## 5 Experiments
### Experimental Result
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{2}{c}{**n = 100**} & \multicolumn{2}{c}{**n = 200**} & \multicolumn{2}{c}{**n = 400**} \\  & **Seed = 1234** & **Seed = 1236** & **Seed = 1234** & **Seed = 1236** & **Seed = 1234** \\ \hline DRL [16] & 50.71 & 49.93 & 71.01 & 69.89 & 204.57 & 235.44 \\ DRLPP [36] & 47.67 & 48.14 & 65.58 & 68.37 & 125.29 & 124.31 \\  & (\(-\)5.99\%) & (\(-\)3.61\%) & (\(-\)7.73\%) & (\(-\)2.17\%) & (\(-\)38.75\%) & (\(-\)47.32\%) \\ DQL [37] & 48.02 & 47.93 & 65.56 & 69.12 & 138.14 & 129.91 \\  & (\(-\)5.50\%) & (\(-\)4.01\%) & (\(-\)7.68\%) & (\(-\)1.10\%) & (\(-\)32.47\%) & (\(-\)44.79\%) \\ PN/Rollout [20] & 47.94 & 47.94 & 64.58 & 69.07 & 195.59 & 264.01 \\  & (\(-\)5.46\%) & (\(-\)3.99\%) & (\(-\)9.05\%) & (\(-\)1.17\%) & (\(-\)4.40\%) & (12.11\%) \\ PN/Exponential [20] & 49.02 & 48.43 & 65.56 & 66.16 & 118.04 & 128.61 \\  & (\(-\)3.33\%) & (\(-\)3.00\%) & (\(-\)7.67\%) & (\(-\)5.34\%) & (\(-\)42.27\%) & (\(-\)45.33\%) \\  & 48.45 & 66.50 & 66.51 & 180.88 & 181.80 \\  & (\(-\)4.45\%) & 50.81 (1.76\%) & (\(-\)6.37\%) & (\(-\)4.85\%) & (\(-\)11.60\%) & (\(-\)22.77\%) \\ AM/Rollout [24] & 47.21 & 47.53 & 65.19 & 65.30 & 179.96 & 215.53 \\  & (\(-\)6.89\%) & (\(-\)4.81\%) & (\(-\)8.18\%) & (\(-\)6.53\%) & (\(-\)12.04\%) & (\(-\)8.46\%) \\ AM/Exponential [24] & 47.37 & 47.37 & 65.03 & 64.53 & 118.45 & 115.71 \\  & (\(-\)6.59\%) & (\(-\)5.13\%) & (\(-\)8.43\%) & (\(-\)7.63\%) & (\(-\)42.03\%) & (\(-\)50.84\%) \\ AM/Critic [24] & 49.20 & 49.42 & 67.03 & 66.17 & 161.13 & 167.20 \\  & (\(-\)2.98\%) & (\(-\)1.02\%) & (\(-\)5.62\%) & (\(-\)5.32\%) & (\(-\)21.22\%) & (\(-\)29.00\%) \\ Ours/Rollout & **46.82** & **46.74** & **63.21** & **63.99** & 148.98 & 209.61 \\  & **(\(-\)7.66\%)** & **(\(-\)6.39\%)** & **(\(-\)10.97\%)** & **(\(-\)8.44\%)** & (\(-\)27.17\%) & (\(-\)10.97\%) \\ Ours/Exponential & 47.47 & 46.94 & 64.59 & 64.51 & **115.09** & **114.04** \\  & (\(-\)6.39\%) & (\(-\)6.00\%) & (\(-\)9.06\%) & (\(-\)7.69\%) & **(\(-\)43.80\%)** & **(\(-\)51.60\%)** \\ Ours/Critic & 47.30 & 46.92 & 64.52 & 65.91 & 160.99 & 150.41 \\  & (\(-\)6.71\%) & (\(-\)6.02\%) & (\(-\)9.16\%) & (\(-\)5.68\%) & (\(-\)21.29\%) & (\(-\)36.08\%) \\ \hline \hline \end{tabular} The bold numbers refer to the best results, and the values in parentheses represent the percentage difference from the solution of the DRL method.  
\end{table} 
Table 8: The performance of different methods under different seeds and number of nodes. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **Mean \(\pm\) Std** & **Eval Time (ms)** \\ \hline DRL [16] & 235.44 \(\pm\) 0.9367 & 68 \\ DRLPP [36] & 124.31 \(\pm\) 0.4674 & 79 \\ DQL [37] & 129.91 \(\pm\) 0.8394 & 66 \\ PN/Rollout [20] & 264.01 \(\pm\) 1.4585 & 44 \\ PN/Exponential [20] & 128.61 \(\pm\) 0.9889 & 35 \\ PN/Critic [20] & 181.80 \(\pm\) 0.6994 & 39 \\ AM/Rollout [24] & 215.53 \(\pm\) 0.4063 & 34 \\ AM/Exponential [24] & 115.71 \(\pm\) 0.6744 & 43 \\ AM/Critic [24] & 167.20 \(\pm\) 0.5045 & 37 \\ Ours/Rollout & 209.61\(\pm\)0.4295 & 74 \\ Ours/Exponential & **114.04 \(\pm\) 0.0371** & 59 \\ Ours/Critic & 150.41 \(\pm\) 0.3243 & 81 \\ \hline \hline \end{tabular} The bold numbers refer to the best results.  
\end{table} 
Table 9: Eval result for different methods on problem UDVRP400. Figures 6 and 7 present a performance comparison during the training phase on the UDVRP100 and UDVRP200 problems between our proposed method and other models. The graphs include learning curves and average cost comparisons for different baseline strategies, specifically including PGDD (denoted as DD in the figures), the attention model (AM), deep reinforcement learning (DRL), the deep reinforcement-learning algorithm for path-planning (DRLPP), deep q-learning (DQL), and the pointer network (PN). By comparing the iterative performance over 128,000 datasets, it is evident that our method outperforms the other algorithms across various baseline strategies.