We present a solution approach for a multi-trip vehicle routing problem with time windows in which the locations of a prescribed number of depots and the fleet sizes must also be optimized. Given the complexity of the task, we divide the problem into subproblems that are solved sequentially. First, we address strategic decisions, which are solved once and remain constant thereafter. Depots are allocated by solving a p-median problem and fleet sizes are determined by identifying the vehicle requirements of several worst-case demand instances. Then, we address the operational planning aspect: optimizing the vehicle routes on a daily basis to satisfy the fluctuating customer demand. We assign customers to depots based on distance and “routing effort,” and for the routing problem we combine a tailor-made branch-and-cut algorithm with a heuristic consisting of a route construction phase and packing of routes into vehicle trips. Our strategic decision models are robust in the sense that when applied to unseen data, all customers could be visited with the allocated fleet sizes and depot locations. Our operational routing methods are both time and cost-effective. The exact method yields acceptable optimality gaps in 20 min and the heuristic runs in less than 2 min, finding optimal or near-optimal solutions for small instances. Finally, we explore the trade-off between depot and fleet costs, and routing costs to make recommendations on the optimal number of depots. Our solution approach was entered into the 12^th AIMMS-MOPTA Optimization Modeling Competition and was awarded the first prize.

中文翻译：

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具有时间窗、车队规模和站点位置的多行程车辆路径问题的解决方法

我们提出了一种具有时间窗口的多行程车辆路径问题的解决方法，其中还必须优化规定数量的站点的位置和车队规模。考虑到任务的复杂性，我们将问题划分为按顺序解决的子问题。首先，我们解决战略决策，这些决策解决一次，之后保持不变。通过求解p分配仓库- 中值问题和车队规模是通过识别几个最坏情况需求实例的车辆需求来确定的。然后，我们解决运营规划方面：每天优化车辆路线以满足不断变化的客户需求。我们根据距离和“路由工作量”将客户分配到站点，对于路由问题，我们将定制的分支和切割算法与启发式算法相结合，该算法包括路径构建阶段和将路径打包到车辆行程中。我们的战略决策模型是稳健的，因为当应用于看不见的数据时，可以使用分配的车队规模和仓库位置访问所有客户。我们的运营路线选择方法既省时又经济。精确方法在 20 分钟内产生可接受的最优性差距，启发式运行不到 2 分钟，为小实例寻找最佳或接近最佳的解决方案。最后，我们探讨了站点和车队成本以及路线成本之间的权衡，以对站点的最佳数量提出建议。我们的解决方案被纳入了 12^th AIMMS-MOPTA 优化建模竞赛并获得一等奖。