The max‐p‐compact‐regions problem

Abstract The max‐ p ‐compact‐regions problem involves the aggregation of a set of small areas into an unknown maximum number ( p ) of compact, homogeneous, and spatially contiguous regions such that a regional attribute value is higher than a predefined threshold. The max‐ p ‐compact‐regions problem is an extension of the max‐ p ‐regions problem accounting for compactness. The max‐ p ‐regions model has been widely used to define study regions in many application cases since it allows users to specify criteria and then to identify a regionalization scheme. However, the max‐ p ‐regions model does not consider compactness even though compactness is usually a desirable goal in regionalization, implying ideal accessibility and apparent homogeneity. This article discusses how to integrate a compactness measure into the max‐ p regionalization process by constructing a multiobjective optimization model that maximizes the number of regions while optimizing the compactness of identified regions. An efficient heuristic algorithm is developed to address the computational intensity of the max‐ p ‐compact‐regions problem so that it can be applied to large‐scale practical regionalization problems. This new algorithm will be implemented in the open‐source Python Spatial Analysis Library. One hypothetical and one practical application of the max‐ p ‐compact‐regions problem are introduced to demonstrate the effectiveness and efficiency of the proposed algorithm.