统计与数据科学学院知识库

混料试验设计主要研究的是配方配比中的试验设计问题，它是安排混料试验和分析混料数据的一种数学理论和方法，是数理统计学的重要分支。混料试验设计是约束区域上的优化问题，大多数是以最优设计理论为基础的。它在工农业生产、经营管理和科学研究中有着广泛的应用，给企业和社会带来了极大的经济和社会双重效益。在各种不同的混料模型下，基于不同最优准则的最优设计问题一直是该领域的研究热点，近些年来，相关研究人员将稳健设计与最优设计结合起来研究，探讨混料模型与回归模型的稳健最优设计问题成为了新的研究方向。

首先，本文应用R-最优准则理论，结合约束条件的限制探讨了二阶中心多项式模型的R-最优配置问题，并利用R-准则的等价性定理对所得到的最优配置进行了证明。同时，文章针对三阶的混料中心多项式模型，探讨了三分量三阶中心多项式模型的R-最优设计问题，并利用Mathematica软件给出了三分量三阶混料模型R-最优设计的数值解。

然后，本文应用稳健R-最优准则理论，结合约束条件的限制探讨了二阶中心多项式模型的稳健R-最优配置问题，并计算得到了该最优配置解。对于具体的先验测度，给出了三分量二阶混料模型的稳健R-最优配置，之后对于所得到的结果利用稳健R-准则的等价性定理，并结合多元函数的条件极值定理，证明了该稳健最优配置就是稳健R-最优设计。最后对R-最优设计与稳健R-最优设计的效率进行了比较分析。

Mixture experiment design mainly studies the experimental design problem in the formula ratio, which is a mathematical theory and method to arrange mixture experiment and analyze mixture data, and is an important branch of mathematical statistics. Mixture experiment design is an optimization problem in the constraint region, most of which is based on optimal design theory. It is widely used in industrial and agricultural production, management and scientific research, and has brought great economic and social benefits to enterprises and society. The optimal design problem based on different optimal criteria has always been a research hotspot in this field under different mixing models. In recent years, relevant researchers combine robust design with optimal design, and it has become a new research direction to discuss the robust optimal design of mixture model and regression model.

Firstly, the R-optimal configuration of the second-degree central polynomial model is discussed by using R-optimal criterion theory, and the obtained optimal configuration is proved by using the equivalence theorem of R-optimal criterion. At the same time, this thesis discusses the R-optimal design problem of the third-degree central polynomial model of the concrete component, and gives the numerical solution of the R-optimal design of the three-component third-degree polynomial model by using Mathematica software.

Secondly, based on the theory of robust R-optimal criterion, the robust R-optimal configuration of the second-degree central polynomial model is discussed, and the optimal configuration solution is obtained. For the specific prior measure, the robust R-optimal configuration of the three-component second-degree mixture model is given. Then, the robust R-optimal configuration is proved to be the robust R-optimal design by using the equivalence theorem of the robust R-criterion and the conditional extremum theorem of multivariate functions. Finally, the efficiency of R-optimal design and robust R-optimal design is compared and analyzed.