Some Advanced SPC Tools 1. Cumulative Sum Control (Cusum) Chart For the data shown in Table 9-1, the x chart can be generated.

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1 Some Advanced SP Tools 1. umulatve Sum ontrol (usum) hart For the data shown n Table 9-1, the x chart can be generated. However, the shft taken place at sample #21 s not apparent. 92

2 For ths set samples, we can see that startng from sample 21, the center has shfted from 1 to 11. However, ths shft may be dffcult to capture usng standard Shewhart control chart. umulatve Sum ontrol (usum) hart wll be more powerful to detect ths small but sustaned change of the mean. Let be the target for the process mean, x j s the average of the jth sample, then the cumulatve sum control chart s formed by plottng the quantty of Developng the chart: ( x j ) j1 x be the th observaton on the process Assume s known or can be estmated Accumulate dervatons from the target above the target wth one statstc, Accumulate dervatons from the target below the target wth another statstc, and are one-sded upper and lower usums, respectvely. The statstcs are computed as follows: max[, x ( K ) 1] max[,( K ) x 1] startng values are:. K s the reference value (allowance or slack value) If ether statstc exceeds a decson nterval H, the process s consdered to be out of control. It s reasonable to use H = 5. 93

3 K s often chosen halfway between the target and the out-of-control value of the mean 1 that we are nterested n detectng Shft s expressed n standard devaton unts as 1 1 K 2 2 In Example 8-1, we have 1, n 1, 1, we are nterested n detectng a shft of one. Then Out-of-control value of the mean s , K 1/ 2 and H 5 5. The equatons for the statstcs are: 1 max[, x 1.5 1] max[, 9.5 x 1] Snce and x , we have 1 max[, ] 1 max[, ].5 For the second sample, we have 2 max[, x ] max[, ] 2 max[, 9.5 x2 1 ] max[, ] 1.56 It contnues to calculate other usum values as shown next. 94

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5 2. Exponentally Weghted Movng Average ontrol hart Smlar to usum chart, the exponentally weghted movng average (EWMA) chart has been developed also for detectng small shft. It s constructed usng: z x ( 1 ) z 1 where 1 s a constant. z (the known process mean) or z x The centerlne and control lmts for the EWMA control chart are: 2 UL= L [1 (1 ) ] (2 ) enter Lne= 2 LL= L [1 (1 ) ] (2 ) where L s the wdth of the control lmts 2 As gets larger, the term [1 (1 ) ] approaches unty. Ths ndcates that after the EWMA control chart has been runnng for several tme perods, the control lmts wll approach steady-state values gven by UL= L (2 ) enter Lne= LL= L (2 ) We normally use and L 3. 96

6 Example 8.2. For the data n Table 9-1, use. 1 and L = 2.7. We know that =1 and =1. For the frst observaton of x 1 =9.45, the frst value of the EWMA s: z1 x1 ( 1 ) z.1(9.45).9(1) and z2 x2 ( 1 ) z1.1(7.99).9(9.945) and so on. The control lmts for the frst perod s 2 UL= L [1 (1 ) ] (2 ) 1 2.7(1) 1.27 LL= [1 (1.1) (2.1) 2(1) ] For perod 2, they are UL=1.36 LL=9.64 And so on. When t s stablzed, the control lmts are: UL= L =1.62 (2 ) enter Lne= =1 LL= L =9.38 (2 ) 97

7 Summary The desgn parameters of the chart are L and. The parameters can be chosen to gve desred ARL performance. In general, works well n practce. L = 3 works reasonably well (especally wth the larger value of. L between 2.6 and 2.8 s useful when.1 EWMA performs well aganst small shfts but does not react to large shfts as quckly as the Shewhart chart. 3. Varable ontrol harts for Short Producton Runs x and R charts Use devaton from nomnal (DNOM) nstead of the measured varables on the control chart M the -th actual sample measurement,t s the nomnal value, then x M T s the devaton from nomnal 98

8 the process standard devaton s approxmately the same for all parts works best when the sample sze s constant an ntutve approach when the nomnal specfcaton s the desred target value of the process. Example. For Part A, T A =5 and for Part B, T B =25. For x chart: UL= x A R LL= x A R For R chart: UL= D R LL= D R

9 4. Attrbutes harts for Short Producton Runs Use standardze attrbutes control charts All centered at wth UL and LL beng +3 and -3, respectvely 1

10 5. Modfed ontrol Lmts For 6 sgma processes. 6. SP wth Autocorrelated Data The process data may not be ndependently dstrbuted These data may be correlated as the process or equpment has ts own features Example The autocorrelaton can be measured by the functon: k ov( xt, x V ( x ) t tk ), k,1,2,... where ov( x t, x t k ) s the covarance of observatons that are k tme perods apart, V x ) s the varance of the random varable. ( t The autocorrelaton s usually estmated by the sample autocorrelaton functon: 11

11 nk We always have r 1. ( xt x)( xtk x) t1 rk, k,1,2,..., K n 2 ( x x) t 1 t For the data gven n Fgure 9.6, we can calculate the values of r k, for dfferent lag k shown below: In general, we need to use the followng chart n dealng wth real data related to SP problems: 12

12 7. SP wth Adaptve Samplng o A control chart that the samplng nterval and/or the sample sze may change dependng on the results o The area between the upper and lower lmts of the control chart may be dvded nto dfferent zones, such as: LL w L w UL o When the sample statstc falls between w and w, use standard samplng, otherwse, one may use shorter samplng nterval or larger sample sze. 8. SP and Engneerng Process ontrol SP has a long hstory of successful use n dscrete parts manufacturng In contnuous processes, engneerng process control (EP) schemes are often used to reduce varablty EP schemes are based on adjustment of some manpulatable process varable for process compensaton and regulaton SP s appled where we assume that the process can be brought nto statstcal control Statstcal control means only stable random varaton s observed around the process target Once n control, a process wll stay there for a relatvely long perod of tme However, n some ndustral settngs, process may have a tendency to drft or wander away from target There s consderable nterest n combnng these two strateges that s, enhance EP to enable detecton of assgnable cause-type dsturbances The need for ntegratng EP and SP s often questoned 13

13 Statstcal framework for EP s parameter estmaton Statstcal framework for SP s smlar to hypothess testng However, reducton of varablty s a common objectve of both strateges ontrol chart s not always best method for reducng varablty about a target Engneerng control theory accomplshes process adjustment through Predctng next observaton on process Manpulatng some other varable to affect process output Knowng effect of manpulatable varable ontrasted wth SP where process adjustment s taken only on out-ofcontrol sgnal, EP makes no attempt to dentfy assgnable causes Substantal process mprovement may occur f feedback control s used wth control charts for statstcal process montorng Some refer to combned EP and SP systems as algorthmc SP 14