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

Wei Lin Teoh*, Jia Kit Chong, Michael Boon Chong Khoo, Philippe Castagliola, W. C. Yeong

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

37 Citations (Scopus)


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 (nS) or (ii) large sample size (nL) 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 = nL significantly outperforms the optimal EWMA X chart, while the former is comparable to the latter when n1 = nS.

Original languageEnglish
Pages (from-to)121-134
Number of pages14
JournalQuality and Reliability Engineering International
Issue number1
Early online date21 Apr 2016
Publication statusPublished - 1 Feb 2017


  • average run length
  • Markov chain
  • median run length
  • percentile of the run-length distribution
  • variable sample size (VSS) X¯ chart

ASJC Scopus subject areas

  • Safety, Risk, Reliability and Quality
  • Management Science and Operations Research


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