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The p-Compact-regions Problem
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DOI：10.1111/gean.12038.png)
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The p-compact-regions problem involves the search for an aggregation of n atomic spatial units into p-compact, contiguous regions. This article reports our efforts in designing a heuristic framework-MERGE (memory-based randomized greedy and edge reassignment)-to solve this problem through phases of dealing, randomized greedy, and edge reassignment. This MERGE heuristic is able to memorize (ME of MERGE) the potential best moves toward an optimal solution at each phase of the procedure such that the search efficiency can be greatly improved. A dealing phase grows seeded regions into a viable size. A randomized greedy (RG of MERGE) approach completes the regions' growth and generates a feasible set of p-regions. The edge-reassigning local search (E of MERGE) fine-tunes the results toward better objectives. In addition, a normalized moment of inertia (NMI) is introduced as the method of choice in computing the compactness of each region. We discuss in detail how MERGE works and how this new compactness measure can be seamlessly integrated into different phases of the proposed regionalization procedure. The performance of MERGE is evaluated through the use of both a small and a large p-compact-regions problem motivated by modeling the regional economy of Southern California. We expect this work to contribute to the regionalization theory and practice literature. Theoretically, we formulate a new model for the family of p-compact-regions problems. The novel NMI introduced in the model provides an accurate, robust, and efficient measure of compactness, which is a key objective for p-compact-regions problems. Practically, we developed the MERGE heuristic, proven to be effective and efficient in solving this nonlinear optimization problem to near optimality.
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