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An Algorithm and a Graphical Approach for Short Run Processes

where M is a number that is increased geometrically at each iteration. At each iteration, we use a BFGS unconstrained minimization algorithm with Goldstein-Armijo linę searches (see Luenberger, 1989). A barrier term was added to avoid descent directions that would violate the bounds on n, k, or f

Since the function CC(X) + MP(X) is in generał not convex in (n,kj), the following algorithm is suggested for finding the best local minimum within a region of interest:

1. For n = 1 to n = 8 do the following:

2. For each value of n, say n₀, optimize the k and f variables by running the BFGS routine from each point on a grid of starting values (n₀,k,f). For the rangę of parameter values analyzed in the numerical experiments presented by Del Castillo (1992), a grid of values (k,J) that provides good performance is to vary k from 0.5 to 3.5 in steps of 0.5 and to vary/from 1 to 41 in steps of 8. (In generał, larger values of/ should be tried for larger run lengths T.)

3. Keep the best design solution (n₀,k,f) for each n₀. If not all the values of n have been tried, go to step 1.

4. Select the design (n,k,J) which gives the minimum over all values of n tested in the linę search.

For short runs, smali values of n are used in practice (usually control charts for individuals are used, i.e., n = 1 ) so we only perform the linę search in n for the rangę (1,8). The algorithm is thus a linę search in n coupled with a grid of Penalty-Barrier trials that are solved using a BFGS method in the k and /variables. The rangę of parameter values tested by Del Castillo (1992) are realistic so the grid suggested in step 3 of the algorithm will work in most of the cases in practice. Grid search methods have been used successfiilly in the optimization of cost models for the design of control charts (see, for example, Montgomery, 1991, and Ladany, 1973). It was observed that a finer grid in k and / does not alter in a significant way the Solutions obtained. Therefore, the results in the next section are not in doubt due to convexity issues. The Appendix lists a PASCAL program that implements the algorithm described above.

Graphical Selection of a Feasible Control Chart Design

Equations [2] and [5] can be used to develop a simple graphical approach for finding a feasible X chart design. The method shows graphically where the feasible region lies so that a user can choose a design from this region. The procedurę is appealing if estimating the cost parameters