BAT Algorithm-Based ANN to Predict the Compressive Strength of Concrete—A Comparative Study

Abstract

The number of effective factors and their nonlinear behaviour—mainly the nonlinear effect of the factors on concrete properties—has led researchers to employ complex models such as artificial neural networks (ANNs). The compressive strength is certainly a prominent characteristic for design and analysis of concrete structures. In this paper, 1030 concrete samples from literature are considered to model accurately and efficiently the compressive strength. To this aim, a Feed-Forward (FF) neural network is employed to model the compressive strength based on eight different factors. More in detail, the parameters of the ANN are learned using the bat algorithm (BAT). The resulting optimized model is thus validated by comparative analyses towards ANNs optimized with a genetic algorithm (GA) and Teaching-Learning-Based-Optimization (TLBO), as well as a multi-linear regression model, and four compressive strength models proposed in literature. The results indicate that the BAT-optimized ANN is more accurate in estimating the compressive strength of concrete.

1. Introduction

Civil engineers have always been interested in estimating the properties of concrete as a composite material by utilizing analytical models, as well as investigating the effect of each component of the mix design on its properties. The first step in all rehabilitation projects is to obtain information about the current conditions of the structure and its analysis. In this regard, using field experiments to perform this evaluation is very important. In these projects, destructive experiments are used to achieve more accurate results, which involve high costs and structure destruction [1]. The application of artificial neural networks and evolutionary optimization algorithms to determine concrete’s compressive strength has received a lot of attention in recent years [2].

Many studies on the use of artificial neural networks (ANNs) to assess the compressive strength of concrete (f’_c) have been conducted in recent decades. Artificial neural network models have proved to be superior for the determination of concrete compressive strength in Alto Sulcis Thermal Power Station in Italy [3], in-place concrete strength estimation to facilitate concrete form removal and scheduling for construction [4], prediction of compressive strength of concrete subject to lasting sulfate attack [5], determination of low-, medium-, and high-strength concrete strength [6], accurate assessment of compressive strength of large-volume fly ash concrete [7], approximating compressive strength of concrete based on weight and ultrasonic pulse velocity [8], compressive strength prediction of high performance concrete [9], and compressive strength estimation of concrete containing various amounts of furnace slag and fly ash [10].

Artificial neural networks are also utilized in modelling high-strength concrete specimens due to the non-linearity of the parameters. Yeh [11,12] predicted high-performance concrete compressive strength using ANNs by performing a series of tests. Artificial neural networks have been more accurate than models based on regression analysis. Öztaş et al. [13] determined the compressive strength of high-strength concrete using ANNs and 187 samples for modelling. The parameters of cover, the water to binder ratio, water content, fine aggregate ratio, fly ash content, air-entraining agent, superplasticizer, and silica fume replacement were employed as input parameters. The results confirmed that an ANN could forecast the compressive strength of high-strength concretes satisfactorily.

Numerous articles have then been published on the use of optimization algorithms with ANNs. In [14], an ANN combined with the metaheuristic Krill Herd algorithm was used to provide satisfactory results in terms of estimation of the mechanical properties of alkali-activated mortar mixes (AAMs). Behnood et al. [15] estimated the compressive strength of silica fume concrete using an ANN as an optimization problem. To find a simple ANN model with acceptable error, they proposed a new multi-objective optimization method called the Multi-Objective Grey Wolves Optimization (MOGWO) method. Sensitivity analysis was also performed to evaluate the final ANN model’s ability to predict silica fume concrete compressive strength by changing the strength variables. Nazari et al. [16] determined the compressive strength of concrete using titanium dioxide (TiO₂) nanoparticles. A genetic algorithm (GA) was used to adjust the weights of the network. The results showed that optimization increases the accuracy of the model. Bui et al. [17] considered the combination of ANNs with firefly algorithms to predict the compressive and tensile strength of high-performance concrete. The results indicate the high performance and accuracy of the proposed model. Sadowski et al. [18] used the colonial competition algorithm to determine the compressive strength of concrete. Their study considered the feasibility of using the algorithm to learn ANN parameters and compared the proposed model with the GA-based ANN. The results confirmed that the ANN combined with the colonial competition algorithm resulted in the least prediction error. Duan et al. [19] used the colonial competition algorithm to determine the compressive strength of recycled aggregate concrete. Zhou et al. [20] utilized ANNs and adaptive neuro-fuzzy inference systems to estimate the compressive strength of hollow concrete block masonry prisms. The analysis was based on 102 data points and showed that the proposed models have excellent prediction with negligible error rates. Armaghani et al. [21] developed neuro-fuzzy systems to predict concrete compressive strength.

The current study concentrates on boosting the accuracy of an ANN model by applying the BAT optimization algorithm. To this aim, the dataset has been collected from UCI (University California Irvine) machine learning repository which includes 1030 experimental results collected from peer-reviewed articles. Section 2 delivers the essential background on ANNs and the BAT algorithm. The experimental model establishment and training are described in Section 3. Section 4 presents detailed results and assesses the efficiency of the model based on comparative analyses.

2. Background

2.1. Artificial Neural Networks

Artificial neural networks, a form of data processing systems, are algorithms that simulate biological neurons’ function. ANNs improve their performance by learning from data in the training step [22,23]. The main assumptions of an artificial neural network model are as follows [24,25]:

• Data are handled in specific entities called nodes.
• Links relay signals between nodes.
• The weight assigned to each link indicates the strength of that link.
• Nodes calculate their outputs by applying activation functions to input data.

The Feed-Forward network is an ANN model in which the connections of its constituent units do not form a cycle. This network is different from a recurrent neural network, given that data move only in one direction, the direction of which is forward. Data initially starts from input nodes and passes through hidden layers to the output nodes [26]. Figure 1 shows an example of a feed-forward network. Typically, data in ANN are split into three distinct subgroups [23]:

• Training: this subgroup of data is used to train the ANN, and learning occurs through examples, similar to the human brain. The training sessions are repeated until the acceptable precision of the model is achieved.
• Validation: this subset determines the extent of training of the model and estimates model features such as classification errors, mean error for numerical predictions, etc.
• Testing: This subgroup can confirm the performance of the training subset developed in the ANN model.

2.2. BAT Algorithm

By utilizing their sophisticated echolocation facilities, bats can avoid obstacles and detect prey. Utilizing the time interval between a pulse emission and its echo, they develop a three-dimensional depiction of their surrounding [27]. Inspired by this behaviour of bats, Yang [28] developed the BAT algorithm. In the algorithm idealization it is assumed that:

• bats use echolocation, and they can discern between prey and surroundings;
• at any given location x_i, they fly randomly with velocity v_i and contingent upon the location of prey they adjust their rate of pulse emission;
• the loudness of the emitted pulse ranges from A₀ to a minimum value of A_min.

Firstly, the BAT algorithm initializes a random population of bats, and then updates their frequencies using Equation (1) [29]:

$$f_i=f_{min}+\left(f_{max}-f_{min}\right)\beta$$

(1)

where f_i is the i-th bat frequency, f_min is the min frequency, f_max is the max frequency, and β is a random quantity between 0 and 1. The location and velocity of bats are revised according to:

$$V_i^{t+1}=V_i^t+\left(x_i^t-{\mathrm{x}}^{*}\right)f_i$$

(2)

$$x_i^{t+1}=x_i^t+V_i^{t+1}$$

(3)

where $V_i^t$ is the i-th bat velocity at recurrence t, x_i^t is the i-th bat position at recurrence t, and $x^{*}$ is the global best position. Then, the procedure shifts some bats to a vicinity of the top global location as:

$$x_{new}=x_{old}+\epsilon A^t$$

(4)

where A denotes loudness and ε is a random quantity between 0 and 1. The criterion for accepting the new position of each bat is a cost value less than the previous iteration. The algorithm then revises the pulse rate and loudness using Equations (5) and (6):

$$A_i^{t+1}=\alpha A_i^t$$

(5)

$$r_i^{t+1}=r_i^0\left(1-\exp \left(-\gamma t\right)\right)$$

(6)

where α is a constant typically selected between zero and one, r⁰_i is the initial pulse rate and γ is a constant.

This algorithm can be utilized to train an ANN. In the present application, the weights and biases of the network are considered as the position vector of a bat, and therefore each bat represents a vector of weights of an artificial neural network. The cost function is the prediction error of the network. The final solution of the bat algorithm results in a trained network [29].

3. Methods and Materials

3.1. Dataset

The dataset utilized in this study was used to follow the schematic procedural steps proposed in Figure 2. More precisely, according to [12], it consists of 1030 concrete compressive strength test results from various sources. The influencing parameters on the concrete compressive strength are cement, blast furnace slag, fly ash, water, superplasticizer, coarse aggregate, and fine aggregate age.

Table 1. Descriptive statistics of samples.

Statistical Index	Unit	Min	Min	Average	Standard Deviation	Mode	Median
Cement	Kg/m3	540	102	281.17	104.51	425	272.9
Blast Furnace Slag	Kg/m3	359.4	0	73.90	86.28	0	22
Fly Ash	Kg/m3	200.1	0	54.19	64.00	0	0
Water	Kg/m3	247	121.75	181.57	21.36	192	185
Superplasticizer	Kg/m3	32.2	0	6.20	5.97	0	6.35
Coarse Aggregate	Kg/m3	1145	801	972.92	77.75	932	968
Fine Aggregate	Kg/m3	992.6	594	773.58	80.18	594	779.51
Age	day	365	1	45.66	63.17	28	28
Concrete compressive strength	MPa	82.60	2.33	35.82	16.71	33.40	34.44

gives the descriptive statistics of samples from [12]. The selected target value is the 28-day compressive strength of concrete.

Following Figure 3, where the histogram for the 28-day compressive strength for concrete specimens is proposed, it can be observed that 780 samples have compressive strength values ranging from 10 to 50 MPa.

If the ANN input variables have different ranges, the training process can suffer from adverse impacts, such as the optimization divergence of algorithm and increased training time [30]. Using Equation (7), each variable was hence normalized into the range from −1 to 1, that is [31]:

$$Y_n=\frac{2\left(Y-Y_{min}\right)}{Y_{max}-Y_{mix}}-1$$

(7)

where Y is the original value of the variable, Y_n is the normalized value, Y_max is the max, and Y_min is its min value. Table 1 shows the minimum and maximum values and the target values of concrete compressive strength used for each of the eight input parameters. It is worth noting that the ANN will undergo training using the normalized data. Therefore, it is essential to feed the network with normalized variables when using the ANN to predict new values and un-normalize the data (i.e., transferring them into their original range, the network outputs).

3.2. Performance Measures

Statistical measures are employed to determine the model accuracy. Using the statistical index helps choose the best model with the least error and select the model with the most generalizability.

The statistical measures employed in evaluating the accuracy of different topologies are Mean Error (ME), Mean Absolute Error (MAE), Mean Squared Error (MSE), and Root Mean Squared Error (RMSE) [32]:

$$ME=\frac{1}{n}{\displaystyle \sum }_{i=1}^n\left(P_i-O_i\right)$$

(8)

$$MAE=\frac{1}{n}{\displaystyle \sum }_{i=1}^n\left|P_i-O_i\right|$$

(9)

$$MSE=\frac{1}{n}{\displaystyle \sum }_{i=1}^n{\left(P_i-O_i\right)}^2$$

(10)

$$RMSE={\left[\frac{1}{n}{\displaystyle \sum }_{i=1}^n{\left(P_i-O_i\right)}^2\right]}^{\frac{1}{2}}$$

(11)

where $P_i$ and $O_i$ represent predicted and observational data, and n represents the number of cases.

3.3. Experimental Model Generation Utilizing ANNs and BAT Algorithm

Eight input factors, influencing the concrete compressive strength, are used in the suggested model. Thus, trained artificial neural networks have eight nodes in the input layer and one node in the output layer (Figure 2). Networks with one or two hidden layers have been used for modelling. A network can have a very high performance in the training phase but may not show the same accuracy in the testing phase. Hence, it is better to randomly bifurcate the data and with a suitable ratio for each stage [33]. To this end, the data were randomly split into two groups to reduce overfitting effects. For training, 70% of data (721 samples) were used and the remaining 30% (309 pieces) were used to test network performance. For an artificial neural network model, the number of hidden layers and the total number of nodes in the hidden layers depends on the problem [34]. Accordingly, trial-and-error was used to obtain the ideal topology (i.e., the topology that best represents the data). A common formula for the total number of nodes in an ANN is given in Equation (12) [35]:

$$N_H\le 2N_I+1$$

(12)

where $N_H$ is the number of neurons in the hidden layer and $N_I$ is the number of inputs to the network. Since there are eight factors, the equation indicates that the number of hidden layer nodes must be less than 17. Thus, different architectures have a maximum of two hidden layers and a maximum of 17 trained neurons. In mode one, the network hidden layer has a maximum of 17 neurons in the hidden layer. In mode two, the hidden layer has one to nine neurons. A total of 89 different architectures were selected for training. The topologies used are given in

Table 2. Trained ANN topologies.

Num	Topology	Num	Topology	...	Num	Topology	Num	Topology
1	1-1	9	2-1	...	65	9-1	73	1
2	1-2	10	2-2	...	66	9-2	74	2
3	1-3	11	2-3	...	67	9-3	75	3
4	1-4	12	2-4	...	68	9-4	76	4
...	...	...	...	...	...	...	...	...
7	1-7	15	2-7	...	71	9-7	88	16
8	1-8	16	2-8	...	72	9-8	89	17

Note: n₁-n₂ format for topologies denotes n₁ neurons in the first hidden layer, and n₂ neurons in the second hidden layer.

.

The tanh function was picked as the transfer function of the nodes in the hidden layer in all the ANNs. Meanwhile, the identity relation was selected as the transfer function of the nodes in the output layer. The bat algorithm was then used to refine the parameters of the artificial neural network to result in the least predictive error. The bat algorithm and ANNs were developed using MATLAB [36]. The hyperparameters of the bat algorithm used for training the 89 ANN topologies are given in

Table 3. Hyperparameters of the bat algorithm.

Hyperparameter	Value	Hyperparameter	Value
Population Total	100	Max Generations	200
Loudness	0.9	Pulse Rate	0.5
Min Freq.	0	Max Freq.	2
Alpha	0.99	Gamma	0.01

.

4. Results

4.1. Experimental Model Assessment

The number of used models was set to 89. In addition, one-layer and two-layer artificial neural networks were used. The bat optimization algorithm is used to refine the weights of the network. The transfer function for all networks is the tanh function. Among the models trained to determine concrete compressive strength, four models were selected as the best models based on parameters presented in Section 3.2. The error metrics of these models in the training phase are given in

Table 4. Error metrics of top four networks on training data.

Num	Network Designation	Training
		MSE	ME	MAE	RMSE
1	ANN-BAT-1L (4)	28.471	0.000	3.989	5.336
2	ANN-BAT-2L (3-2)	28.543	0.000	4.018	5.343
3	ANN-BAT-2L (8-5)	10.928	0.000	2.448	3.306
4	ANN-BAT-2L (7-4)	16.001	0.000	2.895	4.000

Note: ANN-BAT-(m)L (n₁-n₂) format denotes m hidden layers, n₁ neurons in the first hidden layer and n₂ neurons in second hidden layer.

. The test results of the models are given in

Table 5. Error metrics of top four networks on testing data.

Num	Network Designation	Testing
		MSE	ME	MAE	RMSE
1	ANN-BAT-1L (4)	37.146	−0.147	4.674	6.095
2	ANN-BAT-2L (3-2)	37.496	0.148	4.739	6.123
3	ANN-BAT-2L (8-5)	40.130	−0.546	3.828	6.335
4	ANN-BAT-2L (7-4)	27.624	−0.664	3.847	5.256

Note: ANN-BAT-(m)L (n₁-n₂) format denotes m hidden layers, n₁ neurons in the first hidden layer and n₂ neurons in second hidden layer.

.

According to Table 4 and Table 5, the ANN-BAT-2L (7-4) network has the lowest MSE, ME, MAE, and RMSE indices. For this network, the $R^2$ value in the training and testing phases is equal to 0.9395 and 0.9134, respectively, supporting the model’s high precision. The test phase values in MSE, ME, MAE, and RMSE indices are equal to 27.624, −0.664, 3.847 and 5.256, respectively, indicating high modelling precision in forecasting the concrete’s compressive strength. Error metrics for samples are given based on the original range of variables.

To visualize the accuracy of ANN-BAT-2L (7-4), the experimental model predicted values versus their values from the experiment are shown in Figure 4 and Figure 5 for the train and test data, respectively. The predictions of the network are near the identity function values, which suggests that the network is highly accurate.

4.2. Comparison with Other Methods

To assess the accuracy of the model trained using bat optimization, three models have also been trained using other methods. Two models are ANNs trained with the genetic algorithm (GA) and Teaching-Learning-Based-Optimization (TLBO). The third is an MLR model.

The genetic algorithm is an optimization technique based on genetics and natural selection theories. It commences by generating a population of individuals and evolves them under specific selection, cross-over, and mutation rules to minimize the cost function. In this paper, individuals were the parameters of the ANN, and the final solution was the trained network.

Inspired by the teaching-learning process, the TLBO algorithm generates a population of students and designates the one with least cost to be the teacher. The remaining students then learn from the teacher, i.e., move toward the teacher’s position in solution space. In the next phase, students learn by interacting with each other, i.e., a given student (solution) interacts randomly with another student and if the second student has more knowledge (has lower cost), the first student moves towards the second [37]. In the present study, the parameters of ANN were designated as students, and the final iteration of the TLBO algorithm resulted in a trained ANN.

4.2.1. Genetic Algorithm and Teaching-Learning-Based-Optimization Models

The 89 topologies presented in Table 2 were used to train ANNs using the GA and TLBO to find the best ANN topology. The hyperparameters of these two algorithms are listed in

Table 6. GA and TLBO hyperparameters.

Name	Parameter	Value	Parameter	Value
Genetic Algorithm	Max generations	100	Crossover (%)	50
	Recombination (%)	15	Crossover method	single point
	Lower Bound	−1	Selection Mode	1
	Upper Bound	+1	Population Size	150
Teaching Learning Base Optimization	Lower Bound	−1	Max Interaction	50
	Upper Bound	+1	Population Size	150

The TLBO algorithm-trained ANN with 5-6 topologies and the GA-trained ANN with the 3-5 topologies have the highest performance, as shown by the statistical indices in Table 7.

.

The hyperparameters of these models were set by trial-and-error. Their values are provided in

Table 7. Error metrics of GA and TLBO on training and testing data.

Topology	Train				Test
	ME	MAE	MSE	RMSE	ME	MAE	MSE	RMSE
ANN-GA-2L (3-5)	0.04	4.13	30.35	5.51	0.25	4.17	28.44	5.33
ANN-TLBO-2L (5-6)	0.16	3.59	23.68	4.87	−0.14	4.02	31.87	5.65

Note: ANN-(A)-(m)L (n₁-n₂) format denotes algorithm A, m hidden layers, n₁ neurons in the first hidden layer and n₂ neurons in second hidden layer.

. These two networks that are optimized using GA and TLBO algorithms have a much higher prediction error compared to bat-trained neural networks. To visualize the performance of GA and TLBO, the predicted values of the experimental model versus their values from the experiments are shown in Figure 6 and Figure 7 for test data.

4.2.2. Multi Linear Regression Model

As suggested by Nikoo et al. [38], an MLR model was developed using the Minitab software as an easy-to-use classical model [39]. According to the model, each factor influence can be estimated by examining the regression coefficient values [40,41,42]. The resulting regression equation is as follows:

$$\begin{gathered} f^{\prime }c=-23.2+\left(0.11979\times \mathrm{C}\right)+\left(0.1038\times \mathrm{BFS}\right)+\left(0.0879\times \mathrm{FA}\right)-\left(0.1503\times W\right)+\left(0.2907\times S\right)+ \\ \left(0.01803\times CA\right)+\left(0.0202\times FAg\right)+\left(0.11423\times A\right) \end{gathered}$$

(13)

In Equation (13), the parameters are:

C: Cement	W: Water
BFS: Blast Furnace Slag	S: Superplasticizer
FA: Fly Ash	CA: Coarse Aggregate
Fag: Fine Aggregate	f’c: compressive strength
A: Age

The statistical metrics of the MLR model are given in

Table 8. MLR model error metrics for training and testing data.

Topology	Training				Testing
	ME	MAE	MSE	RMSE	ME	MAE	MSE	RMSE
MLR	0.00	8.08	106.49	10.32	−0.02	8.52	108.91	10.44

, and Figure 8 depicts the results obtained for observed versus predicted test data.

4.2.3. Comparison on All Data

The ANN-BAT-2L (7-4) is compared with the GA- and TLBO-based ANNs to validate its accuracy on all data. The MLR model is also employed as a statistical model for comparison. The results are given in

Table 9. Statistics of ANN-BAT, ANN-GA, ANN-TLBO and MLR models.

Type	Network Designation	ME	MAE	MSE	RMSE
All Data	ANN-BAT-2L(7-4)	−0.199	3.181	19.488	4.414
	ANN-GA-2L(3-5)	0.10	4.14	29.77	5.46
	ANN-TLBO-2L(5-6)	0.07	3.72	26.13	5.11
	MLR	−0.01	8.22	107.21	10.35

.

Table 9 highlights that the ANN-TLBO model performs better than the ANN-GA model, and the MLR model has the weakest results. However, the ANN-BAT model offers the highest accuracy in determining the compressive strength of concrete of all models. The comparison between observed and predicted compressive strength of all models are shown in Figure 9, Figure 10, Figure 11 and Figure 12.

4.2.4. Comparative Analysis with Models Proposed in Literature

The developed experimental model is compared to four proposed models in literature using the same sample data. The comparison is over all data and is not divided into training and testing groups. The description of models used is given in

Table 10. Description of models proposed in literature.

Author	Model	Reference
A.H. Gandomi et al.	Genetic-Simulated Annealing	[43]
J.-S. Chou et al.	Support Vector Machines	[44]
Jui-Sheng et al.	Least Squares Support Vector Machines	[45]
D.-K. Bui et al.	Firefly Algorithm combined Artificial Neural Network	[17]

, and the error metrics of various models are given in

Table 11. Error metrics of models proposed in literature, and bat-based ANN (ordered by R²).

Model	R2	MAE
Present (ANN-BAT-2L (7-4))	0.93	3.18
D.-K. Bui et al. (2018)	0.90	3.41
J.-S. Chou et al. (2013)	0.88	4.24
Jui-Sheng et al. (2016)	0.88	5.62
A.H. Gandomi et al. (2013)	0.81	5.48

. As it can be seen from Table 11, the ANN-BAT-2L (7-4) model outperforms the other four models proposed in literature.

4.3. Predictive Model and ANN Weights

The best model presented in this study is ANN-BAT-2L (7-4). To calculate the output of this model manually, the matrices of network parameters are needed. The network input must be scaled using Equation (7) into −1 to 1 range, and the predicted value must then be unscaled into its original range. The input is an 8 × 1 vector called ${\mathrm{a}}_1$, where the eight parameters are cement, blast furnace slag, fly ash, water, superplasticizer, coarse aggregate, and fine aggregate age respectively. The following equations can be utilized to produce the ANN-BAT-2L (7-4) model predictions:

$${\mathit{a}}_2=\tan \mathrm{h}\left({\mathit{\vartheta }}_1^T\times {\mathit{a}}_1+b_1\right)$$

(14)

$${\mathit{a}}_3=\tan \mathrm{h}\left({\mathit{\vartheta }}_2^T\times {\mathit{a}}_2+b_2\right)$$

(15)

$$f_c^{predict\left(normalized\right)}=\tan \mathrm{h}\left({\mathit{\vartheta }}_3^T\times {\mathit{a}}_3+b_3\right)$$

(16)

$$f_c^{predict}=\frac{f_c^{predict\left(normalized\right)}+1}{2}\times \left(f_{m\mathit{a}x}-f_{min}\right)+f_{min}$$

(17)

where ${\mathit{a}}_j$ and ${\mathit{\vartheta }}_j$ matrices represent the outputs and weights of layer j respectively and $b_j$ vector represents its biases. tanh is the hyperbolic tangent function, and T superscript represents transpose operator. f_predict is the predicted value of compressive strength, and f_max and f_min are the min and max compressive strengths of data given in Table 1.

Table 12. Weight and bias values of the ANN-BAT-2L (7-4) model.

${\mathit{\vartheta }}_{\mathbf{1}}$								${\mathit{b}}_{\mathbf{1}}$
−0.8061	−0.2178	−0.2297	−0.4198	−0.5889	−0.3034	−0.2999	−0.1035	−0.8156
0.2371	−0.6965	−0.1132	1.3410	1.5170	0.3782	0.0238	0.1807	0.6200
−6.3017	−2.6506	−2.4931	2.5263	2.0438	−0.8616	−4.4602	−1.1676	−1.1071
0.0226	−0.0670	−0.1191	0.0505	0.0342	0.0304	−0.0781	3.6208	4.6903
−6.9203	−18.4075	−3.0575	−27.3813	−10.0966	−11.8482	−7.9640	−1.1299	−16.3967
31.2215	−7.9121	−19.6231	−0.0551	14.0536	−15.0847	−12.6117	−2.1349	−4.3244
0.6362	−3.2611	4.4076	−5.9958	4.4666	−4.3309	−1.5010	−8.0783	1.1720
${\mathit{\vartheta }}_{\mathbf{2}}$								${\mathit{b}}_{\mathbf{2}}$
−1.6512	−0.7434	0.3050	0.1381	0.1699	−0.1258	−0.1164		−2.0524
6.6296	2.6327	−1.7307	13.1062	−0.6463	−0.0506	1.1651		−9.5413
33.0428	23.3462	3.0604	−21.0571	−1.8540	0.2323	−19.2491		−12.8245
1.0129	0.6331	−0.6631	9.3516	−0.7377	0.4773	0.3696		−8.0055
${\mathit{\vartheta }}_{\mathbf{3}}$								${\mathit{b}}_{\mathbf{3}}$
12.8658	0.4640	0.2489	0.4572					11.7091

provides the weights and biases of the neural network.

5. Conclusions

The use of accurate models plays a vital role in the design and analysis of civil engineering structural members and systems. In this regard, the present study focused on the prediction of the compressive strength of concrete based on efficient and accurate ANNs. The learning of the ANN parameters was done using the bat optimization algorithm by using 1030 experimental results published literature. Several ANN models were in fact trained (89 in total) using the bat algorithm. The top performing model was then compared with networks trained using GA and TLBO algorithms, MLR model, and four proposed compressive strength models published in literature. The main results can be summarized as follows:

• The top-performing bat-based ANN model, ANN-BAT-2L (7-4), yielded a mean squared error of 27.624 on testing data.
• Due to its simplicity, a classical MLR model was presented for predicting compressive strength; however, it is less accurate than the proposed ANN-BAT model.
• The top-performing bat algorithm-based ANN was compared with ANNs trained using GA and TLBO algorithms. The top models based on these algorithms were ANN-GA-2L (3-5) and ANN-TLBO 2L (5-6); however, they were less accurate than the ANN-BAT-2L (7-4) model. The next best performing ANN was the TLBO-based, followed by GA-based, and the MLR model.
• The top-performing bat algorithm-based ANN was compared with four predictive models proposed in literature for compressive strength of concrete. The bat-based ANN outperformed all four.
• The network parameters, i.e., weights and biased of the ANN-BAT-2L (7-4) model were provided in tabular format for manual calculation of network prediction. Thus, for desired and new concrete samples, the compressive strength can be estimated by providing the presented formulas with sample inputs.