# 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  
In this section, the shielding problem described in Section 3.1 is optimized using both the classical NSGA-II and the dual-sorting NSGA-II methods. The mutation strategy described in Case 1 was applied. In the multilayer flat plate shielding problem, each generation comprised 20 solutions, with a total of 50 evolutionary generations. A 6-bit random number was used for thickness encoding, with the thickness of each layer determined by Equation (4):  
\[h_{i}=H\frac{r_{i}}{\sum_{m=1}^{6}r_{m}} \tag{4}\]  
where \(h_{i}\) represents the thickness of the \(i\)th layer, \(H\) denotes the total thickness of the six-layer shielding, and \(r_{i}\) corresponds to the thickness encoding of the \(i\)th layer. To reduce the influence of stochastic effects arising from the genetic algorithm's selection and mutation mechanisms, five independent optimization runs were performed for both the classical NSGA-II and the dual-sorting NSGA-II methods, ensuring that each run commenced from an identical initial population. The results indicate that each run successfully converged and produced the optimal solution set (i.e., the Pareto front). Moreover, similar evolutionary processes and optimization outcomes were observed across all five runs. The distribution of shield solutions throughout a representative optimization process is depicted in Figure 4. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Material** & **Lithium** & **Polyethylene** & **Boron** & **Aluminum** & **Iron** & **Lead** \\ \hline Density g/cm\({}^{3}\) & 0.535 & 0.95 & 2.34 & 2.70 & 7.86 & 11.34 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Selectable materials for shielding optimization. Figure 4: Distribution of shield solutions throughout a representative optimization process for (**a**) the classical NSGA-II method and (**b**) the dual-sorting NSGA-II method. As shown in the figure, by the seventh generation, the population distributions of the two optimization methods exhibit a notable divergence. The classical NSGA-II method produces shield solutions that are densely concentrated in the central region, remaining relatively distant from the Pareto front. In contrast, the dual-sorting NSGA-II method, influenced by crowding distance settings, shifts the solution set toward lower dose levels, facilitating a more rapid convergence. By generation 10, the solution distribution of the classical NSGA-II method begins to move toward the Pareto front, with some solutions approaching it. However, the dual-sorting NSGA-II method achieves a closer alignment with the Pareto front, forming a more complete and continuous solution set across the density range of 0.8 g/cm\({}^{3}\) to 3.6 g/cm\({}^{3}\). By generation 20, most solutions from both methods converge toward the Pareto front. However, solutions obtained via the classical NSGA-II method still exhibit a noticeable gap from the optimal curve, whereas those from the dual-sorting NSGA-II method are predominantly concentrated along the Pareto front, covering a broader solution space as the evolution progresses. By generation 50, the shield solutions obtained by both methods are largely positioned along the optimal solution curves, making it increasingly difficult to discern significant differences in distribution between the two approaches based solely on solution positioning. To further compare the solution capabilities of the classical NSGA-II and the proposed dual-sorting NSGA-II methods on the Pareto front, we examined the shielding configuration with the lowest dose value identified in each of five independent optimization runs and computed the average performance. For the classical NSGA-II method, the average density of the lowest-dose solutions was 1.82 g/cm\({}^{3}\), with an average total dose of 1.18 \(\times\) 10\({}^{-10}\) uSv/n. A representative shielding design consisted of a composite structure: 174.1 mm polyethylene + 35.5 mm iron + 88.8 mm polyethylene + 1.6 mm lead.