统计与数据科学学院知识库

混料试验设计是一种重要的试验设计方法，也是数理统计学的一个重要研究领域，在生物化学、工程、材料和农学等学科领域有广泛的应用。它是通过混料函数来求解的。在实际生产中，许多因素和条件都是不确定的，为满足产品质量和生产效率的要求，往往需要进行多次试验。混料试验设计就是在满足产品质量要求和生产效率要求的前提下，以最小化试验次数为目标，通过对混料函数进行优化来确定最佳混料方案的一种方法。其目的即是在一定的约束条件下，以最小的费用和最大的收益为目标函数，在试验过程中对各个因素进行调整，得到最优的试验设计。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试验设计的一个重要研究课题。

第一，本文应用混料试验的思想，推导讨论了两类常见混料模型：低阶Scheffé中心多项式模型和混料K模型的V-最优配置问题。在约束条件的限制下，基于V-最优准则理论，计算得到了两类模型的V-最优观测的一般解析表达式，同时利用V-准则的等价性定理对三分量二阶混料中心多项式模型、二分量二阶混料K模型所得到的最优配置进行了证明。最后对具体分量的两类低阶混料模型的V-最优设计的效率进行了对比分析。

第二，本文应用R-最优准则理论，结合约束条件的限制探讨了多响应多项式模型的R-最优配置问题。计算得到了由两个Scheffé二阶混料模型构成的多响应模型的R-最优配置，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。最后利用R-准则的等价性定理及乘积设计的理论，证明了由Scheffé一阶和二阶混料模型构成的多响应混料模型的R-最优配置就是R-最优设计。

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.