A New Perspective on the Fundamental Concept of Rational Subgroups

Summary When control charts are used to monitor processes to detect special causes, it is usually assumed that a special cause will produce a sustained shift in a process parameter that lasts until the shift is detected and the cause is removed. However, some special causes may produce a transient shift in a process parameter that lasts only for a short period of time. Control charts are usually based on samples of n ≥ 1 observations taken with a sampling interval of fixed length, say d. The rational-subgroups concept for process sampling implies that sampling should be done so that any change in the process will occur between samples and affect a complete sample, rather than occur while a sample is being taken, so that only part of the sample is affected by the process change. When using n > 1, the rational-subgroups concept seems to imply that it is best to take a concentrated sample at one time point at the end of the sampling interval d, so that any process change will occur between samples. However, if the duration of a transient shift is less than d, then it appears that it might be beneficial to disperse the sample over the interval d, to increase the chance of sampling while this transient shift is present. We investigate the question of whether it is better to use n > 1 and either concentrated or dispersed sampling, or to simply use n = 1. The objective of monitoring is assumed to be the detection of special causes that may produce either a sustained or transient shift in the process mean μ and/or process standard deviation σ. For fair comparisons, it is assumed that the sampling rate in terms of the number of observations per unit time is fixed, so that the ratio n/d is fixed. The best sampling strategy depends on the type of control chart being used, so Shewhart, exponentially weighted moving average (EWMA), and cumulative sum (CUSUM) charts are considered. For each type of control chart, a combination of two charts is investigated; onechart designed to monitor μ, and the other designed to monitor σ. The conclusion is that the best overall performance is obtained by taking samples of n = 1 observations and using an EWMA or CUSUM chart combination. The Shewhart-type chart combination with the best overall performance is based on n > 1, and the choice between concentrated and dispersed sampling for this control chart combination depends on the importance attached to detecting transient shifts of duration less than d.