Control Charts and the Efficient Allocation of Sampling Resources

Abstract

Control charts for monitoring the process mean μ and process standard deviation σ are often based on samples of n > 1 observations, but in many applications individual observations are used (n = 1). In this article we investigate the question of whether it is better, from the perspective of statistical performance, to use n = 1 or n > 1. We assume that the sampling rate in terms of the number of observations per unit time is fixed, so using n = 1 means that samples can be taken more frequently than when n > 1. The best choice for n depends on the type of control chart being used, so we consider Shewhart, exponentially weighted moving average (EWMA), and cumulative sum (CUSUM) charts. For each type of control chart we investigate a combination of two charts, one chart designed to monitor μ and the other designed to monitor σ. Most control chart comparisons in the literature assume that a special cause produces a sustained shift in a process parameter that lasts until the shift is detected. We also consider transient shifts in process parameters, which are of a short duration, and drifts in which a parameter moves away from its in-control value at a constant rate. We evaluate control chart combinations using the expected detection time for the various types of process changes and a quadratic loss function. When a signal is generated, it is important to know which parameters have changed, so the ability of control chart combinations to correctly indicate the type of parameter change is also evaluated. Our overall conclusion is that it is best to take samples of n = 1 observations and use an EWMA or CUSUM chart combination. The Shewhart chart combination with the best overall performance is based on n > 1, but this combination is inferior to the EWMA and CUSUM chart combinations on almost all performance characteristics (the exception being simplicity). This conclusion seems to contradict the conventional wisdom about some of the advantages and disadvantages of EWMA and CUSUM charts relative to Shewhart charts.