# Edge-Driven Multiple Trajectory Attention Model for Vehicle Routing Problems
## 4 Experiments
### Generalization Analysis  
In practical applications, the number of nodes in problems can vary significantly, making it impractical to train a model from scratch for each possible node configuration. Therefore, models trained on given problems should exhibit generalization capabilities, allowing it to perform well on instances with different node scales. To this end, we analyze the generalization ability of E-MTAM on TSP, CVRP, and OP problems, with AM serving as the baseline. Specifically, the performance of models trained on problems with 20 and 50 nodes is evaluated across problems with 20, 50, and 100 nodes, as presented in Table 3.  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \multirow{2}{*}{Model} & \multicolumn{2}{c}{TSP20} & \multicolumn{2}{c}{TSP50} & TSP100 \\ \cline{2-7}  & Length & Opt.gap & Length & Opt.gap & Length & Opt.gap \\ \hline AM, TSP20 & 3.86 & 0.78\% & 5.95 & 4.57\% & 9.21 & 18.69\% \\ AM, TSP50 & 3.88 & 1.31\% & 5.78 & 1.58\% & 8.32 & 7.22\% \\ E-MTAM, TSP20 & **3.84** & **0.26\%** & 5.92 & 4.04\% & 8.89 & 14.56\% \\ E-MTAM, TSP50 & 3.87 & 1.04\% & **5.71** & **0.35\%** & **8.11** & **4.51\%** \\ \hline \multirow{2}{*}{Model} & \multicolumn{2}{c}{CVRP20} & \multicolumn{2}{c}{CVRP50} & CVRP100 \\ \cline{2-7}  & Length & Opt.gap & Length & Opt.gap & Length & Opt.gap \\ \hline AM, CVRP20 & 6.40 & 4.92\% & 11.26 & 8.48\% & 18.25 & 16.61\% \\ AM, CVRP50 & 6.57 & 7.70\% & 10.99 & 5.88\% & 16.52 & 5.56\% \\ E-MTAM, CVRP20 & **6.14** & **0.66\%** & 11.17 & 7.61\% & 17.99 & 14.95\% \\ E-MTAM, CVRP50 & 6.53 & 7.05\% & **10.46** & **0.77\%** & **16.37** & **4.60\%** \\ \hline \multirow{2}{*}{Model} & \multicolumn{2}{c}{OP20} & \multicolumn{2}{c}{OP50} & \multicolumn{2}{c}{OP100} \\ \cline{2-7}  & Prize & Opt.gap & Prize & Opt.gap & Prize & Opt.gap \\ \hline AM, OP20 & 5.27 & 2.23\% & 14.03 & 13.34\% & 23.70 & 28.59\% \\ AM, OP50 & 4.86 & 9.83\% & 15.76 & 2.66\% & 28.19 & 15.06\% \\ E-MTAM, OP20 & **5.35** & **0.74\%** & 14.23 & 12.11\% & 25.42 & 23.41\% \\ E-MTAM, OP50 & 4.93 & 8.53\% & **16.06** & **0.80\%** & **31.03** & **6.51\%** \\ \hline \hline \end{tabular}
\end{table}
Table 3: Generalization results on TSPs, CVRPs and OPs against AM.  
Figure 4: The average tour length convergency curves on (**a**) CVRP20, (**b**) CVRP50 and (**c**) CVRP100. The horizontal dashed line represents the best result obtained by LKH.  
The experimental results demonstrate that our E-MTAM consistently outperforms AM in terms of generalization ability across problems of all scales. Specifically, when models trained on TSP20 are tested on TSP50, E-MTAM achieves a reduction in the optimality gap by **0.53%** compared to AM. This reduction reaches **0.87%** and **1.23%** on CVRP and OP, respectively. Notably, although models trained and tested on the same problem scale perform better than those on different scales, models trained on 50 nodes typically outperform those trained on 20 nodes when confronted with larger problem sizes. In tests on TSP100, CVRP100, and OP100, the best optimality gaps of **4.51%**, **4.60%**, and **6.51%**, respectively, are all achieved by E-MTAM trained on 50 nodes. These results underscore the exceptional generalization capability of the proposed method, highlighting its effectiveness in addressing the complex routing problems encountered in real-world applications.