9.1 The Cumulative Sum Control Chart


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1 Learnng Objectves 9.1 The Cumulatve Sum Control Chart Basc Prncples: Cusum Control Chart for Montorng the Process Mean If s the target for the process mean, then the cumulatve sum control chart s formed by plottng the quantty j j1 C ( x ) (1) aganst the sample. C s called the cumulatve sum up to and ncludng the th sample. The cumulatve sum control charts are more effectve than the Shewhart charts for detectng small process shfts. It has applcaton n
2 chemcal and process ndustres where the sample sze s one. If the process mean remans at target value, the cumulatve sum defned n (1) s a random walk wth mean. If mean shfts upward to some value 1 >, a upward or postve drft wll develop n cumulatve sum C. Conversely, f mean shfts downward to some value < 1, then a downward or negatve drft wll develop n C. For detals, consder data n Table 9.1 (page 41), Shewhart control chart n Fgure 9.1 (page 41) and cumulatve sum control chart n Fgure 9. (page 43). There are two ways to represent cusum (1) Tabular (or algorthmc) () V mask. However, we wll mostly concentrate on Tabular cusum.
3 9.1. The Tabular or Algorthmc Cusum for Montorng the Process Mean Let x be the th observaton on the process and dstrbuted as normal wth mean and varance. We also assume that the as the target value for the qualty characterstc x. The Tabular Cusum C max[, x ( K) C ] () 1 C max[, ( K) x C ] (3) 1
4 where the startng values are C C. The statstcs C and one sded upper and lower cusums respectvely. C are called K s usually called the reference value (or allowance value) and often chosen halfway between the target value and outofcontrol value of the mean 1 that want to detect quckly. Thus f 1 (or 1 ), then 1 K (4) If ether C or C exceed the decson nterval H, the process s consdered to be out of control. The reasonable estmate of H s H 5. Consder data n Table 9.1 (page 41). We have the followng nformaton 1, , k 1/, H 5 The Cusum calculatons are presented n Table 9. (page 45), whch show that the upper sde cusum at perod 9 s C > 5, we would conclude that the process s out of control at that pont. The counter N records the number of consecutve perods snce the uppersde cusum C rose above the value of zero. Snce, N 7 at perod 9, we conclude that the process was last n control at perod 9 7, therefore, the shft lkely occured between perod and 3. Cusum Status Charts Plottng the values of C and C versus the sample number. Fgure 9.3a shows the cusum status chart for the data n Example 9.1. Each of the vertcal bar represents the value of C and C n perod. One should also
5 plotted the observatons x for each perod on the cusum status chart as the sold dots. The cusum s partcularly useful n determnng the assgnable causes by just count backsword from the outofcontrol sgnal to the tme perod when the cusum lfted above the zero to fnd the frst perod followng the process shft. The counters N and N are used n ths capacty. Estmaton of the process mean followng the shft C K, f C > H N C ˆ K, f C H > (5) N Fgure 9.3(a) Cusum status chart for example 9.1(manual chart) Recommendaton for Cusum Desgn Some general recommendaton for selectng H and K based on several research studes are gven here. Defne H h and K k, where s the SD of the sample varable used n formng cusum. Usng h 4,5, k 1/, wll provde a cusum that has good ARL propertes aganst a shft of about 1 n the process mean. To llustrate how well the recommendaton of h and k work, please read page 4849, see the followng Table 9.3 and Table 9.4.
6 One sded Segmund (1985), approxmaton of ARL e ARL b b 1 (6) * * where k for the upper onesded cusum C, k for the lower * ( 1 ) onesded cusum C, b h and. For, use ARL b. * The quantty represents the shft n the mean, n the unts of, for whch * ARL s to be calculated. If, we calculate ARL and for *, we calculate ARL 1. The ARL for two sded cusum s ARL (7) ARL ARL
7 * Example, page 49: Suppose we have, k 1/, h 5, set, then 1/, b We have ( 1/) e ( 1/ ) ARL 938. ( 1/) By the symmetry ARL ARL Thus the ncontrol ARL for two sded cusum s ARL Ths s very close to the true ARL value of 465 shown n Table 9.3. * However, f mean shfts by, then, 1.5 for the upper onesded * cusum, and,.5 for the lower onesded cusum. Then usng equatons (6) and (7), we have ARL3.89. The exact value shown n Table 9.3, whch s The Standardzed Cusum Let x be the th observaton on the process and dstrbuted as normal wth mean and varance. Then x y (8) s the standardzed value of x. Then the standardzed twosded cusum are C max[, y k C ] (9) 1 C max[, y k C 1] (1) where the startng values are C C. There are two advantages of standardzed cusum 1. Many cusum charts have the same values of h and k.. Can be used for controllng the process varablty
8 9.1.8 A Cusum for Montorng Process Varablty Let x be the th observaton on the process and dstrbuted as normal wth mean and varance. Let x y be the standardzed value of x. Then Kawkns (1981, 1993) suggests the followng standardzed quantty y.8 (11).349 The statstc are senstve to both mean and varance changes. Snce the ncontrol process the dstrbuton of approxmately N (,1), the twosded standardzed scale cusum are gven below S max[, k S ] (1) 1 S max[, k S ] (13) 1 where the startng values are S S. If the process standard devaton ncreases, the values of S wll ncrease and exceed h, where as f the standard devaton decreases, the values of S wll ncrease and exceed h. Therefore, the nterpretaton of standardze scale cusum s smlar to cusum to mean Ratonal Subgroups For n > 1, replace x by x and by x. n The VMask Procedure An alternatve to the use of a tabular cusum s the Vmask control scheme proposed by Barnard (1959). The Vmask s appled to successve values of the cusum statstc C y y C j 1 j1 ( x ) where y s the standardzed observaton y. A typcal Vmask s shown n Fgure 9.5.
9 9. The Exponentally Weghted Movng Average (EWMA) Control Chart The EWMA control chart s also a good alternatve to the Shewhart control chart when we are nterseted n detectng a small shft. The performance of EWMA CC s approxmately equvalent to that of the cumulatve sum CC The EWMA Control Chart for Montorng the Process Mean The EWMA s defned as z x (1 ) z 1 (14) where < 1 s a constant and the startng value s z. The EWMA z s a weghted average of all prevous sample means and can be expressed as 1 j (1 ) j (1 ) (15) j z x z j The weght (1 ) decrease geometrcally wth the age of sample mean. If., then the weght assgned to the current sample mean s. and the
10 weghted gven to the precedng means are.16,.18,.14 and so forth. See Fgure 9.6, page 4. If the observatons x are ndependent random varable wth varance, then the varance of z s [1 (1 ) z ] (16) The control lmts for EWMA Control Chart are UCL L Center lne LCL L [1 (1 ) [1 (1 ) ] ] (17)
11 Example 9., page 48 The EWMA calculatons for Data n Table 9.1, wth.1, L.7 and 1 have been presented n Table 9.1, page 41 and the correspondng EWMA control chart s shown n Fgure 9.7. The EWMA control chart sgnals at observaton 8, so we could conclude that the process s out of control. The EWMA control chart has extensve applcaton n tme seres analyss.
12 9.. Desgn of an EWMA Control Chart Several studes on EWMA suggest that 1. the values of n the nterval.5.5 work well n practce. A good rule of thumb s to use smallest to detect smaller shft. Western Electrc rule use.4.. L 3 (the usual threesgma lmts) works well. Average run length (ARL) for dfferent combnatons of and L are gven n Table Ratonal Subgroups The EWMA control chart s often used for the ndvdual measurements. However, f n > 1, replace x wth x, wth x n equaton (17). n
13 9.3 The Movng Average Control Chart Let x, x, 1 be a set of observatons. Then the movng average of span (wndow) w at tme s defned as xw 1 x M w The frst w 1 observatons x x 1 M ; 1,, w 1. Usually the value of w 3,4,5. If more than 5, less senstve to locate the trend. The varance of M s 1 1 V ( M ) V x ( j ) w w w j w1 j w1 Thus f denotes the target value of the mean used as the center of the control chart, then the threesgma control lmts are 3 UCL w Center lne 3 LCL w Then plot each M for each of wth the control lmts n (19). (18) (19)
14 Here we set up a movng average (MA) control chart for data n Table 9.1, wth w 5. The calculatons are shown n Table 9.14 and correspondng control chart s shown n Fgure 9.8, page 49. No ponts exceed the control lmts. The MA control chart s not as effectve aganst small shft as ether cusum or the EWMA. However, very easy to mplement compare to others. Fgure 9.8: Movng Average control chart wth w5, Example 9.3 Exercse 9.4, page 43 Exercse 9.5, page 43 Exercse 9.7, page 431 Exercse 9.1, page 431 Exercse 9.13, page 431
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