## 5 Improved NSGA-II Algorithm
### Optimization Results and Analysis  
In order to verify the performance of the improved NSGA-II algorithm, this experiment employs classical test functions, such as the ZDT series, to evaluate the algorithm's performance prior to and following the implementation of enhancements. The algorithm parameters were set as follows: the population size is 200; the mating pool size is 100; the crossover rate is 0.9; the mutation rate is 0.1; the crossover and mutation distribution indexes are both 20; and the number of iterations is 500. As illustrated in Figure 10, the improved NSGA-II demonstrates superior performance compared to the conventional NSGA-II in terms of the Pareto front. The improved NSGA-II exhibits enhanced convergence and a solution that is closer to the theoretically optimal solution. Additionally, the  
Figure 9: Flowchart of the improved NSGA-II algorithm.  
distribution of the solution is more uniform and covers a wider range, indicating that the improved NSGA-II has accelerated convergence and higher solution accuracy.  
The comparison of the Pareto frontiers of the objective function optimized by the NSGA-II and INSGA-II algorithms is presented in Figure 11. The improved NSGA-II algorithm (INSGA-II) exhibits notable advantages in multi-objective optimization through the initialization of the good point set population and the selection of parents based on linear ranking. The good point set approach ensures diversity and coverage of the solution by rationally distributing the initial population, thereby generating a more uniform Pareto front and accelerating the convergence of the algorithm. The selection of parents based on linear ranking increases the selection pressure, thereby increasing the likelihood of superior individuals being selected. This approach balances global search and local exploitation and improves the convergence speed and stability. These enhancements facilitate the convergence of the INSGA-II algorithm to the Pareto front in a more expeditious and uniform manner, reduce the number of iterations, circumvent premature convergence, and enhance the overall optimization performance of the algorithm.  
Figure 12 shows the evolution of the average objective value of the multi-optimization objective over the course of the multi-generation evolution of the INSGA-II algorithm. The vertical axis represents the average objective value, while the horizontal axis represents the number of population iterations per generation. The objective value of the objective function \(S_{1}\) remains relatively stable and has reached the global optimum near 10 generations. The objective function \(S_{2}\) exhibits a rapid decline before 50 generations, followed by a gradual  
Figure 10: Test function comparison: (**a**) ZDT1, (**b**) ZDT2, (**c**) ZDT3, and (**d**) ZDT6.  
stabilization, indicating that \(S_{2}\) demonstrates superior optimization efficacy at the outset and is in proximity to the global optimum. In consideration of the convergence of the multi-objective function, the rate of convergence, and the optimal solution, this paper proposes 50 generations as the termination condition for the INSG-II algorithm's population iteration.