Control charts for process monitoring have traditionally been designed and evaluated under the assumption that observations on the process output at different times are independent. However, autocorrelation may be present in many processes, and may have a strong impact on the properties of control charts. This paper investigates CUSUM control charts for monitoring the process mean for the situation in which observations from the process can be modeled as an AR(1) process plus an additional random error. CUSUM charts based on plotting the residuals from model forecasts, or on plotting the original observations, are considered. CUSUM charts based on the original observations perform as well as CUSUM charts of residuals, except in the case in which the level of autocorrelation is high and the shift in the process mean is large. A method for designing the CUSUM chart of the observations in the presence of autocorrelation is given. The CUSUM charts are compared to Shewhart and EWMA charts based on the residuals or on the original observations. The CUSUM and EWMA charts perform similarly in terms of the ability to detect shifts in the process mean.Key Words: Autocorrelated Observations, Autoregressive Moving Average Model, Average Run Length, Cumulative Sum Control Charts, Exponentially Weighted Moving Average Control Charts, First Order Moving Average Model, Shewart Control Charts, Statistical Process Control.

*By* **Chao-wen Lu, Chaoyang University of Technology,
WuFeng, Taichung County, Taiwan, ROC** and **Marion R.
Reynolds, Jr., Virginia Polytechnic Institute and State University,
Blacksburg, VA 24061**

**Introduction**

CONTROL charts are widely used in many industries to monitor processes with the objective of improving process quality and productivity. The statistical properties of control charts have traditionally been evaluated under the assumption that observations from the process at different times are independent random variables. However, the observations from many processes exhibit autocorrelation that may be the result of dynamics that are inherent to the process. Autocorrelation is more likely to be observed in processes when observations are closely spaced in time.

Control charts that have been designed under the assumption of independent observations can have properties much different that expected when applied in monitoring a process with significant autocorrelation. For example, positive autocorrelation can produce severe negative bias in traditional estimators of the process standard deviation, and this bias produces control limits that are much tighter than desired. Tight control limits, combined with autocorrelation in the observations being plotted, can result in an average false alarm rate much higher than expected or desired. A very high false alarm rate will cause process personnel to waste effort in unproductive searches for special causes. This can lead to a loss of confidence in the control chart, and even to process monitoring being discontinued. Thus, autocorrelation should not be ignored when designing control charts, because failure to properly account for autocorrelation can greatly reduce or eliminate the effectiveness of control charts.

Two general approaches to dealing with autocorrelation in process monitoring have been advocated and studied in recent years. One approach forecasts each observation from previous observations, and then computes the forecast error (or residual) after each observation is obtained. These residuals are then plotted on standard control charts. The forecast can be based either on a general forecasting technique, such as an EWMA that is applied without fitting a model, or on fitting a time series model and then using the optimal forecast for the fitted time series model.

A second approach to dealing with autocorrelation uses standard control charts that are based on the original observations, but adjusts the control limits and the techniques for estimating process parameters to account for the autocorrelation. References to some of the literature on these two approaches are given in the recent paper by Lu and Reynolds (1999a).

Several questions arise in deciding on an approach to use for process monitoring in any particular application. One question involves whether it is better to use the first approach above and plot residuals on a control chart, or use the second approach and plot the original observations. A second question is how the design parameters of the control chart are determined once an approach has been chosen. The successful application of either approach requires knowledge of the level of autocorrelation in the process. Thus, a third question deals with the estimation of process parameters associated with autocorrelation. There are many different models that have been used to model autocorrelation in processes. In any particular application, the answers to the three questions posed above depend on the process model that is appropriate for this application, and on the level of autocorrelation within this model.

The objective of this paper is to investigate CUSUM control charts based on residuals or on the original observations. This investigation is done for the case of processes that can be modeled as an AR(1) process plus an additional random error. The performance of CUSUM charts is studied for this model and compared to the performance of Shewhart and EWMA charts.

The performance of CUSUM control charts in the presence of autocorrelation has been studied in a number of contexts. See, for example, Johnson and Bagshaw (1994), Bagshaw and Johnson (1975), Harris and Ross (1991), Yashchin (1993), Superville and Adams (1994), Tseng and Adams (1994), Runger, Willemain, and Prabhu (1995), Schmid (1997), Vander Weil (1996), VanBrackle and Reynolds (1997), and Timmer, Pignatiello, and Longnecker (1998). Our paper differs from other work in several respects. We do a systematic investigation of CUSUM charts of observations and of residuals, including determination of the optimal reference value, for the AR(1) plus error model. We believe that this process model is general enough to fit a wide variety of processes encountered in applications, yet it is simple enough to be easy to explain and to fit to process data. We compare CUSUM charts to EWMA charts when each chart has been optimized to detect a particular shift in the process mean. The evaluations and comparisons of charts are based on the expected time to detect shifts in the mean, where we account for the fact that the control statistics of the charts may not be at their initial values when the shift in the process mean occurs.

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