Delivering a parcel from the distribution hub to the customer's doorstep is called the \textit{last-mile delivery} step in delivery logistics. In this paper, we study a hybrid {\it truck-drones} model for the last-mile delivery step, in which a truck moves on a predefined path carrying parcels and drones deliver the parcels. We define the \textsc{online drone scheduling} problem, where the truck moves in a predefined path, and the customer's requests appear online during the truck's movement. The objective is to schedule a drone associated with every request to minimize the number of drones used subject to the battery budget of the drones and compatibility of the schedules. We propose a 3-competitive deterministic algorithm using the next-fit strategy and 2.7-competitive algorithms using the first-fit strategy for the problem with $O(\log n)$ worst-case time complexity per request, where $n$ is the maximum number of active requests at any time. We also introduce \textsc{online variable-size drone scheduling} problem (OVDS). Here, we know all the customer's requests in advance; however, the drones with different battery capacities appear online. The objective is to schedule customers' requests for drones to minimize the number of drones used. We propose a $(2\alpha + 1)$-competitive algorithm for the OVDS problem with total running time $O(n \log n)$ for $n$ customer requests, where $\alpha$ is the ratio of the maximum battery capacity to the minimum battery capacity of the drones. Finally, we address how to generate intervals corresponding to each customer request when there are discrete stopping points on the truck's route, from where the drone can fly and meet with the truck.

翻译：将包裹从配送中心运送到客户家门口的过程在物流配送中被称为\textit{最后一公里配送}。本文研究了用于最后一公里配送的混合{\it 卡车-无人机}模型，其中卡车沿预定路径行驶并携带包裹，无人机负责配送包裹。我们定义了\textsc{在线无人机调度}问题：卡车沿预定路径移动，客户请求在卡车行驶过程中在线出现。目标是为每个请求调度一架无人机，在无人机电池预算约束和调度兼容性条件下，最小化使用的无人机数量。我们提出了一种基于next-fit策略的3-竞争确定型算法，以及基于first-fit策略的2.7-竞争算法，每个请求的最坏情况时间复杂度为$O(\log n)$，其中$n$是任意时刻的最大活跃请求数。我们还引入了\textsc{在线变容量无人机调度}问题（OVDS）。在该问题中，所有客户请求预先已知，但具有不同电池容量的无人机在线出现。目标是为无人机调度客户请求以最小化使用的无人机数量。我们为OVDS问题提出了一种$(2\alpha + 1)$-竞争算法，总运行时间为$O(n \log n)$，其中$n$为客户请求数，$\alpha$为无人机最大电池容量与最小电池容量之比。最后，我们讨论了当卡车路径上存在离散停靠点（无人机可从该点起飞并与卡车会合）时，如何为每个客户请求生成对应的调度区间。