The construction of D- and I-optimal designs for mixture experiments with linear constraints on the components

Mixture experiments in which linear constraints are imposed on the components of the mixture are used extensively in practice. The problem of constructing designs which are in some sense optimal for this experimental setting is not straightforward. More specifically, the design space is a polytope embedded in a regular simplex. In the present paper, a new approach to the problem, which builds on the fact that points in the polytope can be represented as convex combinations of the vertices of that polytope, is introduced. Some theory underpinning this idea and rooted in the notion of barycentric coordinates is developed and algorithms for the construction of exact and approximate D- and I-optimal designs for the Scheffé model are delineated. The methodology is illustrated by means of examples involving three- and four-component mixtures.