## 3 Model Construction
### Establish Model Objective Functions and Constraints
#### 3.2.1. Superstructure of the MSIMPLTS-MLOO Model  
The optimization objective of the superstructure upper structure in the MSIMPLTS-MLOO model is to maximize the task benefit objective function. In large-scale target scenarios, task benefit is closely related to the importance of the tasks. By maximizing task benefit, high-priority tasks can be prioritized, thereby ensuring the effective allocation and utilization of resources across the entire system and improving the overall performance and benefit maximization of the MSIMPLTS system. In this section, the task benefit objective function is defined according to the upper structure of the MSIMPLTS-MLOO model, with its mathematical expression shown in Equation (1):  
\[\text{max}f_{1}(x_{ijo})=\sum_{i=1}^{N_{T}}\sum_{j=1}^{S_{N}}\sum_{o=1}^{N_{o} }x_{ijo}\times Efficiency_{ijo} \tag{1}\]  
where \(x_{tso}\) denotes the decision variable of the \(sat_{j}\) performing the \(Target_{i}\) in \(orbit_{o}\), and \(Efficiency_{ijo}\) denotes the task benefit of \(sat_{j}\) performing the \(Target_{i}\) in \(orbit_{o}\).  
The constraints to be considered for the MSIMPLTS-MLOO model superstructure include the satellite power-on time, maximum number of satellite observations, and target mission uniqueness constraints, with mathematical expressions as follows:  
1. Satellite power-on time constraints: Satellite power-on time is limited by task requirements and the satellite's status. The satellite activation time constraint is set to ensure that observation tasks are carried out during the satellite start-up hours' active time. \[x_{ijo}\cdot Time_{\text{min}}^{j}\leq x_{ijo}\cdot(Time_{et}^{i}-Time_{ st}^{i})\leq x_{ijo}\cdot Time_{\text{max}}^{j},\forall i\in Task,\forall j\in Sat, \forall o\in Orb,\] (2) where \(Time_{\text{min}}^{j}\) and \(Time_{\text{max}}^{j}\) denote the shortest and longest power-on times of satellite \(j\), respectively. \(Time_{et}^{i}\) and \(Time_{st}^{i}\) denote the end and start observation times of the satellite for the target task \(i\), respectively.
2. Maximum number of satellite observation constraints: In the task mission model, satellite capacity is defined by the maximum number of observation tasks each satellite can perform within a unit of time. This ensures that each satellite operates within its capacity when executing observation tasks. If the assigned tasks exceed the satellite's observation capacity, a penalty function is applied. \[obs_{max}=\left\{\begin{array}{ll}0\,,\,n_{ijo}\leq obs_{max}\\ \beta\sum\limits_{i=1}^{N_{T}}Penalty_{i},n_{ijo}>obs_{max}\end{array}\right.,\forall i\in Task,\forall j\in Sat,\forall o\in Orb,\] (3) where \(n_{ijo}\) denotes the number of current observation missions, \(obs_{max}\) denotes the maximum number of observations within the capacity of the satellite, \(Penalty\) denotes the penalty function, and \(\beta\) denotes the weighting coefficients.
3. Target mission uniqueness constraints: Target mission uniqueness means that each target task can only be completed by one satellite, preventing the same task from being redundantly executed by multiple satellites, which could lead to resource waste and task conflicts. \[\sum_{i=1}^{N_{T}}\sum_{j=1}^{S_{N}}\sum_{o=1}^{N_{o}}x_{ijo}\leq 1.\] (4)