The Capacitated Vehicle Routing Problem (CVRP) consists of finding the cheapest way to serve a set of customers with a fleet of vehicles of a given capacity. While serving a particular customer, each vehicle picks up its demand and carries its weight throughout the rest of its route. While costs in the classical CVRP are measured in terms of a given arc distance, the Cumulative Vehicle Routing Problem (CmVRP) is a variant of the problem that aims to minimize total energy consumption. Each arc’s energy consumption is defined as the product of the arc distance by the weight accumulated since the beginning of the route.

The purpose of this work is to propose several different formulations for the CmVRP and to study their Linear Programming (LP) relaxations. In particular, the goal is to study formulations based on combining an arc-item concept (that keeps track of whether a given customer has already been visited when traversing a specific arc) with another formulation from the recent literature, the Arc-Load formulation (that determines how much load goes through an arc).

Both formulations have been studied independently before – the Arc-Item is very similar to a multi-commodity-flow formulation in Letchford and Salazar-González (2015) and the Arc-Load formulation has been studied in Fukasawa et al. (2016) – and their LP relaxations are incomparable. Nonetheless, we show that a formulation combining the two (called Arc-Item-Load) may lead to a significantly stronger LP relaxation, thereby indicating that the two formulations capture complementary aspects of the problem. In addition, we study how set partitioning based formulations can be combined with these formulations. We present computational experiments on several well-known benchmark instances that highlight the advantages and drawbacks of the LP relaxation of each formulation and point to potential avenues of future research.

容量车辆路径问题 ( CVRP ) 包括寻找最便宜的方式来为具有给定容量的车队的一组客户提供服务。在为特定客户服务时，每辆车都会满足其需求，并在其其余路线中承载其重量。虽然经典CVRP中的成本是根据给定的弧距来衡量的，但累积车辆路径问题 ( CmVRP ) 是该问题的一种变体，旨在最小化总能耗。每条弧线的能量消耗定义为弧线距离与自路线开始以来累积的权重的乘积。

这项工作的目的是为CmVRP提出几种不同的公式并研究它们的线性规划 (LP) 松弛。特别是，目标是研究基于弧项概念（跟踪在遍历特定弧时是否已经访问过给定客户）与最近文献中的另一种公式，弧负荷公式（这决定了通过弧线的负载量）。

这两种公式之前都曾独立研究过——Arc-Item 与 Letchford 和 Salazar-González (2015) 中的多商品流公式非常相似，而 Fukasawa 等人已经研究了 Arc-Load 公式。(2016) – 他们的 LP 放松是无与伦比的。尽管如此，我们表明将两者结合的公式（称为 Arc-Item-Load）可能会导致明显更强的 LP 松弛，从而表明这两个公式捕获了问题的互补方面。此外，我们研究了如何将基于集合分区的公式与这些公式结合起来。我们在几个著名的基准实例上进行了计算实验，这些实验突出了每个公式的 LP 松弛的优缺点，并指出了未来研究的潜在途径。