Multi-modal transportation planning for multi-commodity rebalancing under uncertainty in humanitarian logistics

•This study focuses on commodity rebalancing and transportation planning with combined distances.•A bi-objective stochastic mixed-integer nonlinear programming model is proposed for the problem.•A linearization approach is developed and an adaptive E-constraint method is applied to obtain the Pareto-optimal solutions.•A solution strategy is proposed to handle large-sized problems.•The trade-off between the objectives and sensitivity analysis of key parameters with managerial insights are obtained. Multi-commodity rebalancing plays a critical role before and during the attack of large-scale disasters. In practice, some relief centers can be out of reach from the ground for vehicles due to the road disruption. Accordingly, alternative transportation systems are essential to maximize fairness and minimize the total transportation time, simultaneously. However, little study has reported on this issue for humanitarian logistics. To address it, a bi-objective stochastic optimization model is proposed to rebalance and transport commodities with the multi-modal transportation system. This work first linearizes the model and then applies an adaptive augmented E-constraint method to obtain a number of Pareto-optimal solutions. Furthermore, a case study of an emergency event is carried out, of which the computational results indicate its decision making effectiveness. Lastly, sensitivity analysis on critical parameters is conducted and the trade-off between the objectives is also analyzed to provide valuable managerial insights.