Multi-Objective Infeasibility Diagnosis for Routing Problems Using Large Language Models

In real-world routing problems, users often propose conflicting or unreasonable requirements, which result in infeasible optimization models due to overly restrictive or contradictory constraints, leading to an empty feasible solution set. Existing Large Language Model (LLM)-based methods attempt to diagnose infeasible models, but modifying such models often involves multiple potential adjustments that these methods do not consider. To fill this gap, we introduce Multi-Objective Infeasibility Diagnosis (MOID), which combines LLM agents and multi-objective optimization within an automatic routing solver, to provide a set of representative actionable suggestions. Specifically, MOID employs multi-objective optimization to consider both path cost and constraint violation, generating a set of trade-off solutions, each encompassing varying degrees of model adjustments. To extract practical insights from these solutions, MOID utilizes LLM agents to generate a solution analysis function for the infeasible model. This function analyzes these distinct solutions to diagnose the original infeasible model, providing users with diverse diagnostic insights and suggestions. Finally, we compare MOID with several LLM-based methods on 50 types of infeasible routing problems. The results indicate that MOID automatically generates multiple diagnostic suggestions in a single run, providing more practical insights for restoring model feasibility and decision-making compared to existing methods.