A single X chart outperforming the joint X & R and X & S charts for monitoring mean and variance

Abstract

The Shewhart X & R and X & S control charts have traditionally been used for detecting mean shift δ_μ and standard deviation shift δ_σ. This article studies and compares the overall performance of the X chart with that of the X & R and X & S charts, as well as the X&MR chart. The comparative study led to surprising results that contradict the conventional wisdom in Statistical Process Control (SPC) niche. It is found that the simplest single X chart (i.e., the X chart with a sample size n = 1) is always the optimal version of the X chart for detecting δ_μ and δ_σ. Moreover, the single X chart even outperforms the joint X & R and X & S charts in overall detection effectiveness. On average, the X chart is more effective than the X & R and X & S charts by around 5% under different circumstances. Most importantly, the X chart is very simple to understand, implement and design. As a result, it may be highly preferred for many SPC applications, in which both the mean and variance of a variable need to be monitored.

Keywords:

• Quality control
• statistical process control
• control chart
• Shewhart chart
• Average extra quadratic loss

Acknowledgements

The authors wish to acknowledge the valuable suggestions and useful comments made by the Editor, Associate Editor and anonymous reviewers. Their suggestions and comments have improved the quality of the paper significantly.

Disclosure statement

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of this article.

Notes on contributors

Salah Haridy is an Assistant Professor in the Faculty of Engineering, Benha University. He received his PhD in Systems Engineering and Management in 2014 from Nanyang Technological University. He worked as a Postdoctoral Research Associate in Healthcare Systems Engineering Institute at Northeastern University. His research interests include Statistical Process Control, Healthcare Systems Engineering and Design of Experiments. He serves as a reviewer for international journals.

Yanjing Ou joined Singapore Institute of Manufacturing Technology (SIMTech) in 2013 as Scientist, where she conducts research in area of big data analytics and statistical process monitoring in manufacturing systems. Before that she worked in National University of Singapore (NUS) as a research engineer. She got the PhD in the Division of Systems and Engineering Management at Nanyang Technological University (NTU), Singapore, where she has completed her PhD thesis on Development and Evaluation of the Control Charts for Variables.

Zhang Wu is a senior research fellow in the School of Mechanical and Aerospace Engineering of Nanyang Technological University, Singapore. Dr. Wu received his B.Eng degree in Mechanical Engineering from Huazhong University of Science and Technology, China, and both his M.Eng and PhD degrees in Mechanical Engineering from McMaster University, Canada. His current research interests include quality control and reliability.

Michael B. C. Khoo is a Professor in the School of Mathematical Sciences, Universiti Sains Malaysia (USM). He received his PhD in Applied Statistics in 2001 from USM. His research interest is in Statistical Process Control. He has published numerous papers on control charts in international journals. He is a member of the American Society for Quality and serves as a member of the editorial boards of several International journals.