When Is Heterogeneous Distance-Decay Facility Location Tractable? A Structural Classification, Exact Methods, and a Real-World Study

2026-07-18

AI Summary

This study addresses the continuous planar facility location problem under heterogeneous distance decay, where demand point values diminish at distinct rates with distance, unifying clustering, median, and coverage objectives within a single framework. It presents the first tractability classification for objective functions incorporating heterogeneous decay and elucidates how truncation operations disrupt concavity. The authors propose a tight mixed-integer programming (MIP) formulation alongside an efficient heuristic combining force-directed gradient descent and large neighborhood search. Computational experiments demonstrate that the discrete MIP solves instances with up to 500 points in seconds with near-zero LP relaxation gaps, while the heuristic yields solutions within 0.5% of optimality on average—substantially outperforming existing approaches. Validation on real-world urban delivery data reveals that neglecting decay heterogeneity can incur up to 9.7% demand loss.

Abstract

We study continuous planar facility location in which a demand point's captured value decays with distance, with the per-point decay scale varying across points. This heterogeneity is ubiquitous yet underexploited, and one nearest-facility objective unifies decay, clustering, and median goals, containing k-means, the Weber/p-median problem, and maximum covering as special cases. We make four contributions. (i) A tractability classification: the discrete objective is always monotone submodular, so the (1-1/e) greedy guarantee holds regardless of decay shape or heterogeneity, and the continuous cooperative objective is concave if and only if the decay is concave in distance; the clip max(0,d) in common coverage specifications is what destroys concavity, and the classification is tight. (ii) An exact discrete method: the candidate-discretized maximum-cover MIP has an empirically tight LP relaxation (~0% gap) and is solved by branch-and-bound in seconds for n <= 500. (iii) A force-as-gradient / large-neighborhood-search heuristic, within 0.5% of the discrete optimum, that outperforms the (1-1/e) greedy, Cooper-style alternating location-allocation, particle swarm optimization, and weighted k-means (30/30 per-instance wins at K=30, p<10^-9) and is competitive with bespoke solvers on k-means, Weber/p-median, and shape-demand instances. (iv) A real-world study: on 592,667 urban-delivery orders, ignoring the calibrated decay variation loses up to 9.7% of captured demand and relocates facilities by up to 37% of the map; a retail dataset calibrates the decay as exponential with scale R ~ 1.4 km.

Problem

• facility location
• distance decay
• heterogeneity
• submodular optimization
• continuous optimization

Innovation

• heterogeneous distance decay
• submodular optimization
• concave facility location
• exact MIP method
• large-neighborhood search