9.5 Fast Initial Response (FIR) for Cusum Charts

Transcription

1 9.5 Fast Initial Response (FIR) for Cusum Charts The situation may arise where the researcher is concerned that the process may be in the out-of-control state at start-up or when the process is restarted after an adjustment has been made to the process. The standard cusum procedure may be slow in detecting a shift in the process that is present immediately upon start-up or restart. Lucas and Crosier (1982) suggested a modification to the standard cusum procedure that would decrease the response time required to detect an out-of-control signal that is present immediately upon start-up or restart. In the computational tabular form, a standard cusum has both the high and low cusum initially set to 0 (C 0 = C + 0 = 0). The suggested modification is to initially set both the high and low cusum to some specified positive, nonzero constant c (C 0 = C + 0 = c) between 0 and decision interval H. This is called the headstart value for a fast initial response (FIR) cusum. Lucas and Crosier suggested setting c = H 2. The design of the FIR cusum is similar to the design of the standard cusum. The two major differences between the two designs are: 1. Rather than using the ARL table for a standard cusum design, one must use a table of ARL s for the FIR cusum design. 2. Rather than using C 0 = C + 0 = 0 in the formulae of the computational form, one should use C 0 = C + 0 = H/2. Or, instead of using tables, you can also just utilize the headstart FIR option in the cusumarl function in SAS. If the process is in the in-control state initially, the cusums C + i and C i will go to zero in a small number of samples. If the process is in the out-of-control state initially, then one of the cusums will exceed H after a small number of samples. Once the FIR cusum scheme has been set up, one can proceed with quality control testing in the same manner as with the standard cusum scheme. The benefit of the FIR cusum procedure is that the decrease in the out-of-control ARL 1 is relatively large, while the decrease in the in-control ARL 0 is relatively small. This implies that an out-of-control signal can be detected in fewer runs with the FIR than without it. To adjust for the slight decrease in the in-control ARL 0, H can be adjusted to give the desired in-control ARL 0. To use a FIR cusum to generate the tabular cusum in PROC CUSUM in SAS, insert as one of the cusum options HEADSTART=nn where nn = H/2 (or any other headstart value). 195

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3 GENERATING FIR CUSUM ARLs h k delta fir arl1 arl2 h k delta fir arl1 arl DM LOG; CLEAR; OUT; CLEAR; ; ODS LISTING; OPTIONS LS=72 PS=54 NODATE NONUMBER; *** DESIGNING A CUSUM WITH A FIR; DATA firarl; DO h = 4 TO 8 BY 2; DO k =.25 TO.5 BY.25; DO delta = 0 to 2 by.5; DO fir = 0 to h/2 by H/2; arl1 = CUSUMARL( onesided,delta,h,k,fir); arl2 = CUSUMARL( twosided,delta,h,k,fir); END; END; END; END; OUTPUT; PROC PRINT DATA=firarl; ID h k delta fir; TITLE GENERATING FIR CUSUM ARLs ; RUN; 197

4 SAS code for FIR example using piston ring data DM LOG; CLEAR; OUT; CLEAR; ; * ODS PRINTER PDF file= C:\COURSES\ST528\SAS\fir1.pdf ; ODS LISTING; OPTIONS LS=78 PS=500 NONUMBER NODATE; DATA piston; DO sample=1 TO 25; DO item=1 TO 5; n= item + 5*(sample-1); INPUT diameter = diameter+70; OUTPUT; END; END; LINES; ; SYMBOL VALUE=dot WIDTH=3; PROC CUSUM DATA=piston; XCHART diameter*sample= 1 / mu0=74 sigma0=.005 h=4.0 k=0.5 delta=1.0 dataunits headstart = 2 outtable = qsum tablesummary tablecomp haxis=0 TO 25 vaxis=0 TO.035 by.005; INSET ARL0 ARLDELTA H K SHIFT / POS = se; LABEL diameter= Diameter Cusum sample = Piston Ring Sample ; TITLE FIR CUSUM for Piston-Ring Diameters ; DATA qsum; set qsum; h=4; k=.5; sigma=.005; aim=74; ** enter values **; xbar=_subx_; n=_subn_; hsigma=h*sigma/sqrt(_subn_); ksigma=k*sigma/sqrt(_subn_); retain cusum_l 0 cusum_h 0; 198

5 fir = 2*sigma/sqrt(_subn_); if sample=1 then do; cusum_l = fir; cusum_h = fir; END; if (-hsigma < cusum_l < hsigma) then do; cusum_l = cusum_l + (aim - ksigma) - xbar; if cusum_l < 0 then cusum_l=0; END; if (-hsigma < cusum_h < hsigma) then do; cusum_h = cusum_h + xbar - (aim + ksigma); if cusum_h < 0 then cusum_h=0; END; if max(cusum_l,cusum_h) ge hsigma then do; if (cusum_l ge hsigma) then do; flag= lower ; output; END; if (cusum_h ge hsigma) then do; flag= upper ; output; END; cusum_l=fir; cusum_h=fir; END; else output; PROC PRINT DATA=qsum; ID sample; var xbar n fir cusum_l hsigma cusum_h flag; TITLE FIR CUSUM with Reset after Signal ; PROC CUSUM DATA=piston; XCHART diameter*sample= 1 / mu0=74 sigma0=.005 h=4.0 k=0.5 delta=1.0 dataunits headstart=2 scheme=onesided tablesummary tablecomp haxis=1 TO 25; INSET ARL0 ARLDELTA H K SHIFT / POS = n; LABEL diameter= Diameter Cusum sample = Piston Ring Sample ; TITLE Upper One-Sided FIR CUSUM ; PROC CUSUM DATA=piston; XCHART diameter*sample= 1 / mu0=74 sigma0=.005 h=4.0 k=0.5 delta=-1.0 dataunits headstart=2 scheme=onesided tablesummary tablecomp haxis=1 TO 25; INSET LABEL H K SHIFT / POS = nw; diameter= Diameter Cusum sample = Piston Ring Sample ; TITLE Lower One-Sided FIR CUSUM ; run; 199

6 FIR CUSUM with Reset after Signal sample xbar n fir cusum_l hsigma cusum_h flag upper upper lower upper upper upper FIR CUSUM for Piston-Ring Diameters 200

7 Upper One-Sided FIR CUSUM Lower One-Sided FIR CUSUM 201

8 Lower One-Sided FIR CUSUM Cumulative Sum Chart Summary for diameter sample Sample Size Mean Std Dev Cusum Decision Interval Upper One-Sided FIR CUSUM Cumulative Sum Chart Summary for diameter sample Sample Size Mean Std Dev Cusum Decision Interval