Construction of D-Optimal Split-Plot Designs for the Second-Degree Kronecker Model Mixture Experiments in the Presence of Process Variables

Abstract

Practical problems in mixture experiments are usually associated with the investigation of mixture of <i>m</i> ingredients, which are assumed to influence the response through the proportions in which they are blended together. Mixture experiments are modeled using Scheffe’ models or Kronecker models whichever that is applicable. In such problems, the response of mixture experiments may also be affected by the conditions under which the mixture in conducted. This creates a shift in the blending characteristics of the mixture ingredients hence affecting the end product hence the need for inclusion of these conditions during modeling of mixture experiments. The objective of this study is to construct D-optimal designs for mixture experiments in the presence of process variables. In order to achieve this, first, a combined model of the second-degree Kronecker model for mixture experiments and the second-degree polynomial in the process variables in developed. The D-optimal designs are constructed using a Monte Carlo algorithmic approach in the AlgDesign of the R-packages. The designs constructed in this study were augmented with two replications of a level of the process variable. The D-optimal designs are evaluated using their D-optimal values and their relative D-efficiencies. The results of this study illustrate the existence of two alternate designs; one replicating at (-1, -1) and (-1, 1) in Table 2 and the other replicating at (0, 0) in Table 1. In conclusion the results of this study indicate that a design replicating at different levels of the process variable performs better than the one replicating at the overall centroid.