A Generalized Likelihood Ratio Control Chart for Monitoring the Process Mean Subject to Linear Drifts

Abstract

This article considers the problem of monitoring a normally distributed process variable when a special cause may produce a time-varying linear drift in the mean. The design and application of a generalized likelihood ratio (GLR) control chart for drift detection are evaluated. The GLR drift chart does not require specification of any tuning parameters by the practitioner and has the advantage that, at the time of the signal, estimates of both the change point and the drift size are immediately available. An equation to accurately approximate the control limit is provided. The performance of the GLR drift chart is compared with that of other control charts such as a standard cumulative sum chart and a cumulative score chart designed for drift detection. We also compare the GLR chart designed for drift detection with the GLR chart designed for sustained shift detection because both of them require only a control limit to be specified. In terms of the expected time for detection and in terms of the bias and mean squared error of the change-point estimators, the GLR drift chart has better performance for a wide range of drift rates relative to the GLR shift chart when the out-of-control process is truly a linear drift.

GLR Charts for Simultaneously Monitoring a Sustained Shift and a Linear Drift in the Process Mean

“…The GLR-D chart for monitoring a linear drift Xu et al (2013) evaluated a GLR chart designed to detect a linear drift in µ (called the GLR-D chart). Zou et al (2009) also proposed a GLR chart for monitoring a linear drift in µ.…”

“…There are differences between these two charts as follows. First, Zou et al (2009) assumed that the change point τ * is always an integer value, whereas Xu et al (2013) assumed that τ * is a value satisfying τ ≤ τ * < τ + 1. Second, the GLR chart proposed by Zou et al (2009) is a one-sided chart for detecting drifts in only one direction, whereas the GLR-D chart proposed by Xu et al (2013) is a two-sided chart that can detect drifts in either direction.…”

“…Xu et al (2013), however, used m 1 = 400 and m 2 = 4, because these values would give good performance for the GLR-D chart over a wide range of drift rates. The GLR-D chart will give a signal at sampling time…”

“…GLR charts have some advantages that the size of the parameter change does not need to be specified and these charts have been shown to be very effective in a wide variety of settings in SPC applications. Recent investigations of this chart include Hawkins et al (2003), Runger and Testik (2003), Capizzi and Masarotto (2008), Zou et al (2009), Reynolds and Lou (2010), Huang et al (2012Huang et al ( , 2013, Xu et al (2012Xu et al ( , 2013, Wang and Reynolds (2013), and Reynolds et al (2013).…”

“…Recent investigations of linear drifts include Reynolds and Stoumbos (2001), Runger and Testik (2003), and Fahmy and Elsayed (2006). More recently, Zou et al (2009) and Xu et al (2013) proposed GLR charts for detecting a linear drift in µ and evaluated the performance of these GLR charts.…”