Optimal Run Order for Order-of-Addition Experiments

Order-of-addition experiments have emerged as a cornerstone in modern experimental design, yet the critical role of run-order has been entirely neglected in the literature. This oversight is surprising, given that the run order can significantly influence the cost, efficiency, and validity of the experiment. Certain run orders are inherently more economical and effective, while suboptimal orders may introduce unnecessary complexities or compromise results. To ensure experimental integrity, an optimal run order must minimize factor level changes, especially when transitions between levels are costly—and safeguard against the detrimental effects of time trends, which can skew factor effect estimates and undermine conclusions. Despite its importance, the relationship between run order and order-of-addition experiments remains poorly understood. Our research addresses this critical gap by deriving the theoretical lower bound on the total number of factor level changes for ordered order-of-addition designs. We introduce novel construction methods that achieve this minimum, ensuring both cost efficiency and practicality. Additionally, we develop a rigorous framework to eliminate the bias introduced by linear time trends, proposing robust methods to construct time trend-free ordered designs. These methods lead to the establishment of optimal ordered designs that excel under both criteria. The proposed designed orders go beyond theoretical elegance—they are robust to lurking variables in the external environment, making them invaluable for real-world applications. This study not only shines the spotlight on an overlooked dimension of order-of-addition experiments but also delivers solutions that redefine how these experiments should be conducted. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.