Design of the Shewhart Median Scheme with the Percentile-Based Approach

In this article, we examine the design of the Shewhart median scheme using the recently proposed percentile-based (PL) approach and compare it with the average run length (ARL) and median run length (MRL) approaches. The PL approach allows the design of control schemes where practitioners can ensure the desired conditions on the in-control (IC) and out-of-control (OOC) run length (RL) performances satisfied with chosen probabilities. In other words, using the PL approach, we can guarantee the IC and OOC RL performances with desired probabilities. In addition, we examine the IC and OOC performances of the Shewhart median scheme based on the ARL, MRL, and PL approaches in detail. We conclude that the PL approach significantly improves the IC performance of the Shewhart median scheme compared to the ARL approach, without sacrificing much OOC performance when d = 1.5. Moreover, the MRL method can be treated as a particular case of the PL approach, i.e., the MRL is the 50th percentile of the RL distribution.