00139 >858430acb91880687589de10557be5

Simpson & Keats

Economic-Statistical Approach Using Trade-OfT Analysis

In this section, we make comparisons with statistically-constrained optimization and unconstrained optimization, followed by a cost-ARL trade-ofT analysis. In constrained optimization, the ARL (or, equivalently Type I and Type II error) constraints, established a priori, may lead to a solution far from the fiat response surface. Furthermore, the selection of constrained values is often an arbitrary matter. For example, a particular in-control ARL value may be selected because subjectively, below that value there is a detrimental effect on employee morale. A priori ARL constraints ignore the cost penalties associated with extremely high in-control or extremely Iow out-of-control constraints. Even when an experienced control chart designer is available to determine "good" values, it is not elear how the cost function behaves near those design values. It is possible that slightly relaxed statistical performance constraints can result in greatly reduced costs.

As the cost surface tends to be relatively fiat in the region near the minimum cost per cycle, we will show that there are many design altematives within only a few percent of this value. These design altematives tend to offer a wide rangę of ARL1 and ARL2 combinations. As cost inereases fractionally, designs resulting in longer ARL1 and shorter ARL2 values are available. The examples which follow show the dynamics of the design characteristics near optimal cost. We use, once morę, the two examples of the previous section, that of Lorenzen and Vance from (1986a) paper and that of Montgomery (1991, page 420).

LV Example The Golden Section and grid searches discussed earlier were run on a 486 personal Computer in about 30 seconds, computing not only the global minimal cost, but also the local minimal costs for given n and H values. The Golden Section search was used to locate the minimal cost sampling interval given fixed values of the other design parameters n, H, and k. The sample size ranged from one to 12. The decision interval values ranged from 0.5 to 6.5 in 0.5 inerements, resulting in 156 different combinations of n and H. For each shift scenario, the resulting designs were ordered by cost and plotted. Table 9 shows the design alternatives closest to the optimal value for the 1.25 shift case.

Notice that deviations from optimal cost are very smali and in some cases the ARL pairs offer significant performance improvement. For instance, the minimal cost ARL2 can be decreased from 7.0 to 4.7 while nearly doubling the ARL1 (393.9 to 736.0), for only a 1.3% inerease in cost. The ARL2 can be reduced even further to 3.3, keeping the ARL1 at 393.9 by inereasing cost

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