A heuristic scheme for multivariate set partitioning problems with application to classifying heterogeneous populations for multiple binary attributes

We provide a novel heuristic approach to solve a class of multivariate set partitioning problems in which each item is characterized by three attribute values. The scheme first identifies a series of orderings of the items and then solves a corresponding sequence of shortest path problems. We provide theoretical findings on the structure of an optimal solution that motivate the design of the proposed heuristic scheme. The proposed algorithm runs in polynomial-time and is independent of the number of groups in the partition, making it more efficient than existing algorithms. To measure the performance of our solutions, we construct bounds for special instances which allow us to provide optimality gaps. We conduct an extensive numerical experiment in which we solve a large number of problem instances and show that our proposed approach converges to the global optimal solution in the vast majority of cases and in the case it does not, it yields very low optimality gaps. We demonstrate our findings with an application in the context of classifying a large heterogeneous population as positive or negative for multiple binary attributes as efficiently as possible. We conduct a case study on the screening of three of the most prevalent sexually transmitted diseases in the United States. The resulting solutions are shown to be within 2.6% of optimality and lead to a 26% cost saving over current screening practices.