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Coordinating Drop-Off Locations and Pickup Routes: A Budget-Constrained Routing Perspective

Maria Albareda-Sambola, Víctor Blanco, Yolanda Hinojosa

TL;DR

The paper introduces the location-routing problem with Drop-Offs and Budget Constraints (DOBC), which jointly decides drop-off locations and a feasible single-vehicle route under a budget. It develops a flexible mixed-integer linear programming framework on an extended graph that replicates pickup visits and includes design, flow, and load-splitting variables, balanced by a convex combination of routing and transportation costs via $\alpha$. A two-phase branch-and-cut algorithm separates the exponential connectivity constraints on demand, enabling exact solutions for meaningful instance sizes. Extensive computational experiments on synthetic data illustrate feasibility conditions, the impact of visit-bounds, and the effect of $\alpha$ on solution quality, offering practical guidance for planning deployments. The work lays a foundation for extensions to uncertainty and multi-vehicle settings, highlighting the DOBC's potential for real-world last-mile and biogas supply-chain applications.

Abstract

We introduce in this paper a new variant of a location routing problem, to decide, the number and location of drop-off points to install based on the demands of a set of pick-up points, according to a given set-up budget for installing drop-off points. A single vehicle is in charge for all pick-up and drop-off operations, and the solution cost is associated with its route, which must also be decided. We provide a general and flexible mathematical optimization based approach for solving the problem that has some peculiarities to assure that the demand is adequately picked up, that some pickup points can be visited multiple times, that the capacity of the vehicle is respected, or that the vehicle is capable to implement the path or tour in the obtained solution. We report the results of a extense battery of experiments to validate our proposal on synthetic instances, and provide some insighs on the usefulness of our approach in practical applications.

Coordinating Drop-Off Locations and Pickup Routes: A Budget-Constrained Routing Perspective

TL;DR

The paper introduces the location-routing problem with Drop-Offs and Budget Constraints (DOBC), which jointly decides drop-off locations and a feasible single-vehicle route under a budget. It develops a flexible mixed-integer linear programming framework on an extended graph that replicates pickup visits and includes design, flow, and load-splitting variables, balanced by a convex combination of routing and transportation costs via . A two-phase branch-and-cut algorithm separates the exponential connectivity constraints on demand, enabling exact solutions for meaningful instance sizes. Extensive computational experiments on synthetic data illustrate feasibility conditions, the impact of visit-bounds, and the effect of on solution quality, offering practical guidance for planning deployments. The work lays a foundation for extensions to uncertainty and multi-vehicle settings, highlighting the DOBC's potential for real-world last-mile and biogas supply-chain applications.

Abstract

We introduce in this paper a new variant of a location routing problem, to decide, the number and location of drop-off points to install based on the demands of a set of pick-up points, according to a given set-up budget for installing drop-off points. A single vehicle is in charge for all pick-up and drop-off operations, and the solution cost is associated with its route, which must also be decided. We provide a general and flexible mathematical optimization based approach for solving the problem that has some peculiarities to assure that the demand is adequately picked up, that some pickup points can be visited multiple times, that the capacity of the vehicle is respected, or that the vehicle is capable to implement the path or tour in the obtained solution. We report the results of a extense battery of experiments to validate our proposal on synthetic instances, and provide some insighs on the usefulness of our approach in practical applications.
Paper Structure (17 sections, 2 theorems, 10 equations, 8 figures, 6 tables)

This paper contains 17 sections, 2 theorems, 10 equations, 8 figures, 6 tables.

Table of Contents

  1. Literature Review
  2. The single vehicle location-routing problem with Drop-Offs and Budget Constraints
  3. Mathematical Optimization Model for the DOBC
  4. The extended graph
  5. Variables
  6. Objective Function
  7. Constraints
  8. Connectivity Constraints
  9. The model
  10. Special versions of the problem
  11. Symmetry breaking constraints
  12. A two-phase Branch-and-Cut approach
  13. Computational Study
  14. Results for the small-size instances
  15. Large-size instances
  16. ...and 2 more sections

Key Result

Theorem 2.1

The DOBC is NP-hard.

Figures (8)

  • Figure 1: Instance for Example \ref{['ex:0']} (left) and a feasible solution (right).
  • Figure 2: Feasible solutions (non necessarily optimal) the different versions of the problem.
  • Figure 3: Performance profile by the instance size ($n$), with CPU time in log scale
  • Figure 4: Performance profiles of the $(1,1)$ -DOBC_C and $(\infty,1)$-DOBC_C models for feasible instances (left) and $(\infty,k)$-DOBC_P for $k \in \{1,2,3\}$ (right). CPU time in log scale.
  • Figure 5: Performance profile by the vehicles capacities ($\rho$ values). CPU time in log scale.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Example 2.1
  • Theorem 2.1
  • proof
  • Proposition 3.1
  • proof
  • Example 3.1