V-and R-optimal Designs Of Mixture Experiments

Designs of mixture experiments is an important experimental designs methods,and also an important research field of mathematical statistics.It has been widely used in biochemistry,engineering,materials and agronomy.It is solved by the mixing function.In the actual production,many factors and conditions are uncertain,in order to meet the requirements of product quality and production efficiency,often need to carry out multiple tests.Experimental designs methods is a method to determine the best mixing scheme by optimizing the mixing function on the premise of meeting the requirements of product quality and production efficiency,aiming at minimizing the number of tests.Its purpose is to adjust various factors in the process of the test under certain constraints,with the minimum cost and maximum income as the objective function,and get the optimal experiment designs.Optimal designs problems based on different optimality criteria have been an important area of research in experimental designs under different mixture models.Firstly,using the idea of mixture experiments,two kinds of common mixing models are derived and discussed: the low-order Scheffe central polynomial model and the second-order K model.Under the constraint conditions,based on the theory of V-optimal criterion,the general analytical expressions of V-optimal observations for two types of models are obtained.Meanwhile,the equivalence theorem of V-criterion is used to prove the optimal configurations of the three-component second-order central polynomial model and the two-component second-order K model.Finally,the efficiency of V-optimal design of two types of low-order mixture models with specific components is compared and analyzed.Then,This article applies the R-optimality criterion theory and explores the R-optimal allocation problem of a multi-response polynomial model with constraints.The R-optimal allocation of a multi-response model composed of two Scheffé second-order mixed models is computed,and the optimality of the obtained allocation is proved using the equivalence theorem of the R-criterion.Finally,the equivalence theorem of R-criterion is used to prove that the R-optimal configuration of the multi-response mixing model composed of Scheffe’s first-order and second-order mixing models is the R-optimal design.