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Economic design of memory-type control charts: The fallacy of the formula proposed by Lorenzen and Vance (1986)

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Abstract

Memory-type statistical control charts, such as exponentially weighted moving average (EWMA) and cumulative sum (CUSUM), are broadly-used statistical feedback policies for detecting small quality changes in univariate and multivariate processes. Many papers on economic-statistical design of these control charts used the general formula proposed by Lorenzen and Vance (Technometrics 28(1):3–10, 1986) as a semi-closed-form expression of the long-run average quality cost. Contrary to popular opinion, this paper argues that this old formula is not correct for memory-type control charts and shows how the formula can be corrected by using concepts such as conditional average run lengths (ARLs), mean of ARLs (MARL), and average number of false alarms (ANFA). The paper also proposes a simulation method as an alternative to directly estimate the cost function, which can be easily adapted for nonstandard assumptions. The results for the EWMA, multivariate EWMA, and CUSUM control charts indicate that the correct computation of the objective function results in significantly different optimal designs, which implies that the old formula is not an acceptable approximation for memory-type control charts. A numerical study is also conducted to compare the numerical efficiency and stability of the simulation method and the computational procedure based on the corrected formula. The required codes are provided.

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Notes

  1. This code can be used only for non-commercial purposes provided that the source is properly cited.

  2. These codes can be used only for non-commercial purposes provided that the source is properly cited.

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Authors and Affiliations

  1. Department of Industrial Engineering and Management Systems, Amirkabir University of Technology, Tehran, Iran

    Amir Ahmadi-Javid

  2. Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, Canada

    Mohsen Ebadi

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  1. Amir Ahmadi-Javid
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  2. Mohsen Ebadi
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Correspondence to Amir Ahmadi-Javid.

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Appendix A: MATLAB code for MEWMA

Appendix A: MATLAB code for MEWMA

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Ahmadi-Javid, A., Ebadi, M. Economic design of memory-type control charts: The fallacy of the formula proposed by Lorenzen and Vance (1986). Comput Stat 36, 661–690 (2021). https://doi.org/10.1007/s00180-020-01019-6

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