Economically Optimum Design of a Synthetic Xbar Chart

This paper proposes an economic model for the synthetic chart. The synthetic chart is an integration of the chart and the <i>CRL</i> chart. A simplified algorithm to obtain the optimal parameters of the synthetic chart which minimizes the expected cost function is introduced. Numerical examples based on different values of input parameters are given, and sensitivity analyses of the parameters are performed. The input parameters which have a significant impact on the cost and choice of optimal parameters of the synthetic chart are identified. The effect of incorrect estimation of the input parameters is also investigated, and it is found that the optimal design parameters are quite robust to changes in the input parameters, except the shift parameter. It is also shown that if the chart cannot be operated at the economically optimal level, there is still a large choice of parameters to choose from which does not result in a large increase in cost. All the above analyses and results are based on numerical examples and verified through simulation over a wide range of parameter values. Comparisons are also performed among the synthetic Shewhart and EWMA charts. Based on the numerical examples and simulation over a wide range of parameter values, it is shown that the synthetic chart has better economic performance than the other two control charts under most conditions.