Performance comparison of the exact run-length distribution between the run sum Xbar and EWMA Xbar charts

The run sum (RS) Xbar and exponentially weighted moving average (EWMA) Xbar charts are very effective in detecting small and moderate process mean shifts. The design of the RS Xbar and EWMA Xbar charts based on the average run length (ARL) alone, can be misleading and confusing. This is due to the fact that the run-length distribution of a control chart is highly right-skewed when the process is in-control or slightly out-of-control; while that for the out-of control
cases, the run-length distribution is almost symmetric. On the other hand, the percentiles of the run-length distribution provide the probability of getting a signal by a certain number of samples. This will benefit practitioners as the percentiles of the run-length distribution give comprehensive information regarding the behaviour of a control chart. Accordingly, this paper provides a thorough study of the run-length properties (ARL, standard deviation of the run
length and percentiles of the run-length distribution) for the RS Xbar and EWMA Xbar charts. Comparative studies show that the EWMA X chart outperforms the RS X charts for detecting small mean shifts when all the control charts are optimized with respect to a small shift size. However, the RS Xbar charts surpass the EWMA Xbar chart for all sizes of mean shifts when all the control charts are optimized with respect to a large shift size.