A multiobjective edge-based learning algorithm for the vehicle routing problem with time windows

The multiobjective vehicle routing problem with time windows has attracted much attention in recent decades. Until now, various metaheuristic methods have been proposed to solve the problem. However, designing effective methods is not trivial and heavily depends on experts' knowledge. As a research hotspot in recent years, a few deep reinforcement learning methods have been tried to solve the multiobjective vehicle routing problem with symmetric distance and time matrices. However, due to the complex traffic conditions, the travel distance and time between two nodes are probably asymmetric in real-world scenarios. This article introduces a multiobjective edge-based learning algorithm (MOEL) to tackle this issue. In this method, a single neural network model is established and trained to approximate the whole Pareto front of the problem. The edge features, including travel distance and time matrices, are fully learned and used to construct high-quality solutions. MOEL is compared against three state-of-the-art deep reinforcement learning methods (MODRL/D-EL, PMOCO, EMNH) and five metaheuristic methods (NSGA-II, MOEA/D, NSGA-III, MOEA/D-D, MOIA). Experimental results on the real-world instances indicate that MOEL significantly outperforms all competitors, improving IGD by up to 99.80% and HV by up to 62.84%. In addition, MOEL achieves a maximum runtime reduction of 88.65% compared to the deep reinforcement learning methods, highlighting its efficiency and effectiveness for solving the problem.