Principles of radiation-shielding design

The primary objective of reactor radiation shielding is to design a shielding system composed of multi-layer and multi-material structures that surround the core facilities within the reactor, ensuring that the personnel are in an environment where the radiation dose meets the specified limits. This is achieved by utilizing suitable shielding materials and geometric configurations combined with multi-layered and multi-component structures to shield the radiation sources. The selection and arrangement of materials in each layer are based on their specific shielding capabilities and nuclear interactions with radiation particles, with the aim of minimizing radiation levels to the maximum possible extent.

The principles governing the design of radiation shielding are based on the interactions between the radiation particles and materials. Radiation predominantly comprises neutrons and gamma rays. Neutron interactions include absorption and scattering, which is further categorized into elastic and inelastic scattering. For neutron shielding, the secondary gamma rays generated during inelastic scattering reactions must be considered. The interactions of gamma rays include Compton scattering, photoelectric absorption, and pair production. Typically, neutron shielding requires materials containing heavy and light isotopes, such as light water, boron carbide, and polyethylene [17, 18]. Conversely, gamma-ray shielding requires materials with high-Z isotopes, including lead, concrete, and steel [19].

Many-objective mathematical model for radiation-shielding design

Many-objective optimization in practical applications aims to achieve multiple optimized objectives within specific constraints. However, owing to the inherent conflicts between these objectives, optimizing one objective is typically achieved at the expense of deteriorating the others. As a result, a unique optimal solution is elusive; instead, a set of Pareto-front solutions consisting of non-dominated individuals is obtained [20]. Radiation-shielding design presents a typical many-objective optimization problem in nuclear engineering. The goal is to minimize the mass and volume of the total shielding system while ensuring that the external-radiation dose satisfies the ALARA principle. However, weight, volume, and radiation dose are conflicting objectives that cannot be optimized simultaneously. In this study, we focus on minimizing the total weight, total volume, and axial and radial radiation doses outside the shielding layers. These objectives are influenced by various parameters, including the thickness of the shielding layer, materials used, and the composition of the materials used. Additionally, constraints are imposed on the fast-neutron, thermal-neutron, and gamma-ray flux rates outside the shielding layers during the optimization process. Reactor radiation-shielding design involves many objectives, parameters, and constraints. To address these challenges, the mathematical model in Eq. (1) is proposed: $\{\begin{array}{l}\min F\left(X\right)={\left(F_{\text{R}}\left(X\right)\mathrm{,}{F}_{\text{W}}\left(X\right)\mathrm{,}{F}_{\text{V}}\left(X\right)\right)}^T\hfill \\ s\mathrm{.}t\mathrm{.}\{\begin{array}{l}{F}_{\text{R}}\left(X\right)=R_{\text{N}}\left(X\right)+R_{\text{P}}\left(X\right)\le R_0\hfill \\ F_{\text{W}}\left(X\right)={\displaystyle \sum _{m=1}^M}{V}_m\cdot {\rho }_m\le W_0\hfill \\ F_{\text{V}}\left(X\right)={\displaystyle \sum _{m=1}^M}{V}_m\le V_0\hfill \\ {\text{Φ}}_{\text{N}}\left(X\right)\le {\text{Φ}}_{{\text{N}}_0}\hfill \\ {\text{Φ}}_{\text{P}}\left(X\right)\le {\text{Φ}}_{{\text{P}}_0}\hfill \\ X=\left(x_1\mathrm{,}{x}_2\mathrm{,}{x}_3\mathrm{,}\dots \mathrm{,}{x}_{n-1}\mathrm{,}{x}_n\right)\mathrm{,}X\in \Re \hfill \\ L_i\le x_i\le U_i\left(i=\mathrm{1,}\mathrm{2,}\dots \mathrm{,}n\right)\hfill \\ \mathrm{.}\hfill \end{array}\hfill \end{array}$ (1) Based on the optimization model (1), where X is the decision variable in the shielding design (geometry thickness, material type, material composition); the values $L_i$ and $U_i$ represent the lower and upper bounds of the variables, respectively; the vector space containing all decision variables is represented by X; ${\text{Φ}}_{\text{N}}\left(X\right)$ and ${\text{Φ}}_{\text{P}}\left(X\right)$ represent the neutron and photon flux rates outside the shielding layer, respectively, while ${\text{Φ}}_{{\text{N}}_0}$ and ${\text{Φ}}_{{\text{P}}_0}$ are their corresponding constraint values; $V_m$ represents the volume of the m-th shielding layer, and ${\rho }_m$ represents the corresponding density of the shielding material denoted by $m$. $W_m$ is the mass of the shielding layers; $R_{\text{N}}$ and $R_{\text{P}}$ represent the neutron and photon dose external to the shielding layer, respectively; $F_{\text{R}}\left(X\right)$, $F_{\text{W}}\left(X\right)$, and $F_{\text{V}}\left(X\right)$ represent the radiation dose of the shielding layer and the total weight and total volume of the shielding system, respectively; $R_0$, $W_0$, and $V_0$ represent the constraints for the radiation dose, weight, and volume, respectively; $F(X)$ is the radiation-shielding design objective vector, and the minimized value is expected to be optimized for each dimension of the objective vector.

Encoding methods of radiation-shielding model

In recent years, the development of radiation-shielding designs has progressed significantly, and a series of studies combining evolutionary algorithms with various types of shielding designs have emerged. However, the decision vector in this type of research only covers the shielding structure and shielding-material-type changes, and no material-composition-optimization research has been previously carried out. In this study, the reactor-shielding model is simulated for the many-parameter problem in shielding optimization using both binary encoding and real-number encoding, which can be encoded to characterize the shielding-model geometry, material type and material composition, and coupled with a variety of evolutionary algorithms for calculation.

Fig. 2 illustrates the encoding process for the simple shielding scheme. First, the thicknesses of the shielding layers and the corresponding index numbers for the shielding materials are obtained. The composition is determined for materials with variable compositions, such as borated and lead-borated polyethylene. Second, the above parameters are encoded in real or binary numbers. The composition content, which is subject to fixed-sum constraints, is transformed using spherical-coordinate conversion [21], as in Eq. (2) prior to encoding. Finally, the encoding design is completed for all shielding layers. The purpose of encoding is to combine the shielding model with the evolutionary algorithm such that the model parameters can be optimized by the algorithm, and the final shielding scheme can be obtained for the reference of the designer. $\begin{array}{l}{x}_i={\left(\sin {\theta }_1\cdot \sin {\theta }_2\mathrm{...}\sin {\theta }_{n-1}\right)}^2\mathrm{,}i=1\\ x_i={\left(\cos {\theta }_{i-1}\cdot \sin {\theta }_i\mathrm{...}\sin {\theta }_{n-1}\right)}^2\mathrm{,}i=\mathrm{2,}\mathrm{...,}n-1\\ x_i={\left(\cos {\theta }_{n-1}\right)}^2\mathrm{,}i=n\end{array}$ (2)

Fig. 2Full-screen View

(Color online) Example of the encoding method for a radiation-shielding model

Scheme of radiation-shielding optimization with many-objective evolutionary algorithm

In this study, shielding-optimization methods are proposed by combining many-objective evolutionary algorithms with particle-transport calculation software that can optimize the reactor primary-shielding structures, material types, and material compositions to obtain optimal shielding-design schemes through an automated process. These methods are embedded into the multi-functional radiation-transport simulation platform (MOSRT) developed by the NEAL team [22, 23].

A schematic of the many-objective optimization of the radiation-shielding design is shown in Fig. 3, and the steps are described as follows.

Fig. 3Full-screen View

(Color online) Schematic of radiation-shielding design with many-objective evolutionary algorithm

(1) Preprocessing of the initial shielding model is performed using the MOSRT software. First, the CAD model of the reactor geometry is modeled with MOSRT, and the model is automatically converted into a particle-transport calculation model. The shielding regions to be optimized are labeled. Finally, the initial running parameters are set, including the maximum number of iterations g, population size N, and optimization-objective number M.

(2) The evolutionary algorithm is selected. In this study, the genetic and ABC algorithms are invoked as evolutionary algorithms to optimize the weight, volume, and region-specific radiation-dose rate of the reactor radiation-shielding model.

(3) The decision-vector encoding method (real-number encoding or binary encoding) is chosen based on the requirements, and the initial parent population $P_{g=1}$ is generated.

(4) The initial offspring population $Q_{g=1}$ is generated based on the parent-population characteristics. The GA uses selection, crossover, and mutation to generate the offspring population, and the bee-colony algorithm uses employed, onlooker, and scout bees to update the offspring population. The parent population $P_g$ is merged with the offspring population $Q_g$ to obtain the g-th generation combined population $C_g$. The population is decoded and converted into a particle-transport input card to solve the objective values. The MOSRT has a built-in shielding-calculation module integrated with the Monte Carlo method (MC) [24].

(5) A fast non-dominated sorting strategy is applied to the merged population, and each algorithm uses the reference-point selection strategy [25] to generate the new parent population $P_{g+1}$. A performance-comparison analysis is conducted simultaneously with conventional methods that use crowding-distance strategy selection [26]. Specific strategies are explained in Sect. 3.5.

(6) Steps (4) and (5) are repeated if the maximum number of iterations is not reached, and the parent population $P_{g+1}$ is continuously updated; otherwise, step (7) is implemented.

(7) The optimization process is completed, and the optimization results of each algorithm are thoroughly compared and analyzed.

Evolutionary strategy based on genetic algorithm

The GA [27] is a classical evolutionary strategy for simulating inheritance and evolution in nature. It searches for the solution space of optimization problems through operations such as selection, crossover, and mutation. GAs based on different evaluation strategies exhibit significant differences in performance when solving various problems [28]. The following section explains the basic operations of GAs.

Selection: In this study, a binary tournament-selection operation is used, where two individuals (the individual in the paper means a specific shielding scheme) are taken from the current population at a time, the dominance relationship between the two individuals is judged, and the individual in the dominant position is selected for subsequent crossover and mutation. If two individuals do not dominate each other, one individual is randomly selected for the subsequent genetic steps. The determination of the dominance relationship will be explained in Sect. 3.4.1.

Crossover: The crossover operation mimics the process of hybridization in biological evolution. It selects the chromosomal segments (genes) of two or more parent individuals for exchange to generate new offspring. In this study, we use a single-point crossover operation for GAs with binary encoding and a simulated binary crossover operation for GAs with real-number encoding.

Mutation: The mutation operation introduces a degree of randomness to the algorithm by changing some gene values on individual chromosomes to generate new individuals. The mutation operation can increase the diversity of the population and prevent it from falling into a locally optimum solution. The mutation operation can randomly select some genes of an individual to be changed, or mutate each gene on the chromosome with a particular probability. In this study, we use the bit-flip mutation operation for GAs with binary encoding and polynomial mutation operations for GAs with real number encoding.

Evolutionary strategy based on artificial bee-colony algorithm

The ABC [29, 30] algorithm is a global optimization technique based on swarm intelligence. It draws inspiration from the foraging behavior of bee colonies, where individual bees perform specific tasks and share information to collectively find the optimal solution to a problem. This study builds on the fundamental ABC algorithm and introduces improvements by employing two methods: a differential evolution search [31] and golden sine search [32]. The following sections explain the search strategies of the algorithm.

Employed bee search: In this phase, a number of hired bees consistent with the population size will be dispatched to search for historical food sources (solution space). In this study, the bee population searches using a differential-search method based on an elite strategy. Individuals in the Pareto optimal set (the Pareto optimal set will be explained in Sect. 3.4.2.) are randomly used as bootstrap terms, and the differential evolution operator is used to generate new solutions, as shown in Eq. (3): $X^t=X^{t1}+R\left(X^{t1}-X^{t2}\right)+R\left(X^{t1}-X^{t3}\right)\mathrm{,}$ (3) where $X^{t1}$, $X^{t2}$, and $X^{t3}$ are three mutually dissimilar solutions selected from the Pareto optimal set and R are random numbers between [0, 1].

The elite-guided search mechanism is an exploitation operation that unavoidably reduces the diversity of the population, and may cause the algorithm to converge locally. To better balance the global-search and local-exploitation capabilities, this study proposes a solution-space search method based on the golden sine function: $\begin{array}{l}{X}^t=X_i\left|\sin \left(r_1\right)\right|+r_2\cdot \sin \left(r_1\right)\cdot |c_1{X}^{t1}-c_2{X}_i|,\hfill \\ c_1=a{g}_{\text{s}}+b\left(1-g_{\text{s}}\right),\hfill \\ c_2=a\left(1-g_{\text{s}}\right)+b{g}_{\text{s}}.\hfill \end{array}$ (4) In Eq. (4), X_i represents the i-th individual, $r_1$ and $r_2$ are random numbers belonging to $[\mathrm{0,2}\pi ]$ and $[\mathrm{0,}\pi ]$ respectively, $c_1$ and $c_2$ are introduced as golden section coefficients in the position-update formula, respectively, and $a$ and $b$ are initial values set to $-\pi$ and $\pi$, respectively. The golden section number $g_{\text{s}}$ is $\left(\sqrt{5}-1\right)/2$.

The employed bees use Eq. (3) and Eq. (4) for searching with equal probabilities to balance the diversity and local-search capability of the population.

Onlooker bee search: The onlooker bees perform repeated optimization on the excellent food sources based on the roulette-wheel selection. The specific optimization operations are the same as those of the employed bees, and the onlooker-bee evaluation function is as follows: $fi{t}_i=\frac{dom}{FoodNumber}\mathrm{,}$ (5) where $dom$ represents the number of solutions in which the i-th feasible solution dominates among all feasible solutions, and $FoodNumber$ represents the total number of food sources.

The roulette-wheel selection probability is defined as $p_i=\frac{fi{t}_i}{{\displaystyle \sum _{n=1}^{FoodNumber}}fi{t}_n}\mathrm{.}$ (6) Food sources with higher probabilities have a greater probability of being explored in the subsequent bee stage, thereby facilitating the algorithm's efficient exploration of the space for excellent solutions.

Scout bee search: A food source is randomly selected from the last Pareto layer, its decision vector is randomized, and its corresponding objective values are reset.

Evaluation method of shielding scheme based on Pareto domination

Pareto-set-selection strategy

When offspring are generated, the parent set $P_g$ and resulting offspring set $Q_g$ are combined into a set $C_g$. The schemes at the smaller non-dominant level of the set are selected to enter the next generation until all schemes at level $F_l$ are selected to $P_{g+1}$, such that $P_{g+1}$ is equal in size to $P_g$. If all the individuals at level $F_l$ are selected to $P_{g+1}$, the next-generation population $P_{g+1}$ is larger than $P_g$. Therefore, a certain strategy is required to select individuals at level $F_l$. In this study, crowding distance and reference-point selection strategies are used. The details are as follows.

3.5.2

Selection strategy based on reference point

In the reference-point selection strategy [34, 35], the reference points are first predefined in the normalized hyperplane and the number of reference points should be close to the population size, which is defined by Eq. (12): $RefCount=\left(\begin{array}{c}H+M-1\\ H\end{array}\right)=C_{H+M-1}^H\mathrm{,}$ (12) where M denotes the number of optimization objectives and H denotes the number of divisions. A schematic of the reference points in the case of three objectives and six divisions is shown in Fig. 4. The purpose of generating reference points is to generate a set of reference vectors in the objective space, through which schemes in the population can be associated to ensure that the diversity of the population is maintained during subsequent evolution.

Fig. 4Full-screen View

(Color online) Reference points are shown on a normalized hyperplane for a three-objective problem with p =6

We define $S_g$ as the set consisting of all schemes in the $F_1-F_l$ layers and adaptively normalize all schemes in $S_g$ to the hyperplane. First, we define the ideal point $z=\left(z_1^{\min }\mathrm{,}{z}_2^{\min }\mathrm{,}\mathrm{...,}{z}_M^{\min }\right)$, where $z_i^{\min }$ is the minimum value of the i-th objective. Next, the objective values of each scheme in $S_g$ are transformed with respect to the ideal point as $F_i^{\text{'}}\left(X\right)=F_i\left(X\right)-z_i^{\text{min}}$. Subsequently, the extreme point ($z_i^{\max }$) on each (i-th) objective axis is determined, and the hyperplane is formed based on the M extreme points. Finally, the intercept $a_i$ of each objective axis is obtained and the normalized objective value is obtained from Eq. (13). $\begin{array}{l}{F}_i^n\left(X\right)=\frac{F_i^{\text{'}}\left(X\right)}{a_i}\hfill \end{array}$ (13) The normalized scheme in $S_g$ in the hyperplane is associated with a reference point. First, the reference points are connected to the ideal points to form a reference vector. Second, the Euclidean distance between each individual and the reference vector is computed. Finally, each individual is associated with the nearest reference vector.

After associating the reference points with the scheme in $S_g$, the number of times the first l - 1 layers of the schemes are associated with each reference point ${\rho }_j$ is counted. Let $J_{\min }=\left\{j:\arg {\min }_j{\rho }_j\right\}$ be the set of reference points with the smallest ${\rho }_j$. Select the reference point $\overline{j}\in J_{\min }$.

If the selected ${\rho }_{\overline{j}}=0$, no scheme is associated with $\overline{j}$ in the first l - 1 layers. At this time, two choices are available. 1) A scheme in $F_l$ that is associated to this reference point exists. At this time, the individual with the shortest distance from the reference vector Euclidean is selected to join the population, and at the same time, ${\rho }_{\overline{j}}$ is increased by 1. 2) If the reference point $\overline{j}$ does not have a scheme associated with it in $F_l$, then this reference point is skipped.

If ${\rho }_{\overline{j}}>0$, then at least one scheme exists in the first l - 1 layers associated with $\overline{j}$. In this case, the scheme in $\overline{j}$ is chosen randomly to join the population, whereas ${\rho }_{\overline{j}}$ increases by one. The reference-point strategy ensures that the population is well-distributed and can guide the evolution of the population to a uniform Pareto front.

Many-objective optimization for a simple 3D shielding problem

The first problem focuses on the multiple objectives of a simple 3D shielding structure, as shown in Fig. 6(a). The optimization objectives are to determine the total volume and weight of the R1-R10, U1-U5, and L1-L5 shielding layers as well as the radial lateral, axial upward, and axial downward dose rates of the shielding layers. The thickness of each shielding layer in the optimized model ranges from 0.3 cm to 13 cm. The reactor consists of a homogenized core with a height of 167.6 cm and radius of 78.8 cm. The source term is set as a fixed source with a probability distribution based on the Watt-fission spectrum. A comparative analysis of the optimization performance of each algorithm is performed using the average objective value and hypervolume [39] value.

Fig. 6Full-screen View

(Color online) Many-objective optimization results for the simple 3D shielding problem. (a) Schematic of the simple 3D shielding structure. (b) The average values of the volumes for shielding schemes in the Pareto front. (c) The average values of the weights for shielding schemes in the Pareto front. (d) The average values of the axial upward dose rates for shielding schemes in the Pareto front. (e) The average values of the axial downward dose rates for shielding schemes in the Pareto front. (f) The average values of the radial dose rates for shielding schemes in the Pareto front. (g) Hypervolume indicator results

In this problem, GA-based algorithms use binary encoding and ABC-based algorithms use real number encoding. The population size of all algorithms is set to 210, and the number of reference points of the algorithms using the RP strategy is 210 (the number of divisions H=6). The number of MC transported particles is 1.0×10⁷.

For the shielding-optimization process, the time complexity is mainly related to the number of evolutionary generations g and the population size N, where the total time complexity of the GA is $O(gN)$ and that of the ABC algorithm is $O(2gN)$. To ensure that the tests are conducted under identical conditions, the number of evolutionary generations of the GA-based algorithms is set to 100, whereas that of ABC-based algorithm is 50.

As observed in the average objective values shown in Fig. 6(b-f), the algorithms based on the reference-point strategy show excellent optimization results in terms of the weight and volume objectives. Compared with the conventional crowding-distance strategy, the average volume and weight values for the shielding schemes in the final generation of RP-NSGA are only 24.5% and 14.5% of those of CD-NSGA, and the average volume and weight values of RP-MOABC are only 17.3% and 9.77% of those of CD-MOABC. Regarding the dose rates in each region, the results in the final generation of each strategy are not significantly better or worse; the distribution of the results is relatively close, and the algorithm with relatively better performance is RP-NSGA. Overall, based on the average objective values, the algorithms using the reference-point strategy exhibit better convergence results.

To further evaluate the comprehensive performance of each algorithm scientifically, the hypervolume (HV) is chosen as the quantified factor to measure the convergence and distribution of the shielding schemes. The HV metric is a widely used performance evaluation method for engineering optimization problems, and does not require a real Pareto front to judge the strengths and weaknesses of a set of population schemes. The HV metric is calculated as shown in Eq. (14): $\begin{array}{l}HV=\delta \left({\displaystyle \underset{i=1}{\overset{S_i}{\cup }}}{v}_i\right)\mathrm{.}\hfill \end{array}$ (14) The combined performance of the algorithms can be determined by calculating the sum of the values of the hypervolume $v_i$ in space for all the schemes in the population. The larger the combined value of the hypervolume, the better the performance. Fig. 6(g) shows that the optimization performance from high to low follows the order of RP-MOABC, RP-NSGA, CD-MOABC, and CD-NSGA. The algorithms based on the reference-point strategy are comprehensively better than the conventional crowding-distance strategy for many-objective optimization problems.

Constrained many-objective optimization for a complex 3D shielding problem

The second problem focuses on conducting a constrained many-objective optimization for the complex shielding structures shown in Fig. 7(a). For the constraint objectives, the thermal neutron-flux rate on the outermost shielding layer of the primary shielding system must be less than 1.0×10⁵ n/(cm²·s), the fast neutron-flux rate must be less than 1.0×10³ n/(cm²·s), and the gamma-ray energy-flux rate must be less than 6.0×10⁶ MeV/(cm²·s). The optimization objectives include the total volume and weight of shielding layers R1–R8 and U1–U3, as well as the dose rates in the radial and axial upward directions of the shielding layers. The variable parameters are the shielding-layer thickness, shielding-layer materials, and partial material compositions (boron polyethylene, lead boron polyethylene, tungsten alloy). The thickness of each shielding layer in the optimized model ranges from 15 cm to 80 cm. The reactor consists of a homogenized core with a height of 167.6 cm and radius of 78.8 cm. The source term is set as a fixed source with a probability distribution based on the Watt-fission spectrum.

Fig. 7Full-screen View

(Color online) Constrained many-objective optimization results for complex shielding structure. (a) Schematic of the complex shielding structure. (b) Improvements of the optimized objectives with RP-NSGA. (c) Improvements of the optimized objectives with RP-MOABC. (d) Improvements of the optimized objectives with CD-NSGA. (e) Improvements of the optimized objectives with CD-MOABC

In this problem, GA- and ABC-based algorithms use real-number encoding. The population size of all the algorithms is set to 100, and the number of reference points of the algorithms using the RP strategy is 84 (the number of divisions H=6). The number of MC transported particles is 2.0×10⁸. For computational efficiency, eight AMD EPYC 7H12 CPUs (Dual 128 core, 2.6 GHz) are used for massive parallel computation.

A heat map of the optimization ratio is plotted for the last generation of optimized shielding schemes compared to the initial shielding scheme, where the optimization ratio is defined in Eq. (15): $\begin{array}{l}\text{improvements}=\frac{\left(F_{\text{initial}\mathrm{,}j}-F_{i\mathrm{,}j}\right)}{F_{\text{initial}\mathrm{,}j}}\times 100\%\hfill \\ i=\mathrm{1,}\mathrm{...,}Nj=\mathrm{1,}\mathrm{...,}M\hfill \end{array}\mathrm{.}$ (15) In Eq. (15), i represents the individual index, N represents the population size, j represents the dimension of the objective values, M represents the total number of objective values, and F represents the corresponding objective values. A higher value of the optimization ratio indicates a better optimization of the corresponding objective. Fig. 7(b-e) shows that the optimization effect of each method is obvious in the three objective dimensions of volume, weight, and radial dose rate. However, for the top-side dose rate, some of the schemes show a degree of degradation, but they still achieve a better scheme set compared to the initial shielding scheme. The optimization performances of RP-NSGA and RP-MOABC in each objective dimension are better than those of CD-NSGA and CD-MOABC, which proves that the algorithms proposed in this study are feasible for many-objective optimization problems.

In practical engineering shielding-design problems, the final design scheme can be selected from a Pareto set of optimization schemes based on specific requirements. In this study, the miniaturization schemes obtained by the RP-NSGA algorithm (Shield #1) and RP-MOABC algorithm (Shield #2) are selected for demonstration. The shielding structure is shown in Fig. 8(b-c), the dose field distribution is shown in Fig. 8(d-e), and the specific parameters and optimization effects are listed in Tables 1, 2, 3 and 4. The dose rates are computed with the MagicMC Monte Carlo code using the global variance-reduction method [40-43] with a statistical error of less than 5%.

Fig. 8Full-screen View

(Color online) Schematic of typical shielding schemes after optimization. (a) Initial shielding structure. (b) Miniaturization scheme of RP-NSGA. (c) Miniaturization scheme of RP-MOABC. (d) Dose field distribution of the RP-NSGA miniaturization scheme. (e) Dose field distribution of the RP-MOABC miniaturization scheme