A multi-objective optimization for a two-level location-routing problem: a case study of biomass supply chains

This study proposes a two-level location routing model (TLRP) with a multi-objective function to minimize both transportation costs and time in biomass supply chains (BSCs). The model is initially formulated as a mixed-integer nonlinear program (MINLP) and then linearized to a mixed-integer linear program (MILP) with a well-defined second-level routing structure. To overcome computational limitations, a two-phase heuristic algorithm is introduced. The performance of both the MILP and heuristic models was evaluated using small BSC case studies. Multi-objective methods, including weighted sum and epsilon-constraint, were employed to explore trade-offs between cost and time. The heuristic achieved comparable results for single objectives within the BSC context, offering solution times up to 74\% faster on average. While the MILP model provides more fine-grained trade-offs between cost and time, it requires significantly longer computation times. This study offers valuable tools for BSC network optimization: the MILP model for optimal solutions and a heuristic for efficient implementation in large-scale BSCs. Additionally, the proposed framework and heuristic approach show promise for addressing similar challenges in broader TLRP applications.