## 3 Method
### Compositions of the Neural Network
#### 3.3.1 Static Embedding Layer  
The static embedding layer is used to embed the static elements, including time window information and requirement information, into a high-dimensional vector space.  
Figure 4: Comparison between global attention and local attention. Under the same circumstances, the global attention mechanism needs more computation. However, it tends to generate unreasonable task orders, leading some tasks to be abandoned, so its decoding step number is less. (**a**) Global attention. At the first step, \(t_{4}\) is selected from all the tasks to be executed. \(t_{1}\) and \(t_{2}\) are deleted since they have no VTWs in the following orbits, and \(t_{3}\) and \(t_{5}\) are the remaining tasks. At the second step, \(t_{5}\) is selected, and \(t_{3}\) is deleted. The total step number is 2. (**b**) Local attention. At the first step, \(t_{1}\), \(t_{2}\), \(t_{3}\), and \(t_{5}\) have VTWs in the orbits \(o_{1}\) and \(o_{3}\), and \(t_{2}\) is selected from them to be executed. At the second step, \(t_{1}\), \(t_{3}\), and \(t_{5}\) are still available in these two orbits, and \(t_{3}\) is selected from them, causing the scheduling in \(o_{1}\) to be finished. At the third step, \(t_{1}\), \(t_{4}\), and \(t_{5}\) with VTWs in the next two orbits need to be scheduled, and \(t_{1}\) is selected. At the last step, only \(t_{4}\) and \(t_{5}\) have VTWs in the last orbit \(o_{4}\). \(t_{5}\) is selected, and \(t_{4}\) is deleted. The total step number is 4.  
The static elements have the following characteristics: first, the number of tasks is variable; second, the input task sequence does not have an apparent sorting order. However, the features of the time window information and the requirement information are distinct. The VTW number of different tasks is not a fixed value, and the time window information of a task is a time-related sequence. The requirement information of every task is a vector with fixed dimensions. Therefore, the static embedding layer must be able to process task window information and requirement information separately.  
The structure of the static embedding layer is shown in Figure 5. For \(t_{i}\), its time window set \(WX_{i}\) is embedded through a fully connected network (\(W_{w}\)) and an LSTM network (\(L_{w}\)) successively, and a vector \(we_{i}\) is obtained. Meanwhile, its requirement vector \(R_{i}\) is embedded through a fully connected network (\(W_{r}\)), and a vector \(re_{i}\) is obtained. Then, the concatenated form of \(we_{i}\) and \(re_{i}\) is further processed through a fully connected network (\(W_{s}\)). The concatenating operator is denoted by \(\text{Concat}(\cdot,\cdot)\). The final embedding output of \(T\) is denoted as \(SE=\{se_{i}|i=1,\cdots,|T|\}\). The calculating process of \(se_{i}\) is formulated as follows:  
\[we_{i} = L_{w}\big{(}W_{w}(WX_{i})\big{)}, \tag{22}\] \[re_{i} = W_{r}(R_{i}),\] (23) \[se_{i} = W_{s}\big{(}\text{Concat}(we_{i},re_{i})\big{)}, \tag{24}\]  
where \(W_{w}\), \(L_{w}\), and \(W_{s}\) are learnable parameters of the corresponding networks. With the static embedding layer, the proposed neural network can handle the inputs of variable dimensions.