# CubeSat Mission Scheduling Method Considering Operational Reliability
## 4 CubeSat Scheduling Model
### Restrictions  
#### 4.3.1 Task Execution Constraints  
During an orbital cycle of CubeSat operation in orbit, all tasks are subject to work window constraints, execution count constraints, work time constraints, and work cycle constraints.  
\[\sum_{t=1}^{w_{j}^{min}-1}x_{j,t}=0,\forall j \tag{7}\]  
\[\sum_{t=w_{j}^{max}+1}^{T}x_{j,t}=0,\forall j \tag{8}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Scaling**} & \multicolumn{2}{c}{**Consistency Check**} & \(\lambda_{1}\) & \(\lambda_{2}\) & \(\lambda_{3}\) \\ \hline \(a_{12}\) & \(a_{13}\) & \(a_{23}\) & & & \\ \hline \multirow{3}{*}{2} & \multirow{3}{*}{2} & \multirow{3}{*}{1} & \(\lambda_{max}=2\) & & & \\  & & & \(CI=0\) & 0.5 & 0.25 & 0.25 \\ \cline{1-1}  & & & \(CR=0\) & & & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Calculation results of weight coefficients.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Scaling** & **Meaning** & **Explanation** \\ \hline
1 & The former and the latter have the same importance & Two activities contribute equally to the objective \\
3 & The former is slightly more & Experience and judgment slightly favor one \\  & important than the latter & activity over another \\
5 & The former is obviously more & Experience and judgment strongly favor one \\  & important than the latter & activity over another \\
7 & The former is strongly more & An activity is strongly favored and its \\  & important than the latter & dominance demonstrated in practice \\
9 & The former is extremely more & The evidence favoring one activity over another \\  & important than the latter & is of the highest possible order of affirmation \\
2,4,6,8 & Intermediate values between & When compromise is needed \\  & the two adjacent judgments & & \\ \hline \hline \end{tabular}
\end{table}
Table 3: The Saaty Rating Scale [32].  
Constraint (7) means that in each period before \(w_{j}^{min}\), the task will not be executed; constraint (8) states that during each period from \(w_{j}^{max}+1\) to the end of the orbit, the task will not be executed.  
\[y_{j}^{min}\leq\sum_{t=w_{j}^{min}}^{w_{j}^{max}}\phi_{j,t}\leq y_{j}^{max},\forall j \tag{9}\]  
Constraint (9) means that for each orbit, all tasks of the CubeSat should be between the minimum activation times and maximum activation times.  
\[\sum_{l=t}^{t+t_{j}^{min}-1}x_{j,l}\geq t_{j}^{min}\phi_{j,t},\forall t\in \Big{\{}1,\ldots,T-t_{j}^{min}+1\Big{\}},\forall j \tag{10}\]  
\[\sum_{l=t}^{t+t_{j}^{max}}x_{j,l}\leq t_{j}^{max},\forall t\in\Big{\{}1, \ldots,T-t_{j}^{max}\Big{\}},\forall j \tag{11}\]  
\[\sum_{l=t}^{T}x_{j,l}\geq(T-l+1)\phi_{j,t},\forall t\in\Big{\{}T-t_{j}^{min}+ 2,\ldots,T\Big{\}},\forall j \tag{12}\]  
Constraint (10) means that each time the task is started, the continuous execution time is greater than \(t_{j}^{min}\); constraint (11) means that in any time window of size \(t_{j}^{max}\), the continuous execution time of the task is less than \(t_{j}^{max}\); and constraint (12) is used to constrain the task started after \(T-t_{j}^{min}+1\). If it is started, it will be executed until the end of the orbit to ensure that the minimum execution time constraint is met.  
\[\sum_{l=t}^{t+p_{j}^{min}-1}\Big{(}1-\phi_{j,t}\Big{)}\geq p_{j}^{min}-1,\forall t \in\Big{\{}1,\ldots,T-p_{j}^{min}+1\Big{\}},\forall j \tag{13}\]  
\[\sum_{l=t}^{t+p_{j}^{max}-1}\Big{(}1-\phi_{j,t}\Big{)}\leq p_{j}^{max}-1, \forall t\in\Big{\{}1,\ldots,T-p_{j}^{max}+1\Big{\}},\forall j \tag{14}\]  
Constraint (13) states that in any time window of size \(p_{j}^{min}\), the task will not be activated more than once; constraint (14) means that in any time window of size \(p_{j}^{max}\), the task is activated at least once.