This paper addresses the vehicle routing problem with time windows and stochastic demands (VRPTWSD). The problem is modeled as a two-stage stochastic program with recourse, in which routes are designed in the first stage and executed in the second. A failure occurs if the load of the vehicle is insufficient to meet the observed demand of a customer, implying recourse actions to recover feasibility. We consider the classical recourse policy where reactive trips to the depot are made in case of failures and a fixed rule-based recourse policy where, in addition, preventive trips are allowed. These recourse actions delay the vehicle and may cause further failures if the arrival times on the remaining customers of the planned route do not satisfy their time windows. An additional recourse action is used to service the customers whose time windows would be violated in the planned routes. We propose an Integer L-shaped algorithm considering the mentioned recourse actions. To the best of our knowledge, this is the first tailored exact approach for the VRPTWSD. Computational experiments using 112 benchmark instances evaluate the performance of this algorithm as well as the quality of the stochastic problem solutions. The results indicate significant savings in the solutions when using the fixed rule-based policy and round-trip recourse actions instead of the classical policy. Additionally, the algorithm performed better with the fixed rule-based policy, solving to optimality all instances with up to 34 customers, and taking less time on instances that were solved to optimality with both policies.

本文解决了具有时间窗和随机需求 (VRPTWSD) 的车辆路径问题。该问题被建模为具有追索权的两阶段随机程序，其中路线在第一阶段设计，在第二阶段执行。如果车辆的负载不足以满足观察到的客户需求，则会发生故障，这意味着采取追索行动来恢复可行性。我们考虑经典的追索政策，其中在发生故障的情况下对仓库进行反应性旅行，以及基于固定规则的追索政策，此外还允许预防性旅行。如果计划路线的其余客户的到达时间不满足他们的时间窗口，这些追索行动会延迟车辆并可能导致进一步的故障。额外的追索行动用于服务在计划路线中其时间窗口将被违反的客户。考虑到上述追索行为，我们提出了一个整数 L 形算法。据我们所知，这是第一个为 VRPTWSD 量身定制的精确方法。使用 112 个基准实​​例的计算实验评估了该算法的性能以及随机问题解决方案的质量。结果表明，当使用基于固定规则的策略和往返追索操作而不是经典策略时，解决方案会显着节省。此外，该算法在使用基于固定规则的策略时表现更好，解决了具有多达 34 个客户的所有实例的最优性，并且在使用两种策略解决到最优性的实例上花费的时间更少。