Conditional median run length performance of the synthetic X ¯ chart with unknown process parameters

Synthetic‐type charts are efficient tools for process monitoring. They are easy to design and implement in practice. The properties of these charts are usually evaluated under the assumption of known process parameters. This assumption is sometimes violated in practice, and process parameters have to be estimated from different phase I data sets collected by different practitioners. This fact causes the between‐practitioners variability among the properties of the synthetic‐type charts designed for each practitioner. In fact, the shape of the run length distribution of the synthetic‐type charts changes with the mean shift size. As a good alternative, the median run length ( M R L ) metric is argued to evaluate the properties of different control charts. In this paper, the M R L is used as a measure of the syntheticX ¯ chart's performance, and the conditional M R L properties of the syntheticX ¯ chart with unknown process parameters are investigated. Both the average M R L ( A M R L ) and the standard deviation of M R L ( S D M R L ) are used together to investigate the chart's properties when the process parameters are unknown. If the available number of phase I samples is not large enough to reduce the variability of the in‐control M R L values to an acceptable level, a bootstrap‐type approach is suggested to adjust the control limits of the syntheticX ¯ chart and to further prevent many unwanted lower in‐control M R L values.