Sum of defects (SOD) chart for joint monitoring of the two parameters of a zero-inflated Poisson process

Rapid technological advancements and automation in today's world have led to many high quality manufacturing processes, where most of the products are free from defects. The defects data from high quality processes are commonly modeled by twoparameter zero- inflated Poisson (ZIP) distribution. Most of the existing control charting procedures for ZIP processes are developed assuming that all the manufactured units are inspected one by one, and most often separate control charts are proposed for monitoring the two parameters of a ZIP process. However, joint monitoring of the two parameters in a single chart can offer significant operational advantages because one needs to focus on a sole chart and a single charting statistic. Again, in many manufacturing set ups 100% inspection may not always be feasible. Recently, a few one-chart schemes with group inspection of small samples are reported in literature, and all these schemes require estimation of the two parameters based on accumulated samples till the current sampling stage for computation of the monitoring statistics. This accumulation of samples introduces several limitations in these schemes. In this paper, probability mass function and the cumulative distribution function (cdf) of the sum of defects (SOD) in a sample of size are defined, and a control chart for SOD is proposed. Both the parameters of a ZIP process can be jointly monitored using the SOD chart. Again, since no accumulation of the samples is required for application of the SOD chart, it is free from all limitations of the existing one-chart schemes with group inspection. Two case studies based on past data are presented. The performance of the SOD chart is found to be very encouraging.