Should exponentially weighted moving average and cumulative sum charts be used with Shewhart limits?

Traditional Shewhart control charts are usually considered effective for detecting large shifts in process parameters, but ineffective for detecting small shifts. For detecting small parameter shifts, it is much better to use exponentially weighted moving average (EWMA) control charts or cumulative sum (CUSUM) control charts, but these charts are not considered as effective as Shewhart charts, for large parameter shifts. It is frequently recommended that EWMA or CUSUM charts be used in combination with a Shewhart chart to gain the benefits of both types of charts, so that both small and large shifts can be detected quickly. Here we consider the problem of process monitoring when a continuous process variable is being observed and the objective is to detect small or large shifts in either the process mean mu or the process standard deviation sigma. In this situation it is customary to use a combination of two control charts, one chart designed to monitor mu, and the other designed to monitor sigma. For this situation, the best EWMA or CUSUM chart for monitoring mu is based on sample means, and the best chart for monitoring a is based on squared deviations from target. An EWMA or CUSUM chart combination based on sample means and squared deviations from target is very effective for detecting small or large shifts in mu or alpha. We show that this type of combination is more effective in terms of overall performance than other combinations that do not include the chart based on squared deviations from target and generally are at least as effective as any of the combinations that include the Shewhart chart. Thus we conclude that it is not really necessary to use a Shewhart chart with an EWMA or CUSUM chart to obtain the best overall performance, but it is necessary to use the EWMA or CUSUM chart based on squared deviations from target.