Robust multi-objective optimization for multimodal transportation routing under uncertainty using a robustified Gray Wolf Optimization framework

Uncertain freight demand and travel time undermine nominal multimodal routing. This study develops a robust multi-objective framework to minimize transportation expense, time, and carbon emissions while accounting for mixed time windows, transport and transfer capacities, objective bounds, and relative robust regret. A robustified Grey Wolf Optimization framework (R-GWO) integrates scenario evaluation, conservatism control, feasibility checks, and regret penalties into candidate ranking while retaining canonical GWO position updates. Tests use 100 common scenarios and 20 matched runs on a 35-node network under equal budgets. R-GWO achieves the lowest mean robust fitness (11.7225), expense (146,579.93 CNY), and worst regret (1.2243). Its fitness is significantly lower than PSO and AFO, but not GA after Holm correction. Expense is significantly lower than all comparators, while time is longer and emissions do not differ significantly. Ablation identifies relative-regret control as the strongest contributor; sensitivity tests indicate stable route-mode decisions across tested preferences and robustness settings.