System Engineering and Productivity

System Engineering and Productivity

Simultaneous Monitoring of Multivariate Process Mean Vector and Covariance Matrix with Limited Phase I Reference Observations Using Multivariate GLR and Maximum Control Charts

Articles in Press

Document Type : Research Paper

Authors
Fezzeh Abanavaz Kordehmahin 1
Mohammad Reza Maleki 2
Hossein Eghbali 3
1 Ph.D. Student, Department of Industrial Engineering, University of Eyvanekey, Semnan, Iran
2 Corresponding author: Assistant Professor, Industrial Engineering Group, Golpayegan College of Engineering, Isfahan University of Technology, Golpayegan, Iran
3 Assistant Professor, Department of Industrial Engineering, University of Eyvanekey, Eyvanekey, Iran
10.22034/sep.2026.2088164.1483
Abstract
In many real-world applications of statistical quality control, product quality is described by multiple correlated quality characteristics, where the occurrence of an assignable cause often leads to simultaneous changes in process mean vector and the covariance matrix. On the other hand, in Phase I analysis, due to reasons such as high sampling costs, limited time during process start-up, and short-run or customized production, a sufficient number of reference samples for estimating distribution parameters and determining control limits is often unavailable. Therefore, this paper focuses on simultaneous monitoring of mean vector and the covariance matrix under the condition that only a single reference sample is available in Phase I. To this end, a bootstrap-based algorithm is first developed to estimate the mean vector, the covariance matrix, and the control limits of the multivariate generalized likelihood ratio (MGLR) and the multivariate maximum charts in Phase I. Subsequently, online Phase II approaches are designed to evaluate the effect of estimation errors on the performance of the mentioned control charts. The results of extensive Monte Carlo simulations show that the proposed bootstrap algorithm provides accurate estimates of the distribution parameters and control limits despite relying on only a single reference sample. The findings also indicate that as the number of bootstrap samples increases, the indices related to the accuracy of parameter estimation gradually move closer to their target values. Furthermore, the analysis of the average run length (ARL), standard deviation of run length (SDRL), and median run length (MRL) under different shift scenarios demonstrates that estimation error affects the performance of both the multivariate generalized likelihood ratio and the multivariate maximum charts in Phase II.

Highlights

  • Developing a bootstrap-based algorithm to estimate the mean vector and covariance matrix using a single reference sample. 
  • Computing the control limits of multivariate generalized likelihood ratio and multivariate maximum charts based on a single reference sample. 
  • Designing Phase II approaches to assess the effect of parameter estimation error on run-length properties of the proposed simultaneous monitoring methods.

Keywords
Mean Vector
Covariance Matrix
Multivariate Maximum
Multivariate Generalized Likelihood Ratio
Bootstrap Method
Subjects

Copyright © Fezzeh Abanavaz Kordehmahin, Mohammad Reza Maleki, Hossein Eghbali

 

License

This article is released under the Creative Commons Attribution (CC BY 4.0) license. Anyone is free to copy, share, translate, and adapt this article for any purpose, whether commercial or non-commercial, as long as proper citation is given to the authors and original publication.

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System Engineering and Productivity

Articles in Press, Accepted Manuscript
Available Online from 30 May 2026

  • Receive Date 07 April 2026
  • Revise Date 17 May 2026
  • Accept Date 30 May 2026
  • First Publish Date 30 May 2026
  • Publish Date 30 May 2026