## 6 Results and Discussions
### Performance Comparison on TSP Problem  
The Traveling Salesman Problem (TSP) serves as a benchmark for permutation-based optimization over a static, symmetric graph. It is well-known, smooth in fitness landscape, and does not feature the dynamic, temporal, or resource coupling challenges present in AGV scheduling. As such, it provides an ideal contrast to assess how heuristic design strategies generalize from complex, real-world-inspired tasks to well-structured classical problems. Table 3 presents the comparative results across all methods on the TSP benchmark. Algorithm 3 shows an example of the designed heuristics for next node selection. ```
0: Current node \(i\), destination node \(j\), unvisited set \(U\), distance matrix \(d(i,j)\)
0: Next node \(k^{*}\) to visit
1:\(d_{i}\gets d(i,U)\)
2:\(d_{j}\gets d(j,U)\)
3:progress\(\gets 1-\frac{d(i,j)}{\max d}\)
4:\(\alpha\gets 0.7-0.3\cdot\mathit{progress}\)
5:\(\beta\gets 0.3+0.3\cdot\mathit{progress}\)
6:if pheromone matrix \(\tau\) not initialized then
7: initialize \(\tau(i,u)\gets 1\) for all \(i,u\)
8:endif
9:\(\tau_{i}\leftarrow\tau(i,U)\)
10:\(w\gets d_{i}/\sum d_{i}\)
11:\(H\leftarrow-\sum w\cdot\log w\)
12:\(C\leftarrow\frac{1}{1+|d_{i}-\mathrm{median}(d_{i})|}\cdot(1+H)\)
13:if recent visit history exists then
14:\(R\leftarrow\) recent nodes
15:\(\mathit{penalty}\gets 0.15\cdot|\kappa_{U\in R}\)
16:else
17:\(\mathit{penalty}\gets 0\)
18:endif
19:for all\(u\in U\)do
20:\(s(u)\leftarrow\alpha\cdot\frac{1}{d_{i}(u)}+\beta\cdot\frac{d_{i}(u)}{d_{i}(u)} +(0.25+0.1\cdot\mathit{progress})\cdot\tau_{i}(u)+(0.15+0.1\cdot(1-\mathit{ progress}))\cdot C(u)-\mathit{penalty}(u)\)
21:endfor
22:\(k^{*}\leftarrow\arg\max_{u\in U}s(u)\)
23: Append \(k^{*}\) to visit history
24:\(\tau(i,k^{*})\leftarrow\tau(i,k^{*})\cdot(1.1+0.2\cdot\mathit{progress})\)return\(k^{*}\)
```  
**Algorithm 3** TSP Next-Node Selection Heuristic Example  
**Core Performance Metrics. ** EoH-MR again achieves the best peak score of \(-\)**6.3967**, marginally outperforming EoH-M (\(-\)6.4494) and EoH (\(-\)6.6840). While the improvement margin is smaller than in the AGV setting, it demonstrates the transferability of the framework across domains. FunSearch and HillClimb obtain best scores of \(-\)6.8240 and \(-\)6.7326, respectively, trailing the LLM-driven methods. This suggests that while these baselines can handle well-structured domains to some extent, they still fall short in refining high-quality solutions in a generalizable way. Figure 4 confirms this trend, where EoH-MR is shown to outperform all other methods on best score. **Stability and Robustness. ** All EoH variants maintain high success rates (\(\geq\)0.96), with EoH-M and EoH-MR achieving a perfect 1.0 success rate. Coefficient of variation and robustness scores are comparable across methods, though EoH-MR consistently outperforms in robustness score (2.3804) and standard deviation, indicating consistent quality even under stochastic prompt perturbations or evaluator variability. HillClimb achieves a robustness score of 1.3808, with slightly lower reliability than FunSearch (2.0820), both of which fall short of EoH variants. Moreover, EoH-MR achieves perfect success rates, reinforcing its reliability. **Efficiency Metrics. ** The performance-efficiency trade-off is again evident. EoH, with only 19 LLM calls and 347.44 s, is extremely lightweight and delivers strong baseline performance. EoH-MR and EoH-M, though more computationally demanding (2743.37 and 1492.92 s, respectively), provide marginal but meaningful gains in quality and exploration. Figure 4: Performance metrics for the TSP optimization task. (**Left**): Best tour cost obtained by each method (less negative is better). (**Middle**): Score range across generations indicating diversity. (**Right**): Success rate of valid solution generation.