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Figure 2. Illustration of SCE flowchart. nique treats the universal search as a bottom-up population evolution. The SCE combines the advantages of various techniques including competitive evolution, controlled random search, complex shuffling and a simplified version of Nelder-Mead algorithm [51]. Figure 2 illustrates the flowchart of the SCE algorithm.

Figure 2 Illustration of SCE flowchart. nique treats the universal search as a bottom-up population evolution. The SCE combines the advantages of various techniques including competitive evolution, controlled random search, complex shuffling and a simplified version of Nelder-Mead algorithm [51]. Figure 2 illustrates the flowchart of the SCE algorithm.

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Figure 1. Correlation matrix of the dataset (points are data and purple lines show their trend and column charts are histogram). Moreover, Figure 1 depicts the correlation matrix of the dataset. The scatter plots of each two variables are depicted along with their histogram. The last row of this matrix shows the slump (on the y-axis) versus the input parameters (on the x-axis). It can be observed that slump was more correlated to the water content (correlation index = 0.47).
Figure 1. Correlation matrix of the dataset (points are data and purple lines show their trend and column charts are histogram). Moreover, Figure 1 depicts the correlation matrix of the dataset. The scatter plots of each two variables are depicted along with their histogram. The last row of this matrix shows the slump (on the y-axis) versus the input parameters (on the x-axis). It can be observed that slump was more correlated to the water content (correlation index = 0.47).
Table 1. Statistical analysis of the used dataset.
Table 1. Statistical analysis of the used dataset.
Figure 3. Obtained objective functions for different complexities of the ensembles. A multi-layer perceptron (MLP) model is selected as the basic ANN tool for this study. This model is conventionally trained by the Levenberg—Marquardt (LM) algorithm [63]. The VSA, MVO, and SCE metaheuristic algorithms are applied to this network for optimizing its performance. Achieving this objective requires adjusting the weights and biases of the MLP. Therefore, these parameters are considered as the variables of the problem. After creating the VSA-MLP, MVO-MLP, and SCE-MLP hybrid models, the models were fed by the training data. Since the VSA, MVO, and SCE are population-based algorithms, a sensitivity analysis is carried out to find the best population size for each algorithm. To this end, nine different values, namely 10, 25, 50, 75, 100, 200, 300, 400, and 500 were tested for this parameter. Meanwhile, the objective function was set to be the RMSE to measure the error at each iteration. The models were implemented for 1000 iterations. The objective functions obtained for each population size are shown in Figure 3. According to Figure 3, the best accuracy of the VSA-MLP, MVO-MLP, and SCE-MLP was yielded for the population sizes of 300, 200, and 75, respectively. Figure 4a—c illustrate the convergence behavior of these models.
Figure 3. Obtained objective functions for different complexities of the ensembles. A multi-layer perceptron (MLP) model is selected as the basic ANN tool for this study. This model is conventionally trained by the Levenberg—Marquardt (LM) algorithm [63]. The VSA, MVO, and SCE metaheuristic algorithms are applied to this network for optimizing its performance. Achieving this objective requires adjusting the weights and biases of the MLP. Therefore, these parameters are considered as the variables of the problem. After creating the VSA-MLP, MVO-MLP, and SCE-MLP hybrid models, the models were fed by the training data. Since the VSA, MVO, and SCE are population-based algorithms, a sensitivity analysis is carried out to find the best population size for each algorithm. To this end, nine different values, namely 10, 25, 50, 75, 100, 200, 300, 400, and 500 were tested for this parameter. Meanwhile, the objective function was set to be the RMSE to measure the error at each iteration. The models were implemented for 1000 iterations. The objective functions obtained for each population size are shown in Figure 3. According to Figure 3, the best accuracy of the VSA-MLP, MVO-MLP, and SCE-MLP was yielded for the population sizes of 300, 200, and 75, respectively. Figure 4a—c illustrate the convergence behavior of these models.
Figure 4. Convergence behavior of the elite models of (a) VSA-MLP, (b) MVO-MLP, and (c) SCE-MLP.
Figure 4. Convergence behavior of the elite models of (a) VSA-MLP, (b) MVO-MLP, and (c) SCE-MLP.
In this phase, the RMSE of the typical ANN (i.e., the LM-MLP) was 5.5533. This value was decreased to 5.0287, 4.6631, and 5.2971 for the VSA-MLP, MVO-MLP, and SCE-MLP respectively. This demonstrates the efficiency of the used metaheuristic algorithms in optimizing the performance of the ANN. In addition, the reduction of the MAE ranged from 4.4964 for the LM-MLP to 4.0273, 3.8046, and 4.2097 for the VSA-MLP, MVO-MLP, and SCE-MLP, respectively, which indicates that the hybrid ANNs acquired a more reliable understanding of the relationship between the slump and concrete effective factors.
In this phase, the RMSE of the typical ANN (i.e., the LM-MLP) was 5.5533. This value was decreased to 5.0287, 4.6631, and 5.2971 for the VSA-MLP, MVO-MLP, and SCE-MLP respectively. This demonstrates the efficiency of the used metaheuristic algorithms in optimizing the performance of the ANN. In addition, the reduction of the MAE ranged from 4.4964 for the LM-MLP to 4.0273, 3.8046, and 4.2097 for the VSA-MLP, MVO-MLP, and SCE-MLP, respectively, which indicates that the hybrid ANNs acquired a more reliable understanding of the relationship between the slump and concrete effective factors.
Figure 7. Computation time required for implementing the hybrid models. As was inferred, utilizing the VSA, MVO, and SCE metaheuristic techniques for esting (18.83, 13.14, and 36.01% reduction o the best prediction accuracy, followed by the largest time (around 4686 s), followed training the ANN resulted in better accuracy 4.61% reduction of RMSE and 10.43, 15.39, and 6.38% red f RMSE the computation time taken for implementing the pro operating system at 2.5 GHz and 6 Gigs of RAM). for bo E and of MAE) phases. A comparison between the potential of MVO outperformed the VSA and SCE in training t th the training (9.45, 16.03, and uction of MAE, respectively) and 7.00, 10.02, and 32.90% reduction these algorithms reveals that the he ANN. This is while the SCE presented the VSA and MVO. Also, Figure 7 compares posed hybrid algorithms (on the Based on Figure 7, the VSA-MLP took by the MVO, which took around 2801 s. The fastest ensemble was the SCE-MLP with about 867 s. 3.4. Slump Predictive Model
Figure 7. Computation time required for implementing the hybrid models. As was inferred, utilizing the VSA, MVO, and SCE metaheuristic techniques for esting (18.83, 13.14, and 36.01% reduction o the best prediction accuracy, followed by the largest time (around 4686 s), followed training the ANN resulted in better accuracy 4.61% reduction of RMSE and 10.43, 15.39, and 6.38% red f RMSE the computation time taken for implementing the pro operating system at 2.5 GHz and 6 Gigs of RAM). for bo E and of MAE) phases. A comparison between the potential of MVO outperformed the VSA and SCE in training t th the training (9.45, 16.03, and uction of MAE, respectively) and 7.00, 10.02, and 32.90% reduction these algorithms reveals that the he ANN. This is while the SCE presented the VSA and MVO. Also, Figure 7 compares posed hybrid algorithms (on the Based on Figure 7, the VSA-MLP took by the MVO, which took around 2801 s. The fastest ensemble was the SCE-MLP with about 867 s. 3.4. Slump Predictive Model
Figure 8. Importance analysis of the dataset.
Figure 8. Importance analysis of the dataset.
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