Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 1, Dubln (Sesson STS41) p.2996 The Max-CUSUM Chart Smley W. Cheng Department of Statstcs Unversty of Mantoba Wnnpeg, Mantoba Canada, R3T 2N2 smley_cheng@umantoba.ca Keoagle Thaga Department of Statstcs Unversty of Botswana Prvate Bag UB75 Gaborone Botswana thagak@mopp.ub.bw Abstract Control charts have been wdely used n ndustres to montor process qualty. We usually use two control charts to montor the process. One chart s used for montorng process mean and another for montorng process varablty, when dealng wth varables data. A sngle Cumulatve Sum (CUSUM) control chart capable of detectng changes n both mean and standard devaton, referred to as the Max-CUSUM chart s proposed. Ths chart s based on standardzng the sample means and standard devatons. Ths chart uses only one plottng varable to montor both parameters. The proposed chart s compared wth other recently developed sngle charts. Comparsons are based on the average run lengths. The Max- CUSUM chart detects small shfts n the mean and standard devaton qucker than the Max-EWMA chart and the Max chart. Ths makes the Max-CUSUM chart more applcable n modern producton process where hgh qualty goods are produced wth very low fracton of nonconformng products and there s hgh demand for good qualty.
Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 1, Dubln (Sesson STS41) p.2997 1. Introducton Control charts are basc and most powerful tools n statstcal process control (SPC) and are wdely used for montorng qualty characterstcs of producton processes. The frst types of control charts were developed by Shewhart n the 192s and ever snce, several new charts have been developed n an effort to mprove ther capablty to quckly detect a shft of the process from a target value. The statstcal control chart, generally wth3 σ acton lmts and 2σ warnng lmts, s the longest establshed statstcal form of graphcal tool. The control chart statstcs are plotted by smply plottng tme on the horzontal axs and a qualty characterstc on the vertcal axs. A qualty characterstc s regarded to be n an ncontrol state f the statstc falls wthn the acton lmts of the chart and out-of-control f the statstc plots outsde the acton lmts. The dsadvantage of the Shewhart control charts s that they only use the nformaton about the process contaned n the last plotted pont and nformaton gven by the entre sequence of ponts. Ths has led to varous addtonal rules about runs of ponts above the mean. Attempts to ncorporate the nformaton from several successve results have resulted n charts based on some form of weghted mean of past results. In partcular the Arthmetc Runnng Mean has been used n some nstances by Ewan (3). One of the charts developed as an mprovement to the Shewhart chart s the cumulatve sum (CUSUM) chart developed by Page n 1954. Ths technque plots the cumulatve sums of devatons of the sample values from a target value aganst tme. The Shewhart control chart s effectve f the magntude of the shft s 1.5σ to 2σ or larger (Montgomery (21)). The CUSUM charts are hghly recommended by Marquardt (1984) for use n the U.S ndustry snce they can detect small changes n the dstrbuton of a qualty characterstc and thus mantan tght control over a process. An mportant feature of the CUSUM chart s that t ncorporates all the nformaton n the sequence of sample values by plottng the cumulatve
Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 1, Dubln (Sesson STS41) p.2998 sums of the devatons of the sample values from a target value. If there s no assgnable cause of varaton, the two-sded CUSUM chart s a random walk wth a drft zero. If the process statstc shfts, a trend wll develop ether upwards or downwards dependng on the drecton of the shft and thus f a trend develops, a search for the assgnable causes of varaton should be taken. The magntude of a change can be determned from the slope of the CUSUM chart and a pont at whch a change frst occurred s the pont where a trend frst developed. The ablty to show a pont at whch changes n the process mean began makes the CUSUM chart a vable chart and also helps to quckly dagnose the cause of changes n the process. There are two ways to represent CUSUM charts: the tabular (or algorthmc) CUSUM chart and the V-mask form of the CUSUM chart. We shall dscuss the tabular CUSUM chart n ths artcle. The CUSUM chart for the change n the mean and the standard devaton for varables data have been extensvely studed and two separate plots generated to assess for a shft from the targeted values. Constructng ndvdual charts for the mean and standard devaton s very cumbersome and sometmes tedous and Hawkns (3) suggested plottng the two statstcs on the same plot usng dfferent plottng varables. Ths produces a chart that s somewhat complcated to nterpret and s congested wth many plottng ponts on the same chart. Other charts have been developed wth an effort to propose sngle charts to montor both the mean and the standard devaton of the process such as those suggested by Cheng and Sprng (8), Domangue and Patch (1), Chao and Cheng (6), Chen and Cheng (8) and Cheng and Xe (9). The major objectve of ths paper s to develop a sngle CUSUM chart that smultaneously montors both the process mean and varablty by usng a sngle plottng varable. Ths chart s capable of quckly detectng both small and large shfts n the process mean and/or standard devaton and s also capable of handlng cases of varyng sample szes.
Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 1, Dubln (Sesson STS41) p.2999 2. The New Control Chart Let X = X 1, K, X n, = 1, 2, K denote a sequence of samples of sze n taken on a qualty characterstc X. It s assumed that, for each, X 1, K, X n are ndependent and dentcally dstrbuted observatons followng a normal dstrbuton wth means and standard devatons possbly dependng on, where ndcates the th group. Let µ and σ be the nomnal process mean and standard devaton prevously establshed. Assume that the process parameters µ and σ can be expressed as µ = µ aσ and σ = bσ, where a= and b = 1 when the process s n control, otherwse the process has changed due to some assgnable causes. Then a represents a shft n the process mean and b a shft n the process standard devaton and b >. Let X = ( X 1 K X n and n ) n j= 1 ( X j X 2 S = be the mean n1 and varance for the th sample respectvely. The sample mean X and sample varance 2 S are the unformly mnmum varance unbased estmators for the correspondng populaton parameters. These statstcs are also ndependently dstrbuted as do the sample values. These two statstcs follow dfferent dstrbutons. The CUSUM charts for the mean and standard devaton are based on X and S respectvely. statstcs: To develop a sngle CUSUM chart, we defne the followng ) 2 ( X ) µ Z = n (1) σ Y =Φ 1 ( n1) S H 2 σ 2 ; n1, where Φ ( z) = P( Z z), for Z ~ N (, 1) the standard normal dstrbuton. (2) 1 Φ s the nverse of the standard normal cumulatve ds-
Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 1, Dubln (Sesson STS41) p.3 trbuton functon, and H ( w; p) = P( W w p) for W 2 ~ p χ the chsquare dstrbuton wth p degrees of freedom. The functons Z and Y are ndependent and when the process varance s at ts nomnal value, Y follows the standard normal dstrbuton. The CUSUM statstcs based on Z and Y are gven by C = max[, Z k C 1], (3) C = max[, Z k C 1], (4) and S = max[, Y v S 1], (5) S = max[, Y v S 1], (6) respectvely, where C and S are startng ponts. Because Z and Y follow the same dstrbuton, a new statstc for the sngle control chart can be defned as M = max[ C, C, S, S ] (7) The statstc M wll be large when the process mean has drfted away from µ and/or when the process standard devaton has drfted away from σ. Small values of M ndcate that the process s n statstcal control. Snce M s are non-negatve, they are compared wth the upper decson nterval only. The average run length (ARL) of a control chart s often used as the sole measure of performance of the chart. The ARL of the chart s the average number of ponts that must be plotted before a pont plots above or below the decson nterval. If ths happens, an out-of-control sgnal s ssued and a search for an assgnable cause(s) of varaton must be mounted. A chart s consdered to be more effcent f ts ARL s smaller than those of all other competng charts when the process s out of control and the largest when the process s n control. The out-of-control sgnal s ssued when ether the mean or the standard devaton or both have shfted from ther target values. Therefore the plan (the sample sze and control lmts) s chosen so
Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 1, Dubln (Sesson STS41) p.31 that the ARL s large, when the process s n control and small when the process s out of control. Cox (9) suggested that the crtera for a good chart are acceptable rsks of ncorrect actons, expected average qualty levels reachng the customer and expected average nspecton loads. Therefore the n-control ARL should be chosen so as to mnmze the frequency of false alarms and to ensure adequate response tmes to genune shfts. For a predetermned n-control ARL, for quckly detectng shfts n the mean and varablty, an optmal combnaton of h and k s determned whch wll mnmze the out-of-control ARL for a specfed change n the mean and standard devaton, where h s the decson nterval and k s the reference value of the chart. The proposed chart s senstve to changes n both mean and standard devaton when there s an ncrease n the standard devaton and s less senstve when the standard devaton shfts downwards. Ths phenomenon has been observed for other charts based on the standardzed values Domangue and Patch (1). 3. Desgn of a Max-CUSUM chart We use the statstc M to construct a new control chart. Because M s the maxmum of four statstcs, we call ths new chart the Max-CUSUM chart. Monte Carlo smulaton s used to compute the n control ARL for our Max-CUSUM chart. For a gven n-control ARL, and a shft for the mean and/or standard devaton ntended to be detected by the chart, the reference value k s computed as half the shft. For these values (ARL, k), the value of the decson nterval (h) follows. For varous changes n the process mean and/or standard devaton, each ARL value s also obtaned by usng 1 smulatons. Table 1 gves the combnatons of k and h for an n-control ARL fxed at 25. We assume that the process starts n an n-control state wth mean zero ( µ = ) and standard devaton of one ( ( σ = 1 ) and thus the ntal value of the CUSUM statstc s set at
Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 1, Dubln (Sesson STS41) p.32 zero. For example f one wants to have n-control ARL of 25 and to guard aganst 3σ ncrease n the mean and 1.25σ ncrease n the standard devaton,.e., a = 3 and b = 1.25, the optmal parameter values are h = 1.215 and k = 1.5. These shfts can be detected on the second sample,.e., the ARL s approxmately two. A good feature of the Max-CUSUM chart s that smaller shfts n the process mean detected much faster than n the sngle Shewhart chart (Max chart) as seen n the next secton. Table 1 shows that small values of k wth large values of h result n quck detecton of small shfts n mean and/or standard devatons. If one wants to guard aganst 3σ ncrease n the mean and 3σ ncrease n the standard devaton, the value of h = 1.22 and the value of k = 1.5. But for a1σ ncrease n mean and 1.25σ ncrease n standard devaton, h = 4.51 and the value of k decreases to k =.5. The Max-CUSUM scheme s senstve to both small and large shfts n both mean and standard devaton. A.25σ ncrease n the process mean reduces the ARL from 25 to about 53 and a 1.25σ ncrease n the process standard devaton wth a.25 σ ncrease n the process mean reduces the ARL from 25 to about 41 runs. If both parameters ncrease by large values, the ARL s reduced to 2. Thus the ncrease wll be detected wthn the second sample. For example, an 3σ ncrease n both parameters wll be detected wthn the second sample. Another alternatve method of assessng the performance of the CUSUM chart s to fx the values of h and k and calculate the ARL s for varous shfts n the mean and/or standard devaton. Ths s dsplayed n Table 2. The value of k =.5 and thus we want to detect a 1σ shft n the mean and h = 4.51. Ths combnaton gves an n-control ARL = 25. From Table 2 t can be concluded that, even when the chart s desgned to detect a 1σ shft n the process,
Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 1, Dubln (Sesson STS41) p.33 t s senstve to both small and large shfts n the mean and/or standard devaton. Table 1: (k, h) combnatons and the correspondng ARL for the Max-CUSUM chart wth ARL = 25. ARL = 25 a b Parameter..25.5 1. 1.5 2. 2.5 3. 1..5 1.25 1.5 2. 2.5 3. 4. h k ARL h k ARL h k ARL h k ARL h k ARL h k ARL h k ARL h k ARL 4.51.5 25.21 4.51.5 91.3 4.51.5 82.42 4.51.5 41.84 4.51.5 18.81 4.51.5 11.78 4.51.5 8.63 4.51.5 5.85 8.572.125 53.21 8.572.125 67.33 8.572.125 41.1 8.572.125 34.92 8.572.125 23.97 8.572.125 18.16 8.572.125 14.32 8.572.125 1.16 6.161.25 22.4 6.161.25 25. 6.161.25 18.98 6.161.25 16.37 6.161.25 12.75 6.161.25 1.22 6.161.25 8.49 6.161.25 6.38 4.51.5 7.99 4.51.5 8.24 4.51.5 7.4 4.51.5 6.93 4.51.5 5.95 4.51.5 5.11 4.51.5 4.53 4.51.5 3.8 2.981.75 4.56 2.981.75 4.43 2.981.75 4.21 2.981.75 4.11 2.981.75 3.77 2.981.75 3.37 2.981.75 3.15 2.981.75 2.87 2.13 1. 2.76 2.13 1. 2.62 2.13 1. 2.73 2.13 1. 2.72 2.13 1. 2.61 2.13 1. 2.55 2.13 1. 2.47 2.13 1. 2.36 1.554 1.25 1.96 1.554 1.25 1.84 1.554 1.25 1.91 1.554 1.25 1.89 1.554 1.25 1.87 1.554 1.25 1.86 1.554 1.25 1.84 1.554 1.25 1.81 1.22 1.5 1.5 1.22 1.5 1.3 1.22 1.5 1.47 1.22 1.5 1.43 1.22 1.5 1.43 1.22 1.5 1.42 1.22 1.5 1.39 1.22 1.5 1.37 Table 2: ARL s for the Max-CUSUM chart wth h = 4.51 and k =.5. a b..25.5 1. 1.5 2. 2.5 3. 1. 1.25 1.5 2. 2.5 3. 4. 25.21 82.42 41.84 18.81 11.78 8.63 5.85 69.66 36.97 24.8 13.72 9.55 7.4 5.31 29.33 19.58 15.2 1.39 7.88 6.41 4.84 7.99 7.4 6.93 5.95 5.11 4.53 3.8 4.95 4.93 4.86 4.59 4.26 3.96 3.49 3.44 3.44 3.42 3.37 3.33 3.28 3.25 2.68 2.64 2.61 2.59 2.55 2.47 2.44 2.24 2.21 2.17 2.12 2.4 1.94 1.88
Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 1, Dubln (Sesson STS41) p.34 4. Comparson wth other Procedures In ths secton, the performance of the Max-CUSUM chart s compared wth those of several other charts used for qualty montorng. Most of the CUSUM charts developed are desgned to montor the mean and standard devaton separately, even the combned CUSUM charts developed montor these parameters separately n the same plots. Ths s done by plottng the charts usng dfferent plottng varables for the means and standard devatons, and then calculatng ARLs separately for each parameter. The ARL for the chart wll be taken as the mnmum of the two. The new chart (Max- CUSUM) s compared wth the omnbus CUSUM chart proposed by Domanque and Patch (1), the Max chart by Chen and Cheng (8) and the Max-EWMA chart by Cheng and Xe (9). Table 4 shows the ARL's for the Max-CUSUM chart and the omnbus CUSUM chart developed by Domangue and Patch (1) for shfts shown n Table 3. For varous changes n the mean and/or standard devaton, we have calculated the ARL's for the Max CUSUM chart and compared them wth those gven by Domangue and Patch (1) n Table 4. The Max-CUSUM chart performs better than the omnbus CUSUM chart for all shfts snce ts ARL's are smaller than those of the omnbus chart. The Max-CUSUM chart s also easy to plot and read as compared to the omnbus CUSUM chart snce t plots only one plottng varable for each sample. Table 3: Level of shfts n mean and standard devaton consdered. Label µ σ S 1 S 2 S 3 S 4 S 5 S 6.75 1.5.75 1. 1. 1. 1.2 1.4 1.3 1.2
Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 1, Dubln (Sesson STS41) p.35 Table 4: ARL s of the Max-CUSUM chart and the omnbus CUSUM chart k = 1 h = 1.279 α =.5 n = 1 Scheme S 1 S 2 S 3 S 4 S 5 S 6 Omnbus CUSUM Max-CUSUM 37. 9.2 7. 3.1 5.4 26.1 21.5 15.5 15.7 6.3 13. 5. In Table 5 we compare the Max-CUSUM chart wth the Max chart. The Max-CUSUM chart s more senstve for small shfts n the mean than the Max chart and there s no sgnfcant dfference n the performance of these charts at larger shfts even though the Max chart has slghtly low ARLs for large shfts. Ths s a major mprovement n the CUSUM scheme as exstng CUSUM charts are less senstve to large shfts n the process mean and/or standard devaton. Table 6 shows the performance of the Max-CUSUM chart and Max-EWMA chart for n-control ARL = 25. Both charts are senstve to small and large shfts n the mean and/or standard devaton wth the Max-EWMA chart performng better than the Max- CUSUM chart for both small and large shfts. These two charts use only one plottng varable for each sample and have good procedures of ndcatng the source and drecton of shfts n the process. Table 5: ARL for Max-CUSUM chart and the Max chart. ARL = 25 n = 4 Max-CUSUM a b..25.5 1. 25.21 69.66 29.33 1.25 82.42 36.97 19.58 1.5 41.84 24.8 15.2 2. 18.81 13.72 1.39 3. 8.63 7.4 6.41 Max Chart a 1. 2. 3...25.5 1. 2. 3. 7.99 3.44 2.24 25. 143.8 49.3 7.2 1.2 1. 7.4 3.44 2.21 34.3 27.2 15.9 4.9 1.3 1. 6.93 3.42 2.17 9.8 8.9 6.9 3.5 1.3 1. 5.95 3.37 2.4 2.9 2.8 2.6 2.1 1.3 1.1 4.53 3.28 1.94 1.4 1.4 1.4 1.3 1.2 1.1
Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 1, Dubln (Sesson STS41) p.36 Table 6. ARL for Max-CUSUM chart and the Max-EWMA chart ARL = 25 Max-CUSUM Max-EWMA a a b..25.5 1. 2. 3...25.5 1. 2. 3. 1. 1.25 1.5 2. 3. 25.21 82.42 41.84 18.81 8.63 69.66 36.97 24.8 13.72 7.4 29.33 19.58 15.2 1.39 6.41 7.99 7.4 6.93 5.95 4.53 3.44 3.44 3.42 3.37 3.28 2.24 2.21 2.17 2.4 1.94 25. 17.8 6.3 2.5 1.7 24.6 12.3 5.7 2.5 1.6 8.6 7.1 4.5 2.3 1.6 2.9 2.9 2.5 1.8 1.5 1.1 1.2 1.2 1.2 1.2 1. 1. 1. 1. 1.1 5. Chartng Procedures The chartng procedure of a Max-CUSUM chart s smlar to that of the standard upper CUSUM chart. The successve CUSUM values, M s are plotted aganst the sample numbers. If a pont plots below the decson nterval, the process s sad to be n statstcal control and the pont s plotted as a dot pont. An out-of-control sgnal s gven f any pont plots above the decson nterval and s plotted as one of the characters defned below. The Max-CUSUM chart s a combnaton of two two-sded standard CUSUM charts. The followng procedure s followed n buldng the CUSUM chart: 1. Specfy the followng parameters; h, k, δ and the n-control or target value of the mean µ and the nomnal value of the standard devatonσ. 2. If µ s unknown, use the sample grand average X of the data to estmate t, where X = ( X X ) / 1 L m m. If σ s unknown, use R d 2 or S c4 to estmate t, where R= ( R 1 L Rm ) / m s the average of the sample ranges and S = ( S 1 L S m ) / m s the average of the sample standard devatons, and d 2 and c 4 are statstcally determned constants. 3. For each sample compute Z and Y.
Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 1, Dubln (Sesson STS41) p.37 4. To detect specfed changes n the process mean and standard devaton, choose an optmal (h, k) combnaton and calculate C, C, S and S. 5. Compute the M s and compare them wth h, the decson nterval. 6. Denote the sample ponts wth a dot and plot them aganst the sample number f M h. 7. If any of the M s s greater than h, the followng plottng characters should be used to show the drecton as well as the statstc(s) that s plottng above the decson nterval: () If C > h, process mean. plot C. Ths shows an ncrease n the () If C > h, plotc. Ths ndcates a decrease n the process mean. () If S > h, plot S. Ths shows an ncrease n the process standard devaton. (v) If S > h, plot S. Ths shows a decrease n the process standard devaton. (v) If both C > h and S > h, plot B. Ths ndcates an ncrease n both the mean and the standard devaton of the process. (v) If C > h and S > h, plot B. Ths ndcates an ncrease n the mean and a decrease n the standard devaton of the process. (v) If C > h (v) If and S > h, plot B. Ths ndcates a decrease n the mean and an ncrease n the standard devaton of the process. C > h and S > h, plot B. Ths shows a decrease n both the mean and the standard devaton of the process.
Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 1, Dubln (Sesson STS41) p.38 8. Investgate the cause(s) of the shft for each out-of-control pont n the chart and carry out the remedal measures needed to brng the process back nto an n-control state. 6. An Example A Max-CUSUM chart s appled to real data obtaned from DeVor, Chang and Sutherland (2). The data s for measurements of the nsde dameter of the cylnder bores n an engne block. The measurements are made to 1/1, of an nch. Samples of sze n = 5 are taken roughly every half hour, and the frst 35 samples are gven n Table 7. The actual measurements are of the form 3.525, 3.522, 3.524 and so on. The entres gven n Table 7 provde the last three dgts n the measurements. Table 7: Cylnder dameter data Sample X 1 X 2 X 3 X 4 X 5 Sample X 1 X 2 X 3 X 4 X 5 1 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 25 22 25 23 22 22 25 2 25 22 2 22 22 22 23 198 22 2 24 2 187 22 198 24 192 198 24 24 22 2 29 24 24 27 198 2 217 23 2 195 28 2 198 25 22 198 25 22 195 22 24 22 25 198 2 2 23 19 2 21 22 23 24 25 26 27 28 29 3 31 32 33 34 35 27 2 23 24 26 24 23 23 2 2 26 24 2 23 26 23 26 194 24 194 198 24 23 2 2 2 194 24 2 2 194 27 23 2 2 2 Suppose based on past experence, an operator wants to detect a 1 σ shft n the mean, that s a = 1 and a 2 σ shft n the standard devaton, that s b = 2 wth an n-control ARL = 25, the
Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 1, Dubln (Sesson STS41) p.39 correspondng decson nterval from Table 1 s h = 2.475 and the reference value s k =.5. The chart s developed as follows: The nomnal mean µ s estmated by X and σ s estmated by S / c4. The sample produced the followng estmates X = 2.25 and S / c 4 = 3.31 The Max-CUSUM chart n Fgure 1 whch plots all the 35 observatons shows that several ponts plot above the decson nterval. Sample number 6 shows an ncrease n the standard devaton. After ths sample, samples 7 and 8 also plot above the decson nterval. However these ponts show that the standard devaton s decreasng towards the n-control regon. Due to very hgh value of the CUSUM statstc for the standard devaton n sample 6, the successve cumulatve values at samples 7 and 8 shows a value above the decson nterval even though the standard devaton values correspondng to these samples are n control. We therefore nvestgate the cause of hgher varablty at sample number 6. Accordng to DeVor; Chang and Sutherland (2), ths sample was taken when the regular operator was absent, and a relef, nexperenced operator was n charge of the producton lne and thus could have affected the process. Sample number 11 also plots above the decson nterval. Ths pont corresponds to an ncrease n the mean. Ths corresponds to a sample taken at 1: P.M. when producton had just resumed after lunch break. The machnes were shut down at lunch tme for tool changng and thus these tems were produced when the machnes were stll cold. Once the machnes warmed up, the process settled to an n-control state. Ths shows that the shft n the mean was caused by the machne tune-up problem. Sample 16 also plots above the decson nterval ths shft shows an ncrease n the standard devaton. Accordng to DeVor; Chang and Sutherland (2), ths sample corresponds to a tme when an nexperenced operator was n control of the process. In addton to the above mentoned ponts whch also plotted above the
Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 1, Dubln (Sesson STS41) p.31 control lmt n the Shewhart chart, Max chart and EWMA chart, sample 34 plots above the decson nterval. Ths pont corresponds to a decrease n the standard devaton. The Shewhart S chart plotted ths value close to the lower control lmt but wthn the acceptable area. Table 1 show that the Max-CUSUM chart s very senstve to small shft and thus sgnals for ths small decrease n the standard devaton. When these four samples are removed from the data, new estmates for the mean and standard devaton were computed, gvng the followng; X = 2.8 and S / c4 = 3.2. The revsed chart s shown n Fgure 2. The chart plots only one pont above the decson nterval. Ths pont correspondng to sample 1, ths shows an ncrease n the mean. Ths pont corresponds to a sample that was taken at 8: A.M. Ths corresponds roughly to the start up of the producton lne n the mornng, when the machne was cold. Once the machne warmed up, the producton returns to an n-control state. When sample 1 s removed from the data, we re-calculate the estmates and obtan X =.93 and S / c4 = 3.6. The Max- CUSUM chart for ths new data s shown n Fgure 3. All the ponts plot wthn the decson nterval showng that the process s ncontrol. Fgure 1: The frst Max-CUSUM control chart for the cylnder dameter data
Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 1, Dubln (Sesson STS41) p.311 Fgure 2: The second Max-CUSUM control chart for the cylnder dameter data Fgure 3: The thrd Max-CUSUM control chart for the cylnder dameter data 7. Concluson The ARL for ths chart reduces as the shft ncreases. One dsadvantage of the standard CUSUM chart s that t does not quckly detect a large ncrease n the process parameters and thus s not recommended for large ncrease n both mean and varablty. A good feature of the Max-CUSUM chart developed here s ts ablty to quckly detect both small and large changes n both the process mean and the process varablty. Another advantage of the Max- CUSUM s that we are able to montor both the process center and
Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 1, Dubln (Sesson STS41) p.312 spread by lookng at one chart. The performance of the proposed Max-CUSUM s very compettve n comparson wth the Max chart and the Max-EWMA chart. References Brook, D. and Evans, D. A. (2). An Approach to the Probablty Dstrbuton of CUSUM Run Length. Bometrka 59, 539549. Champ, C. W. and Woodall, W. H. (1987). Exact Results for Shewhart Control Charts Wth Supplementary Runs Rules. Technometrcs, 29, 393-399. Chao, M. T. and Cheng, S. W. (6). Semcrcle Control Chart for Varables Data. Qualty Engneerng, 8(3), 441-446. Chen, G. and Cheng, S. W. (8). Max-chart: Combnng X-Bar Chart and S Chart. Statstca Snca 8, 263-271. Cheng, S. W. and Sprng, F. A. (). An Alternatve Varable Control Chart: The Unvarate and MultvarateCase. Statstca Snca, 8, 273-287. Cox, M. A. A. (9). Toward the Implementaton of a Unversal Control Chart and Estmaton of ts Average Run Length Usng a Spreadsheet. Qualty Engneerng, 11, 511-536. Domangue, R. and Patch, S. C. (1). Some Omnbus Exponentally Weghted Movng Average Statstcal Process Montorng Schemes. Technometrcs 33, 299-313. DeVor, R. E., Chang, T. and Sutherland, J. W. (2). Statstcal Qualty Desgn and Control. Macmllan, New York. Ewan, W. D. (3). When and How to Use CUSUM Charts. Technometrcs, 5, 1-22. Gan, F. F. (3). The Run Length Dstrbuton of a Cumulatve Sum Control Chart. Journal of Qualty Technology 25, 25-215. Hawkns, D. M. (1981). A CUSUM for Scale Parameter. Journal of Qualty Technology 13, 228-231.
Int. Statstcal Inst.: Proc. 58th World Statstcal Congress, 1, Dubln (Sesson STS41) p.313 Hawkns, D. M. (2). A Fast Approxmaton for Average Run Length of CUSUM Control Charts. Journal of Qualty Technology 24, 37-43. Hawkns, D. M. (3). Cumulatve Sum Control Chartng: An Underutlzed SPC Tool. Qualty Engneerng, 5, 463-477. Hawkns, D. M. and Olwell, D.H. (8). Cumulatve Sum Charts and Chartng for Qualty Improvement. Sprnger, New York. Lucas, J. M. (1982). Combned Shewhart-CUSUM Qualty Control Schemes. Journal of Qualty Technology 14, 51-59. Lucas, J. M. and Croser, R. B. (2). Fast Intal Response for CUSUM Qualty Control Scheme: Gve Your CUSUM a Head Start. Technometrcs, 42, 12-17. Lucas, J. M. and Saccucc, M. S. (). Exponentally Weghted Movng Average Control Schemes: Propertes and Enhancements. Technometrcs, 32, 1-12. Marquardt, D. W. (1984). New Techncal and Educatonal Drectons for Managng Product Qualty. The Amercan Statstcan, 38, 8-14. Montgomery, D. C. (21). Introducton to Statstcal Qualty Control. 4th Edton, John Wley & Sons, Inc., New York. Page, E. S. (1954). Contnuous Inspecton Schemes. Bometrka, 41, 1-115. Xe, H. (9). Contrbuton to Qualmetry. PhD. thess, Unversty of Mantoba, Wnnpeg, Canada.