs rules–based EWMA at ARL0 = 500



N. ABBAS ET AL. the runs rules–based CUSUM scheme II (with WL = 4.8 and AL = 5.11) and the runs rules–based EWMA scheme 2/3 (with l = 0.1) are presented in Figure 1. In Figure 2, comparison of the proposed chart and the ACUSUM are provided, whereas Figure 3 contains the ARL curves for the proposed chart, the classical CUSUM and the FIR CUSUM (with head start Co = 1 and Co = 2). Figures 1–3 clearly show the best performance of the proposed chart against the other methods.

4. Illustrative example In this section, we demonstrate the application of the proposed PM chart. Also, the classical EWMA and the classical CUSUM charts are included in the example to validate the superiority of the proposal. For this purpose, we have generated two data sets of 40 and 30 observations, respectively, such that in data set 1 the first 20 observations are generated from N(0, 1)(i.e. the in-control situation) and the second set of the 20 observations from N(0.5, 1) (i.e. out-of-control situation having a shift of 0.5s (small shift)). Similarly, in data set 2, the first 20 observations are generated from N(0, 1) and the second set of 10 observations from N(1.5, 1) (i.e. out-of-control situation having a shift of 1.5d(moderate shift)). PM statistics for the proposed chart, EWMA statistics with l = 0.25 and CUSUM statistics with k = 0.5 are calculated. To fix the ARL0 at 500, we have used c = 1.267 for the PM chart, L = 2.998 for the classical EWMA and h = 5.09 for the classical CUSUM. The graphical displays of the proposed PM, EWMA and CUSUM charts are presented in the Figures 4–6 for data set 1 and in Figures 7–9 for data set 2, respectively. From Figure 4, we can see that the proposed chart gives out-of-control signals at samples 35, 36, 37, 38, 39 and 40, thus giving a total of six out-of-control signals. Figure 5 shows that the classical EWMA control chart gives one out-of-control signal at sample 38, and Figure 6 depicts that the classical CUSUM control chart gives out-of-control signal at samples 38, 39 and 40, thus giving three

Proposed

Adaptive CUSUM ( =4, h=4.348)

Adaptive CUSUM ( =2, h=5.105)

80

ARL

60

40

20

0

0.25

0.5

0.75

1

1.5

2

3

4

δ Figure 2. ARL curves for proposed chart and adaptive CUSUM at ARL0 = 400

Proposed

CUSUM

FIR CUSUM (Co=1)

FIR CUSUM (Co=2)

80

ARL

60

40

20

0

0.25

0.5

0.75

1

1.5

2

δ

363

Figure 3. ARL curves for proposed chart, classical CUSUM and FIR CUSUM at ARL0 = 168



N. ABBAS ET AL. 4

PM statistics

UCL

LCL

3 2 1 0 -1 -2 -3 -4 1

3

5

7

9

11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Sample Number Figure 4. Graphical display of the proposed PM chart for data set 1

EWMA statistic

1.5

UCL

LCL

1 0.5 0 -0.5 -1 -1.5 1

3

5

7

9

11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Sample Number Figure 5. Graphical display of the classical EWMA chart for data set 1

CUSUM statistic

h

6 5 4 3 2 1 0 1

3

5

7

9

11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Sample Number Figure 6. Graphical display of the classical CUSUM for data set 1

364

signals. An upward shift occurred after sample 20, which is detected by the proposed chart more quickly than the EWMA and the CUSUM. It illustrates the ability of the proposed chart to quickly detect small shifts in the process. The situation is not much different in data set 2, where the PM chart detects the shift at samples 23, 24, 25, 26, 27, 28, 29 and 30 (cf. Figure 7). Classical EWMA and CUSUM detect the shift at samples 27, 28, 29 and 30 (cf. Figures 8 and 9). Copyright © 2012 John Wiley & Sons, Ltd.


N. ABBAS ET AL. 4

PM statistic

UCL

LCL

3 2 1 0 -1 -2 -3 -4 1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

23

25

27

29

Sample Number Figure 7. Graphical display of the proposed PM chart for Data set 2

2

EWMA statistic

UCL

LCL

1.5 1 0.5 0 -0.5 -1 -1.5 1

3

5

7

9

11

13

15

17

19

21

Sample Number Figure 8. Graphical display of the classical EWMA for data set 2

CUSUM statistic

12

h

10 8 6 4 2 0 1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

Sample Number

Figure 9. Graphical display of the classical CUSUM for data set 2



365

In both small and moderate shifts, we see that the proposed chart detects the shift more quickly than the others and the number of signals given by the proposed chart is also greater than the classic ones. These outcomes are exactly in accordance with the findings of Section 3.

N. ABBAS ET AL.

5. Summary and conclusions Natural variations are a permanent part of any process, and by monitoring and timely identifying sources of unnatural variations, the quality of the output of a process can be improved. Control charts are the most important tool to monitor a process. To keep an eye on the location of the process, two main types of charts, namely, Shewhart-type control charts and memory control charts, like EWMA and CUSUM, are available. The former are recommended for larger shifts (3 ≤ d ≤ 5) whereas the latter are good at detecting small and moderate shifts (e.g. 0.25 ≤ d ≤ 1.5). In this article, we have proposed another type of memory control chart, the so-called PM control chart. The performance of the new chart is evaluated in terms of ARLs, and we have compared the performance of proposed control chart with different existing memory control charts. Comparisons revealed that the newly proposed chart performs very good and outperforms its competitors for small and moderate shifts but also show good performance for the large shifts.

Acknowledgements The authors are thankful to the reviewer(s) and editor for the useful comments to improve the initial version of the article. The authors also owe a great debt of gratitude to Dr Ronald J. M. M Does, General Manager, Institute for Business and Industrial Statistics, University of Van Amsterdam, the Netherlands, for his invaluable suggestions and guidance for the revision and improvement of this manuscript. Finally, the author Muhammad Riaz is indebted to King Fahd University of Petroleum and Minerals Dhahran Saudi Arabia for providing excellent research facilities through project SB111008.

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Shewhart WA. Economic Control of Quality of Manufactured Product, New York. Reprinted by ASQC: Milwaukee, 1980, 1931. Montgomery DC. Introduction to Statistical Quality Control (6th edn). John Wiley & Sons: New York, 2009. Riaz M, Abbas N, Does RJMM. Improving the Performance of CUSUM Charts. Quality and Reliability Engineering International 2011; 27(4):415–424. Abbas N, Riaz M, Does RJMM. Enhancing the Performance of EWMA Charts. Quality and Reliability Engineering International 2011; 27(6):821–833. Page ES. Continuous Inspection Schemes. Biometrika 1954; 41:100–115. Roberts SW. Control Chart Tests Based on Geometric Moving Averages. Technometrics 1959; 1:239–250. Crowder SV. A Simple Method for Studying Run-Length Distribution of Exponentially Weighted Moving Average Charts. Technometrics 1987; 29:401–407. Lucas JM, Crosier RB. Fast Initial Response for CUSUM Quality-Control Scheme. Technometrics 1982; 24:199–205. Jiang W, Shu L, Apley DW. Adaptive CUSUM procedures with EWMA-based shift estimators. IIE Transactions 2008; 40(10):992–1003. Lucas JM, Saccucci MS. Exponentially Weighted Moving Average Control Schemes: Properties and Enhancements. Technometrics 1990; 32:1–12. Capizzi G, Masarotto G. An Adaptive Exponentially Weighted Moving Average Control Chart. Technometrics 2003; 45:199–207. Roberts SW. A Comparison of Some Control Chart Procedures. Technometrics 1966; 8(3):411–430. Sheu SH, Lin TC. The Generally Weighted Moving Average Control Chart for Detecting Small Shifts in the Process Mean. Quality Engineering 2003; 16(2):209–231. Palm AC. Tables of Run Length Percentiles for Determining the Sensitivity of Shewhart Control Charts for Averages with Supplementary Runs Rules. Journal of Quality Technology 1990; 22:289–298. Shmueli G, Cohen A. Run-Length Distribution for Control Charts with Runs and Scans Rules. Communications in Statistics: Theory and Methods 2003; 32:475–495. Antzoulakos DL, Rakitzis AC. The Modified r out of m Control Chart. Communication in Statistics: Simulations and Computations 2008; 37(2):396–408.

Authors' biographies Nasir Abbas got his M.Sc. in Statistics from the Department of Statistics Quaid-i-Azam University Islamabad Pakistan in 2009; M.Phil in Statistics from the Department of statistics Quaid-i-Azam University Islamabad Pakistan in 2011 and currently he is pursuing his Ph.D. in Statistics from the Institute of Business and Industrial Statistics University of Amsterdam The Netherlands. Additionally, he is also serving as Assistant Census Commissioner in Pakistan Bureau of Statistics. His current research interests include Quality Control particularly control charting methodologies. Raja Fawad Zafar did his B.Sc. in Statistics from the Punjab University in 2008 and M.Sc. in Statistics from the Department of Statistics Quaid-i-Azam University Islamabad in 2011. Currently he is pursuing his M.Phil in statistics from the Department of Statistics Quaid-iAzam University Islamabad. His research interest areas are Statistical Quality Control and Time Series Analysis.

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Muhammad Riaz earned his B.Sc. with Statistics and Mathematics as major subjects from the Government Gordon College Rawalpindi, University of the Punjab Lahore Pakistan in 1998; M.Sc. in Statistics from the Department of Mathematics and Statistics Quaid-i-Azam University Islamabad Pakistan in 2001; M.Phil in Statistics from the Department of Mathematics and statistics, Allama Iqbal Open University Islamabad Pakistan in 2006 and Ph.D. in Statistics from the Institute if Business and Industrial Statistics University of Amsterdam The Netherlands in 2008. He served as a Statistical Officer in the Ministry of Food, Agriculture and Livestock, Islamabad, Pakistan during 2002–2003, as a Staff Demographer in the Pakistan Institute of Development Economics, Islamabad, Pakistan during 2003–2004, as a Lecturer in the Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan during 2004–2007, as an Assistant Professor in the Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan during 2007–2010. He is serving as an Assistant Professor



N. ABBAS ET AL. in the Department of Mathematics and Statistics, King Fahad University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia from 2010–Present. His current research interests include Statistical Process Control, Non-Parametric techniques and Experimental Designs. Zawar Hussain did his M.Sc. in Statistics from the Department of Statistics Islamia University Bahawalpur in 1998 and his Ph.D in Statistics from the Department of Statistics Quaid-i-Azam University Islamabad Pakistan in 2009. At present he is serving as an Assistant Professor of Statistics at the Department of Statistics Quaid-i-Azam University Islamabad Pakistan. His research interest areas are: Quality Control, Randomized Response Models, Bayesian Inference and Sampling Techniques.

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