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决策相互交织时，由此产生的优化问题对计算构成重大挑战。例如多车场车辆路径问题（MDVRP）需要在配送前将客户分配到车场，以及能力受限的选址路径问题（CLRP）需首先确定车场位置。处理这类层次化问题的一种直接做法是将高层决策与复杂的车辆路径决策解耦。对于每个高层决策候选方案，我们可通过评估其底层车辆路径问题来验证候选方案的优劣。由于该方法需要多次求解车辆路径问题，在多数情况下被视为不切实际。本文提出一种名为"基于神经代价预测器的遗传算法（GANCP）"的新型深度学习方法，以应对该挑战并简化算法开发。对于每个高层决策候选方案，我们利用预训练的图神经网络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通过学习求解能力受限车辆路径问题的开源软件包HGS-CVRP的目标值。数值实验表明，这种简化方法在MDVRP和CLRP上均能高效生成高质量解，并有望加速复杂层次化问题的算法开发进程。我们提供了基于文献标准基准实例的计算结果评估。