A Semi-Bayesian Method for Shewhart Individual Control Charts

Abstract

Shewhart control limits for individual observations are traditionally based on the average of the moving ranges. The performance of this control chart behaves quite well if the underlying distribution is normal and the sample size is greater than 250. Under non-normality it is recommended to use control charts based on non-parametric statistics. The drawback of these individual control charts is that at least 1,000 observations are needed to obtain appropriate results. In this paper we propose an alternative individual control chart which behaves quite well under non-normality for moderate sample sizes in the range of 250 through 1,000 observations. To apply this control chart one starts with an initial guess for the density function of the characteristic under study. Based on this initial guess and the observed data a density function can be derived by means of an approximation with Bernstein polynomials. The in-control and out-of-control performance of the proposed control chart and the traditional control charts are studied by simulation. If the initial guess is appropriate, then for non-normal data and moderate sample sizes in the order of 250 through 1,000 observations, the new method performs better than the individual control charts based on the average of the moving ranges or based on non-parametric statistics. So for these sample sizes we have tried to close the gap.

Key Words:

• Bernstein polynomials
• non-parametrics
• quality control
• semi-Bayesian
• statistical process control

Additional information

Notes on contributors

M. B. Thijs Vermaat

Thijs Vermaat Master’s Degree in Econometrics and Operations Research at the University of Groningen in 2002 and a Master’s Degree in Statistics at the same university in 2003. Currently he is a PhD student at the University of Amsterdam and a consultant in industrial statistics at the Institute for Business and Industrial Statistics. His research interests are control charts for nonstandard situations. He is a member of ENBIS.

Ronald J. M. M. Does

Ronald J.M.M. Does received a MSc degree (cum laude) in Mathematics at the University of Leiden in 1976 and a PhD degree in Statistics at the same university in 1982. From 1981–1989, he worked at the University of Maastricht, where he became Head of the Department of Medical Informatics and Statistics. In 1989 he joined Philips Electronics as a senior consultant in Industrial Statistics. Since 1991 he is Professor of Industrial Statistics at the University of Amsterdam. In 1994 he founded the Institute for Business and Industrial Statistics, which operates as an independent consultancy firm within the University of Amsterdam. His current research activities are control charts for nonstandard situations and Lean Six Sigma. He is a member of the editorial boards of Quality Engineering and Quality Technology and Quantitative Management. He is director of ENBIS, vice president publications of ISBIS, Senior Member of ASQ, and elected member of the ISI.