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.
