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.

翻译：具有二维装载约束（2L-CVRP）及后进先出（LIFO）规则的车辆路径问题在实际应用和算法设计上均面临重大挑战。该问题的复杂性源于两个NP难问题：车辆路径问题（VRP）与二维装箱问题（2D-BPP）。尽管已有大量启发式方法被提出以应对其复杂性，但针对精确算法的研究相对较少。为填补这一空白，本文提出一种融合先进机器学习技术的精确算法，具体而言是一种注意力机制与循环机制相结合的新颖方法。该集成方法在多种问题实例上，将当前最先进精确算法的求解速度中位数提升了29.79%。此外，所提算法成功求解了标准测试集中的一个开放算例，充分展示了引入机器学习模型所带来的显著性能提升。代码发布于 https://github.com/xyfffff/NCG-for-2L-CVRP。