Two-Echelon Vehicle Routing Problems (2E-VRPs) describe the distribution of goods via two echelons of vehicles and sets of transshipment facilities. They have received growing attention in academic research and their applications are becoming increasingly widespread in modern last-mile logistics systems. This paper introduces a new and extensive variant of the 2E-VRP that combines important real-world features such as time windows, mixed pickup and delivery demand, vehicle range constraints and multiple trips for vehicles on the second echelon. We refer to it as the Two-Echelon Multi-Trip Vehicle Routing Problem with Deliveries, Pickups and Time Windows (2E-MT-VRP-PTW). We present a compact exact formulation that solves small instances to optimality in reasonable time. We present an efficient and customized matheuristic to solve medium and large instances. The solution method integrates an exact formulation for the first-echelon routing into an adaptive large neighborhood search framework for the second-echelon routing. The performance of the matheuristic is evaluated on modified benchmarks from the literature. The heuristic shows strong performance in solution quality and runtime when compared to the exact formulation. Sensitivity analyses over key demand characteristics are performed.

中文翻译：

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具有混合取货和送货需求和时间窗的两级多行程车辆路径问题的数学方法

两梯队车辆路径问题（2E-VRP）描述了通过两梯队车辆和转运设施进行的货物分配。它们在学术研究中受到越来越多的关注，并且在现代最后一英里物流系统中的应用也越来越广泛。本文介绍了 2E-VRP 的一种新的广泛变体，它结合了重要的现实世界特征，例如时间窗口、混合取货和送货需求、车辆范围限制以及第二梯队车辆的多次行程。我们将其称为具有送货、取货和时间窗的两级多行程车辆路径问题 (2E-MT-VRP-PTW)。我们提出了一个紧凑的精确公式，可以在合理的时间内解决小实例的最优问题。我们提出了一种高效且定制的数学方法来解决中型和大型实例。该求解方法将第一梯队路由的精确公式集成到第二梯队路由的自适应大邻域搜索框架中。数学性能是根据文献中修改后的基准进行评估的。与精确公式相比，启发式方法在解决方案质量和运行时间方面表现出强大的性能。对关键需求特征进行敏感性分析。