Abstract In this paper, we introduce an extended version of hub location problem, called bi-objective hierarchical multimodal hub location problem to simultaneously minimize the overall system-wide costs and the maximum delivery time. This problem is distinct from the classic hub location problem in designing a hierarchical multimodal hub-and-spoke network involving multiple transportation modes, multi-class hubs and corresponding layers. Combining cost and time dimensions, we first propose a bi-objective mixed-integer linear programming to model this problem formally with diverse flow balance constraints. We then show that the proposed model can be efficiently solved by a reformulation approach based on the e-constraint method for only small instances. Hence, we develop two heuristics, a variable neighborhood search algorithm and an improved non-dominated sorting genetic algorithm-II to obtain high-quality Pareto solutions for realistic-sized instances. We further illustrate the application of the proposed model to provide decision support for cargo delivery systems. Finally, we conduct extensive numerical experiments based on Turkish network to demonstrate the superiority of the proposed solution methods compared to the standard non-dominated sorting genetic algorithm-II. The statistical results confirm the efficacy of the developed heuristic algorithms by adopting the Wilcoxon test.

中文翻译：

---

货物运输系统中双目标分层多模式枢纽位置问题的启发式算法

摘要在本文中，我们引入了枢纽位置问题的扩展版本，称为双目标分层多模式枢纽位置问题，以同时最小化整个系统的整体成本和最大交付时间。该问题不同于设计涉及多种交通方式、多类枢纽和相应层的分层多模式枢纽辐射网络中的经典枢纽位置问题。结合成本和时间维度，我们首先提出了一个双目标混合整数线性规划，用不同的流量平衡约束对该问题进行正式建模。然后我们表明，可以通过基于 e 约束方法的重构方法有效地解决所提出的模型，仅适用于小实例。因此，我们开发了两种启发式方法，一种可变邻域搜索算法和一种改进的非支配排序遗传算法-II，为现实大小的实例获得高质量的帕累托解。我们进一步说明了所提出的模型为货物运输系统提供决策支持的应用。最后，我们基于土耳其网络进行了广泛的数值实验，以证明与标准的非支配排序遗传算法-II 相比，所提出的解决方案方法的优越性。统计结果通过采用 Wilcoxon 检验证实了开发的启发式算法的有效性。我们基于土耳其网络进行了广泛的数值实验，以证明与标准的非支配排序遗传算法-II 相比，所提出的解决方案方法的优越性。统计结果通过采用 Wilcoxon 检验证实了开发的启发式算法的有效性。我们基于土耳其网络进行了广泛的数值实验，以证明与标准的非支配排序遗传算法-II 相比，所提出的解决方案方法的优越性。统计结果通过采用 Wilcoxon 检验证实了开发的启发式算法的有效性。