Joint Economically Optimal Design of X\bar and R Control Charts

In this paper, we develop an expected cost model for a process whose mean is controlled by an X\bar chart and whose variance is controlled by an R chart. The expected cost comprises the fixed and variable costs of sampling, the cost of investigating and correcting the process when at least one control chart indicates that the process parameters have shifted, and the cost of producing defective units. We use a search procedure to determine the sample size, interval between samples and control limits for both charts that minimize the expected cost. Optimal solutions to numerical examples are presented. A sensitivity analysis of the model is performed. In addition, we find the optimal interval between samples and the expected cost for several examples with large shifts in the mean and variance where Shewhart's heuristic design is used in place of the optimal design. Comparison of the expected cost of the optimal design to the expected cost of Shewhart's design shows an increase in expected cost of only 0.4 to 8.2 percent for the latter design. But other situations are discussed and examples presented which indicate that the optimal design is preferred.