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McWilliams

Introduction

Control charts are widely used to maintain statistical control of a manufacturing process which is subject to possible disruptions due to the occurrence of events called "assignable causes." For example, in the case of an X -chart, a random sample of size n is taken from the process output every h time units. The process is thought to be out of control due to the occurrence of an assignable cause if the sample mean X falls outside of the rangę p ± L (ct/ Vn ), where p represents the process mean and a the standard deviation when in control. Traditional control charts use purely statistical criteria to determine the choice of n, h, and L, choosing values which satisfy a specified "false alarm" or Type I error probability a and a specified power 1 - p of detecting a given shift in the process mean. Duncan (1956) gives generał guidelines for X -chart design.

An altemative approach to control chart design is to directly incorporate economic factors. Costs related to the control process are determined, and control chart parameters n, h, and L are chosen to minimize total expected hourly costs. Relevant costs might include sampling and charting costs, false alarm costs, costs of detecting an assignable cause, and the cost of allowing the process to continue in the presence of an uncorrected assignable cause.

Economic control chart models presented in past research articles can generally be classified according to whether the ąuality characteristic of interest is measured on a continuous or a discrete scalę (variables control charts vs. attributes control charts) and to whether the process is considered to be subject to single or multiple assignable causes. Montgomery (1980) contains an extensive literaturę survey of models published prior to that datę. Examples of single assignable cause models can be found in Duncan (1956), Gibra (1971), or Chiu (1974) (variables control charts) and in Chiu (1975) or Gibra (1978) (attributes control charts). Lorenzen and Vance (1986) present a unified single assignable cause model with a common notation, and most earlier models can be expressed as specific cases of the unified model. McWilliams (1989) shows that the Lorenzen-Vance model is in generał not sensitive to the assumption of exponentially distributed in control times, while Baneijee and Rahim (1988) present a variation involving Weibull in control times and nonuniform sampling intervals.

Examples of multiple assignable cause models include Duncan (1971) (variables control charts) and Montgomery, Heikes, and Mańce (1975) or Chiu (1976) (attributes control charts). Morę recently, Tagaras and Lee (1988)