00163 8bd8ef6e7bb29535cb91c8bcb2b620

164

McWilliams

Constrained Optimization

As discussed in the introduction, our goal is to find control chart parameters which minimize hourly cost while maintaining an acceptably smali probability of an overly long out of control time in any specific cycle. We therefore constrain the search for the least-cost control chart parameters by

specifying that n*, h*, and L* must be chosen so that an upper percentile (arbitrarily chosen as the 95th percentile and denoted p₀ 95) of the distribution

of T_out, the random out of control time, is acceptably smali. Morę specifically, a value T_max is chosen and (1) is minimized subject to the constraint Pq 95 £

Tmax

Imposing this constraint requires knowledge of the distribution of Tout- This time period can be broken down into three random components, assumed to be independent: Si, the time interval between the occurrence of the assignable cause and the next sampling point; S2, the time interval between that sampling point and the sampling point at which the out of control State is detected (equal to zero if detected at the first opportunity); and S3, the time reąuired to locate and correct the assignable cause. Notę that E{ S3} = Ti + T2 in the Lorenzen-Vance notation. The distributions of Si and S2 are determined by existing model assumptions, while that of S3 reąuires a new assumption. Following Gibra (1971), we assume that S3 follows an Erlang distribution with shape parameter r and scalę parameter 0, i.e., a gamma distribution with an integer-valued shape parameter. This distribution allows for a variety of shapes, leading to agreement with empirical data in many cases.

It is straightforward to show that Sj has density fimction

fl(0 =

Xe^Xt e^^h - 1

(3)

with mean and variance Ah

Ul =

he e^-1

2

^CT1

hV^h

(e^^h - l)²

(4)

Letting p represent the probability that the out of control State is