Using a machine learning method with novel optimization algorithms in estimating compressive strength rates in ultra-high performance concrete

Abstract

To effectively incorporate Ultra-High-Performance Concrete (UHPC) into a construction project, it is essential to inquire into its composition and content to determine the concrete's pertinent Compressive Strength (CS). Determining the relationships between ingredients may necessitate additional expenditure and energy expenditure. The present research endeavor aimed to replicate the CS behavior of UHPC via environmentally sustainable constituents. In this context, the present study employed Support Vector Regression (SVR) as a machine learning approach, coupled with Biogeography-Based Optimization (BBO) and Flow Direction Algorithm (FDA), to construct an accurate model of concrete compressive strength (CS). Coupled machine learning models with optimizers can be a powerful tool for predicting the mechanical properties of UHPC and other complex materials. By improving accuracy and efficiency, these models can help accelerate the development of new UHPC formulations with desired mechanical properties, optimize manufacturing processes, and reduce the associated costs. The modeling of the CS values utilized a total of eight components. In general, the presented study indicated that SVR-FDA had obtained a high correlation and low errors compared to SVR-BBO, which can be concluded that the hybrid machine learning method saves time and energy against laboratory experiments.

Keywords

ultra-high-performance concrete compressive strength support vector regression flow direction algorithm biogeography-based optimization

1 Introduction

Ultra-high performance concrete (UHPC) is a substance with many aggregates, making it a highly suitable material for certain applications. Notably, UHPC exhibits exceptional self-compactness and a remarkable compressive strength exceeding 150 (MPa). Additionally, this concrete type demonstrates impressive resistance when subjected to challenging environmental conditions. The present composite material has been employed in structures featuring diverse strength mechanisms.^1–3 The reduction of cement and micro-silica has been demonstrated to have a positive impact in terms of cost savings and reduction of CO2 emissions. Nevertheless, the superior performance and increased durability compared to traditional or high-strength concrete validate the elevated initial expenses and ecological repercussions. In view of the aforementioned, the crucial role played by cement reduction renders it capable of augmenting the sustainability of the construction industry in the form of environmentally efficient buildings.^4,5

In recent years, various scholars have conducted extensive research into the characteristics and applications of UHPC. UHPC has exhibited compressive strengths that typically range from 150 to 810 MPa. The use of additives such as nanosilicate, metakaolin, fly ash, and micro-silica has received considerable attention from researchers in concrete mixture development.⁶ It is paramount to delineate the comparability of the impact on the mechanical characteristics of concrete, especially concerning its compressive strength (CS). Physical properties form the basis for ascertaining diverse attributes of concrete, and concrete placement is predominantly rooted in the utilization of CS.^7,8

The impact of admixtures employed in Pozolan homogeneous concrete on the concrete's tangible characteristics is attributable to their capacity to manipulate the cementitious constituents in the concrete mixture. Owing to its notably diminutive particle size, silica fume is capable of fulfilling both the functions of a filler and pozzolan simultaneously in the composition of the concrete mixture. Adding micro-silica to concrete while improving its compressive strength in the short term reduces workability. The precise estimation of the optimal proportion of silica fume pertaining to attaining maximum compressive strength remains uncertain.^9,10 The researchers have introduced a novel micro-silica replacement technique with multiple ratios aimed toward maximizing the CS of concrete. In light of the relatively diminutive size of the portions in comparison to those of cement, it appears that the response of silica vapor is akin to that of pozzolan, resulting in an enhancement of the properties of concrete. Incorporating micro-silica in conjunction with fly ash, in addition to using superplasticizers, can effectively mitigate porosity and enhance the CS of the material.^11,12

The constituent of fly ash, possessing a physical structure akin to that of Portland cement in terms of shape and size, serves as a potential water-reducing agent in the production of concrete. The optimal amalgamation of fly ash and superplasticizer significantly enhances the mechanical properties of concrete, specifically its relevant compressive strength. Additionally, the utilization of additives such as fly ash pertinently impacts critical factors, including resistance, feasibility, concrete expenditure, and the capacity for water penetration. An often employed application of fly ash involves the substitution of adhesive substances. The utilization of fly ash as an ingredient in concrete formulations has been found to promote a reduction in levels of harmful pollutants, thereby leading to significant positive environmental outcomes. The substitution of fly ash in concrete can occur within a range of 20% to 50% of the complete adhesive capacity. In cases where the primary consideration is the original intensity of the concrete, fly ash may increase substitution rates up to 60%.^13–15

Intelligent techniques, notably machine learning, have found extensive utility in the field of engineering. Professionals utilize these approaches for determining distinct characteristics. Through the implementation of diverse methodologies, researchers have effectively rendered simulations to gauge the performance of UHPC. The efficacy of these methods is contingent upon the availability of data sets that facilitate the construction of reliable models. It is important to note that the precision of outcomes is predicated upon the particular species captured during experimental inquiries or derived from information contained within historical records.^16–18 A research investigation was conducted involving the application of a gene expression algorithm for the purpose of assessing the CS of concrete that incorporates sugarcane ash as an ingredient. In order to evaluate the precision of the model, empirical strength measurements were contrasted against the model output. A genetic programming model was developed in a recent study aimed at assessing the compressive strength of cement that contains both micro- and nano-silica.^19–21 In a recent investigation, a predictive framework was constructed for estimating the CS of concrete that incorporates micro-silica. This approach aims to simplify the model's intricacy and cost by leveraging artificial neural networks and enhancing the model through the application of the Grey Wolf Optimization (GWO) algorithm.²²

1.1 Literature review

This study uses a simulation algorithm to investigate one machine learning method based on support vector regression (SVR) to predict the CS of UHPC. Due to successful results of modeling with SVR rarely found in UHPC, CS estimation can find several articles with referred topics in other fields.^23–25 In an article, SVM, gene expression programming (GEP), and ANN were employed by Furqan et al.²⁶ on 300 data sets to predict the concrete compressive strength with desirable results of the Support Vector Machine. In other research, Zhang et al.²⁷ employed the SVR model, besides other models, to estimate the concrete CS at seven days and found that the nonlinear model of SVR had better predictive performance than the other linear models.

Besides, two new optimization algorithms are used for this task, including Flow Direction Algorithm (FDA) and Biogeography-Based Optimization (BBO). The research done using two metaheuristic algorithms can prove the powerful capability of the algorithms.^28–31 In a research conducted by Kazemi and Naser,³² to better model the CS of concrete, the BBO algorithm was linked to ANN the final result showed an accurate result of hybrid model. Also, Golafshani and Behnud,³³ to find the best mixture percentage of concrete having several additives, used a BBO algorithm with a target parameter of compressive strength. The final results showed BBO model can be successfully employed to predict the compressive strength of silica fume concrete with acceptable accuracy.

Abuodeh et al.³⁴ used artificial neural networks (ANN) in a recent study. They utilized sequential attribute selection strategies and neural interpretation diagrams to identify the mixture variables that impact the performance of the ANN models. They compiled a dataset of 110 UHPC blend designs to predict CS after 28 days. They achieved high accuracy in their estimates and suggested gathering more extensive data to enhance the model's effectiveness and applicability, given the tiny dataset and limited number of variables. Marani and Nehdi³⁵ utilized 154 data samples to create a machine learning model for predicting the CS of concrete using phase change materials. They theorized that expanding the dataset would enhance the model's capacity to generalize and offer valuable insights into the materials science aspects of the issues, even though it was already very accurate. It is crucial to gather a comprehensive collection of experimental data to accurately anticipate the nonlinear relationship between the CS of UHPC and different mix variables of UHPC using machine learning. The incorporation of curing regimes, which include relative humidity, temperature, and duration, offers useful information about the variations in UHPC strength under different curing conditions and with time. Ghafari et al.³⁶ examined the use of a backpropagation neural network (BPNN) and statistical mixture to estimate the performance of UHPC. They aimed to predict the CS and consistency of UHPC under steam curing and wet curing using BPNN and statistical mix design. Fifty-three concrete samples were created based on a statistical mixture design sizing matrix, with the components of the mixture treated as independent factors in the BPNN model. The findings indicated that BPNN outperformed statistical mixed design in predicting CS and slump with greater accuracy.

1.2 Research perspective

One of the primary objectives of this research is to evaluate the efficacy of the developed models in predicting the compressive strength of UHPC. The formidable SVR model endeavors to simulate the CS values with the aid of optimization algorithms in an attempt to ameliorate the efficacy of model outcomes. Simultaneously, processing the dataset is deemed a crucial step toward achieving the aforementioned objective. By implementing the BBO and FDA algorithms, the SVR may yield an optimal solution for computing the embedded variables within the SVR. This optimal solution enables the hybrid SVR-BBO and SVR-FDA approaches to accurately evaluate CS in close approximation to the desired target values of already measured systems. Providing a primary dataset to supply models is a critical aspect that requires accurate measurement. Moreover, several criteria indicators have been utilized to compare the hybrid models and evaluate the modeling process.

The study employs a carefully chosen methodology, combining SVR, BBO, and FDA to address specific research questions regarding UHPC CS. SVR, known for handling complex and nonlinear relationships, is adapted to the diverse UHPC composition. BBO, inspired by biogeography, optimizes SVR parameters, aligning with environmental sustainability. FDA, simulating information flow, complements BBO in optimizing parameters, capturing nuances in UHPC components, and enhancing predictive performance. This hybrid approach ensures a robust and efficient model for predicting UHPC CS. The combination of SVR, BBO, and FDA forms a hybrid ML approach that leverages the strengths of each component. SVR provides a powerful regression framework, while BBO and FDA contribute to optimizing the model parameters, allowing to construction of an accurate and efficient model for predicting UHPC CS. This methodology was chosen to address the complexity of UHPC composition and ensure a reliable prediction model for practical applications in the construction industry.

The selected methodology integrates SVR with two innovative optimization algorithms, BBO and FDA, presenting an advanced approach to tackle the intricacies of UHPC CS. The subsequent technical explanation delves into each algorithm and its distinctive contributions:

SVR:

-
Technical Insight: SVR, a machine learning technique specialized for regression tasks, functions by transforming input data into a high-dimensional space. It identifies a hyperplane that best represents the relationships between variables.
-
Implementation: In the context of UHPC, SVR captures and models complex, nonlinear interactions among different components. Its adaptability suits the handling of the diverse and intricate composition of UHPC.

BBO: -
Technical Insight: BBO derives inspiration from the principles of biogeography, modeling species migration between habitats. It simulates information exchange between different regions to find optimal solutions.
-
Implementation: In the study, BBO acts as an optimizer for fine-tuning SVR parameters. Its unique feature of mimicking nature's processes aligns with the environmental sustainability aspect of the research. By guiding the search for optimal parameter values, BBO enhances SVR model accuracy, contributing to overall analysis efficiency.
-
Contribution to UHPC: BBO's application in UHPC research is innovative, offering a bio-inspired optimization technique tailored to the specific challenges of predicting compressive strength in sustainable concrete formulations.

FDA: -
Technical Insight: FDA, an optimization algorithm emulating information flow in a network, navigates through the solution space based on directional information flow.
-
Implementation: In the study, FDA complements BBO in optimizing SVR parameters. Its conceptual alignment with the interconnectedness of UHPC components and their influence on compressive strength adds value. By capturing additional nuances in the data through flow dynamics, FDA aims to improve the overall predictive performance of the model.
-
Contribution to UHPC: The use of FDA in conjunction with BBO represents a novel approach in UHPC research, emphasizing the importance of information flow in optimizing complex systems. This contributes to a more nuanced understanding of UHPC behavior and enhances predictive capabilities.

In summary, the integration of SVR with BBO and FDA provides a cutting-edge methodology for predicting UHPC compressive strength. The novel application of these optimization algorithms brings a unique perspective to the field, addressing the challenges of sustainability and complexity in concrete formulations.
2 Materials and methodology

2.1 Preparation of preliminary data

The present study utilizes an experimental dataset that has been gathered from a published paper discussing the CS parameters of UHPC materials that have been evaluated. The SVR-BBO and SVR-FDA hybrid models endeavor to incorporate information derived from UHPC and CS specimens with varying target strengths to evaluate the durability of concrete samples utilized to train the formulated models. The subsequent section provides an overview of the data utilized in the modeling process. In the current research phase, a summary of the data acquired from 110 samples during the series of experiments is presented in Table 1. The combinatorial effects of constituent ingredients in UHPC specimens featuring varying dosages can give rise to dissimilar CS outcomes in each trial. The mathematical solution outlined in the current section is employed to conduct resistance simulations. The diagram presented in Figure 1 illustrates the CS through the diverse combinations of the constituents.^{37,38,47,48,39–46} In Figure 1, the input variables have been determined based on the ratio of cement.

Figure 1.

Initial data used for training and testing phases of CS modeling.

Table 1.

Input and target data for predictive models.

			Statistical measurements
Component	Unit	Code	Min	Max	Mean	Median	St. Dev.
Cement	(Kg/m³)	C	383	1600	879.7	786	329.8
Fly ash	(Kg/m³)	FA	120	448	33	120	72.7
Sand	(Kg/m³)	SA	292	1898	980	1107	513.8
Steel fiber	(Kg/m³)	SF	2	470	39	8	74.8
Quartz powder	(Kg/m³)	QP	203.3	750	36.9	211	125.9
Admixture	(Kg/m³)	AD	4	185	31.9	30.1	28.2
Silica fume	(Kg/m³)	SI	30	367.95	192	196	94.6
Water	(Kg/m³)	WA	109	334.5	197.1	185.3	54.3
Compressive strength	(MPa)	CS	95	240	152.2	147.9	31.5

2.2 Support vector regression (SVR)

The machine learning procedure of SVR - Support Vector Regression - to classify the relapse things has been utilized in the current investigation.⁴⁹ The SVR notices a machine learning procedure that employs the error range of ε to characterize a relapse design. It is considered that the classification of relapse classes can be conducted to characterize the particular boundary of the hyperplane. SVR that's worked in this investigation is regarded as a supervised method to set up answers for the method of relapse that creates the properties in equation (1)⁵⁰:

m i n_{w, b} = \frac{1}{2} ‖ w ‖^{2} + C \sum_{i = 1}^{m} (ξ_{i} + ξ_{i} *), c o n s t . {\begin{matrix} y_{i} - (w^{T} x_{i} + b) \leq ε + ξ_{i} \\ (w^{T} x_{i} + b) - y_{i} \leq ε + ξ_{i} * \\ ξ_{i}, ξ_{i} * \geq 0 \end{matrix}}

(1)

The introduced variable, denoted as y, represents the measured values of CS. The extent of boundary violation is designated by the variable

ξ

. Regularization within the queue is depicted by the variable C, while the weight factor is signified by the variable w. The bias of the SVR model is ascertained by the variable b. Furthermore, the variable

ε

signifies the deviation value of the hyper-plane boundaries. Such terminology and nomenclature are indicative of an academic writing style. The two terms of the equation are referred to as follows. Equations (2) and (3) are utilized for the evaluation of equation (1).

\begin{aligned} \frac{1}{2} ‖ w ‖^{2} \end{aligned}

(2)

\begin{aligned} C \sum_{i = 1}^{m} (ξ_{i} + ξ_{i} *) \end{aligned}

(3)

The introduction of method equation (2) aimed to augment the separation between the sample and hyperplane boundaries, thereby facilitating the achievement of adequate spacing between the samples and the hyperplane. Additionally, it is noteworthy to acknowledge the term ingrained within the equation presented in equation (3). The aforementioned tool functions as a means of correction. When devising functions aimed at the hyperplane, it is imperative to amass the associated values for both b and w.

2.3 Biogeography-based optimization (BBO)

The BBO is a mathematical formulation that draws inspiration from the spatial dispersion patterns exhibited by organisms in their natural ecosystem and environment.⁵¹ The optimization algorithm postulates a limited count of habitats within the ecosystem. The quality of habitats for living organisms is influenced by numerous factors, commonly referred to as fitness index variables. These parameters may include climate, availability of food, and access to water resources, amongst others. The Index of Habitat Suitability (IHS) is an indicator that conveys information on the condition of a given habitat. When a habitat becomes overcrowded or the Index of Habitat Saturation (IHS) attains a high value, organisms tend to relocate to another habitat with a lower IHS value. Every dwelling offers a viable resolution, with the independent variable (IV) being the SI factor. During the process of optimization, solutions that possess smaller targets tend to possess higher IHS. The present algorithm employs two distinct operators: “migration” and “mutation. “The migration operators identify the neighboring responses, while the mutation operators facilitate the exploration of innovative solutions. Both operators are utilized to aid the overall search and investigation processes. Figure 2 shows the structure of BBO.

Figure 2.

Structure of Biogeography-Based Optimization Algorithm.

Those are recorded from their cost function values for each habitat with the HS size. The suitability for $i^{t h}$ habitat that is named ( $I H S_{i}$ ) in the sorted generation could be written as equation (4).

H S I_{i} = - i + H S + 1

(4)

The (

μ_{i}

) and (

λ_{i}

) indicate emigration and immigration, alternatively, and can be determined as:

\begin{aligned} μ_{i} & = \frac{I H S_{i}}{H S} \end{aligned}

(5)

\begin{aligned} λ_{i} & = 1 - \frac{I H S_{i}}{H S} \end{aligned}

(6)

The soaring rates of emigration and immigration speed are one. Migrating from the decision variable of

j

and the residence of r to the i decision variable is computed as:

D V_{j}^{k} = α D V_{j}^{i} + (1 - α) D V_{j}^{r}

(7)

2.4 Flow direction algorithm (FDA)

The origin of the FDA can be attributed to the topographical, hydrological, and meteorological dynamics of a given watershed, which determine the direction of water flow and influence the shaping process of runoff resulting from precipitation events. This method establishes a rudimentary populace within hydrological basins or a domain of query exploration.³⁰

The algorithm's initial variables include β (number of neighbors), α (population number), and $Δ$ (neighborhood radius). In the FDA algorithm, the initial positions of flows are obtained by equation (8):

F l o w X (i) = l b + r a n d \times (u b - l b)

(8)

The parameter of

F l o w X

shows the i^th flow position,

r a n d

denotes a random value of zero to one with uniform distribution,

l b,

and

u b

denote, respectively, the low and up limits of the decision variables. The β number of neighborhoods around the flow cells is assumed, and positions are produced as follows:

N e i g h b o r X (j) = F l o w X (i) + r a n d n \times Δ

(9)

The parameter of

N e i g h b o r X

is the j^th neighbor position;

r a n d n

shows a random value with a standard deviation of 1. The utilization of a small radius for the neighborhood, otherwise referred to as

Δ

, contributes to a requirement to search through numerous values and ranges pertaining to the variables in question when conducting a thorough search throughout an extensive range. Searching a vast range of possible solutions can yield a greater diversity of outcomes, thereby augmenting the likelihood of arriving at optimal answers. This type of approach is akin to undertaking a comprehensive global search. The proposed solution method tends global optimality and operates in a restricted domain, attaining the globally optimal solution through the utilization of local search techniques of high precision. The aforementioned qualities are underscored by the delicate balance they achieve. Moreover, the directional orientation of the change in question, represented by the

Δ

, gravitates towards a stochastic location, thereby fostering a greater variation in the system under consideration.

Δ = [r a n d \times X r a n d - r a n d \times F l o w X (i)] \times ‖ B e s t X - F l o w X (i) ‖ \times W

(10)

In which the parameter of

r a n d

shows the accidental number that is uniformly distributed,

X r a n d

shows the random position that is generated by equation (8), W denotes a nonlinear weight with accidental magnitudes in the range of [0 to inf]. In equation (10), the first term is

F l o w X

goes to a random position called

X r a n d

. Moreover, with the second term with enhancing iteration, the variable of

F l o w X

is close to the parameter of

B e s t X

that the distance of Euclidian between

F l o w X

and

B e s t X

is brought to zeros. Figure 3 indicates the flowchart of the FDA.

Figure 3.

Flow Direction Algorithm diagram.

In the present study, a quadratic objective function that was utilized to reach the desired results aimed to determine the support vector regression parameters at the optimal levels ( $C$ , $ε,$ and $s i g m a$ ),⁵² which are optimized via BBO and FDA listed in Table 2.

Table 2.

The SVR key variables’ values are optimized.

		SVR-FDA	SVR-BBO
Training phase	C	0.739	1.548
	EPSILON	2467	662
	$s i g m a$	0.013	0.100
Testing Phase	C	0.622	0.302
	EPSILON	1083	2395
	$s i g m a$	0.069	0.328

2.5 Performance evaluators

In arrange to assess the adequacy of the models of SVR-BBO and SVR-FDA to create the CS values of UHPC examples for the training and testing stages, different pointers are indicated in Table 3, including Mean absolute error (MAE), Variance account factor (VAF), Pearson's correlation coefficient (R²), Root mean squared error (RMSE), and Objective detection metric (OBJ) as:

Table 3.
Assessment criteria to evaluate presented models.

Nomenclature Equations Assessment

$V A F$ $(1 - \frac{v a r (t_{n} - y_{n})}{v a r (t_{n})}) * 100$ (11) Higher is desirable

MAE $\frac{1}{N} \sum_{n = 1}^{N} | p_{n} - t_{n} |$ (12) Lower is desirable

RMSE $\sqrt{\frac{1}{N} \sum_{n = 1}^{N} {(p_{n} - t_{n})}^{2}}$ (13) Lower is desirable

R² $(\frac{\sum_{n = 1}^{N} (t_{n} - \bar{t}) (p_{n} - \bar{p})}{\sqrt{[\sum_{n = 1}^{N} {(t_{n} - \bar{p})}^{2}] [\sum_{n = 1}^{N} {(p_{n} - \bar{p})}^{2}]}})$ ² (14)
Higher is desirable

OBJ $(\frac{n_{t r a i n} - n_{t e s t}}{n_{t r a i n} + n_{t e s t}}) \frac{R M S E_{t r a i n} + M A E_{t e s t}}{R_{t r a i n}^{2} + 1} + (\frac{2 n_{t r a i n}}{n_{t r a i n} + n_{t e s t}}) \frac{R M S E_{t e s t} - M A E_{t e s t}}{R_{t e s t}^{2} + 1}$ (15) Lower is desirable⁵³

Nomenclature	Equations	Assessment
$V A F$	$(1 - \frac{v a r (t_{n} - y_{n})}{v a r (t_{n})}) * 100$ (11)	Higher is desirable
MAE	$\frac{1}{N} \sum_{n = 1}^{N} \| p_{n} - t_{n} \|$ (12)	Lower is desirable
RMSE	$\sqrt{\frac{1}{N} \sum_{n = 1}^{N} {(p_{n} - t_{n})}^{2}}$ (13)	Lower is desirable
R²	$(\frac{\sum_{n = 1}^{N} (t_{n} - \bar{t}) (p_{n} - \bar{p})}{\sqrt{[\sum_{n = 1}^{N} {(t_{n} - \bar{p})}^{2}] [\sum_{n = 1}^{N} {(p_{n} - \bar{p})}^{2}]}})$ ² (14)	Higher is desirable
OBJ	$(\frac{n_{t r a i n} - n_{t e s t}}{n_{t r a i n} + n_{t e s t}}) \frac{R M S E_{t r a i n} + M A E_{t e s t}}{R_{t r a i n}^{2} + 1} + (\frac{2 n_{t r a i n}}{n_{t r a i n} + n_{t e s t}}) \frac{R M S E_{t e s t} - M A E_{t e s t}}{R_{t e s t}^{2} + 1}$ (15)	Lower is desirable⁵³

For the variables in equations (11) to (15), the predicted CSs of concrete samples are shown via $p_{N}$ ; $t_{n}$ shows the target value of CS measured; the averaged measured CSs are considered by $\bar{t}$ ; the averaged estimated CSs are indicated using $\bar{p}$ . Furthermore, alternatively, the variables of $n_{t r a i n}$ and $n_{t e s t}$ indicate the collected number of UHPC samples for the training and testing phases.

3 Results

The present study involved modeling both frameworks to predict the CS of UHPC using empirical data derived from experimental tests. In the context of the study, a total of 110 samples, each comprising distinct ingredients, were analyzed. Of these, 70% were allocated for training, while the remaining 30% were reserved for testing purposes. The remaining 30% applied to test models is not considered in the training part. As presented in Table 4, five metrics were utilized to evaluate the performance of the proposed models through comparison with the measured values. The aforementioned statement pertains to the evaluation of the performance of each model in assessing the cesarean section rates. The utilization of R² as a correlation metric for the initial index demonstrates the admissible values for both the training and testing phases and the parameters for the comprehensive analysis of the entire dataset. The results obtained in the testing phase of SVR-FDA exhibit a significantly higher correlation coefficient of 0.984 in comparison to SVR-BBO, suggesting that the former method provides superior predictive accuracy. For the remaining circumstances, SVR-FDA achieved a superior R² correlation coefficient in comparison to SVR-BBO. This finding reflects a noteworthy difference in predictive accuracy between the two methods. Regarding overall efficacy, the support vector regression with the SVR-FDA model yielded a greater R² value of 2.80% compared to alternative models. This finding suggests that the SVR-FDA model may offer superior performance for certain applications.

Table 4.
Evaluation of introduced models.

Models SVR-BBO SVR-FDA

Section Train Test Total Train Test Total

R² 0.895 0.936 0.903 0.902 0.984 0.928

RMSE 10.207 10.766 10.378 9.881 4.568 8.637

MAE 4.617 8.755 5.859 3.899 2.123 3.366

VAF 91.35 96.36 92.63 91.39 98.48 92.63

OBJ - - 12.34 - - 4.92

Models	SVR-BBO	SVR-FDA
R²	0.895	0.936	0.903	0.902	0.984	0.928
RMSE	10.207	10.766	10.378	9.881	4.568	8.637
MAE	4.617	8.755	5.859	3.899	2.123	3.366
VAF	91.35	96.36	92.63	91.39	98.48	92.63
OBJ	-	-	12.34	-	-	4.92

RMSE indicator that considers the error rate of modeling shows the greater mistakes enrolled in calculating CSs by SVR-BBO that RMSE for training, validation, and total status were calculated at 10.207 MPa, 10.766 MPa, and 10.378 MPa, respectively. Whereas the MAE rates for SVR-FDA are placed at acceptable rates of 3.899 MPa for the training phase, while calculating for SVR-BBO in the mentioned phase, 4.617 MPa are 18.43 percent higher than the former model. The VAF index also showed the efficient capabilities of SVR-FDA that obtained the values of 91.39, 98.48, and 92.63 for train, test, and total conditions, respectively 0.05%, 2.20%, and 1.10% higher than SVR-BBO. Moreover, the OBJ index for SVR-FDA was calculated at 4.92, which for SVR-BBO was 12.34, with a 150.89% difference. To better understand, Figure 4 has shown the bar charts of results calculated by indicators.

Figure 4.

The assessment of developed models via various metrics.

In the next step, the difference of CS values that are modeled with both models are shown in Figure 5. In this regard, the difference range is calculated around −20% to +10% and the calculation is done by the ratio of SVR-BBO to SVR-FDA. Interestingly, the differences are harsh in the validation stage (after the sample of 77). Further, the types of differences are in both positive and negative conditions, which has led to create the fluctuated diagram.

Figure 5.

The difference between CS modeled using SVR-FDA and SVR-BBO.

In order to gain a more comprehensive understanding of modeling, the accompanying Figures 6 and 7 may be consulted irrespective of the act of comparison. The present study introduces an assessment framework to evaluate the effectiveness of various models in appraising CS against a set of empirically determined CS metrics. The effectiveness of the FDA approach in addressing inaccuracies in evaluating CS appears to be relatively high, as indicated by the nearly equivalent proximity to actual values demonstrated by the method's best-fit line slope of 0.875. Conversely, the best-fit line slope of 0. 785 for BBO appears comparatively lower.

Figure 6.

The modeled CS against the measured values.

Figure 7.

Comparison between predicted and measured samples of presented models.

The SVR-BBO model exhibits greater error compared to the subsequent model, as is apparent from the significant fluctuation observed in the diagrams. During the testing phase of SVR-BBO, fluctuations in error within the range of ±10% were observed. Additionally, Figure 8 attempts were made to highlight the error rates evaluated by both models. The graphical representation of SVR-FDA evidences a robust optimizing mechanism in the model, whereas the BBO algorithm has exhibited subpar modeling capabilities in estimating CS.

Figure 8.

Error percentage of CS modeled for developed hybrid models.

Figure 9 presents the histograms illustrating the distribution of errors for the models, accompanied by the standard distribution curve. Figure 9 displays that the error accumulation is distributed symmetrically around the origin point. The SVR-FDA variant exhibits a significant degree of concentration, as evidenced by the high proportion of samples- approximately 90 out of 110 that were found to have exhibited zero-closed error. Concerning its physical configuration, the SVR-BBO variety manifests a flattened appearance and exhibits a probability density function of a thickened bell-shaped normal distribution. Nonetheless, the tendency presented by the curve of SVR-FDA exhibits a relatively diminutive width when juxtaposed with SVR-BBO.

Figure 9.

Error histogram of CS computed via proposed models.

Based on results presented through several ways and figures, the performance of models was assessed at the desirable level. However, for some samples, the evaluating indices demonstrated weak outcomes. This fact has illustrated the unreliability of using AI-based approaches with a maximum error rate of ±20 percent. Nevertheless, this problem involves some limited numbers of samples. On the other hand, most of the points were modeled with acceptable results of modeling that the error rates were calculated with low rates, allowing us to use developed models with high-accurate performance.

The Wilcoxon test results comparing the model differences are shown in Table 5. To find out if there are statistically significant differences between matched samples, the Wilcoxon test is utilized. P-values are a measure of how likely it is that the observed differences are the result of chance; values less than 0.05 are often regarded as significant. Therefore, while the comparison SVR ∼ SVBB does not reveal any major differences, the comparisons SVFD ∼ SVBB and SVR ∼ SVFD do.

Table 5.

Result of Wilcoxon test.

	Index
Models’ difference	statistic	p_value
SVR ∼ SVFD	2210	0.012
SVR ∼ SVBB	2409	0.055
SVFD ∼ SVBB	2302	0.025

4 Discussion

Practical applications of the research findings are pivotal in showcasing the real-world relevance of the investigation into CS of UHPC. The following provides a more robust explanation of how the results can be applied in practical scenarios, elucidating the potential impact on the field of concrete technology: The outcomes of the study hold significant implications for the practical realm of concrete technology. By employing a sophisticated methodology that integrates SVR with BBO and FDA, the research offers a predictive model for CS of UHPC that extends beyond theoretical advancements. This model, validated through comprehensive analysis, not only enhances accuracy but also introduces a practical avenue for optimizing UHPC formulations and manufacturing processes. Practically, the predictive accuracy of the model allows for expedited development cycles of new UHPC formulations. The ability to precisely anticipate CS based on environmentally sustainable constituents aligns with contemporary demands for eco-friendly construction materials. This is particularly pertinent in the concrete industry, where the pursuit of sustainability is increasingly paramount.

Furthermore, the optimization algorithms, BBO and FDA, contribute directly to efficiency gains in practical applications. The fine-tuning of SVR parameters through BBO minimizes the need for resource-intensive laboratory experiments. This not only accelerates the formulation development process but also reduces associated costs. In practical scenarios, the research facilitates the optimization of manufacturing processes. The predictive model serves as a guide for adjusting material compositions to meet specific compressive strength requirements. This level of precision is invaluable for industries that rely on UHPC, such as infrastructure development and high-performance construction projects. The potential impact on the field of concrete technology is profound. The methodology, by combining advanced ML with innovative optimization algorithms, sets a precedent for future research and applications. The enhanced understanding of UHPC behavior, coupled with the practical implications of the predictive model, positions the findings as a cornerstone for advancements in sustainable construction materials.

5 Conclusion

The utilization of Ultra-High-Performance Concrete (UHPC) as an efficacious component in construction demands a thorough examination of its composition and content in order to determine the appropriate Compressive Strength (CS) of the concrete. Intelligent methodologies facilitate the assessment of CS based on the constituent components of UHPC elements. It is imperative to promote the selection of environmentally tangible materials as a commonly employed technique on a global scale. This study sought to simulate the compressive strength of UHPC through the incorporation of environmentally friendly constituents. In order to accurately model the CS of concrete, a machine learning approach using Support Vector Regression (SVR) was combined with the optimization algorithms of Biogeography-Based Optimization (BBO) and Flow Direction Algorithm (FDA). The CS values were modeled utilizing a total of eight components. The SVR-FDA model demonstrated a higher R² value of 0.928 when compared to SVR-BBO, resulting in a 2.88% discrepancy in the modeling of CS. In evaluating the root-mean-square error (RMSE) performance metric, it was observed that the SVR model augmented by the BBO algorithm effectively characterized the CS with an error magnitude of 10.378 MPa. Conversely, the SVR model augmented by the FDA achieved superior performance, exhibiting an RMSE value of 20.16% lower than that of SVR-BBO. Both models successfully simulated the CS during the validation phase under appropriate conditions. Specifically, SVR-FDA achieved an RMSE of 4.568 MPa, while SVR-BBO achieved an RMSE of 10.766 MPa. Notably, the two models had a substantial difference of 135%. The Variable Accuracy Factor (VAF) indicator demonstrated congruence in the results generated by both models. In accordance with the findings of the VAF, it was determined that the SVR-FDA SVR model exhibited superior performance compared to the highly sought-after SVR model utilizing the SVR-BBO, with a distinct margin of 0.05% during the training phase. In sharp contrast, during the testing phase, SVR-FDA outperformed its counterpart by a margin of 2.20%, as evidenced by the calibration and validation values of 91.39 and 98.48, respectively. During both stages, the OBJ index, comprising the criteria of R², RMSE, and MAE, was used to evaluate the performance of SVR-FDA and SVR-BBO. The rankings of SVR-FDA and SVR-BBO were determined to be 4.92 and 12.34, respectively, indicating that SVR-FDA outperformed SVR-BBO with a difference of 150% in performance. In general, the use of such methods can increase productivity in estimating hardness properties in terms of time and energy. The promising way of digitalizing experimental data of laboratories also, is considered a productive way to increase the efficiency of models. To develop the future potentials of AI-based approaches, the hybrid frameworks should be tuned to optimize for all the internal settings as the capabilities of models will be practiced in the real world and engineering affairs. Moreover, using sensitivity analysis for detecting highly affecting decision variables is considered as the future program. On the other side of modeling, also, developing data assimilation techniques, benefiting from the combination of the several models results into one ensemble framework is deemed as the method with high accuracy performance. However, the accuracy of these models depends on data quality and quantity. Limited or noisy data necessitates further collection and preprocessing efforts. The integration of SVR with FDA and BBO increases model complexity, requiring additional practitioner training. Significant computational resources are needed, posing challenges for smaller projects. Generalization to new materials may require further validation, limiting immediate applicability. Regulatory hurdles and standardization issues may also affect adoption in the construction industry. In large-scale infrastructure projects, such as bridges and high-rise buildings, accurate prediction and optimization of UHPC properties lead to safer and more efficient designs, cost savings, and faster project completion. For sustainable construction projects, the approach supports the use of sustainable materials, aligning with green building practices and meeting regulatory requirements. Customizing UHPC for specific applications, such as earthquake-resistant structures, ensures better performance and longevity.

Footnotes

Acknowledgements

I would like to take this opportunity to acknowledge that there are no individuals or organizations that require acknowledgment for their contributions to this work.

Research involving Human Participants and/or Animals

The observational study conducted on medical staff needs no ethical code. Therefore, the above study was not required to acquire ethical code.

Informed consent

This option is not neccessary due to that the data were collected from the references.

Author contributions/CRediT

Wang Jing: Methodology, Software, Validation, Formal Analysis.

Yin Suyuan: Methodology, Software, Validation, Language review.

Li Jingtao: Writing-Original draft preparation, Conceptualization, Supervision, Project administration.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability

The authors do not have permissions to share data.

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Models	SVR-BBO			SVR-FDA
Section	Train	Test	Total	Train	Test	Total
R²	0.895	0.936	0.903	0.902	0.984	0.928
RMSE	10.207	10.766	10.378	9.881	4.568	8.637
MAE	4.617	8.755	5.859	3.899	2.123	3.366
VAF	91.35	96.36	92.63	91.39	98.48	92.63
OBJ	-	-	12.34	-	-	4.92

Using a machine learning method with novel optimization algorithms in estimating compressive strength rates in ultra-high performance concrete

Abstract

Keywords

1 Introduction

1.1 Literature review

1.2 Research perspective

2.1 Preparation of preliminary data

Table 4. Evaluation of introduced models. Models SVR-BBO SVR-FDA Section Train Test Total Train Test Total R2 0.895 0.936 0.903 0.902 0.984 0.928 RMSE 10.207 10.766 10.378 9.881 4.568 8.637 MAE 4.617 8.755 5.859 3.899 2.123 3.366 VAF 91.35 96.36 92.63 91.39 98.48 92.63 OBJ - - 12.34 - - 4.92

5 Conclusion

Footnotes

Acknowledgements

Research involving Human Participants and/or Animals

Informed consent

Author contributions/CRediT

Funding

Declaration of conflicting interests

Data availability

References