## 3 Model Construction
### Establish Model Objective Functions and Constraints
#### 3.2.2. Mesostructure of the MSIMPLTS-MLOO Model  
Based on the objective function of maximizing the task benefit in the superstructure upper structure of the MSIMPLTS-MLOO model, the mesostructure is set to maximize the task allocation rate as the optimization objective, which ensures that as many target tasks as possible are allocated through the consideration of the observation time window, and the overall efficiency of the MSIMPLTS system and resource utilization are improved. This section defines the task allocation rate according to the mesostructure of the MSIMPLTSMLOO model. The mathematical expression of the objective function of the task allocation rate is shown as follows:  
\[\max f_{2}(x_{ijo})=(\sum_{i=1}^{N_{T}}\sum_{j=1}^{S_{N}}\sum_{o=1}^{N_{e}}x_{ijo }\Big{/}\sum_{i=1}^{N_{T}}Target_{i})\times 100\% \tag{5}\]  
The mesostructure of MSIMPLTS-MLOO model needs to consider constraints, including side-swing angle conversion time constraints, task urgency constraints, and task visible time window constraints. The mathematical expressions of each constraint constructed in this section are shown in Equations (6)-(8).  
1. Side-swing angle conversion time constraint: The side-swing angle conversion time refers to the time required for a satellite to adjust its observation angle while in orbit to perform different tasks. This constraint ensures that there is sufficient time between consecutive tasks. \[\sum_{i=1}^{N_{T}}\sum_{j=1}^{S_{N}}\sum_{o=1}^{N_{e}}x_{ijo}\times(et_{k}^{S} -st_{k}^{S})\geq Time_{trans}^{j},\forall i\in Task,\forall j\in Sat,\forall o \in Orb,\] (6) where \(Time_{trans}^{j}\) denotes the satellite side-swing angle conversion time.
2. Task urgency constraints: Task urgency indicates the level of urgency for a target task within the designated task planning period. This constraint requires the model to prioritize urgent tasks, ensuring that high-priority tasks are not delayed due to the execution of other tasks. \[\sum_{i=1}^{N_{T}}\sum_{j=1}^{S_{N}}\sum_{o=1}^{N_{e}}x_{ijo}\times Time_{during }^{j}\times\big{(}Efficiency_{ijo}/Num(w_{Targeti})\big{)}\leq Time_{rv}^{j}, \forall w_{gadij}^{Targetj}\in W_{satj}^{Targeti},\] (7) where \(Time_{re}^{i}\) denotes the prescribed time for the target task \(i\), and \(Num(w_{Targeti})\) denotes the number of visible time windows for the target task \(i\).
3. Task visible time window constraint: Task visible time window refers to the time interval during which a target task is visible to satellite resources. This constraint ensures that task assignments adhere to visibility conditions, guaranteeing that the tasks are executed correctly during the satellite's visibility period. \[w_{gsafj}^{Targetj}=<st_{k}^{i,S},et_{k}^{i,S}>_{,}(et^{i}-st_{k}^{i})-(et_{k} ^{i,S}-st_{k}^{i,S})\geq 0,\forall w_{Targeti}\in W_{Target},w_{Targeti}=<st^{S},et ^{S}>.\] (8) where \(st^{i}\) and \(et^{i}\) denote the start and end of the visible time window \(w_{Targeti}\) of the target mission \(i\), respectively, and \(st_{k}^{i,S}\) and \(et_{k}^{i,S}\) denote the start and end of the observation time window \(g\) of the satellite \(j\) for the target mission, respectively.