Optimal slope designs for second degree kronecker model mixture experiments with application in blending of selected fruits

Abstract

Response surface methodology is a set of techniques that includes setting up a series of experiments that yields adequate and reliable measurements of the response of interest, determine a model that best fits the data collected from the experimental design chosen and determine the optimal settings of the experimental factors that produce the maximum (or minimum) value of response. The aim of the study was to investigate D- and A- optimal slope designs in the second degree Kronecker model for mixture experiments with assumptions that the errors are independent and with constant variance. The objectives of study were to obtain: equivalence relation that serve as the necessary and sufficient condition for the existence of optimal slope designs; optimal slope designs for the D- and A-optimality criteria and numerical optimal weighted centroid designs and to demonstrate the practical use of generated design in analysis of data obtained from a designed experiment on fruit blending. The equivalence relation was proved using matrix algebra. Support points, elementary centroid designs, coefficient, moment, information and slope matrices, were used to derive optimal designs. D- and A-optimal designs were employed to generate numerical optimal designs. The data collected from the designed experiment were analyzed using SAS (Version 8) software. As a result, the study was able to obtain generalized optimal slope design for a mixture experiment with at least two ingredients. The Kronecker models fitted to the data from the experiment on fruit blending explained the variation adequately well with coefficients of determination 98.2, 96.3 and 96.67 percent for the blend of two, three and four ingredients respectively. Kronecker model with the weighted centroid design is very economical considering the few support points that are necessary for a particular number of ingredients experiment. In conclusion, the findings of this study strongly supports the use of the form of the Kronecker model discussed to analyze the response surfaces for mixture experiments. The study therefore highly recommends use of these models to describe juice qualities that depend on variations in mixture amounts.