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478

Del Castillo

ATS =

T

/o -p.)

<y T

or

¹⁵¹

Ineąuality [5] indicates that the ATS has to be smaller than some fraction y of the total production run time. This implies that on the average, out of control conditions will be detected when we have processed y% of the total lot size. Notę, however, that the average time to signal used in [5] may be referred to as the nominał ATS, sińce the production run time can be exceeded before any signal occurs. It is easy to show that the average time T to signal is in fact given by

and therefore we can still use [5] for simplicity.

Minimization Algorithm

For minimizing Eąuation [1] subject to constraints [2]-[5], a nonlinear constrained optimization algorithm must be used. Saniga (1989) used the Generalized Reduced Gradient algorithm for solving economic constrained designs. We have investigated the use of penalty methods. The ill-conditioning typical of penalty methods can be avoided using the partial conjugate approach proposed by Luenberger (1989). This method is well suited for our problem sińce the number of active constraints is quite smali. The penalty function we have used is

P(X) = (Power* -1 + P.Y +(<*-«J* +(1/_Y -/(1 -P.)f +(«/-TP)*

where X = (n,k,J) ■ Then, unconstrained optimization techniąues are needed to minimize

CC(X) + MP(X)