Heuristic algorithms for the bi-objective hierarchical multimodal hub location problem in cargo delivery systems

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.