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Economic Control Chart Models with Cycle Duration Constraints

present a multiple assignable cause approach which calls for diiferent types of corrective action depending on the perceived severity of the process disruption.

A potential problem with the economic design approach is that resulting designs may exhibit unacceptable behavior with respect to criteria other than long-run cost minimization. For example, Woodall (1985) points out that (i) some economic designs have unreasonably high false alarm probabilities; and (ii) as economic models are generally designed around detection of a single extreme shift in the process parameter of interest, they may not have adequate power to detect smaller but nonetheless important shifts. Rather than reject the economic approach on this basis, a solution is to introduce one or morę constraints into the optimization process. As a recent example, Saniga (1989) considers the economic design of a joint X and R chart subject to constraints on the false alarm probability, the power at various levels of shift in the process parameter(s), and the average time to signal (ATS) an expected shift. Saniga calls the resulting design an economic statistical design and States that it reflects the best features of both the economic and statistical approaches, as costs are contained while maintaining Iow process variability and long-term ąuality. In generał, any number of constraints can be incorporated into the economic model, with a pure optimization problem being replaced by a nonlinear programming problem. Examples presented by Saniga and in this paper illustrate many cases where statistical behavior can be considerably improved under the constrained approach while realizing minimal economic penalties.

We consider a constraint on the length of time during which the process is allowed to remain in an out of control State, denoted T₀ut- Iⁿ choosing control chart parameters to minimize expected hourly cost, the economic control chart approach can only assure satisfactory process behavior in the long run. There is no guarantee of acceptable short term behavior with respect to an individual “ąuality cycle,” namely the time between the start of successive in control production cycles. This may cause problems in practice, particularly considering the current trend towards JIT or "stockless" production Systems. For example, consider a process whose output serves as direct inputs to other processes. These downstream processes may have to shut down for lack of input materials if the original process experiences an unusually long out of control period in a specific cycle, with a subseąuent increase in the number of nonconforming items produced or a lengthy down time period.

Costs resulting from problems of this naturę are difficult to incorporate into the economic control chart model's expected cost function. Therefore, rather than modify the cost function, we consider constrained approaches which minimize cost while retaining control over indiyidual cycles. Ideally, we would