When vehicle routing decisions are intertwined with higher-level decisions, the resulting optimization problems pose significant challenges for computation. Examples are the multi-depot vehicle routing problem (MDVRP), where customers are assigned to depots before delivery, and the capacitated location routing problem (CLRP), where the locations of depots should be determined first. A simple and straightforward approach for such hierarchical problems would be to separate the higher-level decisions from the complicated vehicle routing decisions. For each higher-level decision candidate, we may evaluate the underlying vehicle routing problems to assess the candidate. As this approach requires solving vehicle routing problems multiple times, it has been regarded as impractical in most cases. We propose a novel deep-learning-based approach called Genetic Algorithm with Neural Cost Predictor (GANCP) to tackle the challenge and simplify algorithm developments. For each higher-level decision candidate, we predict the objective function values of the underlying vehicle routing problems using a pre-trained graph neural network without actually solving the routing problems. In particular, our proposed neural network learns the objective values of the HGS-CVRP open-source package that solves capacitated vehicle routing problems. Our numerical experiments show that this simplified approach is effective and efficient in generating high-quality solutions for both MDVRP and CLRP and has the potential to expedite algorithm developments for complicated hierarchical problems. We provide computational results evaluated in the standard benchmark instances used in the literature.

翻译：当车辆路径决策与更高层级的决策相互交织时，所产生的优化问题对计算能力提出了重大挑战。例如多车场车辆路径问题（MDVRP）中需在配送前将客户分配到车场，以及带容量约束的选址路径问题（CLRP）中需首先确定车场位置。针对此类层级问题的直接简化思路是将高层级决策与复杂的车辆路径决策分离。对于每个高层级决策候选方案，可通过求解其底层的车辆路径问题来评估方案的优劣。然而这种方法需要反复求解车辆路径问题，在大多数情况下被视为不切实际。本文提出一种基于深度学习的新方法——遗传算法结合神经代价预测器（GANCP），以应对该挑战并简化算法开发流程。对于每个高层级决策候选方案，我们利用预训练的图神经网络直接预测底层车辆路径问题的目标函数值，而无需实际求解路径问题。特别地，所提出的神经网络通过学习HGS-CVRP开源软件包（用于求解带容量约束车辆路径问题）的目标值进行训练。数值实验表明，这种简化方法在生成MDVRP和CLRP的高质量解方面高效且有效，并具有加速复杂层级问题算法开发的潜力。我们在文献中使用的标准基准实例上提供了计算结果评估。