# Compressive Strength Prediction of Self-Compacting Concrete-A Bat Optimization Algorithm Based ANNs

Authors: Amir Andalib; Babak Aminnejad; Alireza Lork
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

This article examines the feasibility of using bat-trained artificial neural networks (ANNs) to predict the compressive strength of self-compacting concrete (SCC). The nonlinear behavior of SCC challenges traditional modeling techniques. Therefore, this work takes advantage of the superior predictive performance of ANNs coupled with the bat algorithm. A database of 205 SCC samples collected from the literature is used to develop the ANN model. The correctness of the bat-based neural network model is then substantiated by contrasting its performance with that of the particle swarm optimization and teaching-learning-based optimization algorithms employed to train a neural network model. The statistical indices indicate the superior performance of the bat-based ANN model. In addition, a sensitivity analysis was carried out to determine the effects of various input parameters on the compressive strength of SCC.


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## Body

## 1. Introduction

Developed in Japan in 1988, self-compacting concrete (SCC) flows under its weight, solidifies inside the formwork, and is easier to produce. However, SCC can be costly, with an estimated cost of around two to three times the cost of conventional concrete. Researchers have utilized various admixtures such as fly ash, limestone filler, and ground clack brick to lower the cost of production [1]. Because of the intricate interaction of concrete mixture proportions with the mechanical and rheological properties of SCC [2], and to facilitate the time-consuming process of determining the optimum mix design, researchers have recommended several treatments in the literature; for instance, the rheological model [3], multivariate linear regression analysis [4], and numerical methods [5].Numerous studies have been conducted on applying soft computing techniques, particularly artificial neural networks (ANNs), to examine self-compacting concrete properties. ANN models have been shown to be superior for determining the properties of fresh and hardened self-compacting concrete [1], compressive strength estimation of SCC [6], predicting the performance of self-compacting concrete mixtures [7], modeling the mechanical properties of fiber-reinforced SCC [8], approximating the strength of SCC containing polypropylene fiber and mineral additives [9], foreseeing the ingredients of SCC [10], assessment of compressive strength of self-compacting concrete with high volume fly ash [11], estimating the strength of SCC mixtures with mineral additives [12], forecasting the workability of SCC [13], approximating the strength of SCC containing bottom ash [14], estimating the initial and final setting times of SCC containing mineral admixtures [15], modeling the strength of admixture-based SCC [16], and estimating the properties of SCC containing fly ash [17].Ghorpade and Koneru [18] used the pattern recognition neural network to predict the compressive strength grades of self-compacting concrete (SCC) effectively with experimental data obtained from the laboratory. Moreover, in this study, they created pattern recognition neural network model to classify output as compressive strength grade of SCC from inputs [18]; Song et al. [19] investigated machine learning algorithms and supervised ML algorithms, such as decision tree (DT), artificial neural network (ANN), genetic engineering programming (GEP), and bagging regressor (BR) to predict the compressive strength of concrete with fly ash admixture [19]; Awoyera et al. [20], focused on modeling the strength characteristics of geopolymer self-compacting concrete GSCC by addition of mineral admixtures, by using both genetic programming (GEP) and the artificial neural networks (ANN) [20]; In this study, Koneru and Ghorpade [21] used artificial neural network (ANN) to predict output strength characteristics of experimentally based workable SCC mixtures mineral admixtures, namely, metakaolin, micro silica, and fly ash [21]; Pazouki et al. [22] used the firefly optimization algorithm (FOA) and a hybrid model of the radial basis function neural network (RBFNN) (FOA + RBFNN) for the prediction of the compressive strength of self-compacting concrete (SCC) containing CFFA [22]; Golafshani et al. [23] used adaptive neuro-fuzzy inference system (ANFIS) algorithms and artificial neural network to develop predictive models for the CS of normal concrete (NC) and high-performance concrete (HPC) containing fly ash and blast furnace slag hybridized with grey wolf optimizer [23]; Shariati et al. [24] used a metaheuristic algorithm known as grey wolf optimizer (GWO) combined with an extreme learning machine (ELM) and a novel hybrid ELM-GWO model to predict the compressive strength of concrete [24]; Jesika et al. [25] used eleven algorithms for the shear strength prediction of steel fiber reinforced concrete beams. In this study, data-driven 11 machine learning methods are employed to predict the shear capacity of SFRC beams. Reference [25]; Behnood and Golafshani [26] used ANN with an optimizer as multiobjective grey wolves (MOGW) to predict the mechanical properties of the silica fume concrete. In this study, estimating the compressive strength of silica fume concrete by using the ANN method was considered a two-objective optimization problem. These two objectives were the accuracy and the complexity of the developed ANN models [26]; Salimbahrami and Shakeri [27] used an artificial neural network (ANN), decision tree (DT), support vector machine (SVM), and linear regression (LR) algorithms for the prediction of concrete compressive strength [27]; Getahun et al. [28] used the artificial neural network (ANN) to predict the compressive strength and tensile strength of waste concrete incorporating rice husk ash (RHA) and reclaimed asphalt pavement (RAP) as partial replacements of Portland cement and virgin aggregates, respectively [28]; Ling et al. [29] used support vector machine (SVM) model optimized by using k-fold cross validation to predict the compressive strength of concrete in marine nature and compared the result with artificial neural network (ANN) and decision tree (DT) models [29]; Yaseen et al. used extreme learning machine (ELM) to predict the compressive strength of foamed concrete [30].Soft computing techniques offer abundant opportunities to improve model uncertainty. This study concentrates on enhancing the prediction accuracy of artificial neural network models by using the bat algorithm. To this end, the model was developed by using an experimental database of 205 samples gathered from the literature. This study focuses primarily on compressive strength estimation for SCCs. Section2 offers the necessary background on artificial neural networks and the bat optimization algorithm. Section 3 discusses the empirical model construction and learning process, while Sections 4 and 5 provide the discussion and results.


## 2. Background

### 2.1. Artificial Neural Networks

As nature-inspired computational models, artificial neural networks (ANNs) mimic the behavior of biological neurons. They are composed of entities called neurons that are arranged in layers. In a fully connected feed-forward artificial neural network, each neuron in a middle layer, called the hidden layer, is linked to nodes in the preceding and ensuing layers. The network input is fed to the input nodes of the input layer, and its neurons in the output layer determine the network output. The linkages between nodes have weights that signify the effect of the first neuron on the second neuron. Upon receiving a signal from connected neurons, a given neuron calculates its output by summing the weighted signals and applying a function entitled the activation function. The output of that neuron is then fed to neurons in subsequent layers, and so on. The value assigned to these weights ultimately determines the output and, therefore, the predictive accuracy of the network. In the feed-forward ANN, data are fed to neurons in the input layer; and then, by calculating their outputs, the neurons in the layer are fed to the neurons in the following layer. Neural networks must be trained using an optimization algorithm based on the training data. The training process constitutes adjusting the network’s weights to minimize the network’s predictive error on training data. Figure1 demonstrates a typical feed-forward ANN. This specific architecture (i.e., the configuration) is obtained later in this paper as the optimum architecture [31, 32].Figure 1
The optimal ANN identified in this study has 11 input nodes, 14 hidden layer nodes, and one output node in the output layer.The ANN models’ accuracy is judged based on their predictive error on datasets not met through the training phase. These data are called testing data. The division of data into training and testing subsets ensures the generalizability of the network and avoids a well-known phenomenon called overfitting. An ANN model is usually initialized by random weights and is then trained. Gradient-based and metaheuristic approaches are the two optimization algorithms utilized to train the network. The gradient-based methods use gradients to adjust the weight. However, they are quick and well-known for getting stuck in regional extrema. On the other hand, metaheuristic methods do not get trapped in regional extrema, but they are not as fast. They provide valuable solutions by exploring and exploiting the solution space; however, the result is not automatically the global extremum [33–35].


### 2.2. Bat Algorithm

Bats, or more specifically, microbats, can develop a three-dimensional model of their surroundings. Using the time difference between pulse emission and echo, bats use echolocation to detect prey and avoid obstacles. The bats’ fascinating behavior inspired Yang [36] to develop the bat algorithm. This algorithm idealizes the behavior of bats by assigning each bat in a population of bats, location xi, and velocity vi. The loudness of the signal ranges from A0 to Amin. The location vector of each bat denotes the parameters of the optimization problem. Initially, the bat algorithm generates a bat population by assigning them random locations. Then, it uses the following equation to determine the frequency of each bat j [37, 38]:(1)fj=fminimum+fmaximum−fminimumβ,where the minimum and maximum frequencies, fminimum and fmaximum, are among the algorithm’s hyperparameters. fj is the frequency of the jth bat, and β is a random number generally taken between zero and one. The bats’ location and velocity are then adjusted based on the following equations [39]:(2)Vjt+1=Vjt+xjt−x∗fj,xjt+1=xjt+Vjt+1,where xjt is the position of the jth bat in the epoch t, x∗ is the global best position, and Vjt is the velocity of the jth bat in the epoch t. A small number of bats are moved close to the global optimum by using [38, 39](3)xf=xi+εAt,where xi is the initial location, and xf is the final location. A is loudness, a hyperparameter of the algorithm, and ε is a random number between 0 and 1. The final location of a given bat is accepted only if the value of the cost function is smaller at the final location. The pulse rate and loudness of each bat are then updated by using [39](4)Ajt+1=αAjt,rjt+1=rj01−exp,−γt,where rj0 is the commencing pulse level, and α and γ are hyperparameters of the algorithm. The value of α is usually taken between zero and one.An ANN can be trained by using the bat optimization algorithm by treating the weights of a given network as the location vector of a bat. The predictive error of the ANN is defined as the cost function. The best performing bat is selected as the solution of the bat algorithm when the bat algorithm meets its stopping criterion by either completing a set number of iterations or ceasing to improve accuracy in multiple iterations in a row. The location vector of the bat represents the weights of the trained network [36, 38].


## 2.1. Artificial Neural Networks

As nature-inspired computational models, artificial neural networks (ANNs) mimic the behavior of biological neurons. They are composed of entities called neurons that are arranged in layers. In a fully connected feed-forward artificial neural network, each neuron in a middle layer, called the hidden layer, is linked to nodes in the preceding and ensuing layers. The network input is fed to the input nodes of the input layer, and its neurons in the output layer determine the network output. The linkages between nodes have weights that signify the effect of the first neuron on the second neuron. Upon receiving a signal from connected neurons, a given neuron calculates its output by summing the weighted signals and applying a function entitled the activation function. The output of that neuron is then fed to neurons in subsequent layers, and so on. The value assigned to these weights ultimately determines the output and, therefore, the predictive accuracy of the network. In the feed-forward ANN, data are fed to neurons in the input layer; and then, by calculating their outputs, the neurons in the layer are fed to the neurons in the following layer. Neural networks must be trained using an optimization algorithm based on the training data. The training process constitutes adjusting the network’s weights to minimize the network’s predictive error on training data. Figure1 demonstrates a typical feed-forward ANN. This specific architecture (i.e., the configuration) is obtained later in this paper as the optimum architecture [31, 32].Figure 1
The optimal ANN identified in this study has 11 input nodes, 14 hidden layer nodes, and one output node in the output layer.The ANN models’ accuracy is judged based on their predictive error on datasets not met through the training phase. These data are called testing data. The division of data into training and testing subsets ensures the generalizability of the network and avoids a well-known phenomenon called overfitting. An ANN model is usually initialized by random weights and is then trained. Gradient-based and metaheuristic approaches are the two optimization algorithms utilized to train the network. The gradient-based methods use gradients to adjust the weight. However, they are quick and well-known for getting stuck in regional extrema. On the other hand, metaheuristic methods do not get trapped in regional extrema, but they are not as fast. They provide valuable solutions by exploring and exploiting the solution space; however, the result is not automatically the global extremum [33–35].


## 2.2. Bat Algorithm

Bats, or more specifically, microbats, can develop a three-dimensional model of their surroundings. Using the time difference between pulse emission and echo, bats use echolocation to detect prey and avoid obstacles. The bats’ fascinating behavior inspired Yang [36] to develop the bat algorithm. This algorithm idealizes the behavior of bats by assigning each bat in a population of bats, location xi, and velocity vi. The loudness of the signal ranges from A0 to Amin. The location vector of each bat denotes the parameters of the optimization problem. Initially, the bat algorithm generates a bat population by assigning them random locations. Then, it uses the following equation to determine the frequency of each bat j [37, 38]:(1)fj=fminimum+fmaximum−fminimumβ,where the minimum and maximum frequencies, fminimum and fmaximum, are among the algorithm’s hyperparameters. fj is the frequency of the jth bat, and β is a random number generally taken between zero and one. The bats’ location and velocity are then adjusted based on the following equations [39]:(2)Vjt+1=Vjt+xjt−x∗fj,xjt+1=xjt+Vjt+1,where xjt is the position of the jth bat in the epoch t, x∗ is the global best position, and Vjt is the velocity of the jth bat in the epoch t. A small number of bats are moved close to the global optimum by using [38, 39](3)xf=xi+εAt,where xi is the initial location, and xf is the final location. A is loudness, a hyperparameter of the algorithm, and ε is a random number between 0 and 1. The final location of a given bat is accepted only if the value of the cost function is smaller at the final location. The pulse rate and loudness of each bat are then updated by using [39](4)Ajt+1=αAjt,rjt+1=rj01−exp,−γt,where rj0 is the commencing pulse level, and α and γ are hyperparameters of the algorithm. The value of α is usually taken between zero and one.An ANN can be trained by using the bat optimization algorithm by treating the weights of a given network as the location vector of a bat. The predictive error of the ANN is defined as the cost function. The best performing bat is selected as the solution of the bat algorithm when the bat algorithm meets its stopping criterion by either completing a set number of iterations or ceasing to improve accuracy in multiple iterations in a row. The location vector of the bat represents the weights of the trained network [36, 38].


## 3. Methods and Materials

The general outline of the approach taken in this work is given in Figure2. This section details the components shown in Figure 2.Figure 2
Outline of the paper.


### 3.1. Dataset

The dataset used in this work was published by Asteris and Kolovos [2] on self-compacting concrete (SCC) from diverse and distinctive sources. It encompasses 205 SCC compressive strength results [40–47]. The 11 influencing parameters of the compressive strength of self-compacting concrete are cement (C), limestone powder (LP), fly ash (F), ground granulated blast furnace slag (GGBFS), silica fume (SF), rice husk ash (RHA), coarse aggregate (CA), fine aggregate (FA), water (W), new generation superplasticizers (SP) as chemical admixtures, and viscosity modifying admixtures (VMA). The target quantity is the 28-day compressive strength of the self-compacting concrete specimens [2, 48]. The compressive strength is measured in MPa, and the inputs are in kg/m3. Table 1 shows the descriptive statistics of the gathered Figure 3 dataset.Table 1
Descriptive attributes of the collected dataset.
Input factorMinimumMaximumMeanStandard deviationMedianRangeCement110600349.2293.43337.5490Limestone Powder027225.6760.780272Fly ash0440106.3694.01110440GGBS033017.3952.010330Silica fume025014.9133.450250RHA02006.5524.290200Coarse Aggregate5001600772.35175.36768.881100Fine Aggregate3361135827.93144.33836799Water94.5250179.2727.65180155.5SP022.55.964.355.9722.5VMA01.230.140.3101.23Figure 3
Shows the histogram for specimens’ 28-day compressive strength.The correlation matrix of the input variables is portrayed in Figure4. The inputs do not highly correlate, and initially, these parameters seem adequate for further analysis.Figure 4
Correlation matrix of experimental data.Divergence and long training time can arise from feeding the ANN with input variables with various ranges [34]. Input and output data were normalized into a span of −1 to 1 utilizing (6) to mitigate the adverse effects [48](5)Yn=2Y−YminYmax−Ymix−1,where Yn is normalized quantity, Ymax denotes maximum, and Ymin denotes the minimum quantity. Y denotes the initial “unnormalized” value of the parameter. The minimum and maximum values for each parameter are the ones given in Table 1. It is noteworthy that since ANN is trained on normalized data, to utilize its model, it is necessary to provide the ANN with normalized values of parameters and “unnormalize” the network’s outputs [32].


### 3.2. Performance Measures

Performance measures will ultimately be employed to determine the predictive accuracy of the models. The goal is to choose the model with the most predictive accuracy while demonstrating high generalizability. Therefore, the descriptive characteristics utilized to gauge the error metrics of different models are defined. They are mean error (ME), mean absolute error (MAE), mean squared error (MSE), root mean squared error (RMSE), average absolute error (AAE), model efficiency (EF), and variance account factor (VAF) defined as [49](6)ME=1n∑i=1nPi−Oi,MAE=1n∑i=1nPi−Oi,MSE=1n∑i=1nPi−Oi2,RMSE=1n∑i=1nPi−Oi21/2,AAE=∑i=1nOi−Pi/Oin×100,EF=1−∑i=1nPi−Oi2∑i=1nO¯i−Oi2,VAF=1−varOi−PivarOi×100.


### 3.3. Experimental Model Formation Utilizing Neural Networks and the BAT Algorithm

As stated in Section3.1, 11 parameters affect the compressive strength of self-compacting concrete (SCC). So, the artificial neural network (ANN) models will have 11 nodes in the entry layer and one node in the output layer, as shown in Figure 1. As detailed in Section 2.1, the overfitting phenomenon can lead to ANN models with small error values in training but unacceptable generalizability. Therefore, to mitigate the adverse effects of this phenomenon, as advocated in previous studies [32, 35], the dataset was randomly split into two groups: 70% (144 samples) for training and 30% (61 samples) for testing data. This led to choosing the generalizable networks.The layer count and the overall number of nodes in an ANN’s hidden layer are problem-dependent [37]. Therefore, a trial-and-error method was taken. As a rule of thumb, the total count of hidden layer nodes of the network, NH, must satisfy the following inequality [50]:(7)NH≤2NI+1,where NI denotes the count of input nodes. Given that the number of affecting factors is 11, the count of hidden layer nodes is recommended to be less than 23. Various designs with one and two hidden layers and a maximum of 23 neurons were trained. A total of 155 distinct designs were trained, as shown in Table 2. The tan h function was chosen as the hidden layer’s transfer function and the identity function as the output layer’s.Table 2
Various configurations of trained ANNs.
NumTopology (2L)NumTopology (2L)…NumTopology (2L)NumTopology (1L)11-1122-1…12212-1133121-2132-2…12312-2134231-3142-3…12412-3135341-4152-4…12512-41364………………………101-10212-10…13112-1015422111-11222-11…13212-1115523MATLAB R2021, a [51] software was employed to develop the ANNs and bat optimization procedure. The optimum values of hyperparameters for the bat optimization utilized to learn the models are provided in Table 3.Table 3
Hyperparameters of bat optimization.
HyperparameterValuePopulation size100Loudness0.9Minimum frequency0Alpha0.99Max generations200Pulse rate0.5Maximum frequency2Gamma0.01


## 3.1. Dataset

The dataset used in this work was published by Asteris and Kolovos [2] on self-compacting concrete (SCC) from diverse and distinctive sources. It encompasses 205 SCC compressive strength results [40–47]. The 11 influencing parameters of the compressive strength of self-compacting concrete are cement (C), limestone powder (LP), fly ash (F), ground granulated blast furnace slag (GGBFS), silica fume (SF), rice husk ash (RHA), coarse aggregate (CA), fine aggregate (FA), water (W), new generation superplasticizers (SP) as chemical admixtures, and viscosity modifying admixtures (VMA). The target quantity is the 28-day compressive strength of the self-compacting concrete specimens [2, 48]. The compressive strength is measured in MPa, and the inputs are in kg/m3. Table 1 shows the descriptive statistics of the gathered Figure 3 dataset.Table 1
Descriptive attributes of the collected dataset.
Input factorMinimumMaximumMeanStandard deviationMedianRangeCement110600349.2293.43337.5490Limestone Powder027225.6760.780272Fly ash0440106.3694.01110440GGBS033017.3952.010330Silica fume025014.9133.450250RHA02006.5524.290200Coarse Aggregate5001600772.35175.36768.881100Fine Aggregate3361135827.93144.33836799Water94.5250179.2727.65180155.5SP022.55.964.355.9722.5VMA01.230.140.3101.23Figure 3
Shows the histogram for specimens’ 28-day compressive strength.The correlation matrix of the input variables is portrayed in Figure4. The inputs do not highly correlate, and initially, these parameters seem adequate for further analysis.Figure 4
Correlation matrix of experimental data.Divergence and long training time can arise from feeding the ANN with input variables with various ranges [34]. Input and output data were normalized into a span of −1 to 1 utilizing (6) to mitigate the adverse effects [48](5)Yn=2Y−YminYmax−Ymix−1,where Yn is normalized quantity, Ymax denotes maximum, and Ymin denotes the minimum quantity. Y denotes the initial “unnormalized” value of the parameter. The minimum and maximum values for each parameter are the ones given in Table 1. It is noteworthy that since ANN is trained on normalized data, to utilize its model, it is necessary to provide the ANN with normalized values of parameters and “unnormalize” the network’s outputs [32].


## 3.2. Performance Measures

Performance measures will ultimately be employed to determine the predictive accuracy of the models. The goal is to choose the model with the most predictive accuracy while demonstrating high generalizability. Therefore, the descriptive characteristics utilized to gauge the error metrics of different models are defined. They are mean error (ME), mean absolute error (MAE), mean squared error (MSE), root mean squared error (RMSE), average absolute error (AAE), model efficiency (EF), and variance account factor (VAF) defined as [49](6)ME=1n∑i=1nPi−Oi,MAE=1n∑i=1nPi−Oi,MSE=1n∑i=1nPi−Oi2,RMSE=1n∑i=1nPi−Oi21/2,AAE=∑i=1nOi−Pi/Oin×100,EF=1−∑i=1nPi−Oi2∑i=1nO¯i−Oi2,VAF=1−varOi−PivarOi×100.


## 3.3. Experimental Model Formation Utilizing Neural Networks and the BAT Algorithm

As stated in Section3.1, 11 parameters affect the compressive strength of self-compacting concrete (SCC). So, the artificial neural network (ANN) models will have 11 nodes in the entry layer and one node in the output layer, as shown in Figure 1. As detailed in Section 2.1, the overfitting phenomenon can lead to ANN models with small error values in training but unacceptable generalizability. Therefore, to mitigate the adverse effects of this phenomenon, as advocated in previous studies [32, 35], the dataset was randomly split into two groups: 70% (144 samples) for training and 30% (61 samples) for testing data. This led to choosing the generalizable networks.The layer count and the overall number of nodes in an ANN’s hidden layer are problem-dependent [37]. Therefore, a trial-and-error method was taken. As a rule of thumb, the total count of hidden layer nodes of the network, NH, must satisfy the following inequality [50]:(7)NH≤2NI+1,where NI denotes the count of input nodes. Given that the number of affecting factors is 11, the count of hidden layer nodes is recommended to be less than 23. Various designs with one and two hidden layers and a maximum of 23 neurons were trained. A total of 155 distinct designs were trained, as shown in Table 2. The tan h function was chosen as the hidden layer’s transfer function and the identity function as the output layer’s.Table 2
Various configurations of trained ANNs.
NumTopology (2L)NumTopology (2L)…NumTopology (2L)NumTopology (1L)11-1122-1…12212-1133121-2132-2…12312-2134231-3142-3…12412-3135341-4152-4…12512-41364………………………101-10212-10…13112-1015422111-11222-11…13212-1115523MATLAB R2021, a [51] software was employed to develop the ANNs and bat optimization procedure. The optimum values of hyperparameters for the bat optimization utilized to learn the models are provided in Table 3.Table 3
Hyperparameters of bat optimization.
HyperparameterValuePopulation size100Loudness0.9Minimum frequency0Alpha0.99Max generations200Pulse rate0.5Maximum frequency2Gamma0.01


## 4. Results

As stated in the previous section, 155 architectures of networks with one or two hidden layers were learned. The tanh function was used as the transfer function for the hidden layers in all networks, whereas the identity function was used as the transfer function for the output layer. In what follows, the ANNs are designated by ANN-(alg)-nL (n1 − n2), in which “alg” is the optimization algorithm employed, n is the number of hidden layers, and n1 and n2 denote the node count in the first and second hidden layers, correspondingly.Among the 155 developed networks for predicting the compressive strength of self-compacting concrete (SCC), the top five were picked according to the value of mean squared error (MSE). The networks and error metrics on the training set are presented in Table4, and the error metrics on the testing set are presented in Table 5.Table 4
Error metrics of the leading five networks for the training set.
Network labelMSERMEMAERMSEANN-BAT-1L (14)24.1900.9730.0073.6624.918ANN-BAT-2L (13-3)37.6150.9580.0014.4386.133ANN-BAT-2L (7-10)27.5370.9700.0084.1385.248ANN-BAT-1L (8)41.5690.954−0.0044.8476.447ANN-BAT-2L (9-2)128.8350.8480.0009.46511.351Table 5
Error metrics of the leading five networks for the testing set.
Network labelMSERMEMAERMSEANN-BAT-1L (14)34.7830.9692.3884.5075.898ANN-BAT-2L (13-3)55.9550.9400.6225.9197.480ANN-BAT-2L (7-10)62.4710.934−1.4026.2357.904ANN-BAT-1L (8)69.2480.9250.6036.2398.322ANN-BAT-2L (9-2)114.2290.887−1.6428.95510.688Concerning Tables4 and 5, the ANN-BAT-1L (14) model demonstrates the smallest MSE, MAE, and RMSE values. The coefficient of determination of the modelR2 for training and testing data is 0.947 and 0.939, respectively, denoting the network model’s accuracy. The network’s MSE, ME, MAE, and RMSE values are 34.783, 2.388, 4.507, and 5.898, respectively, showing the network’s accuracy in predicting SCC’s compressive strength.To demonstrate the precision of ANN-BAT-1L (14) visually, the experiment versus forecasted compressive strengths is depicted in Figures5–7 for the training, testing, and all data, respectively. The model’s outputs are close to the y = x line, which confirms the model’s high accuracy.Figure 5
Experimental vs. forecasted compressive strengths for ANN-BAT-1L (14) model for training data.Figure 6
Experimental vs. forecasted compressive strengths for ANN-BAT-1L (14) model for testing data.Figure 7
Experimental vs. forecasted compressive strengths for ANN-BAT-1L (14) model for all data.


## 5. Comparison with Other Metaheuristic Algorithms

The developed model is compared with ANN models trained by using particle swarm optimization (PSO) and the teaching-learning-based-optimization (TLBO) algorithm to evaluate its accuracy.


### 5.1. Particle Swarm Optimization (PSO)

The particle swarm optimization algorithm defines individual agents, called particles, and alters their positions to close in on the optimum by changing the positions of distinct particles. The final position of each particle is the linear combination of vectors connecting the present location of the particle to its local best location and the global best location obtained in the current epoch of the algorithm. In this study, to train ANNs by using the PSO algorithm, the weights of ANN models were treated as the positions of various particles, and each particle represented a different trained network [52]. The algorithm’s hyperparameters were discovered by examination and are presented in Table 6.Table 6
Particle swarm optimization hyperparameters.
HyperparameterValueLower bound−1Upper bound+1Coefficient (C1)2Swarm size200Max iterations150Coefficient (C2)2


### 5.2. Teaching-Learning-Based-Optimization

The algorithm for teaching-learning-based optimization attempts to replicate the teaching approach. Initially, it creates a set of students and labels the individual with the most negligible cost as the teacher. The teacher then teaches the rest of the students, adjusting their positions relative to the teacher’s location. The subsequent stage consists of students learning by interrelating with others (i.e., each individual works randomly with another student, and if the latter’s position is better, the former advances toward the latter) [53]. In this study, to train ANNs by using the TLBO algorithm, the weights of ANN models were treated as the positions of various students; therefore, each student represented a different trained network. The last epoch of the TLBO algorithm leads to a trained ANN. The algorithm’s hyperparameters were found by examination and are presented in Table 7.Table 7
Teaching-learning-based-optimization hyperparameters.
HyperparameterValueLower bound−1Upper bound+1Population size50Max iterations150


### 5.3. Comparison of All Models

The experiment versus forecasted values of compressive strength of PSO and TLBO algorithms are depicted in Figures8 and 9, respectively, to visually represent the performance of top-performing ANN models based on PSO and TLBO algorithms. The graphs are produced based on all the data.Figure 8
Experimental vs. forecasted compressive strengths for ANN-PSO-2L (10-8) model for all data.Figure 9
Experimental vs. forecasted compressive strengths for ANN-TLBO-1L (8) model for all data.Among the different tested architectures, ANN-PSO-2L (10-8) and ANN-TLBO-1L (8) showed the highest performance in their groups. The top-performing models from each group of training algorithms are selected, and their error metrics are given in Tables8 and 9 for the training and testing data, respectively.Table 8
Error metrics of top ANNs trained by using BAT, PSO, and TLBO methods on training data.
Network labelMEMAEMSERMSEAAEEFVAF% (%)ANN-BAT-1L (14)0.0073.66224.1904.9180.080.9292ANN-TLBO-1L (8)0.014.8243.246.580.100.8888ANN-PSO-2L (10-8)0.284.7748.636.970.090.8787Table 9
Error metrics of top ANNs trained by using BAT, PSO, and TLBO methods on testing data.
Network labelMEMAEMSERMSEAAEEFVAF% (%)ANN-BAT-1L (14)2.3884.50734.7835.8980.060.9797ANN-TLBO-1L (8)−2.065.8264.668.040.100.9090ANN-PSO-2L (10-8)−1.555.4474.688.640.090.8889As shown in Tables8 and 9, the bat-based ANN-BAT-1L (14) outperforms both the PSO- and TLBO-based models and is, therefore, selected for further analysis. A visual comparison of the predictions of the three models is given in Figure 10.Figure 10
Prediction comparison of all models by using testing data.The Taylor diagram represents a graphical sketch of the capacity of each studied model based on the root-mean-square-centered difference, the correlation coefficient, and the standard deviation. The Taylor diagram of the three models is given in Figure11 [54].Figure 11
Taylor diagram visualization of different models.


### 5.4. Sensitivity Analysis, Predictive Model, and ANN Weights

Because the ANN-BAT-1L (14) outperforms other developed networks, a sensitivity analysis was carried out on this network to establish the comparative impact of each of the 11 input quantities on the compressive strength of self-compacting concrete. For this purpose, Lek’s profile method of sensitivity analysis [55, 56] was applied. The relative importance and impact of explanatory quantities on the response were computed and depicted in Figure 12. Regarding Figure 12, the top three influencing parameters are cement content with 0.22 influence, GGBFS with 0.18 influence, and RHA with 0.17 influence. According to the ANN-BAT-1L (14) model, the least influential parameters on compressive strength are VMA and FA, with 0.02 and 0.00 influence, respectively.Figure 12
Relative impact of factors on the compressive strength of self-compacting concrete.The values of the model parameters are required to manually calculate the output of the top-performing ANN-BAT-1L (14) model. The inputs should be normalized by using equation (7); the prediction should then be “unnormalized” into its initial scope. The 11 × 1 vector denoted by a1 gives the influencing parameters [32](8)a2=tanhϑ1×a1+b1Toutput=tanhϑ2×a2+b2,where aj and ϑj signify the outputs and weights of layer j, respectively; and bj vector signifies the biases. The weights and biases of the ANN are given in Table 10.Table 10
The weight parameters of the ANN-BAT-1L (14) network.
ϑ1b1b20.018−0.328−0.5370.0930.3790.491−0.6370.7100.8940.124−0.236−0.5540.3260.2980.709−0.5160.610−0.309−0.3880.3650.4690.395−0.1520.495−0.9170.3480.390−0.6360.5320.3460.356−0.078−0.242−0.1590.066−0.089−0.041−0.161−0.740−0.1420.3020.045−0.424−0.835−0.356−0.2840.4140.7890.0360.375−0.044−0.284−0.0360.4680.4420.0630.2450.003−0.1180.5521.275−0.8580.9690.6390.622−0.522−0.827−0.322−0.2110.1420.2380.294−0.150−0.694−0.458−0.1640.023−0.129−0.826−0.012−0.3810.013−0.078−0.0260.266−1.1600.448−1.1590.528−0.560−0.334−0.791−0.171−0.4580.0090.4760.320−0.2010.8970.141−0.7720.042−0.0160.6750.268−0.680−0.4010.0410.184−0.5110.243−0.6451.356−0.248−0.083−0.6330.048−0.014−0.243−0.627−0.0060.1740.0810.2500.5480.5190.2710.360−0.2250.643−0.413−0.2200.0610.3500.2110.020−0.399−0.3440.5540.4050.197−0.155−0.4741.1020.189−0.0450.7590.3490.5250.012−0.081−0.598−0.0831.1340.311−0.141−0.301−0.2260.609−0.1630.042−0.612−0.6500.306−0.296−0.205−0.7780.020−1.020ϑ2−0.840−0.315−0.006−0.5351.040−0.425−0.055−1.0720.7961.212−0.4570.8470.695−0.739


## 5.1. Particle Swarm Optimization (PSO)

The particle swarm optimization algorithm defines individual agents, called particles, and alters their positions to close in on the optimum by changing the positions of distinct particles. The final position of each particle is the linear combination of vectors connecting the present location of the particle to its local best location and the global best location obtained in the current epoch of the algorithm. In this study, to train ANNs by using the PSO algorithm, the weights of ANN models were treated as the positions of various particles, and each particle represented a different trained network [52]. The algorithm’s hyperparameters were discovered by examination and are presented in Table 6.Table 6
Particle swarm optimization hyperparameters.
HyperparameterValueLower bound−1Upper bound+1Coefficient (C1)2Swarm size200Max iterations150Coefficient (C2)2


## 5.2. Teaching-Learning-Based-Optimization

The algorithm for teaching-learning-based optimization attempts to replicate the teaching approach. Initially, it creates a set of students and labels the individual with the most negligible cost as the teacher. The teacher then teaches the rest of the students, adjusting their positions relative to the teacher’s location. The subsequent stage consists of students learning by interrelating with others (i.e., each individual works randomly with another student, and if the latter’s position is better, the former advances toward the latter) [53]. In this study, to train ANNs by using the TLBO algorithm, the weights of ANN models were treated as the positions of various students; therefore, each student represented a different trained network. The last epoch of the TLBO algorithm leads to a trained ANN. The algorithm’s hyperparameters were found by examination and are presented in Table 7.Table 7
Teaching-learning-based-optimization hyperparameters.
HyperparameterValueLower bound−1Upper bound+1Population size50Max iterations150


## 5.3. Comparison of All Models

The experiment versus forecasted values of compressive strength of PSO and TLBO algorithms are depicted in Figures8 and 9, respectively, to visually represent the performance of top-performing ANN models based on PSO and TLBO algorithms. The graphs are produced based on all the data.Figure 8
Experimental vs. forecasted compressive strengths for ANN-PSO-2L (10-8) model for all data.Figure 9
Experimental vs. forecasted compressive strengths for ANN-TLBO-1L (8) model for all data.Among the different tested architectures, ANN-PSO-2L (10-8) and ANN-TLBO-1L (8) showed the highest performance in their groups. The top-performing models from each group of training algorithms are selected, and their error metrics are given in Tables8 and 9 for the training and testing data, respectively.Table 8
Error metrics of top ANNs trained by using BAT, PSO, and TLBO methods on training data.
Network labelMEMAEMSERMSEAAEEFVAF% (%)ANN-BAT-1L (14)0.0073.66224.1904.9180.080.9292ANN-TLBO-1L (8)0.014.8243.246.580.100.8888ANN-PSO-2L (10-8)0.284.7748.636.970.090.8787Table 9
Error metrics of top ANNs trained by using BAT, PSO, and TLBO methods on testing data.
Network labelMEMAEMSERMSEAAEEFVAF% (%)ANN-BAT-1L (14)2.3884.50734.7835.8980.060.9797ANN-TLBO-1L (8)−2.065.8264.668.040.100.9090ANN-PSO-2L (10-8)−1.555.4474.688.640.090.8889As shown in Tables8 and 9, the bat-based ANN-BAT-1L (14) outperforms both the PSO- and TLBO-based models and is, therefore, selected for further analysis. A visual comparison of the predictions of the three models is given in Figure 10.Figure 10
Prediction comparison of all models by using testing data.The Taylor diagram represents a graphical sketch of the capacity of each studied model based on the root-mean-square-centered difference, the correlation coefficient, and the standard deviation. The Taylor diagram of the three models is given in Figure11 [54].Figure 11
Taylor diagram visualization of different models.


## 5.4. Sensitivity Analysis, Predictive Model, and ANN Weights

Because the ANN-BAT-1L (14) outperforms other developed networks, a sensitivity analysis was carried out on this network to establish the comparative impact of each of the 11 input quantities on the compressive strength of self-compacting concrete. For this purpose, Lek’s profile method of sensitivity analysis [55, 56] was applied. The relative importance and impact of explanatory quantities on the response were computed and depicted in Figure 12. Regarding Figure 12, the top three influencing parameters are cement content with 0.22 influence, GGBFS with 0.18 influence, and RHA with 0.17 influence. According to the ANN-BAT-1L (14) model, the least influential parameters on compressive strength are VMA and FA, with 0.02 and 0.00 influence, respectively.Figure 12
Relative impact of factors on the compressive strength of self-compacting concrete.The values of the model parameters are required to manually calculate the output of the top-performing ANN-BAT-1L (14) model. The inputs should be normalized by using equation (7); the prediction should then be “unnormalized” into its initial scope. The 11 × 1 vector denoted by a1 gives the influencing parameters [32](8)a2=tanhϑ1×a1+b1Toutput=tanhϑ2×a2+b2,where aj and ϑj signify the outputs and weights of layer j, respectively; and bj vector signifies the biases. The weights and biases of the ANN are given in Table 10.Table 10
The weight parameters of the ANN-BAT-1L (14) network.
ϑ1b1b20.018−0.328−0.5370.0930.3790.491−0.6370.7100.8940.124−0.236−0.5540.3260.2980.709−0.5160.610−0.309−0.3880.3650.4690.395−0.1520.495−0.9170.3480.390−0.6360.5320.3460.356−0.078−0.242−0.1590.066−0.089−0.041−0.161−0.740−0.1420.3020.045−0.424−0.835−0.356−0.2840.4140.7890.0360.375−0.044−0.284−0.0360.4680.4420.0630.2450.003−0.1180.5521.275−0.8580.9690.6390.622−0.522−0.827−0.322−0.2110.1420.2380.294−0.150−0.694−0.458−0.1640.023−0.129−0.826−0.012−0.3810.013−0.078−0.0260.266−1.1600.448−1.1590.528−0.560−0.334−0.791−0.171−0.4580.0090.4760.320−0.2010.8970.141−0.7720.042−0.0160.6750.268−0.680−0.4010.0410.184−0.5110.243−0.6451.356−0.248−0.083−0.6330.048−0.014−0.243−0.627−0.0060.1740.0810.2500.5480.5190.2710.360−0.2250.643−0.413−0.2200.0610.3500.2110.020−0.399−0.3440.5540.4050.197−0.155−0.4741.1020.189−0.0450.7590.3490.5250.012−0.081−0.598−0.0831.1340.311−0.141−0.301−0.2260.609−0.1630.042−0.612−0.6500.306−0.296−0.205−0.7780.020−1.020ϑ2−0.840−0.315−0.006−0.5351.040−0.425−0.055−1.0720.7961.212−0.4570.8470.695−0.739


## 6. Conclusion

The compressive strength of self-compacting concrete (SCC) samples was modeled by using an artificial neural network (ANN) model coupled with the bat optimization algorithm. To this end, 205 samples collected from published studies were utilized. The evaluation of the outcomes implies(1)
The learned ANN-BAT-1L (14) network forecasted the compressive strength more precisely than other networks. The model’s mean squared error and coefficient of determination on the test set were 34.783 and 0.939, respectively;(2)
A particle swarm optimization algorithm-based ANN model was also learned, and the top-performing model of this class was ANN-PSO-2L (8-10), with a coefficient of determination of 0.879;(3)
A teaching-learning-based-optimization ANN model was also learned, and the top-performing model of this class was ANN-TLBO-1L (8), with a coefficient of determination of 0.893;(4)
A sensitivity analysis was performed by using the ANN-BAT-1L (14) model, and the results suggested that the top three influencing parameters are cement content with 17.0% influence, coarse aggregate with 15.1% influence, and ground granulated blast furnace slag (GGBFS) with 14.0% influence;(5)
The weights and biases of the ANN-BAT-1L (14) were offered to facilitate the manual calculation of the network output.



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*Source: 8404774-2022-09-22.xml*