We consider a combined truck-drone delivery problem with stochastic truck travel times and soft time windows. A fleet of homogeneous trucks and drones are deployed in pairs to provide delivery services to customers. Each drone can be launched from and retrieved to its truck multiple times, and in each flight, a drone can serve one or more customers. Our objective is to determine the truck routes and drone flights that minimise the total cost, including time window violation penalties. We formulate this problem into a two-stage stochastic model with recourse action in the second stage to optimise the truck waiting time at each node. We approximate the stochastic model with a large-scale mixed-integer program using the sample average approximation (SAA) framework, which is computationally intractable. To this end, we propose a hybrid metaheuristic approach that incorporates SAA. The waiting times of each truck obtained in the planning phase are optimal against the sampled or estimated travel times along the entire route, but the actual values are known only once the truck has returned to the depot. To this end, we reformulate the second-stage model in a rolling-horizon manner, which can be easily implemented and efficiently solved in the execution phase. Extensive numerical experiments demonstrate the strong performance of the proposed metaheuristic approach and rolling-horizon model. The results also highlight the clear benefits of the stochastic modelling approach over its deterministic counterpart, with a pronounced reduction in the total cost in various scenarios.

中文翻译：

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具有软时间窗口和随机卡车行驶时间的多次访问无人机辅助路径问题


我们考虑了具有随机卡车行驶时间和软时间窗口的卡车-无人机组合交付问题。由同质卡车和无人机成对部署的车队为客户提供送货服务。每架无人机都可以多次从卡车上发射和回收，在每次飞行中，无人机可以为一个或多个客户提供服务。我们的目标是确定将总成本降至最低的卡车路线和无人机飞行，包括时间窗口违规处罚。我们将这个问题表述为一个两阶段随机模型，第二阶段采取追索行动，以优化每个节点的卡车等待时间。我们使用样本平均近似 （SAA） 框架用大规模混合整数程序来近似随机模型，这在计算上是难以处理的。为此，我们提出了一种包含 SAA 的混合元启发式方法。在规划阶段获得的每辆卡车的等待时间与沿整条路径的采样或估计行驶时间相比是最优的，但只有在卡车返回站点后才能知道实际值。为此，我们以滚动视野的方式重新构建了第二阶段模型，该模型可以在执行阶段轻松实现并有效解决。广泛的数值实验证明了所提出的元启发式方法和滚动视界模型的强大性能。结果还突出了随机建模方法相对于确定性建模方法的明显优势，在各种情况下，总成本显著降低。