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

arXiv: https://arxiv.org/abs/2406.12454 · HTML: https://mlsanity.com/paper/2406.12454/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 39 references that are in the map.
- First posted on arXiv 18 Jun 2024.

## Builds on (most cited first, 14 of 14)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Attention, Learn to Solve Routing Problems!](https://mlsanity.com/paper/1803.08475/) (2018)
- [Machine Learning for Combinatorial Optimization: a Methodological Tour d'Horizon](https://mlsanity.com/paper/1811.06128/) (2018)
- [POMO: Policy Optimization with Multiple Optima for Reinforcement Learning](https://mlsanity.com/paper/2010.16011/) (2020)
- [Sym-NCO: Leveraging Symmetricity for Neural Combinatorial Optimization](https://mlsanity.com/paper/2205.13209/) (2022)
- [A Deep Reinforcement Learning Framework For Column Generation](https://mlsanity.com/paper/2206.02568/) (2022)
- [Machine-learning-based arc selection for constrained shortest path problems in column generation](https://mlsanity.com/paper/2201.02535/) (2022)
- [Enhancing Column Generation by a Machine-Learning-Based Pricing Heuristic for Graph Coloring](https://mlsanity.com/paper/2112.04906/) (2021)
- [A Reinforcement-Learning-Based Multiple-Column Selection Strategy for Column Generation](https://mlsanity.com/paper/2312.14213/) (2023)
- [Exact Solution Techniques for Two-dimensional Cutting and Packing](https://mlsanity.com/paper/2004.12619/) (2020)
- [Deep Reinforcement Learning for Demand Driven Services in Logistics and Transportation Systems: A Survey](https://mlsanity.com/paper/2108.04462/) (2021)

## Abstract (arXiv)

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.