## 5 Improved SM-EMOEAs for MSIMP
### Diversity Maintenance and Quality Assessment of Solutions  
The diversity maintenance and quality assessment of solutions are important processes when evaluating the performance of SM-EMOEAs [131; 132]. The diversity maintenance of solutions aims to improve the prediction accuracy of the surrogate model by providing richer and more diverse training data for the model through a diverse solution set. Diversity maintenance ensures that the solutions are uniformly distributed across the whole target space, preventing the algorithm from focusing on certain regions too early and thus preventing it from falling into a local optimal solution. The diversity solution enables broader coverage of the solution space, helping the surrogate model to capture global problem characteristics. This facilitates the model in finding reasonable near-optimal solutions more quickly during the evolutionary process, preventing the algorithm from getting trapped in local optima and accelerating the optimization process [133]. Solution quality refers to the superiority of candidate solutions in multi-objective optimization, typically assessed by how closely they approach the global optimal region in the solution space. It reflects the performance of SA-EMOEAs in balancing multiple optimization objectives, such as task benefits and response time, while evaluating the algorithm's search efficiency. High-quality solutions indicate that candidate solutions not only achieve high objective values but also demonstrate strong stability and diversity in both the global exploration and local exploitation of the solution space [134]. In addition, the consistent occurrence of high-quality solutions indicates that the surrogate model can approximate the true objective function more accurately, thus effectively guiding the optimization search process [135]. At present, diversity maintenance methods for solutions include computing the crowding distance, using distribution metrics, performing multi-model collaboration, and implementing diversity-preserving operations, as shown in Figure 8.  
1. The crowding distance is used to measure the distance between the current solution and its neighboring solutions in the target space. The candidate solutions are ranked based on their fitness values for the objective functions, followed by calculating the crowding distance for each solution. A larger crowding distance indicates that the solution is located in a sparsely populated region of the objective space, where there are fewer neighboring solutions. Conversely, a smaller crowding distance suggests that the solution is in a densely populated region with more neighboring solutions in the objective space. A larger crowding distance indicates that there are fewer solutions in the current region while a smaller crowding distance indicates that there are more solutions in the current region. Through calculating the crowding distance, the density  
Figure 8: Diversity maintenance methods for solutions. of each solution in the target space can be effectively evaluated, thus providing a reference for the selection of the next generation of solutions and ensuring the uniform distribution of the solutions in the target space [136; 137]. 2. Distribution metrics are used to assess the uniformity of distribution and the coverage of a solution set in the target space. The specific method is to cluster the solutions using a clustering algorithm, where each cluster centroid represents a subset of the solutions, reflecting the characteristics of each subset in the solution space. The clustering centroids are preferentially selected as part of the next generation of solutions to ensure the uniform distribution of the solutions. Distribution metrics can be used in conjunction with the crowding distance to jointly maintain the solution diversity through different metrics [138; 139]. 3. Diversity-preserving operations aim to introduce and maintain diversity in the population during the optimization search process to prevent the algorithm from falling into a local optimum.