Multi-stage facility location and transshipment design for coal railway transport systems: A LLM-augmented optimization approach

Coal remains a dominant energy source in many countries, underscoring the need to improve transport efficiency. The spatial separation between production and consumption necessitates long-distance transportation and tightly coordinated distribution. However, current coal rail transportation is often challenged by transportation losses, demand backlogs, and demand uncertainty, which collectively undermine supply chain reliability. To address these challenges, this paper develops a two-stage stochastic mixed-integer linear programming model that integrates multi-stage distribution center location and transshipment network design to minimize operating costs including fixed DC opening, transportation, transshipment, and inventory holding, while maximizing revenue through meeting demand. To enhance solution quality and computational efficiency, a variable neighborhood search heuristic enhanced by a large language model (LLM@VNS) is developed to adaptively recommend neighborhood perturbation operators and their execution sequences. Numerical experiments indicate that LLM@VNS matches Gurobi's optimal solutions on small-scale instances. For large instances where Gurobi hits the time limit, LLM@VNS delivers 1.36% superior solutions and reduces CPU time by 77.62%, whereas benchmark heuristics exhibit higher optimality gaps up to 11.67%. Although LLM guidance does not always yield substantial improvements, it still provides a modest improvement in the objective value and a 12.72% increase in computational efficiency, demonstrating the effectiveness of LLM@VNS in guiding the search toward higher-quality solutions.