This paper provides a systematic overview of machine learning methods applied to solve NP-hard Vehicle Routing Problems (VRPs). Recently, there has been great interest from both the machine learning and operations research communities in solving VRPs either through pure learning methods or by combining them with traditional handcrafted heuristics. We present a taxonomy of studies on learning paradigms, solution structures, underlying models, and algorithms. Detailed results of state-of-the-art methods are presented, demonstrating their competitiveness with traditional approaches. The survey highlights the advantages of the machine learning-based models that aim to exploit the symmetry of VRP solutions. The paper outlines future research directions to incorporate learning-based solutions to address the challenges of modern transportation systems.

Many exact algorithms, heuristics, and metaheuristics have been proposed to solve the Vehicle Routing Problem with Drones, which involves using a fleet of trucks and drones to fulfil customer orders in last-mile delivery. In this study, the problem is formulated using the Markov Decision Process, and a Reinforcement Learning (RL) based solution is proposed. The proposed RL model is based on an attention-encoder and a recurrent neural network-decoder architecture. This approach enhances coordination by determining which vehicles should visit specific customers and where vehicles can rendezvous, effectively leveraging drones and reducing the overall completion time. The RL model has demonstrated competitive performance compared to benchmark algorithms through extensive experiments.