Optimising Variable Sample Size X-bar Chart through Median Run Length with Estimated Process Parameters

The classic charting procedures for designing the estimated process parameters-based variable sample size (VSS) X-bar chart rely on the average run length (ARL) criterion. Nevertheless, variations in the number of Phase-I samples and sample size, as well as the magnitude of the process mean shift affect the skewness of the run-length distribution for a control chart. Hence, we claim that the ARL can be a misleading metric when adopted in the estimated process parameters-based control charts. Instead, examining percentiles of the run-length distribution, which focus on the run-length behaviour, are more realistic and intuitive. From this point of view, this paper aims to develop two new optimal VSS X-bar charts using estimated process parameters, by minimising the (i) median run length (MRL) and (ii) expected MRL criteria, for known and unknown shift-size cases, respectively. Besides, the 5th and 95th percentiles are computed to closely examine the variability of the run length. In this paper, two VSS schemes that involve estimated process parameters are investigated extensively, i.e., the first sample size can be either small or large. Various practically manageable Phase-I sample sizes and magnitudes of process mean shift are implemented in the optimal design of the proposed charts. The results ascertain that the proposed optimal VSS X-bar charts based on estimated process parameters not only provide a comprehensible interpretation for quality practitioners, but also give a low false-alarm rate. The proposed optimal charts are illustrated using real data from a wafer substrate manufacturing company.