The statistical optimal design of Shewhart control charts with supplementary stopping rules
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Monitoring production processes to assure product quality has been a major problem in industrial engineering and quality control. These processes are subject to shifts to "out-of-control" states resulting in the production of nonconforming items. In general, desirable control schemes are those that require a small number of samples (small run length) for the detection and correction of these shifts in the process meanwhile providing a large in-control average run length (A.R.L.);In this study, a method for designing control schemes for X-bar charts is presented and discussed. This method can be used in conjunction with economical-design methods for these charts or as an alternative to these methods when estimates of cost parameters are not available. Our method is based exclusively on the average run length properties of the different control schemes and it seeks the minimization of the out-of-control A.R.L. (by the appropriate selection of the control limits) for a given in-control A.R.L;A Markov-chain based method is presented for obtaining the exact A.R.L. properties of Shewhart control charts with supplementary stopping rules. Computer code is given for obtaining the transient states and the transition probabilities of the Markov chain representation of several control schemes. Computer programs and algorithms (based on penalty-function methods and inverse parabolic interpolation) to find the optimal control limits for various control schemes are given, as well. For practitioners, tables and nomographs, giving the control-limit combinations that minimize the out-of-control A.R.L. for different values of the in-control A.R.L. and for several control schemes, are provided. An optimal control scheme is compared to a commonly used scheme and their performance, under different out-of-control situations, is evaluated using Monte Carlo simulation techniques.