## 2 DEOS Mission Scheduling Model
### Scheduling Problem Modeling  
#### 2.2.1 Model Constrains  
On the basis of the above symbolic parameters, the constraints considered in this paper include:  
1. Visible window constraints. The satellite payload must be visible to target, and the window duration must not be less than the mission observation time. \[ \begin{split}\forall i\in\{1,\ \dots,n\},j\in\{1,\ \dots,m\},st_{i}^{j}\in St,\exists w^{k}\in W\\ \mathit{sbt}_{i}{}^{j}\geq\mathit{wbt}^{k},\mathit{set}_{i}{}^{j} \leq\mathit{wet}^{k},\mathit{Dur}_{j}\leq\mathit{wdt}^{k}\end{split} \] (1)
2. Slew angle constraints. During task switch, the slew angle cannot exceed the maximum slew angle of satellite. \[ \begin{split}\forall i\in\{1,\ \dots,n\},j\in\{1,\ \dots,m\}\\ \mathit{sa}_{i}{}^{j}\leq\mathit{sa}_{i}\end{split} \] (2)
3. Task preparation time constraints. The interval between two tasks (the time interval from the end of the previous task to the beginning of the next task) must not be less than the payload preparation time. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Notation** & **Definition** & **Notation** & **Definition** \\ \hline \multirow{3}{*}{\(T\)} & \(T=\{t_{1},t_{2}\ \dots\ t_{n},\ \dots\ \}\), where \(T\) represents the set of targets, \(m\) represents the serial number of targets. & \(\mathit{Req}\) & \(\mathit{Req}=\{\mathit{req}_{1},\mathit{req}_{2}\ \dots\ \mathit{req}_{j},\ \dots\ \}\), where \(\mathit{Req}\) represents the set of observation type requests, \(j\) represents the corresponding target serial number. \\ \hline \multirow{3}{*}{\(Num\)} & \(Num=\{num_{1},\ num_{2}\ \dots\ num_{i},\ \dots\ \}\), where \(Num\) & \multirow{3}{*}{\(\mathit{Prio}\)} & \(\mathit{Prio}=\{\mathit{prio}_{1},\mathit{prio}_{2}\ \dots\ \mathit{prio}_{m},\ \dots\ \}\), where \(\mathit{Prio}\) \\  & represents the number of tasks that are executed by the satellite, \(i\) represents the corresponding satellite serial number. & \multirow{3}{*}{\(\mathit{Prio}\)} & \(\mathit{Prio}=\{\mathit{prio}_{1},\mathit{prio}_{2}\ \dots\ \mathit{prio}_{m},\ \dots\ \}\), where \(\mathit{Prio}\) \\  & & & represents the set of targets priority, \(m\) represents the serial number of targets. \\ \hline \multirow{3}{*}{\(Sun\)} & \(Sum(l)\) represents the minimum sun elevation angle in \(l\) window period. & \(maxP\) & \(\maxP\) is the initial power of a satellite. \\ \hline \multirow{3}{*}{\(TR\)} & \(TR=\{tr_{1},\ r_{2}\ \dots\ tr_{m},\ \dots\ \}\), where \(TR\) represents the set of targets’ resolution requirement, \(m\) represents the serial number of targets. & \(SR\) & \(SR=\{sr_{1},sr_{2}\ \dots\ sr_{n},\ \dots\ \}\), where \(SR\) represents the set of satellites’ resolution, \(n\) represents the serial number of targets. \\ \hline \multirow{3}{*}{\(X\)} & \(X\)/ represents the decision variables of satellites, \(j\) represents corresponding target serial number, \(i\) represents corresponding satellite serial number. & \multirow{3}{*}{\(SunAng\)} & \multirow{3}{*}{\(SunAng_{i}\) represents the minimum sun elevation angle of the \(i\)-\(th\) satellite carrying an optical payload. \\  & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: \(Cont.\)\[\forall i\in\{1,\ \ldots,n\},j\in\{1,\ \ldots,m\},st_{i}^{j},st_{i}^{j+1} \in St \tag{3}\] \[\mathit{sbt}_{i}^{j+1}-\mathit{set}_{i}^{j}\geq\mathit{Pre}_{i}^{j+1}\] \[\mathit{Pre}_{i}^{j+1}=(\mathit{sa}_{i}^{j+1}-\mathit{sa}_{i}^{j})/ \mathit{Sas}_{i}\]
4. Energy constraints. The accumulation of energy consumed by a single satellite to execute tasks and switch tasks cannot exceed the upper limit of the energy storage of the satellite. \[\forall i\in\{1,\ \ldots,n\},j\in\{1,\ \ldots,m\}\] (4) \[\sum_{j=1}^{\mathit{num}_{i}}P_{i}^{j}\leq\mathit{pow}_{i}\] \[P_{i}^{j}=\mathit{Dur}_{i}^{j}\times\mathit{Pt}_{i}+\mathit{Pre}_{i}^{j} \times\mathit{PS}_{i}\]
5. Storage capacity constraints. The accumulation of storage consumed by a single satellite to execute tasks cannot exceed the upper limit of the storage capacity of satellite.