Symmetry breaking in mixed integer linear programming formulations for blocking two-level orthogonal experimental designs

Abstract

Two-level orthogonal designs play an important role in industrial screening experiments, in which the primary goal is to identify the treatment factors with the largest main effects on the output of a process or the performance of a product. Often, the experimental tests suggested by the orthogonal designs cannot be performed on a single day or using a single batch of raw material. This causes day-to-day or batch-to-batch variation in the experimental results, and necessitates the use of orthogonal blocking patterns. These blocking patterns ensure that the factors’ main effects can be estimated independently from the day-to-day or batch-to-batch variation. Finding an optimal orthogonal blocking pattern for an orthogonal design is a major challenge. Recently, mixed integer linear programming has been used for that purpose. While this approach is promising, computational experiments have indicated it is quite slow. We show how to speed up the mixed integer linear programming approach by adding symmetry breaking constraints to the formulation, and study the usefulness of an asymmetric representatives formulation. In other words, we introduce state-of-the-art symmetry breaking approaches in optimal experimental design. We perform extensive computational experiments to test which combinations of symmetry breaking constraints work best. Throughout, we consider two kinds of symmetry: symmetry due to the fact that the blocks can be relabeled without affecting the quality of the blocking pattern, and symmetry due to replicated test combinations.