This study extends the multi-depot vehicle scheduling problem with time windows (MDVSPTW) to the case of electric vehicles which can recharge at charging stations located at any point of the service operation area. We propose a mixed-integer nonlinear model for the electric bus multi-depot vehicle scheduling problem with time windows (EB-MDVSPTW). Our formulation considers not only the operational cost of vehicles, but also the waiting times. In addition, it explicitly considers the capacity of charging stations and prohibits the simultaneous charging of different vehicles at the same charger. Chargers are modeled as task nodes of an extended network and can be placed at any location utilizing the charging infrastructure of a city instead of using only bus-dedicated chargers. Further, we linearize the MINLP formulation of the EB-MDVSPTW by reformulating it to a mixed-integer linear program (MILP) that can be solved to global optimality. Because EB-MDVSPTW is NP-Hard, we also introduce valid inequalities to tighten the search space of the MILP and we investigate the trade-off between the compactness and the tightness of the problem in benchmark instances with up to 30 trips. In the numerical experiments, we show that the valid inequalities reduce the problem’s compactness by increasing up to three times the number of constraints, but, at the same time, improve tightness resulting in computational time improvements of up to 73% in 20-trip instances. The implementation of our exact approach is demonstrated in a toy network and in the benchmark instances of Carpaneto et al. (1989).

本研究将带时间窗的多站点车辆调度问题 (MDVSPTW) 扩展到可以在位于服务运营区域任意点的充电站充电的电动汽车的情况。我们针对带时间窗的电动公交多站点车辆调度问题提出了混合整数非线性模型 (EB-MDVSPTW)。我们的公式不仅考虑了车辆的运营成本，还考虑了等待时间。此外，它明确考虑了充电站的容量，禁止不同车辆在同一充电器上同时充电。充电器被建模为扩展网络的任务节点，可以放置在利用城市充电基础设施的任何位置，而不是仅使用公交车专用充电器。进一步，我们通过将 EB-MDVSPTW 的 MINLP 公式重新公式化为可以解决全局最优性的混合整数线性规划 (MILP)，将其线性化。因为 EB-MDVSPTW 是 NP-Hard，我们还引入了有效的不等式来收紧 MILP 的搜索空间，并且我们研究了最多 30 次行程的基准实例中问题的紧凑性和紧密性之间的权衡。在数值实验中，我们表明有效的不等式通过增加多达三倍的约束数量来降低问题的紧凑性，但同时提高了紧密性，从而在 20 次行程实例中导致计算时间缩短高达 73% . 我们的精确方法的实现在玩具网络和 Carpaneto 等人的基准实例中得到了证明。（1989）。