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McWilliams

To understand the wide rangę of cost penalties associated with reducing g^, consider the standard deviation values ai, 02, 03, and o_out

presented in Tables 3A and 3B. By reducing h and choosing parameter combinations which increase p we can substantially reduce ai and 02, while <33 is a fixed input parameter. In fact, specification of <33 establishes a lower bound for a_out. The opportunity to reduce p_Q with a corresponding

reduction in a_out, is therefore greatest when cti and 02 contribute heavily to a_out Comparing Cases 3, 5, and 6, we see that this occurs in Cases 3 and 6 but not in Case 5. In Case 5, virtually all variability in T₀ut is due to variability in S3, so there is little opportunity for improvement.

Regarding the impact of increasing the Erlang shape parameter r, the results of Case 1 are as expected. As the variability in S3 decreases, the constraint is easier to meet so total cost decreases. Other values such as the sample size n and the intersample interval h do not exhibit consistent behavior in this limited study.

As mentioned in the introduction, other constraints such as statistical constraints can be added to the model. To illustrate this, notę that in Example 6 the Type I error probability a is, at best, eąual to 0.032, leading to an average run length (ARL) when in-control of 31.25. Suppose that this ARL value is considered to be unacceptably Iow and that a minimum value of 100 is desired, leading to the reąuirement a ^ 0.01. Tables 5 A and 5B illustrate the result of adding this constraint to each Example 6 case originally considered in Tables

3B and 4B. The constraint on a forces L = 2.58. Corresponding changes include an increase in n, presumably to maintain power which in tum keeps p₀ 95 within limits, and a minimal increase in total cost. Values for the

intersample interval h and the power p are relatively insensitive to the addition of the new constraint. Control chart parameters for this example were found

using IMSL® routine NCONF.

Routine NCONF could also be used to introduce other constraints involving control chart performance measures, such as power or percentiles of Tout calculated at other possible shift levels in the process parameter. We do not provide results for such an analysis, but do investigate the performance of our original constrained designs at shifts other than the expected shift. This is illustrated in Table 6 for Examples 1 and 6. As expected, both power and the percent of time in control increase as the severity of the shift increases, while the 95th percentile ofT_out decreases. If any performance measure shown on