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A Branch-and-Price Algorithm for Fast and Equitable Last-Mile Relief Aid Distribution

Mahdi Mostajabdaveh, F. Sibel Salman, Walter J. Gutjahr

TL;DR

This paper tackles post-disaster relief logistics by modeling a bi-objective vehicle routing problem that jointly plans routes and delivery quantities from a single depot to shelters under scarce supplies. It introduces a branch-and-price algorithm with route-delivery columns, a GRASP-enhanced pricing subproblem, and valid inequalities to efficiently compute optimal delivery allocations while balancing total travel time and equity via a $Gini$-based measure. The authors derive structural properties of optimal deliveries, enabling arithmetic computation of amounts per route, and demonstrate substantial performance gains over a commercial MIP solver on real-world-like Van and Kartal data, including a 34% improvement in distribution equity. Practically, the approach supports faster, fairer decision-making in the critical early post-disaster phase, with insights into the trade-offs between efficiency and fairness under varying time constraints and a plan for extending to split deliveries and alternative equity formulations.

Abstract

The distribution of relief supplies to shelters is a critical aspect of post-disaster humanitarian logistics. In major disasters, prepositioned supplies often fall short of meeting all demands. We address the problem of planning vehicle routes from a distribution center to shelters while allocating limited relief supplies. To balance efficiency and equity, we formulate a bi-objective problem: minimizing a Gini-index-based measure of inequity in unsatisfied demand for fair distribution and minimizing total travel time for timely delivery. We propose a Mixed Integer Programming (MIP) model and use the $ε$-constraint method to handle the bi-objective nature. By deriving mathematical properties of the optimal solution, we introduce valid inequalities and design an algorithm for optimal delivery allocations given feasible vehicle routes. A branch-and-price (B&P) algorithm is developed to solve the problem efficiently. Computational tests on realistic datasets from a past earthquake in Van, Turkey, and predicted data for Istanbul's Kartal region show that the B&P algorithm significantly outperforms commercial MIP solvers. Our bi-objective approach reduces aid distribution inequity by 34% without compromising efficiency. Results indicate that when time constraints are very loose or tight, lexicographic optimization prioritizing demand coverage over fairness is effective. For moderately restrictive time constraints, a balanced approach is essential to avoid inequitable outcomes.

A Branch-and-Price Algorithm for Fast and Equitable Last-Mile Relief Aid Distribution

TL;DR

This paper tackles post-disaster relief logistics by modeling a bi-objective vehicle routing problem that jointly plans routes and delivery quantities from a single depot to shelters under scarce supplies. It introduces a branch-and-price algorithm with route-delivery columns, a GRASP-enhanced pricing subproblem, and valid inequalities to efficiently compute optimal delivery allocations while balancing total travel time and equity via a -based measure. The authors derive structural properties of optimal deliveries, enabling arithmetic computation of amounts per route, and demonstrate substantial performance gains over a commercial MIP solver on real-world-like Van and Kartal data, including a 34% improvement in distribution equity. Practically, the approach supports faster, fairer decision-making in the critical early post-disaster phase, with insights into the trade-offs between efficiency and fairness under varying time constraints and a plan for extending to split deliveries and alternative equity formulations.

Abstract

The distribution of relief supplies to shelters is a critical aspect of post-disaster humanitarian logistics. In major disasters, prepositioned supplies often fall short of meeting all demands. We address the problem of planning vehicle routes from a distribution center to shelters while allocating limited relief supplies. To balance efficiency and equity, we formulate a bi-objective problem: minimizing a Gini-index-based measure of inequity in unsatisfied demand for fair distribution and minimizing total travel time for timely delivery. We propose a Mixed Integer Programming (MIP) model and use the -constraint method to handle the bi-objective nature. By deriving mathematical properties of the optimal solution, we introduce valid inequalities and design an algorithm for optimal delivery allocations given feasible vehicle routes. A branch-and-price (B&P) algorithm is developed to solve the problem efficiently. Computational tests on realistic datasets from a past earthquake in Van, Turkey, and predicted data for Istanbul's Kartal region show that the B&P algorithm significantly outperforms commercial MIP solvers. Our bi-objective approach reduces aid distribution inequity by 34% without compromising efficiency. Results indicate that when time constraints are very loose or tight, lexicographic optimization prioritizing demand coverage over fairness is effective. For moderately restrictive time constraints, a balanced approach is essential to avoid inequitable outcomes.
Paper Structure (22 sections, 2 theorems, 17 equations, 6 figures, 6 tables, 2 algorithms)

This paper contains 22 sections, 2 theorems, 17 equations, 6 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Literature review
  3. Relief aid distribution
  4. Equity in humanitarian logistics
  5. Problem definition and mathematical models
  6. Mathematical model
  7. Representation of inequity
  8. Characteristics of the optimal solution
  9. Exact solution method
  10. Multi-objective approach
  11. Route-delivery based formulation
  12. Column generation
  13. Pricing subproblem
  14. Generating initial columns
  15. Branch and Price
  16. ...and 7 more sections

Key Result

Proposition 1

If $\frac{D_k}{D} \le \frac{Q}{C}$ for all $k \in K$, then the optimal solution of (xObj) -- (xCon) is given by which constitutes perfect equity.

Figures (6)

  • Figure 1: Example of three vehicle routes (shown with different line types) delivering to shelters from a single depot. Numbers above each shelter show delivered amounts and corresponding demands, with demand satisfaction varying: 100% for the two rightmost shelters, but only 50% for the leftmost.
  • Figure 2: A flowchart representing the main components of our branch-and-price algorithm and their interactions.
  • Figure 3: Map of Van province: the demand nodes are represented by red dots, the road junction points by green triangles, and the depot by a blue star.
  • Figure 4: Map of Istanbul's Kartal district highlighting selected shelter locations (marked in green) and the central relief aid distribution center (marked in yellow).
  • Figure 5: Pareto front of four selected instances
  • ...and 1 more figures

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2