## 1 Introduction  
Satellite imaging mission planning (SIMP) refers to the appropriate scheduling of a single satellite to effectively complete an Earth observation mission. During the planning process, it is necessary to not only consider the coverage of satellites for multiple observation targets but also address the potential failure or delay of the observation mission. The efficiency and reliability of the entire mission could potentially be impacted by unforeseen circumstances, such as malfunctions in individual satellite equipment [1, 2]. In contrast, multi-satellite imaging mission planning (MSIMP) can achieve the efficient and coordinated execution of Earth observation missions through rational scheduling and optimal allocation of multiple satellite resources (including multiple satellites with different orbits, imaging capabilities, and observational coverage), thus improving imaging coverage and accuracy. By reducing repeated observations and optimizing the time window of the imaging mission, MSIMP can effectively shorten the observation time required for the whole mission, avoid unnecessary resource occupation and waste, and further enhance the flexibility and response speed of the system when facing dynamic tasks. Thus far, MSIMP has been widely used in many fields, such as environmental monitoring, agricultural management, urban planning, and so on, and has become a focal area of research [34]. However, the MSIMP based on traditional optimization algorithms faces additional challenges stemming from three main factors: the continuous increase in the number of satellites in orbit, the explosive growth in the demand for observation targets, and the numerous uncertainties encountered by satellites during mission execution. Specifically, MSIMP involves a variety of coupled constraints and optimization objectives, such as visible time windows, satellite energy limitations, remote sensor tuning time, and so on, as well as maximizing the imaging coverage while minimizing the imaging time [5; 6]. This means that determining the optimal MSIMP approach requires a high-dimensional and complex solution space. Simultaneously, the MSIMP solution usually requires extensive simulations and evaluations to determine the optimal path, imaging time, and mission assignment scheme for each satellite. However, each evaluation is computationally expensive, which not only reduces the responsiveness of the satellite planning system but also increases the delay rate of the planning system [7]. Therefore, there is an urgent need to find a method that reduces the solution space and drastically reduces the number of actual evaluations to improve the overall efficiency of the system. The proposed surrogate model expensive multi-objective evolutionary algorithms (SMEMOEAs) provide an effective approach to overcome the above challenges. First, the SMEMOEAs exploit the natural selection and genetic mechanisms of evolutionary algorithms to continuously explore and optimize the solution space, aiming to find the Pareto optimal solution set that can satisfy multiple optimization objectives simultaneously. Second, the SM-EMOEAs use surrogate models to approximate the objective function and reduce the direct assessment of the objective function. In this case, the surrogate model is also well suited to addressing the impacts of dynamic factors on MSIMP, and can better predict and respond to changes in the planning environment through the collection of and training on historical and current data [8; 9; 10; 11; 12]. These advantages make SM-EMOEAs a key approach to the MSIMP challenge. To date, to the best of our knowledge, no researchers have systematically summarized this subject. Therefore, this work focuses on the application of surrogate model expensive multi-objective evolutionary algorithms based on the MSIMP problem. The main contributions are as follows.