Ever since their presentation in the late 80s, self-compacting concrete (SCC) has been well received by researchers. SCC can flow under their weight and exhibit high workability. Nonetheless, their nonlinear behavior has made the prediction of their mix properties more demanding. Furthermore, the complex relationship between mixed proportions and rheological and mechanical properties of SCC renders their behavior prediction challenging. Soft computing approaches have been shown to optimize and reduce uncertainties, and therefore in this paper, we aim to address these challenges by employing artificial neural network (ANN) models optimized using the grey wolf optimizer (GWO) algorithm. The optimized model proved to be more accurate than genetic algorithms and multiple linear regression models. The results indicate that the four most influential parameters on the compressive strength of SCC are the cement content, ground granulated blast furnace slag, rice husk ash, and fly ash.
A proper self-compacting concrete mix design requires balancing two conflicting objectives: deformability and stability, that is, acceptable rheological behavior and appropriate mechanical characteristics. So, the proportions of available materials, minerals, and admixtures must be considered. The optimum balance of coarse and fine aggregates and chemical admixtures ensures the greater cohesiveness of self-compacting concrete. External variations such as changes in the production process of cement and mineral additives and the type of aggregates can trigger significant variations in the properties of fresh self-compacting concrete. To minimize such external variants, the use of industrial derivatives and mineral additives in the manufacturing of lightweight self-compacting concrete has been the focus of many scientists [
Due to the complex relationship between mixed proportions and rheological and mechanical properties of SCC, researchers have proposed numerous treatments in the literature. Some researchers have used statistical models such as linear regression to predict the compressive strength of SCC [
Zhang et al. developed a random forest model based on the beetle antenna search algorithm to predict the compressive strength of SCC [
There are several opportunities in soft computing to develop new models aiming to optimize and reduce uncertainties. Mirjalili introduced the grey wolf optimizer metaheuristic algorithm, which is inspired by grey wolves’ social hierarchy and has successfully outperformed other metaheuristic algorithms [
This study emphasizes enhancing the prediction model by training an ANN using the GWO algorithm. The proposed ANN-GWO model addresses all significant mix design parameters that influence SCC’s compressive strength. The accuracy of the model is confirmed against 205 samples of a database from the literature. Section
Overview of the research.
Artificial Neural Networks (ANNs) are among the most dynamic areas of current research [
The present ANN architecture comprising 11 input neurons, 7 neurons in the first hidden layer, 13 neurons in the second hidden layer, and a single neuron in the output layer.
The basis of ANNs is as follows [ Data are processed in units called neurons (or nodes). The signals between nodes are transmitted over the connection lines. The weight associated with each connection indicates its power. An activation function is applied to the weighted input (plus a bias value) of each neuron to determine its output [
A feed-forward network is a type of ANN in which the connection between its constituent units does not form a cycle, and information flows from the input nodes through the hidden layers to the output nodes [
The weights of an ANN are generally set randomly, thus, resulting in different output values in the network. The network’s weights and biases are adjusted and optimized in a training process to reduce the model error to minimize the model error [
Native to the remote areas of Eurasia and North America, grey wolves are members of the Canidae family and are at the top of the food chain. Proposed by Mirjalili et al. in 2014, the grey wolf optimizer (GWO) algorithm is inspired by the strict sovereignty hierarchy of these animals [
The grey wolves typically live in groups of 5 to 12, and their social dominance hierarchy consists of four groups: alphas, betas, deltas, and omegas. Alpha grey wolves are the pack’s leaders and dictate things like the decision for hunting, place to sleep, time to wake up, and so on [
Grey wolves also have strict hunting rituals. The main stages of their hunting are as follows [ Pursuing and advancing toward the prey. Circling and harassing the target to make it stop moving. Attacking the prey [
If the position of the prey is assumed to be
The grey wolf optimizer starts by randomly generating a wolf population and then ranking them to increase values of the cost function used to evaluate solutions. Then, GWO labels the top, second, and third solutions as alpha, beta, and delta, respectively, to simulate the social hierarchy of the grey wolves. The algorithm designates the remaining population of wolves as omega that follows the alphas, betas, and deltas. Considering that we are unaware of the exact location of the prey in the search space, we assume that the top three wolves have better knowledge of the location of the potential prey. Accordingly, in GWO, the hunting process is driven by alpha, beta, and delta. So GWO modifies equations (
Update rule of grey wolf optimizer [
When using the GWO algorithm for ANN, each wolf’s position vector represents the weights and biases of the neural network. Therefore, in terms of dimension, it equals the weights and biases of the network. The prediction error of the network is defined as the cost function. After the GWO has completed its set number of iterations, the position vector of the wolf with the least cost (the alpha) is selected as the trained network’s final weights and biases.
The database in this paper uses the dataset obtained by Asteris and Kolovos [
Descriptive statistics of the experimental data.
Input parameter | Min | Max | Mean | Standard deviation | Median | Range | Standard error | Average deviation |
---|---|---|---|---|---|---|---|---|
Cement | 110 | 600 | 349.22 | 93.43 | 337.5 | 490 | 6.53 | 72.48 |
Limestone powder | 0 | 272 | 25.67 | 60.78 | 0 | 272 | 4.25 | 41.82 |
Fly ash | 0 | 440 | 106.36 | 94.01 | 110 | 440 | 6.57 | 78.60 |
GGBS | 0 | 330 | 17.39 | 52.01 | 0 | 330 | 3.63 | 30.19 |
Silica fume | 0 | 250 | 14.91 | 33.45 | 0 | 250 | 2.34 | 22.16 |
RHA | 0 | 200 | 6.55 | 24.29 | 0 | 200 | 1.70 | 11.88 |
Coarse aggregate | 500 | 1600 | 772.35 | 175.36 | 768.88 | 1100 | 12.25 | 124.16 |
Fine aggregate | 336 | 1135 | 827.93 | 144.33 | 836 | 799 | 10.08 | 105.95 |
Water | 94.5 | 250 | 179.27 | 27.65 | 180 | 155.5 | 1.93 | 20.49 |
SP | 0 | 22.5 | 5.96 | 4.35 | 5.97 | 22.5 | 0.30 | 3.33 |
VMA | 0 | 1.23 | 0.14 | 0.31 | 0 | 1.23 | 0.02 | 0.22 |
The histogram of the 28-day compressive strength of SCC specimens is depicted in Figure
Distribution of compressive strength in the experimental data.
During ANN training, different input variable ranges can negatively affect the resulting model, including the divergence of the optimization algorithm and increased training time [
The minimum and maximum values used for each of the 11 input parameters and the target value of compressive strength are provided in Table
When evaluating the model results, it is essential to define the measures by which the model’s performance and accuracy will be evaluated. The objective used will be the best fitness value (or lowest cost) on test data. The reason for gauging model performance based on test data is to choose the model with the highest generalization capability.
The statistical indices, including 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), are utilized to assess the performance of different topologies. They are defined as follows [
According to Section
ANNs are inclined toward the overfitting phenomenon, meaning that the trained ANN is a high performer (i.e., with minimum error on training data) but fails to perform well on the test data, which are unavailable during the training process. As recommended in literature [
In ANNs, the number of hidden layers and their neurons varies based on the problem [
Considering that the number of effective parameters equals 11, equation (
Artificial neural network architectures trained.
Num | Topology | Num | Topology | ... | Num | Topology | Num | Topology |
---|---|---|---|---|---|---|---|---|
1 | 1-1 | 14 | 2-1 | ... | 144 | 12-1 | 157 | 13-1 |
2 | 1-2 | 15 | 2-2 | ... | 145 | 12-2 | 158 | 13-2 |
3 | 1-3 | 16 | 2-3 | ... | 146 | 12-3 | 159 | 13-3 |
4 | 1-4 | 17 | 2-4 | ... | 147 | 12-4 | 160 | 13-4 |
... | ... | ... | ... | ... | ... | ... | ... | ... |
12 | 1-12 | 25 | 2-12 | ... | 155 | 12-12 | 168 | 13-12 |
13 | 1-13 | 26 | 2-13 | ... | 156 | 12-13 | 169 | 13-13 |
The ANN training process (i.e., adjusting the weights and biases) is a minimization problem. The optimal solution to the problem is choosing the weights and biases that minimize the network error (cost function). The grey wolf optimizer (GWO) algorithm (see Section
MATLAB [
Different variables in the GWO algorithm.
Parameter | Value |
---|---|
Max generations | 500 |
Search agents | 30 |
As mentioned previously, 169 different ANN architectures with two hidden layers were trained using the GWO algorithm. The hyperbolic tangent function was the common activation function of the hidden layers in all the networks. The output layer’s activation function was chosen as the identity function. For simplicity, in the following sections, we will refer to ANNs as ANN 2
The top 10 models among the 169 trained models were selected based on their MSE values. The ANNs and their training and testing data results are provided in Tables
Statistical indexes of the top 10 models in the neural networks on training data.
Network designation | ME | MAE | MSE | RMSE | AAE | EF | VAF (%) |
---|---|---|---|---|---|---|---|
ANN-GWO (7-13) | 0.387 | 3.53 | 27.55 | 5.249 | 0.08 | 0.95 | 0.95 |
ANN-GWO (12-4) | −0.420 | 4.37 | 33.17 | 5.760 | 0.11 | 0.94 | 0.94 |
ANN-GWO (9-6) | −0.120 | 4.21 | 36.31 | 6.025 | 0.10 | 0.93 | 0.93 |
ANN-GWO (8-3) | 0.002 | 3.72 | 40.34 | 6.352 | 0.08 | 0.93 | 0.92 |
ANN-GWO (7-3) | 0.713 | 4.99 | 46.48 | 6.818 | 0.11 | 0.91 | 0.91 |
ANN-GWO (11-4) | −0.390 | 5.54 | 51.50 | 7.177 | 0.12 | 0.90 | 0.90 |
ANN-GWO (7-2) | 0.171 | 5.43 | 57.41 | 7.577 | 0.12 | 0.89 | 0.89 |
ANN-GWO (7-5) | 0.550 | 6.09 | 65.40 | 8.087 | 0.13 | 0.88 | 0.88 |
ANN-GWO (11-9) | 0.597 | 5.79 | 67.50 | 8.216 | 0.14 | 0.87 | 0.87 |
ANN-GWO (12-12) | −0.280 | 6.46 | 73.13 | 8.551 | 0.14 | 0.86 | 0.86 |
Statistical indexes of the top 10 models in the neural networks on testing data.
Network designation | ME | MAE | MSE | RMSE | AAE | EF | VAF (%) |
---|---|---|---|---|---|---|---|
ANN-GWO (7-13) | −0.47 | 3.43 | 27.10 | 5.206 | 0.07 | 0.94 | 0.94 |
ANN-GWO (12-4) | −0.38 | 4.66 | 44.88 | 6.699 | 0.09 | 0.90 | 0.90 |
ANN-GWO (9-6) | −0.42 | 3.33 | 22.16 | 4.708 | 0.06 | 0.95 | 0.95 |
ANN-GWO (8-3) | 0.18 | 3.37 | 27.65 | 5.258 | 0.06 | 0.94 | 0.94 |
ANN-GWO (7-3) | 0.25 | 4.89 | 41.75 | 6.462 | 0.09 | 0.90 | 0.90 |
ANN-GWO (11-4) | −0.57 | 4.63 | 38.49 | 6.204 | 0.09 | 0.91 | 0.91 |
ANN-GWO (7-2) | 0.43 | 4.63 | 41.33 | 6.429 | 0.09 | 0.91 | 0.90 |
ANN-GWO (7-5) | 0.83 | 5.97 | 78.57 | 8.864 | 0.11 | 0.82 | 0.82 |
ANN-GWO (11-9) | 0.27 | 5.15 | 47.22 | 6.871 | 0.09 | 0.89 | 0.89 |
ANN-GWO (12-12) | −1.28 | 5.17 | 53.29 | 7.30 | 0.09 | 0.88 | 0.88 |
RMSE values for the two-hidden-layer ANN for (a) training data and (b) testing data.
According to Tables
Figure
RMSE for the top three models for GWO-ANN models.
Experimental versus predicted values of SCC’s compressive strength for the ANN-GWO (7-13) model using training and testing data. (a) Training and (b) testing.
Experimental versus predicted values of SCC’s compressive strength for the ANN-GWO (7-13) model using all data.
The floating bar chart of the prediction errors of the ANN 2L (7-13) on testing data is depicted in Figure
Floating bar chart of errors of ANN-GWO (7-13) model using testing data.
Two separate models were developed to assess the grey wolf optimizer algorithm’s performance on training an ANN for predicting SCC compressive strength. For this purpose, an ANN was trained using the genetic algorithm and a multivariable linear regression model.
The 169 architectures provided in Table
The GA parameters used for the ANN-GA 2L (11-9) training were determined by trial and error (see Table
Genetic algorithm parameters.
Parameter | Value |
---|---|
Max generations | 150 |
Recombination (%) | 15 |
Crossover (%) | 50 |
Crossover method | Single point |
Floating bar chart of errors of ANN-GA (7-13) model using testing data.
Compressive strength experimental versus predicted values for the ANN-GA (7-13) model using testing data.
For simplicity, a multiple linear regression (MLR) model [
In equation (
Floating bar chart of errors of multiple linear regression model using testing data.
Compressive strength experimental versus predicted values for the multiple regression model using testing data.
The statistical indices of MAE, MSE, EF, and VAF for ANN-GWO 2L (7-13), ANN-GA 2L (11-9), and multiple linear regression models based on the training and testing data are provided in Tables
Training data statistics of the ANN-GWO, ANN-GA, and multiple linear regression models.
Network designation | ME | MAE | MSE | RMSE | AAE | EF | VAF (%) |
---|---|---|---|---|---|---|---|
ANN-GWO (7-13) | 0.387 | 3.53 | 27.55 | 5.25 | 0.08 | 0.95 | 95 |
ANN-GA (11-9) | 0.180 | 12.24 | 238.04 | 15.43 | 0.27 | 0.45 | 45 |
Multiple linear regression | 0.210 | 9.64 | 135.42 | 11.64 | 0.19 | 0.69 | 69 |
Testing data statistics of the ANN-GWO, ANN-GA, and multiple linear regression models.
Network designation | ME | MAE | MSE | RMSE | AAE | EF | VAF (%) |
---|---|---|---|---|---|---|---|
ANN-GWO (7-13) | −0.47 | 3.43 | 27.10 | 5.206 | 0.07 | 0.94 | 94 |
ANN-GA (11-9) | 2.38 | 13.92 | 296.79 | 17.23 | 0.39 | 0.45 | 46 |
Multiple linear regression | −0.57 | 8.68 | 104.96 | 10.24 | 0.20 | 0.80 | 80 |
For a visual comparison of the three models, the Taylor diagram plot is shown in Figure
Comparison between models using normal distribution.
Prediction comparison of all models using testing data.
Given that the ANN-GWO 2L (7-13) model offers superior performance compared to the ANN-GA 2L (11-9) and multiple linear regression models, we conducted a sensitivity analysis to measure the relative influence of each of the 11 input parameters on SCC’s compressive strength. For this purpose, we implemented the profile method suggested by Lek [
Figure
The relative influence of the input parameters on SCC’s compressive strength.
Since the relative importance of FA (fine aggregate), W (water), LP (limestone powder), and VMA (viscosity modifying admixtures) is small (less than 5%), they were removed from the model, and all 169 architectures were trained with 7 inputs instead of 11 as in the previous models. The statistics of the top 3 models selected based on small MSE for training and testing data are given in Tables
Statistics of top 3 seven-input ANN-GWO models on training data.
Network designation | ME | MAE | MSE | RMSE | AAE | EF | VAF (%) |
---|---|---|---|---|---|---|---|
ANN-GWO (11-4) | −0.65 | 8.44 | 114.99 | 10.72 | 0.18 | 0.73 | 74 |
ANN-GWO (7-5) | −0.42 | 10.15 | 151.70 | 12.32 | 0.21 | 0.65 | 65 |
ANN-GWO (11-9) | −0.69 | 8.46 | 115.76 | 10.76 | 0.17 | 0.73 | 73 |
Statistics of top 3 seven-input ANN-GWO models on testing data.
Network designation | ME | MAE | MSE | RMSE | AAE | EF | VAF (%) |
---|---|---|---|---|---|---|---|
ANN-GWO (11-4) | 0.00 | 9.59 | 178.12 | 13.35 | 0.22 | 0.67 | 67 |
ANN-GWO (7-5) | 0.21 | 11.08 | 213.68 | 14.62 | 0.27 | 0.60 | 60 |
ANN-GWO (11-9) | 1.16 | 10.38 | 220.29 | 14.84 | 0.25 | 0.59 | 59 |
The top-performing ANN-GWO (11-4) model has 114.99 and 178.12 for MSE of training and testing, respectively. Compared to 27.55 and 27.10 training and testing MSE of the 11-input ANN-GWO (7-13) model, these values are higher and indicate less accuracy than 7-input models. Although 7-input models are simpler, they will not be considered in the following sections.
This study’s top two empirical models are ANN-GWO 2L (7-13) and multiple linear regression (MLR) models. MLR can be efficiently utilized; however, the ANN-GWO 2L (7-13) model would not be applicable unless the source code is provided. Hence, the weights and biases of the ANN-GWO model are provided in this section. As noted previously, the ANNs input data for each variable must be initially normalized by equation (
This paper aimed to predict the compressive strength of self-compacting concrete (SCC). To this end, a total of 205 experimental results were obtained from the literature. The grey wolf optimizer (GWO) algorithm was employed to train the artificial neural network models. The following points were concluded: For predicting the compressive strength of self-compacting concrete, ANN-GWO 2L (7-13) model provides superior performance compared to other ANN models. The model’s Mean Squared Error and efficiency using test data were 27.10 and 0.94, respectively. A less accurate, but easy to implement, multiple regression model is provided to simplify the prediction process. Benchmarking the ANN-GWO 2L (7-13), ANN-GA 2L (11-9), and multiple regression models portrays the ANN-GWO’s superior accuracy,followed by the multiple regression and ANN-GA models. The sensitivity analysis on the ANN-GWO 2L (7-13) model suggests that the most influential parameters on compressive strength of self-compacting concrete are cement content, ground granulated blast furnace slag, rice husk ash, and fly ash. The fine aggregate content was the least influential factor. The weights and biases of the best performing model, ANN-GWO 2L (7-13), are extracted, and a predictive matrix-based equation is developed for predicting the compressive strength of self-compacting concrete.
The datasets are available in Table
The authors declare no conflicts of interest regarding this paper.