# An Improved Dual-Sorting NSGA-II Method for Optimal Radiation Shielding Design
## 2 Improved NSGA-II Methodology with Dual-Sorting Approach
### O-Based Sorting Method  
In combinatorial optimization, optimal radiation shield solutions often exhibit concave characteristics with respect to their objective values. Specifically, for a Pareto-optimal solution curve \(y=F(x)\) defined by two objectives, \(x\) and \(y\), any straight line \(y=f(x)\) connecting two points \((x_{1},\ \ y_{1})\) and \((x_{2},\ \ y_{2})\) on the curve must satisfy Equation (1). \[F(x)<f(x),\ x\in[x_{1},x_{2}] \tag{1}\]  
Building on the concave characteristics, a novel dominance determination method is introduced, considering a shielding solution \(S_{k}:(x_{k},\ \ y_{k})\) and two arbitrary solutions, \(S_{k-i}:(x_{k-i},\ \ y_{k-i})\) and \(S_{k+j}:\left(x_{k+j},\ \ y_{k+j}\right)\), located on either side of \(S_{k}\). If the straight line \(y=f_{1}(x)\) connecting \(S_{k-i}\) and \(S_{k+j}\) satisfies the condition \(y_{k}>f_{1}(x_{k})\), then \(S_{k}\) does not lie on the Pareto-optimal solution curve and is dominated by \(S_{k-i}\) and \(S_{k+j}\). Conversely, if this condition is not met, \(S_{k}\) is considered a non-dominated solution. Based on this dominance determination method, the fitness value \(O\) is introduced to determine the hierarchical ranking of the shielding solution within the population. The detailed computational procedure can be found in Reference [13]. Figure 1: Dual-sorting NSGA-II algorithm structure. (**a**) Flowchart of the dual-sorting NSGA-II method, showing the overall evolutionary process, including population initialization, dual-sorting, offspring generation, and population updating. (**b**) Pseudocode of the proposed dual-sorting NSGA-II algorithm, illustrating the integration of \(R\)-based and \(O\)-based sorting mechanisms along with genetic operations. To further enhance the \(O\)-value sorting algorithm, this study developed a novel crowding distance calculation method, as illustrated in Figure 2.  
For shield solutions \(S_{k,O=1}\) with \(O=1\), the crowding distance \(I_{o}(S_{k,O=1})\) is computed using the relative distance between two adjacent solutions, which is the same as the classical approach, as provided in Equation (2):  
\[I_{o}(S_{k,O=1})=dis(S_{k-1,O=1},S_{k+1,O=1})=|\frac{x_{k+1}-x_{k-1}}{x_{n}-x_ {min}}|_{+}|\frac{y_{k+1}-y_{k-1}}{y_{1}-y_{min}}| \tag{2}\]  
where \(S_{k-1,O=1}\) and \(S_{k+1,O=1}\) are the solutions adjacent to \(S_{k,O=1}\) on the left and right, respectively, among all solutions with \(O=1\). The solutions \((x_{min},\ y_{1})\) and \((x_{n},\ y_{min})\) are the two endpoints of the Pareto curve. For shielding solutions \(S_{l,O>1}:(x_{l},\ y_{l})\) with \(O>1\), the crowding distance \(I_{o}(S_{l,O>1})\) is instead determined by the relative distance to the optimal solution curve, as described in Equation (3):  
\[I_{o}(S_{l,O>1})=(1-\frac{y_{l}}{f_{k}(x_{l})})\cdot\frac{1}{dis(S_{k,O=1},\ S_{k+1,O=1})} \tag{3}\]  
where \(S_{k,O=1}\) and \(S_{k+1,O=1}\) are neighboring shield solutions with objective values \(x_{k}<x_{l}\) and \(x_{k+1}>x_{l}\), respectively. The function \(y=f_{k}(x)\) represents the straight line passing through solutions \(S_{k,O=1}\) and \(S_{k+1,O=1}\). The crowding distance \(I_{O}(S_{l,O>1})\) not only indicates the relative proximity of the shield solution \(S_{l,O>1}\) to the optimal solution curve but also signifies its potential contribution to refining the optimal solution upon evolution and inclusion in the optimal solution set. In the newly proposed non-dominated sorting algorithm based on fitness \(O\), all solutions are initially sorted in ascending order according to their \(O\)-values. Subsequently, solutions with the same \(O\)-value are further ranked in descending order based on their crowding distance \(I_{o}\). This dual-sorting strategy effectively distinguishes and prioritizes solutions with \(O=1\)--which are closest to the Pareto front (termed optimal solutions)--from solutions with \(R=1\) and \(O>1\) (termed suboptimal solutions), ensuring that optimal solutions are placed at the forefront of the population \(C_{l}\).