Economic-Statistical Design of Multivariate Control Charts Using Quality Loss Function

When a control chart is applied to monitor a production process, three test parameters should be determined: the sample size, the sampling interval between successive samples, and the control limits or critical region of the chart. In this paper, we develop the procedure to carry out the economic-statistical design of multivariate control charts by using a quality loss function for monitoring the process mean vector and covariance matrix simultaneously; i.e., to determine economically the optimum values of the three test parameters so that the statistical constraints (including the requirements of type I error probability and power) of the control chart can be satisfied. The test statistic

_2_nL is used to develop this procedure and the cost model is established based on the cost function developed by Montgomery and Klatt and the multivariate quality loss function presented by Kapur and Cho. A numerical example is provided to illustrate the solution procedure of the design and then the effects of cost parameters on the optimal design are studied.