On the Performance of Geometric Charts with Estimated Control Limits

The control chart based on geometric distribution (geometric chart) has been shown to be competitive to por npcharts for monitoring proportion nonconforming, especially for applications in high quality manufacturing environment. However, implementing a geometric chart often assumes the process parameter to be 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 process parameter needs to be estimated. It is shown that the estimated control limits create dependence among the monitoring events. Analytical approximation is derived to compute shift detection probabilities and run length distributions. It is found that, when there is no shift in proportion nonconforming, the false alarm probability increases as the sample size decreases and the effect can be significant even with sample size as large as 10,000. However, the in-control average run length is only affected mildly. On the other hand, when there is a process shift, the out-of-control average run length can be significantly affected by the estimated control limits, even with very large sample sizes. 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.