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McWilliams

parameters such as the sample size, time between samples, and control limits are chosen to minimize an expected cost function rather than being selected according to morę traditional heuristic approaches. A variet> of articles followed Duncan's and presented variations on this basie concept. Montgomery (1980) presented a comprehensive literaturę review of materiał appearing prior to that datę. In 1986 Lorenzen and Vance published a "unified" model which established a generał approach to economic control charts. Most earlier single assignable cause models can be expressed as special cases of this unified model. Morę recently, Baneijee and Rahim (1988) presented a variation involving Weibull distributed in control times and nonuniform sampling intervals, McWilliams (1989) showed that the Lorenzen-Vance model is not sensitive to the assumption of exponentially distributed in control times, while Saniga (1989) and McWilliams (1992b) presented models where expected cost is minimized subject to one or morę constraints.

The proliferation of models published between 1956 and 1986 may leave the practitioner in a State of confusion regarding how the models relate and which is appropriate for the intended application. To address this, I provide a table and comments which relate earlier np- and X -chart models to the Lorenzen-Vance (1986) unified model. I also provide advice on algorithms and an approximation formula which can be used to improve the efficiency of Computer programs written to determine economic control chart parameters. This approximation is seen to be considerably morę accurate than those proposed in earlier articles.

The Lorenzen-Vance "Unified" Model

Consider a process which is initially in control and is subject to the occurrence of a single assignable cause. The random length of the in control period is assumed to have an exponential distribution with mean \/A. The control charting procedurę involves taking a sample of n observations from the process output every h hours. A search for the assignable cause is undertaken if the calculated process measurement, for example the sample mean or the number of nonconforming items obtained, exceeds a control limit. These limits can be expressed in terms of L, the number of standard deviations above or below the process center linę. In the case of a sampling by attributes, limits can morę naturally be expressed in terms of an acceptance number d.

In the Lorenzen-Vance model, control chart design parameters n*, L (or d*), and h* are selected as the values of n, L (or d), and h which minimize the expected hourly cost function: