The Use of Probability Limits of COM–Poisson Charts and their Applications

Abstract

The conventional c and u charts are based on the Poisson distribution assumption for the monitoring of count data. In practice, this assumption is not often satisfied, which requires a generalized control chart to monitor both over‐dispersed as well as under‐dispersed count data. The Conway–Maxwell–Poisson (COM–Poisson) distribution is a general count distribution that relaxes the equi‐dispersion assumption of the Poisson distribution and in fact encompasses the special cases of the Poisson, geometric, and Bernoulli distributions. In this study, the exact k‐sigma limits and true probability limits for COM–Poisson distribution chart have been proposed. The comparison between the 3‐sigma limits, the exact k‐sigma limits, and the true probability limits has been investigated, and it was found that the probability limits are more efficient than the 3‐sigma and the k‐sigma limits in terms of (i) low probability of false alarm, (ii) existence of lower control limits, and (iii) high discriminatory power of detecting a shift in the parameter (particularly downward shift). Finally, a real data set has been presented to illustrate the application of the probability limits in practice. Copyright © 2012 John Wiley & Sons, Ltd.