Right Answers, Costly Models: The Efficiency Gap in LLM-based Optimization Modeling

Optimization problems are pervasive in sectors from manufacturing and distribution to healthcare. However, most such problems are still solved heuristically by hand rather than optimally by state-of-the-art solvers because the expertise required to formulate and solve these problems limits the widespread adoption of optimization tools and techniques. We introduce a Large Language Model (LLM)-based system designed to formulate and solve (mixed integer) linear programming problems from their natural language descriptions. Our system can develop mathematical models, write and debug solver code, evaluate the generated solutions, and improve efficiency and correctness of its model and code based on these evaluations. OptiMUS is designed as a productivity tool for optimization practitioners who understand the problem domain and can describe it precisely, but seek to accelerate the modeling and implementation workflow. OptiMUS-0.3 utilizes a modular structure to process problems, allowing it to handle problems with long descriptions and complex data without long prompts. Experiments demonstrate that OptiMUS-0.3 outperforms direct-prompting baselines by over 43% on easy and 18% on hard instances. It remains competitive with fine-tuned specialist models on benchmark problems, and outperforms them on real-world case studies (28.6% vs. 0%) where fine-tuned models fail to generalize. Ablation studies show that modular architecture with error correction is central to these gains. A key finding is that system architecture is a stronger driver of performance than model capability. Structured decomposition with targeted error correction enables weaker models to match stronger models under naive prompting, and remains competitive with fine-tuned specialist models without retraining costs.

LLM-based agents are increasingly deployed to solve optimization problems, yet existing benchmarks evaluate them on pre-structured mathematical formulations that bypass the most critical challenge: translating complex business requirements into correct models and solve efficiently. We introduce Opti-Agent-Bench, an end-to-end benchmark that evaluates Large Language Models (LLMs) across the complete optimization R&D pipeline, from understanding business-language descriptions through mathematical modeling, algorithm selection, and code implementation, to solution report generation. Our design rests on three pillars: (1) businesssemantic authenticity with anti-template traps that defeat pattern matching; (2) modular evaluation with cross-module consistency checking across Problem Understanding, Formal Modeling, Implementation, and Reporting; and (3) the ORAC bi-level validity framework that simultaneously ensures task quality and scoring integrity. Across several industrialscale tasks spanning integer programming, robust optimization, stochastic programming, and non-convex optimization, we expose critical failure modes of current models, including constraint omission, model-code inconsistency, and report-implementation divergence, that remain invisible under conventional single-metric evaluation.

Large language models can translate business descriptions into optimization models, but executable code may misrepresent constraints or objectives. A solver can then return an optimal solution to the wrong problem. Even when the solution satisfies the intended operating rules, a better plan may exist. For organizations that repeatedly use optimization modeling, an LLM-based framework should produce accurate formulations at low cost and, ideally, run locally. We study how to verify improvements and allocate attempts across LLMs that differ in price and capability. We develop OSCAR (Optimization modeling by Simulator, Coder, And Reviewer), which uses an offline Simulator certified against labeled decision examples to compare candidates and continues searching beyond feasibility. We model the search for the next certified improvement as sequential decisions under unobserved difficulty: which LLMs to call and when to stop. In a simplified known-prior setting, we give conditions under which cost-ordered escalation is optimal. For general menus, we derive a prior-free competitive guarantee. On five benchmark problems, OSCAR achieves 95% to 100% accuracy at the reported settings using two small open-weight LLMs, each deployable locally on a single GPU. Their single-attempt accuracies average 29% and 48%. In five runs per problem, Codex and Claude Code incur average token costs 3.1 and 5.8 times OSCAR's, respectively. OSCAR supports open-weight models locally or in the cloud, depending on budget and confidentiality requirements. Firms should maintain labeled decision examples of feasible and infeasible decisions to clarify plain-language operating rules. OSCAR follows these labels when an LLM's interpretation conflicts with them. As LLM capabilities and prices change, OSCAR's simple operating rules and adjustable settings help firms adapt their model choices and benefit from these advances.