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McWilliams

constrain the solution by placing an upper bound on T₀ut- This is not possible due to the random naturę of the process. We can, however, constrain an upper percentile of the distribution of T₀ut> such as the 95th percentile. We refer to such a constraint as a cycle duration constraint. This constraint provides assurance that the probability of an unacceptably long out of control period is smali, and is consistent with Woodall’s (1985) suggestion of specifying, under the statistical design approach, a percentage point of the out of control run length distribution. A similar approach, expressed in terms of the number of nonconforming items produced during a ąuality cycle rather than the out of control time, was originally proposed by Gibra (1971). We refine and generalize Gibra's concept, deriving exact distributional results, and illustrate the value and wide applicability of the constrained approach through a variety of numerical examples.

The logie behind the cycle duration constraint differs from that of Saniga's ATS constraint in several ways. The cycle duration constraint deals with the entire out of control period, while the ATS constraint focuses on the time to signal the presence of an assignable cause. Morę significantly, the cycle duration constraint is expressed in terms of an upper percentile of the out of control time distribution rather than an average. Under the ATS constraint there is no control over the possibility of an unacceptably long out of control time during a particular ąuality cycle - we only know that behavior is acceptable on average. With the cycle duration constraint, the probability of an unacceptably long out of control time is maintained at user-specified limits such as 0.05 or 0.01. Finally, Saniga proposes using the ATS constraint in conjunction with a collection of statistical constraints which consider process parameter shifts other than the nominał "expected" shift, while the focus of this article is on a single expected shift and a single constraint. Notę, however, that the cycle duration constraint can easily be supplemented by additional constraints involving other parameter shift levels or other performance measures. This is illustrated in the Examples section.

The Lorenzen-Vance unified economic control chart model is summarized in the following section. Subseąuent sections contain the derivations needed to apply the constrained optimization approach, the computational methods used, and a wide variety of numerical examples. These examples illustrate situations where the 95th percentile of the distribution of out of control times can be substantially reduced at very little cost. The examples are followed by a comparison of our exact results, for numerical examples presented by Gibra (1971), to those obtained using Gibra's approximate optimization procedurę.