A model is developed for optimum choice of the operation parameters for X-bar - R (or X-bar - S) charts, which minimizes the ratio of sampling costs to detection speed of shifts in the process mean or increases in the process dispersion. Three alternative problem formulations are allowed: minimization of the average time to signal subject to a sampling cost constraint; minimization of the sampling cost subject to a constraint on the average times to signal; and multi-objective optimization of both the average time to signal and the sampling cost. Through optional additional constraints, several practical variants of the problem are admitted. The procedure avoids the complexity of usual economic design models, and methods for determining the values for the few input and specification parameters required are given in detail. An example shows that the procedure is easy to apply. All these features increase its applicability for a wide range of typical practical situations.
Statistical process control; control charts; semi-economic design; optimization; multi-objective
