Variable sampling interval (X)over-bar charts in the presence of correlation

Traditional applications of control charts use fixed sampling interval (FSI) charts in which the time interval between samples is fixed. Recently, more efficient variable sampling interval (VSI) charts have been developed in which the next observation is taken sooner than usual if there is an indication that the process is operating off the target value. It has been found that VSI charts can detect process shifts faster than FSI charts. The underlying assumption behind most statistical process control charts is that the observations from the process are independent. However, there are many practical situations in which the observations cannot be treated as independent, particularly if they are closely spaced in time as may be the case with a VSI chart. In this paper successive values of the process mean are modeled using a first-order autoregressive time series model, and observations are modeled as the mean plus a random error. A Markov process model is used to study the properties of the FSI and VSI (X) over bar charts under this model. Numerical results show that correlation between successive means has a significant effect on the properties of both FSI and VSI (X) over bar charts. When correlation is present, the VSI (X) over bar chart will detect process shifts faster than the FSI (X) over bar chart, but for high correlation there is little difference between the performance of the two charts.