Control Charts for Process Dispersion in Statistical Process Monitoring

在統計過程控制中用於控制過程方差的控制圖的設計研究

Student thesis: Doctoral Thesis

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Supervisors/Advisors
Award date12 Oct 2020

Abstract

Statistical process monitoring (SPM) is a statistical method designed to detect variation in manufacturing and nonmanufacturing processes. A control chart is a graphical tool for SPM, employed to detect changes in the process over time. Control charts are implemented in two phases (Phase I and II). In Phase I, the process parameters are often estimated from a historical dataset before monitoring begins. The estimated parameters are employed to set up the control chart's limits for Phase II online monitoring. Future observations are compared against the control limits, and any observations that fall outside the limits are said to be out-of-control. Control charts are generally proposed for the monitoring of process location and dispersion parameters. The location charts are used to monitor for changes in the process mean, while the dispersion charts are employed to monitor for changes in the process variability. It is usually advised to control process variability before monitoring the location parameter of the process. As an increase in the process variability would decrease process performance, while a decrease in the process dispersion would enhance the process behavior. The focus of this thesis is on monitoring the process dispersion of univariate and multivariate processes using control charts.

In Chapter 2, we provide guidelines for practitioners on how observations should be arranged for setting up effective multivariate dispersion charts. We developed three multivariate dispersion charts based on grouped observations. We compare their performance with dispersion chart for multivariate individual observations. We find that monitoring methods based on individual observations are the quickest in detecting shifts in the process dispersion. Also, we compare multivariate dispersion charts based on overlapping and nonoverlapping subgroups and find that the charts based on overlapping subgroups show the best performance when monitoring with subgroup data. The effect of subgroup size for the monitoring methods based on grouped observations is also investigated.

Chapter 3 is motivated by the conclusion in chapter 2 that monitoring methods based on individual observations outperform methods based on grouped observations. In this chapter, we review the literature on various control charts for monitoring the covariance matrix of a process where data are collected as individual observations. More than 200 research articles published from 1987 to 2019 on multivariate control charts are studied, and thirty relevant articles are selected for review. The selected articles focus on monitoring the covariance matrix and are designed for continuous multivariate individual observations from normal and non-normal distributions. Methods for count processes, multinomial data and other discrete data are excluded. The selected articles are grouped into five categories. It is observed that less research has been done on CUSUM, high-dimensional, and non-parametric type control charts for monitoring the covariance matrix of a process. Suggestions for future research are provided.

Chapter 4 examines the performance of the exponentially weighted moving average (EWMA) control charts for normally and non-normally distributed data when the parameters are estimated from Phase I. In this chapter, we consider univariate process data. This study focuses on a comparison of the EWMA control charts based on the sample variance (S^2), sample standard deviation (S), and the logarithm of the sample variance ln\left(S^2\right). The three EWMA dispersion charts are compared based on Phase I estimates under normally distributed data as well as non-normally distributed data. It is shown that the EWMA chart based on S^2 is least influenced by estimation error under normally distributed data. However, it is shown that the EWMA chart based on ln\left(S^2\right) is the most robust to non-normality, and the chart is recommended to be used in practice when it is not certain that the normality assumption is exactly satisfied.

Finally, Chapter 5 extends the findings in chapter 4 to multivariate charts. We introduce a new robust multivariate control chart for monitoring the process dispersion of individual observations. The proposed control chart is developed by applying a logarithm to the diagonal elements of the covariance matrix. Then, this vector is incorporated into an EWMA statistic. The performance of the proposed control chart is compared with the most popular alternatives. The mean of the conditional average run length (ARL), based on Phase I estimates, is employed as a performance measure. Simulation studies show that the proposed control chart outperforms the existing procedures when there is an overall decrease in the process variance. Also, the proposed chart is the most robust to non-normality among the compared parametric charts.

We hope that the proposed methodologies in this thesis will aid practitioners in their applications of dispersion charts for monitoring process variability.