Optimal constrained design of control charts using stochastic approximations

Abstract

In statistical process monitoring, control charts typically depend on a set of tuning parameters besides its control limit(s). Proper selection of these tuning parameters is crucial to their performance. In a specific application, a control chart is often designed for detecting a target process distributional shift. In such cases, the tuning parameters should be chosen such that some characteristic of the out-of-control (OC) run length of the chart, such as its average, is minimized for detecting the target shift, while the control limit is set to maintain a desired in-control (IC) performance. However, explicit solutions for such a design are unavailable for most control charts, and thus numerical optimization methods are needed. In such cases, Monte Carlo-based methods are often a viable alternative for finding suitable design constants. The computational cost associated with such scenarios is often substantial, and thus computational efficiency is a key requirement. To address this problem, a two-step design based on stochastic approximations is presented in this paper, which is shown to be much more computationally efficient than some representative existing methods. A detailed discussion about the new algorithm’s implementation along with some examples are provided to demonstrate the broad applicability of the proposed methodology for the optimal design of univariate and multivariate control charts. Computer codes in the Julia programming language are also provided in the supplemental material.

Keywords:

• numerical optimization
• simultaneous perturbations
• statistical process monitoring
• stochastic approximations
• tuning parameters

Acknowledgements

The authors are grateful to the editor and two anonymous reviewers for their many constructive comments and suggestions, which improved the quality of the paper significantly.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The authors confirm that the data supporting the findings of this study are available within the article and its supplementary materials.

Additional information

Funding

This work was supported by UNIPD under the Grant DOR2021.

Notes on contributors

Daniele Zago

Daniele Zago is a current Ph.D. student in Statistics at the University of Padua since 2021. He obtained his Bachelor’s degree in Statistics for Technology and Science and his Master’s degrees in Statistical Sciences from the University of Padua. His main research interests revolve around fundamental issues in practical applications of Statistical Process Control and optimization.

Giovanna Capizzi

Dr. Giovanna Capizzi is a full professor of statistics at the University of Padua. She earned her PhD in Statistics from the University of Padua in 1992. Dr. Capizzi’s main research interest is in Statistical Process Monitoring, and she has made significant contributions to the field. She has published extensively in several international peer-reviewed journals, including Statistics and Computing, Technometrics, Journal of Quality Technology, IIE Transactions, and Quality Engineering. Dr. Capizzi serves as an associate editor of Technometrics since 2013, and she is a member of the editorial board of the Journal of Quality Technology since 2014.

Peihua Qiu

Dr. Peihua Qiu is currently serving as the Dean’s Professor and Founding Chair of the Department of Biostatistics at the University of Florida. With a Ph.D. in statistics from the University of Wisconsin at Madison, dr. Qiu has a distinguished career spanning several institutions, including the University of Minnesota. He has contributed to various research areas, such as jump regression analysis, image processing, statistical process control, survival analysis, dynamic disease screening, and spatio-temporal disease surveillance. Dr. Qiu has authored two research monographs and published over 150 research papers in peer-reviewed journals. He holds fellowships in prominent scientific associations, including the American Association for the Advancement of Science (AAAS) and the American Statistical Association (ASA). Dr. Qiu has also served as associate editor for leading statistical journals such as Journal of the American Statistical Association, Biometrics, and Technometrics. He was the editor of the flagship statistical journal Technometrics during 2014-2016.