Minimum-loss design of x-bar charts for correlated data

When the x-bar chart is used to monitor a production process, three parameters should be determined: the sample size, the sampling interval between successive samples, and the control limits for the chart. In 1956, Duncan presented a cost model to determine the three parameters, which is called the economic design of x-bar charts. In 1995, Alexander et al. combined Duncan’s cost model with the Taguchi’s quality loss function to present a loss model for determining the three parameters. When designing an x-bar chart, one usually assumes that the measurements within a sample are independent; however, this assumption may not be true for some processes. In this paper, we develop the minimum-loss design of x-bar charts for correlated measurements within a sample by incorporating the Taguchi’s quality loss function. An example of orange juice production process is presented to illustrate the solution procedure. From the results of sensitivity analysis, we find that as the measurements in the sample are positively correlated, highly correlated data result in a smaller sample size and a frequent sampling interval; however, as the measurements in the sample are negatively correlated, highly correlated data yield a smaller sample size and a narrower control limits.