Adaptive Order-of-Addition Experiments via the Quick-Sort Algorithm

The order-of-addition (OofA) experiment has received a great deal of attention in the recent literature. The primary goal of the OofA experiment is to identify the optimal order in a sequence of <i>m</i> components. All the existing methods are model-dependent and are limited to small number of components. The appropriateness of the resulting optimal order heavily depends on (a) the correctness of the underlying assumed model, and (b) the goodness of model fitting. Moreover, these methods are not applicable to deal with large <i>m</i> (e.g., m≥7). With this in mind, this article proposes an efficient adaptive methodology, building upon the quick-sort algorithm, to explore the optimal order without any model specification. Compared to the existing work, the run sizes of the proposed method needed to achieve the optimal order are much smaller. Theoretical supports are given to illustrate the effectiveness of the proposed method. The proposed method is able to obtain the optimal order for large <i>m</i> (e.g., m≥20). Numerical experiments are used to demonstrate the effectiveness of the proposed method.添加顺序（order-of-addition, OofA）实验在近期学术文献中受到广泛关注。该类实验的核心目标是在由m个组分构成的序列中识别最优添加顺序。现有方法均依赖模型设定，且仅适用于组分数量较少的场景。所得最优顺序的合理性高度依赖于两点：一是底层假设模型的正确性，二是模型拟合效果。此外，此类方法无法处理组分数量较大的情形（例如m≥7）。有鉴于此，本文提出了一种基于快速排序（quick-sort）算法的高效自适应方法，无需指定模型即可探索最优添加顺序。与现有研究相比，本文所提方法达成最优顺序所需的试验次数显著更少。本文给出了理论支撑以阐明所提方法的有效性。该方法可适用于组分数量较大的场景（例如m≥20）。本文通过数值实验验证了所提方法的有效性。