Optimal Designs of the Variable Sample Size ¯xChart Based on Median Run Length and Expected Median Run Length

The variable sample size (VSS) X chart, devoted to the detection of moderate mean shifts, has been widely investigated under the context of the average run-length criterion. Because the shape of the run-length distribution alters with the magnitude of the mean shifts, the average run length is a confusing measure, and the use of percentiles of the run-length distribution is considered as more intuitive. This paper develops two optimal designs of the VSS X chart, by minimizing (i) the median run length and (ii) the expected median run length for both deterministic and unknown shift sizes, respectively. The 5th and 95th percentiles are also provided in order to measure the variation in the run-length distribution. Two VSS schemes are considered in this paper, that is, when the (i) small sample size (n_S) or (ii) large sample size (n_L) is predefined for the first subgroup (n₁). The Markov chain approach is adopted to evaluate the performance of these two VSS schemes. The comparative study reveals that improvements in the detection speed are found for these two VSS schemes without increasing the in-control average sample size. For moderate to large mean shifts, the optimal VSS X chart with n₁ = n_L significantly outperforms the optimal EWMA X chart, while the former is comparable to the latter when n₁ = n_S.