The Revised m-of-k Runs Rule Based on Median Run Length

Abstract

Runs rules are used to increase the sensitivity of the Shewhart control chart in detecting small and moderate process mean shifts. Most of the charts incorporating runs rules are designed based on the average run length (ARL). It is known that the shape of the run length distribution changes according to the magnitude of the shift in the process mean, ranging from highly skewed when the process is in-control to approximately symmetric when the shift is large. Since the shape of the run length distribution changes with the magnitude of the shift in the mean, the median run length (MRL) provides a more meaningful explanation about the in-control and out-of-control performances of a control chart. In this article, we propose the design of the revised m-of-k runs rule based on MRL. In addition, the standard deviation of the run length (SDRL) of the revised m-of-k rule will also be studied. The revised m-of-k runs rule, suggested by Antzoulakos and Rakitzis (2008), was originally designed based on ARL. The Markov chain technique is employed to obtain the MRLs. Compared with the standard chart, the MRL results show that the revised rules give better performances for small and moderate mean shifts, while maintaining the same sensitivity towards large mean shifts. The MRL results are in accordance with the results obtained by Antzoulakos and Rakitzis (2008), where the rules are designed based on ARL.

Keywords:

• Average run length (ARL)
• Inner limit
• Markov chain
• Median run length (MRL)
• Outer limit
• Runs rules
• Standard deviation of the run length (SDRL)

Mathematics Subject Classification:

• 62P30
• 60J22

Acknowledgment

This research is supported by the Universiti Sains Malaysia (USM) Fellowship, no. 1001/441/CIPS/JHEA/AUPE001.