## 3 Model Construction
### Establish Model Objective Functions and Constraints
#### 3.2.3. Understructure of the MSIMPLTS-MLOO Model  
Based on the comprehensive consideration of the output scheme of the superstructure and mesostructure, the undersructure takes the optimization of the task response time as the optimization objective, so that as many target tasks as possible can be observed in a limited time by minimizing the response time to meet the requirement of a fast task response speed of the MSIMPLTS system. The undersructure of the MSIMPLTS-MLOO model builds the task response time objective function, and the mathematical expression is shown in Equation (9):  
\[\min f_{3}(x_{ijo})=\sum_{i=1}^{N_{T}}\sum_{j=1}^{S_{N}}\sum_{o=1}^{N_{e}}x_{ ijo}\cdot\sum_{i=1}^{N_{T}}Time_{res}^{i},\forall i\in Task,\forall j\in Sat, \forall o\in Orb \tag{9}\]  
The constraints to be considered in the undersructure of the MSIMPLTS-MLOO model include the neighboring task observation time window, observation duration, and task execution order constraints. The mathematical expressions for constructing each constraint are shown in Equations (10)-(12).  
Neighboring task observation time window constraints: When the observation time windows of neighboring missions overlap, the satellite is unable to perform more than one mission at the same time. To avoid resource conflicts, the constraint ensures that there is no overlap in the time windows of neighboring missions on satellite resources in the same time frame.  
\[Time_{\text{min}}^{j}\leq st_{k}^{i-1,g}\leq et_{k}^{i-1,g}\leq et_{k}^{i-1,g} \leq st_{k}^{i,g}\leq et_{k}^{i,g}\leq st_{k}^{i+1,g}\leq Time_{\text{min}}^{j},\forall w_{gsatj}^{Targetj}\in W_{satj}^{Targeti},w_{gsatj}^{Targetj}=<st_{k}^{ i,g},et_{k}^{i,g}>, \tag{10}\]  
where \(Time_{\text{min}}^{j}\) and \(Time_{\text{max}}^{j}\) denote the shortest and longest on-time of the satellite, respectively.
2. Observation duration constraints: Observation duration refers to the total time a satellite spends performing all assigned target tasks. This constraint ensures that task allocation remains within the satellite's capacity and that the satellite's observation time does not exceed the maximum operating time for a single orbital pass. \[\sum_{i=1}^{N_{T}}\sum_{j=1}^{S_{N}}\sum_{o=1}^{N_{o}}x_{ijo}\times(Time_{ trans}^{j}+Time_{et}^{i}-Time_{st}^{i})\leq Time_{orb}^{o},\forall o\in Orb,\] (11) where \(Time_{orb}^{o}\) indicates the maximum operating time of the satellite on a single orbital lap.
3. Task execution order constraint: In cases where there is a precedence relationship between tasks, this constraint requires that a given task \(i\) must be performed before another task \(i+1\), meaning the start time of the task \(i+1\) must be earlier than the end time of the task \(i\). \[\sum_{i=1}^{N_{T}}\sum_{j=1}^{S_{N}}\sum_{o=1}^{N_{o}}x_{i+1 jo}\times st^{i+1}- Time_{trans}^{j}-(\sum_{i=1}^{N_{T}}\sum_{j=1}^{S_{N}}\sum_{o=1}^{N_{o}}x_{ijo} \times et^{i})\geq 0,\forall i\in Task,\forall j\in Sat,\forall o\in Orb,\] (12) In summary, the total objective function of the MSIMPLTS-MLOO model can be expressed as a weighted sum of the objective functions from the superstructure, mesostructure, and understructure, reflecting the overall goal of multi-layer optimization. The mathematical expression for the MSIMPLTS-MLOO model's objective function is shown in Equation (13): \[F(x_{ijo})=\alpha f_{1}(x_{ijo})+\beta f_{2}(x_{ijo})-\chi f_{3}(x_{ijo}).\] (13) where, \(\alpha\), \(\beta\), and \(\chi\), respectively, are the proportional scaling factors of task benefit, task allocation rate, and task response time. Satellite controllers can adaptively adjust the weights according to the preference demand. This work does not consider the preference situation in the experiment, so the proportionality factor is set to \(\mu=\lambda=\tau=\beta=1\). The weighted solution of the optimization objective can transform the multi-objective problem into a single-objective problem.