A design parameter-free geometric Kullback-Leibler information control chart for monitoring Bernoulli processes

Abstract

This paper considers a design parameter-free Kullback-Leibler information (KLI) control chart for monitoring the nonconforming proportion p of Bernoulli processes in Phase II. The chart statistic is derived from the Kullback-Leibler information based on a geometric distribution. Unlike conventional charts for Phase II, the proposed chart does not require design parameters. The users only need to determine a desired in-control average run length. The performance of conventional control charts deteriorates if the prespecified optimal design parameter is not appropriate for the actual size of the shift in the monitored quality characteristic. A design parameter-free approach prevents this situation, and the time needed to find the optimal design parameter can be saved as well. The performance of charts is evaluated by the average number of observations to signal under a specific shift in p, and the penalized relative mean index assesses the overall performance. The proposed chart has the best overall performance for detecting an upward shift in p compared with the cumulative count of conforming (CCC-r), Bernoulli cumulative sum (CUSUM), Bernoulli exponentially weighted moving average (EWMA), Bernoulli generalized likelihood ratio (GLR), geometric CUSUM, and binomial GLR charts. The KLI chart has decent performance for detecting a downward shift in p if the size of the shift is small. For detecting a possible shift in p to both sides, the overall performance of the KLI chart is better than those of the charts.