The Fleet Size and Mix Vehicle Routing Problem (FSMVRP) is a prominent variant of the Vehicle Routing Problem (VRP), extensively studied in operations research and computational science. FSMVRP requires simultaneous decisions on fleet composition and routing, making it highly applicable to real-world scenarios such as short-term vehicle rental and on-demand logistics. However, these requirements also increase the complexity of FSMVRP, posing significant challenges, particularly in large-scale and time-constrained environments. In this paper, we propose a deep reinforcement learning (DRL)-based approach for solving FSMVRP, capable of generating near-optimal solutions within a few seconds. Specifically, we formulate the problem as a Markov Decision Process (MDP) and develop a novel policy network, termed FRIPN, that seamlessly integrates fleet composition and routing decisions. Our method incorporates specialized input embeddings designed for distinctdecision objectives, including a remaining graph embedding to facilitate effective vehicle employment decisions. Comprehensive experiments are conducted on both randomly generated instances and benchmark datasets. The experimental results demonstrate that our method exhibits notable advantages in terms of computational efficiency and scalability, particularly in large-scale and time-constrained scenarios. These strengths highlight the potential of our approach for practical applications and provide valuable inspiration for extending DRL-based techniques to other variants of VRP.

翻译：车队规模与混合车辆路径问题（FSMVRP）是车辆路径问题（VRP）的一个重要变体，在运筹学和计算科学领域得到了广泛研究。FSMVRP要求同时确定车队构成和行驶路线，这使其在短期车辆租赁和按需物流等现实场景中具有高度适用性。然而，这些要求也增加了FSMVRP的复杂性，带来了重大挑战，尤其是在大规模和时间受限的环境中。本文提出了一种基于深度强化学习（DRL）的方法来求解FSMVRP，该方法能够在几秒内生成近似最优解。具体而言，我们将该问题建模为马尔可夫决策过程（MDP），并开发了一种新颖的策略网络，称为FRIPN，该网络无缝集成了车队构成和路径决策。我们的方法结合了为不同决策目标设计的专用输入嵌入，包括一个剩余图嵌入，以促进有效的车辆使用决策。我们在随机生成的实例和基准数据集上进行了全面的实验。实验结果表明，我们的方法在计算效率和可扩展性方面表现出显著优势，尤其是在大规模和时间受限的场景中。这些优势凸显了我们方法在实际应用中的潜力，并为将基于DRL的技术扩展到其他VRP变体提供了有价值的启示。