Economical operation of the Cpm control chart for monitoring process capability index

Process capability indices are widely used to provide the evaluation measure of a process. Especially, the process capability index Cpm, which is defined by the range of the process standard specification limits and the deviation from a target value, is called the Taguchi index. Boyles has investigated the statistical characteristics of the estimator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\hat{C}_{pm}$\end{document}, and also proposed a technique for the Cpm control chart. Since the process capability index Cpm is based on the concept of the Taguchi’s quality loss, the process capability index Cpm already includes an economical concept. In this article, we evaluate an operating cost consisting of the sampling cost, the sample cost, and the quality loss of failing to detect an out-of-control state when the Cpm control chart is used. Then, we derive an optimal operating plan by sample size and sampling interval in order to minimize the ceiling value of the operating cost based on the min–max criterion.