Delivery of goods into urban areas constitutes an important issue for logistics service providers. One of the most talked-about developments in recent years has been the potential use of unmanned aerial vehicles, or drones, for transporting packages, food, medicine, and other goods. Delivery by drones offers new possibilities, but also induces new challenging routing problems. In this paper, we address and extend the so-called Parallel Drone Scheduling Traveling Salesman Problem. Basically, in this problem, deliveries are split between a vehicle and one or several drones. The vehicle performs a classical delivery tour from the depot, while the drones are constrained to perform back and forth trips. The objective is to minimize the completion time. We extend the problem by considering several vehicles. We call it Parallel Drone Scheduling Multiple Traveling Salesman Problem. We propose a hybrid metaheuristic for its solution. The procedure starts by building a giant tour visiting all customers. Then, the giant tour is split in order to determine a set of vehicles tours (each vehicle tour following the order defined by the giant tour) and a set of customers assigned to drones. Thirdly, an improvement step move customers between vehicles or between vehicles and drones. We also propose a Mixed Integer Linear Programming formulation and a simple branch-and-cut approach. The proposed approach is validated via an experimental campaign on instances taken from the CVRPLIB¹ library. Computational experiments comparing several variants of the hybrid metaheuristic give some insights on this drone delivery system.

将货物运送到城市地区是物流服务提供商的一个重要问题。近年来最受关注的发展之一是无人驾驶飞行器或无人机的潜在用途，用于运输包裹、食品、药品和其他货物。无人机交付提供了新的可能性，但也引发了新的具有挑战性的路线问题。在本文中，我们解决并扩展了所谓的并行无人机调度旅行商问题。基本上，在这个问题中，交付是在车辆和一架或多架无人机之间分配的。车辆从仓库执行经典的交付之旅，而无人机则被限制来回执行。目标是最小化完成时间。我们通过考虑几种车辆来扩展问题。我们称之为并行无人机调度多个旅行推销员问题。我们为其解决方案提出了一种混合元启发式算法。该程序首先建立一个访问所有客户的大型旅游。然后，将巨人之旅拆分，以确定一组车辆之旅（每个车辆之旅遵循巨人之旅定义的顺序）和分配给无人机的一组客户。第三，改进步骤在车辆之间或车辆与无人机之间移动客户。我们还提出了混合整数线性规划公式和简单的分支切割方法。提议的方法通过对从 CVRPLIB ¹库中获取的实例进行的实验活动进行验证。比较混合元启发式的几种变体的计算实验为这种无人机交付系统提供了一些见解。