Machine Learning-Assisted Sustainable Mix Design of Waste Glass Powder Concrete with Strength–Cost–CO₂ Emissions Trade-Offs

Abstract

Glass powder, a non-degradable waste material, offers significant potential to reduce cement consumption and carbon emissions in concrete production. However, existing mix design methods for glass powder concrete (GPC) fail to systematically balance economic efficiency, environmental sustainability, and mechanical performance. To address this gap, this study proposes an AI-assisted framework integrating machine learning (ML) and Multi-Objective Optimization (MOO) to achieve a sustainable GPC design. A robust database of 1154 experimental records was developed, focusing on five key predictors: cement content, water-to-binder ratio, aggregate composition, glass powder content, and curing age. Seven ML models were optimized via Bayesian tuning, with the Ensemble Tree model achieving superior accuracy (R² = 0.959 on test data). SHapley Additive exPlanations (SHAP) analysis further elucidated the contribution mechanisms and underlying interactions of material components on GPC compressive strength. Subsequently, a MOO framework minimized unit cost and CO₂ emissions while meeting compressive strength targets (15–70 MPa), solved using the NSGA-II algorithm for Pareto solutions and TOPSIS for decision-making. The Pareto-optimal solutions provide actionable guidelines for engineers to align GPC design with circular economy principles and low-carbon policies. This work advances sustainable construction practices by bridging AI-driven innovation with building materials, directly supporting global goals for waste valorization and carbon neutrality.

1. Introduction

Globally, the large-scale construction of buildings and infrastructure has led to a continuous increase in cement consumption, resulting in substantial carbon emissions [1]. According to statistical data, the cement industry accounts for 7–8% of global greenhouse gas emissions [2]. Furthermore, inefficiencies in technology and management during conventional cement production processes result in additional CO₂ emissions [3]. Meanwhile, approximately 130 million tons of glass waste are generated worldwide annually, with only 21% being recycled [4]. The majority of waste glass is disposed off in landfills, causing not only resource wastage but also significant environmental risks [5]. Studies have revealed that waste glass processing generates various pollutants such as particulate matter and heavy metals, contaminating surrounding areas [6]. Modern architectural demands impose increasingly stringent requirements on concrete performance, particularly in high-seismic-risk zones [7]. Traditional concrete structures frequently suffer catastrophic failures due to material brittleness, insufficient ductility, or joint inadequacies [8]. These costly lessons underscore the urgent need to develop high-performance, high-toughness concrete. To synergistically achieve performance enhancement and carbon emission reduction objectives, modern concrete technology widely employs supplementary cementitious materials such as fly ash, slag powder, and silica fume for partial cement replacement [9]. However, as primarily industrial byproducts, these materials exhibit inherent limitations: diverse sourcing, supply volatility, impurity content variability, and heterogeneous pozzolanic reactivity [10]. These factors pose critical challenges for SCM quality control, consequently impeding the development of unified and reliable mix design specifications. Glass powder leverages its excellent pozzolanic activity to address environmental challenges by reducing cement and aggregate consumption, thereby minimizing the exploitation of natural resources and mitigating environmental pollution [11,12]. Glass powder concrete (GPC) incorporates processed recycled glass waste as a supplementary cementitious material [13]. This process significantly enhances concrete sustainability [14]. The manufacturing process of waste glass powder is illustrated in Figure 1. Furthermore, optimizing glass powder content in GPC improves compressive strength and enhances concrete durability [15,16,17,18]. Replacing up to 30% of cement in certain concrete formulations not only addresses glass waste recycling challenges but also reduces carbon emissions, offering innovative solutions for municipal solid waste management and sustainable construction practices [19].

Currently, research on the mechanical properties of GPC has been gaining increasing attention. Ref. [20] investigated the long-term performance of GPC in large-scale field applications. Ref. [21] explored the synergistic mechanism of crystalline admixtures and glass powder incorporation on the impermeability of recycled concrete. Ref. [22] analyzed the effect of glass powder on the compatibility of polynaphthalene sulfonate (PNS) or polycarboxylate (PC)-based superplasticizers with ordinary Portland cement. Ref. [23] evaluated the influence of immersion time of glass powder in water before mixing it with the other concrete ingredients on the fresh and hardened properties of concrete. As the glass powder content increases, the compressive strength, dynamic elastic modulus, and impermeability of GPC gradually improve, alongside a significant improvement in toughness [24]. Experimental studies further confirmed that the use of 20% glass powder content seems to be the best in terms of the reduction in the creep [25]. Ref. [26] examined the impact of glass powder on cement hydration and demonstrated its contribution to enhancing hydration kinetics. The aforementioned studies comprehensively elucidate the influence mechanisms of glass powder on the fundamental properties of GPC. However, to promote the application of GPC, it is essential to first achieve accurate calculation of the core mechanical property of concrete, which is compressive strength [27]. Numerous studies have been conducted on the compressive strength of concrete. Ref. [28] predicted concrete compressive strength values based on image processing results. Ref. [29] developed an effective non-destructive testing method to assess concrete compressive strength under wet–dry cycling conditions. Traditional studies on compressive strength of CPC predominantly rely on experiments, which suffer from inherent limitations such as prolonged duration and high costs [30,31]. Ref. [32] established a mechanical prediction model for GPC using experimental data, revealing functional relationships among tensile, flexural, and compressive strengths. But the systematic influence of compositional parameters on strength formation mechanisms remains unquantified, although studies have confirmed significant correlations between GPC compressive strength and parameters including cement content, curing age, gravel content, sand content, and water content [33,34,35]. However, due to the complex interactions between multiple components, there is currently no single general equation that can accurately capture the correlation between CS and this series of influencing factors [36,37].

In recent years, data-driven methods, such as machine learning (ML) and deep learning (DL), have demonstrated revolutionary potential in civil engineering [38,39,40,41,42,43,44,45,46,47,48,49,50], particularly in predicting concrete mechanical properties, due to their powerful nonlinear fitting capabilities. Artificial neural networks exhibit exceptional performance in predicting concrete fracture toughness by capturing nonlinear correlations among sensitive parameters such as cement content, achieving high consistency with experimental data [51]. For slump optimization of radiation-resistant serpentine concrete, a hybrid optimization strategy integrating support vector machines with simulated annealing–genetic algorithms successfully increased the slump to 166 mm (error < 5%), demonstrating the high robustness of ML models under small-sample conditions [52]. Ref. [53] reduced compressive strength prediction errors to 2.02 MPa through a fusion of deep learning and ultrasonic testing, highlighting the potential of multimodal data integration. Various ML models have been employed to predict the comprehensive mechanical properties of recycled aggregate concrete [54,55,56]. ML models have also shown outstanding predictive accuracy (with determination coefficients R² > 0.95) for concrete containing fly ash as SCM [57,58,59]. However, insufficient model interpretability continues to constrain its potential for engineering implementation. SHAP-based interpretable modeling framework and optimization strategy effectively enhance model transparency and decision traceability in complex material systems, significantly elevating the trustworthiness of AI-driven materials design methodologies [60]. These findings further validate the efficacy of ML models in concrete mechanical property prediction. Diverse technical approaches have been developed for predicting GPC compressive strength. Ref. [61] systematically revealed the influence patterns of parameter combinations on compressive strength using univariate, bivariate, and multivariate regression models. Prediction models based on support vector machines and extreme gradient boosting algorithms exhibit excellent generalization capabilities (R² > 0.9), confirming the superior performance of ML models in GPC compressive strength prediction [62]. Ref. [63] innovatively integrated the sparrow search algorithm with ML models, achieving significantly enhanced prediction accuracy compared to baseline models. While significant breakthroughs have been made in GPC compressive strength prediction, current research predominantly focuses on single mechanical property prediction, neglecting synergistic mix design that balances sustainability and cost-effectiveness, thereby limiting its engineering applicability in green construction.

Concrete mix design constitutes a technical process grounded in materials science principles, systematically determining quantitative proportional relationships among components to fulfill requirements for fresh-state workability and hardened-state comprehensive properties [64]. Its core objective lies in establishing optimized multiphase system solutions that integrate material efficiency, economic viability, and structural reliability. Precise mix design not only provides fundamental assurance for mechanical strength development and structural safety of concrete, but also serves as a critical pathway for regulating its long-term durability [65]. Simultaneously, this design governs the rheological behavior of fresh concrete, profoundly influencing construction efficiency, casting compactness, and ultimate surface quality formation [66]. It must be emphasized that improper mix design or deviation from specified performance targets triggers multifaceted detrimental consequences. Primarily, it compromises fundamental mechanical properties, inducing degradation of structural load-bearing capacity and potentially catastrophic failure [67]. Secondary effects include markedly exacerbated concrete permeability, accelerating the ingress and transport kinetics of aggressive agents [65]. This initiates cascading reactions: accelerated destruction of steel reinforcement’s passive film and corrosion-induced expansion, accompanied by the risks of concrete cover delamination. Concurrently, it severely diminishes resistance to freeze–thaw cycles and chemical degradation, ultimately jeopardizing structural service life.

Traditional concrete mix design follows industry standards, determining initial proportions based on the water–cement ratio law and empirical aggregate gradation formulas, followed by trial mixing validation [68]. However, this approach primarily targets conventional cement–aggregate systems and fails to quantify the physicochemical heterogeneity of solid waste materials, resulting in significant limitations in mechanical performance prediction and durability control for solid waste-based concrete design [69]. Through orthogonal experimental design methods, a Multi-Objective Optimization (MOO) study of concrete mix proportions was conducted, and a quantitative mapping relationship between material properties and component parameters was systematically established [70,71]. In depth analysis reveals that both water-to-binder ratio and glass powder replacement rate significantly influence GPC compressive strength, with optimal ranges identified as 0.33–0.34 for water-to-cement ratio and 30% for glass powder replacement [72,73]. The integration of response surface methodology with MOO algorithms offers innovative approaches for concrete mix design [74,75,76]. Although experimental-based mix design is effective, the required number of samples and experiments grows exponentially when multiple mixture parameters or their values are considered as optimization variables, resulting in substantially increased resource consumption and time costs [77].

MOO has demonstrated significant advantages in concrete mix design as an effective approach for resolving conflicting objectives [78]. With the extensive application of ML models, intelligent MOO systems using ML as surrogate models have been widely adopted in concrete mix design. Refs. [79,80] developed an ML-MOO hybrid optimization framework for manufactured sand concrete, achieving coordinated optimization between mesoscopic parameters and macroscopic properties. Refs. [81,82,83] optimized recycled aggregate concrete mix proportions using compressive strength and multiple factors as objectives, employing TOPSIS to identify optimal solutions from the Pareto frontier. Refs. [84,85] developed an ML-based framework that systematically addresses sustainable design of high-performance concrete. Refs. [86,87,88] integrated Bayesian-optimized ML models with MOO for mix design, substantially improving computational efficiency. Refs. [89,90] comprehensively evaluated strength–factor trade-offs, providing design-specific solutions through Pareto-optimal sets. Refs. [91,92] systematically explored low-carbon and cost-effective concrete mixtures under carbon emission constraints. However, systematic research remains scarce on balancing the triple objectives of compressive strength, unit cost, and CO₂ emissions in GPC through intelligent algorithms. Especially for the complex coupling relationship between the important components that affect the compressive strength of GPC, the existing methods have not established an effective MOO coordination mechanism, and it is impossible to maximize the triple benefits of “compressive strength–unit cost–sustainability”.

Based on these findings, this study proposes a model to evaluate the trade-off relationships between GPC compressive strength, cost, and CO₂ emissions, with the main contents as follows: First, we established the most comprehensive database to date, containing 1154 datasets for training and testing ML models (six input features and one output), with detailed data distribution analysis. Seven ML models were introduced and optimized using Bayesian optimization for hyperparameter tuning. The prediction accuracy of Bayesian-optimized ML models was then compared using four evaluation indicators, with hyperparameter optimization results presented. After model performance evaluation, SHAP analysis and Partial Dependence Plot were applied to the best-performing model for interpretability analysis, identifying key features influencing compressive strength. Finally, the optimized ML model served as a surrogate to establish a multi-performance evaluation model for GPC, analyzing strength–CO₂ emissions–cost trade-offs, with the Topsis method determining optimal mix proportions for different design strengths. Therefore, the main contents of this paper are shown in Figure 2.

The synergistic application of ML and MOO to GPC mix design represents a pivotal technological paradigm shift. Compared to traditional cement–aggregate systems, the compressive strength of GPC exhibits pronounced characteristics of multivariate interactions and nonlinearity. Consequently, conventional regression models struggle to accurately characterize its complex mapping relationships. Furthermore, conducting systematic orthogonal experiments for novel material systems like GPC often encounters significant challenges in terms of resource and time expenditure. The ML-MOO framework, grounded in data-driven modeling, effectively deciphers the intricate relationships between raw material constituents and compressive strength, thereby overcoming inherent difficulties in mix design for novel material systems. Ultimately, this approach enables the simultaneous optimization of GPC production costs and environmental impact (carbon emissions) under the constraint of meeting target strength requirements.

2. Methodology

This section systematically delineates the database construction process, encompassing the definition of data sources and establishment of screening criteria. Subsequently, a comprehensive analysis is conducted on the performance characteristics and applicability of the selected diverse machine learning models. Furthermore, detailed elaboration is provided on the implemented hyperparameter optimization strategy (Bayesian optimization). Finally, the quantitative metrics selected for model performance evaluation and their underlying rationale are explicitly articulated.

2.1. Database Establishment and Analysis

By compiling experimental GPC data from the literature, we established a comprehensive database containing 1154 samples, with detailed specifications provided in

Table 1. Detailed information of the database.

Ref.	CE (kg/m3)	W (kg/m3)	GA (kg/m3)	S (kg/m3)	GP (kg/m3)	AGE (d)	CS (MPa)
[93]	221.25–295	132.75	620	502	0–737.75	28	25.32–34.2
[94]	280–400	160–175	976–1200	720–808	0–70	7–90	19.7–60
[95]	315–477	167–173	998–1008	0–792	0–914	7–91	17.7–33.4
[16]	262.5–450	157.5–175	992–1084	661–723	0–112.5	7–56	25.3–56
[96]	379.11	151.64	1107	588.45–799.92	0–191.98	7–90	23.8–46.5
[97]	380	201	1020	572–715	0–143	3–28	26.9–45.9
[98]	297.9–490	178.74–200.9	776.31–848	822.51–956.64	0–73.5	28–180	41.22–67.56
[99]	341–426	143.22–178.92	1128	713	0–85	7–90	36–58.87
[100]	182–372	99–263	1039–1096	391–873	0–182	1–365	2.4–76.3
[101]	200	146–174	880	900	0–90	7–28	13–29.9
[102]	75.9–379.5	184	1160	740	0–264	2–180	0.5–40.4
[103]	313.85–369.23	156.93–184.62	1200	516.92–646.15	0–129.23	7–60	19.2–32.8
[104]	457.66	205.95	1199.74	385.04–550.06	0–165.02	7–28	15.34–26.88
[105]	245–350	147	1120–1128	735–767	0–105	3–90	12–63
[106]	218.2–375	206	979	761	0–131.25	3–84	3.16–26.5
[21]	248.5–355	163	1185	625	0–106.5	3–180	16.2–41.68
[107]	320–400	200	1188	594	0–80	28	26.28–36.06
[108]	400	200	594–1188	594	0–594	28	28.09–38.87
[109]	280–350	112–140	1281–1293	643–649	0–70	7–91	20.01–45.04
[110]	240–312	141	1063	799	0–72	7–270	20.91–50.11
[111]	210–300	215	1200	600	0–90	2–90	8.22–28.01
[112]	152–380	185	825	913–960	0–228	7–365	22.91–62.23
[113]	333.75–445	215	1005	625	0–111.25	7–365	27.02–61.27
[114]	378–578	185	1048	0–741	0–741	7–90	45.82–95.82
[115]	275–355	185	1290–1300	312.5–670	0–335	3–365	10.93–59.58
[116]	152–380	185	825	907–960	0–228	7–91	26.14–63.14
[117]	255.85–320	160	1020.94	782.26	0–64.15	7–91	24.87–52.19
[118]	152.75–354	164.5–166.38	1246–1313	563–729	0–123.9	7–120	10.2–39.3
[119]	170–340	170	1224–1247	560–720	0–170	7–28	10.4–37.8
[120]	372–465	200	1030	715	0–93	7–90	25–65

. The literature included in this database adhered to specific criteria.

① Portland cement, possessing a consistent strength rating of 42.5 MPa, was utilized.

② Glass powder served as the sole supplementary cementitious material incorporated into the solid waste.

③ GPC was produced and subsequently cured under controlled laboratory conditions in strict adherence to applicable standards.

④ References were carefully selected from authoritative journals to ensure the accuracy of data.

To ensure metrological consistency and statistical reliability of the dataset, this study implements a systematic preprocessing protocol: First, dimensional equivalence conversion standardizes material content units to the baseline “kg/m³”, eliminating numerical deviations arising from unit heterogeneity. Second, compressive strength outliers are identified via Tukey’s boxplot method and removed to ensure data rationality. Finally, the duplicate literature samples undergo identification and aggregation using curing-age weighted averaging, constructing a non-redundant, structured dataset to mitigate informational interference and overfitting risks during model training.

This study selected six key sample features influencing GPC compressive strength as input variables [33,34,35]: cement content (CE), water content (W), gravel content (GA), sand content (S), glass powder content (GP), and curing age (Age), with concrete compressive strength (CS) as the output variable. CE, W, GA, and S constitute the fundamental parameters governing conventional concrete performance. The incorporation of GP as a SCM introduces dimensions of sustainability and performance modification. Age quantifies the temporal evolution of material properties.

Table 2. Statistical information of database.

Data Type	Mean	Standard Deviation	Minimum	Maximum
CE	329.1448	80.0057	75.9	578
W	174.9617	22.3842	99	263
GA	1106.909	133.2771	594	1313
S	654.9478	152.055	0	960
GP	89.52323	119.5525	0	914
Age	53.58059	74.85395	1	365
CS	34.37651	14.43581	0.5	95.82

presents the mean values, standard deviations, and minimum and maximum values for all variables employed in this study. The data distribution is illustrated in Figure 3. The cement content primarily ranged between 300 and 400 kg/m³, water content clustered around 180 kg/m³, coarse aggregate was mostly distributed above 1000 kg/m³, sand content showed highest frequency in the 600–800 kg/m³ range, Age was predominantly short durations, and CS values exhibited a skewed distribution concentrated between 20 and 40 MPa.

2.2. ML Models

This study employs seven machine learning models—Ensemble Tree (ET), Gaussian Process Regression (GPR), Decision Tree (DT), Kernel Ridge Regression (KRR), Artificial Neural Network (ANN), Support Vector Machine (SVM), and Efficient Linear (EL)—to comparatively evaluate their accuracy in predicting the compressive strength of GPC. The objective is to systematically assess their adaptability under conditions of small sample size, high-dimensional features, and nonlinear threshold effects through a multi-paradigm modeling framework. Comparative analysis of multiple performance evaluation metrics ultimately provides theoretical foundations and modeling paradigms for mix proportion optimization.

2.2.1. Ensemble Tree (ET)

Ensemble Trees (ET) enhances model accuracy and robustness by constructing multiple decision trees and combining their predictions. In this process, each tree is trained independently as a weak learner, with predictions aggregated through voting (for classification) or averaging (for regression) to produce the final output. The model’s strengths include handling diverse data types, resistance to overfitting, and strong generalization capability. But it may underperform with high-dimensional sparse data and exhibits sensitivity to minor training data variations.

2.2.2. Gaussian Process Regression (GPR)

The prediction process of Gaussian Process Regression (GPR) begins with selecting an appropriate kernel function to capture similarities between data points, followed by estimating model hyperparameters using training data. Subsequently, a covariance matrix and covariance vector are constructed, which are then used to form a Gaussian distribution for new test points. The advantages of this model include providing uncertainty estimates for predictions, excellent performance in small-sample data fitting, and kernel function flexibility that adapts to various data distributions. However, GPR has high computational complexity, particularly with large datasets where prediction times may be prolonged, and hyperparameter tuning requires substantial computational resources.

2.2.3. Decision Tree (DT)

The construction process of Decision Tree (DT) models begins from a root node, recursively selecting optimal features and thresholds to split the dataset and form internal nodes. This process repeats for each resulting data subset until predefined stopping criteria are met, followed by pruning the generated tree to ultimately form a complete decision tree where each leaf node represents a classification outcome or regression prediction. The model’s advantages include simplicity and intuitiveness, strong interpretability, capability to handle both categorical and numerical data, and insensitivity to missing values. However, DT models are prone to overfitting and exhibit weak generalization capability.

2.2.4. Kernel Ridge Regression (KRR)

The core of Kernel Ridge Regression (KRR) lies in utilizing kernel functions to measure the influence of neighboring data points for estimating the dependent variable value at a given point. The kernel function assigns weights to observations based on their distance to the target point, with closer points receiving higher weights. This approach excels at handling nonlinear data by employing the kernel trick to avoid direct computation in high-dimensional space, thereby controlling computational complexity. However, kernel methods still exhibit high computational complexity and memory requirements, while kernel function and parameter selection often rely on empirical knowledge, making parameter tuning challenging.

2.2.5. Artificial Neural Network (ANN)

Artificial Neural Networks (ANNs) are widely employed to solve complex problems using nonlinear data-driven models. By mimicking the connections and interactions of biological neurons in the human brain, ANN can learn intricate nonlinear relationships and possess powerful feature extraction capabilities. The model’s strengths include strong fitting capacity, automatic learning of feature representations from data, parallel computing capability, and fast training/prediction speeds. However, neural networks are prone to overfitting, require substantial amounts of data and regularization techniques, and involve complex training processes.

2.2.6. Support Vector Machine (SVM)

The Support Vector Machine (SVM) model can be applied to both data classification (binary classification) and regression problems. In regression tasks, SVM aims to identify a function that minimizes prediction errors on given data points. SVM demonstrates strong performance with small samples, exhibits excellent generalization capability, handles high-dimensional data effectively, and maintains relatively low computational complexity. However, SVM shows lower training efficiency with large-scale datasets, requires complex implementation for multi-class classification, and lacks interpretability in its decision-making process.

2.2.7. Efficient Linear (EL)

The Efficient Linear (EL) model assumes a linear relationship between output and input features, determining the optimal linear combination through learned parameters. Compared to conventional linear regression models, the EL model incorporates more sophisticated considerations and optimizations in computational efficiency, feature processing, and regularization. However, this model exhibits poor fitting capability for nonlinear data, cannot handle complex relationships, and is highly sensitive to feature selection—improper feature choices may significantly degrade model performance.

2.3. Bayesian Optimization

Bayesian optimization is a probability model-based global optimization method that, compared to random search and grid search, can more intelligently adapt to hyperparameter relationships and spatial structures while achieving higher efficiency through reduced computational load. Figure 4 illustrates the general workflow of Bayesian optimization. Its core principle is that the optimized objective function only requires specified inputs and outputs without knowledge of internal relationships. The posterior distribution of the objective function is continuously updated by incrementally adding sample points until it closely approximates the true distribution [121]. The most critical components in Bayesian optimization are the establishment of surrogate models and the construction of acquisition functions. This study employs Gaussian processes as the surrogate model, where the distribution of the objective function is inferred from observed data through Gaussian processes. The acquisition function drives the optimization process. This study uses expected improvement (EI) per second as the acquisition function, whose fundamental concept involves calculating the expected improvement at potential parameter locations given the current optimum, then selecting the point with maximum expected improvement as the next evaluation location. The mathematical formulation of the expected improvement criterion is shown in Equation (1).

$$EI(x)=E{\left[f(x)-f\left(x^{+}\right)\right]}^{+}={\displaystyle {\int }_{\max \left[f(x)-f\left(x^{+}\right),0\right]}p}(f(x)\mid D)df(x)$$

(1)

where x+ represents the value corresponding to the current best-known solution, and f(x) is the predicted value of the objective function by the Gaussian Process.

2.4. Performance Evaluation Indicators

To accurately evaluate the performance and predictive accuracy of each ML model, this study employs four standard regression indicators for model assessment [122]. The indicators include the following: coefficient of determination (R²), Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and Mean Absolute Error (MAE), with their mathematical formulations detailed in

Table 3. Formulas for calculating four indices.

Index	Formula
R2	$R^2=1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}}{{\displaystyle \sum _{i=1}^n{\left(y_i-\overline{y}\right)}^2}}}$	(2)
MSE	$MSE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}$	(3)
RMSE	$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}}$	(4)
MAE	$MAE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|y_i-{\widehat{y}}_i\right|}$	(5)

.

Where y_i is the experimental value, ${\stackrel{\wedge }{y}}_i$ is the predicted value, $\overline{y}$ is the mean of experimental values, and n is the number of datasets.

3. ML Model Training Results and Analysis

This section presents the training results of ML models based on Bayesian optimization. The dataset was partitioned into training and test sets at an 8:2 ratio, where algorithms learn and adapt from the training set while the test set evaluates model generalization capability. To prevent overfitting and obtain a more accurate performance estimation, ten-fold cross-validation was implemented on the training set.

3.1. Hyperparameter Optimization Results

Hyperparameters primarily govern the search space of learning algorithms and assist models in avoiding underfitting and overfitting [123]. Model hyperparameters are external configuration variables, with each ML model typically associated with a specific set including neural network depth, minimum leaf size, and learning rate. These parameters must be predetermined before training, either manually or algorithmically, and their selection directly determines model performance.

Different ML models require distinct hyperparameter types and search ranges due to their varying internal architectures and algorithmic complexities. The selected hyperparameters and their search ranges for each model are specified in

Table 4. Hyperparameter optimization results of 7 ML models.

ML Model	Hyperparameter	Search Range	Optimal Hyperparameters
ET	Ensemble Method	[Bag, LSBoost]	LSBoost
	Minimum Leaf Size	[1–418]	9
	Number of Learners	[10–500]	491
	Learning Rate	[0.001–1]	0.11128
GPR	Sigma	[0.0001–147.6161]	134.3861
	Basis Function	[Constant, Zero, Linear]	Constant
	Kernel Scale	[0.001–1000]	0.21846
	Standardize Data	[True, False]	True
DT	Minimum Leaf Size	[1–418]	2
KRR	Learner	[SVM, Least Squares Kernel]	Least Squares Kernel
	Kernel Scale	[0.001–1000]	877.3523
	Regularization Strength	[1.192 × 10−6–1.1962]	4.8953 × 10−6
	Extension Dimension	[100–10,000]	556
	Standardize Data	[True, False]	True
ANN	Fully Connected Layers	[1–3]	3
	First Layer Size	[1–300]	293
	Second Layer Size	[1–300]	1
	Third Layer Size	[1–300]	252
	Regularization Strength	[1.192 × 10−8–119.6172]	0.00010326
SVM	Box Constraint	[0.001–1000]	0.10065
	Kernel Function	[0.001–1000]	Linear
	Epsilon	[0.013141–1314.1216]	0.04278
EL	Learner	[SVM, Least Squares]	Least Squares

. Bayesian optimization was employed for hyperparameter tuning with the iteration number set to 100 and no time constraints imposed. Figure 5 illustrates the iterative optimization process across ML models. The ET model demonstrated the most rapid MSE reduction among all models, achieving superior performance with fewer iterations, which indicates its fast convergence and excellent predictive capability.

Table 4 presents the search ranges and optimal hyperparameter values for each ML model during Bayesian optimization. The optimized ET model employs LSBoost as its ensemble method, which constructs regression trees by minimizing the sum of squared errors to enhance predictive accuracy. As a specialized gradient boosting algorithm, LSBoost fits the residuals from previous iterations when using squared error loss, effectively improving model performance through this algorithmic framework [124]. For the GPR model, the kernel scale determines the covariance function’s rate of variation, while data standardization enhances model stability and accuracy. Minimum leaf size serves as a crucial hyperparameter for the DT model, governing model complexity and overfitting risk. Regularization strength prevents overfitting in the KRR model, with the expansion dimension determining model complexity. The layer sizes in the ANN model determine model capacity and expressive power, directly impacting prediction accuracy. For the SVM model, both box constraint and kernel function significantly influence classification boundaries and performance—the former controls misclassification tolerance while the latter determines data mapping and classification capability [125]. The least squares kernel was selected as the optimal hyperparameter due to its balanced trade-off between computational efficiency and model performance.

3.2. Performance Evaluation of the Model

The seven ML models mentioned in Section 2.2 were employed to predict GPC compressive strength, with both training and test set prediction results visualized in Figure 6. Most models demonstrated optimal predictive performance within the medium range of 20–80. Prediction accuracy declined in extreme value ranges (0–20 and 80–100), where data points deviated from the diagonal. This likely occurs because extreme values are typically scarce in datasets, limiting models’ ability to adequately learn their characteristic patterns during training. The ET, GPR, and DT models showed strong agreement between predicted and actual values, with data points tightly clustered along the diagonal for both training and test sets. This indicates that these models possess superior predictive accuracy and generalization capability overall. The Bayesian-optimized ET model performed best, with most predictions within ±10% error margins and only some extreme values exceeding ±20%. The KRR, ANN, SVM, and EL models showed relatively poorer agreement, with more scattered data points and larger prediction deviations in certain ranges. The EL model performed worst, significantly exceeding ±20% error margins despite Bayesian optimization, and exhibited overfitting on the test set. This stems from its inherent linearity assumption between independent and dependent variables, whereas GPC compressive strength actually demonstrates complex nonlinear relationships with input factors. Moreover, the model’s minimization of squared errors during optimization exacerbated the overfitting.

Table 5. Results of 4 precision evaluation indexes of ML models.

ML Model	R2		RMSE		MSE		MAE
	Train	Test	Train	Test	Train	Test	Train	Test
ET	0.954	0.959	3.157	2.949	9.967	8.698	2.033	2.007
GPR	0.927	0.924	3.998	4.011	15.982	16.091	2.561	2.616
DT	0.802	0.848	6.578	5.675	43.272	32.203	4.149	4.078
KRR	0.736	0.711	7.600	7.815	57.767	61.068	5.530	5.719
ANN	0.558	0.556	9.824	9.698	96.503	94.056	7.488	7.506
SVM	0.542	0.558	9.991	9.727	99.814	94.607	7.516	7.210
EL	0.356	0.050	11.866	14.182	140.801	201.119	8.836	10.716

presents the performance indicators of seven ML models on both training and test sets across four evaluation criteria. Lower values of RMSE, MSE, and MAE indicate better model performance, while for R², values closer to one demonstrate superior overall performance. The ET model achieved the highest R² values of 0.954 (training) and 0.959 (test) among all models, along with the lowest RMSE, MSE, and MAE values, demonstrating both high predictive accuracy and strong generalization capability, thereby confirming its superior predictive performance. The GPR model ranked second only to ET, with R² values of 0.927 (training) and 0.924 (test). In contrast, the EL model yielded an R² of merely 0.050 on the test set, indicating virtually no predictive capability on unseen data. The remaining models (DT, KRR, ANN, and SVM) demonstrated intermediate performance between ET and EL, each with distinct strengths and weaknesses.

Figure 7 presents boxplots of C_s-pre/C_s-exp ratios for both training and test sets across all models. The ET model demonstrates ratios more closely clustered around one and exhibits smaller interquartile ranges in both datasets. Excessively high C_s-pre/C_s-exp ratios indicate substantial overprediction of compressive strength, potentially compromising structural safety in practical GPC applications, while excessively low ratios reflect severe underestimation that may lead to material overuse and increased costs. In this study, the Bayesian-optimized ET model outperformed all other models across all evaluation indicators, establishing it as the prime candidate for GPC compressive strength prediction.

3.3. Interpretability Analysis of the ET Model

3.3.1. SHAP Analysis

Given the superior performance of the ET model, this study employs SHAP for model interpretability analysis. SHAP values precisely quantify each feature’s contribution to individual predictions. The bee swarm plot aggregates SHAP values for all input features across the dataset, with each point illustrating how specific features influence the model’s output prediction for particular samples. Figure 8a displays the SHAP bee swarm plot for the ET model, revealing that Age exhibits the widest value range, indicating its most significant impact on model predictions. Higher feature values correspond to higher SHAP values, demonstrating positive contributions to compressive strength. At elevated levels, CE shows greater influence than Age, suggesting that CE becomes more critical than Age for higher compressive strength development. S also positively affects compressive strength, though less prominently than Age and CE. Both W and GA demonstrate negative correlations with compressive strength. GP shows a weak negative correlation with compressive strength, attributable to its lower pozzolanic effects compared to cement and slower reaction kinetics. Increased GP reduces early-stage hydration products, thereby inhibiting early strength development [126].

Figure 8b presents a global feature importance bar plot, which considers all samples and calculates the mean absolute SHAP values for each feature, demonstrating their average impact on model predictions. Age and CE emerged as the most critical features influencing GPC compressive strength, with mean SHAP values around five, consistent with concrete strength development mechanisms where CE and Age are known dominant factors. W, GA, and S were identified as important features (mean SHAP: 1–3). The water-to-binder ratio directly affects strength, as reflected by W’s negative correlation—higher water content markedly reduces GPC compressive strength. While GA and S influence compressive strength, their impact is generally less significant than Age, CE, and water-to-binder ratio [63].

3.3.2. Partial Dependence Plot (PDP)

The PDP for CS of GPC in Figure 9, derived from six input features, corroborates with SHAP-based feature importance analysis, unequivocally revealing nonlinear influence patterns of key factors. In-depth analysis demonstrates that CE within 300–480 kg/m³ exhibits the most substantial contribution to CS enhancement. Beyond this threshold, however, strength progression enters a plateau phase, primarily because calcium silicate hydrate gels from cement hydration have sufficiently filled the matrix pores, achieving relative densification where further cement addition yields negligible densification benefits. While W generally demonstrates negative correlation with CS, anomalous strength enhancement occurs specifically within the 180–190 kg/m³ range. This phenomenon is hypothesized to originate from optimized hydration kinetics of CE and GP under such water content conditions. Aggregate constituents exhibit markedly distinct mechanisms in influencing concrete compressive strength. Specifically, GA achieves peak strength enhancement near 1050 kg/m³, indicating a critical optimal content threshold, whereas S demonstrates a persistent monotonic positive correlation with strength improvement. This complex behavior originates from the gradation characteristics’ modulation of pore structure—optimized gradation effectively reduces porosity by enhancing aggregate–paste interfacial bonding strength and dense skeleton effects, thereby substantially elevating macroscopic compressive performance. GP significantly enhances CS at lower levels, yet exhibits precipitous decline near 200 kg/m³, indicating a critical threshold beyond which GPC strength is substantially compromised, necessitating stringent content optimization. Furthermore, Age influence manifests as follows: most pronounced CS improvement occurs during early curing, driven predominantly by cement hydration, whereas later stages show sustained, though gradual, strength gain attributable to GP’s pozzolanic reactivity and slow-burning micro-aggregate effects.

The feature of importance results from SHAP analysis aligns well with established concrete material mechanisms. The model successfully identified key strength-influencing features and appropriately weighted their relative importance. GP’s contribution was moderately reflected in SHAP values, indicating the model’s capability to recognize its performance effects. PDPs delineate average trends and inflection thresholds across the entire dataset. These methods mutually cross-validate as follows: plateau regions in PDP correspond to high-density clusters with near-zero contributions in SHAP analysis, while anomalous drops in PDP align with samples exhibiting negative SHAP values. This synergy prevents the obscuring of individual variations by averaging effects while preventing isolated cases from distorting global trends.

4. Multi-Objective Optimization (MOO) Model for GPC Mix Design

4.1. Establishment of MOO Model

Cost and CO₂ emissions represent two critical factors in concrete production. The developed model employs minimization of cost per cubic meter and CO₂ emissions as objective functions to identify optimal GPC mix proportions for different design strengths, as formally expressed in Equation (6). Both cost and CO₂ emissions are calculated by multiplying material quantities with their respective unit prices and emission factors, with specific values provided in

Table 6. The values of each material component.

Input	Cement	Water	Gravel	Sand	GP
Cost ($/kg)	0.113	0.00072	0.01	0.019	0.087
CO2 (kg/m3)	0.927	0.00013	0.022	0.00987	0.1664
Density (kg/m3)	3150	1000	2500	2650	2450

[89,127,128,129]

This study established four constraint types to define the search space for optimal mix proportions: strength constraints, variable constraints, volumetric constraints, and water-to-binder ratio constraints. Strength constraints ensure that the predicted compressive strength of the GPC mix meets or exceeds the target design strength, serving as a fundamental requirement for practical applicability. Variable constraints restrict each material’s quantity within the extreme values from the database. Volumetric constraints require the summed volumes of all materials to equal one; however, considering source-induced variations [89], this study adopted a range of 0.9–1.1. Water-to-binder ratio constraints limit the water-to-cementitious materials ratio to database extremes. The specific constraint expression is shown in Equation (7).

The NSGA-II algorithm was employed for MOO based on the defined objective functions and constraints. As shown in Figure 10a, the algorithm introduces a crowding distance metric to quantify population density around specific points, eliminating those with excessively small distances to maintain population diversity. This facilitates selection, crossover, and mutation across the entire search space, with the crowding distance calculation given by Equation (8). To prevent the loss of high-quality individuals during evolution, the algorithm implements an elitist non-dominated sorting strategy [130]. Specifically, the parent population is combined with offspring generated through crossover and mutation, followed by non-dominated sorting of the merged population. This effectively preserves elite individuals from the parent population, ensuring superior genetic information propagates to subsequent generations [131]. The detailed procedural steps are illustrated in Figure 10b.

The algorithm parameters were configured as follows: population size is 200, maximum generations are 100, crossover function is scattered crossover, selection function is stochastic uniform, crossover probability is 0.8, and mutation rate is 0.1.

$$Objective function \left\{\begin{array}{l}\min C{O}_2\\ \min \mathrm{Cos}t\end{array}\right.$$

(6)

$$s.t.\left\{\begin{array}{l}C{S}_{\mathrm{t}\arg \mathrm{et}}\le CS\\ Q_{i,\min }\le Q_i\le Q_{i,\max }\\ 0.9\le V_{GPC}={\displaystyle \frac{Q_{CE}}{D_{CE}}}+{\displaystyle \frac{Q_W}{D_W}}+{\displaystyle \frac{Q_{GA}}{D_{GA}}}+{\displaystyle \frac{Q_S}{D_S}}+{\displaystyle \frac{Q_{GP}}{D_{GP}}}\le 1.1\\ r_{W/B,\min }\le {\displaystyle \frac{Q_W}{Q_{CE}+Q_{GP}}}\le r_{W/B,\max }\end{array}\right.$$

(7)

where Q_i is the content of component i; Q_i,min, and Q_i,max are the minimum and maximum contents of CE, W, GA, S, and GP; V_GPC is the unit volume of GPC; Q_CE, Q_W, Q_GA, Q_S, and Q_GP are the amounts of CE, W, GA, S, and GP; D_CE, D_W, D_GA, D_S, and D_GP are the unit weights of CE, W, GA, S, and GP; r_W/B and

$$i_d={\displaystyle \sum _{j=1}^m\left(\left|f_j^{i+1}-f_j^{i-1}\right|\right)}$$

(8)

where i_d is the crowding distance of point i; f_jⁱ⁺¹ is the j-th objective function value at point i + 1; and f_jⁱ⁻¹ is the j-th objective function value at point i − 1.

4.2. Pareto Frontier Analysis

In optimization design, conflicting objectives exist where improving one objective may deteriorate another. The Pareto optimization serves as a fundamental tool to address this challenge [132]. The Pareto frontier visualization intuitively displays objective trade-offs, transforming complex multi-objective conflicts into 2D or 3D representations that facilitate identification of practically optimal solutions.

Figure 11 illustrates the Pareto frontier formation process, where all yellow points represent feasible solutions—achievable cost and CO₂ emission combinations under given constraints. The red curve comprises Pareto-optimal solutions, with green points highlighting selected optimal solutions that achieve the best balance between cost and emissions. The utopia point represents the ideal scenario where all objective functions simultaneously reach their optimal values, which is generally unattainable in practice. In this study, since both CO₂ emissions and cost are minimized, the utopia point is situated near the origin of the coordinate axes. To identify the most suitable optimal solution from all non-dominated alternatives, the Topsis method was employed to evaluate each solution’s performance score. Normalization was applied to cost and emission metrics using Equation (9) to eliminate dimensional differences. Equal weights of 0.5 were assigned to both cost and CO₂ emission criteria to ensure balanced consideration of these two objectives in the decision-making process.

The Topsis method determines the positive ideal solution (PIS) and negative ideal solution (NIS), where the PIS represents the optimal value for each criterion (representing the theoretically best alternative), while the NIS corresponds to the worst value for each criterion (representing the theoretically worst alternative) [81]. The distances of each alternative from the PIS and NIS are then calculated, with the specific formulas given in Equations (10) and (11). Based on these distances, the Relative Closeness (RC) index of each alternative is computed using Equation (12), which quantifies its proximity to the PIS—a higher RC value indicates a better alternative. This evaluation method fully utilizes raw data information and accurately reflects performance differences among alternatives.

$$r_{ij}={\displaystyle \frac{x_{ij}}{\sqrt{{\displaystyle \sum _{i=1}^n{x}_{ij}^2}}}}$$

(9)

where x_ij is the original value of the i-th scheme under the j-th criterion and r_ij is the normalized value.

$$D_i^{+}=\sqrt{{\displaystyle \sum _{j=1}^m{\left(v_{ij}-v_j^{+}\right)}^2}}$$

(10)

$$D_i^{-}=\sqrt{{\displaystyle \sum _{j=1}^m{\left(v_{ij}-v_j^{-}\right)}^2}}$$

(11)

where D_i⁺ is the distance from the i-th solution to the positive ideal solution; D_i⁻ is the distance from the i-th solution to the negative ideal solution.

$$C_i={\displaystyle \frac{D_i^{-}}{D_i^{+}+D_i^{-}}}$$

(12)

where C_i represents the RC of the i-th solution.

4.3. Optimal Mix Design Results

Building upon the established model and evaluation methodology, this study analyzes the cost–CO₂ emission trade-offs at varying design strengths (28-day Age) and identifies optimal concrete mix proportions for each strength grade via Topsis. Figure 12 displays Pareto frontiers for 20–70 MPa compressive strengths (10 MPa intervals), where color-coded points denote Topsis scores and red stars indicate optimal solutions for target strengths. A clear negative correlation emerges between cost and CO₂ emissions. Both cost and CO₂ emissions of optimal solutions increase with design strength, primarily because high-strength concrete requires more cement—the dominant factor influencing both CO₂ emissions and cost.

Table 7. The optimal mixture scheme with different design strengths.

Design Strength (MPa)	Optimal Mix Ratio					Pareto Front		W/B	Replacement Rate
	Cement (kg)	Water (kg)	Gravel (kg)	Sand (kg)	GP (kg)	Cost ($/m3)	CO2 Emissions (kg)
15	75.90	185.22	1053.61	464.88	230.50	48.14	136.51	0.605	75.23%
20	138.60	187.39	1043.51	480.21	198.60	52.64	189.25	0.556	58.90%
25	180.63	188.97	1047.20	464.62	185.32	55.97	225.93	0.516	50.64%
30	208.87	156.61	1034.84	552.16	173.20	59.62	250.68	0.410	45.33%
35	237.50	142.37	1091.40	587.59	158.60	62.80	276.38	0.359	40.04%
40	271.46	136.12	1049.43	615.83	148.70	65.89	305.57	0.324	35.39%
45	314.18	130.09	993.88	663.81	140.00	70.31	342.97	0.286	30.83%
50	352.40	133.95	844.28	707.32	126.31	72.80	373.27	0.280	26.39%
55	376.50	126.01	848.64	773.65	113.25	75.67	394.18	0.257	23.12%
60	412.56	116.53	889.09	829.36	123.91	80.71	419.18	0.217	23.10%
65	451.40	119.46	735.49	837.08	116.70	84.45	462.35	0.210	20.54%
70	487.28	118.05	665.82	872.96	136.70	90.27	497.73	0.189	21.91%

presents the optimal mix proportions determined by Topsis for design strengths ranging from 15 to 70 MPa. The results demonstrate an increasing trend in cement content while glass powder replacement ratio gradually decreases to approximately 20%. This occurs because lower-strength concrete contains less cement and higher porosity, where glass powder’s micro-aggregate effect more significantly improves microstructure, enhancing compactness and strength [16]. In high-strength concrete with higher cement content, glass powder’s micro-aggregate effect becomes less pronounced due to lower porosity, offering limited strength enhancement through pore-filling. This indicates that high-strength applications require sufficient cement content and controlled glass powder replacement to ensure adequate strength. For lower strengths, higher replacement ratios can be adopted to improve cost-effectiveness. The water-to-binder ratio is a critical parameter—lower ratios reduce hardened concrete porosity, enhancing compactness and strength [133]. The designed, progressively decreasing water-to-binder ratios align with practical requirements. In summary, the developed MOO model effectively guides GPC mix design by identifying optimal proportions that simultaneously minimize cost and CO₂ emissions across all target strength grades.

5. Limitations and Future Research

This study investigates the potential of integrating ML and MOO algorithms for mix design optimization in GPC. Results demonstrate that this approach significantly enhances design efficiency by effectively learning complex nonlinear relationships and co-optimizing conflicting objectives, establishing a data-driven framework for developing high-performance, low-environmental-impact GPC. Core advantages include reduced experimental iteration costs, global Pareto-optimal solution identification, and advancement of the critical sustainability goal of valorizing glass waste. Nevertheless, practical implementation faces certain limitations: Firstly, model performance critically depends on high-quality, comprehensive training datasets, yet data acquisition is complicated by heterogeneity in glass powder sources and concrete system complexity. Secondly, the black-box nature of mainstream ML models limits mechanistic interpretability, potentially undermining confidence in engineering decisions. Thirdly, engineering feasibility of optimized solutions requires rigorous physical validation, while model generalizability is constrained by material and process variability.

Future research should prioritize establishing standardized GPC databases, developing interpretable hybrid modeling methods (e.g., integrating physical mechanisms with data-driven approaches), and implementing closed-loop “optimization–validation–feedback” design workflows. Despite limitations, ML-MOO exhibits indispensable value as an intelligent design tool for advancing GPC toward low-carbon and high-performance solutions, though its successful deployment necessitates cross-disciplinary collaboration and continuous technological refinement.

6. Conclusions

This study developed a performance evaluation model for GPC to analyze the relationships between compressive strength, cost, and CO₂ emissions. A comprehensive database of 1154 GPC compressive strength records was constructed, with Bayesian optimization applied to tune hyperparameters of seven ML models for performance comparison. SHAP analysis was employed to interpret the ML models, overcoming their “black-box” nature and revealing feature interactions. An MOO model was then established with four constraints and two objectives to identify optimal mix proportions at different design strengths through cost-CO₂ emissions trade-off analysis. The main findings are as follows:

(1)

Among Bayesian-optimized ML models, performance ranked as ET > GPR > DT > KRR > ANN > S-VM > EL. The ET model achieved superior accuracy with R² = 0.954 (0.959), RMSE = 3.157 (2.949), MSE = 9.967 (8.698), and MAE = 2.033 (2.007) on test (training) sets.

(2)

SHAP-based interpretability analysis revealed mechanistic insights: compressive strength showed positive correlations with Age, cement, and sand, but negative correlations with gravel, water, and GP. Age and cement emerged as dominant features, while GP’s moderate SHAP values confirmed its contributive role in concrete performance.

(3)

The developed MOO model incorporated dual objectives (emissions/cost) and four constraints (strength requirement, variable bounds, volumetric, and water-binder ratio constraints), solved via NSGA-II to effectively balance cost and emissions per m³ across strength grades. Strength constraints ensured that the predicted compressive strength met or exceeded the target design strength, which is crucial for practical applications. Topsis-based selection yielded optimal mixes that simultaneously minimized economic and environmental impacts.