The p-Regions Problem

The p-regions problem involves the aggregation or clustering of n small areas into p spatially contiguous regions while optimizing some criteria. The main objective of this article is to explore possible avenues for formulating this problem as a mixed integer-programming (MIP) problem. The critical issue in formulating this problem is to ensure that each region is a spatially contiguous cluster of small areas. We introduce three MIP models for solving the p regions problem. Each model minimizes the sum of dissimilarities between all pairs of areas within each region while guaranteeing contiguity. Three strategies designed to ensure contiguity are presented: (1) an adaptation of the Miller, Tucker, and Zemlin tour-breaking constraints developed for the traveling salesman problem; (2) the use of ordered-area assignment variables based upon an extension of an approach by Cova and Church for the geographical site design problem; and (3) the use of flow constraints based upon an extension of work by Shirabe. We test the efficacy of each formulation as well as specify a strategy to reduce overall problem size.