AI 中文总结
研究异构距离衰减设施选址问题，给出可解性分类，提出精确离散方法和力梯度/大邻域搜索启发式算法，通过实际应用研究表明忽略衰减变化会有需求损失和设施重定位，校准后衰减为指数形式。
AI 中文摘要
我们研究连续平面设施选址问题，其中需求点的捕获值随距离衰减，且各点的衰减尺度不同。这种异构性普遍存在但未被充分利用，一个最近设施目标统一了衰减、聚类和中位数目标，k均值、Weber/p中位数问题和最大覆盖问题都是其特殊情况。我们有四项贡献：（i）可解性分类：离散目标总是单调次模的，因此（1 - 1/e）贪心保证成立；连续合作目标当且仅当衰减在距离上是凹的时才是凹的，常见覆盖规范中的clip max(0,d)破坏了凹性，且该分类是紧密的。（ii）一种精确离散方法：候选离散化最大覆盖混合整数规划有经验上紧密的线性规划松弛（差距约为0%），对于n <= 500可在数秒内通过分支定界法求解。（iii）一种力梯度/大邻域搜索启发式算法，与离散最优解的差距在0.5%以内，优于（1 - 1/e）贪心算法、Cooper式交替定位分配算法、粒子群优化算法和加权k均值算法（在K = 30，p < 10^-9时，每个实例有30/30的胜率），并且在k均值、Weber/p中位数和形状需求实例上与定制求解器具有竞争力。（iv）一项实际应用研究：在592,667个城市配送订单上，忽略校准后的衰减变化会损失高达9.7%的捕获需求，设施重新定位最多可达地图的37%；一个零售数据集将衰减校准为指数形式，尺度R约为1.4公里。
英文摘要
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.
