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Well-designed multi-modal transportation networks are crucial for our connected world. For instance, the excessive construction of railway tracks in China, at speeds up to 350 km/h, makes it necessary to consider the interaction of rail with air transportation for network design. In this study, we propose a model for an air/rail multi-modal, multiple allocation hub location problem with uncertainty on travel demands. Our model is unique in that it integrates features from the existing literature on multi-modal hub location problem (including hub-level capacities, link capacities, direct links, travel cost and time, transit costs and uncertainty), which have not been considered simultaneously, given its high computational complexity. We formulate this model with$O(n^{4})$variables and show that the implementation of a Benders decomposition algorithm is inherently hard, because of the cubic number of variables in the master problem. Furthermore, we derive an iterative network design algorithm and additional improvement strategies: MMHUBBI which resolves a restricted problem by the solver CPLEX and MMHUBBI-DIRECT which re-designs the transportation network by a heuristic. Our evaluation on real-world dataset for Chinese domestic transportation shows that MMHUBBI provides a significant speed-up on all instances, compared to using CPLEX, while obtaining near-optimal solutions. MMHUBBI-DIRECT further reduces the runtime/memory usage but provides solutions with worse quality. We believe that our study contributes towards the design of more realistic multi-modal hub location problems.
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