Abstract
Mixture experiments are special type of response surface designs where the factors under study are proportions of the ingredients of a mixture. In response surface designs the main interest of the experimenter may not always be in the response at individual locations, but the differences between the responses at various locations is of great interest. Most of the studies on estimation of slope (rate of change) have concentrated in Central Composite Designs (CCD) yet mixture experiments are intended to show the response for all possible formulations of the mixture and to identify optimal proportions for each of the ingredients at different locations. Slope optimal mixture designs for third degree Kronecker model were studied in order to obtained optimal formulations for all possible ingredients in simplex centroid. Weighted Simplex Centroid Designs (WSCD) and Uniformly Weighted Simplex Centroid Designs (UWSCD) mixture experiments were obtained in order to identify optimal proportions for each of the ingredients formulation. Derivatives of the Kronecker model mixture experiment were used to obtain Slope Information Matrices (SIM) for four ingredients. Maximal parameters of interest for third degree Kronecker model were considered. D-, E-, A-, and T- optimal criteria and their efficiencies for both WSCD and UWSCD third degree Kronecker model were obtained. UWSCD was found to be more efficient than WSCD for almost all the points in the simplex designs, therefore recommended for more optimal results in mixture experiments.
Highlights
Response surface methodology (RSM) is a collection of mathematical and statistical tools or techniques that are useful for modeling and analysis of problems in which a response of interest is influence by several ingredients and the objective is to optimize this response, Montgomery (2001)
In many applications of response surface methodology, good estimation of the derivatives of the response function is as important as estimation of the Cheruiyot Kipkoech et al.: Slope Optimal Designs for Third Degree Kronecker Model Mixture Experiments mean response
From the work of Hader and Park (1978), Huda and Al-Siha (1999) and Huda (2006), it is clear that most of the work has been done on central composite designs there was a need to extend the concept of slope to mixture experiments third degree Kronecker, this method was used for proper identification of the ingredients ratio that leads to an optimal response
Summary
Response surface methodology (RSM) is a collection of mathematical and statistical tools or techniques that are useful for modeling and analysis of problems in which a response of interest is influence by several ingredients and the objective is to optimize this response, Montgomery (2001). Response Surface Methodology is an important subject in the statistical design and analysis of experiments. In many applications of response surface methodology, good estimation of the derivatives of the response function is as important as estimation of the Cheruiyot Kipkoech et al.: Slope Optimal Designs for Third Degree Kronecker Model Mixture Experiments mean response. From the work of Hader and Park (1978), Huda and Al-Siha (1999) and Huda (2006), it is clear that most of the work has been done on central composite designs there was a need to extend the concept of slope to mixture experiments third degree Kronecker, this method was used for proper identification of the ingredients ratio that leads to an optimal response. J =1 f (t) f (t)t dη η for the third degree Kronecker model has all entries homogeneous in
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