Strength property estimation of high-performance concrete with novel admixtures using optimal regression-based approaches

Abstract

Concrete is known as one of the most important materials in the world. Concrete composites consisting of cement, water, aggregates, and some additives are used to improve the properties of concrete. These concrete have a certain compressive strength that can be increased depending on the type of concrete. In line with these ideas, high-performance concrete (HPC) has been produced, which can have high compressive strength by adding materials such as fly ash, silica fume, etc. This type of concrete is used in bridges, dams, and special constructions. However, obtaining the mixture design of HPC is problematic and complex, for this reason, the machine learning methods can make it easy to achieve the output by saving time and energy. This study has used support vector regression (SVR) to predict the compressive strength of HPC. Moreover, this study provided two meta-heuristic algorithms for obtaining suitable and optimized results, which are contained the artificial hummingbird algorithm (AHA) and Sine Cosine Algorithm (SCA). The model by coupling with algorithms created the hybrid method in the framework of SVR-AHA and SVR-SCA. Furthermore, some criteria indicators have been used for determining the most desirable hybrid model, which is included coefficient of correlation (R²), root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and weight absolute percentage error (WAPE). As a result, the AHA algorithm could have a more satisfying association model with the SVR model, and the results were RMSE = 2.00 (MPa), R² = 98.59%, MAE = 0.717 (MPa), MAPE = 1.22 (MPa), and WAPE = 0.114 (MPa).

Keywords

High-performance concrete sine cosine algorithm artificial hummingbird algorithm support vector regression compressive strength

1 Introduction

Concrete material is the most commonly utilized material for infrastructure, sidewalks, and buildings [1]. Regular concrete usually includes coarse aggregate (CA), Normal Portland cement (NPC), water (w), and sand (S). Much attention has been paid to the other materials’ usage in concrete mixes to create more durable concrete. Decreasing the amount of carbon dioxide (CO2) emitted is one of the key benefits of using these materials [2]. Cement factories account for about 5–7% of the world’s CO2 emissions. The usage of supplemental cement material as an alternative to OPC in concrete mixes can decrease the CO2 amount emitted in connection with its product [3]. In addition, although NCs have many usages in the construction industry, has the disadvantages of low machinability, low durability, and low strength in special applications like sidewalks, high-rise buildings, and long bridges. Highly performance concrete (HPC), which typically includes SCM such as Silica fume (Sf) and fly ash (Fa), can improve concrete performance and reduce issues related to the NC’s usage [4–6].

HPC is generally associated with superior compressive strength (CS) and lower porosity values than NC. The CS’s value is able to be acquired either by the predictive model or experimental testing and is a needed parameter in design code. Predictive models can fast generate design data that need saving time, energy, and cost. In-situ strength information can also furnish detail on when to build and remove scaffolds. Over the last 20 years, many researchers have proposed predictive models and observed formulas for predicting CS from NC [7–9]. Nevertheless, estimating HPC’s CS using these models has been questioned because of differences in mixing ratios and curing properties. Therefore, a predictive model specially designed to predict the HPC’s CS is needed to get a reliable estimated CS value. In addition, developing reliable models requires a detailed dataset containing various potential mix ratios and conditions [10, 11].

Machine learning (ML) techniques are utilized in many engineering disciplines because they can be prioritized, optimized, planned, and predicted. In the concrete industry, ML technology has been employed primarily to model the different freshness and hardening properties of different sorts of concrete, like high-performance concrete (HPC) [12–15]. The ML methods used to estimate concrete’s C.s contain artificial neural networks (ANN) [16], Adaptive Neuro-Fuzzy Inference System (ANFIS) [17], support vector machines (SVM) [18], genetic programming (GP) [19], Biogeography Based Programming (BBP) [20], classification and regression Tree (CART) [21], etc.

Regarding HPC’s CS, use SVM to estimate mixtures’ CS, including Sf and Fa. Other researchers have also utilized SVMs to predict the HPC’s C.s, including various kinds of supplemental cement materials like Sf, Fa, copper slag, and nano-silica [22–25]. Developing equations that are easier to understand than other methods like SVM is an advantage of SVM technology [26]. Because of the superior performance of the hierarchical structures to the flat structures, a combination of regression and classification techniques was employed to improve HPC’s CS predictions [27]. Regression and hierarchical classification have been reported to be superior to traditional forecasting methods like ANN and SVM. SVMs allow a group of datasets that share common characteristics. This can improve the reliability of the models associated with some datasets [28].

Traditional optimization algorithms supply better prediction results than numerical or statistical methods but only locally find the optimal solution. In addition, several factors (hardening, mixing, mixing ratio, etc.) can affect the concrete mixture’s CS [29, 30]. A large number of hidden neurons are needed to explain the interactions between the influencing factors, which can lead to the problem of overfitting. Several processes, like hybrid methods, have been introduced to cover the lack of classical optimization algorithms [31].

In this study, the regression method of support vector machine i.e., SVR has been used to predict one of the mechanical properties of HPC containing CS. The prediction process has performed via experimental laboratory samples. Furthermore, for improving the accuracy of the model and obtaining a suitable result the related model has coupled with two meta-heuristic optimization algorithms. The algorithms can improve the predicted results to be close to experimental outputs, which are consisting of the artificial hummingbird algorithm (AHA) and Sine Cosine Algorithm (SCA). In addition, the result of hybrid models has been compared with each other by some criteria indicators, where the results are explained in the next sections.

2 Materials and methodology

As mentioned, SVR has been analyzed to model the HPC’s compressive strength composites. Moreover, two optimization algorithms were utilized to compute the optimal number of neurons in the hidden layer and the applied propagation rate. Investigations on the Sine Cosine Algorithm (SCA) and Artificial hummingbird algorithm (AHA) have indicated that these algorithms provide favorable and suitable responses. Furthermore, scholars and researchers have utilized the SCA and AHA optimizers as applicable and useful methods for optimizations in various disciplines. Two-hybrid frameworks, SVR-AHA and SVR-SCA, were generated by combining the SVR’s main model with the proposed optimizer.

2.1 Data gathering

The dataset used to allow the model is from the literature. The dataset includes cement (C.e), fly ash (F.a), Silica fume (S.f), coarse aggregate (C.a), total aggregate (T.a), High rate water reducing agent (H.rwra), water (W), Curing time (Cu), and compressive strength (C.s). The dataset is divided into two parts, training and testing. The statistical characteristics are shown in Table 1. In addition, Fig. 1 illustrates the input variables of the model and the spectrum of their effect on the compressive strength of HPC.

Fig. 1

The range of Dataset component and their effects on the com-pressive strength.

Table 1

The statistical features of Dataset components

Statistical features	Dataset components
	Cement	Fly ash	Silica-fume	Coarse aggregate	Total aggregate	High-rate water reducing agent	Water	Curing age	Compressive strength
Max	500	275	43.3	1157	1867	13	205	180	107.8
Min	394	0	0	1086	1668	0	118.2	28	24
Mean	418.381	101.077	21.8428	1133.57	1805.78	4.47190	165.638	68.5714	64.0323
St. Dev	40.13686	66.00055	17.87788	24.40491	63.2487	3.0936	28.703	49.985	15.355

2.2 Support Vector Regression (SVR)

Support Vector Machine (SVM) is one of the learning models for classification and regression. Regression SVM is specifically called support vector regression. Support vector regression (SVR) by employing suitable nonlinear or linear kernel functions.

2.2.1 Linear SVR

Support vector regression utilizes the same linear regression multipliers as support vector machines, but unlike the SVM, SVR sets the allowable margin (ɛ) as indicated in Fig. 2 to a single and approximate value.

Fig. 2

SVR linear form [32].

The support vector regression is represented in Equation 1 utilizing a linear multiplier function. $f (x) = xu + c$ (1)

f (x): Represents the input space

x, u, c: Constant, representing the vector product

An indication according to the error function in Equation 2 can be minimized if the target is y_i. $\begin{matrix} \min . \frac{1}{2} ∥ m ∥^{2} \\ {\begin{matrix} y_{i} - (x . u + c) ⩽ ɛ + ξ_{j} \\ (x . u) + c - z ⩽ ɛ + \underset{j}{ξ^{*}} \\ ξ_{j}, \underset{j}{ξ^{*}} ⩾ 0 \end{matrix} \end{matrix}$ (2)

2.2.2 Non-linear SVR

As indicated in Fig. 3, nonlinear support vector regression utilizes a nonlinear kernel function to method feature space training data.

Fig. 3

SVR non-linear form [32].

Standard SVR employing Equation 3 is applied after processing the training data in the feature space.

$\begin{matrix} \max {\begin{matrix} \frac{1}{2} \sum_{j, i = 1}^{n} (a_{j} - a_{j}^{*}) (a_{i} - a_{i}^{*} h) (y_{j}, y_{i}) \\ - ɛ \sum_{j = 1}^{n} (a_{j} - a_{j}^{*}) + \sum_{j = 1}^{n} y_{j} (a_{j} - a_{j}^{*}) \end{matrix} \\ 0 ⩽ a_{j}, a_{j}^{*} ⩽ T \end{matrix}$ (3)

$a_{j}, a_{j}^{*}$ : multiplier Lagrange.

2.3 Hybrid sine cosine algorithm based SVR (SCA-SVR)

The sine cosine algorithm (SCA) corresponds to the mathematical properties of the sine and cosine functions of the recently extended metaheuristic algorithms. Mirjalili extended this algorithm in 2015 and 2016 [33]. Alternatively, human investigative methods begin with an optimization method from a random initial solution set. These random solutions are iteratively generated by the objective functions produced and utilized by the optimization operators’ set that form the basis of the optimization procedure performed. There is no guarantee that a solution will be available all at once, as human research methods will probabilistically try to find the optimal value for the problem. Nevertheless, using multiple probabilistic solutions and optimizations process increases the likelihood of achieving global optimizations [33].

Like the same population-based aggregation algorithms, the SCA starts with a randomly assigned set of solutions. The use of the following equation is updated for each solution: $y_{j, t + 1} = P sin (c) | D y_{best} - y_{j, t} |$ (4) $y_{j, t + 1} = P cos (c) | D y_{best} - y_{j, t} |$ (5)

In SCA, the above two equations are employed in the following: $y_{j, t + 1} = {\begin{matrix} y_{j, t} + P sin (c) | D y_{best} - y_{j, t} | if rand < 0.5 \\ y_{j, t} + P cos (c) | D y_{best} - y_{j, t} | otherwise \end{matrix}$ (6)

y_j,t+1 and y_j,t: define the j^th iteration’s solution (t + 1) and t

c: Vector utilized to determine the moment’s direction in the current solution. It may be in the direction towards y_best and outside of y_best.

y_best: The most suitable solution in the solution set.

rand: Random numbers are evenly distributed during the pause (0, 1).

P: A random vector that can determine the search space around the region of the current solution.

D: Weighted vector y_best, with emphasis on explore (D > 1) and inflection (D < 1). Vector D also avoids premature convergence at the end of the iteration.

y_best and y_j,t are within and outside the search space. The P parameter is also useful for exploring and employing the proper balance and exploring space between them. In the first half of the maximum number of iterations, factor P is search-only, and in the second half of the maximum number of iterations, it is dedicated to the use of available search space. The vector (P) can be mathematically expressed as: $P = 2 - 2 (\frac{t}{R})$ (7)

R: Indicates the maximum number of repetitions specified as the SCA cancellation criterion.

The vector rand is useful for transitions from sine functions to inverse and cosine functions.

Figure 4 indicates the range of the sine and cosine functions for parameter P.

Fig. 4

The sine and cosine functions’ range with parameter P [34].

2.4 Hybrid Artificial hummingbird algorithm base SVR(AHA-SVR)

The Artificial Hummingbird Algorithm (AHA) [35] is an optimization method that simulates hummingbird feeding and flight. Hummingbirds are the smallest birds in the world, measuring 7 to 13 cm in length. The unique bird, the hummingbird, is found only in the United States and Central America, where most of these species live due to tropical and subtropical conditions. This bird can fly at speeds up to 45 km / h. Moreover, the beat rate of fast wings can be suspended in the air, ranging from about 12 beats per second for the largest seed to more than 80 beats for the smallest seed. Hummingbirds’ staple food is composed of nectar, and they must rely on feeding to maintain high body metabolism.

Foraging for hummingbirds includes two feeding stages. The first is the self-discovery phase. At this stage, hummingbirds can seek their cognitive behavior without interacting with other populations. Essentially, a person’s cumulative explore experience follows the cognitive manner of the selective explore method. The guidance exploration stage is the second phase. Hummingbirds can not only seek experience but can also utilize many dominant individuals as guidance.

In algorithmic form, AHA can be divided into three main models:

1) Guiding Foraged

There are three flight manners: axial flight, diagonal flight, and omnidirectional flight, which are employed in this feeding model for feeding. These three flight motions are indicated in Fig. 5 in 3D space. The following equation can simulate these guided foraging and feeding sources:

Fig. 5

The demeanor of the three flights in 3D space [35].

$g_{i} (t + 1) = y_{j, ta} (t) + d . r . (y_{j} (t) - Y_{j, ta} (t)) d$ (8)

y_j,ta (t): indicates the target food source’s location

y_j (t): show the factor of guiding

y_j (t): at (t) the j^th food source’s location

The following update on the j^th food source position is: $y_{Bj} (t) = {\begin{matrix} y_{j} (t) f (y_{j} (t)) ⩽ f (g_{i} (t + 1)) \\ g_{i} (t + 1) f (y_{j} (t)) > f (g_{i} (t + 1)) \end{matrix}$ (9)

f (g_i (t + 1)) shows the values of matching for g_i (t + 1) and y_j (t)

f (y_j (t)) indicates the values of matching for y_j (t)

2) Foraged of territory

The following equation expresses the local feeding of hummingbirds in a territorial foraging method: $g_{i} (t + 1) = y_{j, ta} (t) + f . r . (y_{j} (t)) f$ (10)f shows the factor of territory

3) Foraged of migrating

In the following, the equation for hummingbird feeding is defined: $g_{wor} (t + 1) = ld + c . (ud - ld)$ (11)

g_wor defines food sources with the worst population of nectar supplements.

c is a factor of random.

ud and ld show the limit ranges of lower and upper.

2.5 Evaluation criteria

This study evaluated the SVM model’s results using several criteria like the coefficient of resolution (R²), root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and weight absolute percentage error (WAPE). R² is an important issue in regression analysis. This is solved by ranging from 0 to 1 as the square of the correlation (R) between the predicted and actual results. A high R² value indicates a good correlation between the predicted and actual values. RMSE measures the mean squared difference between the SVM model’s actual and predicted outputs, while MAE measures the mean error between them. In contrast to R², lower RMSE, and MAE values indicate the SVM’s more suitable performance. The quantities R², RMSE, MAE, MAPE, and WAPE are described in the following: $R^{2} = {(\frac{\sum_{i = 1}^{n} (p_{i} - \bar{p}) (t_{i} - \bar{t})}{\sqrt{[\sum_{i = 1}^{n} {(p_{i} - \bar{t})}^{2}] [\sum_{i = 1}^{n} {(t_{i} - \bar{t})}^{2}]}})}^{2}$ (12) $RMSE = \sqrt{\frac{1}{n} \sum_{i = 1}^{n} {(t_{i} - p_{i})}^{2}}$ (13) $MAE = \frac{1}{n} \sum_{i = 1}^{n} | t_{i} - p_{i} |$ (14) $MAPE = \frac{100}{n} \sum_{i}^{n} \frac{| p_{i} |}{| A_{i} |}$ (15) $WAPE = \max [\frac{| p_{i} - t_{i} |}{p_{i}}]$ (16)

In the above equations, p_i shows the predicted value, t_i shows the observed value, $\bar{p}$ and $\bar{t}$ determine the mean value of predicted and observed value, respectively, and n indicates the number of samples.

3 Results and discussion

This article uses the support vector regression (SVR) model employed for laboratory modelling and estimation to predict the compressive strength of concrete in high-performance concrete (HPC) and obtain fair and acceptable results. SVR uses the optimized Artificial Hummingbird Algorithm (AHA) and Sine Cosine Algorithm (SCA). Based on previous statements, the optimizers mentioned are among the most recent and innovative algorithms, which conclude the error rate and, therefore, the potency of every optimizer in the following. Moreover, the modelling example has two stages including training and testing, whose output is represented according to each optimizer’s inputs. 70% of the samples belong to the tensile section and the remaining 30% to the test section.

The approach is taken for integrating the SVR model with the optimization techniques involved utilizing the optimization techniques to determine the number of neurons and spread rate in the SVR network. These variables, which define the network structure (excluding the activation function), were considered optimization parameters, while the SVR itself was used as the cost function. In each iteration of the optimization process, the variables were determined by the optimizers and fed to the cost function. The cost rate, which was evaluated using the Root Mean Square Error (RMSE), as compared with other cost rates to find the minimum RMSE over the range of iterations. The values of the variables corresponding to the minimum RMSE were considered the optimal structure of the SVR network. To ensure a fair comparison between the two hybrid models, the same number of iterations with the same domain of variables were used.

A step-by-step explanation of the SVR formulation, including the equations and formulas used at each step:

Step 1: Data Preparation

Collect the training dataset consisting of input feature vectors (x_i) and corresponding target values (y_i).

Step 2: Kernel Function Selection

Choose a suitable kernel function, such as the Gaussian radial basis function (RBF), to compute the similarity between input feature vectors.

The RBF kernel function is used in SVR and is defined as: $\begin{matrix} K (x_{i}, y_{i}) = \exp (- γ x_{i} - {y_{i}}^{2}) \\ γ = - 1 / 2 \times sigm a^{2} \end{matrix}$ (17)

Step 3: Define the Epsilon (ɛ)

Epsilon (ɛ) is a hyperparameter that determines the width of the ɛ-insensitive tube around the predicted values.

The ɛ-insensitive tube allows for deviations within ɛ without incurring a loss penalty.

The SVR formulation aims to minimize deviations larger than ɛ.

The value of ɛ depends on the specific problem and desired trade-off between model complexity and accuracy.

Step 4: Formulate the Optimization Problem

The SVR optimization problem seeks to minimize the objective function while adhering to the constraints.

The objective function is defined as: $\begin{matrix} min \frac{1}{2} ∥ ω ∥^{2} + C \sum_{i = 1}^{m} (ξ_{i} + ξ_{i}^{*}) \\ s . t . {\begin{matrix} y_{i} - (ω^{T} x_{i} + b) ⩽ ɛ + ξ_{i} \\ (ω x_{i} + b) - y_{i} ⩽ ɛ + ξ_{i}^{*} \\ ξ_{i}, ξ_{i}^{*} ⩾ 0 \end{matrix}} \end{matrix}$ (18)

In this objective function, $\frac{1}{2} ∥ ω ∥^{2}$ represents the norm (length) of the weight vector ω, C is the regularization parameter, and ξ_i and $ξ_{i}^{*}$ are slack variables that capture deviations beyond the ɛ-insensitive tube.

Step 5: Solve the Optimization Problem

Utilize optimization algorithms such as quadratic programming or the Sequential Minimal Optimization (SMO) algorithm to solve the formulated optimization problem and obtain the Lagrange multipliers (α_i).

These Lagrange multipliers correspond to the support vectors, which are the training samples that lie within or on the margin.

Step 6: Compute the Bias Term

Using the support vectors and their corresponding Lagrange multipliers (α_i), compute the bias term (intercept) of the SVR model.

The bias term is given by: $b = (1 / N_{s}) * Σ (y_{i} - Σ α_{i} * y_{i} * K (x_{i}, x_{s})),$ (19) where N_s is the number of support vectors and x_srepresents the support vectors.

Step 7: Make Predictions

Once the Lagrange multipliers (α_i) and the bias term (b) are obtained, use the SVR model to make predictions on new, unseen data points.

Compute the predicted value (f (x)) for a new input feature vector x using: $f (x) = Σ α_{i} * y_{i} * K (x_{i}, x) + b$ (20)

By following these steps and applying the equations and formulas, it can be constructed and trained an SVR model to make accurate predictions compressive strength.

The values computed by optimization algorithms for Epsilon, C and Sigma are presented in Table 2.

Table 2

The key parameters of SVR model determiened by Novel optimization algorithms

Model	SVR Parameters
	Epsilon	C	Sigma
AHA-SVR	0.2154	1000	0.1
SCA-SVR	2.5860	359.64	0.13568

Based on Table 3, the SVR model is converged with two optimizers including AHA and SCA, to create a combined SVR-SCA and SVR_AHA model. Each hybrid model contains two sections: train and test. The output is indicated with R², and Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and Worst Absolute Percentage Error (WAPE) parameters. The data in Table 3 indicates that the results acquired with the presented model show that the AHA optimizer can be combined with the SVR model to perform more suitable and more appropriate results than SCA. R² has the best value in the test section of SVR-AHA with 98.59%, and the weakest performance in R² belongs to the SVR-SCA in the train section with a 97.39% value. RMSE is the minimum and maximum values of 2.00 and 2.87 and reached the SVR-AHA of the test section with the fewest errors and the SVR-SCA of the test section with the most errors. The MAE in the SVR-AHA test section is 0.717, which is the smallest error in this parameter, but the largest error belongs to the SVR-SCA test section, numbered 2.52. The MAPE is equal to 1.22 in the test section of SVR-AHA, which is the most suitable in this modelling, and the lowest value is 4.12, which belongs to the SVR-SCA in the test section. As well as other parameters, the hybrid model SVR-AHA has the best performance with a value of 0.013 in the test section, and on the other hand, the weakest performance for the SVR-SCA model is in the test section with a value of 0.04.

Table 3

The results of the models are considered to evaluate their accuracy

Metrics	Models
	SVR-AHA		SVR-SCA
	Train	Test	Train	Test
Correlation coefficient (R²)	0.9788	0.9859	0.9739	0.9777
Root Mean Square Error (RMSE)	2.1297	2.0026	2.3767	2.8768
Mean Absolute Error (MAE)	0.8602	0.7177	1.2714	2.5286
Mean Absolute Percentage Error (MAPE)	1.3543	1.2205	2.0869	4.1272
Weight Absolute Percentage Error (WAPE)	0.0133	0.0114	0.0197	0.0402

Figure 6 illustrates the results of models considering the evaluation metrics of two-hybrid models in the train and test section. In the comparison of the metrics related to the models in the training section, it can be seen that in R², the models have shown similar behavior and the values are close to each other. In RMSE and MAE, the SVR-AHA model has a difference of 10% and 32%, respectively, compared to SVR-SCA. In addition, in MAPE and WAPE, the combined SVR-AHA model has been able to obtain better values so that it has a difference of 35% and 32% with SVR-SCA, respectively. On the other hand, the SVR-AHA test section, by improving its performance, has obtained more appropriate values than the training section, which shows that this improvement shows the good training of the samples in the training section. In general, in both sections, the SVR-AHA hybrid model has obtained better values than SVR-SCA.

Fig. 6

The results of models considering the evaluation metrics of two-hybrid models in the train and test section.

Figure 7 shows a model’s correlation graph outputs and measurements of two-hybrid models. In these figures, the centerline has a coordinate of X = Y and the linear fit describes the midline of all samples. Two lines are drawn in the +10X% and -10X%, which are indicated the over and underestimated points. For each model, the figures indicate two parameters, R² and RMSE, which determine to be in a series and desperation of points, respectively. As illustrated in Fig. 7-a, The points related to learning in SVR-AHA have a certain amount of dispersion, so in one sample it has been observed to underestimate, but by improving its performance, i.e. increasing the value of R² and reducing the RMSE, the dispersion in the points has reached its minimum. Also, there is a difference between linear fit and centerline. On the other hand, in Fig. 7-b, which characterizes the performance of SVR-SCA, it can be seen that in the learning section, the points are scattered and underestimated. In addition, in the validation section, this dispersion and the difference between the linear fit and centerline lines have increased, which indicates a weak performance. The weak performance in the validation section shows that the sample was not properly trained in the learning section.

Fig. 7

The accuracy correlation of present models.

Figure 8 shows the error percentage distribution diagram of the present models. The error of related models has been examined, which are split between 70 and 30 percent of samples in training and testing sections, respectively. In the learning part of SVR-AHA, the error obtained was equal to approximately 14%, which remained the same value in the validation part, with the difference that the dispersion of errors has decreased compared to learning. On the other hand, for SVR-SCA, its behavior has been similar to SVR-AHA, but it has obtained a greater error, which is equal to approximately 18%. In general, it can be concluded that SVR-AHA has been able to perform more desirable than SVR-SCA.

Fig. 8

The error percentage distribution diagram of present models.

4 Conclusion

Concrete material is the most commonly utilized material for infrastructure, sidewalks, and buildings. The compressive strength (CS) value can be acquired either by the predictive model or experimental testing and is a needed parameter in design code. Predictive models can fast generate design data that need saving time, energy, and cost. For this reason, this paper intended to use one of the machine learning models to predict the mechanical properties of high-performance concrete (HPC). The proposed model is the regression form of support vector machine (SVM), which is called support vector regression (SVR). In addition, in order to optimize the results of the model and increase the accuracy, which reduces the error, two meta-heuristic algorithms, are Artificial Hummingbird Algorithm (AHA) and Sine Cosine Algorithm (SCA), were used. The optimizers combined with the corresponding model formed a hybrid method that was in the framework of SVR-AHA and SVR-SCA. The hybrid models were validated by laboratory samples extracted from the published article. This verification was carried out in two sections, training, and testing, which were allocated 70% and 30% of the samples, respectively. In addition, some evaluator metrics were used to compare the two hybrid models, including coefficient of correlation (R²), root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and weight absolute percentage error (WAPE). The results were as follows:

The highest value of R² in both sections was obtained by SVR-AHA.

For RMSE, SVR-AHA was able to obtain the most suitable value compared to SVR-SCA, so they differed by 10% in training and 30% in testing.

In general, it was concluded from the obtained results that the AHA optimizer was able to obtain a more desirable combination with SVR with high accuracy.

Moreover, can be concluded that machine learning methods can be suitable for predicting the mechanical properties of HPC.

Footnotes

Acknowledgment

Appendix A

CS	B	FA	MS	SP	CA	TA	W	Age
90.4	500	125	0	9.25	1086	1769	150	56
51.43	394	78.8	39.4	3.94	1145	1861	197	28
99.6	500	75	0	7.5	1086	1786	150	90
56	400	60	0	4.4	1157	1847	160	28
64.29	394	78.8	39.4	3.94	1145	1861	118.2	28
60.7	410	82	20.5	0	1132	1710	205	56
46.5	394	118.2	43.3	3.94	1145	1861	157.6	28
60.8	400	100	0	4.8	1157	1817	160	56
70.2	400	100	0	4.8	1157	1817	160	180
55.7	410	61.5	0	0	1132	1721	205	56
59.1	400	160	20	6	1157	1793	160	90
35.6	410	184.5	0	0	1132	1681	205	28
97.7	500	0	0	7.5	1086	1810	150	180
77.79	394	78.8	43.3	3.94	1145	1861	118.2	28
52.9	400	220	0	5.5	1157	1778	160	90
57.3	410	0	20.5	0	1132	1737	205	28
63.2	400	220	0	5.5	1157	1778	160	180
53.7	410	102.5	0	0	1132	1708	205	90
49.23	394	118.2	35.5	3.94	1145	1861	157.6	28
72.4	400	80	20	5.5	1157	1819	160	90
47.84	394	118.2	39.4	3.94	1145	1861	197	28
70.5	400	0	0	4	1157	1867	160	90
63.7	400	180	0	5.2	1157	1791	160	180
48.7	410	164	20.5	0	1132	1684	205	90
49.73	394	118.2	35.5	3.94	1145	1861	197	56
59.9	394	98.5	43.3	3.94	1145	1861	157.6	28
49.16	394	78.8	35.5	3.94	1145	1861	197	28
62.16	394	118.2	35.5	3.94	1145	1861	118.2	56
85.8	500	100	25	9.25	1086	1772	150	56
107.8	500	125	0	9.25	1086	1769	150	180
54.62	394	78.8	35.5	3.94	1145	1861	197	56
45.79	394	59.1	39.4	3.94	1145	1861	197	28
71.43	394	78.8	39.4	3.94	1145	1861	118.2	56
62.23	394	78.8	43.3	3.94	1145	1861	197	28
99.3	500	0	25	8	1086	1805	150	180
66.2	400	100	0	4.8	1157	1817	160	90
69.71	394	98.5	39.4	3.94	1145	1861	118.2	28
52.63	394	118.2	39.4	3.94	1145	1861	157.6	28
52.84	394	118.2	43.3	3.94	1145	1861	118.2	28
44.76	394	118.2	35.5	3.94	1145	1861	197	28
52.9	410	82	20.5	0	1132	1710	205	28
85.4	500	125	0	9.25	1086	1769	150	28
68.49	394	59.1	43.3	3.94	1145	1861	118.2	56
54.1	410	184.5	0	0	1132	1681	205	90
38.4	410	164	20.5	0	1132	1684	205	28
59.81	394	118.2	39.4	3.94	1145	1861	118.2	28
54.45	394	98.5	43.3	3.94	1145	1861	197	28
97.7	500	225	0	10.5	1086	1736	150	180
52.86	394	78.8	43.3	3.94	1145	1861	197	28
43.9	400	180	0	5.2	1157	1791	160	28

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