# Edge-Driven Multiple Trajectory Attention Model for Vehicle Routing Problems
Furthermore, these approaches often struggle to generate sufficiently diverse solution trajectories. A broader range of candidate routes could enhance exploration within the solution space, potentially yielding better routing outcomes. However, current approaches typically rely on a single strategy trained by a single decoder, and this limited variability restricts the model's ability to explore more optimal routing configurations. To address the aforementioned issues and limitations, we propose a novel Edge-Driven Multiple Trajectory Attention Model (E-MTAM) for vehicle routing problems, building on the existing encoder-decoder framework. First, we emphasize the integration of edge information by incorporating both edge and node embeddings as inputs to the encoder. This allows the model to capture comprehensive edge feature representations alongside node features. During the encoding phase, we enhance the multi-head attention mechanism driven by edge information, thereby improving the encoder's capacity to model graph topology and its associated relational data. Next, in the decoding phase, we incorporate dynamic visitation information into static graph embeddings via a mask, allowing graph embeddings to be input into the decoder to evolve in real time in accordance with the visited graph, thereby endowing the decoder with an enhanced capacity to represent dynamic information. Third, our model employs a multi-decoder structure, where the decoders are identical but with non-shared parameters. We introduce regularization loss to encourage these decoders to generate diverse trajectories, which enhances the exploration capacity within the solution space and increases the likelihood of obtaining superior results. Finally, extensive experimental evaluations across three distinct VRP variants demonstrate that our proposed E-MTAM significantly outperforms a range of heuristic algorithms and DRL-based models, highlighting its effectiveness in addressing the complexities of VRP solutions. Our results indicate that E-MTAM not only effectively addresses the conventional TSP but also proves applicable to other types of vehicle routing problems (CVRP and OP), showcasing its potential for application in real-world scenarios. In general, the main contributions of this work are as follows:  
* We propose a model called E-MTAM for VRPs, which effectively incorporates edge information to drive the node encoding process, thereby fully leveraging the rich edge features embedded within the graph topology. * We combine visitation information with graph embeddings to integrate dynamic information into static graph embeddings, enabling the decoder to perceive real-time changes in the visited graph at each time step. * We employ a multi-decoder framework and introduce a regularization term to encourage the decoders to generate diverse trajectories, thereby learning distinct routing strategies and fostering a more comprehensive exploration of the solution space. * We conduct extensive experiments on three distinct VRP variants, and the results demonstrate that our E-MTAM model consistently outperforms a broad spectrum of alternative methods, underscoring its superior performance and robustness.