Calculation of the run length distribution for Bayesian control chart

Control charts are powerful tools for monitoring processes in manufacturing and other industries to improve product or service quality. The average run length (ARL) is used as an important performance measure when designing control charts. In recent years, development of control charts with varying sampling interval (VSI) has attracted lots of attention. It has been shown that these charts give faster detection of a shift when compared with the conventional, constant sampling interval control charts. In this paper, we derive formulas for the run length (RL) distribution and for the ARL for a Bayesian control chart with two sampling intervals. When the posterior probability that the process is out of control (PO) exceeds a warning limit, the longer sampling interval is changed to a shorter one, and when the PO exceeds a control limit, the process is stopped and a search for an assignable causes of variation begins. Numerical example is provided to illustrate the computation of RL distribution and ARL developed for the Bayesian control chart with two sampling intervals.