A review and extensions of sample size determination for estimating process precision and loss with a designated accuracy ratio

Abstract

Numerous process capability indices, including Cp, Cpk, Cpm, and Cpmk, have been proposed in the manufacturing industry providing numerical measures of process capability based on various criteria. The index Cp provides a measure of process precision, which reflects the consistency of product quality. The index Cpm, also called the Taguchi index, essentially measures process loss. Lower confidence bounds estimate the minimum process capability conveying critical information regarding product quality, which is essential to quality assurance. Existing research works have focused on constructing lower confidence bounds, but investigation on the sample sizes required for a specified accuracy ratio of the estimation has been comparatively neglected. The sample size determination is important, as it is directly related to the cost of the data collection plan. In this paper, we review some existing formulas for lower confidence bounds that can be used to determine the sample size required for a given estimating accuracy ratio. We then present a different approach, with efficient MATLAB programs, to obtain the sample sizes using the UMVUE (uniformly minimum variance unbiased estimator) of Cp, and the MLE (maximum likelihood estimator) of Cpm. We also provide tables of the sample size information for the engineers/practitioners to use for their in-plant applications.