Benders Decomposition for Multimodal Facility Location Allocation Problem Considering Capacity Levels and Uncertainty

Freight movement from various cities of origin in China, through consolidation centers and frontier ports, to different destinations in Europe within the China Railway Express logistics network is studied in this paper. The problem is formulated as a multi-capacity and multi-mode facility location-allocation problem with stochastic demand and delivery time which is modeled as a distributionally robust optimization problem. The objective is to minimize the total cost, which includes facility construction, transportation, and time delay costs. Given the stochastic nature of destination demand and transportation time, historical data is utilized to construct the ambiguity set of these stochastic parameters. Then, the proposed distributionally robust optimization problem is transformed into a two-stage deterministic optimization problem using probability and duality theory. An enhanced Benders decomposition algorithm is developed to solve the transformed problem that integrates several valid inequalities, multi-cut subproblem reformulation, and Pareto-optimal cuts to improve the performance of the algorithm. The computational experiments demonstrate that this improved Benders decomposition algorithm significantly outperforms the widely-used Gurobi solver in terms of solving speed. Finally, consolidation centers with different capacity levels are established in Xi’an, Urumqi, Chongqing, Shenyang, and Hohhot, and the corresponding transportation routes are given.