00146 &5dd89e4c9f1d1c9b9ade934fbf44bb

147

Optimization and Sensitivity Analysis

Table 10. Comparison of X and CUSUM Control Chart ARLs

Shift	N=f CUSUM Shewhart		-- CUSUM Shewhart		-- CUSUM Shewhrr>
P	3me	370		3X4	370
0.25	124.4	281.2	47.0	184.3	2C>.	133.2
0.5	36.4	155.2	12P	60.7	8.4	33.4
0.75	16.3	81.2	6.7	225	4.8	10.8
1	10.0	43.9	4.6	9.8	3.4	4.5
1.25	7.1	25.0	3.5	5.0	2.7	2.4
1.5	5.5	15.0	2.9	2.9	2.2	1.6
1.75	4.5	9.5	2.5	2.0	2.0	1.2
2	3.9	6.3	2.2	1.5	1.8	1.1
2.25	3.4	4.4	2.0	1.2	1.6	1.0
2.5	3.0	3.2	1.9	1 1	1.4	1.0

Conlrol Chart Design ParametBrs

Decision Int Rei Value

CUSUM 4.788*    0.50

Shev\hart 3.00

♦Selected so that the in-control CUSUM ARL matched the in-control Shewhart 3 sigma ARL.

The design parameters used to compare each procedurę (L = 3 for the X chart; and h = 4.788, k = 0.5 for the CUSUM) are within the rangę of values commonly found in industry. The decision interval value of 4.788 was selected so that the resulting CUSUM chart ARL1 would be close to the X chart ARL1 for L = 3. As such, valid comparisons can be madę between the two control charts design average run lengths for different levels of process shift. For the individuals' case (n = 1), the CUSUM control chart outperforms the X chart in all ARL2 cases. As the sample size increases, the X outperforms the CUSUM for large shifts. The obvious ąuestion that follows is: which chart is morę economical for monitoring a process when sample sizes are allowed to vary and all the process cost penalties are considered. Another approach is reąuired to answer this ąuestion.

We developed a program to determine optimal cost designs for the two control chart procedures. The program performs a search on the control chart design parameters, the sample size and the sampling interval to find the optimal combination of values that minimize hourly cost. The design parameters include the control interval width (L) for the X chart, and the decision interval (H) and reference value (k) for the CUSUM chart. As in the earlier studies, the optimization routine contains a grid search on the design parameters and sample size, and a Golden section search on sampling interval