The bernoulli CUSUM chart for detecting decreases in a proportion

Abstract

This article considers the Bernoulli CUSUM chart for detecting a decrease in the proportion p of nonconforming items when a continuous stream of Bernoulli observations from the process is available. The properties of the Bernoulli CUSUM chart can be obtained using a Markov chain model. However, in some cases, the number of transient states in the Markov chain may be so large that it is not feasible to work directly with the transition matrix. This article provides a solution to the equations used in defining the Markov chain so that properties of the chart can be obtained without explicitly using the transition matrix. This solution allows the practitioner to determine the control limit required to give a specified in-control performance. A simple example is used to show that using the Bernoulli CUSUM chart is a better option than the standard practice of artificially grouping the observations into samples of size n > 1 and using a Shewhart chart. Copyright © 2012 John Wiley & Sons, Ltd.