Multi-Objective Optimization Design of FRP Reinforced Flat Slabs under Punching Shear by Using NGBoost-Based Surrogate Model

Abstract

Multi-objective optimization problems (MOPs) in structural engineering arise as a significant challenge in achieving a balance between prediction accuracy and efficiency of the surrogate models, which are conventionally adopted as mechanics-driven models or numerical models. Data-driven models, such as machine learning models, can be instrumental in resolving intricate structural engineering issues that cannot be tackled through mechanics-driven models. This study aims to address the challenges of multi-objective optimization punching shear design of fiber-reinforced polymer (FRP) reinforced flat slabs by using a data-driven surrogate model. Firstly, this study employs an advanced machine learning model, namely Natural Gradient Boosting (NGBoost), to predict the punching shear resistance of FRP reinforced flat slabs. The comparisons with other machine learning models, design provisions and empirical theory models illustrate that the NGBoost model has higher accuracy in predicting the punching shear resistance. Additionally, the NGBoost model is explained with Shapley Additive Explanation (SHAP), revealing that the slab’s effective depth is the primary factor affecting the punching shear resistance. Then, the formulated NGBoost model is adopted as a surrogate model in conjunction with the Non-Dominated Sorting Genetic Algorithm-II (NSGA-II) algorithm for multi-objective optimization design of FRP reinforced flat slabs subjected to punching shear. Through a case study, it is demonstrated that the Pareto-optimal set of the punching shear resistance and cost of the FRP reinforced flat slabs can be successfully obtained. By discussing the effects of design parameter changes on the results, it is also shown that increasing the slab’s effective depth is a relatively effective way to achieve higher punching shear resistance of FRP reinforced flat slabs.

1. Introduction

Fiber-reinforced polymers (FRPs) possess properties including a light weight, high strength and corrosion resistance. Among them, CFRP has higher mechanical and better fatigue and corrosion resistance while the durability of BFRP and GFRP may decrease in long-term alkaline environments [1,2,3]. In order to enhance the longevity of structures in a corrosive environment, FRP reinforcement can be employed as an alternative to steel bars in reinforced concrete structures [4,5,6]. In this regard, FRP reinforced concrete slabs can be served as prospective supplements to traditional reinforced concrete (RC) structures, owing to their superior durability and resistance to corrosions [7]. As for RC flat slab structures, particularly at the joints, they are susceptible to brittle punching failures, resulting in continuous collapse incidents (Figure 1) [8,9,10]. When it comes to FRP reinforced flat slab structures, since FRP reinforcement has a lower elasticity modulus and ductility compared to steel bars, they are more prone to brittle failure under punching shear [11]. Therefore, the punching shear resistance prediction and design optimization are the two most critical aspects in the research of FRP reinforced flat slabs, which are also prerequisites for the application of FRP reinforced concrete structures.

For the purpose of avoiding punching shear failure in FRP reinforced concrete flat slabs, researchers have conducted extensive experimental and theoretical studies. Matthys et al. [12] discovered that the punching shear strength of FRP reinforced slabs and RC slabs was similar under the same flexural stiffness. Bouguerra et al. [13] found that the thickness of FRP reinforced concrete slabs and the concrete’s compressive strength had significant impacts on their punching shear capacity. Several researchers, including El Gandour et al. [14], El Gamal et al. [15] and Matthys et al. [12] modified the punching shear resistance models of FRP flat slabs based on the various theories. Nguyen et al. [16] put forward a punching shear strength model based on fracture mechanics, which took the size effect and effective slab-depth ratio into consideration. Ospina et al. [17] established a model that considered eccentric shear stress to determine the stiffness and punching shear resistance of FRP reinforced concrete slabs.

Since the structural design needs to take both the safety, usually represented as structural resistance, and cost into consideration, two or more conflicting optimization objectives such as these can be summarized as multi-objective optimization problems (MOPs) [18,19,20]. There can be only one set of equilibrium solutions for MOPs, described as the Pareto-optimal set, because it is impossible to achieve the optimal states for all objectives simultaneously [21]. Evolutionary algorithms (EAs) can obtain multiple optimal solutions and reduce the total computation time. Hence, it is considered as a viable strategy for addressing multi-objective optimization problems [22,23,24]. The common EAs used for multi-objective optimization include Particle Swarm Optimization (PSO) [25], Simulated Annealing (SA) [26], Sand Cat Swarm Optimization (SCSO) [27] and so on. Among these evolutionary optimization algorithms, the Non-Dominated Sorting Genetic Algorithm-II (NSGA-II) [28] stands out as one of the most influential and advanced multi-objective genetic algorithms. This algorithm has become one of the most prospective algorithms in multi-objective optimization problems due to its simplicity and effectiveness [29,30]. The fast non-dominated sorting approach proposed with NSGA-II simplifies the complexity of the non-dominated order. With the incorporation of the crowding operator and elitist strategy, NSGA-II demonstrates perfect performance in multi-objective evolutionary optimization [31].

Conventionally, two types of models can be adopted as surrogate models in multi-objective optimization. The first type is theoretical-derivation-based mechanical models, which have the advantages of straightforward expressions and high computational efficiency. However, their prediction accuracies are usually low due to the difficulties of considering all the influencing factors and the introduction of simplifications when predicting complex structural behaviors [32]. The second type is based on numerical models such as a nonlinear finite elements analysis (FEA), which simulates the complex structural behavior and combines with optimization algorithms to obtain accurate optimization results. Steven et al. [33] developed the evolutionary structure optimization (ESO) method to optimize beams of a three-story RC frame structure. Li and Xie [34] developed the bidirectional evolutionary structural optimization (BESO) method for composite structures with varying material properties under tension and compression. However, the large computational requirement of FEA encounters some obstacles in MOP applications, especially for complex structures [35].

Machine learning (ML) approaches in structural performance prediction have been rapidly developed [36,37,38,39,40]. Due to the high prediction accuracy, good generalization performance and high efficiency, ML is particularly well suited for complex multi-objective optimization problems with the ability to achieve global optimization [41,42]. Therefore, an ML-surrogate-based multi-objective optimization design of FRP reinforced slabs is proposed in this study. The flowchart is given in Figure 2 as follows: firstly, this paper applies an advanced ML model, namely Natural Gradient Boosting (NGBoost), to predict the punching shear resistance of FRP reinforced flat slabs; secondly, the SHAP interpretable method is applied for influencing the factor analysis; thirdly, an NSGA-II-based multi-objective optimization for FRP reinforced flat slabs subjected to punching shear is proposed, in which the established NGBoost model is utilized as the surrogate model; and finally, the optimization results are examined to understand the impact of different design parameters. The innovations of this paper are as follows: 1, introduced innovative data-driven Natural Gradient Boosting (NGBoost) for predicting punching shear resistance of FRP reinforced flat slabs; 2, integrated an NGBoost-based surrogate model with Non-Dominated Sorting Genetic Algorithm-II (NSGA-II) for efficient multi-objective optimization, which obtains the Pareto-optimal set of punching shear resistance and cost for FRP reinforced flat slabs.

2. ML Model for Punching Shear Resistance Prediction

2.1. Experimental Dataset

A punching shear experimental dataset of 154 FRP reinforced slabs was gathered from relevant papers, which covers almost all currently published test data of FRP reinforced slabs. The complete dataset is provided in Appendix A. Through literature review [12,43,44], the shape of the column ($s$), area of the column ($A$), effective depth of the slab ($h_0$), cylinder concrete compressive strength ($f_c^{\prime }$), FRP reinforcement’s elasticity modulus ($E_f$) and FRP reinforcement ratio (${\rho }_f$) are identified as the primary factors that influence the punching shear resistance of FRP reinforced concrete slabs ($V$). The statistical values of the six input parameters and one output parameter are presented in

Table 1. Statistical values of dataset.

Design Parameters	$\mathit{s}$	$\mathit{A}/{\mathbf{cm}}^2$	${\mathit{h}}_0/\mathbf{mm}$	${\mathit{f}}_{\mathit{c}}^{\prime }/\mathbf{MPa}$	${\mathit{E}}_{\mathit{f}}/\mathbf{GPa}$	${\mathit{\rho }}_{\mathit{f}}/\mathit{\%}$	$\mathit{V}/\mathbf{kN}$
Maximum	3	2025	284	118	230	3.76	1600
Minimum	1	4.91	45	22.16	28.4	0.15	45
Mean	/	740.65	164.50	43.51	71.15	0.98	365.96
Standard deviation	/	489.54	119.5	16.38	36.37	0.61	267.65

s = 1, 2 and 3 represent the shape of a column as square, circular and rectangular, respectively.

. Figure 3 shows the distribution of six input parameters and one output parameter. It can be seen that all parameters except $s$ exhibit continuous distribution characteristics, which verifies that the samples selected in this paper have no significant deviation and are representative. The data size employed for ML modeling adheres to a widely recognized consensus, which asserts that the quantity of data should exceed 10 times the number of input parameters [45]. In this study, the dataset contains 154 experimental results and six input parameters; hence, the data size meets the requirement of ML modeling.

By splitting the dataset into a training set and a testing set with an 8:2 ratio, it ensures an adequate number of training sets for model training and learning; meanwhile, the testing set is utilized to assess performance on new data and validate generalization ability of the model. From Table 1, noticeable variations can be observed in the numerical values of the different input parameters. Therefore, it becomes essential to normalize the dataset by scaling the input parameters within the range of [0, 1] to strengthen the convergence and precision of the model. The normalization is represented as

$$x_i^{\ast }=\frac{x_i-\underset{1\le j\le m}{\min }\left\{x_j\right\}}{\underset{1\le j\le m}{\max }\left\{x_j\right\}-\underset{1\le j\le m}{\min }\left\{x_j\right\}}$$

(1)

where $x$ represents the parameter before processing; $x^{\ast }$ represents the normalized parameter; $m$ stands for the sample sizes; $i$ is the $i$-th input parameter.

2.2. Natural Gradient Boosting (NGBoost)

NGBoost is a newly developed boosting approach [46], which applies a natural gradient-based boosting method composed of three parts: base learners, probability distribution types and scoring rules (Figure 4). This method directly acquires the full probability distribution in the output space, enabling probability prediction to measure uncertainty. It is shown that NGBoost provides eminent prediction performance in both deterministic and stochastic problems [47,48]. Figure 5 depicts the modeling process of NGBoost.

In Section 2.1, the dataset for modeling is preliminarily established. The NGBoost algorithm is a derivative of the gradient boosting framework, which is an ensemble learning method that combines multiple base learners together to achieve better predictive performance. A decision tree is the common choice for the base learner, because it can effectively adapt to complex data patterns and improve predictive performance through the ensemble learning [49]. Unlike previously proposed ML methods, NGBoost can further obtain the probability distribution of the predicted results by utilizing a natural gradient instead of a conventional gradient. The definition of the natural gradient ${\tilde{\nabla }}_{\theta }$ is shown in Equation (2):

$$\begin{array}{c}{\tilde{\nabla }}_{\theta }S\left(\theta ,y\right)=\underset{\epsilon \to 0}{\lim }\underset{d:D_S\left(P_{\theta },P_{\theta +d}\right)=\epsilon }{are\max }S\left(\theta +d,y\right)\\ ={\mathcal{I}}_S{\left(\theta \right)}^{-1}{\nabla }_{\theta }S\left(\theta ,y\right)\end{array}$$

(2)

where $S$ represents the scoring rule; $D_S\left(P_{\theta },P_{\theta +d}\right)$ is the divergence of the scoring rule; $P_{\theta }=P\left(\theta |x,y\right)$ is the conditional probabilistic distribution that represents the prediction $y$ for a new input $x$; ${\mathcal{I}}_S\left(\theta \right)$ is the Riemannian metric of the statistical manifold at parameter set $\theta$ [46].

One of the commonly used scoring rules is the maximum likelihood estimation (MLE), which is more suitable for probability predictions in complex resistance mechanisms compared to another commonly used scoring rule called the continuous ranked probability score (CPRS) [50]. The specific expression of the MLE scoring rule is shown in Equation (3) as

$$S\left(\theta ,y\right)=\mathcal{L}\left(\theta ,y\right)=-\log P\left(\theta |x,y\right)$$

(3)

where $\mathcal{L}$ represents the calculation of the MLE scoring rule.

All appropriate scoring rules should satisfy the following equation:

$$S\left(\theta ,y\right)=\mathcal{L}\left(\theta ,y\right)=-\log P\left(\theta |x,y\right)$$

(4)

Therefore, the Kullback–Leibler divergence (KL divergence, i.e., $D_{KL}$) of the MLE scoring rule can be obtained based on Equation (4) [51]:

$$\begin{array}{lll}{D}_{\mathcal{L}}\left(P_{\theta },P_{\theta +d}\right)& =& E_{y\sim P_{\theta }}\left[\mathcal{L}\left(P_{\theta +d},y\right)\right]-E_{y\sim P_{\theta }}\left[\mathcal{L}\left(P_{\theta },y\right)\right]\\ & =& E_{y\sim P_{\theta }}\left(\log \frac{P_{\theta }}{P_{\theta +d}}\right)\\ & =:& D_{KL}\left(P_{\theta },P_{\theta +d}\right)\end{array}$$

(5)

Utilizing the new score and the gradient rule, multiple base learners can be trained through the gradient boosting framework to fit the parameters. The following text provides a description of the precise iteration procedure used with NGBoost. The learning algorithm first estimates a common initial distribution that minimizes the scoring rule $S$ on the sum of response variables for all training samples and becomes the initial predicted parameter ${\theta }^{\left(0\right)}$. At iteration $r$, a set of base learners $f^{\left(r\right)}$ takes input $x$ for predicting the output, which is then scaled based on the stage-specific scaling factors ${\rho }^{\left(r\right)}$ and a common learning rate $\eta$. As a result, after each iteration, the updated prediction parameter $\theta$ can be represented as

$$y\left|x\right.\sim P_{\theta }\left(x\right),\theta ={\theta }^{\left(0\right)}-\eta {\displaystyle \sum _{r=1}^R{\rho }^{\left(r\right)}}\cdot f^{\left(r\right)}\left(x\right)$$

(6)

2.2.1. Other ML Models for Comparison

This section briefly introduces three ML methods and their modeling processes. Their prediction results will be compared with NGBoost in Section 2.2.3.

Random Forest (RF) [52] is an ensemble algorithm, which is composed of several base learners (Figure 6). The RF constructs multiple independent decision trees through random sampling and executes them in parallel to obtain prediction results.

Adaptive Boosting (AdaBoost) [53] is also an ensemble learning algorithm (Figure 7). Firstly, the weight distribution of the training set is initialized. Then, the error rate and weight of each base learner are calculated, and the weight of the training samples is updated, until $n$ base learners are trained. Finally, the prediction model is gained by combining strategies for prediction.

The support vector regression (SVR) [54] model can be simply understood that a spacer band $\epsilon$ is created on the two sides of the linear function and samples falling inside are not taken into account when calculating loss (Figure 8). By minimizing the total loss and maximizing the interval, an optimized model can be obtained.

2.2.2. Model Evaluation

To assess the prediction performance of the proposed models, three evaluation indexes, namely RMSE, MAE and R², are selected in

Table 2. Model evaluation indexes.

Model Evaluation Index	Abbreviation	Equation	Explanations
Root Mean Square Error	RMSE	$\mathrm{RMSE}=\sqrt{\frac{1}{m}{\displaystyle \sum _{i=1}^m{\left(\stackrel{\wedge }{y_i}-y_i\right)}^2}}$	When the predicted value and the true value are in exact agreement, the value of RMSE/MAE is 0, indicating an excellent model. The value of RMSE/MAE rises as the error does as well.
Mean Absolute Error	MAE	$\mathrm{MAE}=\frac{1}{m}{\displaystyle \sum _{i=1}^m\left|\stackrel{\wedge }{y_i}-y_i\right|}$
Coefficient of Determination	R2	${\mathrm{R}}^{{}^2}=1-\frac{{\displaystyle \sum _i{\left(\stackrel{\wedge }{y_i}-y_i\right)}^2}}{{\displaystyle \sum _i{\left(\stackrel{-}{y_i}-y_i\right)}^2}}$	The R2 value, which ranges from 0 to 1, is a measure of the model’s goodness fit. The larger the value, the better the model’s fit.

$m$ stands for the sample size, $\stackrel{\wedge }{y_i}$ represents the predicted value of the sample, $\stackrel{-}{y_i}$ represents the mean value of all samples and $y_i$ represents the true value of the sample.

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2.2.3. Prediction Results

In the case of ML models, the goal of model training is to identify the optimal hyper-parameter within the provided search space. To achieve this objective, the search spaces for four ML models (i.e., SVR, RF, Adaboost and NGBoost) are presented in

Table 3. Hyperparameters of 4 ML models.

ML Models	Optimal Hyper-Parameter
NGBoost	Base = default_tree_learner, Dist = Nomal, Score = MLE learning_rate = 0.2, Estimators = 300
AdaBoost	Max_depth = 5, n_estimators = 76, learning_rate = 0.85
RF	n_estimators = 20, min_samples_split = 2, min_samples_leaf = 1, n_jobs = 1
SVR	Epsilon = 1.26, C = 9923443, Gamma = 10.8

. The value range in this table encompasses the potential values for each hyper-parameter. In order to meticulously construct a prediction model with satisfactory performance, the grid searching method is employed to determine the appropriate hyper-parameters for each ML model. Additionally, a 10-fold cross-validation approach is utilized to evaluate the outcomes of this selection process.

To conduct a more comprehensive evaluation of NGBoost’s prediction accuracy, four empirical models, containing two design codes and two theoretical models, were selected for the comparison. The specifics of the empirical models are displayed in

Table 4. Formulas of calculation of punching shear capacity.

Empirical Models	Formulas
ACI 440.15R-15 [55]	$V_1=0.8\sqrt{f_c^{\prime }}{b}_{0,0.5h_0}\left(u{h}_0\right),u=\sqrt{2{\rho }_f{d}_f+{\left({\rho }_f{d}_f\right)}^2}-{\rho }_f{d}_f,E_c=4700\sqrt{f_c^{\prime }},d_f=\frac{E_f}{E_c}$
JSCE [56]	$V_2={\beta }_{h_0}{\beta }_e{\beta }_r{f}_{pcd}{b}_{0,0.5h_0}{h}_0/{\gamma }_b, {\beta }_e=\sqrt[3]{\left(100{\rho }_f{E}_f/E_s\right)}\le 1.5$ ${\beta }_{h_0}={\left(1000/h_0\right)}^{1/4}\le 1.5, f_{pcd}=0.2{\left(f_c^{\prime }\right)}^{1/2}\le 1.2\mathrm{MPa}, {\beta }_r=1+1/\left(1+0.25b_{0,0.5h_0}/h_0\right)$
El-Ghandour et al. [57]	$V_3=0.33\sqrt{f_c^{\prime }}\times \sqrt[3]{\left(E_f/E_s\right)}{b}_{0,0.5h_0}{h}_0$
Ospina et al. [17]	$V_4=2.77\times \sqrt[3]{\left({\rho }_f{f}_c^{\prime }\right)}\sqrt{\frac{E_f}{E_s}}{b}_{0,0.5h_0}{h}_0$

$b_{0,0.5h_0}$ is the perimeter of the critical section for slabs and footings at a distance of $0.5h_0$ away from the column face; $b_{0,1.5h_0}$ is the perimeter of the critical section for slabs and footings at a distance of $1.5h_0$ away from the column face; $E_s$ is the elastic modulus of steel, with a value of 200 GPa; ${\gamma }_b$ is the safety factor, with a value of 1.3.

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Table 5. Prediction results of models.

ML	Training Set			Testing Set			Complete Dataset
	RMSE	MAE	R2	RMSE	MAE	R2	RMSE	MAE	R2
NGBoost	12.35	6.94	0.99	71.89	53.90	0.93	34.42	16.81	0.98
RF	34.47	22.61	0.98	103.16	74.26	0.85	55.69	34.99	0.96
AdaBoost	32.71	25.20	0.98	113.06	80.13	0.82	60.25	36.02	0.95
SVR	21.18	11.97	0.99	152.91	104.41	0.66	71.12	30.60	0.93
ACI 440.15R-15 [55]	250.84	194.49	0.09	278.02	214.24	0.01	256.54	198.47	0.08
JSCE [56]	152.72	111.74	0.66	165.95	121.24	0.65	155.48	113.65	0.66
El-Ghandour [57]	147.27	105.86	0.59	101.86	71.73	0.80	139.32	98.91	0.73
Ospina [17]	129.23	84.74	0.81	125.48	74.37	0.78	128.48	82.65	0.77

and Figure 9 demonstrate the prediction results of four ML models and four empirical models. It is depicted in Figure 9 that the prediction results generated with RF, Adaboost and NGBoost demonstrate good agreements with the experimental punching shear resistance; all the dots are closely distributed around the best fitting line. It is revealed in Table 5 that the NGboost model has the highest prediction accuracy among RF, Adaboost and NGBoost. In contrast, the SVR model performs poorly on the test set with an R² of 0.66. Additionally, it can be indicated that the Ospina et al. [17] prediction accuracy is the highest among empirical models with an RMSE, MAE and R² of 128.48, 82.65 and 0.77, respectively. However, the predicted accuracy is still lower than the NGBoost model, for which the RMSE, MAE and R² are 34.42, 16.81 and 0.98, respectively. Therefore, NGBoost demonstrates superior performance in predicting the FRP reinforced slabs’ punching shear capacity compared to all the selected ML and empirical models.

2.3. Parameter Sensitivity Analysis Based on SHAP

ML is commonly described as a black-box model that lacks the ability to elucidate the impact of input parameters on output parameters [58,59]. Therefore, this article introduces the Shapley Additive Explanation (SHAP) model for a sensitivity analysis of parameters, which helps explain the behavior of the NGBoost model, and identify which input parameters need attention and adjustment to improve model performance.

SHAP is an additive explanation model inspired by cooperative game theory [60]. The fundamental concept of SHAP is to compute input variables’ marginal contribution, providing explanations for the ‘black-box model’ from both global and local perspectives [61]. The expression of SHAP can be given as

$$y_{pred}^{\left(i\right)}=y_{base}+{\displaystyle \sum _j^{n^{\prime }}f\left(x_{ij}\right)}$$

(7)

where $f\left(x_{ij}\right)$ is the SHAP value of $x_{ij}$, $n^{\prime }$ stands for the size of input variables; $y_{base}$ is the baseline value of the entire model; $y_{pred}^{\left(i\right)}$ is the predicted value of sample $x_i$. Intuitively, $f\left(x_{i1}\right)$ reflects the first input parameter’s contribution to the prediction value $y_{pred}$ in the $x_i$. If $f\left(x_{i1}\right)>0$, it means that the predicted value is positively impacted by the input parameter and vice versa. The major benefit of the SHAP value lies in its ability to indicate the effect of input parameters, both positively and negatively, thus improving the interpretability of the model. Moreover, SHAP provides powerful data visualization functions to display the interpretation results of the prediction, making important contributions to explain complex ML models.

The parameter sensitivity analysis of FRP reinforced flat slabs is given as follows: (1) among all input parameters, the slab’s effective depth $h_0$ has the greatest influence, far exceeding other influencing factors; (2) the second-ranked factor is the area of column $A$, with an importance of about 0.38% of $h_0$; (3) the following influential factors are FRP reinforcement ratio ${\rho }_f$, compressive strength of concrete $f_c^{\prime }$, elasticity modulus of FRP reinforcement $E_f$; (4) the shape of column $s$ has the minimal influence.

The impact range and distribution of input parameters on the FRP reinforced flat slabs’ punching shear capacity are displayed in Figure 10b. The x-axis represents the SHAP value observed using each parameter in a single sample in the dataset. Each point is colored from blue to red to indicate the magnitude of numerical changes from small to large. The parameters are listed on the y-axis in descending order of their significance in influencing the outcome. In Figure 10b, it is shown that when the $h_0$ reaches 400 or above, it almost solely determines the FRP reinforced flat slabs’ punching shear capacity.

3. Multi-Objective Optimization of FRP Reinforced Flat Slabs

3.1. NSGA-II Optimization

Non-Dominated Sorting Genetic Algorithm II (NSGA-II) [28] is a multi-objective genetic algorithm suitable for dealing with multi-objective optimization problems. It incorporates fast non-dominated sorting, crowding distance comparison and an elitist selection strategy into traditional genetic algorithms. The algorithm follows these steps: firstly, a random initial population $N$ is created, and the initial generation offspring population is generated via selection, crossover and mutation operations; secondly, from the second generation onwards, the parent population and the offspring population are combined, and fast non-dominated sorting and crowding distance calculation are performed; thirdly, individuals suitable for selection are chosen to develop a new parent population based on non-dominated relationship and fitness sharing; finally, a new offspring population is created through basic genetic operations. This process is repeated until the termination condition is satisfied. The flowchart of this algorithm is depicted in Figure 11.

3.1.1. Fast Non-Dominated Sorting Approach

In MOPs, dominance is used to describe a solution that is not inferior to another solution in all objective functions. Throughout the population, individuals without dominant solutions are placed in set Rank 1, and then individuals in Rank 1 are removed from the set. The remaining individuals without dominant solutions are placed in set Rank 2 and so forth, until the population level is completely divided. Through fast non-dominated sorting, solutions can be brought nearer to the Pareto-optimal front (Figure 12).

3.1.2. Crowded-Comparison Approach

To achieve a more even distribution of obtained solutions in the objective space and cover the Pareto-optimal front as much as possible, the crowded-comparison approach is introduced. The individual crowding distance is illustrated in Figure 13, and the specific expression is shown in Equation (8):

$$i_d={\displaystyle {\sum }_{j=1}^{m_j}\left(\left|f_j^{i+1}-f_j^{i-1}\right|\right)}$$

(8)

where $i_d$ represents the crowding distance of point $i$, $f_j^{i+1}$ represents the value of the objective function $j$ at point $i+1$ and $f_j^{i-1}$ represents the value of the objective function $j$ at point $i-1$; $m_j$ represents the number of objective functions.

3.1.3. Elitist Strategy

An elitist strategy preserves excellent solutions to accelerate the convergence speed of the algorithm. To begin with, the new population $Q_t$ generated in $t$ is combined with the parent population $P_t$ to form the population $R_t$. Then, non-dominated sorting is performed and non-dominated sets $Z_i$ are filled in new parent population $P_{t+1}$ in rank order until the population size exceeds $N$. At this time, the crowding distance comparison operator is used to ensure that the population size in $P_{t+1}$ remains at $N$. Lastly, a new offspring population $Q_{t+1}$ is created through selection, crossover and mutation (Figure 14).

3.2. Optimization Design of FRP Reinforced Flat Slabs

As mentioned earlier, the high-precision NGBoost prediction model has been successfully established and validated. Subsequently, this model will be used as a surrogate model for NSGA-II to conduct the optimization design for FRP reinforced flat slabs.

We took a set of experimental data [62] from the dataset as an example to conduct structural optimization design. To ensure that the optimized results meet the practical engineering’s requirements, Chinese design code (GB 50010-2010 (2015)) [63] is applied in this paper for parameter value range determination. The range of the parameters are selected as follows: $s=1,2,3$, $100 {\mathrm{cm}}^2\le A\le 400 {\mathrm{cm}}^2$, $100 \mathrm{mm}\le h_0\le 200 \mathrm{mm}$, $0.2\le {\rho }_f\le 1.0$. Since the design parameters such as the size and depth of the slab and columns are the main consideration in this study, the material parameters of the concrete compressive strength and elasticity modulus of FRP reinforcement are chosen as constants as $f_c^{\prime }=28.32\mathrm{MPa}$ and $E_f=45.6\mathrm{MPa}$. The punching shear resistance obtained with the NGBoost model is considered as the objective function $f_1$, while the material cost of the FRP reinforced slabs is deemed as function $f_2$. The calculation of objective function $f_2$ is shown in Equation (9) as

$$f_2\left({\mathrm{cost}}_{FRP}\right)=A\cdot \sqrt{A}\cdot {10}^{-6}\cdot {\mathrm{cost}}_1+1.5\times 1.5\cdot h_0\cdot {\mathrm{cost}}_1+1.5\times 1.5\times h_0\cdot \gamma \cdot {\rho }_f\cdot {10}^{-3}\cdot {\mathrm{cost}}_2$$

(9)

where ${\mathrm{cost}}_1$ represents the price of concrete, with the price of approximately Chinese $\mathrm{RMB} 500{/\mathrm{m}}^3$; $\gamma$ denotes the density of FRP reinforcement ranging from $1250\sim {2100 \mathrm{kg}/\mathrm{m}}^3$ and ${2100 \mathrm{kg}/\mathrm{m}}^3$ is chosen as the representative; ${\mathrm{cost}}_2$ represents the price of FRP reinforcement, which is about 8–12 times that of steel bars with a price of approximately $\mathrm{RMB} \text{40,000}/\mathrm{t}$. The optimization design results of the selected FRP reinforced slab obtained by using NSGA-II are manifested in Figure 15.

Observing Figure 15, two objective functions, namely punching shear resistance and cost, can be found with a wide range of values, indicating that the optimization model has good generalization performance. In addition, all FRP reinforced slab data points are located above the Pareto-optimal set, suggesting that in the dataset, the material cost is higher than the optimization results of the optimization model. Therefore, the proposed optimization framework is effective in reducing the material cost of FRP reinforced slab design. In addition to obtaining the objective function values $f_1$ and $f_2$ of the Pareto solutions, the NSGA-II algorithm can also provide corresponding optimal parameter combination. In order to clearly illustrate the optimized design parameters and the resulting optimizations, three sets of samples among the Pareto solutions of Figure 15 are depicted in

Table 6. Optimization samples.

Pareto Solutions	$\mathbf{Objective} \mathbf{Function} \left[{\mathit{f}}_1,{\mathit{f}}_2\right]$	$\mathbf{Parameters} \left[\mathit{s},\mathit{A},{\mathit{h}}_0,{\mathit{f}}_{\mathit{c}}^{\prime },{\mathit{E}}_{\mathit{f}},{\mathit{\rho }}_{\mathit{f}}\right]$
1	[180, 149]	[1, 240, 131, 28.32, 45.6, 0.20]
2	[467, 262]	[1, 165, 168, 28.32, 45.6, 0.23]
3	[719, 427]	[1, 171, 170, 28.32, 45.6, 0.74]

.

Figure 16 shows the impact of different parameter changes on Pareto solutions. As represented in Figure 16, within the same price range (y-axis), the slab’s effective depth has the most significant influence on enhancing the punching shear resistance among the design parameters. When the cost ranges from RMB 150 to RMB 250, the increase in the area of the column and the reinforcement ratio increases the punching shear capacity by about 60 kN, while the increase in the slab’s effective depth can increase the punching shear capacity by above 100 kN. But when the cost ranges from RMB 250 to RMB 350, the rise in the reinforcement ratio only provides a limited improvement of about 40 kN. In comparison, it can be seen that increasing the slab’s effective depth is the most efficient method to balance economic effects and structural resistance requirements.

The analysis of optimization results reveals which design parameter has the greatest impact on the objective function, as well as which solution can meet the engineering design requirements. NSGA-II-based optimization results help to adjust the engineering design plan to achieve the optimal design goal. The design plan with the lowest cost and best performance can be chosen from the optimized results, or the design plan with the lowest cost can be selected while meeting the reliability requirements. By examining the impact of variations in design parameters on the results of NSGA-II, it is demonstrated that augmenting the effective depth of the slab is a feasible approach to substantially improve the punching shear resistance of flat slabs reinforced with FRP.

4. Conclusions

This paper introduces machine learning-based multi-objective optimization of FRP reinforced flat slabs subjected to punching shear, offering a new perspective on solving the multi-objective structural design optimization problems. Firstly, the NGBoost method is introduced to accurately predict the punching shear capacity of FRP reinforced slabs. Subsequently, the NGBoost model is served as the surrogate model in the multi-objective optimization. Finally, the NSGA-II algorithm is employed for multi-objective structural design optimization. Based on the ML prediction and multi-objective optimization results, conclusions can be summarized as follows:

(1)

The proposed NGBoost model has superior prediction performance in predicting the punching shear resistance of FRP reinforced slabs, which shows the highest R² and lowest RMSE and MAE errors among the selected four ML models and four empirical models. The exceptional accuracy exhibited with NGBoost predictions renders it a highly advantageous surrogate model for multi-objective optimization applications.

(2)

According to the analysis provided with SHAP, the slab’s effective depth ($h_0$) has the greatest influence on the prediction results among all the six input parameters, followed by the FRP reinforcement ratio (${\rho }_f$), the area of the column ($A$), the compressive strength of concrete ($f_c^{\prime }$) and the elasticity modulus of the FRP reinforcement ($E_f$), while the shape of the column ($s$) has the least effect.

(3)

The proposed NSGA-II algorithm, which combines NGBoost as a surrogate model, achieves balance between punching shear capacity and cost. Through an analysis of the effects of design parameter variations on the multi-objective optimization results, it is shown that increasing the slab’s effective depth is a viable strategy for significantly enhancing the punching shear resistance of FRP reinforced flat slabs.