## 4 Proposed Method
### Node Feature Extraction Module  
The module we employ resembles the encoder used in the transformer architecture [30]. The decoder utilizes N attention layers to update embeddings, with each attention layer consisting of two sublayers: multihead attention (MHA) and feed-forward network (FFN) sublayers. The input \(x\) comprises information including demands at distribution points and warehouse locations, time windows, and coordinates, with a dimensionality of \(d_{x}\) (\(d_{x}=7\)). Each input \(x_{i}\) undergoes a linear projection using parameters \(W^{0}\) and \(b^{0}\) to initialize the node embeddings \(h_{i}^{0}\) (\(d_{h}=128\)).  
\[h_{i}^{0}=W^{0}x_{i}+b^{0} \tag{19}\]  
The embeddings are further modified using L attention layers. Each layer \(l\) (\(l=1,2,3,\dots,L\)) comprises two sublayers: a multihead attention (MHA) layer that facilitates communication between the nodes and a fully connected node-wise feed-forward network (FFN) [24]. Each sublayer incorporates a skip connection [31] and makes use of batch normalization (BN) [32].  
\[MHA(Q,K,V)=Concat(head_{1},...,head_{h})W^{M} \tag{20}\]  
\[head_{i}=Attention(QW_{i}^{Q},KW_{i}^{K},VW_{i}^{V}) \tag{21}\]  
where \(W_{i}^{Q}\in\mathbb{R}^{d_{h}\times d_{k}},W_{i}^{K}\in\mathbb{R}^{d_{h} \times d_{k}},W_{i}^{V}\in\mathbb{R}^{d_{h}\times d_{k}},W_{i}^{O}\in\mathbb{R }^{d_{v}\times d_{h}}\), if head \(h=8\), then \(d_{k}=d_{v}=\frac{d_{h}}{h}=16\).  
The feed-forward sublayer calculates projections for each node using a hidden sublayer with a dimension of \(d_{f}=512\) and a ReLU activation function.  
\[\text{FFN}(\hat{t}_{i})=W^{F}\cdot\text{ReLU}(W^{\text{tf}}\hat{t}_{i}+b^{t}) +b^{F} \tag{22}\]  
During the encoding process, we incorporate batch normalization at each step, which involves learnable parameters \(\gamma\) and \(\beta\), and \(\epsilon\) represents a small constant used for numerical stability.  
\[\text{BN}(h_{i})=\text{BN}_{\epsilon,\hat{\rho},\gamma}(h_{i}) \tag{23}\]  
The input sequences to the attention layer undergo parallel processing through both the multihead attention mechanism and the feed-forward neural network.  
\[\hat{h}_{i}=BN^{\ell}\Big{(}h_{i}^{(\ell-1)}+MHA_{i}^{\ell}\Big{(}h_{1}^{(\ell -1)},...,h_{n}^{(\ell-1)}\Big{)}\Big{)} \tag{24}\]  
\[h_{i}^{\ell}=\text{BN}^{\ell}\Big{(}\hat{h}_{i}+\text{FFN}^{\ell}(\hat{h}_{i}) \Big{)} \tag{25}\]  
\begin{table}
\begin{tabular}{l l l} \hline
**Parameter** & **Symbol** & **Description** \\ \hline Node Coordinates & \(x,y\) & Spatial position \\ Node Demand & \(d_{i}\) & Cargo volume (kg) \\ Ideal Time Window & \([L_{i},U_{i}]\) & Optimal delivery interval \\ Tolerance Window & \([L_{i,\text{min}},U_{i,\text{max}}]\) & Extended acceptable time range \\ Edge Distance & \(d_{ij}\) & Euclidean distance (km) \\ Travel Time & \(t_{ij}\) & \(d_{ij}/v_{\text{max}}\) (h) \\ \hline \end{tabular}
\end{table}
Table 4: Key parameters for nodes and edges.  
Residual connections and batch normalization are implemented at each layer to address concerns regarding the preservation of information and the stability of gradients. These strategies together enhance the resilience and efficiency of the model's training process.