Journal of Quality Measurement and Analysis JQMA 12(1-2) 2016, Jurnal Pengukuran Kualiti dan Analisis

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1 Journal of Qualty Measurement Analyss JQMA 12(1-2) 2016, Jurnal Pengukuran Kualt dan Analss PERFORMANCE COMPARISON OF THE EXACT RUN-LENGTH DISTRIBUTION BETWEEN THE RUN SUM X AND EWMA X CHARTS (Perbngan Prestas antara Carta Hasl Tambah Laran X dan EWMA X Berdasarkan Taburan Panjang Laran Tepat) J.K. CHONG 1, W.L. TEOH 1, MICHAEL B.C. KHOO 2, Z.L. CHONG 1 & S.Y. TEH 3 ABSTRACT The run sum (RS) X exponentally weghted movng average (EWMA) X charts are very effectve n detectng small moderate process mean shfts. The desgn of the RS X EWMA X charts based on the average run length (ARL) alone, can be msleadng confusng. Ths s due to the fact that the run-length dstrbuton of a control chart s hghly rght-skewed when the process s n-control or slghtly out-of-control; whle that for the out-ofcontrol cases, the run-length dstrbuton s almost symmetrc. On the other h, the percentles of the run-length dstrbuton provde the probablty of gettng a sgnal by a certan number of samples. Ths wll beneft practtoners as the percentles of the run-length dstrbuton gve comprehensve nformaton regardng the behavour of a control chart. Accordngly, ths paper provdes a thorough study of the run-length propertes (ARL, stard devaton of the run length percentles of the run-length dstrbuton) for the RS X EWMA X charts. Comparatve studes show that the EWMA X chart outperforms the RS X charts for detectng small mean shfts when all the control charts are mzed wth respect to a small shft sze. However, the RS X charts surpass the EWMA X chart for all szes of mean shfts when all the control charts are mzed wth respect to a large shft sze. Keywords: average run length; EWMA X chart; percentles of the run-length dstrbuton; run sum X chart; stard devaton of the run length ABSTRAK Carta X hasl tambah laran (RS) dan X purata bergerak berpemberat eksponen (EWMA) adalah sangat berkesan untuk mengesan anjakan mn proses yang kecl dan sederhana. Reka bentuk carta X RS dan X EWMA berdasarkan panjang laran purata (ARL) sahaja adalah mengelrukan. Hal n kerana taburan panjang laran untuk carta kawalan adalah sangat terpencong ke kanan apabla proses berada dalam kawalan atau hanya sedkt d luar kawalan; manakala bag kes d luar kawalan, taburan panjang laran adalah hampr smetr. Sebalknya, persentl taburan panjang laran memberkan kebarangkalan untuk mendapat syarat dengan blangan sampel yang tertentu. Hal n dapat memanfaatkan para pengamal kerana persentl taburan panjang laran member maklumat yang komprehensf tentang kelakuan carta kawalan. Oleh hal yang demkan, makalah n memberkan kajan yang menyeluruh tentang sfat-sfat panjang laran (ARL, sshan pawa panjang laran dan persentl taburan panjang laran) untuk carta X RS dan X EWMA. Perbngan dalam kajan n menunjukkan bahawa carta X EWMA adalah lebh bak darpada carta X RS untuk mengesan anjakan mn yang kecl apabla semua carta kawalan tu dmumkan berdasarkan saz anjakan yang kecl. Walau bagamanapun, carta X RS adalah lebh bak darpada carta X EWMA bag semua saz anjakan mn apabla semua carta kawalan tu dmum berdasarkan suatu saz anjakan yang besar. Kata kunc: panjang laran purata; carta X EWMA; persentl taburan panjang laran; carta X hasl tambah laran; sshan pawa panjang laran

2 J.K. Chong, W.L. Teoh, Mchael B.C. Khoo, Z.L. Chong & S.Y. Teh 1. Introducton Statstcal Process Control (SPC) s a collecton of statstcal technques to montor control a process. Control charts are appled to montor the qualty characterstcs of products. The foremost objectve of developng a control chart s to quckly detect the occurrence of process shfts; thus, necessary correctve actons can be taken before a large amount of nonconformng unts are beng manufactured (Montgomery 2013). The performance of a control chart s usually based on the average run length (ARL) crteron, whch s long recognsed n SPC feld. Interpretaton of a control chart solely based on the ARL can be msleadng to practtoners (Bschak & Tretsh 2007; Chakrabort 2007; Teoh et al. 2015). Ths s because the shape of the run-length dstrbuton changes accordng to the magntude of shfts. It changes from hghly rght-skewed when the shft s small to nearly symmetrc when the shft s large. In general, the ARL only gves the expected number of samples to sgnal. It does not provde the probablty of gettng an out-of-control sgnal by a certan number of samples. On the other h, the percentles of the run-length dstrbuton s more ntutve as they focus on the behavour of the charts. Several studes have found that the percentles of the run-length dstrbuton provde a comprehensve nterpretaton regardng the exact run length, such as those by Khoo Quah (2002), Khoo (2004), Radson Boyd (2005), Shmuel Cohen (2003), Teoh et al. (2016). Palm (1990) clamed that the 50 th percentle of the run-length dstrbuton (.e. the medan run length, MRL) represents half of the tme. For example, the out-of-control MRL (MRL 1) of 30 ndcates that 50% of all the run lengths are less than 30. Furthermore, the percentles of the run-length dstrbuton provde a practcal gudance regardng the early false alarms spread of the run-length dstrbuton. To have a good understng of a control chart, t s necessary to supplement the ARL wth the percentles of the run-length dstrbuton stard devaton of the run length (SDRL) (Jones 2002). On a dfferent note, the run sum (RS) chart was developed by Roberts (1966) to ncrease the senstvty of the basc chart. The advantage of the RS chart s the great mprovement of the detecton speed, whle mantanng the smplcty of the basc chart. Reynolds (1971) suggested to use the RS control chart wth eght regons, where four regons are on each sde of the centre lne scores are assgned for each regon. Jaehn (1987) suggested that the zone control chart s a specal case of the RS X chart. Davs et al. (1994) proposed a general model of the zone control chart wth fast-ntal-response (FIR) feature. Champ Rgdon (1997) developed the ARL-based RS X chart by usng the Markov chan approach. They concluded that the RS X chart offers better statstcal effcency than the Shewhart X chart wth supplementary runs rules. Besdes, they also ndcated that by addng more regons scores, the RS X chart s compettve n terms of detecton speed, compared to the exponentally weghted movng average (EWMA) X cumulatve sum (CUSUM) X charts. Parkhdeh Parkhdeh (1998) desgned a flexble zone control chart for ndvdual observaton. Agurre-Torres Reyes-Lopez (1999) studed both the RS X R charts. Davs Krehbel (2002) compared the performance of the Shewhart X chart wth runs rules wth that of the zone control chart when lnear trend presents n the process. Acosta-Meja Pgnatello (2010) nvestgated the RS R control chart wth FIR feature for montorng the process dsperson. The RS t chart, whch s more robust aganst changes n the process varance compared to the RS X chart, was studed by Stt et al. (2014). Acosta-Meja Rncon (2014) ntroduced the contnuous RS chart, whch has a centre lne equal to zero scores equal to the stardzed subgroup means. Recently, Teoh et al. (2017) developed the run sum charts for montorng the coeffcent of varaton, whch broaden the chartng capablty to varous scentfc socetal applcatons. 12

3 Performance comparson of the exact run-length dstrbuton between the run sum X EWMA X charts Another type of control chart, called the EWMA chart, was ntroduced by Roberts (1959). A consderable amount of lterature on EWMA control charts have been conducted over the years. Crowder (1987) formulated the EWMA chart propertes by usng ntegral equatons. The mal desgn of the EWMA chart based on ARL was further dscussed by Crowder (1989). Lucas Saccucc (1990) aded the Markov chan approach to compute the run-length propertes of the EWMA control chart. Gan (1993) developed the mal MRL-based EWMA control chart. Stener (1999) proposed the FIR-EWMA control charts wth tme-varyng control lmts. Moreover, Jones (2002) nvestgated the effect of parameters estmaton on the EWMA X chart. Shu et al. (2007) studed the two one-sded EWMA charts for upward downward changes n the process mean. Some recent studes on the EWMA control charts can be found n Abdul et al. (2015), Castaglola et al. (2011), Khan et al. (2016), Khoo et al. (2016) Zhang et al. (2009). To date, none of the exstng lterature compares the RS X EWMA X charts, n terms of the percentles of the run-length dstrbuton. The percentles of the run-length dstrbuton are more credble compared to the ARL, especally when the skewness of the assocated runlength dstrbuton s dfferent for dfferent shfts. Therefore, the objectve of ths paper s to thoroughly examne compare the ARL, SDRL percentles of the run-length dstrbuton for both the RS X EWMA X charts. The structure of ths paper s as follows: Sectons 2 3 ntroduce the RS X EWMA X charts. In Secton 4, the comparson of the ARL, SDRL percentles of the run length dstrbuton are nvestgated. Fnally, some conclusons are drawn n Secton The RS X chart Let {X,1, X,2,, X,n} be the th sample, where = 1, 2, n s the number of observatons. Assume that the j th (for j 1, 2,, n ) qualty characterstc n sample follows a normal dstrbuton,.e., ~ j 2 X N μ0 δσ0, σ0. Here, μ0 s the n-control mean σ 0 s the ncontrol stard devaton. The magntude of a mean shft n multples of the stard devaton s denoted as δ. The process s statstcally n-control when δ 0 out-of-control when δ 0. Also, assume that there s ndependence between wthn samples. The RS X chart s dvded nto k regons above k regons below the central lne (CL). Fgure 1 graphcally shows the k regons RS X chart wth the assocated scores, probabltes control lmts. The nteger scores 0 S1 S2... Sk Sk Sk 1... S2 S1 0 are assgned to each of the regons above below the CL, respectvely. From Fgure 1, the regons probabltes above the CL are R1, R 2,..., R k p 1, p 2,, p k, respectvely; whle the regons probabltes below the CL are R 1, R 2,..., R k p, 1 p 2,, p k, respectvely. Note that CL μ0 k s the number of regons desred. In ths paper, we consder k {4, 7}. Champ Rgdon (1997), Stt et al. (2014), Teoh et al. (2017) also consdered k {4, 7}. The k regons RS X chart at the th sample s defned by the followng upper U lower L cumulatve sums: U 1 S X, f X μ0 U, 0, f X μ0 (1) 13

4 ... J.K. Chong, W.L. Teoh, Mchael B.C. Khoo, Z.L. Chong & S.Y. Teh X UCL k = UCL k 1 Regon R +k S k p +k UCL 3 UCL 2 Regon R +3 S 3 p +3 UCL 1 Regon R +2 S 2 p +2 CL = μ 0 Regon R +1 S 1 p +1 LCL 1 LCL 2 LCL 3 Regon R 1 Regon R 2 Regon R 3 S 1 S 2 S 3 p 1 p 2 p 3 LCL k 1 LCL k = Regon R k S k p k Fgure 1: Graphcal vew of the k regons above k regons below the CL of the RS X chart wth the assocated scores, probabltes control lmts L 1 S X, f X μ0 L, (2) 0, f X μ0 where U0 L0 0 when X t. In Equatons (1) (2), S X s the score functon, where S X R S X = St when X R t = S t, for t 1, 2,, k. The upper UCL1 lower LCL LCL... LCL LCL UCL2... UCLk 1 UCLk control lmts of the k regons RS X chart are computed as k k t σ 0 UCLt μ0 K k 1 n (3) 14

5 Performance comparson of the exact run-length dstrbuton between the run sum X EWMA X charts 3t σ 0 LCLt μ0 K k 1 n, (4) respectvely, where t 1, 2,, k K s the control lmts parameter. The procedure of constructng the k regons RS X chart s as follows (Champ & Rgdon 1997): (1) Determne the number of regons k, scores St, control lmts UCL t LCL t, based on an mal desgn of the chart. (2) Collect a sample, each havng n observatons compute the sample mean, n X X n. 1 (3) Start wth the cumulatve score at 0,.e. (U 0, L 0) = (+0, 0). (4) Accumulate the scores U L correspondng to the regons R +t R t, respectvely, n whch the sample mean, X falls (refer to Equatons (1) (2)). (5) Reset the cumulatve scores of U or L to zero f X falls on the other sde of the CL,.e, X μ 0 (refer to Equaton (1)) or X μ 0 (refer to Equaton (2)), respectvely. (6) Declare the process as out-of-control f a postve cumulatve score U S k or a negatve cumulatve score L S k, where S k s the trggerng score whch serves as a boundary beyond whch the chart wll sgnal an out-of-control. For example, the probabltes p +2 p +3 n regons R +2 R +3, respectvely, (refer to Fgure 1) can be obtaned by usng Equatons (5) (6), as follows (Champ & Rdgon 1997): where p P UCL X UCL σ0 6 σ0 P μ0 K X μ0 K k1 n k1 n 6 3 Φ K Φ 1 δ n k K k 1 δ n, (5) p P UCL X UCL σ0 9 σ0 P μ0 K X μ0 K k 1 n k 1 n 9 6 Φ K Φ 1 δ n k K k 1 δ n, (6) Φ. s the cumulatve dstrbuton functon (cdf) of the stard normal dstrbuton. p The probabltes t, for other values of t, can be computed usng the same method as dsplayed n Equatons (5) (6). Suppose that we have a dscrete-tme Markov chan wth p + 1 states, where states 1, 2,, p are transent the state p + 1 s an absorbng state. Here, p s the number of possble ordered pars (U, L ) when the process s n-control. The transton probablty matrx P of the dscretetme Markov chan has the followng structure: 15

6 J.K. Chong, W.L. Teoh, Mchael B.C. Khoo, Z.L. Chong & S.Y. Teh Q P T 0 r, (7) 1 where Q s the (p p) transent-probablty matrx, 0 = (0, 0,, 0) T r s the (p 1) vector that satsfes r = 1Q1 (.e. the row probabltes must sum to 1), wth 1 = (1, 1,, 1) T. Note that the dmenson generc elements of matrx Q for the k regons RS X chart depend on the choce of the scores {S 1, S 2,, S k}. The matrx Q does not have a general form. Refer to Teoh et al. (2017) for the detaled procedure for obtanng matrx Q. The computaton of the ARL by usng the Markov chan approach s as follows: T ARL ( ) 1 s I Q 1, (8) where s = (1, 0,, 0) T s the vector of ntal probablty havng a unty n the frst element zeros elsewhere; whle I s the (p p) dentty matrx. Also, the computaton of the SDRL by means of the Markov chan approach s calculated as T 2 2 SDRL 2 ( ) ARL ARL s I Q Q1. (9) Let N denotes the run length for the k regons RS X chart. The cdf of N, regons RS X chart s calculated as FN for the k FN T PN ( ) s I Q 1, (10) where 1, 2,.... The 100 th percentles of the run-length dstrbuton can be determned as the value such that (Gan 1993) 1 P N P N, (11) where s n the range of 0 1. For example, the 30 th percentle of the run-length dstrbuton can be obtaned from both Equatons (10) (11) by settng 0.3 n Equaton (11). 3. The EWMA X chart The plottng statstc Z of the EWMA X chart s as follows (Crowder 1989): Z X (1 ) Z 1, for = 1, 2,, (12) where 0 1 s the smoothng constant, X s the th sample mean Z0 0. The upper (UCL) lower (LCL) control lmts of the EWMA X chart are computed as follows: UCL H, (13)

7 Performance comparson of the exact run-length dstrbuton between the run sum X EWMA X charts LCL H, (14) 0 0 where H h n (2 ) wth the multpler h controllng the wdth of the control lmts. The EWMA X chart sgnals an out-of-control stuaton f Z UCL or Z LCL. The ARL SDRL of the EWMA X chart can be computed by usng Equatons (8) (9), respectvely. Smlarly, the percentles of the run-length dstrbuton for the EWMA X chart can be computed from both Equatons (10) (11). Note that the detals of the matrces s Q n Equatons (8), (9) (10) are gven Zhang et al. (2009). 4. Performance Comparson In ths secton, we dscuss the entre run-length dstrbutons of the ARL-based mal RS X mal EWMA X charts wth specfc ranges of {0.00, 0.25, 0.50,, 1.00, 1.50, 2.00} n {3, 5, 7, 9}. Also, throughout ths paper, we specfy a fxed ARL 0 = 500 {0.5, 1.5}, where ARL 0 s the n-control ARL s the desred mean shft, for whch a quck detecton s ntended. Tables 1 to 4 present the entre run-length profles for the 4 7 regons RS X charts when ARL 0 = 500, n{3, 5, 7, 9} {0.5, 1.5} ; whle the correspondng run-length profles of the EWMA X chart are shown n Tables 5 6. From Tables 1 to 4, the mal charts parameters (K, S 1, S 2, S 3, S 4) (K, S 1, S 2, S 3, S 4, S 5, S 6, S 7) of the 4 7 regons RS X charts are obtaned by mnmzng the out-of-control ARL (ARL 1), subject to a desred ARL 0 value. Champ Rgdon (1997) presented the detals of the mzaton algorthm of the RS X chart. Smlarly, Tables 5 6 present the mal chart s parameters (, H) of the EWMA X chart by mnmzng the ARL 1, subject to a desred ARL 0 value. A study by Gan (1993) gves the detals of the mzaton algorthm of the EWMA X chart. Tables 1 to 6 provde suffcent evdence to clarfy that the ARL s a confusng measure. For example, the ARL 0 of 500 only provdes us wth the expected number of samples to sgnal. The ARL 0 does not provde a comprehensve measure regardng the probablty of gettng a false alarm by a certan sample. Therefore, there may exst a rsk that a practtoner falsely nterprets a control chart wth half of the tme that a sgnal wll be detected by the 500 th sample. But n actual scenaro, a false alarm wll be notced earler,.e. by the 348 th sample (the 50 th percentle of the run length dstrbuton s 348) wth half of the tme for all n {3, 5, 7, 9} (see Table 1). For such a case, t s mportant to note that the MRL (.e. the 50 th percentle of the run-length dstrbuton) s a better representatve of central tendency of the run-length dstrbuton compared to the ARL (Chakrabort 2007). If MRL 0 = 500, a practtoner may clam wth 50% certanty that a false alarm wll occur by the 500 th sample. Here, MRL 0 s the n-control MRL. The results n Tables 1 to 6 show that there s a great dfference between ARLs MRLs, especally when the process s n-control ( 0) or f the shft s small; whle ths dfference s small when the shft s large. Ths shows that the shape of the run-length dstrbuton changes accordng to the magntude of shfts,.e. from hghly rght skewed when the shft s small to almost symmetrc when the shft s large. Also, ths mples that for a rght-skewed dstrbuton, the ARL s larger than the MRL; whle the ARL s almost the same as the MRL n an almost symmetrc dstrbuton. Therefore, nterpretaton based on ARL alone s confusng. Ths s because nterpretaton based on ARL for a hghly rght-skewed dstrbuton s surely dfferent 17

8 J.K. Chong, W.L. Teoh, Mchael B.C. Khoo, Z.L. Chong & S.Y. Teh wth that of an almost symmetrc dstrbuton. For a comprehensve understng of a control chart, a practtoner cannot solely depend on the ARL, where the ARL needs to be supplemented wth the MRL percentles of the run-length dstrbuton. The percentles of the run-length dstrbuton allow a practtoner to state wth an exact probablty that a chart wll sgnal by a certan sample, regardless of the shft szes. The computaton of the hgher percentles (.e. 80 th, 90 th 95 th ) of the run-length dstrbuton provdes some crtcal mportant nformaton to practtoners. For nstance, by referrng to Table 4, when n = 9, practtoners wll have 90% confdent that an out-of-control sgnal wll be detected by the 8 th sample. The computaton of the lower percentles (.e. 5 th, 10 th 25 th ) of the run-length dstrbuton when 0, provdes an analyss regardng the early false alarm n a process. For example, n the case of the 4 regons RS X chart wth 1.5 Table 1: Exact ARL, SDRL percentles of the run-length dstrbuton for the 4 regons RS X chart wth mal parameters (K, S1, S2, S3, S4), when n {3, 5, 7, 9}, 0.5 ARL0 = 500 Percentles of the run-length dstrbuton ARL SDRL 5 th 10 th 20 th 25 th 30 th 40 th 50 th 60 th 70 th 75 th 80 th 90 th 95 th n = 3, (K, S 1, S 2, S 3, S 4) = (1.2432, 0, 3, 5, 10) n = 5, (K, S 1, S 2, S 3, S 4) = (1.2432, 0, 3, 5, 10) n = 7, (K, S 1, S 2, S 3, S 4) = (1.2432, 0, 3, 5, 10) n = 9, (K, S 1, S 2, S 3, S 4) = (1.2432, 0, 3, 5, 10)

9 Performance comparson of the exact run-length dstrbuton between the run sum X EWMA X charts n = 5, there s a 10% chance or 0.1 probablty that a false alarm wll occur by the 54 th sample (see Table 2). Accordng to Chakrabort (2007), the dfference between the 5 th 75 th (or 5 th 95 th ) percentles of the run-length dstrbuton descrbes the spread varaton of the run-length dstrbuton. Let us denote as the dfference between the 25 th 75 th (5 th 95 th ) percentles of the run-length dstrbuton. Here,,, 0.25 are the th, 25 th, 75 th 95 th percentles of the run-length dstrbuton. For example, for the 7 regons RS X chart, the value of 0.95 s qute large,.e when 1.5, Table 2: Exact ARL, SDRL percentles of the run-length dstrbuton for the 4 regons RS X chart wth mal parameters (K, S1, S2, S3, S4), when n {3, 5, 7, 9}, 1.5 ARL0 = 500 Percentles of the run-length dstrbuton ARL SDRL 5 th 10 th 20 th 25 th 30 th 40 th 50 th 60 th 70 th 75 th 80 th 90 th 95 th n = 3, (K, S 1, S 2, S 3, S 4) = (1.0810, 0, 0, 1, 2) n = 5, (K, S 1, S 2, S 3, S 4) = (1.0810, 0, 0, 1, 2) n = 7, (K, S 1, S 2, S 3, S 4) = (1.0323, 0, 0, 1, 3) n = 9, (K, S 1, S 2, S 3, S 4) = (1.5451, 0, 0, 1, 1)

10 J.K. Chong, W.L. Teoh, Mchael B.C. Khoo, Z.L. Chong & S.Y. Teh n = 3 0 (see Table 4). Ths llustrates that the run length has a large varaton because of the long rght tal when 0. Generally, the dfference between these two percentles of the run-length dstrbuton decreases as n ncrease. When comparng the control charts performance, a control chart havng the smallest ARL 1, SDRL 1, MRL 1, , s deemed as the best chart. Here, ARL 1, SDRL 1 MRL 1 represent the out-of-control ARL, SDRL, MRL, respectvely. From Tables 1, 3 5 (.e. when 0.5 ), t s obvous that the EWMA X chart has the smallest ARL 1, Table 3: Exact ARL, SDRL percentles of the run-length dstrbuton for the 7 regons RS X chart wth mal parameters (K, S1, S2, S3, S4, S5, S6, S7), when n {3, 5, 7, 9}, 0.5 ARL0 = 500 Percentles of the run-length dstrbuton ARL SDRL 5 th 10 th 20 th 25 th 30 th 40 th 50 th 60 th 70 th 75 th 80 th 90 th 95 th n = 3, (K, S 1, S 2, S 3, S 4, S 5, S 6, S 7) = (1.4687, 0, 1, 2, 3, 4, 5, 8) n = 5, (K, S 1, S 2, S 3, S 4, S 5, S 6, S 7) = (1.3554, 0, 1, 2, 4, 5, 7, 10) n = 7, (K, S 1, S 2, S 3, S 4, S 5, S 6, S 7) = (1.5731, 0, 1, 3, 5, 6, 8, 10) n = 9, (K, S 1, S 2, S 3, S 4, S 5, S 6, S 7) = (1.1952, 0, 0, 1, 2, 3, 4, 6)

11 Performance comparson of the exact run-length dstrbuton between the run sum X EWMA X charts SDRL 1, MRL 1, when. It s followed by the 7 4 regons RS X charts. For example, when 0.5, n , the (ARL 1, SDRL 1, MRL 1, 0.25, 0.95 ) are (29.66, 23.29, 23, 26, 70) (see Table 5) for the EWMA X chart as opposed to (45.89, 41.55, 33, 46, 123) (see Table 3) (51.00, 47.01, 37, 51, 139) (see Table 1) for the 7 4 regons RS X charts, respectvely. For 0.5 (see Tables 1, 3 5), on the other h, the 4 regons RS X chart generally has the fastest detecton speed (.e. the Table 4: Exact ARL, SDRL percentles of the run-length dstrbuton for the 7 regons RS X chart wth mal parameters (K, S1, S2, S3, S4, S5, S6, S7), when n {3, 5, 7, 9}, 1.5 ARL0 = 500 Percentles of the run-length dstrbuton ARL SDRL 5 th 10 th 20 th 25 th 30 th 40 th 50 th 60 th 70 th 75 th 80 th 90 th 95 th n = 3, (K, S 1, S 2, S 3, S 4, S 5, S 6, S 7) = (1.1233, 0, 0, 0, 2, 5, 6, 8) n = 5, (K, S 1, S 2, S 3, S 4, S 5, S 6, S 7) = (1.0527, 0, 0, 0, 0, 2, 3, 5) n = 7, (K, S 1, S 2, S 3, S 4, S 5, S 6, S 7) = (1.2433, 0, 0, 0, 0, 1, 2, 2) n = 9, (K, S 1, S 2, S 3, S 4, S 5, S 6, S 7) = (1.0331, 0, 0, 0, 0, 0, 1, 2)

12 J.K. Chong, W.L. Teoh, Mchael B.C. Khoo, Z.L. Chong & S.Y. Teh smallest ARL 1 value) among all the three charts under comparson when 1.0. However, the SDRL 1 value for the EWMA X chart s generally the smallest compared to the 4 7 regons RS X charts when 1.0. Regardng the MRL 1, , the three charts have about the same performance when 1.0. Let us focus on Tables 2, 4 6 when 1.5. Here, we observe that the EWMA X chart generally has the worst performances compared to the 4 7 regons RS X charts, for all szes of mean shfts when n{5, 7}; whle for n{3, 9}, the 4 regons RS X chart s Table 5: Exact ARL, SDRL percentles of the run-length dstrbuton for the EWMA X chart wth mal parameters (, H), when n {3, 5, 7, 9}, 0.5 ARL0 = 500 Percentles of the run-length dstrbuton ARL SDRL 5 th 10 th 20 th 25 th 30 th 40 th 50 th 60 th 70 th 75 th 80 th 90 th 95 th n = 3, (, H) (0.1090, ) n = 5, (, H) (0.1594, ) n = 7, (, H) (0.2041, ) n = 9, (, H) (0.2445, )

13 Performance comparson of the exact run-length dstrbuton between the run sum X EWMA X charts generally the worst among all the three control charts. Stll nvestgatng Tables 2, 4 6, when n{3, 9}, the ARL 1, SDRL 1, MRL 1, for the 7 regons RS X chart are the smallest among all the three charts under comparson; whle these performance measures are the smallest for the 4 regons RS X chart when n{5, 7}. For 1.0, the 7 regons RS X chart outperforms the 4 regons RS X EWMA X charts, n terms of ARL 1 SDRL 1 (see Tables 2, 4 6). Smlarly, all the three charts have compettve performances, n terms of the MRL 1, , when 1.0. Table 6: Exact ARL, SDRL percentles of the run-length dstrbuton for the EWMA X chart wth mal parameters (, H), when n {3, 5, 7, 9}, 1.5 ARL0 = 500 Percentles of the run-length dstrbuton ARL SDRL 5 th 10 th 20 th 25 th 30 th 40 th 50 th 60 th 70 th 75 th 80 th 90 th 95 th n = 3, (, H) (0.5495, ) n = 5, (, H) (0.7664, ) n = 7, (, H) (0.8834, ) n = 9, (, H) (0.9415, )

14 J.K. Chong, W.L. Teoh, Mchael B.C. Khoo, Z.L. Chong & S.Y. Teh From Tables 1 to 6, when, t s found that the detecton speed varaton of the run-length dstrbuton of the three control charts specally desgned for 0.5 (see Tables 1, 3 5) s faster than those desgned for 1.5 (see Tables 2, 4 6). For example, the (ARL 1, SDRL 1, MRL 1, 0.25, 0.95 ) values for the 7 regons RS X chart are (10.18, 6.70, 8, 7, 20) when 0.5, n (see Table 3). These (ARL 1, SDRL 1, MRL 1,, ) values for the 7 regons RS X chart when 1.5, n are larger than those for 0.5,.e. (18.89, 16.45, 14, 18, 50) (see Table 4). On the contrary, for 1.0, the three control charts specally desgned for 1.5 (see Tables 2, 4 6) surpass those desgned for 0.5 (see Tables 1, 3 5). Ths mples that the control charts mzed based on 0.5 are more effectve n detectng small shfts; whle those mzed based on 1.5 are more sutable for detectng large shfts. 5. Concluson In ths paper, we demonstrate that the ARL s a pecular performance measure. Therefore, dependence on the ARL measure alone s dscouraged. On the contrary, the percentles of the run-length dstrbuton whch provde the exact behavour of the run-length dstrbuton of a control chart, are more ntutve. To have an n-depth knowledge hgh confdence of a control chart, t s necessary to nvestgate a control chart, n terms of the ARL, SDRL percentles of the run-length dstrbuton. Hence, n ths paper, we provde a thorough nvestgaton of the exact run-length propertes of the RS X EWMA X charts. The comparatve results reveal that the 7 regons RS X charts whch are mzed wth respect to 1.5, outshne the correspondng EWMA X chart for all szes of shfts. When all the three control charts are mally desgned wth respect to 0.5, the EWMA X chart s the best n detectng small mean shfts ( ) has the smallest varaton of the run-length dstrbuton for all levels of shft szes. The percentles of the run-length dstrbuton change wth n (see Tables 1 to 6), even though the same value of ARL 0 s attaned. For detectng small mean shfts, t s recommended to desgn a control chart based on a small mal shft sze vce versa. Acknowledgements Ths research s supported by the Unverst Tunku Abdul Rahman, Fundamental Research Grant Scheme (FRGS) no. FRGS/1/2015/SG04/UTAR/02/3. References Abdul H., Jennfer B. & Elena M A new exponentally weghted movng average control chart for montorng the process mean. Qualty Relablty Engneerng Internatonal 31(8): Acosta-Meja C.A. & Pgnatello J.J.Jr The run sum R chart wth fast ntal response. Communcatons n Statstcs-Smulaton Computaton 39(5): Acosta-Meja C.A. & Rncon L The contnuous run sum chart. Communcatons n Statstcs-Theory Methods 43(20): Agurre-Torres V. & Reyes-Lopez D Run sum charts for both X R. Qualty Engneerng 12(1): Bschak D.P. & Tretsch D The rate of false sgnals n X control charts wth estmated lmts. Journal of Qualty Technology 39(1):

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16 J.K. Chong, W.L. Teoh, Mchael B.C. Khoo, Z.L. Chong & S.Y. Teh 1 Department of Physcal Mathematcal Scence Faculty of Scence Unverst Tunku Abdul Rahman Kampar, Perak, MALAYSIA E-mal: cjk_890116@hotmal.com, teohwl@utar.edu.my*, chongzl@utar.edu.my 2 School of Mathematcal Scences Unverst Sans Malaysa Penang, MALAYSIA E-mal: mkbc@usm.my 3 School of Management Unverst Sans Malaysa Penang, MALAYSIA E-mal: tehsyn@usm.my * Correspondng author 26