AI 中文总结
研究容量受限p-选址问题及其扩展，提出带领土覆盖约束的多尺度变体，给出强化ILP公式和RSSV启发式算法，通过实验证明方法的效率、可靠性及对选址规划公平性的影响。
AI 中文摘要
本文研究容量受限p-选址问题（C$p$LP）及其包含公平性考量的扩展问题。在选址科学中，p-中位数问题（$p$MP）是经典模型。C$p$LP包括容量受限p-中位数问题（C$p$MP）及其松弛变体（C$p$MP$^r$），通过纳入容量约束扩展了$p$MP。本文首先引入带领土覆盖约束的C$p$LP（C$p$LP-TC），在此基础上提出多尺度变体（C$p$LP-MTC）。提出了带有效不等式的强化整数线性规划（ILP）公式，还开发了随机抽样空间投票（RSSV）启发式算法。通过基准和实际案例的计算实验证明了其效率和可靠性，也展示了基于公平领土约束对选址规划公平性的影响。
英文摘要
This paper studies the Capacitated $p$-Location Problem (C$p$LP) and its extensions incorporating equity considerations. In location science, the $p$-Median problem ($p$MP) is a classical model that selects $p$ facilities from a finite set of candidates to serve a set of customers while minimizing total allocation costs. The C$p$LP, which includes the Capacitated $p$-Median Problem (C$p$MP) and its relaxed variant (C$p$MP$^r$), extends the $p$MP by incorporating capacity constraints on facilities. We formalize the C$p$LP with Territorial Coverage Constraints (C$p$LP-TC), an extension that enforces equity across spatial units, and generalize it to a multi-scale variant (C$p$LP-MTC) that enforces equity simultaneously across nested spatial scales. We present a strengthened Integer Linear Programming (ILP) formulation with valid inequalities that enables the exact solution of medium-sized instances. To tackle larger problems, we adapt the Random Sampling Spatial Voting (RSSV) heuristic, originally proposed for the $p$MP, into a competitive open-source matheuristic that combines a heuristic reduction phase with the strengthened ILP formulation. The resulting method remains flexible, requires minimal parameter tuning, and is accessible to non-specialist users.Computational experiments on a new open-source benchmark instance set, built from a French regional case study, demonstrate the effectiveness of both the exact and heuristic approaches. Beyond their computational performance, the results quantify the trade-offs between efficiency and territorial equity under different equity constraints, providing practical insights for equitable location planning.
Comments29 pages, 10 figures
