A Two-Stage Cumulative Quantity Control Chart for Monitoring Poisson Processes

Abstract

This paper is concerned with the cumulative quantity control chart (CQC chart) defined based on the gamma random variable that is the quantity of product inspected in order to observe r (≥ 1) nonconformities. A CQC chart with a small value of r has a smaller average run length (ARL), but has lower discriminating power for detecting shifts in the nonconforming rate λ than a CQC chart with a large r. In the present paper, inspired by the concepts of double sampling procedures in acceptance sampling as well as reduced inspection in MIL-STD-105E and the procedures for CSP plans in MIL-STD-1235C, a two-stage CQC chart is proposed aiming at gaining both the advantages of the 1-stage CQC charts with r = 1 and r = 2. The authors apply a rigorous analytic approach to perform sensitivity analysis to compare the discriminating power of CQC charts in detecting change in λ, rather than using the less rigorous approach of numerical verification based on an ac hoc choice of values of parameters. The authors also obtain and compare the analytic expressions for the ARLs of these CQC charts. Economic analysis of the CQC charts is performed. Numerical examples will be given to compare the performance of these control charts in terms of discriminating power (in detecting shift of λ), ARL, and average total cost, and to show that each of these charts could be the best choice in each specific situation. It is also shown that, when the penalty cost due to nonconformities is relatively low, it is optimal not to apply statistical process control at all.