## 5 Experiments
### Experimental Result
Specifically, our approach not only achieves faster convergence during the iterative process but also attains the highest level of convergence efficiency when different baseline strategies are employed. This demonstrates that the proposed algorithm not only offers excellent computational efficiency in handling large-scale data and complex problem scenarios but also maintains high stability and accuracy during the policy evolution process, thereby providing a more efficient and robust solution for the relevant field. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **RC201** & **RC202** & **RC203** \\ \hline DRL [16] & 15.57 & 14.55 & 15.09 \\ DRLPP [36] & 14.84 & 14.79 & 15.41 \\ DQL [37] & 14.97 & 14.97 & 15.26 \\ PN/Rollout [20] & 15.34 & 15.88 & 15.08 \\ PN/Exponential [20] & 15.08 & 15.41 & 15.75 \\ PN/Critic [20] & 15.18 & 15.52 & 15.06 \\ AM/Rollout [24] & 14.61 & 14.98 & 15.05 \\ AM/Exponential [24] & 14.56 & 14.61 & 14.29 \\ AM/Critic [24] & 14.52 & 15.23 & 14.97 \\ PGDD/Rollout & **14.18** & **13.74** & **13.97** \\ PGDD/Exponential & 14.31 & 14.40 & 14.14 \\ PGDD/Critic & 14.53 & 14.42 & 14.37 \\ \hline \hline \end{tabular} The bold numbers refer to the best results.  
\end{table} 
Table 10: Eval results on different methods on RC201, RC202 AND RC203. Figure 5: Averaged eval result of all algorithms on RC201, RC202, RC203. Among them, each color corresponds to a type of method, and different wire styles represent different baselines or different improvement points. Through the integration of a Graph Neural Network (GNN) for extracting the topological characteristics of customer nodes and the fusion of these features with point embeddings, the representational capacity of the algorithm is significantly enhanced. Moreover, the introduction of a dual-decoder mechanism has effectively improved the optimization efficiency and accuracy of route selection. Compared with other state-of-the-art algorithms, the improved model exhibits marked advantages when processing large-scale datasets; not only does it achieve faster solution speeds, but it also demonstrates substantial breakthroughs in accuracy. This indicates that the enhanced algorithm possesses superior convergence and stability when handling large and complex datasets, thereby greatly increasing its adaptability for practical applications. Furthermore, to comprehensively illustrate the performance disparities among the algorithms, Figure 8 displays the mean performance of all algorithms for the UDVRP400. The PGDD algorithm demonstrates superior convergence, suggesting that our model exhibits a greatly enhanced training effect and is suitable for difficult and extensive distribution scenarios. Figure 6: Averaged performance for UDVRP100, which contains learning curves and average cost for PGDD (called DD), attention model (AM), deep reinforcement Learning (DRL), deep reinforcement-learning algorithm for path-planning (DRLPP), deep q-learning (DQL), and pointer network (PN) with different baselines. Among them, each color corresponds to a type of method, and different wire styles represent different baselines or different improvement points. Moreover, Figure 9 illustrates sample solutions obtained by the PGDD model for the UDVRP problem with a total of 100 nodes on RC201. These visualizations provide valuable insight into the heuristics that our model has acquired. The trained model demonstrates the ability to effectively produce many paths from the initial point back to the starting place. Table 11, below, summarizes the Success Rate of PGDD across different problem scales, defined as the percentage of customers served within their tolerance windows. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Algorithm** & **n = 100** & **n = 200** & **n = 400** \\ \hline PGDD & 98.7\% & 96.5\% & 92.3\% \\ AM [24] & 94.2\% & 91.8\% & 87.4\% \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Success rate of PGDD.