# An Improved Dual-Sorting NSGA-II Method for Optimal Radiation Shielding Design
## 1 Introduction  
Radiation shielding design is a fundamental aspect of nuclear safety engineering, playing a critical role in minimizing radiation exposure while ensuring the structural and economic feasibility of shielding systems. The design of radiation shielding inherently involves multiple competing objectives, such as radiation attenuation, material weight minimization, mechanical stability, and cost-effectiveness. Given the complexity of these interrelated factors, traditional trial-and-error approaches based on empirical knowledge are often inadequate for achieving optimal shielding configurations. Consequently, the development of computational optimization techniques has become an essential research focus in radiation protection and shielding design [1, 2, 3, 4, 5, 6]. Multi-objective optimization techniques, particularly evolutionary algorithms, have been widely employed to address the challenges associated with radiation shielding design [7, 8, 9, 10]. Among them, non-dominated sorting-based genetic algorithms, such as multi-objective genetic algorithms (MOGA) [11] and the Non-dominated Sorting Genetic Algorithm II (NSGA-II) [12], have demonstrated remarkable effectiveness in generating diverse trade-off solutions within a single optimization run, without requiring predefined preference settings. Unlike conventional genetic algorithms that rely on multiple iterative computations under varying weight conditions, MOGA and NSGA-II efficiently construct a set of Pareto-optimal solutions, making them particularly well suited for complex multi-objective optimization problems. Compared to MOGA, the NSGA-II algorithm offerssuperior convergence speed in the later stages of evolution and demonstrates enhanced capability in identifying the extrema of objective functions [13; 14], thereby making it especially suitable for radiation shielding design optimization [15; 16; 17; 18]. However, despite its advantages, NSGA-II has inherent limitations that hinder its efficiency in radiation shielding optimization. Monte Carlo methods require extensive computational time for radiation shielding calculations, particularly when dealing with complex coupled models. At the same time, the NSGA-II algorithm encounters convergence difficulties when handling more than three optimization objectives [16; 17; 18]. These two factors limit the performance and practical applicability of NSGA-II in radiation shielding optimization, making it critically important to improve the algorithm's intrinsic search efficiency. One of the primary shortcomings of NSGA-II is its inability to effectively distinguish among solutions with the same non-dominated rank in terms of their relative proximity to the Pareto front. As the optimization progresses, this limitation results in reduced selection pressure, slower convergence rates, and difficulties in achieving high-quality solutions within a reasonable computational time frame [19; 20]. Addressing these shortcomings is essential to enhance the algorithm's performance and practical applicability in radiation shielding design. To overcome these challenges, this study proposes an enhanced dual-sorting NSGA-II method, incorporating a novel dominance determination strategy and an \(O\)-based sorting mechanism. By leveraging the concave characteristics of Pareto-optimal shielding solutions, the proposed approach refines the ranking process and maintains evolutionary pressure toward optimality. This dual-sorting mechanism not only improves convergence speed but also enhances solution diversity, ultimately leading to a more efficient and robust optimization process. This paper is structured as follows: Section 2 presents the methodological framework of the proposed dual-sorting NSGA-II, detailing the integration of the \(O\)-based sorting mechanism with conventional NSGA-II ranking. Section 3 provides a case study on a multilayer shielding optimization problem, comparing the performance of the proposed method with the classical NSGA-II in terms of convergence rate and optimization efficiency.