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Run-length distributions for various statistical process-control charts and techniques for computing them recently have been reported in the literature. The real advantages of knowing the run-length distribution for a process-control chart versus knowing only the associated average-run length of the chart have not been exploited. Our purpose is to use knowledge of the run-length distribution as an aid in deciding if an out-of-control signal is a true signal or merely a false alarm. The ability to distinguish between true and false signals is important, especially in operations where it is costly to investigate the causes of out-of-control conditions. Knowledge of the run-length distribution allows us to compute likelihood ratios, which are simple to calculate and to interpret and which are used to determine the odds of obtaining an out-of-control signal at a particular run length when a shift in the process mean actually has occurred vis-a-vis no such shift. We extend our analysis in a Bayesian sense by incorporating prior information on the distribution of the shift size of the process mean, combined with the likelihood ratio obtained from the run-length distribution, to determine if a shift larger than a critical size has occurred. We give examples for the Shewhart chart, the exponentially weighted moving-average chart, and the special-cause control chart for processes with autocorrelated observations. The examples show that the current recommended usage of the average-run length alone as a guide for determining whether a signal is a false alarm or otherwise can be misleading. We also show that the performance of the traditional charts, in terms of their average-run length, can be enhanced in many instances by using the likelihood-ratio procedure. |
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