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.

翻译：本研究通过利用大型语言模型（LLMs）的能力发现新的启发式算法，为调度和组合优化领域做出了贡献。我们专注于单机总延迟（SMTT）问题，该问题旨在给定处理时间和截止日期的情况下，通过在单个处理器上无抢占地排序n个作业来最小化总延迟。我们开发并基准测试了两种新颖的LLM发现的启发式算法：EDD挑战者（EDDC）和MDD挑战者（MDDC），其灵感来源于著名的最早截止日期（EDD）规则和修正截止日期（MDD）规则。与先前采用较简单基于规则的启发式算法的研究不同，我们使用严格的标准评估我们的LLM发现算法，包括基于SMTT的混合整数规划（MIP）公式得出的最优性差距和求解时间。我们在不同作业规模（20、100、200和500个作业）下，将它们的性能与最先进的启发式算法和精确方法进行比较。对于超过100个作业的实例，MIP和动态规划等精确方法在计算上变得不可行。在多达500个作业的情况下，EDDC相较于经典的EDD规则和文献中另一种广泛使用的算法有所改进。MDDC始终优于传统启发式算法，并在较大和较复杂的实例上保持与精确方法的竞争力。本研究表明，在资源有限但有效配置的情况下，人-LLM协作可以为NP难约束组合优化问题产生可扩展且高性能的启发式算法。