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次的决策相互交织时，所产生的优化问题对计算提出了重大挑战。例如多车场车辆路径问题（MDVRP），其中客户需在配送前分配至车场；以及容量约束选址路径问题（CLRP），其中需首先确定车场的位置。针对此类分层问题，一种简单直接的方法是将高层决策与复杂的车辆路径决策分离。对于每个高层决策候选方案，我们可以通过评估其底层的车辆路径问题来对该候选方案进行评价。由于这种方法需要多次求解车辆路径问题，在大多数情况下被认为是不切实际的。我们提出了一种称为“基于神经成本预测器的遗传算法”（GANCP）的新型深度学习方法，以应对这一挑战并简化算法开发流程。对于每个高层决策候选方案，我们使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，我们提出的神经网络学习了用于求解容量约束车辆路径问题的HGS-CVRP开源软件包的目标值。数值实验表明，这种简化方法能有效且高效地为MDVRP和CLRP生成高质量解，并有望加速复杂分层问题的算法开发进程。我们提供了基于文献标准测试实例评估的计算结果。