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Optimization and Sensitiyity Analysis

Table 9. LV Example with Shift = 1.25

Cost Cost Sampłe Reference Decision Sampling ARL1    ARL2

Increase Size Value lnterval lnterval

$214.74	0.0%	1	0.625	4.5	0.20	736.0	7.8
$214.95	0.1%	1	0.625	5	0.18	1369.8	8.6
$215.32	0.3%	1	0.75	4	0.19	991.1	8.4
$215.50	0.4%	1	0.5	5.5	0.20	755.2	8.1
$215.66	0.4%	1	0.5	6	0.18	1239.7	8.7
$216.07	0.6%	1	0.625	4	0.23	393.9	7.0
$216.09	0.6%	1	0.75	3.5	0.22	468.8	7.4
$216.11	0.6%	1	0.625	5.5	0.17	2542.2	9.4
$216.30	0.7%	1	0.5	5	0.22	458.5	7.4
$216.35	0.8%	1	0.75	4.5	0.17	2088.6	9.4
$216.49	0.8%	1	0.5	6.5	0.17	2029.1	9.4
$217.27	1.2%	1	0.875	3.5	0.18	1094.7	9.0
$217.38	1.2%	2	0.875	3	0.38	456.8	4.1
$217.79	1.4%	2	1	2.5	0.40	357.3	4.0
$217.80	1.4%	2	0.75	3.5	0.38	468.8	4.2
$217.82	1.4%	1	0.625	6	0.16	4705.4	10.2
$217.85	1.4%	1	0.875	3	0.21	456.8	7.7
$218.30	1.7%	2	0.875	3.5	0.33	1094.7	4.7

2.5%.

The plot shown in Figurę 1 is a three dimensional representation of the ARL trade-offs versus cost. The data points for this illustration were obtained by applying the model to the LV example (shift of 0.75). The sampłe size and CUSUM decision interval were varied (n: 1-12, H: 4.0-6.5) to generate the points below. The resulting surface highlights the relative insensitivity of changes in ARL1 to changes in cost. Considerable gain in the in-control run length can be madę for smali cost penalties. Conversely, changes in ARL2 cause steep changes in cost. The costs increase exponentially with decreasing ARL2 for a given ARL1. This asymptotic behavior is also illustrated later in Figurę 6.