# An Improved Dual-Sorting NSGA-II Method for Optimal Radiation Shielding Design
## 3 Results and Discussion
### Comparison of Convergence: Classical NSGA-II vs. Dual-Sorting NSGA-II
This configuration follows well-known shielding design principles, where iron is embedded between polyethylene layers to enhance neutron moderation, and lead is positioned at the rear to absorb secondary gamma photons. In comparison, the dual-sorting NSGA-II method produced shielding solutions with an average density of 2.17 g/cm\({}^{3}\) and an average dose of 1.14 \(\times\) 10\({}^{-10}\) uSv/n. A typical configuration featured a similar composite structure: 184.7 mm polyethylene + 40.0 mm iron + 66.7 mm polyethylene + 8.6 mm lead. These results confirm that both NSGA-II variants are effective in identifying optimal shielding configurations, while the dual-sorting method exhibits improved convergence accuracy and greater consistency in locating near-optimal low-dose solutions. The number of solutions with \(O=1\) (i.e., optimal solutions) in population \(C_{t}\), as well as the number of solutions with \(R=1\) and \(O>1\) (i.e., suboptimal solutions), serve as key metrics for assessing the effectiveness of an optimization method in approximating the Pareto front. Figure 5 presents the evolutionary progression of optimal and suboptimal solutions within population \(C_{t}\) over successive generations. The results demonstrate that the dual-sorting NSGA-II method consistently outperforms the classical NSGA-II method in maintaining a higher count of both optimal and suboptimal solutions beyond generation 15. By the final generation, the Pareto front obtained via the dual-sorting NSGA-II method consists of 29 optimal solutions and 74 suboptimal solutions, whereas the classical NSGA-II method yields only 22.6 optimal solutions and 49 suboptimal solutions. This outcome underscores the superior performance of the dual-sorting NSGA-II approach, which more effectively approximates the Pareto front and facilitates a more refined optimal solution set. For \(t>16\), both optimization algorithms exhibit a predominance of suboptimal solutions over optimal solutions, necessitating an increased selection of optimal solutions for the next-generation parent population \(P_{t+1}\) to sustain evolutionary pressure. In the classical NSGA-II method, the proportion of suboptimal solutions selected for \(P_{t+1}\) gradually increases, as depicted in Figure 6. The inclusion of optimal solutions in \(P_{t+1}\) begins to decline from \(t=14\), and by the later evolutionary stages, optimal and suboptimal solutions are selected in equal proportions, each accounting for 50% of \(P_{t+1}\). As shown in Figures 5 and 6, although both the absolute number and relative proportion of suboptimal solutions in the Pareto front are higher for the dual-sorting NSGA-II method compared to the classical NSGA-II method, the \(O\)-value-based sorting strategy effectively filters out suboptimal solutions, leading to a greater concentration of optimal solutions in \(P_{t+1}\). By \(t=50\), optimal solutions constitute over 80% of \(P_{t+1}\) in the dual-sorting NSGA-II method, demonstrating its ability to maintain selection pressure on high-quality  
Figure 5: Changes in the number of solutions with optimal (\(O=1\)) and suboptimal (\(R=1\) and \(O>1\)) solutions in population \(C_{t}\) with evolutionary generations. Figure 6: Number of (**a**) optimal, (**b**) suboptimal, and (**c**) other solutions in parent \(P_{t+1}\) across generations. solutions. Additionally, in this method, solutions beyond the optimal and suboptimal sets are more likely to be selected for \(P_{t+1}\), thereby improving population diversity and enhancing the overall robustness of the optimization process by preventing premature convergence and maintaining a more comprehensive exploration of the solution space.