A study on the run sum X-bar control chart with unknown parameters

It is well known that the run sum control chart is a simple and powerful statistical process control tool in the monitoring of the process mean. The implementation of the run sum chart is generally based on the assumption that the process parameters are known. However, since the process parameters are usually unknown in practice, they are estimated from an in-control Phase I data set. In this paper, by means of the Markov chain approach, we investigate the effects of parameter estimation on the performance of the run sum X̄ chart with the scores 0, 1, 2 and 4. The results reveal that when the size of the shift and the number of samples from the Phase I process used for the estimation of parameters are both small, the performance of the run sum X̄ chart is significantly deteriorated. Moreover, very large sample sizes are required for the chart with estimated parameters to have a favorable performance like the known parameters case. By virtue of this adverse performance, new charting parameters are proposed for practitioners in the design of the run sum X̄ chart, based on the weights (0, 1, 2, 4) when parameters are estimated. The suggested parameters give a satisfactory performance even when process parameters are estimated from small number of samples.