On the performance evaluation of the proportional control chart

A proportional control chart (P chart), a special case of the PID chart arise by setting the chart parameters k
_I
and k
_D
to zero. From the viewpoint of the operators, the use of the complete PID chart is more complicated than that of the P chart, and would only result in a marginal improvement of the detecting ability when the process autocorrelation is very strong or the mean shifts are very large. In the most of literatures, the average run lengths (ARLs) of a PID chart are obtained by simulation technique and used to evaluate the charting performance. But the simulation technique is quite time consume in the case of longer on-target ARL analysis. In the present paper, a simple unified method based on a finite Markov chain approach for evaluating the performance of P chart is developed. The method is implemented by using discretization of control statistic and the finite Markov chain imbedding (FMCI) technique to investigate the ARL of a P chart when the observations have an AR(1) autocorrelated structure. This simple model may be a reasonable model for many processes encountered in practice. The numerical studies show that the approximation results are consistent with the simulation results.