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Modular Vehicle Routing Problem: Applications in Logistics

Hang Zhou, Yang Li, Chengyuan Ma, Keke Long, Xiaopeng Li

TL;DR

This paper introduces the Modular Vehicle Routing Problem (MVRP), a new logistics routing variant that allows modular vehicles to dock and split en route to form platoons. It develops three-index MILP formulations for the capacitated and time-window variants (CMVRP and MVRPTW) and designs a specialized Tabu Search with a Gantt-chart-based solution representation and tailored merging/relocation operators to efficiently solve large instances. Theoretical bounds relate MVRP to the classical VRP, and empirical results show that MV platooning can reduce total costs by approximately 5.6%, with larger gains possible as platooning capability improves. The study also discusses practical extensions (en-route cargo transfer, electric MV, heterogeneous modules) and outlines directions for future work, including advanced metaheuristics and faster exact methods for large-scale problems.

Abstract

Recent studies and industry advancements indicate that modular vehicles (MVs) have the potential to enhance transportation systems through their ability to dock and split during a trip. Although various applications of MVs have been explored across different domains, their application in logistics remains underexplored. This study examines the use of MVs in cargo delivery to reduce total delivery costs. We model the delivery problem for MVs as a variant of the Vehicle Routing Problem, referred to as the Modular Vehicle Routing Problem (MVRP). In the MVRP, MVs can either serve customers independently or dock with other MVs to form a platoon, thereby reducing the average cost per unit. In this study, we mainly focus on two fundamental types of MVRPs, namely the capacitated MVRP and the MVRP with time windows. To address these problems, we first developed mixed-integer linear programming (MILP) models, which can be solved using commercial optimization solvers. Given the NP-hardness of this problem, we also designed a Tabu Search (TS) algorithm with a solution representation based on Gantt charts and a neighborhood structure tailored for the MVRP. Multi-start and shaking strategies were incorporated into the TS algorithm to escape local optima. Additionally, we explored other potential applications in logistics and discussed problem settings for three MVRP variants. Results from numerical experiments indicate that the proposed algorithm successfully identifies nearly all optimal solutions found by the MILP model in small-size benchmark instances, while also demonstrating good convergence speed in large-size benchmark instances. Comparative experiments show that the MVRP approach can reduce costs by approximately 5.6\% compared to traditional delivery methods. Sensitivity analyses reveal that improving the cost-saving capability of MV platooning can enhance overall benefits.

Modular Vehicle Routing Problem: Applications in Logistics

TL;DR

This paper introduces the Modular Vehicle Routing Problem (MVRP), a new logistics routing variant that allows modular vehicles to dock and split en route to form platoons. It develops three-index MILP formulations for the capacitated and time-window variants (CMVRP and MVRPTW) and designs a specialized Tabu Search with a Gantt-chart-based solution representation and tailored merging/relocation operators to efficiently solve large instances. Theoretical bounds relate MVRP to the classical VRP, and empirical results show that MV platooning can reduce total costs by approximately 5.6%, with larger gains possible as platooning capability improves. The study also discusses practical extensions (en-route cargo transfer, electric MV, heterogeneous modules) and outlines directions for future work, including advanced metaheuristics and faster exact methods for large-scale problems.

Abstract

Recent studies and industry advancements indicate that modular vehicles (MVs) have the potential to enhance transportation systems through their ability to dock and split during a trip. Although various applications of MVs have been explored across different domains, their application in logistics remains underexplored. This study examines the use of MVs in cargo delivery to reduce total delivery costs. We model the delivery problem for MVs as a variant of the Vehicle Routing Problem, referred to as the Modular Vehicle Routing Problem (MVRP). In the MVRP, MVs can either serve customers independently or dock with other MVs to form a platoon, thereby reducing the average cost per unit. In this study, we mainly focus on two fundamental types of MVRPs, namely the capacitated MVRP and the MVRP with time windows. To address these problems, we first developed mixed-integer linear programming (MILP) models, which can be solved using commercial optimization solvers. Given the NP-hardness of this problem, we also designed a Tabu Search (TS) algorithm with a solution representation based on Gantt charts and a neighborhood structure tailored for the MVRP. Multi-start and shaking strategies were incorporated into the TS algorithm to escape local optima. Additionally, we explored other potential applications in logistics and discussed problem settings for three MVRP variants. Results from numerical experiments indicate that the proposed algorithm successfully identifies nearly all optimal solutions found by the MILP model in small-size benchmark instances, while also demonstrating good convergence speed in large-size benchmark instances. Comparative experiments show that the MVRP approach can reduce costs by approximately 5.6\% compared to traditional delivery methods. Sensitivity analyses reveal that improving the cost-saving capability of MV platooning can enhance overall benefits.
Paper Structure (27 sections, 2 theorems, 4 equations, 12 figures, 3 tables, 1 algorithm)

This paper contains 27 sections, 2 theorems, 4 equations, 12 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Applications of MVs
  4. Vehicle Platooning Problem
  5. Research Gap
  6. Problem Definition and Formulation
  7. Problem Definition
  8. MILP Models
  9. Theoretical Bound Compared to the VRP
  10. Tabu Search Algorithm
  11. Solution Representation
  12. Neighborhood Structure
  13. Merging Operators.
  14. Relocation Operators.
  15. Acceleration for Time Window Constraints.
  16. ...and 12 more sections

Key Result

Theorem 1

For a given graph and fleet of MVs, denote the optimal VRP solution as $\pi_{VRP}$ and the optimal MVRPTW solution as $\pi_{MVRP}$. $c(\pi)$ is the cost of the solution $\pi$, $c_{max}$ is the maximum average cost for one MV in the unit distance (i.e., the cost with one MV in the platoon), and $c_{m

Figures (12)

  • Figure 1: An illustration of the docking and splitting operations for MVs.
  • Figure 2: A feasible solution of the MVRP.
  • Figure 3: An example of solution separation.
  • Figure 4: The Gantt chart for solution in Figure \ref{['fig:sol']}.
  • Figure 5: Examples of the merging operators.
  • ...and 7 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Corollary 1