Extensions of D-Optimal Minimal Designs for Mixture Models

Abstract

The purpose of mixture experiments is to explore the optimum blends of mixture components, which will provide desirable response characteristics in finished products. D-Optimal minimal designs have been considered for a variety of mixture models, including Scheffe's linear, quadratic, and cubic models. Usually, these D-Optimal designs are minimally supported since they have just as many design points as the number of parameters. Thus, they lack the degrees of freedom to perform the Lack of Fit tests. Also, the majority of the design points in D-Optimal minimal designs are on the boundary: vertices, edges, or faces of the design simplex. In this dissertation, extensions of the D-Optimal minimal designs are developed to allow additional interior points in the design space to enable prediction of the entire response surface. First, the extensions of the D-Optimal minimal designs for two commonly used second-degree mixture models are considered. Second, the methodology for adding interior points to general mixture models is generalized. Also a new strategy for adding multiple interior points for symmetric mixture models is proposed. When compared with the standard mixture designs, the proposed extended D-Optimal minimal design provides higher power for the Lack of Fit tests with comparable D-efficiency.