Economic and economic statistical design of T² control chart with two adaptive sample sizes

Abstract

The Hotelling's T ² control chart, a direct analogue of the univariate Shewhart X¯ chart, is perhaps the most commonly used tool in industry for simultaneous monitoring of several quality characteristics. Recent studies have shown that using variable sampling size (VSS) schemes results in charts with more statistical power when detecting small to moderate shifts in the process mean vector. In this paper, we build a cost model of a VSS T ² control chart for the economic and economic statistical design using the general model of Lorenzen and Vance [The economic design of control charts: A unified approach, Technometrics 28 (1986), pp. 3–11]. We optimize this model using a genetic algorithm approach. We also study the effects of the costs and operating parameters on the VSS T ² parameters, and show, through an example, the advantage of economic design over statistical design for VSS T ² charts, and measure the economic advantage of VSS sampling versus fixed sample size sampling.

Keywords:

• Hotelling's T ² control chart
• adjusted average time to signal
• variable sample size scheme
• economic statistical design
• Markov chain
• genetic algorithm

Acknowledgements

The authors wish to thank the referee, who suggested a number of changes to earlier drafts of the paper; these subsequently led to an improved paper.