## 5 Improved SM-EMOEAs for MSIMP
### Diversity Maintenance and Quality Assessment of Solutions
The diversity preservation of solutions in the population is achieved by improving the mutation and crossover operations, setting up an external archive to preserve non-dominated solutions, and periodically introducing randomly generated new individuals [140; 141]. 4. Multi-model collaboration combines the advantages of multiple surrogate models through dynamically selecting and updating different surrogate models, guiding the search process to explore the target space more comprehensively by evaluating and exploiting the prediction uncertainty at different stages of the optimization search, and prioritizing the solutions with higher prediction uncertainty for evaluation to maintain the diversity of the solutions [142; 143]. The SM-EMOEA ensures the uniform distribution and comprehensive coverage of the solution in the objective space through the above four methods, which provide richer training data for the surrogate model and improves the overall optimization performance of the algorithm. The assessment of the quality of the solutions in the improved SM-EMOEAs for MSIMP includes both the performance metrics of MSIMP and the performance metrics of the improved SM-EMOEA. The performance measures of MSIMP include maximizing the mission benefit [144], maximizing the task allocation rate [145], maximizing satellite resource utilization [146], minimizing motion disturbances [147], and minimizing the response time [148]. The performance measures of the improved SM-EMOEA include Pareto frontier approximation [149], hypervolume metrics [150], the surrogate model prediction error [151], the average fitness value [152], the maximum fitness value [153], and the minimum fitness value [154]. In the performance metrics of the improved SA-EMOEAs, the Pareto frontier proximity is used to evaluate the distance between the solution set and the Pareto frontier. The Pareto frontier represents the set of non-dominated solutions in the objective space. Smaller proximity to the Pareto frontier indicates that the solution set is closer to the optimal solutions in the objective space, reflecting better algorithm convergence. The hypervolume metric measures the coverage of the solution set in the objective space by calculating the volume formed between the solution set and a reference point. A larger hypervolume value indicates that the solution set covers a wider range in the objective space, demonstrating higher diversity and better solution quality. The surrogate model prediction error is used to assess the accuracy of the surrogate model's approximation of the actual objective function: the smaller the error, the better the performance of the surrogate model. In the improved SA-EMOEAs for MSIMP, fitness values represent the objective function values, such as task benefit and task completion, and are used to assess the quality of the solution set and the performance of the algorithm. Specifically, the average fitness value reflects the overall fitness level of the population, indicating the general quality of the solutions. The maximum fitness value represents the fitness of the best solution in the current population and is used to evaluate the quality of the current optimal solution. The minimum fitness value indicates the fitness of the worst solution in the population, helping to determine whether the population has fallen into a local optimum.