Two perspectives for designing a phase II control chart with estimated parameters: The case of the Shewhart X¯ Chart

Abstract

The impact of parameter estimation on control charts has been studied with great interest in the recent literature. The estimated control limits affect chart performance, often negatively, relative to the known parameter case. Guided by the need to design control charts with a specified in-control performance, so as to avoid excessive false alarms, two major perspectives have been advocated. Under the first, the so-called unconditional perspective, control limits are determined so that the in-control unconditional average run length equals a specified value. However, this perspective does not account for the so-called practitioner-to-practitioner variability inherent in control charts using parameter estimates. Hence, researchers have considered a second perspective, called the conditional perspective, under which the so-called exceedance probability criterion is used to calculate the control limits so that the in-control conditional average run-length is at least equal to a specified value with a given high probability. These perspectives, in turn, have led to adjustments to the traditional control limits, and various methods have been proposed to calculate the adjusted limits. In this article, we consider these two perspectives and examine the performance of the various proposed adjustments to the control limits for the Shewhart $\overline{X}$ chart. We also provide a simple adjustment formula under the conditional perspective that can be used in practice without too many resources, which works well relative to the other methods. In addition, we derive an exact formula for calculating the adjusted limit coefficient (proposed in an earlier work using bootstrapping) which does not require any model fitting. For completeness, the adjusted limits calculated under one perspective are examined under the other in terms of performance. A summary and some practical recommendations are provided.

Keywords:

• bootstrap
• control limit adjustments
• exceedance probability criterion
• false alarm rate
• guaranteed in-control performance
• unconditional and conditional run length and average run length

About the authors

Dr. Jardim is a post-doctoral researcher in the Department of Industrial Engineering. His e-mail address is [email protected].

Dr. Chakraborti is a professor of statistics in the Department of Information Systems, Statistics, and Management Science. His e-mail address is [email protected].

Dr. Epprecht is an associate professor in the Department of Industrial Engineering. He is a member of ASQ. His e-mail address is [email protected].

Acknowledgements

We are grateful to two anonymous reviewers for their insightful comments which improved the presentation.

Funding

This work was supported in part by the CNPq (Brazilian Council for Scientific and Technological Development) through projects 401523/2014-4 (for the second author), and 201172/2016-0 (for the first author). This study was also financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior–Brasil.