Optimal designs of the multivariate synthetic chart for monitoring the process mean vector based on median run length

Abstract

The average run length (ARL) is usually used as a sole measure of performance of a multivariate control chart. The Hotelling's T 2 , multivariate exponentially weighted moving average (MEWMA) and multivariate cumulative sum (MCUSUM) charts are commonly optimally designed based on the ARL. Similar to the case of univariate quality control, in multivariate quality control, the shape of the run length distribution changes in accordance to the magnitude of the shift in the mean vector, from highly skewed when the process is in-control to nearly symmetric for large shifts. Because the shape of the run length distribution changes with the magnitude of the shift in the mean vector, the median run length (MRL) provides additional and more meaningful information about the in-control and out-of-control performances of multivariate charts, not given by the ARL. This paper provides a procedure for optimal designs of the multivariate synthetic T 2 chart for the process mean, based on MRL, for both the zero and steady-state modes. Two Mathematica programs, each for the zero state and steady-state modes are given for a quick computation of the optimal parameters of the synthetic T 2 chart, designed based on MRL. These optimal parameters are provided in the paper, for the bivariate case with sample sizes, n ∈{4, 7, 10}. The MRL performances of the synthetic T 2 , MEWMA and Hotelling's T 2 charts are also compared.

Design and implementation issues for a class of distribution-free Phase II EWMA exceedance control charts

“…Therefore, in this paper we use the median run-length (MRL) to measure the chart performance. This is supported and motivated by the works of several authors including Gan (1994), Radson and Boyd (2005), Khoo et al (2011), andGraham et al (2014). To this end, we set the desired nominal MRL 0 , say 350, meaning that there is at least a 50% chance that the first OOC signal will be witnessed at or before the 350 th sample even though the process is actually IC.…”

Distribution-free Phase II Mann–Whitney control charts with runs-rules

“…These include the standard deviation of the run-length ( ) and other percentiles of the run-length, more specifically, the median run-length ( ), which provides additional and more meaningful information about the in-control and out-of-control performances of control charts, not given by the ARL. The idea of looking at percentiles, in SPC, goes back to Barnard [17] and more recently researchers such as Gan [18], Chakraborti [19] and Khoo et al [20] have advocated the use of percentiles, such as the median, for assessment of chart performance. Tables 3 and 4 give the , ,…”

Design and implementation of CUSUM exceedance control charts for unknown location

“…The idea of looking at percentiles in SPC goes back to Barnard (1959). Recently Khoo et al (2011) have advocated the use of percentiles, such as the median, for assessment of chart performance. The disadvantages of using the ARL are summarized in Appendix A.2.…”

“…Moreover we use the median run-length (MRL), motivated by Khoo et al (2011) and others, as the performance metric in implementing the charts. The MRL has certain practical advantages over the traditional practice of using the ARL.…”