This paper investigates a variant of the heterogeneous fleet vehicle routing problem (HVRP) that incorporates soft time deadlines for customers and allows for tardiness at penalty costs. Distinct vehicle types feature varying fixed usage costs and utilize different road networks, resulting in differences in both travel times and travel costs. The objective is to optimize fleet assignment and vehicle service routes to minimize the total fixed vehicle usage costs, routing variable costs, and tardiness costs, while ensuring each customer is visited exactly once and respecting route duration limits. To address this problem, we introduce three compact formulations: the Miller-Tucker-Zemlin formulation (MTZF), single-commodity flow formulation (SCFF), and two-commodity flow formulation (TCFF), comparing their linear programming (LP) relaxations. Additionally, we propose two new families of valid inequalities, in conjunction with generalized subtour elimination constraints, to strengthen these LP relaxations, integrating them into branch-and-cut solution schemes. The theoretical results on the comparison of formulations and the validity of the proposed inequalities hold also for other HVRPs with limited route duration. Computational experiments demonstrate the superior performance of SCFF and TCFF over MTZF, the effectiveness of the proposed valid inequalities in tightening formulations, and the enhanced computational efficiency achieved by incorporating them. Finally, we explore the impact of depot relocation, varying degrees of urgency in customer requests, and varying fixed vehicle usage costs on optimal solutions.

中文翻译：

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具有软时间截止时间的异构车队车辆路径问题的公式和分支切割算法


本文研究了异构车队车辆路径问题 （HVRP） 的一种变体，该问题为客户纳入了软时间截止日期，并允许以罚款成本计算延迟。不同的车辆类型具有不同的固定使用成本，并使用不同的道路网络，从而导致旅行时间和旅行成本的差异。目标是优化车队分配和车辆服务路线，以最大限度地降低固定车辆总使用成本、路线可变成本和迟到成本，同时确保每个客户只访问一次并遵守路线持续时间限制。为了解决这个问题，我们引入了三种紧凑的公式：Miller-Tucker-Zemlin 公式 （MTZF）、单一商品流公式 （SCFF） 和双商品流公式 （TCFF），比较了它们的线性规划 （LP） 松弛。此外，我们提出了两个新的有效不等式族，结合广义子环消除约束，以加强这些 LP 松弛，将它们整合到分支和切割解方案中。公式比较和所提出的不等式有效性的理论结果也适用于其他路线持续时间有限的 HVRP。计算实验表明，SCFF 和 TCFF 的性能优于 MTZF，所提出的有效不等式在拧紧公式中的有效性，以及通过合并它们实现的增强计算效率。最后，我们探讨了仓库搬迁、客户请求的不同紧迫程度以及不同的固定车辆使用成本对最佳解决方案的影响。