Evaluating mixture-process designs with control and noise variables

Mixture-process experiments involve two distinct types of variables. The mixture variables are varied through their relative proportions and cannot be adjusted independently. The process variables can be varied independently of one another and of the mixture components. When some of the process variables are noise variables, variables that cannot be controlled in normal process operation, we are in a robust design setting. In these situations, it is customary to fit a response model combining mixture, process, and noise variables and to derive a model for the mean response and a model for the variance of the response for this response model. The variance model is directly related to the directional derivative (slope) of the response surface in the directions of the noise variables. We develop expressions for the scaled and unscaled prediction variances of both the mean and slope models. We then demonstrate the use of variance dispersion graphs (VDGs) and fraction of design space (FDS) plots of the prediction variance values to evaluate a variety of competing designs for such settings.