Under-capacitated p-median location problem

Abstract

In many Location Science applications, a key challenge is the fair allocation of scarce resources among clients, especially in contexts such as disaster response, transportation planning, and healthcare systems. This challenge becomes particularly critical under severe capacity shortages, where not all demand can be satisfied, an aspect that has received limited attention in the existing literature. This paper studies the joint problem of locating p facilities from a set of candidates with heterogeneous, limited capacities and allocating indivisible resources (e.g., hospital beds) among clients. We propose a generalized capacitated p-median model that integrates location decisions with a fair allocation of insufficient capacity. Fairness is defined using concepts from bankruptcy theory in cooperative game theory, which are adapted to explicitly represent capacity scarcity. Several optimization models based on bankruptcy-inspired allocation rules are introduced and their theoretical properties are analyzed. We examine how different weights in the objective function affect the trade-offs between efficiency, equity, and network costs, and we compare bankruptcy-based allocations with alternative fairness rules. Extensions to divisible resources (e.g., water) are also discussed, and computational experiments are used to assess the impact of the proposed models on location decisions. The proposed framework is well suited for strategic and tactical planning under static and deterministic settings in which total capacity is insufficient to meet demand. Relative to classical p-median formulations and existing fairness-based benchmarks, the model explicitly represents demand shortfalls through bankruptcy-based allocation rules, leading to substantially fairer allocations while preserving high capacity utilization and competitive network costs. From a managerial perspective, the approach allows decision-makers to embed equity considerations directly into the planning stage, rather than relying on ad hoc rationing after facility locations are fixed. The practical relevance of the framework is illustrated through a real-world case study on food insecurity in Todd County, South Dakota, demonstrating its effectiveness in designing fair and efficient service networks under severe capacity shortages.