Optimal Designs for Multiple Constrained Mixture Experiments

Abstract

In mixture experiments, the response to a mixture (treatment) depends on the proportions x;, i= I, ... , q of the q components constituting the mixture. The natural constraints on these proportions are : q X; 0, i = l,... ,q, :L;x; =I i=l

Let y denote the response variable. Then, the second-order canonical polynomial model is

q q E(y) = L;fJ;X; +:L;fJ;jX;Xj i=l i<j

The forms of canonical polynomials of other different orders can be presented.

In many practical situations additional constraints in the form of lower bounds or upper bounds or both lower and upper bounds are placed on some or all of the component proportions. Such constraints are imposed for physical, chemical or economic reasons especially in experiments on alloys, textile fibers, food products and beverages. The general forms of these additional constraints are

i = l, ... ,q

The objective of this thesis is to construct optimal designs for estimating the coefficients of the considered canonical polynomial model subject to the considered constraints. Different criteria of the optimality will be considered including the D-optimality, which seeks to minimize the determinant of the covariance matrix (L) of the estimators, or the A­ optimality which minimizes the trace of (L). The thesis consists of five chapters. The main characteristics of mixture experiments and the polynomial models associated with them are introduced in Chapter I, which also states the. differences between mixture experiments and factorial experiments.