Conditional design of the Shewhart <i>X̄</i> chart with unknown process parameters based on median run length

Abstract

Numerous researches have been done on the Shewhart X̄ chart based on the average run length (ARL) metric. Since the shape of the run length (RL) distribution changes with the mean shift size, the median run length (MRL) is argued to be a better criterion for evaluating the performance of the Shewhart X̄ chart. Moreover, when the process parameters are unknown, the phase 2 properties of the Shewhart X̄ chart are conditioned on the parameter estimates arising from different practitioners in phase 1. This variability among the estimated process parameters is usually called as the between-practitioners variability. In order to investigate this variability in the conditional MRL values, both the average MRL (AMRL) and the standard deviation of MRL (SDMRL) will be used together in our article. The performance analyses of the MRL-based Shewhart X̄ chart are provided. To prevent too many lower in-control MRL values than the desired one, an appropriate bootstrap approach is adopted to adjust the control limits, and to further balance the in- and out-of-control MRL values of the Shewhart X̄ chart. [Submitted: 26 September 2018; Accepted: 15 January 2022]