## 3 Description of MSIMP
### MSIMP Mathematical Modeling  
MSIMP defines the optimal allocation sequences and visible time window allocation schemes between multiple satellites and the target mission using precise mathematical descriptions. MSIMP models often adopt objective functions and constraints to satisfy the user's mission requirements, as well as multiple environmental requirements [36]. The objective function may involve maximizing the mission benefits, maximizing the mission allocation rate, minimizing resource consumption, minimizing the response time, and so on. The considered constraints include environmental factors, mission conflicts, imaging visible time windows, fuel consumption, and so on [37]. Common MSIMP models include integer planning models [38], constraint satisfaction models [39], dynamic planning models [40], Markov decision models [41], and so on. Figure 4 shows the main factors affecting the MSIMP model, including the target mission set, the satellite resource set, the time window set, and the evaluation set. Figure 4 is a schematic diagram describing the key factors affecting the MSIMP model and the related constraints. The main factors include the target task set, satellite resource set, time window set, and evaluation set, while the constraints involve satellite limitations and target task restrictions. The specific details are as follows:  
The target task set is critical in MSIMP modeling, representing a collection of imaging or observation tasks to be carried out by multiple satellites. These tasks typically involve precise imaging of ground areas or objects. The target task set includes several key attributes, such as the geographic location of the tasks, image requirements, image resolution, and priority information. These attributes directly influence the imaging location, sensor type, and image resolution settings for the satellites. Priority information is used to distinguish the importance and urgency of the tasks, helping to determine the order in which tasks should be executed. These task attributes not only define the complexity and demands of the mission but also impose strict constraints on the optimal scheduling of satellite resources, ensuring that imaging tasks are completed with maximum quality and efficiency within the limits of available satellite resources [42; 43]. The satellite resource set is a crucial element in MSIMP modeling and solution, representing the collection of satellites and all their available resources used to carry out tasks. This set plays a decisive role in the formulation and execution of the MSIMP plan. Key components of the satellite resource set include resource capacity, resource types, storage  
Figure 4: MSIMP mathematical model and its constraints. capabilities, and energy constraints. Resource capacity and energy constraints determine the number of tasks that each satellite can perform and the duration of its operation. Resource types ensure that the satellites can meet the diverse needs of different target tasks, while storage capacity directly impacts the efficiency of data storage and transmission during missions. By optimally allocating the satellite resource set, satellite controllers can maximize resource utilization under limited conditions, improving the success rate and efficiency of mission planning and execution [44,45]. The time window set is a critical component of MSIMP, referring to the collection of time periods during which satellites can image ground targets. These windows are determined by the start time, end time, and duration of target task observations, along with satellite orbital information. Additionally, precise calculation of time windows must consider factors such as sunlight conditions and weather in the target area to ensure imaging quality and the successful execution of the mission. By effectively scheduling time windows, satellite resources can be maximized, resource waste minimized, and the overall success rate of mission planning improved [46,47].