Developing a space-filling mixture experiment design when the components are subject to linear and nonlinear constraints

This article presents a case study of developing a space-filling design (SFD) for a constrained mixture experiment when the experimental region is specified by single-component constraints (SCCs), linear multiple-component constraints (LMCCs), and nonlinear multiple-component constraints (NMCCs). Traditional methods and software for designing constrained mixture experiments with SCCs and LMCCs (using either optimal design or SFD approaches) are not directly applicable because of the NMCCs. A SFD algorithm in the JMP^® software was modified to accommodate the NMCCs; the modification is described in this article. The case study involves high-level waste (HLW) glass that is subject to the formation of nepheline crystals as the glass cools. This can significantly reduce the durability of HLW glass (which is undesirable). The goal of the study was to develop a SFD for the HLW glass compositional region where nepheline may form, and generate data for modeling nepheline formation as a function of HLW glass composition. The HLW glass composition region was specified in terms of eight components with SCCs, two LMCCs, and two NMCCs. The NMCCs were based on a nonlinear logistic regression model for a binary nepheline response that was developed from previous data. This article discusses the HLW glass example, the constraints specifying the experimental composition region, and how an existing algorithm for generating SFDs was modified to accommodate the NMCCs. The methodology discussed in this article can be applied to any example in which the experimental region is specified by one or more nonlinear constraints in addition to linear constraints on mixture components and/or non-mixture variables.

本文针对实验区域由单组分约束（single-component constraints, SCCs）、线性多组分约束（linear multiple-component constraints, LMCCs）与非线性多组分约束（nonlinear multiple-component constraints, NMCCs）共同限定的约束混料试验，开展了空间填充设计（space-filling design, SFD）构建的案例研究。由于存在非线性多组分约束，针对仅含SCCs与LMCCs的约束混料试验的传统设计方法与软件（无论采用最优设计还是空间填充设计思路）均无法直接适用。本文对JMP®软件内置的空间填充设计算法进行了适配非线性多组分约束的改进，并详述了该改进方案。

本案例的研究对象为高放废物（high-level waste, HLW）玻璃：该类玻璃在冷却过程中易生成霞石晶体，此举会显著降低高放废物玻璃的耐久性，属于非期望现象。本研究的目标为：在可能生成霞石晶体的高放废物玻璃组分区域构建空间填充设计，并生成用于建模霞石生成量与高放废物玻璃组分间关联关系的试验数据。该高放废物玻璃的组分区域由8个组分、单组分约束、2项线性多组分约束以及2项非线性多组分约束共同定义；其中非线性多组分约束基于过往试验数据得到的二分类霞石响应非线性逻辑回归模型构建。

本文详述了该高放废物玻璃案例、试验组分区域的约束设定方式，以及现有空间填充设计生成算法的适配性改进方法。本文所提出的方法可推广应用于任意一类试验场景：其试验区域除混料组分的线性约束与/或非混料变量约束外，还包含一项或多项非线性约束。