Two-stage robust multimodal hub network design under budgeted demand uncertainty: A Benders decomposition approach and a case study

The widening use of hub networks in urban agglomeration freight systems requires several actual extensions in conventional hub network design problems. For this purpose, we introduce a two-stage robust multimodal hub network design problem for the urban agglomeration freight system by considering incomplete hub network topology, multiple transportation modes, travel time limit and discuss the uncertainty in the constructed network from the demand point of view. Particularly, we model the demand uncertainty for the considered problem in two different ways. The basic model supposes that interval-budgeted uncertainty set is adopted to characterize uncertain demand, while the expanded model additionally considers possible states of the uncertain demand and weights summation of performances over multiple uncertainty sets, namely state-wise budgeted uncertainty set. By using a min–max criterion, we develop the path-based mixed-integer programming formulations for the proposed problem, which can significantly decrease the number of required integer variables and constraints. To handle large-sized problems, we propose an improved Benders decomposition algorithm, in which the master problem is implemented in a branch-and-bound framework and the subproblem is optimality solved by a customized two-step strategy. In addition to evaluating on the standard CAB, TR and AP datasets, we conduct a real-world case study of the Beijing–Tianjin–Hebei urban agglomeration freight system to explore the effect of incorporating uncertainty and showcase the superior performance of the proposed methods.