Graphical methods for Order-of-Addition experiments

Abstract In an Order-of-Addition (OofA) experiment, the order in which m components are added to a system influences a response. Although much research has been done on optimal OofA experiments, existing methodologies typically assume that all m! orders are possible. However, in many practical examples, there are directed constraints on the pairwise order of components, making some of the m! orders infeasible. These constraints can be represented by a directed acyclic graph (DAG). The goal of the OofA experiment is to find an optimal order, which is equivalent to finding an optimal topological sort of the DAG. A multiplicative algorithm is used to identify approximate ϕ-optimal designs for an arbitrary DAG. Simulated annealing (SA) is proposed as a method to identify efficient exact designs. It is shown that the SA designs have very high efficiency relative to the approximate ϕ-optimal designs. A general procedure is proposed to search for the optimal order on a DAG given the results of an OofA experiment using two popular models. An application to a sequential job scheduling problem is investigated.