Monitoring the shape parameter of a Weibull renewal process

ABSTRACT

This research arose from a challenge faced in real practice—monitoring changes to the Weibull shape parameter. From first-hand experience, we understand that a mechanism for such a purpose is very useful. This article is primarily focused on monitoring the shape parameter of a Weibull renewal process. We derive a novel statistic on the Weibull shape parameter making use of maximum likelihood theory, which is demonstrated to follow an approximately normal distribution. This desirable normality property makes the statistic well suited for use in monitoring the Weibull shape parameter. It also allows for a simple approach to constructing a Shewhart-type control chart, named the Beta chart. The parameter values required to design a Beta chart are provided. A self-starting procedure is also proposed for setting up the Phase I Beta chart. The Average Run Length (ARL) performance of the Beta chart is evaluated through Monte Carlo simulation. A comparison with a moving range exponentially weighted moving average (EWMA) chart from the literature shows that the Beta chart has much better ARL performance when properly designed. Application examples, using both simulated and real data, demonstrate that the Beta chart is effective and makes good sense in real practice.

KEYWORDS:

• Weibull renewal process
• shape parameter
• control chart
• time-between-events
• ARL unbiased

Acknowledgements

The authors are indebted to the anonymous reviewers for their valuable comments and suggestions that have significantly improved the article. A major thank you is also due to Prof. F. Pascual at the Washington State University for providing us with his group's raw ARL data that allowed for a meaningful comparative study. The lead author also appreciates the helpful comments on the article provided by Prof. Guangzhong Li at Sun Yat-sen Business School.

Additional information

Notes on contributors

Cai Wen Zhang

Cai Wen Zhang is an Associate Professor of Management Science in the Business School, Sun Yat-sen University, Guangzhou, People's Republic of China. He received both a master's degree and a Ph.D. in Industrial & Systems Engineering from the National University of Singapore. Prior to joining Sun Yat-sen Business School, he worked as a statistical consultant for 5 years at the Hitachi Global Storage Technologies (Singapore). He has published papers in international peer-reviewed journals including IIE Transactions, European Journal of Operational Research, International Journal of Production Research, International Journal of Production Economics, Computers & Operations Research, Reliability Engineering and System Safety, Quality and Reliability Engineering International, etc. His current research interests include statistical quality control, quality management, reliability, and operations management.

Zhisheng Ye

Zhisheng Ye received a dual bachelor's degree in Material Science & Engineering and in Economics from Tsinghua University in 2008. He received his Ph.D. degree from the National University of Singapore in 2012. He is currently an Assistant Professor in the Department of Industrial & Systems Engineering, National University of Singapore. His research interests include reliability engineering, complex systems modeling, and industrial statistics.

Min Xie

Min Xie is currently a Chair Professor in the Department of Systems Engineering and Engineering Management, City University of Hong Kong. He has been active in areas including quality and reliability engineering and applied statistics for over two decades. He serves as editor or associate editor and is on the editorial board of over 15 international journals. He has served as conference chair of a number of conferences and delivered keynote speeches at many others. He has authored and co-authored seven books and over 200 journal papers.