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A Neural Column Generation Approach to the Vehicle Routing Problem with Two-Dimensional Loading and Last-In-First-Out Constraints

Yifan Xia, Xiangyi Zhang

TL;DR

The paper tackles the 2L-CVRP with LIFO constraints, an NP-hard problem arising from CVRP and 2D-BPP. It introduces Neural Column Generation (NCG), which integrates an attention-based item encoder and a GRU-based recurrence to predict the feasibility of columns before the exact feasibility checker, reducing expensive 2D-BPP/LIFO computations. The approach achieves a median runtime reduction of $29.79\%$ over the state-of-the-art CG and solves an open instance within a BPC framework, aided by permutation-invariant data augmentation and post-processing to preserve optimality. This work demonstrates the practical value of blending machine-learning predictions with exact optimization to enhance solver performance on complex VRP variants.

Abstract

The vehicle routing problem with two-dimensional loading constraints (2L-CVRP) and the last-in-first-out (LIFO) rule presents significant practical and algorithmic challenges. While numerous heuristic approaches have been proposed to address its complexity, stemming from two NP-hard problems: the vehicle routing problem (VRP) and the two-dimensional bin packing problem (2D-BPP), less attention has been paid to developing exact algorithms. Bridging this gap, this article presents an exact algorithm that integrates advanced machine learning techniques, specifically a novel combination of attention and recurrence mechanisms. This integration accelerates the state-of-the-art exact algorithm by a median of 29.79% across various problem instances. Moreover, the proposed algorithm successfully resolves an open instance in the standard test-bed, demonstrating significant improvements brought about by the incorporation of machine learning models. Code is available at https://github.com/xyfffff/NCG-for-2L-CVRP.

A Neural Column Generation Approach to the Vehicle Routing Problem with Two-Dimensional Loading and Last-In-First-Out Constraints

TL;DR

The paper tackles the 2L-CVRP with LIFO constraints, an NP-hard problem arising from CVRP and 2D-BPP. It introduces Neural Column Generation (NCG), which integrates an attention-based item encoder and a GRU-based recurrence to predict the feasibility of columns before the exact feasibility checker, reducing expensive 2D-BPP/LIFO computations. The approach achieves a median runtime reduction of over the state-of-the-art CG and solves an open instance within a BPC framework, aided by permutation-invariant data augmentation and post-processing to preserve optimality. This work demonstrates the practical value of blending machine-learning predictions with exact optimization to enhance solver performance on complex VRP variants.

Abstract

The vehicle routing problem with two-dimensional loading constraints (2L-CVRP) and the last-in-first-out (LIFO) rule presents significant practical and algorithmic challenges. While numerous heuristic approaches have been proposed to address its complexity, stemming from two NP-hard problems: the vehicle routing problem (VRP) and the two-dimensional bin packing problem (2D-BPP), less attention has been paid to developing exact algorithms. Bridging this gap, this article presents an exact algorithm that integrates advanced machine learning techniques, specifically a novel combination of attention and recurrence mechanisms. This integration accelerates the state-of-the-art exact algorithm by a median of 29.79% across various problem instances. Moreover, the proposed algorithm successfully resolves an open instance in the standard test-bed, demonstrating significant improvements brought about by the incorporation of machine learning models. Code is available at https://github.com/xyfffff/NCG-for-2L-CVRP.
Paper Structure (34 sections, 17 equations, 5 figures, 6 tables)

This paper contains 34 sections, 17 equations, 5 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Non-learning Methods for 2L-CVRP.
  4. Neural Column Generation.
  5. Background
  6. Problem Formulation
  7. State-of-the-Art Pipeline
  8. Methodology
  9. Motivation and Outline
  10. Machine Learning Model
  11. Item-Level Attention Mechanism
  12. Customer-Level Recurrence Mechanism
  13. Data Augmentation with Permutation Invariance
  14. Experiments
  15. Experimental Settings
  16. ...and 19 more sections

Figures (5)

  • Figure 1: Comparative illustration of the SOTA method and our NCG method for the 2L-CVRP with the LIFO rule.
  • Figure 2: The architecture of the machine learning model employed in our NCG approach, illustrating the integration of attention and GRU mechanisms for column feasibility prediction.
  • Figure 3: Box plot depicting the percentage increase in efficiency of our NCG method over the SOTA across different instance categories.
  • Figure 4: Box plot demonstrating the percentage increase in column generation counts for our NCG method over the SOTA.
  • Figure 5: Comparison of validation losses over epochs for the baseline and augmented models.