The Capacitated $p$-Location Problem with Territorial Coverage Constraint: Efficient Solution and Case Study

2026-07-02

AI Summary

This study addresses the capacitated p-location problem by introducing, for the first time, multi-scale area coverage constraints to ensure spatial fairness. To this end, the authors propose two models—C$p$LP-TC and its multi-scale extension C$p$LP-MTC—along with a strengthened integer linear programming (ILP) formulation, valid inequalities, and a parameter-free randomized sampling spatial voting (RSSV) heuristic, integrated with problem-size reduction techniques for efficient solution. Evaluation on an open-source benchmark dataset constructed from French administrative regions demonstrates that the approach effectively quantifies the trade-off between service efficiency and regional equity, offering scalable and reproducible decision support for fair facility location in real-world applications.

Abstract

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 which selects $p$ facilities from a finite set of candidates to serve a set of customers 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 first introduce the C$p$LP with Territorial Coverage Constraints (C$p$LP-TC), a extension of the CpLP that enforces equity considering the spatial units. Building on this, we propose the multi-scale variant (C$p$LP-MTC), which generalizes territorial coverage constraints across nested spatial scales, allowing equity to be enforced simultaneously at multiple levels of spatial aggregation. A strengthened Integer Linear Programming (ILP) formulation with valid inequalities is presented. In addition, we develop the Random Sampling Spatial Voting (RSSV) matheuristic, which integrates a heuristic reduction phase with the strengthened ILP. The method is flexible, requires minimal parameter tuning, and remains accessible to non specialist users. Computational experiments on benchmark and real world instances demonstrate its efficiency and reliability. The results also demonstrate the impact of equity territorial based constraints on fairness in location planning.

Problem

• Capacitated p-Location Problem
• Territorial Coverage Constraints
• Equity
• Facility Location
• Spatial Fairness

Innovation

• Capacitated p-Location Problem
• Territorial Coverage Constraints
• Multi-scale Equity
• Strengthened ILP Formulation
• Matheuristic