A median run length-based double-sampling X¯$$ \overline{X} $$ chart with estimated parameters for minimizing the average sample size

The existing control charts with estimated parameters have been widely studied from the perspective of the average run length (ARL). However, when parameters are estimated, the shape and the skewness of the run length distribution change with the magnitude of the mean shift, the number of phase I samples and sample sizes. Therefore, in this paper, we argue that the median run length (MRL) and the average sample size (ASS) have several advantages over the traditional ARL to effectively evaluate the performance of the double-sampling (DS) X¯$$ \overline{X} $$ chart with estimated parameters. Precisely, by correctly accounting for parameter estimation and using the expectation by conditioning approach, we establish a theoretical method for the run length of the DS X¯$$ \overline{X} $$ chart in phase II process monitoring. Also, the MRL-based DS X¯$$ \overline{X} $$ chart with estimated parameters is optimally designed using an optimization algorithm, aiming at minimizing the in-control ASS by subjecting to both the desired in-control and out-of-control MRLs. Most importantly, the proposed optimal MRL-based chart with estimated parameters not only employs a smaller sample size on average, when the process is in-control, but also has a lower false-alarm rate and provides a clearer interpretation to practitioners. The proposed optimal chart with estimated parameters is illustrated with some real data from a tape-and-reel packing process used in a manufacturing company.