Monitoring a Proportion Using CUSUM and SPRT Control Charts

Abstract

In many process monitoring problems, items from the process are classified into one of two categories, usually called defective and nondefective, and the objective is to detect any change in the proportion p of defective items produced by the process. In some applications, there may be a continuous stream of inspection data from the process, as would occur with 100% inspection. In other applications, the inspection data may be in the form of samples taken from the process. The traditional control chart for monitoring p is the Shewhart p-chart. For this chart, the proportion defective for segments or samples of n items is determined and plotted. There are a number of disadvantages to using the Shewhart p-chart. In particular, the p-chart is not efficient for detecting small changes in p, and when the target value is close to 0, the discreteness and skewness of the binomial distribution frequently produces a p-chart with undesirable properties. This paper evaluates and compares several control charts which are alternatives to the Shewhart p-chart. A CUSUM chart for monitoring p can be based on the binomial distribution when items are grouped into segments or samples. A CUSUM chart can also be based on the Bernoulli distribution when inspection results are recorded and used individually as they are obtained. These two CUSUM charts are compared in terms of the time required to detect a shift in p for the case of 100% inspection and for the case of samples of items. When samples are taken from the process, a highly efficient control chart can be based on using sequential sampling at each sampling point. In particular, an SPRT can be applied at each sampling point and the decision of this test used to determine whether to signal or go to the next sampling point. It is shown that this SPRT chart will detect shifts in p much faster than CUSUM charts and the Shewhart p-chart.