Research On Optimal Design And Model-Robust Design For Mixture Experiment

The optimal design of mixture experiments has always been a research hotspot of this kind of experiments,and there have been quite abundant theoretical results so far.This paper studies several special kinds of optimal design problems of mixture experiments,including the algorithm for solving asymptotically optimal design,direct sum design of mixture experiments,product design of rectangular domain in simplex region,and model-robust optimal designs under asymmetric errors.Firstly,due to the influence of mixture constraints,the experimental domain of mixture experiments can only be a regular simplex or an irregular convex polygon(polyhedron)inside the simplex.Therefore,the theoretical optimal design can only be found in some low-order models,and in most cases,the optimal design can be found by algorithm.In this paper,an algorithm based on Fedorov algorithm is constructed to calculate the asymptotically optimal design of mixture experiments.The algorithm first uses the multiplicative algorithm to determine the number of support points of the optimal design,and adds points to each set of points in each iteration and adjusts the measure of each set of points.We prove the convergence of the algorithm in theory,and illustrate the effectiveness of the algorithm through several numerical examples.Secondly,in the research of optimal design of mixture experiments,researchers always considered the whole simplex region.In this paper,we discuss two kinds of -optimal designs on mixture experiment regions.The first kind is the direct sum experimental region of mixture additive model.It is proved that the -optimal design of mixture additive model can be constructed by the direct sum design of its sub-mixture model only when the sufficient condition is satisfied.The second kind is the product experimental region of the rectangular region inside the mixture simplex.Firstly,we discuss the independent variable transformation on the rectangular region inside the simplex region,and then construct the optimal product designs of two rectangular regions by the transformed models.Thirdly,it is generally assumed that the random error of the model is normal distributed or is symmetric in optimal design theory,and there are some nice statistical properties can be obtained when using ordinary least squares estimator.However,when the random error is asymmetric,it is more appropriate to use second-order least squares estimator.In this paper,we study the problem of model-robust optimal designs based on second-order least squares estimator when the random error of the model is asymmetric.We redefine the model-robust -and -optimal criteria under second-order least squares estimator.And based on the general equivalence theorem in optimal design theory,we give out the equivalence theorems and proof processes under the corresponding criterion.Finally,several numerical examples are given to illustrate the influence of different error parameters on the model-robust optimal design.