Economic design of memory-type control charts: The fallacy of the formula proposed by Lorenzen and Vance (1986)
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Economic design of memory‑type control charts: The fallacy of the formula proposed by Lorenzen and Vance (1986) Amir Ahmadi‑Javid1 · Mohsen Ebadi2 Received: 10 April 2019 / Accepted: 15 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 memorytype 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. Keywords Statistical process control · Quality control charts · Simulation-based optimization · Optimal control · Long-run average cost · Average run length (ARL) · EWMA and CUSUM charts
* Amir Ahmadi‑Javid [email protected] 1
Department of Industrial Engineering and Management Systems, Amirkabir University of Technology, Tehran, Iran
2
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, Canada
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1 Introduction The events associated with the transitions of a process from the in-control state to an out-of-control state are referred to as stochastic disorders. The goal of a disorder problem is to design a statistical decision rule (detection procedure or algorithm) that can detect a specific type of disorder effectively (Tartakovsky et al. 2015). In disorder problems like quality control procedures, “stopping times” play a key role in the description of control procedures, based on which one can design optimal control policies (Shiryaev 2019). In statistical process control (SPC), decision rules are called control charts, which are broadly used for monitoring the quality of a production or service process to detect disorders. Control charts can be shown graphically to depict whether a disorder occurs or not. A control chart typically has a center line that represents
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