Cumulative conformance count chart with sequentially updated parameters

The Cumulative Conformance Count (CCC) chart has been used for monitoring processes with a low percentage of nonconforming items. However, previous work has not addressed the problem of establishing the chart when the parameter is estimated with a prescribed sampling scheme. This is a prevalent problem in statistical process control where the true values of the process parameters are not known but it is desired to determine if there have been drifts since process start-up. This situation is also not well-covered by the conventional CCC chart, which generally assumes known process parameters. In this paper, we examine a sequential sampling scheme for a CCC chart that arises naturally in practice and investigate the performance of the chart constructed using an unbiased estimator of the percent nonconforming, p. In particular, we examine the false alarm rate and its intended target as well as deriving the mean and standard deviation of the run length; and compare the performance with that established under a binomial sampling scheme. We then propose a scheme for constructing the CCC chart in which the estimated p can be updated and the control limits are revised so that not only the in-control average run length of the chart is always a constant but it is also the largest which is not the case for the CCC chart even when the true p is known. It is shown that the proposed scheme performs well in detecting process changes, even in comparison with the often utopian situation in which the process parameter, p, is known exactly prior to the start of the CCC chart.