On the performance of geometric charts with estimated control limits

Abstract

The control chart based on the geometric distribution (geometric chart) has been shown to be competitive with p- or np-charts for monitoring the proportion nonconforming, especially for applications in high quality manufacturing environments. However, implementing a geometric chart is often based on the assumption that the in-control proportion nonconforming is known or accurately estimated. For a high quality process, an accurate parameter estimate may require a very large sample size that is seldom available. In this paper we investigate the sample size effect when the proportion nonconforming is estimated. An analytical approximation is derived to compute shift detection probabilities and run length distributions. It is found that the effect on the alarm probability can be significant even with sample sizes as large as 10,000. However, the average run length is only affected mildly unless the sample size is small and there is a large process improvement. In practice, the quantitative results of the paper can be used to determine the minimum number of items required for estimating the control limits of a geometric chart so that certain average run length requirements are met.