A general approach to modeling CUSUM charts for a proportion

This paper considers two CUmulative SUM (CUSUM) charts for monitoring a pro cess when items from the process are inspected and classified into one of t wo categories, namely defective or non-defective. The purpose of this type of process monitoring is to detect changes in the proportion p of items in the first category. The first CUSUM chart considered is based on the binomi al variables resulting from counting the total number of defective items in samples of n items. A point is plotted on this binomial CUSUM chart after n items have been inspected. The second CUSUM chart considered is based on the Bernoulli observations corresponding to the inspection of the individua l items in the samples. A point is plotted on this Bernoulli CUSUM chart af ter each individual inspection, without waiting until the end of a sample. The main objective of the paper is to evaluate the statistical properties o f these two CUSUM charts under a general model for process sampling and for the occurrence of special causes that change the value of p. This model ap plies to situations in which there are inspection periods when n items are inspected and non-inspection periods when no inspection is done. This model assumes that there is a positive time between individual inspection result s, and that a change in p can occur anywhere within an inspection period or a non-inspection period. This includes the possibility that a shift can oc cur during the time that a sample of n items is being taken. This model is more general and often more realistic than the simpler model usually used t o evaluate properties of control charts. Under our model, it is shown that there is little difference between the binomial CUSUM chart and the Bernoul li CUSUM chart, in terms of the expected time required to detect small and moderate shifts in p, but the Bernoulli CUSUM chart is better for detecting large shifts in p. It is shown that it is best to choose a relatively smal l sample size when applying the CUSUM charts. As expected, the CUSUM charts are substantially faster than the traditional Shewhart p-chart for detecti ng small shifts in p. But, surprisingly, the CUSUM charts are also better t han the p-chart for detecting large shifts in p.