## 6 Results and Discussions  
### Performance Comparison on AGV Scheduling Problem  
The AGV scheduling problem represents a highly dynamic, resource-constrained optimization task involving coordinated part transport via autonomous ground vehicles and drones. Key complexities include heterogeneous agent capabilities, hard energy constraints, dynamic part arrivals, limited processing slots, stochastic delays from traffic jams, and non-negligible failure probabilities. These attributes make the problem substantially harder than classical routing tasks, especially for approaches without domain-aware adaptability. Table 2 summarizes the comparative performance of the proposed EoH-based frameworks and two baselines (FunSearch and HillClimb) on this task. **Core Performance Metrics. ** The proposed EoH-MR achieves the best objective score of \(-\)**159,306.0, improving substantially over EoH-M (\(-\)184,051.25), the original EoH (\(-\)184,620.0), and the strongest baseline HillClimb (\(-\)187,813.33). This performance gap of over **15%** in total cost demonstrates the superiority of EoH-MR in navigating the large, noisy solution space of the AGV task. Notably, while HillClimb is competitive in structured domains like TSP, it is not well-suited for scenarios involving multi-agent energy-aware scheduling under real-time stochasticity. FunSearch, though more robust than HillClimb, also fails to match the effectiveness of LLM-driven co-evolution in this domain. As shown in Figure 3, EoH-MR demonstrates a significant performance advantage over other algorithms, reaching the lowest cost among all methods. FunSearch and HillClimb perform notably worse, confirming their limitations under complex scheduling constraints. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Metric Category** & **FunSearch** & **HillClimb** & **EoH** & **EoH-M** & **EoH-MR** \\ \hline \hline Core Performance Metrics & & & & & \\ \hline Best Score (Peak) & \(-\)189,158.5714 & \(-\)187,813.3333 & \(-\)184,620.0 & \(-\)184,051.25 & \(-\)**159,306.0** \\ Std Dev of Scores (Overall) & 26,395.5240 & 33,752.3249 & 17,748.3088 & 26,740.5316 & 40,589.8931 \\ \hline Stability and Robustness Metrics & & & & & \\ \hline Success Rate & 0.3182 & 0.2273 & **0.4839** & 0.3137 & 0.3580 \\ Stability Score & 1.1547 & 1.1338 & 1.0997 & 1.1426 & **1.2494** \\ \hline Efficiency Metrics & & & & & \\ \hline Total LLM Calls & 20 & 20 & **40** & 64 & 100 \\ Total Tokens Used & 27,247 & 30,900 & **70,349** & 142.597 & 306,075 \\ Total Evaluator Calls & 66 & 66 & **93** & 153 & 243 \\ Total Computation Time (s) & 1136.06 & **1033.19** & 1071.9302 & 1815.0409 & 3177.5700 \\ \hline Solution Diversity Metrics & & & & & \\ \hline Total Unique Scores & 21 & 15 & 45 & 48 & **87** \\ Score Range & 10,179.7619 & 19,562.6667 & 51,470.5667 & 78,425.7333 & **131,009.8333** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comprehensive comparison results of AGV-drone scheduling task using deepseek-chat. Figure 3: Performance metrics for the AGV-drone scheduling task. (**Left**): Best objective score achieved by each method (less negative is better). (**Middle**): Score range indicating solution diversity. (**Right**): Stability score reflecting performance consistency across runs. **Stability and Robustness. ** The baseline EoH achieves the highest success rate (0.4839), suggesting that it produces feasible solutions more reliably. This is partly due to its simpler evolution strategy, which favors safe sampling. However, it also shows lower overall stability and higher coefficient of variation, reflecting inconsistent solution quality. In contrast, EoH-MR, despite a slightly lower success rate (0.3580), achieves the best normalized stability score (1.2494). This robustness is a result of its memetic refinement via CMA-ES, which tunes promising heuristics toward more repeatable and interpretable behaviors. This is corroborated by Figure 3, which shows EoH-MR reaching the highest stability score (1.2494), indicating the most consistent performance across runs, despite its complexity.