Some Advances in Mixture Designs Applied to Orthogonal Blocks

Abstract

When we formulate a mixture like juice, medicine, food etc, we have very little knowledge about our final product. In mixture experiments the product is a mixture of several ingredients. So, we need to maximize the product performance by using the optimum proportions of the ingredients. Mixture experiments are very useful in handling such optimum proportions. We perform a mixture experiment to answer the questions raised about the finished product. The literature review of all types of optimal designs revealed that optimality could be achieved on the boundary of the simplex. Such optimal designs were constituted of binary blends, except the common centroid in each block. Hence the orthogonally blocked optimal mixture designs did not formulate a complete mixture. With the compromise on the efficiency of designs, nearly optimal orthogonally blocked mixture designs in three and four components were proposed for the Scheffé quadratic mixture model. We propose nearly D- , A- and E-optimal mixture designs for three and four components in two blocks, under Latin squares based orthogonal blocking scheme for Scheffé quadratic mixture model, quadratic K-model, Becker’s quadratic homogeneous models and for Darroch and Waller’s quadratic mixture model. The robustness of nearly D- A- and E- optimal designs for a particular value of shrinkage parameter s is observed. We have addressed the properties of D-optimal designs for five components in two orthogonal blocks, for Darroch and Waller’s quadratic mixture model, based upon Latin squares and F-squares orthogonal blocking schemes. In real life situation sometimes the total amount of the mixture also affects the response, say amount of fertilizers used. In existing literature D-optimal mixture component- amount designs, including blocks with orthogonal Latin squares and F-squares, were constructed by projection. Such designs in two and in three components were composed of binary mixture blends. The construction of nearly D-optimal mixture component-amount designs were not addressed with reference to F-squares based orthogonal blocking scheme. We construct F-squares based orthogonally blocked nearly D-optimal component-amount designs in two and three components from orthogonally blocked mixture component-amount designs obtained via projections of orthogonally blocked F-square designs. Recently in literature it was verified that when the initial (q-1)-dimensional unit spherical orthogonally blocked response surface designs (like Box Behnken and Central Composite Design) were transformed into a (q-1)-dimensional ellipsoidal restricted region, then the resulting q-component mixture designs were also orthogonally blocked. We have verified the same issue by using some other unit spherical orthogonally blocked designs as an initial response surface design. The idea of slope-rotatability in axial directions (SRIAD) and over all directions (SROAD) is mostly addressed in literature for different response surface designs. Not much work so far has been done for slope-rotatable designs in mixture experiments. We have derived the necessary and sufficient conditions for slope-rotatability in axial directions and over all directions for the quadratic K- model. Some new measures of Slope-Rotatability for unconstrained and constrained mixture regions are introduced, using Gini Mean Difference method. Further a measure of slope-rotatability over all directions is established for quadratic K- model. More on we have tried to compare different loss functions for Bayesian control in mixture models. Although the last two chapters do not use orthogonally blocked mixture designs but still carry some new research issues related to mixture experiments. Some research work from chapter three and chapter nine has been published.