Exact economic design of X¯ charts for general time in-control distributions

Abstract

Economic design of control charts seeks to choose the parameters of control charts that will minimize the cost. In an economic model, the time to the occurence of an assignable cause is usually assumed to follow an exponential distribution. In addition to this assumption, certain approximations are used in the optimization procedures. This paper extends the current economic design for X¯ charts in two directions. First, economic design is considered for a general distribution for the time to the occurrence of an assignable cause. The method is specialized to several distributions such as Weibull, lognormal, folded-normal, folded-logistic, and gamma. Second, an exact method to obtain the optimal design parameters is developed. Using Duncan's examples, the optimal designs obtained by exact method are compared for different distributions. These are also compared with the approximate optimal designs given by Duncan. This comparison shows that the optimality of the design for X¯ charts is insensitive to the distributional assumption of the time to the occurrence of an assignable cause for small values of the sampling interval.