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McWilliams

specify constraints regarding the time during which the process parameter has shifted and production continues (Si + S2 or S] + S2 + Ti, depending on the model). Also, we did not include the time required to take a sample (nE) in our out of control time calculation as in most examples this time is either assumed negligible or is relatively short. If desired, this time element could be accounted for in the development of the distribution of T_out.

Numerical Optimization Methods

Determination of optimal control chart parameters involves minimization of the expected cost function over both discrete and continuous values. The sample size n is discrete in all examples considered, and for p-chart models the control limit L is morę conveniently expressed in terms of the rejection number R, which is also discrete. We identified plausible ranges for discrete parameters, used numerical search routines to minimize cost with respect to continuous parameters (h for the p-chart examples, L and h for the X -chart examples) for each discrete value within the plausible rangę, and then choose as the solution the parameter values yielding the overall minimum cost. A morę sophisticated search procedurę, based on the Fibonacci seąuence, is described in Chiu (1975) and can be used to reduce the number of function evaluations reąuired to reach a solution. Since Computer time was costless to the author and our routines in generał ąuickly converged to optimum values (using a VAX 8650), we saw no need to implement this morę complex procedurę. The FORTRAN program used to generate all example results is avai labie from the author on reąuest.

The techniąues and Computer search routines used for expected cost function minimization over continuous parameters are summarized in Table 1. The initial approach was to use the Nelder-Mead algorithm (see Nelder and Mead (1965) or Himmelbrau (1972)) in a FORTRAN subroutine deveIoped at

Los Alamos National Laboratory. In addition, sińce the IMSL FORTRAN subroutines are available at many Computer installations, we explored the use of these routines for minimizing cost using default search parameter settings. We

used IMSL routine UVMIF for unconstrained p-chart examples, UMINF for unconstrained X -chart examples.

For constrained p-chart examples, once the discrete sample size and rejection number are determined, there is no minimization over the continuous parameter h but rather a single choice: h must be selected to satisfy Pq 95 =

Tmax- The value for h can be found using a simple equation-solving