## 7 Conclusions  
We proposed a novel LLM-driven memetic framework that integrates large language models into heuristic search to address limitations of traditional AHD and neural meta-heuristics. By combining generative exploration with local refinement, our method achieves superior adaptability and performance across diverse combinatorial optimization tasks. The framework leverages diversity-aware sampling and adaptive feedback to maintain a dynamic balance between exploration and exploitation, reducing reliance on hand-crafted operators or costly retraining. Moreover, there are several research directions for future work. First, extending the framework to handle real-time optimization and online learning scenarios could enhance its applicability in dynamic environments. Second, incorporating domain-specific constraints more explicitly into the generative process may improve solution feasibility and interpretability. Finally, further study into alignment techniques for LLM-generated heuristics, such as reinforcement learning with human feedback or symbolic reasoning, could yield deeper integration between neural and symbolic approaches in automated algorithm design. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Metric Category** & **TSP** & **ODE** \\ \hline \hline \multicolumn{3}{l}{Core Performance Metrics} \\ \hline \hline Best Score (Peak) & \(-6.2209\) & \(-0.0654\) \\ Std Dev of Scores (Overall) & \(2.8032\) & \(11.4508\) \\ \hline \hline \multicolumn{3}{l}{Stability and Robustness Metrics} \\ \hline \hline Success Rate & \(1.0000\) & \(0.8750\) \\ Stability Score & \(1.7227\) & \(0.4887\) \\ \hline \hline \multicolumn{3}{l}{Efficiency Metrics} \\ \hline \hline Total LLM Calls & \(135\) & \(83\) \\ Total Tokens Used & \(242,201\) & \(149,925\) \\ Total Evaluator Calls & \(73\) & \(49\) \\ Total Computation Time (s) & \(51,490.72\) & \(4533.44\) \\ \hline \hline \multicolumn{3}{l}{Solution Diversity Metrics} \\ \hline \hline Total Unique Scores & \(45\) & \(24\) \\ Score Range & \(17.1948\) & \(36.29\) \\ Solution Diversity Score & \(0.6164\) & \(0.5675\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The EoH-MR results of TSP and ODE 1D tasks using gemini-2.5-flash. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Metric Category** & **gpt-4o** & **Gemini-2.5-Flash** & **Claude-Sonnet-4** \\ \hline \hline \multicolumn{3}{l}{Core Performance Metrics} \\ \hline \hline Best Score (Peak) & \(-177,796.0\) & \(-168,067.5\) & \(-114,012.0\) \\ Std Dev of Scores (Overall) & \(23,492.5412\) & \(17,715.6189\) & \(34,029.9269\) \\ \hline \hline \multicolumn{3}{l}{Stability and Robustness Metrics} \\ \hline \hline Success Rate & \(0.4650\) & \(0.6870\) & \(0.8085\) \\ Stability Score & \(1.1286\) & \(1.0971\) & \(1.1693\) \\ \hline \hline \multicolumn{3}{l}{Efficiency Metrics} \\ \hline \hline Total LLM Calls & \(147\) & \(128\) & \(53\) \\ Total Tokens Used & \(427,447\) & \(458,264\) & \(216,936\) \\ Total Evaluator Calls & \(243\) & \(246\) & \(141\) \\ Total Computation Time (s) & \(2221.0811\) & \(7541.3417\) & \(1964.7157\) \\ \hline \hline \multicolumn{3}{l}{Solution Diversity Metrics} \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The EoH-MR results of AGV-drone scheduling task using other LLM variants. **Author Contributions:** Conceptualization, F.Q. and T.W. ; methodology, F.Q. and T.W. ; software, F.Q. and T.W. ; validation, F.Q. and T.W. ; formal analysis, R.Z. ; writing--original draft preparation, F.Q. and T.W. ; writing--review and editing, R.Z. and M.L. ; visualization, F.Q. and T.W. ; supervision, M.L. ; project administration, R.Z. ; funding acquisition, M.L. All authors have read and agreed to the published version of the manuscript. **Funding:** This research received no external funding. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** The data presented in this study are available on request from the corresponding author due to the privacy consideration.