The paper introduces a stochastic optimization-based approach to address the logistics for the timely delivery of aid packages to disaster-affected areas utilizing a fleet of drones when the set of demand points is unknown. The major problem addressed is to locate a set of drone take-off platforms so that with a specified probability $\alpha$, the maximum total disutility (or cost) under all realizations of the demand locations is minimized. A set of discrete scenarios defines the uncertainty set of the demand points. A Chance Constrained Programming (CCP) formulation is developed to select a set of platform locations whose disutility distribution produces minimum $\alpha$ percentile. For each platform location set, and each demand scenario, the total disutility is defined as the total delivery time for serving the demand points plus a penalty for unvisited demand points. For every set of drone platform locations, referred to as a candidate combination of platforms, the resultant disutility distribution is obtained by solving a space-time drone scheduling subproblem for all possible demand scenarios. The drone scheduling subproblem optimally schedules and sequences a set of trips for each drone so that the total disutility is minimized. Owing to the computational complexity of the proposed approach, an approximation method is developed that decomposes the problem into three tractable stages. The first stage identifies a set of most preferable platform combinations. The second stage develops an approximation algorithm based on a greedy approach to mitigate the extensive computational requirements for solving the large number of drone scheduling subproblems. The last stage builds upon the properties of a Sample Average Approximation (SAA) method and of the CCP formulation to select the optimum set of platforms. Finally, the performance of the proposed stochastic approach is evaluated through a series of computational experiments and a case study of Central Florida. The results reveal interesting insights and demonstrate the effectiveness of the proposed logistics system for drone delivery of humanitarian aid packages.

本文介绍了一种基于随机优化的方法来解决物流问题，以便在未知需求点的情况下利用无人机机队及时向受灾地区提供援助包。解决的主要问题是定位一组无人机起飞平台，以便以指定的概率$\alpha$，在需求位置的所有实现下的最大总负效用（或成本）被最小化。一组离散情景定义了需求点的不确定性集。开发了机会约束规划 (CCP) 公式以选择一组平台位置，其负效用分布产生最小$\alpha$百分位。对于每个平台位置集和每个需求场景，总负效用被定义为服务需求点的总交付时间加上未访问的需求点的惩罚。对于每组无人机平台位置，称为平台的候选组合，通过求解一个时空无人机调度子问题得到最终的无用分布适用于所有可能的需求场景。无人机调度子问题为每架无人机优化调度和排序一组行程，从而最大限度地减少总的负效用。由于所提出方法的计算复杂性，开发了一种近似方法，将问题分解为三个易于处理的阶段。第一阶段确定一组最优选的平台组合。第二阶段开发一种基于贪心方法的近似算法，以减轻解决大量无人机调度子问题的大量计算需求。最后一个阶段建立在样本平均近似 (SAA) 方法和 CCP 公式的属性之上，以选择最佳平台集。最后，通过一系列计算实验和佛罗里达州中部的案例研究评估了所提出的随机方法的性能。结果揭示了有趣的见解，并证明了拟议的物流系统在无人机运送人道主义援助包裹方面的有效性。