00167 108af5b296acd20f6b353d6f081757

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McWilliams

minimization routine requires a cost function evaluation, find the corresponding h (once again using an equation-solving routine) which satisfies Po 95 ⁼ ^max Option (2) was only used after first verifying that the

unconstrained solution did not satisfy the constraint.

Ali approaches ąuickly lead to optimal values in most cases considered. We did find, in the case of constrained X -chart examples, that the altemative "search within a search" procedurę converged morę consistently than the nonlinear programming routine and prefer its use. In cases where convergence problems were observed with NCONF, the search was being conducted over near-optimal parameter values but the routine's convergence criteria is not satisfied. This would probably change with adjustments to the

criteria. This was not attempted as the purpose of using the IMSL routines was to determine how successful a user who did not wish to delve into the details of the optimization process would be in finding optimal parameters.

The unconstrained optimization procedures invariably lead to Solutions similar to those reported in the articles from which the numerical examples were taken. The only source of comparison for constrained optimization is the example taken from Gibra (1971), and our results differ considerably. This is discussed in a later section.

Examples

To illustrate the impact of the constrained optimization approach we apply it to a variety of numerical examples found in the economic control chart literaturę. We examine the impact, on control chart parameters and hourly costs, of imposing successively morę restrictive constraints on the distribution of T_out. The wide variety of examples considered illustrates the generał applicability of the constrained approach.

Initial example input parameters are Iisted in Table 2. Cases 1 through 5 represent attribute control chart (p-chart) models taken from Lorenzen and Vance (1986), Chiu (1975), Duncan (1978), and Gibra (1978). Cases 6 and 7 represent X -chart models from Duncan (1956) and Gibra (1971). Gibra originally specified cost values Ci = W = 0, in which case there is no unconstrained solution as there is no penalty for producing nonconforming items. We have added arbitrary non-zero values for C i and W in Case 7 to obtain an unconstrained solution in order to measure the impact of adding a constraint. Cases 8 and 9 use Gibra's original cost values and compare our constrained solution to that obtained when an approximation