Page 359 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

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344 Simultaneous Monitoring of the Mean, Variance and Autocorrelation
22.1 INTRODUCTION
Statistical process control (SPC) techniques aim to detect as early as possible the
presence of external or special causes of variation affecting a given process. These
special causes of variation usually result in unwanted deviation of important prod-
uct characteristics from the desired or target value. SPC procedures rely mainly on
the assumption that, when the process is affected only by the inherent or common
causes of variation, the process will produce measurements that are independent and
identically distributed over time. Any deviation from such in-control iid behavior is
usually interpreted as an indication of the presence of special causes of variation. This
assumption has been the mainstay of classical SPC techniques.
Recent advances in manufacturing technology, however, are challenging the appli-
1
cability of the in-control iid concept. For example, in continuous chemical processing
and automated manufacturing environments, process measurements are often seri-
ally correlated even when there are no special causes of variation affecting the system.
This autocorrelation violates the in-control independence assumption in many SPC
control charting procedures. Such a violation has a significant impact on the perfor-
mance of the classical SPC procedures. 2,3
The most popular approach for monitoring processes with serially correlated ob-
servations is to model the inherent autocorrelation of the process measurements using
an autoregressive moving average (ARMA) model. To detect changes in the process,
residuals are generated using the chosen model. When the model is appropriately
chosen and well estimated, these residuals approximate an iid behavior. Using this
assumption, we can employ the traditional SPC charts to monitor the residuals. Any
deviation of the residual from iid behavior indicates a change in the process that must
be due to a special cause of variation.
The development of SPC techniques for monitoring autocorrelated processes has
received considerable attention in the quality engineering literature. The focus, how-
ever, is mainly on the detection of the mean shift of the process. The detection of
changes in the variance and autocorrelation structure of the series, which are also
important indicators of the presence of process changes that must be due to a special
cause of variation affecting the system, is often overlooked. There are, however, some
notable exceptions. 4−9
4
Dooley and Kapoor suggested monitoring of the changes in the mean, variance
and autocorrelation structure (MVAS) of the process measurements by simultaneously
2
maintaining a CUSUM, a χ , and an autocorrelation chart on residuals. Yourstone and
5
Montgomery noted that the autocorrelation function (ACF) ‘will detect shifts in the
autocorrelation structure as well as shifts in the mean and variance of the real-time
process quality data’. They suggested real-time monitoring of the first m residual
autocorrelations calculated for the n latest process observations. They call their chart
8
the sample autocorrelation chart or (SACC). Atienza et al. studied the average run
length (ARL) performance of the SACC. They noted that although the SACC can
detect changes in the mean and variance of a series, it does not perform well even in
comparison with the Shewhart control chart (SCC) on residuals. The only advantage
in using the SACC is that it can detect changes in the autocorrelation structure of
a series better than the SCC. Thus, one cannot rely on the SACC to simultaneously
monitor changes in the MVAS of a series. In this chapter, we propose an alternative
to the SACC. Compared to the SACC, the proposed procedure, which is based on the