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Del Castillo

Introduction

There has been an increased recent interest in Statistical Process Control techniąues for short run processes. However, most models for the economic design of control charts are based in the model by Duncan (1956) who assumed the production process operates for an infinite time horizon. While this assumption may be valid for continuous process or mass-production industries, a finite-length production run can have an impact on the optimal economic design of a control chart.

Recently, Del Castillo and Montgomery (1996) presented a model for the optimal-economic design of X charts for short production run processes. Their model converges to Duncan's model if the length of the production run tends to infinity. Their results indicate a strong relation between the length of the production run and the statistical power of the optimal chart design. This relation was found to be a function of the production process assumed, where both a repetitive manufacturing system and a job-shop (non-repetitive) process were considered.

In this paper an algorithm for the constrained optimization of the model by Del Castillo and Montgomery (1996) is presented. Constraints are introduced in the model to deal with some of the problems that economically-designed control charts tend to have, as Woodall (1986) has pointed out. For the case when cost and parameter estimation is impractical, we present a simple graphical method for finding a feasible chart design, i.e., a design that satisfies all the constraints considered.

Model Description

In this section we reproduce the model presented by Del Castillo and Montgomery (1996) for completeness in our discussion. For details about the derivations, refer to Del Castillo (1992) and Del Castillo and Montgomery (1995).

The objective is to minimize the expected total cost per cycle, CC , as a function of the chart design variables: sample size, n , control limits width multiple, k , and number of samples,/, to take during the production run. The production run length T is a given parameter, usually determined from inventory or production planning considerations.

The model is based on the time variables described in Figurę 1. A cycle is defined as min (R, T) where R is the point in time at which the process shifts to the out of control mean of p±5a . The parameter 5 indicates the size of the shift in the mean. Del Castillo (1992) shows that the following relations