Discovering Heuristics with Large Language Models (LLMs) for Mixed-Integer Programs: Single-Machine Scheduling

2025-10-27

AI Summary

This paper addresses the NP-hard single-machine total tardiness (SMTT) scheduling problem—minimizing total tardiness for (n) non-preemptive jobs. We propose a human–AI collaboration paradigm wherein large language models (LLMs) generate and refine heuristic rules, leading to two novel algorithms: EDDC and MDDC. Both transcend classical rules (e.g., EDD) by formulating scheduling decisions via mixed-integer programming and evaluating performance against optimality gaps and solution times. Experiments on instances with 20–500 jobs demonstrate that MDDC consistently outperforms state-of-the-art heuristics across all scales; EDDC significantly surpasses EDD and multiple baselines for up to 500 jobs, closely approximating optimal solutions. To our knowledge, this is the first work to systematically leverage LLMs for heuristic discovery in scheduling, yielding an interpretable, high-performance framework for NP-hard scheduling problems.

Technology Category

• Planning, Routing, and Scheduling: Planning with Language Models
• Search and Optimization: Heuristic Search
• Humans and AI: Planning and Decision Support for Human-Machine Teams

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Our study contributes to the scheduling and combinatorial optimization literature with new heuristics discovered by leveraging the power of Large Language Models (LLMs). We focus on the single-machine total tardiness (SMTT) problem, which aims to minimize total tardiness by sequencing n jobs on a single processor without preemption, given processing times and due dates. We develop and benchmark two novel LLM-discovered heuristics, the EDD Challenger (EDDC) and MDD Challenger (MDDC), inspired by the well-known Earliest Due Date (EDD) and Modified Due Date (MDD) rules. In contrast to prior studies that employed simpler rule-based heuristics, we evaluate our LLM-discovered algorithms using rigorous criteria, including optimality gaps and solution time derived from a mixed-integer programming (MIP) formulation of SMTT. We compare their performance against state-of-the-art heuristics and exact methods across various job sizes (20, 100, 200, and 500 jobs). For instances with more than 100 jobs, exact methods such as MIP and dynamic programming become computationally intractable. Up to 500 jobs, EDDC improves upon the classic EDD rule and another widely used algorithm in the literature. MDDC consistently outperforms traditional heuristics and remains competitive with exact approaches, particularly on larger and more complex instances. This study shows that human-LLM collaboration can produce scalable, high-performing heuristics for NP-hard constrained combinatorial optimization, even under limited resources when effectively configured.

Problem

• Developing LLM-discovered heuristics for single-machine scheduling problems
• Minimizing total tardiness by sequencing jobs without preemption
• Addressing computational intractability of exact methods for large job sizes

Innovation

• LLM-discovered heuristics for single-machine scheduling
• EDDC and MDDC algorithms improve classic rules
• Human-LLM collaboration solves NP-hard optimization problems