研发新型材料的试验设计研究

Experimental design plays a key role in the process of industrial innovation. Researchers want to obtain new products with high quality, high production and low cost through some experiments. For example, researchers hope to get the best production process and stable quality of polyformaldehyde by using the method of experimental design. It is an optimization problem for the cases with multivariate, multiple responses and unknown response surface. In this project we use sequential designs for such complex problems. Since the response surface is unknown, in the first step of sequential design, we consider a new type of space-filling design which balances uniformity and maximin distance. We will show the relationships among measure of uniformity, maximin distance and other design criteria, and give some new construction methods for such type of design based on their theoretical results. Moreover, for improving the reliability of analysis result, one needs some addition experiments such that the data from the two steps can be used for cross-validation. Then, the important variables can be screened out. Therefore, it is worth to study the selection method for addition experiments in the second step of sequential design. We will study the construction method and the corresponding theory of sequential design, which may have higher estimation efficiency than the common used sequential designs such as the central composite design. The research result of this project will also be used for problems of parameter optimization in other projects.

试验设计在工业创新过程中发挥了很大的作用。研究人员希望通过试验达到优质、高产和低消耗。例如，研究人员希望通过试验设计的方法得到聚甲醛的最佳生产工艺，且使产品质量稳定，其为多因素、多响应且真实模型未知的寻优问题。针对此类复杂问题，本项目考虑使用序贯设计方法。由于真实模型未知，在序贯设计的第一阶段，我们考虑一种兼顾均匀性和极大极小距离的空间填充设计。为此，我们首先研究距离准则和均匀性准则与其它设计准则之间关系，并基于这些理论结果给出构造这类设计的新方法。同时为了提高分析结果的可靠度，需要再安排一批追加试验，使得前后两批数据可以交叉验证，从而筛选出重要变量。因此在序贯设计的第二阶段，即追加设计点的选取方法是值得研究的问题。我们将研究序贯设计的构造方法，并讨论其相关理论，并希望该序贯设计比常用的序贯设计例如中心组合设计有更高的估计效率。该项目的研究结果也将应用于其它项目中的模型参数优化问题。

试验设计在工业创新过程中发挥了很大的作用。研究人员希望通过试验达到优质、高产和低消耗。本项目的研究成果如下：(1)空间填充准则之间的关系。用理论和实例说明了距离准则和均匀性准则之间存在密切联系。(2) 空间填充设计的新的构造算法，尤其适用于大设计矩阵，比如试验次数达到几百、几千甚至上万。其构造算法具有快速高效性。(3) 给出了多种序贯设计的方法。例如，二水平和四水平的正交列复合而成的正交列复合设计，以及扩充均匀设计等。(4) 均匀性度量。给出超球体内的均匀性度量，并研究了低维投影均匀性。(5) 代表点理论。其考虑给定一维或多维分布函数后，寻找 n 个点使其很好地表示分布函数。我们给出了新的衡量准则，并给出相应的构造算法。这些研究结果发展了为空间填充设计与序贯设计等方面的理论。