OPTIMAL DESIGN FOR SECOND-DEGREE KRONECKER MODEL MIXTURE EXPERIMENTS FOR MAXIMAL PARAMETER SUBSYSTEM

Abstract

Products in many disciplines frequently involve blending two or more ingredients together. The design factors in a mixture experiment are the proportions of the components of a blend, and the response variables vary as a function of these proportions making the total and not the actual quantity of each component. This study investigated optimal design for maximal parameter subsystem for second-degree Kronecker model mixture experiments put forward by Draper and Pukelsheim. Based on the completeness result, the investigations was restricted to weighted centroid designs. In mixture model on the simplex an improvement is obtained for a given design in terms of increasing symmetry as well as obtaining a larger moment matrix under the Loewner ordering. These two criteria constitute the Kiefer design ordering. The parameter subsystem of interest K in the study was maximal parameter subsystem which is a subspace of the full parameter space  . In this model the full parameter subsystem was not estimable. By a proper definition of parameter matrix, a maximal parameter subsystem in the model was selected. Canonical unit vectors and the concept of Kronecker products were employed to identify the parameter matrices as well as the information matrices. For the second degree mixture model with two, three, four and m ingredients, a set of weighted centroid designs were obtained for a characterization of the feasible weighted centroid designs for the maximal parameter subsystem. After obtaining the feasible weighted centroid designs the information matrix of the design was computed. Derivations of A-, D- and E-optimal weighted centroid designs were then obtained from the information matrix. The optimality criteria A, D and E were used to compute optimal centroid designs. The results based on maximal parameter subsystem, second degree mixture model with m≥2 ingredient for A-, D- and E-optimal weighted centroid design for K exist for the choice of the coefficient matrix specifically in this study. Optimal weights and values for the weighted centroid designs were numerically computed using Matlab software.