Optimal design of mixture experiments for general blending models

Abstract

Mixture models of the Scheffé polynomial class are standard in several scientific fields. For these models there is a vast literature on the optimal design of experiments to provide good estimates of the parameters with the use of minimal resources. Contrarily, the optimal design of experiments for general blending models, extending the class of Becker, have not been systematically addressed. Nevertheless, there are practical examples where the models relating the response variables, the parameters and the factors including nonlinear blending effects fall into a general form. We propose a general formulation to find continuous and exact D– and A–optimal designs for general blending models. First, we consider designs to estimate the regression coefficients, and then extend the formulations to find locally optimal continuous designs for estimating both the coefficients and the power exponents. The treatment relies on converting the Optimal Experimental Design (OED) problem into an optimization problem of the Nonlinear Programming (or Mixed Integer Nonlinear Programming) class. We apply this approach to quadratic and special cubic general blending models of the H2 class of polynomials introduced by Becker, and to three examples of practical interest in combustion science and in the characterization of fuel properties.