###### Abstract  
The increasing complexity of real-world engineering problems, ranging from manufacturing scheduling to resource optimization in smart grids, has driven demand for adaptive and high-performing heuristic methods. Automatic Heuristic Design (AHD) and neural-enhanced metaheuristics have shown promise in automating strategy development, but often suffer from limited flexibility and scalability due to static operator libraries or high retraining costs. Recently, Large Language Models (LLMs) have emerged as a powerful alternative for exploring and evolving heuristics through natural language and program synthesis. This paper proposes a novel LLM-based memetic framework that synergizes LLM-driven exploration with domain-specific local refinement and memory-aware reflection, enabling a dynamic balance between heuristic creativity and effectiveness. In the experiments, the developed framework outperforms other LLM-based state-of-the-art approaches across the designed AGV-drone scheduling scenario and two benchmark combinatorial problems. The findings suggest that LLMs can serve not only as general-purpose optimizers but also as interpretable heuristic generators that adapt efficiently to complex and heterogeneous domains. automatic heuristic design; combinatorial optimization; heuristic generation; large language models; memetic algorithm 2020 academetic Editor: [PERSON] 16 July 2025 30 July 2025  
on the hyper-heuristics [3; 4], which employ genetic programming or reinforcement learning to evolve modular combinations of low-level operators, yielding interpretable strategies that generalize across domains such as job-shop scheduling and vehicle routing. Building on this foundation, neural-enhanced metaheuristics like DeepACO integrate deep reinforcement learning into classical frameworks, automatically learning heuristic measures that outperform hand-designed counterparts on a variety of combinatorial benchmarks [5]. In parallel, automated algorithm configuration techniques based on Bayesian optimization and multi-armed bandits have shown that careful tuning of metaheuristic templates can match or exceed more expressive GP-based schemes under constrained design spaces [6]. More recently, large language models have been harnessed to explore program and heuristic spaces: FunSearch [7] and QUBE [8] sample and refine candidate programs via best-shot prompting and uncertainty-aware selection; the original Evolution of Heuristics (EoH) framework co-evolves natural-language \"thoughts\" and executable code [9]; and more recent LLM frameworks such as ReEvo [10] and HSEvo [11] maintain exploration-exploitation balance through adaptive feedback loops and diversity metrics. Despite these advances, existing approaches still face issues that hinder their adaptability and efficiency. Hyper-heuristics and neural metaheuristics typically rely on a static library of curated operators or demand expensive retraining to tackle new domains, limiting their scalability and flexibility. LLM-assisted methods can instead generate creative heuristic candidates once the exploration and exploitation are carefully balanced. Memetic algorithms [12; 13] have proven effective at polishing promising solutions with local search to refine the convergence, yet few prior works have applied them to enhance LLM-generated heuristics. The integration of memetic concept into the current pipeline is not trivial, since it requires associated structured mechanisms for leveraging historical success patterns, guiding operator selection, and dynamically adjusting their search directions. This absence of integrated refinement can lead to poor diversity maintenance, difficulty recovering from stagnation, and inefficient parameter tuning.