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The uncertainty of transportation duration between nodes is an inherent characteristic and should be concerned in the routing optimization of the multimodal transportation network to guarantee the reliability of delivery time. The interval number is used to deal with the uncertainty of transportation duration, and the multi-objective robust optimization model is established which covers the transportation duration and the cost. To solve the combinatorial optimization problem of this study, Non-Dominated Sorting Genetic Algorithm-II (NSGA-II) is designed, which integrates the ( μ + λ ) selection method elite retention and the external filing elite retention. Our findings verify the efficiency of the proposed approach by analyzing the diversity, distribution and convergence of the frontier solutions. Finally, near-optimal solutions are obtained with the proposed algorithm in the numerical example. The present study can provide decision reference for multimodal transportation carriers in making transportation plan under uncertainty.

The route problem of multimodal transportation with timetable under uncertainty: multi-objective robust optimization model and heuristic approach