Optimal design of the double sampling X¯ chart with estimated parameters based on median run length

•When parameters are estimated, the run length distribution is highly skewed.•The MRL is a more meaningful optimal design criterion to rely on compared to ARL.•We propose the MRL-based DS X¯ chart with known and estimated parameters.•A new optimization algorithm for minimizing MRL1(δopt) is developed.•New optimal chart parameters for practical number of Phase-I samples are given.

The double sampling (DS) X¯ chart when the process parameters are unknown and have to be estimated from a reference Phase-I dataset is studied. An expression for the run length distribution of the DS X¯ chart is derived, by conditioning and taking parameter estimation into account. Since 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, it is shown that the traditional chart’s performance measure, i.e. the average run length, is confusing and not a good representation of a typical chart’s performance. To this end, because the run length distribution is highly right-skewed, especially when the shift is small, it is argued that the median run length (MRL) provides a more intuitive and credible interpretation. From this point of view, a new optimal design procedure for the DS X¯ chart with known and estimated parameters is developed to compute the chart’s optimal parameters for minimizing the out-of-control MRL, given that the values of the in-control MRL and average sample size are fixed. The optimal chart which provides the quickest out-of-control detection speed for a specified shift of interest is designed according to the number of Phase-I samples commonly used in practice. Tables are provided for the optimal chart parameters along with some empirical guidelines for practitioners to construct the optimal DS X¯ charts with estimated parameters. The optimal charts with estimated parameters are illustrated with a real application from a manufacturing company.