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Multi-vehicle routing problem with soft time windows (MVRPSTW) is anindispensable constituent in urban logistics distribution systems. Over thepast decade, numerous methods for MVRPSTW have been proposed, but most arebased on heuristic rules that require a large amount of computation time. Withthe current rapid increase of logistics demands, traditional methods incur thedilemma between computational efficiency and solution quality. To efficientlysolve the problem, we propose a novel reinforcement learning algorithm calledthe Multi-Agent Attention Model that can solve routing problem instantlybenefit from lengthy offline training. Specifically, the vehicle routingproblem is regarded as a vehicle tour generation process, and anencoder-decoder framework with attention layers is proposed to generate toursof multiple vehicles iteratively. Furthermore, a multi-agent reinforcementlearning method with an unsupervised auxiliary network is developed for themodel training. By evaluated on four synthetic networks with different scales,the results demonstrate that the proposed method consistently outperformsGoogle OR-Tools and traditional methods with little computation time. Inaddition, we validate the robustness of the well-trained model by varying thenumber of customers and the capacities of vehicles.

Multi-Vehicle Routing Problems with Soft Time Windows: A Multi-Agent Reinforcement Learning Approach