Abstract This paper addresses the vehicle routing problem with time windows and multiple deliverymen (VRPTWMD) under uncertain demand as well as uncertain travel and service times. This variant is faced by logistics companies that deliver products to retailers located in congested urban areas, where service times are relatively long compared to travel times, and depend on the number of deliverymen assigned to each route. Differently from traditional variants, these service times show high variability, requiring an appropriate way of handling the related uncertainty. We extend two mathematical formulations to represent the VRPTWMD under uncertainty, using the robust optimization paradigm with budgeted uncertainty sets, and developed effective exact solution methods for solving each of them. The first formulation is a robust vehicle flow model solved by a tailored branch-and-cut algorithm that resorts to 1- and 2-path inequalities that we show how to effectively separate. The second formulation is a set partitioning model, for which we propose a branch-price-and-cut algorithm that relies on a robust resource-constrained elementary shortest path problem. The results of computational experiments using instances from the literature and risk analysis via a Monte Carlo simulation show the importance of incorporating uncertainties in the VRPTWMD, and indicate the sensitivity of decisions as well as cost and risk to the level of uncertainty in the input data.

中文翻译：

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具有时间窗和多个送货员的鲁棒车辆路径问题的精确方法

摘要 本文解决了需求不确定以及出行和服务时间不确定的情况下具有时间窗口和多个送货员（VRPTWMD）的车辆路径问题。向位于拥挤城市地区的零售商交付产品的物流公司面临这种变体，与旅行时间相比，这些地区的服务时间相对较长，并且取决于分配给每条路线的送货员数量。与传统变体不同，这些服务时间表现出高度可变性，需要以适当的方式处理相关的不确定性。我们扩展了两个数学公式来表示不确定性下的 VRPTWMD，使用具有预算不确定性集的稳健优化范式，并开发了有效的精确求解方法来解决它们中的每一个。第一个公式是一个鲁棒的车辆流模型，它通过定制的分支和切割算法解决，该算法采用我们展示了如何有效分离的 1 和 2 路径不等式。第二个公式是一个集合分区模型，为此我们提出了一个分支价格和切割算法，该算法依赖于一个强大的资源受限的基本最短路径问题。使用文献实例和通过蒙特卡罗模拟进行风险分析的计算实验结果表明在 VRPTWMD 中纳入不确定性的重要性，并表明决策的敏感性以及成本和风险对输入数据中的不确定性水平。为此，我们提出了一种依赖于稳健的资源受限的基本最短路径问题的分支价格和切割算法。使用文献实例和通过蒙特卡罗模拟进行风险分析的计算实验结果表明在 VRPTWMD 中纳入不确定性的重要性，并表明决策的敏感性以及成本和风险对输入数据中的不确定性水平。为此，我们提出了一种依赖于稳健的资源受限的基本最短路径问题的分支价格和切割算法。使用文献实例和通过蒙特卡罗模拟进行风险分析的计算实验结果表明在 VRPTWMD 中纳入不确定性的重要性，并表明决策的敏感性以及成本和风险对输入数据中的不确定性水平。