Multi-Objective Mix Proportion Optimization of Basalt Fiber-Reinforced Concrete Considering Cost and Carbon Emission Constraints

Abstract

Basalt fiber-reinforced concrete (BFRC) exhibits superior mechanical performance, durability, and environmental benefits, making it a promising material for promoting green and low-carbon construction. This study develops a novel multi-objective mix design optimization method for BFRC under cost and carbon emission constraints, presents a framework that considers tensile strength (f_t) as a core design objective, and establishes high-precision strength prediction models via gene expression programming (GEP). Material cost and carbon emission functions were formulated based on market data, while compressive strength (f_c) and tensile strength (f_t) prediction models were established using using GEP implemented in MATLAB 2018b with seven mix design variables, including cement dosage, aggregate parameters, and basalt fiber (BF) characteristics (diameter, length, and dosage). Multiple constraints covering material quantities, mix ratios, workability, and density were incorporated into the optimization model, which was solved via the non-dominated sorting genetic algorithm II (NSGA-II). The method identifies the optimal cement dosage, aggregate proportions, and BF dosage to maximize tensile strength (f_t) while minimizing cost and carbon emissions. Computational results suggest that within the target strength range of 30–60 MPa, the proposed design yields reductions of 10–20% in carbon emissions and 12–18% in costs compared to conventional methods, offering potential advantages for sustainable construction. Unlike existing multi-objective studies, which focus on compressive strength, this work addresses critical factors of tensile strength (f_t) and prediction inaccuracy, proposing a systematic low-carbon design framework for potential BFRC applications.

1. Introduction

Basalt fiber-reinforced concrete (BFRC) is produced by incorporating specific amounts of basalt fiber (BF) into ordinary concrete. The synergistic interaction between BF and the cement matrix enhances the mechanical properties of the concrete, including its tensile strength (f_t), flexural strength, crack resistance, impact resistance, ductility, and toughness [1,2,3]. Basalt fiber (BF) is a continuous fiber produced from natural basalt. It exhibits beneficial characteristics, such as high strength, good thermal insulation, high- and low-temperature resistance, and corrosion resistance. Moreover, it is an environmentally friendly, low-carbon green material [4]. Accordingly, applications of BFRC have become a research focus in infrastructure and civil engineering [5,6,7,8].

Traditional concrete mix design methods primarily emphasize strength and workability while neglecting economic and environmental aspects, such as cost and carbon emissions. As a result, their application often leads to high expenses and large carbon footprints [5,9]. In recent years, various optimization-based mix design strategies have been developed to balance performance, economy, and sustainability. Celik et al. [10] designed self-compacting concrete using a ternary binder system of cement, fly ash, and limestone powder, adjusting replacement ratios and aggregate gradations under a fixed water-to-binder ratio. However, their method did not consider cost and carbon emissions, resulting in costly and carbon-intensive mixtures. Zhang et al. [11] proposed a comprehensive multi-objective optimization framework integrating machine learning models (ANN, Random Forest, and XGBoost) with metaheuristic algorithms (MOALO, MOPSO, and NSGA-II) to simultaneously optimize concrete mixture proportions for cost, CO₂ emissions, and compressive strength. Their results demonstrated that the MOALO-XGBoost hybrid model achieved superior optimization performance with a lower computational cost compared to conventional methods. Dabbaghi et al. [12] developed a life cycle assessment multi-objective optimization framework for sustainable lightweight aggregate concrete, combining deep belief networks with evolutionary algorithms to effectively balance cost, CO₂ emissions, energy consumption, and compressive strength. Campos et al. [13] applied regression analysis and ANOVA to identify key strength-related factors and determined optimal proportions using response surface methodology. They later proposed a sustainable high-strength concrete mix design based on particle packing optimization, which reduced cement use and CO₂ emissions while maintaining performance [14]. Rath et al. [15] employed the packing density method, incorporating fly ash and pond ash as partial replacements for cement and sand, thereby improving packing density and reducing material consumption, cost, and emissions. Marani et al. [16] developed a tabular generative adversarial network (TGAN) for predicting the compressive strength of ultra-high-performance concrete, demonstrating the potential of advanced machine learning techniques in concrete mix optimization. Building upon these advances, Kaveh and Sabzi [17] employed a water cycle algorithm, imperialist competitive algorithm, and teaching-learning-based optimization for sustainable fly ash concrete, achieving optimal trade-offs between cost, CO₂ emissions, and compressive strength. Nunez et al. [18] developed a hybrid machine learning model combining artificial neural networks with genetic algorithms for recycled aggregate concrete mixture optimization, achieving effective material consumption and carbon footprint reductions while maintaining mechanical performance.

Nevertheless, when applied to BFRC, existing approaches face two major limitations: (1) tensile strength has not yet been incorporated as a design objective, and (2) there is a lack of accurate predictive models for compressive and tensile strengths. Existing models, which are often based on least squares regression, capture only partial linear or low-order nonlinear relationships between strength and influencing factors. As a result, these models cannot fully describe the complex, high-order nonlinear mechanical behavior of fiber-reinforced concrete, resulting in limited accuracy, with maximum relative errors reaching around 30% [19,20,21]. Recent studies have made significant progress in machine learning-based strength prediction for fiber-reinforced concrete. Zheng et al. [22] developed a novel hybrid machine learning model combining XGBoost with Bayesian optimization for BFRC compressive strength prediction, achieving substantially higher accuracy than traditional regression methods. However, their study primarily focused on compressive strength (f_c) prediction alone, without integrating tensile strength (f_t) as a design objective or incorporating cost and carbon emission constraints into a unified optimization framework.

To address these issues, we propose a multi-objective optimization approach for BFRC mix design. The proposed method integrates compressive strength (f_c), tensile strength (f_t), cost, carbon emissions, and workability as performance objectives, establishes a total-cost function that accounts for both material costs and carbon taxes, and incorporates accurate strength prediction models developed via GEP. The genetic algorithm is then applied to optimize key variables, including cement dosage, aggregate proportions, and fiber characteristics, thereby constructing a systematic low-carbon design framework. Results suggest that the proposed method can be used to improve mechanical performance and reduce cost and carbon emissions, establishing a theoretical basis for sustainable BFRC design.

2. Optimization of Concrete Mix Proportion

2.1. Objective Function

According to GB/T 51366-2019 “Standard for Building Carbon Emission Calculation” [19], the carbon emissions of building materials are determined as the sum of emissions resulting from production and transportation. The carbon emissions during the production stage are calculated as follows [23]:

$$C_{JC}={\displaystyle \sum _{i=1}^n{M}_i{F}_i}$$

(1)

where C_JC denotes carbon emissions resulting from the production of building materials (kg CO₂e), M_i represents the consumption of the i-th primary building material, and F_i refers to its corresponding carbon emission factor, defined as the amount of CO₂ equivalent emitted per unit of the material (kg CO₂e/unit).

The carbon emissions arising from the transportation of building materials are determined as follows [19]:

$$C_{ys}={\displaystyle \sum _{i=1}^n{M}_i{D}_i{T}_i}$$

(2)

where C_ys denotes the carbon emissions resulting from the production of building materials (kg CO₂e), D_i denotes the transportation distance of the i-th primary building material (km), and T_i denotes the carbon emission factor per unit weight and transportation distance under the transportation mode of the i-th building material (kg CO₂e/(t·km)).

Equation (2) indicates that carbon emissions during the transportation stage are primarily determined by the transportation distance and mode of transport. The carbon emission factor for transportation (0.104 kg CO₂e/(t·km) for 18-ton trucks) is much smaller than that for production (735 kg CO₂e/t for ordinary cement) [19]. Since this study primarily focuses on the optimization of concrete mix proportions, only the carbon emissions from the production stage are considered, which are further converted into monetary costs. Quantitative justification based on [23] shows that transportation emissions account for less than 0.02% of total carbon emissions and have no significant impact on the results, justifying this simplification in the context of mix proportion optimization. Nevertheless, it is acknowledged that excluding transportation emissions is a model limitation, as actual construction may involve variable transport distances and modes. Construction-phase and end-of-life emissions are also excluded as they are determined by construction practices and demolition strategies rather than mix proportions. Accordingly, the cost function for BFRC mix proportion optimization is defined as the sum of material and carbon emission costs, expressed as

$$C_T={\displaystyle \sum _{i=1}^5{c}_i{P}_i+P_{{CO}_2}}\cdot C{O}_{2i}$$

(3)

where C_T denotes the total cost of preparing concrete (¥); c_i denotes the dosage of the i-th concrete component (kg); P_i denotes the unit price of the i-th concrete component (¥· kg⁻¹), in which i = 1, 2, 3, 4, and 5 corresponds to cement, coarse aggregate, fine aggregate, mixing water, and BF, respectively; P_CO2 denotes the unit price of carbon dioxide equivalent (¥· kg CO₂-eq⁻¹); and CO_2i denotes the carbon emission factor of the i-th concrete component (kg CO₂-eq·kg⁻¹).

It should be noted that f_c is treated as a constraint targeting specific performance classes, such as C30 and C40, rather than an objective to be maximized, ensuring that the optimization results satisfy engineering grade requirements without over-design.

2.2. Constraint Conditions

In the multi-objective mix proportion optimization design of BFRC based on cost and carbon emission constraints, in addition to meeting requirements for strength, cost, carbon emissions, etc., it is also necessary to comply with constraints on material dosage, key proportion parameters, workability, and volume, among others. The strength of BFRC can be controlled according to the concrete’s 28-day compressive strength and tensile strength. The workability of concrete is adjusted by controlling the optimal paste–aggregate volume ratio [24,25]. Constraints on the range of material dosage can be adhered to by controlling the dosages of concrete components such as cement, coarse aggregate, fine aggregate, water, and BF within the ranges specified in the specifications. Volume constraints mainly refer to the absolute volume method constraint, which means that the sum of the volumes of the components of the fiber-reinforced concrete per cubic meter and the air volume should equal one cubic meter. Accordingly, the constraint conditions for the multi-objective mix proportion optimization of the concrete are summarized as follows [9,26,27]:

$$st\left\{\begin{array}{l}{f}_{cl}\le f_c\le f_{cu};f_{tl}\le f_t\le f_{tu};\\ c_{il}\le c_i\le c_{iu};R_{jl}\le R_j\le R_{ju};{\displaystyle \sum _{i=1}^5\frac{c_i}{{\rho }_i}+V_a=1}\end{array}\right.$$

(4)

where R_j represents the j-th proportion parameter, in which j = 1, 2 … 7 correspond to the water–binder ratio, sand ratio, volume ratio of coarse aggregate to paste, paste–aggregate volume ratio, and volume ratio of BF dosage, respectively; f_cl and f_tl represent the 28-day compressive strength and tensile strength of concrete, respectively; c_il and c_iu represent the lower and upper limits of the dosage of the i-th concrete component, respectively; R_jl and R_ju represent the lower and upper limits of the j-th proportion parameter, respectively; ρ_i is the density of the i-th concrete component; and V_a represents the air volume in concrete.

2.3. GA Optimization

Multi-objective optimization is solved using the NSGA-II with real-valued encoding. Following De Jong’s systematic GA study [28] and standard NSGA-II implementation [29], the parameters are set as follows: population size, 100; generations, 500; crossover probability, 0.9; mutation probability, 0.1; distribution indices, η_c = η_m = 20. Tournament selection, simulated binary crossover, and polynomial mutation are employed. The algorithm runs 10 times with random seeds 1--10 to ensure robustness. The NSGA-II algorithm generates a Pareto front of non-dominated solutions, enabling explicit trade-off analysis among competing objectives without prior weighting [29]. The NSGA-II algorithm has been successfully applied in various concrete optimization studies. Zhang et al. [11] demonstrated its effectiveness in the multi-objective optimization of concrete mixture proportions, achieving simultaneous optimization of cost, CO₂ emissions, and compressive strength through hybridization with machine learning models. Their comparative analysis showed that the NSGA-II provides competitive performance with better convergence characteristics compared to other multi-objective evolutionary algorithms such as MOPSO and MOALO.

3. Prediction Models for Compressive and Tensile Strengths of BFRC

3.1. Gene Expression Programming

Gene Expression Programming is employed to establish strength prediction models. Unlike artificial neural networks, random forests, and gradient boosting, which function as black-box models without explicit mathematical expressions, GEP generates transparent algebraic equations that can be integrated directly into multi-objective optimization frameworks. Although a direct benchmark on the current dataset is not provided, Akin et al. [30] demonstrated that GEP achieves accuracy comparable to that of ANNs, with superior interpretability for concrete strength modeling. The formulation of the objective function plays a pivotal role in the process of optimizing the BFRC mix design. Optimization is conventionally realized by constructing predictive models for compressive and tensile strengths, thereby enabling a quantitative characterization of the relationship between mixture constituents and the resulting mechanical performance. Akin et al. [30] identified a significant nonlinear relationship between the mechanical properties of concrete and its components. Moreover, it is difficult to establish a high-precision prediction model when using the traditional least squares method for multiple regression analysis. To this end, this study adopts the GEP method to establish a model for predicting the strength of BFRC. Ferreira proposed the GEP in 2001. This evolutionary algorithm efficiently generates mathematical models by integrating the chromosomal structure of genetic algorithms and the tree-like expression ability of genetic programming while simulating the evolutionary process of biological genes. The GEP method was fully validated by some scholars [30], published in Advances in Civil Engineering, for its high prediction accuracy and reliability, and has been successfully applied to concrete compressive strength modeling, laying a solid foundation for this study.

The core premise of the GEP is to represent candidate prediction models (mathematical expressions) as expression trees (ETs). For example, the expression tree of the mathematical expression $\sqrt{(\mathrm{a}+b)\times (c-d)}$ is shown in Figure 1a, where “Q” represents the square root function.

As shown in Figure 1a, the expression tree depicts the phenotype of an individual in the GEP. Reading the tree sequentially from left to right and from top to bottom facilitates the deduction of its chromosomal genes, which can then be represented as follows:

$$\begin{array}{cccccccc}0& 1& 2& 3& 4& 5& 6& 7\\ \mathrm{Q}& \times & +& -& \mathrm{a}& \mathrm{b}& \mathrm{c}& \mathrm{d}\end{array}$$

(5)

Subsequently, the gene chromosome shown in Equation (5) is transformed into a new gene chromosome through genetic operations such as selection, crossover, and mutation, and its form is assumed to be

$$\begin{array}{ccccccccccc}0& 1& 2& 3& 4& 5& 6& 7& 8& 9& 0\\ \mathrm{Q}& \times & +& \times & \mathrm{a}& \times & \mathrm{Q}& \mathrm{a}& \mathrm{a}& \mathrm{b}& \mathrm{a}\end{array}$$

(6)

The expression tree corresponding to Equation (6) is shown in Figure 1b. The corresponding mathematical expression is $\sqrt{(\mathrm{a}+\mathrm{a}\times \mathrm{b})\times (\sqrt{\mathrm{a}}\times \mathrm{a})}$. Subsequently, the coefficient of determination (R²), mean squared error (MSE), and root mean squared error (RMSE) are used to evaluate the fitting performance of this expression. If the result is unsatisfactory, new gene chromosomes will be generated through genetic operations such as selection, crossover, and mutation, and iterative optimization will be performed continuously until a mathematical expression with better performance is obtained.

In Akin et al.’s study [30], cement, metakaolin, coarse and fine aggregates, and water were selected as input parameters, while the 28-day compressive strength served as the output variable. A highly accurate predictive model for concrete compressive strength was established by employing GEP. The results indicated that this model surpassed conventional regression methods in predictive performance, effectively characterizing the nonlinear interactions between input and output variables and demonstrating the superior capability of GEP in concrete performance prediction.

3.2. Prediction Models for Compressive and Tensile Strengths

3.2.1. Data Collection

In this study, experimental data on the mechanical behavior of BFRC were collected from publicly accessible domestic and international sources to construct predictive models for compressive and tensile strengths. Data were screened from peer-reviewed studies based on complete mix parameters and standardized curing conditions. Outliers were removed using Grubbs’ test at a 0.05 significance level, and inconsistent records were excluded by cross-checking specimen dimensions to ensure reliability. Notably, part of the dataset has been published in reputable SCI journals [22]. The resulting dataset includes 314 compressive and 293 tensile records involving seven input parameters and two output variables. A statistical summary of the dataset is provided in

Table 1. Collected dataset for compressive and splitting tensile tests.

Authors	Cement/ (kg/m3)	Coarse Aggregate/(kg/m3)	Fine Aggregate/(kg/m3)	Water/(kg/m3)	Fiber Diameter /(mm)	Fiber Length /(mm)	Fiber Dosage /(%)	Compressive Test		Splitting Tensile Test
								fc/(MPa)	Number	ft/(MPa)	Number
Sun X [31]	271	1416	694	130	17.4	6, 12	0–0.5	25.5–37.2	11	2.2–2.51	11
Wang K [32]	392	1211	672–682	196	15	18	0–0.35	39.71–48.75	8	2.59–3.01	8
Zhou H [33]	330	1389.6	540.5	139	12	12	0–0.6	39.1–41.23	7	3.36–4.21	7
Chen W [34]	480	1069	656	210	15	30	0–0.35	45.4–55.0	7	4.0–5.0	7
Wang M [35]	264–740	642–1540	507–1194	112–301	13–30	6–30	0–0.6	23.33–71.1	281	2.2–9.8	260
Total									314		293

.

Wang M [35] initially collected 346 compressive test results and 289 splitting tensile test results. After reviewing the sources of these data and removing those with incorrect sources, the author finally retained 281 compressive test results and 260 splitting tensile test results.

To better understand the relationships among variables and their influence on performance indicators, the BFRC compressive strength dataset was analyzed. Figure 2 presents a scatterplot matrix illustrating the relationships between input features and the response variable and the pairwise relationships among the input features. This visualization reveals correlations and potential distribution patterns.

3.2.2. Prediction Model

The collected dataset was partitioned into training and external test sets at a ratio of 8:2. Five-fold cross-validation was employed during GEP training to prevent overfitting, with model selection based on the lowest average validation RMSE. Additionally, to ensure model generalizability, we evaluated the final selected models on a reserved 20% external validation set consisting of 63 compressive and 59 tensile records that were entirely excluded from the training process. The GEP models achieved R² = 0.93 and RMSE = 2.41 MPa on this independent test data, confirming robust predictive performance beyond the training domain. The GEP modeling was implemented using code written by the authors in Matlab 2018b, which has been thoroughly validated using concrete strength prediction examples [30]. Model performance was evaluated using the RMSE, MAE, R², and relative error percentage, with a ±20% error band adopted as the practical applicability criterion. This section describes the use of GEP to develop models for predicting the compressive and tensile strengths of BFRC. Based on the component parameters of BFRC defined in

Table 2. Design variables for the concrete mix.

Parameters Notation	Parameters	Constant Notations	Constant Values
x1	Cement	G1C1	7.088
x2	Coarse aggregate	G1C5	8.305
x3	Fine aggregate	G1C0	5.322
x4	Water	G2C5	−7.574
x5	Fiber diameter	G2C4	9.311
x6	Fiber length	G2C8	−5.181
x7	Fiber dosage	G3C0	−6.896
fc	28-day compressive strength	G3C6	−0.630
ft	28-day tensile strength	G4C9	−2.799
-	-	G4C1	−0.306

(denoted as input variables x₁, x₂, x₃, x₄, x₅, x₆, x₇), and with the compressive strength f_c as the output response variable, a model for the nonlinear mapping relationship between the component parameters and the compressive strength was established. Its implicit function form can be written as

$$f_c(x)=f(x_1,x_2,x_3\cdots x_7)$$

(7)

Table 2 presents the optimal coefficients evolved by the GEP modeling, where positive values (e.g., G1C1 = 7.088) enhance strength while negative values (e.g., G2C5 = −7.574) reflect inhibitory effects, such as excessive water dosages. These dimensionless scaling factors were substituted into Equations (8) and (13) to establish the explicit prediction models.

The key parameters of the GEP algorithm selected in this study are shown in

Table 3. GEP algorithm parameters.

Parameter Definition	Compressive Strength Values/Designations	Tensile Strength Values /Designations
Number of generations	100,000~800,000	100,000~800,000
Number of chromosomes	80	80
Function set	+, −, ×, /, x2, sin, cos	+, −, ×, /, x2, tan, sin, cos, e, ln
Number of genes	4	5
Head size	12	12
Linking function	+	+
Mutation rate	0.00138	0.00138

, including the number of generations and the number of chromosomes. The accuracy of the prediction model was improved by introducing trigonometric and exponential functions.

The GEP parameters were selected based on established guidelines [30], balancing model complexity with generalization. Convergence was monitored via fitness stabilization, typically achieved within 50,000–80,000 generations. The gene expression tree structure of the model for predicting the compressive strength (f_c) of BFRC, established using the GEP method, is shown in Figure 3. The undetermined parameters in Figure 3 are listed in Table 2, and the corresponding explicit mathematical expression is

$$f_{\mathrm{c}}=f_{\mathrm{c}1}+f_{\mathrm{c}2}+f_{\mathrm{c}3}+f_{\mathrm{c}4}$$

(8)

$$f_{\mathrm{c}1}=7.088\cdot \sin (x_3-x_2+0.141\cdot x_4-1.560)$$

(9)

$$f_{\mathrm{c}2}=\sqrt{-1.797x_4\cdot \cos (-0.132\cdot x_2)+x_1}$$

(10)

$$f_{\mathrm{c}3}=-6.896\cdot \sin \left({({\displaystyle \frac{0.397}{x_3}}-x_1)}^2\right)$$

(11)

$$f_{\mathrm{c}4}=\sqrt{x_1+0.306\cdot x_2+x_6+\sin (-0.357\cdot x_1)}$$

(12)

Similarly, the model for predicting the tensile strength f_t of BFRC, developed using the GEP modeling, produces the following explicit mathematical expression:

$$f_{\mathrm{t}}=f_{\mathrm{t}1}+f_{\mathrm{t}2}+f_{\mathrm{t}3}+f_{\mathrm{t}4}+f_{\mathrm{t}5}$$

(13)

$$f_{\mathrm{t}1}=\cos \left(\tan \left(\cos (2.376\cdot x_1+x_4-{\displaystyle \frac{x_3}{x_4}}-8.666)\right)\right)$$

(14)

$$f_{\mathrm{t}2}=\ln \left(\tan (-6.245\cdot x_2)-6.924\cdot x_2-x_7+x_4\right)$$

(15)

$$f_{\mathrm{t}3}={\displaystyle \frac{1}{x_6/x_5-2\cdot x_2}}\cdot \left(e^{\tan (x_6)}+14.987\right)$$

(16)

$$f_{\mathrm{t}4}={\displaystyle \frac{1}{x_6}}\cdot \left(\tan \left(\tan \left(\cos \left({\displaystyle \frac{x_5+4.8069}{\tan (x_5)}}\right)\right)\right)\right)$$

(17)

$$f_{\mathrm{t}5}={\displaystyle \frac{1}{\left(0.348-0.563/x_6\right)\left(x_5+\cos (x_1)-2.963\right)}}$$

(18)

Figure 4 presents the scatter distribution of predicted versus experimental values for the 28-day compressive strength (f_c) of BFRC, along with a comparison between predicted and experimental data for tensile strength (f_t) Figure 4a indicates that the f_c model achieves high accuracy (RMSE = 2.02), with all data points falling within a ±20% error band, and relative errors ranging between 10% and 16%. Figure 4b shows that the tensile strength model exhibits minor absolute errors (RMSE = 1.66), with most data points falling within the ±20% error band and relative errors controlled within 20%. The MAE and R² are also reported to ensure comprehensive validation beyond the RMSE.

The combined f_c and f_t validation results demonstrate that the model results have minor prediction errors and a high level of agreement with the experimental data. The model effectively captures the fiber’s strengthening effect and exhibits overall excellent performance. It also exhibits reliable predictive accuracy and good engineering applicability, meeting the requirements for BFRC strength prediction.

4. Performance Comparison and Analysis of BFRC Mix Proportion Schemes

Taking a specific project as an example, and based on this project’s design requirements, including bearing capacity, it is necessary to prepare BFRC with design strengths of 30 MPa, 40 MPa, 50 MPa, and 60 MPa. For each target grade, optimization maximizes f_t while minimizing cost and carbon emissions, achieving the specified strength without over-design. The concrete slump at approximately 200 mm can be controlled by adjusting the paste–aggregate volume ratio (the ratio of paste volume to aggregate volume) within a range of 33:67 to 35:65 [36]. The unit price of carbon dioxide tax is set at 0.07 ¥·kg⁻¹ [37]; the air volume is taken as 1% [9]. The dose ranges of the concrete components were determined according to JGJ 55—2011 “Specification for Mix Proportion Design of Ordinary Concrete” [9], GB/T 50476-2019 “Standard for Durability Design of Concrete Structures” [26], and GJ/T 221-2010 “Technical Specification for Application of Fiber-Reinforced Concrete” [27]. These components include cement (c₁), coarse aggregate (c₂), fine aggregate (c₃), water (c₄), and basalt fiber (BF, c₅). In addition, the ranges for key mix proportion parameters were defined, including the water–binder ratio (R₁), sand ratio (R₂), coarse aggregate-to-paste volume ratio (R₃), paste-to-aggregate volume ratio (R₄), and fiber volume fraction (R₅). The specific values of these parameters are summarized in

Table 4. Materials, dosage ranges, and scale constraints for concrete per unit volume.

Component	Density/ (kg·m−3)	Price/ (¥·kg−1)	Carbon Emission Factor/(kg CO2e·kg−1)	Lower Limit/ (kg)	Upper Limit/(kg)	Proportion Parameter	Lower Limit	Upper Limit
Cement c1	3150	0.3	0.735	100	500	R1	0.26	0.55
Coarse Aggregate c2	2500	0.3	2.18 × 10−3	700	1810	R2	0.29	0.61
Fine Aggregate c3	2650	0.14	2.51 × 10−3	500	1260	R3	1.55	4.92
Water c4	1000	0.006	1.68 × 10−4	100	260	R4	0.49	0.54
Fiber c5	2700	30	2.2	1	5	R5	0.05	0.3

Note: ¥ denotes Chinese Yuan (CNY).

.

Based on the material information, dosage ranges, and mix proportion parameter constraints shown in Table 4, and combined with Equations (1)–(18), a multi-objective mix proportion optimization model for BFRC considering cost and carbon emission constraints was established. The optimization model was solved using a genetic algorithm (GA), and the mix design results are presented in

Table 5. Comparative analysis of mix proportion per unit volume of concrete for different methods.

Strength	Method	c1 /(kg)	c2 /(kg)	c3 /(kg)	c4 /(kg)	BF Diameter /(mm)	BF Length /(mm)	BF Dosage /(Mass Fraction, %)	Total Cost /(¥)	Carbon Emission Cost/(¥)	ft /(MPa)	fc /(MPa)
30 MPa	M1	362.2	941.2	630.2	189.10	0.016	15	0.1	553.87	238.47	5.70	36.98
	M2	339.4	692.3	868.7	176.6	0.016	12	0.1	578.84	208.45	5.97	37.61
	M3	278.1	658.2	854.5	155.3	0.016	15	0.2	531.21	253.10	5.26	37.61
	M4	284.7	706.5	693.1	157.6	0.016	19	0.1	491.63	212.78	5.32	35.60
40 MPa	M1	425.0	781.0	730.8	167.3	0.016	24	0.2	650.06	316.38	5.85	44.76
	M2	406.5	875.7	551.0	171.7	0.016	20	0.1	565.39	302.32	7.16	47.72
	M3	371.3	672.6	708.7	174.2	0.016	15	0.2	595.60	276.62	8.16	45.64
	M4	343.2	784.8	633.3	151.1	0.016	9	0.1	527.27	255.80	6.10	44.08
50 MPa	M1	588.7	752.8	521.1	185.2	0.013	19	0.2	669.84	436.11	8.51	57.94
	M2	582.7	735.7	502.6	242.7	0.013	15	0.1	578.94	431.41	9.74	57.92
	M3	458.5	384.9	590.1	148.0	0.013	16	0.2	613.15	340.44	7.92	57.72
	M4	416.4	605.4	740.7	135.8	0.013	22	0.1	514.12	309.48	7.63	57.86
60 MPa	M1	626.0	695.0	621.6	151.3	0.013	17	0.2	679.49	463.65	9.43	64.62
	M2	576.6	511.9	664.6	232.8	0.013	18	0.2	613.69	427.06	9.14	64.94
	M3	513.6	1025.0	679.6	111.7	0.013	12	0.1	665.51	381.68	8.84	65.94
	M4	519.5	639.2	686.8	168.4	0.013	18	0.1	553.14	385.20	9.94	65.97

Note: ¥ denotes Chinese Yuan (CNY).

.

Four concrete mix proportion design methods were selected for comparative analysis. Method M1 prioritizes f_c; Method M2 represents a traditional design approach, considering both f_c and f_t [9]; Method M3 employs a multi-objective optimization strategy, taking into account f_c, f_t and cost; Method M4 is based on a multi-objective optimization model with cost and carbon emission constraints, simultaneously balancing f_c, f_t, cost, and carbon emissions. These methods are formally defined in

Table 6. Objective functions and constraints for different mix design methods.

Method	Objective Function	Optimization Strategy
M1	Achieve fc = ftarget	Single-objective (empirical)
M2	Achieve fc ≥ ftarget, maximize ft (empirical)	Trial and error
M3	Maximize ft, minimize CT	Multi-objective: strength + cost
M4	Maximize ft, minimize CT and CJC	Multi-objective: strength + cost + carbon

Note: f_target denotes the target compressive strength, ranging from 30 to 60 MPa. C_T and C_JC represent the total cost and carbon emissions, as defined in Equations (3) and (1), respectively. All methods satisfy f_c ≥ f_target and practical constraints, including workability and dosage.

. The mix proportion results obtained using the four methods are presented in Table 5, and comparisons of total cost and carbon emissions are shown in Figure 5. The ranges used are specified in Table 4.

To quantify the trade-offs between strength, cost, and carbon emissions, a sensitivity analysis was conducted on key mix proportion variables (cement, coarse aggregate, fine aggregate, and BF), as illustrated in Figure 6. As shown in Figure 6a, cement dosage exhibits the highest sensitivity coefficients for both total cost and carbon emissions, indicating that it is the dominant factor controlling the cost–emissions trade-off. Moderate sensitivity was found for the BF dosage, while aggregate dosages have negligible effects. In contrast, Figure 6b reveals that the effect of cement dosage dominates f_c, and the effect of BF dosage dominates f_t, which is consistent with the mechanistic behavior of BFRC. These results confirm that optimizing the balance between cement and BF dosages is the core strategy to achieve synergistic improvements in f_c, f_t, cost, and carbon emissions.

To further verify the rationality of simplifying transportation emissions in the carbon emission model, carbon emission composition and sensitivity analyses were carried out for the four mix proportion schemes, as illustrated in Figure 7. The results confirm that transportation emissions account for less than 0.02% of the total carbon emissions and have negligible effects on the optimization results, justifying the simplification of transportation emissions in the carbon emission model.

Figure 6 reveals that cement dosage has the strongest effect on cost and carbon emissions, with a sensitivity exceeding 0.8, while basalt fiber preferentially affects tensile strength at 0.65 compared to 0.21 for compressive strength. This validates the decision to reduce cement to decrease emissions while employing fiber bridging to improve tensile performance.

It can be seen from Table 5 and Figure 6 that with the increase in concrete strength, the cement dosage, total cost, and carbon emissions determined by the four methods increase accordingly, which is consistent with existing research results [11,12]. On this basis, this study further analyzes the correlation characteristics among the cost, carbon emissions, and mechanical properties of BFRC.

(1)

Correlation between Strength and Carbon Emissions: As the target f_c increases from 30 MPa to 60 MPa, the carbon emissions associated with various mix design methods generally increase. For instance, the carbon emissions of Scheme M1 increase from 238.47 kg CO₂-eq to 463.65 kg CO₂-eq, suggesting that high-strength concrete largely comprises materials with high carbon footprints, such as cement.

After incorporating carbon emission constraints and optimizing the mix design, Scheme M4 indicates lower carbon emissions across all strength grades compared to Schemes M1, M2, and M3. For instance, at a target strength of 60 MPa, the carbon emissions of M4 are 385.20 kg CO₂-eq, representing a notable reduction of approximately 16.9% relative to M1.

Within the strength range of 30 to 60 MPa, M4 consistently demonstrates the optimal carbon emission reduction effect. With the premise of ensuring the mechanical properties of concrete, carbon emissions are reduced by 11% to 30%. These carbon emission reductions align with the optimization potential reported by Dabbaghi et al. [12] for lightweight aggregate concrete at 15–25% and Zhang et al. [11] for conventional concrete at 10–20%, demonstrating consistent efficacy across different concrete systems.

The analysis results show that by incorporating carbon emission targets into optimization constraints, the BF dosage and aggregate gradation can be effectively regulated, the use of high-carbon materials can be reduced, and a low-carbon concrete design path balancing the trade-off between performance and environmental impact can be realized.

(2)

Correlation between Compressive Strength and Cost: As the target f_c increases, the unit cost of concrete per cubic meter exhibits an overall upward trend. For example, using the M1 method, the unit cost rises from 553.87 ¥/m³ for C30 concrete to 679.49 ¥/m³ for C60 concrete.

Although the cost increases associated with M3 and M4 reduce significantly in the high-strength range of 50 Mpa and above, these methods still demonstrate superior economic performance. Within the strength grade range of 30 to 60 MPa, cost comparison results indicate that M4 consistently achieves the lowest cost. Compared with M1, M2, and M3, M4 reduces costs by 11% to 19%. These cost reductions are comparable to the 15% reduction achieved by Kaveh and Sabzi [17] for fly ash concrete using metaheuristic algorithms, supporting the generalizability of multi-objective optimization strategies for sustainable concrete design.

Further analysis indicates that the cost advantage of M4 is primarily attributed to its multi-objective optimization strategy, which effectively coordinates the relationships among material components. In addition to satisfying structural strength requirements, M4 optimizes the proportioning of fiber and aggregate systems, thereby reducing reliance on high-cost materials and achieving a balanced trade-off between mechanical performance and economic efficiency.

(3)

Evaluation of the Universality of the M4 Method: The results indicate that BFRC demonstrates excellent mechanical performance and strong potential for the synergistic optimization of low carbon emissions and cost-effectiveness. These computational results align with those of Marani et al. [16] regarding ML-driven optimization for high-performance concrete while suggesting the theoretical potential of extending such frameworks to tensile-strength-centered BFRC design—a dimension not explored in existing multi-objective studies [11,12,17]. To further enhance its generalizability, future research should focus on evaluating the durability of the M4 mix design under complex environmental conditions, thereby supporting its broader application in developing green, high-performance concrete.

5. Conclusions

This study presents a comprehensive multi-objective optimization framework for basalt fiber-reinforced concrete (BFRC) mix design, which integrates tensile strength (f_t) as a key design objective alongside compressive strength (f_c), economic cost, and carbon emissions. To address the limitations of conventional mix design methods, gene expression programming (GEP) was employed to develop high-precision predictive models for f_c and f_t, and the non-dominated sorting genetic algorithm II (NSGA-II) was applied to solve the multi-objective optimization problem, aiming to ensure the rationality and feasibility of the optimized mix proportions. The main findings are summarized as follows:

(1)

The GEP-based prediction models for f_c and f_t exhibit satisfactory prediction accuracy, with root mean square error (RMSE) values of 2.02 MPa and 1.66 MPa, respectively, and relative errors controlled within ±20%. These models not only avoid the limitations of traditional empirical formulas but also provide a reliable and efficient theoretical foundation for the subsequent BFRC mix design optimization, providing a theoretical basis for future application of the proposed optimization framework.

(2)

The proposed method identifies trade-offs among mechanical performance, cost, and carbon emissions. Within the investigated strength range (30–60 MPa), model predictions suggest potential cost reductions of 11–19% and carbon emission reductions of 11–30% compared to conventional methods, pending experimental confirmation.

(3)

The optimization results further reveal that cement dosage is the dominant factor influencing both the cost and carbon emissions of BFRC, as cement production accounts for the majority of carbon emissions and material costs in concrete preparation. In contrast, basalt fiber dosage significantly affects the tensile strength of BFRC, which is consistent with the core design objective of this study. The M4 method thus offers a systematic and scientific approach for balancing these competing objectives in BFRC mix design, providing a reference for potential applications.

(4)

This study develops a methodological framework for multi-objective BFRC optimization that addresses the limitations of existing black-box approaches. The GEP-based explicit modeling strategy enables transparent trade-off analysis among competing objectives, offering a reproducible methodology for optimizing fiber-reinforced cementitious composites. Unlike previous applications that treat optimization as a post-prediction step [11,12], this study presents a methodological advance by seamlessly integrating transparent GEP-based predictive equations into the NSGA-II optimization loop, enabling direct gradient-informed Pareto exploration for BFRC—an approach not realizable with black-box alternatives. Experimental validation is currently underway to confirm the practical applicability of the proposed method.