1. Introduction

Ultra-high performance concrete (UHPC) has emerged as a promising advanced cementitious material in modern civil engineering, owing to its superior mechanical properties and durability^[1,2]. However, the realization of these performance advantages is critically dependent on the precise proportioning of multiple constituents, including cementitious components, fibers, and admixtures. The high-dimensional and strongly coupled nature of the mix proportion space renders conventional empirical and trial-and-error methods ineffective for systematic optimization^[3,4]. On the one hand, localized search strategies struggle to accurately capture the nonlinear mapping relationships between components and performance across the entire variable space, thereby failing to exploit the intrinsic performance potential of UHPC materials. On the other hand, existing design paradigms predominantly emphasize unidimensional improvements in mechanical performance, while paying comparatively limited attention to the simultaneous consideration of multi-dimensional metrics such as global warming potential (GWP) and production cost. This deficiency impedes the fulfillment of practical engineering demands for under the imperatives of green and low-carbon construction^[5,6]. Consequently, developing a mixed design optimization method capable of accurately predicting UHPC performance while simultaneously addressing multiple conflicting objectives holds considerable practical significance for promoting the large-scale engineering application of UHPC^[7].

In recent years, with the interdisciplinary advancement of materials data science and artificial intelligence, machine learning methods have garnered widespread attention in concrete performance prediction and mix proportion optimization^[8-11]. Compared with traditional regression models, machine learning algorithms can autonomously extract the intrinsic relationships between compositional features and target properties from extensive experimental datasets without the need for predefined explicit physical equations, thereby exhibiting distinct advantages in addressing complex multicomponent and highly nonlinear systems such as UHPC^[12]. Existing studies have successfully employed algorithms including random forest (RF) and support vector regression (SVR) to predict UHPC compressive strength^[13,14], flexural strength^[15], elastic modulus^[16], and durability performance^[17,18], achieving varying degrees of improvement in predictive accuracy. However, most of these studies have focused on the construction and evaluation of single-algorithm models, with limited in-depth exploration of the systematic integration of multi-model collaboration and multi-objective optimization decision-making.

Despite the considerable potential demonstrated by machine learning techniques in UHPC materials research, several pivotal challenges persist. First, single-algorithm models are prone to imbalances between bias and variance. For instance, tree-based models have limited extrapolation capabilities, kernel methods are sensitive to hyperparameter selection, and neural networks are prone to overfitting with limited sample sizes. Moreover, the hyperparameter tuning process for these models frequently relies on empirical heuristics and lacks an efficient dynamic update mechanism^[19]. Second, the complementary advantages of different algorithms have not yet been effectively integrated, and the generalization bottleneck of single-algorithm models constrains the reliability of subsequent optimization. Third, existing research has predominantly focused on predictive accuracy and has not extended to embedding high-fidelity predictive models within a multi-objective optimization framework, resulting in a conspicuous disconnection between performance prediction and mix proportion decision-making. UHPC mix design is inherently a high-dimensional, multi-constraint optimization problem, yet current machine learning methods struggle to achieve coordinated optimization across multiple mutually constraining objectives such as mechanical performance, environmental effects, and economic cost. Therefore, establishing an effective linkage between predictive accuracy and decision-making intelligence constitutes a critical bottleneck that urgently demands resolution in this domain.

To address the aforementioned limitations, this study innovatively proposes a multi-objective optimization design framework for UHPC based on Stacking ensemble learning. Stacking ensemble learning and non-dominated sorting genetic algorithm III (NSGA-III) were adopted to enhance surrogate prediction reliability and to handle constrained multi-objective optimization with conflicting performance, environmental, and economic targets. Firstly, a Stacking ensemble predictive model integrating multiple heterogeneous machine learning algorithms was constructed, including RF, SVR, multi-layer perceptron neural network (MLPNN), gradient boosting decision tree (GBDT), extreme gradient boosting (XGBoost), categorical boosting (CatBoost), adaptive boosting (AdaBoost), and light gradient boosting machine (LightGBM), serving as base learners, with linear regression employed as the meta-learner. To fully leverage the strengths of each base learner, various hyperparameter optimization strategies, including random search (RS), Bayesian optimization (BO), particle swarm optimization (PSO), grey wolf optimization (GWO), simulated annealing (SA), and the tree-structured parzen estimators (TPE), were further employed to fine-tune the model. Subsequently, the optimal Stacking model is embedded as the surrogate objective function within the NSGA-III multi-objective optimization framework. With the objectives of maximizing compressive and flexural strength while simultaneously minimizing environmental and economic impacts, a global search for the optimal UHPC mix proportion was performed under actual engineering constraints to obtain a set of Pareto non-dominated solutions. Finally, the technique for order preference by similarity to ideal solution (TOPSIS) method was employed to comprehensively rank the non-dominated solutions, thereby identifying the mix design with the optimal overall performance. The research aims to provide innovative insights into the multi-objective collaborative optimization of UHPC and to promote deeper integration of engineering materials and machine learning technologies.

2. Datasets and Methods

2.1 Dataset construction

This study established a UHPC database by collecting 1,133 samples collected from 95 peer-reviewed publications. To improve data consistency and compatibility among different literature sources, only samples with clearly reported mix proportions, specimen dimensions, curing age, and corresponding mechanical properties were retained. The curing age was fixed at 28 days to reduce the uncertainty caused by age-dependent strength development. In addition, compressive specimen volume (CV) and flexural specimen volume (FV) were introduced as input variables to partially account for the influence of different specimen dimensions on measured strength.

Table 1 presents the input and output parameters of the dataset, along with their units and statistical distribution characteristics. The input parameters consist of the following variables: cement (CE), fly ash (FA), quartz powder (QP), steel slag powder (SSP), slag powder (SLP), limestone powder (LSP), metakaolin (ME), rice husk ash (RHA), silica fume (SF), steel fiber (Fiber), sand (SA), coarse aggregate (CA), water (WA), superplasticizer (SP), water-to-binder ratio (WB), CV and FV. The model output variables are the compressive strength (CS) and flexural strength (FS) of UHPC. Some mineral admixture variables, such as LSP, ME, and RHA, exhibit relatively high skewness because they are selectively used in different literature-derived UHPC mixtures, resulting in a large proportion of zero values.

To eliminate the adverse effects of different feature dimensions and numerical scales on model training, a normalization procedure was applied to the raw dataset. The specific calculation formula is as follows:

(1)$X_{norm}=\frac{X-X_{min}}{X_{max}-X_{min}}$

where X_min and X_max represent the minimum and maximum values within the dataset, respectively, while X and X_norm denote the original and normalized values.

2.2 Feature correlation analysis

To elucidate the interrelationships and distribution characteristics among the feature variables, a matrix plot was employed to visualize the dataset, as illustrated in Figure 1 and Figure 2. In this matrix plot, the kernel density estimation curve for each variable is displayed along the diagonal. The lower triangular region presents pairwise scatter plots accompanied by linear regression lines, whereas the upper triangular region reports the Pearson correlation coefficients, with the background color gradient representing the sign and magnitude of the correlations. In addition to revealing the relationships between mix proportion variables and mechanical properties, this analysis also provides a preliminary assessment of variable redundancy and compatibility in the collected multi-source dataset. It can be found that the absolute values of all pairwise Pearson correlation coefficients among the variables are less than 0.7, indicating the absence of strong pairwise linear correlations and suggesting a low risk of pairwise collinearity^[115]. Therefore, no feature was removed solely on the basis of the pairwise correlation analysis at this stage.

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Figure 1. Feature correlation analysis heatmap for compressive strength dataset. CS: compressive strength; CV: compressive specimen volume; WB: water-to-binder ratio; SP: superplasticizer; WA: water; CA: coarse aggregate; Fiber: steel fiber; SA: sand; SF: silica fume; RHA: rice husk ash; ME: metakaolin; LSP: limestone powder; SLP: slag powder; SSP: steel slag powder; QP: quartz powder; FA: fly ash; CE: cement.

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Figure 2. Feature correlation analysis heatmap for the flexural strength dataset. FS: flexural strength; FV: flexural specimen volume; WB: water-to-binder ratio; CA: coarse aggregate; SA: sand; Fiber: steel fiber; SF: silica fume; RHA: rice husk ash; ME: metakaolin; LSP: limestone powder; SLP: slag powder; SSP: steel slag powder; QP: quartz powder; FA: fly ash; CE: cement.

Furthermore, the steel fiber content exhibits a significant positive effect on both compressive and flexural strength, with Pearson coefficients of 0.31 and 0.45, respectively. This is primarily attributed to the ability of steel fibers to bridge cracks through interfacial bonding forces following matrix cracking, effectively alleviating stress concentration at the crack tip and thereby inhibiting crack propagation. Under flexural loading, this bridging effect is particularly pronounced, significantly delaying the process from initial cracking to ultimate failure. Consequently, flexural strength is more sensitive to variations in steel fiber content.

Conversely, the water-to-binder ratio exerts a consistent negative influence on the mechanical properties of UHPC, with Pearson coefficients of -0.24 and -0.23 for compressive strength and flexural strength, respectively. This inverse relationship is attributed to the fact that a lower water-to-binder ratio reduces the initial porosity of the hardened paste and promotes the formation of a denser microstructure within the hydration products, thereby effectively enhancing the mechanical performance of the matrix.

2.3 Machine learning model

2.3.1 Stacking ensemble model

Stacking is an ensemble learning strategy based on a multi-layer architecture, which typically comprises a two-level learning structure. At the first level, multiple base learners are independently trained on the training dataset to generate predictive outputs. At the second level, these predictions serve as newly constructed input features for training a meta-learner, thereby producing the final ensemble prediction^[116].

This study selected eight heterogeneous algorithms as the base learners to comprehensively capture the diverse patterns within the data. RF is composed of multiple decision trees that process high-dimensional data through random sampling and complete splitting, exhibiting nonlinearity and high robustness^[117]. SVR minimizes structural risk by employing kernel functions and regularization parameters, demonstrating strong performance on small-sample datasets^[118]. The MLPNN employs a feedforward network architecture to approximate complex mappings through nonlinear activation functions^[119]. GBDT uses decision trees as base learners, and iteratively reduces error through gradient boosting, demonstrating strong fitting capabilities for nonlinear relationships and feature interactions^[120]. XGBoost enhances predictive accuracy and computational efficiency through regularization and parallel operations, excelling particularly in handling sparse data and effectively executing classification and regression tasks^[121]. CatBoost builds upon the GBDT framework, employing ordered boosting to mitigate gradient bias and prediction drift, enabling high-precision predictions with strong generalization capability^[122]. AdaBoost iteratively adjusts sample weights to focus on misclassified instances, enabling fast convergence, though it remains sensitive to noisy data and outliers^[123]. LightGBM employs gradient-based one-sided sampling and exclusive feature bundling to achieve rapid model training with low memory consumption, thereby facilitating efficient processing of large-scale datasets^[124].

Furthermore, to evaluate the efficacy of the Stacking ensemble model, RF, SVR, MLPNN, and AdaBoost were selected as benchmark models for comparative analysis. These four models represent four distinct modeling paradigms, specifically bagging ensemble, kernel methods, neural networks, and boosting ensemble, thereby providing sufficient representativeness for controlled comparison. The remaining four base learners were excluded from the control group because they belong to the same gradient boosting framework, thereby avoiding redundancy in the comparative assessment.

2.3.2 Hyperparameter optimization

Hyperparameters are configuration parameters set prior to model training, and their assigned values directly influence both the model architecture and the learning process. Systematically adjusting hyperparameter combinations and comparing the corresponding model performance enables the identification of optimal configurations tailored to a specific dataset, thereby effectively enhancing predictive accuracy. In this study, the dataset was randomly partitioned into training and testing subsets at a ratio of 7:3. The training set was employed for model fitting and hyperparameter tuning, whereas the testing set was retained independently to evaluate the generalization performance of the final model. During the hyperparameter optimization phase, a five-fold cross-validation strategy was applied to the training set to mitigate the variance introduced by a single random data split and to ensure the robustness of the performance evaluation.

Based on this validation framework, six hyperparameter optimization algorithms were introduced and systematically compared. RS explores the parameter space through uniform sampling, independent of objective function modeling, exhibiting robust performance and strong global search capabilities^[125]. BO iteratively selects evaluation points based on the surrogate model, achieving efficient global optimization with fewer samples^[126]. PSO simulates collective intelligence behavior by guiding the search through individual and collective historical optima, achieving fast convergence but remaining prone to local optima^[127,128]. GWO emulates the social hierarchy and hunting behavior of grey wolves, effectively balancing global exploration and local exploitation. It is well-suited for continuous optimization problems, features minimal parameters, and demonstrates robust convergence^[129]. SA is inspired by the thermodynamic annealing processes, avoids local optima by probabilistically accepting suboptimal solutions, and gradually converges to an approximate solution near the global optimum^[130]. TPE is a sequential model-based optimization method that constructs separate probability models for high-performing and mediocre hyperparameters, thereby efficiently guiding the search process and improving overall optimization efficiency^[131].

2.3.3 NSGA-III multi-objective optimization and TOPSIS comprehensive evaluation

The NSGA-III^[132] is a multi-objective optimization algorithm developed from non-dominated sorting genetic algorithm II (NSGA-II)^[133], demonstrating significant advantages in solving high-dimensional objectives and large-scale optimization problems. The core innovation of this algorithm lies in introducing a reference point guidance mechanism. Pre-setting a set of uniformly distributed reference points to guide the evolutionary direction of the population, the algorithm effectively enhances convergence efficiency while maintaining the diversity of the Pareto front^[134,135]. This study employs the NSGA-III algorithm to optimize the mix proportion design of UHPC, addressing challenges such as high-dimensional nonlinearity and multi-objective conflicts inherent in this process.

Although the iterative optimization process generates a set of non-dominated solutions constituting the Pareto front, practical engineering applications require selecting an optimal mix proportion design that satisfies specific preference criteria. Considering the inherent conflicts among multiple optimization objectives within the UHPC system, this study employs the TOPSIS comprehensive evaluation method for final decision-making, as shown in Figure 3. This method calculates the Euclidean distance between each alternative and both the positive and negative ideal solutions, thereby comprehensively evaluating their relative merits^[136]. A normalized relative closeness score closer to 1 indicates that the corresponding alternative is closer to the theoretical optimum solution when balancing multiple conflicting objectives.

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Figure 3. Flowchart of the integrated NSGA-III and TOPSIS framework for UHPC. NSGA-III: non-dominated sorting genetic algorithm III; TOPSIS: technique for order preference by similarity to ideal solution; UHPC: ultra-high performance concrete; GWP: global warming potential; CS: compressive strength; FS: flexural strength.

3. Results and Discussion

3.1 Comparative analysis of predictive performance

The Taylor diagram is a visualization tool for comprehensively evaluating model prediction performance using multiple statistical metrics^[137]. In this diagram, the radial distance from the coordinate origin represents the standard deviation (SD) of the predicted values, the radial distance from the observational reference point reflects the root mean square error (RMSE), and the azimuthal angle corresponds to the correlation coefficient (R). The experimental observation reference point is marked with a red five-pointed star. The optimal model was selected according to its proximity to the experimental reference point in the Taylor diagram shown in Figure 4, together with the R, RMSE, and SD values listed in Table 2 and Table 3.

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Figure 4. Taylor diagrams of the different models for: (a) Compressive strength; (b) Flexural strength. RF: random forest; SVR: support vector regression; MLPNN: multi-layer perceptron neural network; AdaBoost: adaptive boosting; RS: random search; BO: bayesian optimization; PSO: particle swarm optimization; GWO: grey wolf optimization; SA: simulated annealing; TPE: tree-structured parzen estimators.

For the compressive strength dataset (Figure 4a and Table 2), the Stacking-GWO model exhibits the closest proximity to the experimental reference point, indicating the optimal comprehensive predictive performance. Specifically, this model achieves an SD of 16.4329, which closely approximates the observed value (SD = 18.8091), demonstrating its ability to accurately reproduce the dispersion characteristics of the original data. Concurrently, it attains a relatively low RMSE (7.3164) among all evaluated models. Although the R value of Stacking-GWO (0.9242) is marginally lower than that of the Stacking-SA model (R = 0.9271), the superior reproduction of data variance compensates for this slight discrepancy in the comprehensive distance metric of the Taylor diagram, resulting in a closer spatial alignment with the reference point.

For the flexural strength dataset (Figure 4b and Table 3), the Stacking-SA model demonstrates the most prominent performance. This model attains a SD of 5.7901, exhibiting the smallest deviation from the observed value (SD = 6.3620) among all models, alongside the highest R (0.9116) and the lowest RMSE value (2.6178). In contrast, while the Stacking-GWO model yields a relatively high R (0.8977), its SD deviates more substantially from the observed value, resulting in a larger comprehensive distance from the reference point in the Taylor diagram compared with the Stacking-SA model. Consequently, for flexural strength prediction, the Stacking-SA model not only achieves optimal performance between predicted and measured values but also more accurately captures the fluctuation characteristics of the data.

In comparison, the MLPNN model, which represents a single-algorithm model, demonstrates inferior predictive capability across both datasets. For compressive strength prediction, the MLPNN model yields SD = 13.9480, R = 0.7416, and RMSE = 12.6338. For flexural strength prediction, the corresponding metrics are SD = 5.6069, R = 0.7236, and RMSE = 4.5055, all of which are substantially less favorable than those of the various Stacking ensemble models. This pronounced disparity underscores the inherent limitations of individual machine learning models in addressing complex prediction tasks such as material performance assessment, wherein they often struggle to achieve effective bias control and accurate variability reproduction simultaneously. Conversely, the Stacking ensemble strategy, by integrating the outputs of multiple base learners, substantially enhances model generalization capacity and predictive stability. This improvement is visually corroborated by the overall convergence of the ensemble model positions toward the reference point in the Taylor diagrams, further substantiating the feasibility and effectiveness of the ensemble learning methodology employed in this investigation.

3.2 Objective functions

This section establishes the objective functions for compressive strength, flexural strength, GWP, and economic impact, building upon the optimal Stacking learning algorithm identified in Section 3.1. The NSGA-III algorithm was employed for high-dimensional multi-objective optimization, generating a Pareto-optimal solution set through an elite preservation strategy. Subsequently, the TOPSIS comprehensive evaluation model was established based on this set to achieve multi-criteria decision-making for UHPC mix proportion designs across different strength grades.

The Stacking-GWO and Stacking-SA models serve as the optimal objective functions for predicting compressive strength and flexural strength, respectively. During the NSGA-III iteration, each candidate mix proportion was evaluated by the pre-trained Stacking models to obtain the predicted strengths, which were then combined with the calculated GWP and cost as the four objective values for non-dominated sorting and population updating. The mapping relationship between different objective functions and feature parameters is as follows:

(2)$f_{(CS-\mathrm{Stacking}GWO)}=f(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_{10},x_{11},x_{12},x_{13},x_{14},x_{15},x_{16})$

(3)$f_{(FS-ST\text{acking}SA)}=f(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_{10},x_{11},x_{12},x_{13},x_{14},x_{15},x_{17})$

where i ranges from 1 to 17, corresponding to CE, FA, QP, SSP, SLP, LSP, ME, RHA, SF, Fiber, SA, CA, WA, SP, WB, CV and FV, respectively.

The environmental and economic impacts are calculated as the sum of the mass of each constituent material multiplied by its corresponding environmental impact factor or unit cost, with the specific coefficients provided in Table 4.

3.3 Boundary constraints

Before conducting the mix proportion optimization, the decision variables x_i were subject to the following boundary constraints:

(4)$lb\le x_i\le ub$

(5)$lb=[502.1,0,0,0,0,0,0,0,0,0,332,0,126.7,4.2,0.10,64,256]$

(6)$ub=[1046,528,380,196.3,683.91,332,300,230,494,390,1460.3,1075,310.68,54,0.26,64,256]$

To eliminate the potential adverse effects of extreme conditions on the cementitious system, the effective range of cement content was defined as the intersection between the 10th and 90th percentiles of the corresponding cement content data for compressive strength and flexural strength.

To control experimental variables, uniform specimen dimensions and curing conditions were adopted. The specific parameter settings are as follows: Compressive strength specimens were 40 × 40 × 40 mm cubes (CV = 64 cm³), while flexural strength specimens were 40 × 40 × 160 mm prisms (FV = 256 cm³). All specimens were cured for 28 days under standardized conditions.

The WB was calculated using the following formula and was subject to the following constraint:

(7)$0.10\le WB=\frac{x_{13}}{\sum _{i=1}^8{x}_i}\le 0.26$

The total amount of cementitious materials was subject to the following distribution range:

(8)$608\le M_{SME}=\sum _{i=1}^9{x}_i\le 1726$

The total amount of all raw materials was subject to the following distribution range:

(9)$2135\le M_{\text{Total}}=\sum _{i=1}^{14}{x}_i\le 3076$

With reference to the draft revision of GB/T 31387-2015 (draft GB/T 31387-2025^[152]), the strength thresholds listed in Table 5 were adopted in this study as scenario-specific design constraints for the three UHPC strength grades.

3.4 Multi-dimensional performance validation based on NSGA-III and TOPSIS

To evaluate the multi-objective adaptability of UHPC under typical engineering scenarios, a multi-dimensional performance verification was conducted for three different strength grades: UC100, UC120, and UC140. The weights of CS, FS, GWP, and cost were set as 0.05, 0.05, 0.05, and 0.85 for UC100, 0.25, 0.25, 0.25, and 0.25 for UC120, and 0.30, 0.30, 0.20, and 0.20 for UC140, respectively, according to different engineering preferences. The validation utilized the established datasets and the top 100 optimal solutions selected by the TOPSIS decision-making model. The results are shown in Figure 5. For the UC100 strength grade, the GWP range is 414.66-655.42 kg CO₂/m³, with a cost range of 104.15-372.89 USD/m³. In contrast, the GWP of UC140 increases further to 425.35-778.24 kg CO₂/m³, and the cost rises to 382.50-1182.67 USD/m³. This trend primarily stems from mixture adjustments required to achieve higher mechanical properties, particularly the significant increase in crucial components like steel fiber. As an energy-intensive raw material, steel fiber exhibits substantial carbon emissions during production (1.49 kg CO₂ eq) and relatively high market prices (5.00 USD/kg). Consequently, higher fiber content directly drives up both GWP and production cost. Nevertheless, UHPC mixtures across all strength grades still demonstrate a favorable balance of overall performance, confirming the effectiveness of the TOPSIS method in achieving multi-objective synergistic optimization of mechanical properties, environmental impact, and economic cost.

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Figure 5. Pareto distribution of optimization objectives in three typical application scenarios. (Scenario 1, Scenario 2 and Scenario 3 correspond to the strength levels of UC100, UC120 and UC140, respectively). GWP: global warming potential; CS: compressive strength; FS: flexural strength.

3.5 Multi-objective optimization results and performance evolution analysis

The optimized mix proportions and corresponding output objectives for different strength grades under multi-objective optimization are summarized in Table 6. As the strength grade progressively increased from UC100 to UC140, the mix proportions were systematically adjusted to meet the escalating performance demands. Regarding mechanical properties, compressive strength exhibited a clear upward trend, reaching 118.65 MPa, 134.44 MPa, and 143.08 MPa for UC100, UC120, and UC140, respectively. The production cost increased from 104.15 USD/m³ for UC100 to 307.54 USD/m³ for UC120 and 382.50 USD/m³ for UC140. Notably, the GWP exhibited an initial increase followed by a decrease: rising from 484.71 kg CO₂/m³ in UC100 to 622.27 kg CO₂/m³ in UC120, then decreasing to 425.35 kg CO₂/m³ in UC140. This variation primarily stems from the mechanical performance enhancement provided by silica fume: the silica fume content progressively increases from UC100 to UC140, peaking at 301.25 kg/m³. This allows for reduced cement usage while achieving the same target strength, consequently lowering the carbon emissions of the concrete.

The optimization performance of the NSGA-III algorithm was quantitatively evaluated using the hypervolume (HV) and spacing (SP) metrics, as shown in Figure 6. The HV value of the solution set fluctuated and gradually increased with iterations before eventually stabilizing, indicating that the algorithm effectively guided the population toward convergence to the true Pareto front through its elite retention strategy^[143]. Regarding distribution performance, the SP values rapidly decreased with increasing iterations and eventually stabilized, reflecting continuous improvement in the uniformity of the solution set distribution throughout the evolutionary process. This improvement achieved a balanced distribution of non-dominated solutions while maintaining diversity. The comprehensive evaluation based on the TOPSIS method reveals that the relative closeness gradually decreased from 0.9827 for UC100 to 0.7751 for UC120 and 0.6852 for UC140, as shown in Figure 7. This trend suggests that achieving a delicate balance among mechanical performance, environmental impact, and cost control becomes progressively more challenging at higher strength levels, substantially increasing the complexity of the multi-objective synergistic optimization.

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Figure 6. Evolutionary trajectories of the HV and SP metrics under the different strength grades. (a,b) UC100; (c,d) UC120; (e,f) UC140. HV: hypervolume; SP: spacing.

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Figure 7. The ideal solution and relative closeness under the different strength grades.

Compared with studies mainly focused on UHPC performance prediction, the present framework further connects the trained Stacking model with NSGA-III optimization and TOPSIS ranking. In this way, the output is extended from predicted strength values to practical mix design options considering strength, cost, and GWP requirements.

4. Conclusions

This study proposes a multi-objective optimization design framework for UHPC based on Stacking ensemble learning. By integrating the NSGA-III multi-objective optimization algorithm with the TOPSIS comprehensive evaluation method, the framework effectively addresses the limitations inherent in conventional trial-and-error approaches when confronting high-dimensional nonlinear problems. The principal conclusions are summarized as follows:

(1) An experimental database comprising 1,133 UHPC mix proportions and their corresponding mechanical properties was established. Feature correlation analysis revealed that steel fiber content exhibits a significant positive correlation with both compressive and flexural strength, with Pearson coefficients of 0.31 and 0.45, respectively. In contrast, the water-to-binder ratio demonstrates a negative correlation, with Pearson coefficients of -0.24 for compressive strength and -0.23 for flexural strength. These findings indicate that the bridging effect exerted by steel fibers enhances the mechanical performance of the concrete, whereas an increase in the water-to-binder ratio adversely affects the mechanical properties by inducing a more porous microstructure.

(2) Comprehensive evaluation based on Taylor diagrams demonstrates that the Stacking ensemble models substantially outperform individual single-algorithm models in predicting both compressive and flexural strength. The Stacking-GWO model performed optimally in compressive strength prediction (R = 0.9242, SD = 16.4329, RMSE = 7.3164), while the Stacking-SA model performed best in flexural strength prediction (R = 0.9116, SD = 5.7901, RMSE = 2.6178). These results validate the superiority of ensemble learning strategies in enhancing prediction accuracy and robustness.

(3) By embedding the optimal Stacking ensemble models into the NSGA-III and TOPSIS frameworks, multi-objective collaborative optimization of UHPC mix designs was successfully achieved. For the optimization of the three strength grades UC100, UC120 and UC140, the predicted compressive strengths reached 118.65 MPa, 134.44 MPa and 143.08 MPa, respectively, while effectively balancing environmental impact and production cost, thereby providing a scientific basis for decision-making that integrates mechanical performance, environmental sustainability, and economic efficiency for practical engineering applications.

(4) Different from existing studies that mainly focus on performance prediction or algorithmic optimization alone, the methodological contribution of this work lies in establishing a closed-loop data-driven design pathway that integrates interpretable performance prediction, engineering-constrained multi-objective optimization and reliable decision-making.

From a practical perspective, the proposed framework can support the preliminary design and rapid screening of UHPC mixtures by balancing strength requirements, environmental impact, and production cost under engineering constraints. Nevertheless, it should be acknowledged that the optimized UHPC mix designs obtained in this study were not experimentally prepared and tested. Future work will further conduct laboratory verification of the optimized mixtures to evaluate the consistency between predicted and measured performance, thereby improving the robustness, reliability, and engineering applicability of the proposed framework.

Acknowledgements

The authors declare that ChatGPT was used solely for language polishing during manuscript preparation. All research content, including the study design, data analysis, interpretation, figures, and tables, is original and was not generated using AI tools. The authors are responsible for the accuracy and scientific content of the article.

Authors contribution

Zhang W: Investigation, data curation, validation, writing-original draft.

Liu C: Conceptualization, resources, supervision.

Yao Y, Nasr A: Validation, writing-review & editing.

Yang Q: Data curation, validation.

Duan Z: Funding acquisition, supervision.

Conflicts of interest

Not applicable.

Ethical approval

Not applicable.

Consent to participate

Not applicable.

Consent for publication

Not applicable.

Availability of data and materials

Data will be made available on request.

Funding

The authors would like to acknowledge the National Natural Science Foundation of China (Grant No. 52178244) for financial support.

Copyright

© The Author(s) 2026.