The Efficiency of the VSI Exponentially Weighted Moving Average Median Control Chart

Transcription

1 The Efficiency of the VSI Exponentially Weighted Moving Average Median Control Chart Kim Phuc Tran, Philippe Castagliola, Thi-Hien Nguyen, Anne Cuzol To cite this version: Kim Phuc Tran, Philippe Castagliola, Thi-Hien Nguyen, Anne Cuzol The Efficiency of the VSI Exponentially Weighted Moving Average Median Control Chart 24th ISSAT International Conference on Reliability and Quality in Design, Aug 2018, Toronto, Canada <hal > HAL Id: hal Submitted on 10 Sep 2018 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not The documents may come from teaching and research institutions in France or abroad, or from public or private research centers L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés

2 The Efficiency of the VSI Exponentially Weighted Moving Average Median Control Chart Kim Phuc Tran 1, Philippe Castagliola 2, Thi Hien Nguyen 1, and Anne Cuzol 1 1 Laboratoire de Mathématiques de Bretagne Atlantique, UMR CNRS 6205, Université de Bretagne-Sud, Vannes, France 2 Université de Nantes & LS2N UMR CNRS 6004, Nantes, France May 14, 2018 Abstract In the literature, median type control charts have been widely investigated as easy and efficient means to monitor the process mean when observations are from a normal distribution In this work, a Variable Sampling Interval (VSI) Exponentially Weighted Moving Average (EWMA) median control chart is proposed and studied A Markov chain method is used to obtain optimal designs and evaluate the statistical performance of the proposed chart Furthermore, practical guidelines and comparisons with the basic EWMA median control chart are provided Results show that the proposed chart is considerably more efficient than the basic EWMA median control chart Finally, the implementation of the proposed chart is illustrated with an example in the food production process Keywords: EWMA, VSI, Median, Control chart, Order statistics 1 Introduction Statistical Process Control (SPC) is a method of quality control which uses statistical methods in achieving process stability and improving capability through the reduction of variability, see Montgomery [1] It s well known that control charts are the fundamental tool for SPC applications There are numerous types of control charts, the most common ones are the Shewhart control charts, the cumulative sum (CUSUM) control charts and the exponentially weighted moving average (EWMA) control charts The EWMA control charts have a built in mechanism for incorporating information from all previous subgroups by means of weights decreasing geometrically with the sample mean age Thus EWMA type control charts are very effective for the detection of small or moderate process shifts, see Tran et al [2] Their properties and design stategies have been thoroughly investigated by many authors For further details see, for instance, Robinson and Ho [3], kim-phuctran@univ-ubsfr (corresponding author) Hunter [4], Crowder [5], Lucas and Saccucci [6], Tran et al [2] to name a few In recent years, many researchers have focused on developing advanced control charts with various applications in manufacturing and service processes, for example, see Castagliola and Figueiredo [7], Huang [8], Da Costa Quinino et al [9], Tran et al [10], Castagliola et al [11], Tran [12], Tran et al [13] and Tran [14] Among these control charts, median ( X) type charts have been widely investigated as easy and efficient means to monitor the mean The main advantages of median type charts are that they are simpler than mean ( X) charts and that they are robust against outliers, contamination or small deviations from normality, see Castagliola et al [11] In the SPC literature, the EWMA median chart was introduced by Castagliola [15] (EWMA- X) with fast detection of assignable causes Then, a generally weighted moving average median (GWMA- X) control chart has been studied by Sheu and Yang [16] as a continuation to improve the statistical performance of median type control charts When the parameters are estimated, Castagliola and Figueiredo [7] and Castagliola et al [11] developed a Shewhart median chart and a EWMA- X chart, respectively, with estimated control limits to monitor the mean value of a normal process Very recently, Lin et al [17] investigated the performances of the EWMA- X control chart under several distributions As a result, the EWMA- X is always more efficient than the EWMA- X chart in detecting shifts in the process mean if the data follow a heavy-tailed distribution Finally, Tran [18] proposed and studied the Run Rules Shewhart median control charts (RR r,s X charts) It is known that, the EWMA- X control chart suggested by Castagliola [15] is a Fixed Sampling Interval (FSI) control chart By definition, an adaptive control chart involves varying at least one of the chart s parameters, such as the sampling interval or the sample size Variable Sampling Interval (VSI)

3 control charts are adaptive control charts where the sampling intervals vary as a function of what is observed from the process The VSI control charts are demonstrated to detect process changes faster than FSI control charts The idea is that the time interval until the next sample should be short, if the position of the last plotted control statistic indicates a possible out-ofcontrol situation; and long, if there is no indication of a change Most work on developing VSI control charts has been done for the problem of monitoring the mean of the process (see Reynolds [19], Reynolds et al [20] and Castagliola et al [21]) In this paper, we propose a VSI EWMA- X control chart as a logical extension of the control chart developed by Castagliola [15] The goal of this paper is to show how the VSI behaves with respect to the basic EWMA median control chart The rest of this paper proceeds as follows: in Section 2, a brief review of the FSI EWMA- X control chart is provided; Section 3 provides a VSI version of the FSI EWMA- X control chart; in Section 4, the run length performances of proposed chart are defined by using the Markov Chain-based approach; in Section 5, the computational results and the tables reporting the optimal design parameters of the VSI EWMA- X chart are presented Section 6 presents an illustrative example and, finally, some concluding remarks and recommendations are made in Section 7 2 The FSI EWMA- X control chart Let us assume that, at each sampling period i = 1,2,, we collect a sample of n independent random variables {X i,1,,x i,n } We assume that each X i, j follows a normal distribution N(µ 0 + δσ 0,σ 0 ), j = 1,,n, µ 0 is the in-control mean value, σ 0 is the in-control standard deviation and δ is the magnitude of the standardized mean shift If δ = 0 the process is in-control and, when δ 0, the process is out-of-control Let X i be the sample median of subgroup i, ie X i = X i,((n+1)/2) X i,(n/2) + X i,(n/2+1) 2 if n is odd if n is even where {X i,(1),x i,(2),,x i,(n) } is the ordered i-th subgroup In the rest of this paper, without loss of generality, we assume that the sample size n is an odd value Let Z 1,Z 2, be the EWMA sequence obtained from X 1, X 2,, ie for i {1,2,}, (1) Z i = (1 )Z i 1 + X i, (2) where Z 0 = µ 0 and (0,1] is a smoothing constant If the in-control mean value µ 0 and the standard deviation σ 0 are assumed known, the control limits of the EWMA- X chart for the median are simply equal to LCL = µ 0 K 2 σ 0, (3) UCL = µ 0 + K 2 σ 0, (4) where K > 0 is a constant that depends on n and on the desired in-control performance 3 Implementation of the VSI EWMA- X control chart In this section, a VSI version of the FSI EWMA- X control chart described in the previous section is presented (denoted as VSI EWMA- X) The control statistic Z i for the VSI EWMA- X control chart is given by (2) The upper (UCL) and lower (LCL) control limits of the VSI EWMA- X control chart can be easily calculated as: LCL = µ 0 K 2 σ 0, (5) UCL = µ 0 + K 2 σ 0, (6) where K 0 is a constant influencing the width of the control interval For the FSI control chart, the sampling interval is a fixed value h 0 As for the VSI control chart, the sampling interval depends on the current value of Z i A longer sampling interval h L is used when the control statistic falls within region R L = [LWL,UWL] defined as: LWL = µ 0 W 2 σ 0, (7) UWL = µ 0 +W 2 σ 0, (8) where W is the warning limit coefficient of the VSI EWMA X control chart that determines the proportion of times that the control statistic falls within the long and short sampling regions On the other hand, the short sampling interval h S is used when the control statistic falls within the region R S = [LCL,LWL) (UWL,UCL] The process is considered out-ofcontrol and action should be taken whenever Z i falls outside the range of the control limits [LCL,UCL] In order to evaluate the ARL and SDRL of the VSI EWMA- X control chart, we follow the discrete Markov chain approach originally proposed by Brook and Evans [22] In Appendix, the discrete Markov chain approach for VSI EWMA- X control chart is provided 4 Optimal design of the VSI EWMA- X control chart In the literature, the Average Run Length (ARL), defined as the average number of samples before the control chart signals an out-of-control condition or issues a false alarm, and the Average Time to Signal (AT S), which is the expected value of the time between the occurrence of a special cause and a signal from the chart are used as the performance measures of control charts, see Castagliola et al [21] It is well known that, when

4 the process is in-control, it is better to have a large AT S, since in this operating condition a signal represents a false alarm (in this case, the AT S will be denoted as AT S 0 ) On the other hand, after the parameter of the process under control has shifted, it is preferable to have an AT S that is as small as possible (in this case, the AT S will be denoted as AT S 1 ) For a FSI model, the AT S is a multiple of the ARL since the sampling interval h F is fixed Thus, in this case we have the following expression: AT S FSI = h F ARL FSI (9) For a VSI model, the AT S is defined as: AT S VSI = E(h) ARL VSI (10) where E(h) is the expected sampling interval value According to Castagliola et al [21], for VSI type control charts, we need to define them with the same in-control ARL = ARL 0 and the same in-control average sampling interval E 0 (h) For FSI-type control charts, the sampling interval is set equal to h S = h L = h F = 1 time units Then, the in-control expected sampling interval of the VSI chart is set equal to E 0 (h) = 1 time unit to ensure AT S 0 = ARL 0 time unit for both FSI and VSI type control charts The value of h S represents the shortest feasible time interval between subgroups from the process, see Castagliola et al [21] for more details Then, in this paper we will consider the impact on the expected time until detection, using small but non-zero values of h S The design procedure of VSI EWMA- X control chart is implemented by finding out the optimal combination of parameters, K and h L which minimize the out-of-control AT S for predefined values of δ, W, h S, n and AT S 0, ie, the optimization scheme of the VSI EWMA- X consists in finding the optimal parameters, K and h L such that (,K,h L) = argminat S(n,,K,W,h L,h S,δ) (11) (,K,h L ) subject to the constraint E 0 (h) = 1, AT S(n,,K,W,h L,h S,δ = 0) = AT S 0 (12) Similar to Tran and Tran [23], the choice of the optimal combination of parameters generally entails two steps: 1 Find the potential combinations (,K,h L ) such that AT S = AT S 0 and E 0 (h) = 1 2 Choose, among these potential combinations (,K,h L ), the one (,K,h L ) that allows for the best performance, ie the smallest out-of-control AT S value for a particular shift δ In this study, like in Tran and Tran [23], in order to find these optimal combinations (,K,h L ) we simultaneously use a non-linear equation solver coupled to an optimization algorithm (developed with Scicoslab software) 5 Numerical results Optimal designs were obtained for the FSI and VSI EWMA X control charts, for all combinations of δ [05, 2] and n = {3,5,7,9} The sampling interval h F of the FSI charts has been set equal to 1 time unit The shorter time interval h S can assume the following values: 05 and 01 time units The optimal combinations of design parameters (,K,h L ) have been selected by constraining the in-control AT S at the value AT S 0 = 3704 and the in-control expected sampling interval of the VSI chart is set equal to E 0 (h) = 1 To ensure a fair comparison, the ARL 0 of EWMA- X chart is set as 3704 The optimal combinations of design parameters (,K,h L ) of the VSI EWMA- X control chart are presented in Tables 1-4 Some simple conclusions can be drawn from Tables 1-4: n = 3 h S = 05 δ W = 09 W = 06 W = 03 W = 02 W = (00500,16686) (00500,16686) (00500,16686) (00513,16750) (00514,16750) (108,1395) (124,1359) (181,1338) (254,1347) (468,1359) 02 (00500,16686) (00500,16686) (00500,16686) (00500,16686) (00514,16750) (108,500) (124,473) (181,462) (240,464) (468,486) 03 (00514,16750) (00518,16767) (00500,16686) (00517,16763) (00535,1684) (109,261) (126,246) (181,243) (255,249) (468,269) 05 (00989,18090) (01095,18273) (01073,18234) (01046,18190) (01124,18315) (110,118) (128,112) (189,114) (246,118) (449,138) 07 (01605,18883) (01690,18957) (01563,18844) (01742,19000) (01798,19043) (110,69) (128,67) (187,70) (258,76) (439,94) 10 (02743,19557) (02773,19569) (02759,19563) (02783,19572) (02885,19609) (110,40) (129,39) (191,44) (254,50) (431,68) 15 (04746,20017) (04681,20008) (04685,20008) (04745,20016) (04293,19950) (110,22) (128,23) (189,29) (251,35) (426,52) 20 (06883,20195) (06890,20196) (05499,20100) (05498,20100) (04293,19950) (111,16) (130,17) (189,23) (250,29) (426,47) h S = 01 δ W = 09 W = 06 W = 03 W = 02 W = (00500,16686) (00500,16686) (00500,16686) (00500,16686) (00500,16686) (115,1342) (144,1277) (247,1240) (352,1239) (652,1255) 02 (00500,16686) (00500,16686) (00500,16685) (00500,16686) (00500,16686) (115,448) (144,400) (247,380) (352,385) (652,410) 03 (00515,16752) (00515,16753) (00577,1701) (00502,16693) (00643,17240) (115,221) (146,194) (257,190) (351,198) (753,234) 05 (00987,18090) (01140,18339) (01278,18530) (01325,18589) (01394,18669) (117,93) (150,83) (259,85) (390,96) (719,128) 07 (01737,18995) (01952,19154) (02088,19241) (02142,19273) (02269,19344) (118,53) (150,48) (267,54) 381,65) (702,96) 10 (03086,19676) (03239,19721) (03326,19746) (03415,19769) (03661,19829) (119,30) (152,29) (263,37) (374,48) (690,79) 15 (05211,20072) (05218,20072) (05366,20087) (05498,20100) (04025,19903) (118,17) (150,19) (259,29) (370,40) (688,72) 20 (07043,20203) (05498,20100) (05498,20100) (05498,20100) (04025,19903) (119,13) (150,16) (259,27) (370,38) (688,70) Table 1: Optimal couples (,K ) (first row of each block) and values of (h L,AT S 1 )(second row of each block) of the VSI EWMA- X control chart for n = 3 Given the values of δ, n and W, the value of AT S depends on h S In particular, with smaller values of h S, the value of AT S 1 decreases For example, when δ = 01, n = 3, W = 06 we have AT S 1 = 1359 for h S = 05 and AT S 1 = 1277 for h S = 01, see Table 1 For a defined value of h S, it is obvious that when W decreases the length of the long sampling interval h L increases For example, when δ = 01, n = 3, h S = 05 we have h L = 108 for W = 09 and h L = 468 for W = 01, see Table 1

5 The VSI EWMA- X control chart is directly compared to the FSI EWMA- X control chart, to evaluate the impact of the adaptive feature on the statistical performance of the original static chart As expected, the results in Tables 1-4 clearly indicate that the VSI EWMA- X chart is superior to the FSI EWMA- X control chart For example, when δ = 01, n = 3, W = 06 and h S = 05 we have AT S 1 = 1359 for VSI EWMA- X chart and ARL = 1461 for FSI EWMA- X control chart, see Table 3 in Castagliola [15] n = 5 h S = 05 δ W = 09 W = 06 W = 03 W = 02 W = (00500,13341) (00500,13341) (00500,13341) (00500,13341) (00500,13341) (103,1061) (114,1014) (160,980) (205,977) (360,986) 02 (00500,13341) (00500,13341) (00500,13341) (00500,13341) (00500,13341) (103,367) (114,337) (160,321) (205,323) (360,336) 03 (006134,13705) (00619,13719) (00689,13899) (00710,13951) (00733,14002) (104,196) (115,179) (163,173) (211,176) (353,189) 05 (01290,14828) (01388,14921) (01467,14989) (01483,15002) (01526,15036) (104,88) (116,81) (163,80) (215,84) (383,100) 07 (02175,15423) (02304,15479) (02325,15487) (02264,15462) (02400,15517) (105,51) (117,48) (165,50) (212,54) (376,70) 10 (03773,15869) (03721,15860) (03662,15850) (03684,15854) (03804,15874) (105,30) (117,28) (164,32) (221,37) (371,52) 15 (06405,16119) (06369,16117) (06400,16119) (06450,16121) (06579,16127) (105,17) (117,17) (168,22) (219,27) (367,42) 20 (08517,16182) (08540,16182) (08594,16183) (08634,16184) (08728,16185) (105,12) (117,13) (167,18) (219,23) (366,38) h S = 01 δ W = 09 W = 06 W = 03 W = 02 W = (00500,13341) (00500,13341) (00500,13341) (00500,13341) (00500,13341) (106,1027) (125,943) (208,881) (289,876,,3704) (569,891) 02 (00500,13341 (00516,13402) (00500,13341) (00500,13341) (00500,13341) (106,337) (127,282,,3704) (208,255) (289,258,,3704) (569,282) 03 (00695,13915) (00644,13788) (00791,14124) (00837,14212) (00772,14085) (107,172) (127,142) (211,130) (297,135) (553,158) 05 (01573,15075) (01586,15082) (01543,15050) (01799,15225) (01902,15286) (108,72) (131,60) (212,59) (305,66) (603,94) 07 (02618,15596) (02629,15598) (02784,15647) (02264,15462) (03018,15711) (109,41) (131,35) (216,38) (302,46) (592,74) 10 (04059,15914) (04240,15939) (03768,15868) (04404,15960) (04721,15996) (109,23) (132,21) (214,27) (317,37) (584,64) 15 (06616,16129) (06621,16129) (03768,15868) (07056,16146) (07466,16159) (109,14) (131,15) (214,23) (314,33) (579,59) 20 (08505,16182) (08649,16184) (03768,15868) (09103,16189) (09343,16191) (109,11) (131,13) (214,22) (313,32) (579,58) Table 2: Optimal couples (,K ) (first row of each block) and values of (h L,AT S 1 ) (second row of each block) of the VSI EWMA- X control chart for n = 5 n = 7 h S = 05 δ W = 09 W = 06 W = 03 W = 02 W = (00500,11427) (00500,11427) (00500,11427) (00500,11427) (00500,11427) (101,870) (109,818) (146,775) (188,770) (325,777) 02 (00507,11449) (00500,11427) (00500,11427) (00500,11427) (00505,11443) (101,302) (109,270) (146,253) (188,254) (325,265) 03 (00870,12223) (00767,12060) (00860,12208) (00886,12247) (00913,12286) (102,162) (110,144) (149,136) (191,139) (315,149) 05 (01593,12921) (01723,12998) (01819,13049) (01831,13055) (01873,13076) (102,72) (111,65) (149,63) (194,66) (337,80) 07 (02678,13370) (02845,13412) (02847,13413) (02788,13398) (02912,13428) (102,42) (112,38) (151,39) (191,43) (332,57) 10 (04864,13705) (04590,13681) (04507,13673) (04527,13675) (04646,13686) (103,24) (111,23) (150,26) (198,30) (328,43) 15 (07609,13831) (07604,13831) (07643,13831) (07679,13832) (07766,13834) (102,14) (112,14) (153,18) (197,23) (326,35) 20 (09354,13853) (09377,13853) (09419,13853) (09448,13853) (09513,13854) (102,11) (112,12) (153,16) (197,20) (325,33) h S = 01 δ W = 09 W = 06 W = 03 W = 02 W = (00500,11427) (00500,11427) (00500,11427) (00500,11427) (00500,11427) (102,852) (117,758) (182,681) (259,672) (505,685) 02 (00557,11596) (00500,11427) (00505,11443) (00553,11583) (00599,11702) (103,286) (117,229) (182,197) (257,198) (499,218) 03 (00870,12224) (00927,12306) (00986,12383) (01046,12455) (00996,12395) (103,148) (119,115) (187,101) (261,104) (484,123) 05 (01593,12921) (01946,13111) (02161,13201) (02212,13221) (02330,13264) (104,62) (120,49) (194,46) (267,51) (522,76) 07 (03249,13500) (03204,13491) (03368,13521) (03426,13532) (03617,13563) (104,36) (121,29) (191,30) (279,38) (514,61) 10 (04890,13707) (05066,13721) (04709,13692) (05203,13731) (05548,13753) (105,20) (121,18) (190,23) (276,31) (508,54) 15 (07655,13832) (07675,13832) (04709,13692 (08106,13840) (08430,13845) (104,12) (121,13) (190,20) (274,28) (506,51) 20 (09324,13853) (09459,13853) (04709,13692) (00626,11768) (09825,13854) (104,11) (121,12) (190,19) (256,27) (506,51) Table 3: Optimal couples (,K ) (first row of each block) and values of (h L,AT S 1 ) (second row of each block) of the VSI EWMA- X control chart for n = 7 n = 9 h S = 05 δ W = 09 W = 06 W = 03 W = 02 W = (00500,10152) (00500,10152) (00500,10152) (00500,10152) (00500,10152) (100,744) (106,694) (138,643) (175,637) (297,642) 02 (00548,10279) (00524,10220) (00540,10258) (00569,10330) (00598,10395) (101,263) (106,231) (137,212) (174,212) (294,221) 03 (01013,11030)) (00954,10964) (01017,11034) (01048,11066) (01077,11096) (101,141)) (107,123) (139,114) (176,115) (285,124) 05 (01878,11617) (02030,11679) (02141,11719) (02145,11720) (02182,11733) (101,63) (108,55) (142,53) (178,55) (303,67) 07 (03139,11970) (03334,12003) (03318,12000 (02408,11804) (03367,12008) (101,37) (108,33) (141,33) (177,37) (299,48) 10 (05404,12203,) (05410,12203) (05335,12199) (05354,12200) (05456,12206) (101,21) (108,20) (143,22) (181,26) (296,37) 15 (08434,12289) (08442,12289) (08480,12289) (08507,12289) (08570,12290) (101,12) (108,12) (142,16) (181,20) (294,31) 20 (09757,12297) (09772,12297) (09802,12297) (09822,12297) (09859,12297) (101,10) (108,11) (142,14) (180,18) (294,30) h S = 01 δ W = 09 W = 06 W = 03 W = 02 W = (00504,10163) (00500,10152) (00500,10152) (00500,10152) (00500,10152) (101,736) (111,647) (168,556) (236,544) (455,554) 02 (00599,10397) (00598,10397) (00599,10398) (00649,10501) (00711,10616) (101,254) (111,199) (171,163) (232,162) (445,179) 03 (01014,11031) (00964,10975) (01171,11183) (01106,11124) (01027,11044) (101,132) (113,100) (175,83,) (236,85) (435,103) 05 (02224,11747) (02192,11737) (02524,11836) (02359,11789) (02701,11880) (102,56) (113,43) (175,38) (239,43) (461,64) 07 (03240,11987) (03737,12060) (03496,12028) (03916,12082) (04117,12104) (102,33) (115,25) (26,10) (248,32) (455,52) 10 (05801,12223) (05775,12221) (05783,12222) (05896,12227) (06235,12240) (102,18) (114,16) (177,20) (246,27) (451,47) 15 (08417,12289) (08441,12289) (08686,12291) (08829,12292) (09070,12294) (102,11) (114,12) (176,18) (245,25) (450,45) 20 (09741,12297) (09868,12297) (03496,12028) (09941,12297) (09967,12297) (102,10) (114,11) (173,17) (09245,25) (450,45) Table 4: Optimal couples (,K ) (first row of each block) and values of (h L,AT S 1 ) (second row of each block) of the VSI EWMA- X control chart for n = 9 6 Illustrative example In this Section, we discuss the implementation of the VSI EWMA- X chart The context of the example presented here is similar as the one introduced in Castagliola et al [11], ie, a production process of 500 ml milk bottles where the quality characteristic X of interest is the capacity (in ml) of each bottle Like in Castagliola et al [11], we have µ 0 = and σ 0 = In fact, according to the process engineer experience, a shift δ = 05 should be interpreted as a signal that something is going wrong in the production process Thus, for n = 5, δ = 05 and AT S 0 = 3704 the VSI EWMA- X parameters are chosen to be h S = 05, h L = 163, = 01467, K = and W = 03 This yields the following control limits for the VSI EWMA- X chart: LCL = = , UCL = = and the warning control limits for the VSI EWMA- X chart: LWL = = , UWL = = The first 10 subgroups are supposed to be in-control while the last 10 subgroups are supposed to have a lower milk capacity, and thus, to be out-of-control The corresponding sample median values X i and the EWMA sequence Z i for VSI EWMA- X

6 control chart are both presented in Table 5 and plotted in Figure 1 This figure confirms that from sample #15 onwards, the process is clearly out-of-control Zi Sampling interval Total time Phase II (X i, j) X i Z i Table 5: Illustrative Phase II dataset VSI EWMA- X chart t UCL UWL LWL LCL Figure 1: VSI EWMA- X control chart corresponding to Phase II data set in Table 5 7 Concluding remarks In this paper, we have investigated a VSI EWMA- X control chart for monitoring process median We have also studied the statistical properties of the VSI EWMA- X control chart and optimized their parameters for several shift sizes For fixed values of the shift size δ, several tables are provided for presenting the out-of-control AT S 1 corresponding to many different scenarios Also, the numerical comparison with the performance of the EWMA- X control chart proposed by Castagliola [15] shows that the detection ability of the proposed control chart are better than the EWMA- X control chart Thus, the proposed chart can be used as a best alternative method Finally, possible enhancements and future work about VSI EWMA- X control chart include the investigation of the effect of the parameters estimation and of the measurement errors on their statistical properties Appendix The Markov chain approach of Brook and Evans [22] and Lucas and Saccucci [6] is modified to evaluate the Run Length properties of the VSI EWMA- X chart This procedure involves dividing dividing the interval [LCL,UCL] into 2m + 1 subintervals (H j,h j + ], j { m,,0,,+m}, centered at H j = LCL+UCL j where 2 = UCL LCL (2m+1) Each subinterval (H j,h j + ], j { m,,0,,+m}, represents a transient state of a Markov chain If Z i (H j,h j + ] then the Markov chain is in the transient state j { m,,0,,+m} for sample i If Z i (H j,h j + ] then the Markov chain reached the absorbing state (, LCL] [UCL, + ) We assume that H j is the representative value of state j { m,,0,,+m} Let Q be the (2m + 1,2m + 1) sub-matrix of probabilities Q j,k corresponding to the 2m + 1 transient states defined above, ie Q m, m Q m, 1 Q m,0 Q m,+1 Q m,+m Q 1, m Q 1, 1 Q 1,0 Q 1,+1 Q 1,+m Q = Q 0, m Q 0, 1 Q 0,0 Q 0,+1 Q 0,+m Q +1, m Q +1, 1 Q +1,0 Q +1,+1 Q +1,+m Q +m, m Q +m, 1 Q +m,0 Q +m,+1 Q +m,+m By definition, we have Q j,k = P(Z i (H k,h k + ] Z i 1 = H j ) or, equivalently, Q j,k = P(Z i H k + Z i 1 = H j ) P(Z i H k Z i 1 = H j ) Replacing Z i = (1 )Z i 1 + X i, Z i 1 = H j and isolating X i gives ( Q j,k = P X i H ) ( k + (1 )H j P X i H ) k (1 )H j, ( ) ( Hk + (1 )H j = F X n Hk (1 )H j F X ), n where F X ( n) is the cdf (cumulative distribution function) of the sample median X i, i {1,2,}, ie F X (y n) = F β ( ( y µ0 Φ σ 0 δ ) n + 1 2, n ), (13) where Φ(x) is the cdf of the standard normal distribution and F β (x a,b) is the cdf of the beta distribution with parameters (a,b) Here a = b = n+1 2 Let q = (q m,,q 0,,q m ) T be the (2m + 1, 1) vector of initial probabilities associated with the 2m + 1 transient states, where { 0 if Z0 (H q j = j,h j + ] 1 if Z 0 (H j,h j + ] (14) The AT S 1 can be evaluated through the following expression: AT S 1 = q T (I Q) 1 g (15) where g is the vector of sampling intervals corresponding to the discretized states of the Markov chain and the jth element g j of the vector g is the sampling interval when the control statistic is in state j (represented by H j ), ie { hl if LWL H g j = j UWL, (16) otherwise h S

7 The average sampling interval of the VSI EWMA- X chart is given as: E(h) = qt (I Q) 1 g q T (I Q) 1 (17) 1 References [1] DC Montgomery Statistical Quality Control: A Modern Introduction, 7th Edn John Wiley& Sons, Hoboken, New Jersey, 2013 [2] KP Tran, P Castagliola, and G Celano Monitoring the Ratio of Two Normal Variables Using EWMA Type Control Charts Quality and Reliability Engineering International, 32(2): , 2016 [3] PB Robinson and TY Ho Average Run Lengths of Geometric Moving Average Charts by Numerical Methods Technometrics, 20(1):85 93, 1978 [4] JS Hunter The Exponentially Weighted Moving Average Journal of Quality Technology, 18: , 1986 [5] SV Crowder A Simple Method for Studying Run-Length Distributions of Exponentially Weighted Moving Average Charts Technometrics, 29(4): , 1987 [6] JM Lucas and MS Saccucci Exponentially Weighted Moving Average Control Schemes: Properties and Enhancements Technometrics, 32(1):1 12, 1990 [7] P Castagliola and FO Figueiredo The Median Chart with Estimated Parameters European Journal of Industrial Engineering, 7(5): , 2013 [8] CC Huang Max control chart with adaptive sample sizes for jointly monitoring process mean and standard deviation Journal of the Operational Research Society, 65 (12): , 2014 [9] E Da Costa Quinino, L L LL Ho, and A L G Trindade Monitoring the process mean based on attribute inspection when a small sample is available Journal of the Operational Research Society, 66(11): , 2015 [10] KP Tran, P Castagliola, and G Celano Monitoring the Ratio of Population Means of a Bivariate Normal distribution using CUSUM Type Control Charts Statistical Papers, 2016 In press, DOI: /s [11] P Castagliola, P E Maravelakis, and F O Figueiredo The EWMA Median chart with estimated parameters IIE Transactions, 48(1):66 74, 2016 [12] KP Tran The efficiency of the 4-out-of-5 Runs Rules scheme for monitoring the Ratio of Population Means of a Bivariate Normal distribution International Journal of Reliability, Quality and Safety Engineering, 23(5):1 26, 2016 [13] KP Tran, P Castagliola, G Celano, and MBC Khoo Monitoring compositional data using multivariate exponentially weighted moving average scheme Quality and Reliability Engineering International, 2017 In press, DOI: /qre2260 [14] KP Tran Designing of Run Rules t control charts for monitoring changes in the process mean Chemometrics and Intelligent Laboratory Systems, 2018 In press, DOI: /jchemolab [15] P Castagliola An X/R-EWMA Control Chart For Monitoring the Process Sample Median International Journal of Reliability, Quality and Safety Engineering, 8(2): , 2001 [16] SH Sheu and L Yang The Generally Weighted Moving Average Control Chart for Monitoring the Process Median Quality Engineering, 18(3): , 2006 [17] YC Lin, CY Chou, and CH Chen Robustness of the EWMA median control chart to non-normality International Journal of Industrial and Systems Engineering, 25 (1):35 58, 2017 [18] KP Tran Run Rules median control charts for monitoring process mean in manufacturing Quality and Reliability Engineering International, 33(8): , 2017 [19] MR Reynolds Shewhart and EWMA variable sampling interval control charts with sampling at fixed times Journal of Quality Technology, 28(2): , 1996 [20] MR Reynolds, RW Amin, JC Arnold, and JA Nachlas Charts with variable sampling intervals Technometrics, 30(2): , 1988 [21] P Castagliola, G Celano, S Fichera, and F Giuffrida A variable sampling interval s 2 -EWMA control chart for monitoring the process variance International Journal of Technology Management, 37(1-2): , 2006 [22] D Brook and DA Evans An Approach to the Probability Distribution of CUSUM Run Length Biometrika, 59 (3): , 1972 [23] PH Tran and K P Tran The Efficiency of CUSUM schemes for monitoring the Coefficient of Variation Stochastic Models in Business and Industry, 2016 In press, DOI: /asmb2213