Modeling mechanical properties of rubberized concrete using gene expression programming (GEP) and random forest: a comparative study

Abstract

Rubberized concrete is becoming increasingly popular as a sustainable construction material due to its capacity to repurpose waste tyre rubber (WTR) and reduce the use of natural aggregates. However, experimental testing of its mechanical properties is expensive and time-consuming, emphasizing the importance of reliable predictive tools. To solve this issue, this research uses gene expression programming (GEP) and random forest (RF) models to estimate the flexural strength (FS) and split tensile strength (STS) of rubberized concrete. The development of the models were carried out using a comprehensive dataset that included 112 experimental findings and seven main input parameters. Multiple statistical metrics were used to evaluate model performance, such as R, MAE, RMSE, RAE, and RRSE. Although both models correctly predicted the strength trends of rubberized concrete, RF outperformed GEP with testing R² values of 0.9799 for FS and 0.9817 for STS, compared to 0.9183 and 0.8965, respectively. The error analysis revealed that both models’ prediction errors were within acceptable bounds, with RF models having lower error as compared to GEP. Furthermore, the proposed models were compared to previously published prediction models and outperformed them in terms of accuracy and generality. Furthermore, the Shapley Additive Explanations (SHAP) analysis applied to the dataset revealed the relative importance of input parameters on mechanical behavior. The findings show that machine learning-based techniques are successful at predicting the mechanical properties of rubberized concrete and have the potential to enhance environmentally friendly construction through the efficient reuse of WTR.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-40897-6.

Introduction

The building sector is expanding quickly and contributes significantly to greenhouse gas emissions since it uses around 33% of the world’s energy¹. Concrete is frequently utilized in construction for drainage, rehabilitation, and structural purposes². Natural resources are used in its production, and demand for it is increasing continuously. The European Union has placed limitations on the use of virgin aggregates and actively promotes the use of recycled materials in construction to reduce the excessive consumption of natural resources³. To solve this issue, the building industry should adopt the concept of sustainability to avoid excessive consumption trends⁴. As a result, researchers are working hard to find alternatives for aggregates and cement in concrete mixtures to reduce natural resource depletion and energy consumption. Similarly, growing urbanization and technical developments in industries such as transportation, mining, and agriculture have resulted in a significant increase in solid waste generation⁵. By 2050, approximately 70% of the world’s population is expected to live in cities, which will undoubtedly increase transportation demand and vehicle ownership⁶. As a result, the production of waste tyre rubber (WTR) has become a major global environmental issue. Shen et al.⁷ and Siddique et al.⁸ estimate that 1.5 billion tyres are produced annually worldwide. Approximately 1 billion waste tires are created each year in affluent nations; in these areas, the rate of waste tire output is one tire per passenger⁵. On the other hand, there are between 200 and 300 million worn-out tyres in the United States⁹, while Turkey is anticipated to have over 10 million waste tyres¹⁰, and the EU roughly 3.4 million¹¹. By 2030, it is projected that there will be 1.2 billion worn tyres, with approximately 4 billion already buried in landfills around the world. According to the European Association of Tyre and Rubber Businesses, 3.2 million tons of worn tyres were disposed of in 2009. The materials recovered from these tyres were remarkably high, accounting for 96% of the total. Of this total, 18% was reused or reprocessed, 38% was recycled, and 40% was converted into energy¹². Despite these initiatives, illegal tire dumping and landfilling are still common in many developing countries, bringing major dangers such as vector-borne diseases, soil contamination, and fire hazards^13,14.

Fortunately, by incorporating used rubber into products like concrete and asphalt, the construction sector can potentially lessen the impact^15,16. This lowers the amount of waste tyres that would have been dumped into the environment and conserves basic materials. Rubberized concrete, which is environmentally friendly and enhances properties like shock and sound resistance, wear and tear resistance, and thermal properties, improves the energy performance of the structures when waste tyre powder is added in the right amounts to concrete mixtures. This lowers the amount of waste tyres that would have been dumped in the environment and conserves basic materials. Rubberized concrete, which is environmentally friendly and enhances qualities like shock and sound resistance, wear and tear resistance, and thermal properties, improves the energy performance of the structures when waste tyre powder is added in the right amounts to concrete mixtures^17,18. Despite these benefits, the mechanical properties of rubberized concrete, particularly flexural strength (FS) and split tensile strength (STS) are strongly controlled by a variety of interacting parameters such as rubber content, particle size, and mix proportions. Experimental research of these features is both time-consuming and expensive, which encourages the use of predictive modeling tools. However, a few studies have concentrated on rubberized concrete and attempted to simulate its properties through machine learning (ML). For instance, Jingkui Zhang et al.¹⁹ utilized extreme learning machine (ELM) models to forecast 28-day compressive, split tensile, and flexural strengths, incorporating variables such as rubber content, particle size, and polypropylene fiber content, achieving high accuracy and robustness. Another investigation²⁰ employed ML methods to model the mechanical properties of rubberized concrete, including flexural and tensile strengths, highlighting the potential of these techniques in material property prediction. Additionally, Miladirad et al.²¹ research applied machine learning methods, including neuro-fuzzy system (NFS) and artificial neural network (ANN) to model the mechanical properties of rubberized concrete, including flexural and split tensile strengths, highlighting the potential of these techniques in rubberized concrete properties prediction. Ghanizadehe et al.²² used artificial neural networks (ANN) and support vector machines (SVM) to solve the problem of forecasting the compressive strength of rubberized concrete. Similarly, to model the compressive strength of rubberized concrete, Amlashi et al.²³ used four machine learning algorithms: SVM, Multigene Genetic Programming (MGGP), Group Method of Data Handling (GMDH), and Response Surface Methodology (RSM). Out of the three models, they discovered that the SVM model performed better. Amlashi et al.‘s study²⁴ used SVM and adaptive neuro-fuzzy inference system (ANFIS) approaches, which were improved by particle swarm optimization, to evaluate the performance of rubberized concrete. The majority of the research in this corpus focused on methods that use neural networks, which by their very nature lack the interpretability and transparency of any machine learning model. Additionally, there is a greater chance of overfitting when using ANN approaches^25,26. A summary of the machine learning models used to forecast the mechanical characteristics of various concrete kinds is also displayed in Table 1.

Table 1.

A summary of recent studies on ML-based prediction of rubberized concrete mechanical characteristics.

S. no.	Author (year)	ML technique applied	Material	Target properties	Main findings
1	Kovačević et al. (2021)31	Multilayered perceptron artificial neural network (MLP-ANN), Ensemble ANN, RF, Support vector regression (SVR), Gaussian process regression (GPR)	Self-compacting rubberized concrete (SCRC)	Compressive strength (CS)	Ensemble ANN achieved the highest accuracy (MAE ≈ 2.81 MPa, R ≈ 0.96); GPR provided comparable accuracy with lower model complexity
2	Huang et al. (2021)32	ANN ((Levenberg–Marquardt (LM), Bayesian Regularisation (BR), Scaled Conjugate Gradient (SCG) algorithms), MLR	Rubberized concrete	CS (7 & 28 days), FS, TS, elastic modulus (EM)	ANN successfully predicted multiple mechanical properties simultaneously; ANN significantly outperformed MLR, demonstrating the feasibility of ML-aided mix design for rubberized concrete
3	Senthil Vadivel et al. (2024)20	RF, SVM, DT, ANN (with Grid Search CV)	Rubberised concrete (FR & CR replacement)	CS, TS, FS (28-day)	RF models showed superior predictive performance with optimized hyperparameters; ML effectively captured nonlinear relationships between rubber content and strength properties
4	Adamu et al. (2023)33	ANN	Rubberized concrete with CR and supplementary materials	CS, TS, FS, EM	ANN accurately predicted multiple mechanical properties; results highlighted the importance of rubber content and binder composition on strength degradation and enhancement trends
5	Moussa (2020)34	Regression analysis, ANN	Rubberized concrete	Compressive strength, flexural strength, split tensile strength, elastic modulus	ANN significantly outperformed regression models (R² = 0.82–0.96), demonstrating strong capability for predicting multiple mechanical properties of rubberized concrete
6	Gregori et al. (2021)35	Support Vector Machine (SVM), Gaussian Process Regression (GPR)	Rubbercrete (crumb rubber replacing fine aggregates)	Compressive strength	GPR outperformed SVM and was used to develop strength reduction factors (SRF) equations for rubberized concrete mixtures
7	Present study	Gene Expression Programming (GEP), Random Forest (RF)	Rubberized concrete	Flexural strength, split tensile strength	RF exhibited superior predictive accuracy, while GEP provided interpretable equations; SHAP analysis quantified the influence of mix parameters on strength behavior
8	Pal et al. (2023)36	Comparative ML review (incl. XGBoost, SVR, others)	Rubberized concrete & sustainable variants	Compressive strength	CatBoost was the best-performing model
9	Alahmari (2024)37	ANN, GEP, Bagging	Rubberized concrete	Compressive strength	ANN was the most accurate algorithm

The majority of current research is on individual modeling approaches and provides limited systematic comparison between interpretable evolutionary models with ensemble learning methods for forecasting the flexural and split tensile strengths of rubberized concrete, despite the increasing usage of machine learning techniques. Moreover, numerous current research depend on very small or heterogeneous datasets, constrained validation methodologies, and insufficient model-agnostic interpretability assessments, hence limiting their practical engineering usefulness. These restrictions draw attention to a glaring research gap in the creation of reliable, comprehensible, and thoroughly verified predictive models for the mechanical characteristics of rubberized concrete.

Gene Expression Programming (GEP) is a powerful machine learning technique that uses symbolic regression to produce explicit mathematical expressions. It is based on Genetic Algorithms (GA) and Genetic Programming (GP). A potent ensemble learning method called Random Forest (RF) combines several decision trees to increase prediction accuracy and generalization while lowering overfitting^27,28. RF has been used successfully to forecast many concrete parameters, including compressive, tensile, and durability characteristics^29,30. Nonetheless, there exists a lack of research that systematically contrasts GEP and RF models utilizing harmonized multi-source experimental datasets, augmented by explainable artificial intelligence technologies, for forecasting FS and STS of rubberized concrete. To bridge this gap, the current study creates a harmonized experimental database based on multiple published studies and conducts a structured comparative evaluation of GEP and RF models under identical modeling conditions for predicting the FS and STS of rubberized concrete. Key influencing parameters are used as model inputs, and the dataset is split into training and testing subsets. Several statistical indicators and external validation criteria are used to assess the model’s performance. Furthermore, Shapley Additive ExPlanations (SHAP) analysis is used to measure the relative significance of input variables, improving the interpretability and practical usefulness of the model. It is anticipated that the results of this study will offer reliable forecasting instruments and insightful information for the sustainable design and optimal use of rubberized concrete.

Gene expression programming (GEP) approach review

Genetic algorithms (GA) are a probabilistic technique for discovering and optimizing problem solutions using genetic principles³⁸. Cramer developed Genetic Programming (GP) in 1985, which Koza later expanded to incorporate many forms and sizes, boosting its versatility^39,40. nlike fixed-length binary strings, GP uses nonlinear patterns known as parse trees, making it a versatile problem-solving tool. It functions similarly to genetic processes, including mutation, crossing, and reproduction^41,42. Unlike fixed-length binary strings, GP employs nonlinear patterns known as parse trees, giving it an adaptable problem-solving tool. It works similarly to genetic processes including mutation, crossover, and reproduction^42,43. Mutation inhibits premature convergence of the model, ensuring its durability⁴². Figure 1 shows the overall procedure of the GP modeling method.

Fig. 1.

GP Flowchart.

Candida Ferreira⁴⁴ introduced the notion of GEP as an evolutionary approach to computer programming. GEP is notable for its capacity to generate complex, tree-like computer programs or models with qualities akin to live organisms, such as adaptability and learnability. A significant characteristic of the GEP model is the use of simple mathematical expressions developed during the process. This results in accurate predictions and better predicting outcomes⁴⁵. GEP is an evolutionary version of GP that combines fixed-length chromosomes from Genetic Algorithms (GA) and parse trees. GEP has certain variables with GP, but it varies in that it uses fixed-length strings rather than variable-length parse trees. The GEP algorithm separates genotype and phenotype, accelerating the process by just transmitting genotype to following stages. This method avoids the requirement to replicate and modify the whole structure for mutations⁴⁶. Furthermore, GEP produces populations with a single chromosome containing several genes, known as head and tail. This leads to a direct one-to-one link between chromosomal symbols and their associated functions⁴². The chromosomes carry critical information for building causal relationships, which is processed using the Karva language. By supplying the gene sequence, one can determine the specific phenotype, known as Karva expression⁴⁶. Figure 2 depicts the GEP approach in schematic form. This procedure entails randomly producing fixed-length chromosomes for the entire population, known as Expression Trees (ETs). Each chromosome’s fitness is then assessed. The people most ideal for reproduction are chosen, and this process is repeated with fresh individuals over many generations until the best answer is found. Mutation, crossover, and reproduction are the key genetic mechanisms that help the population evolve.

Fig. 2.

GEP algorithm flowchart diagram.

Random forest (RF) programming review

Breiman initially presented Random Forest (RF) regression in 2001²⁸, is a popular ensemble learning technique for classification and regression applications. RF has shown high predictive abilities in modeling nonlinear relationships between input and output variables. Previous research has quantitatively demonstrated that RF can reduce prediction errors compared to classic regression models and single decision-tree techniques^28,47. RF generates many decision trees during the training phase and combines their predictions to provide improved predictive accuracy and robustness, as illustrated in Fig. 3⁴⁸. RF has been employed in a variety of industries, including banking to forecast client responses⁴⁹, medicine/pharmaceutical business⁵⁰, stock market price direction⁵¹, e-commerce⁵², demonstrating its versatility across multiple domains. RF is created by combining many tree predictors, with a random vector influencing each tree’s decision. This vector is sampled independently and uniformly dispersed throughout the forest²⁸. Each tree in the forest is trained on a unique training set, which may produce somewhat varied outcomes⁵³. During the training phase, each tree is built using a randomly selected subset of the training data, known as bootstrapped samples, which means that certain data points may be repeated while others are deleted. Furthermore, RF uses feature randomization to select a random set of features for each tree node split. This approach reduces overfitting while increasing tree variability.

Fig. 3.

Schematic of RF generation and prediction.

RF assesses its training performance using the Out-of-Bag (OOB) error estimation technique, which entails evaluating each tree’s predictions for data points that did not appear in the bootstrap sample⁵⁴. The OOB error provides an unbiased estimate of the model’s generalization error. The final output is obtained by averaging the numerous tree forecasts for regression tasks^55,56.

RF has various advantages, as evidenced by prior quantitative studies. Studies have found that RF retains good predictive accuracy while requiring relatively minimum hyperparameter modification when compared to other machine learning models. Furthermore, the independent creation of individual decision trees allows for parallel processing, which considerably enhances computing efficiency and scalability when dealing with huge datasets and high-dimensional data environments⁵⁷.

Figure 3 depicts the process of the RF model as used in this study. To begin, the generated experimental dataset is separated into training and testing subsets to assess model generalization capabilities. The training dataset is then subjected to bootstrap sampling, which produces several randomized subsets of data. These subsets are utilized to build distinct decision trees, ensuring variation among tree predictors. Random feature selection is performed within each tree during node splitting to reduce tree correlation and improve model robustness. Each decision tree generates an independent forecast based on its bootstrap sample and input variables. The final anticipated answer is generated by aggregating the outputs of all trees using averages. This workflow represents the customized RF framework used in this study, which combines dataset partitioning, bootstrap sampling, random feature selection, and ensemble prediction aggregation to improve predictive accuracy and reduce overfitting when modeling the mechanical properties of rubberized concrete.

Methodology

After specifying the study objectives, a large dataset of reliable experimental investigations was assembled to train and assess the predictive algorithms. The dataset contained 112 data points for both FS and STS. The acquired data was preprocessed, statistically evaluated, and separated into training and testing subsets to ensure balanced representation and consistent model performance. This data splitting technique mitigates model bias and enhances generalization by facilitating independent assessment of model accuracy on novel datasets. The dataset was partitioned into training and testing sections with a 70/30 ratio. The partitioning was executed by random sampling to guarantee an impartial representation of the dataset. Stratified or grouped splitting was not utilized due to the dataset comprising independent experimental findings sourced from various literature. A fixed random seed value of 1 was employed during the creation of the Random Forest model to guarantee the reproducibility of the training process. Two advanced machine learning methods, GEP and RF, were used to create prediction models for evaluating rubberized concrete’s mechanical properties. To attain the best level of accuracy, the models were refined using iterative parameter tuning. The employment of two independent modeling methodologies facilitates a comparison evaluation between evolutionary symbolic regression and ensemble learning procedures, hence enhancing the dependability of model validation. Figure 4 depicts the study’s total workflow, highlighting the important stages of data collection, preprocessing, model construction, performance evaluation using statistical indicators (R, MAE, RMSE, RRSE, RAE), and external validation. In addition to depicting the workflow sequence, Fig. 4 demonstrates the comprehensive validation technique employed in this investigation. The integration of internal statistical measurements with external validation parameters enhances the robustness of performance evaluation and diminishes the risk of overfitting. Moreover, the incorporation of SHAP analysis improves model interpretability by quantifying the impact of various input parameters on strength prediction. This organized multi-phase architecture guarantees that the created models are both precise and transparent, as well as dependable for engineering applications.

Fig. 4.

Flowchart outlining the methodology used in the study.

Collection of data and input parameters selection

The creation of a dependable dataset is critical to the construction of a machine learning model. To build strong models, a comprehensive database was compiled from internationally published publications on rubberized concrete FS^{58,60–69,69} and STS^{58,60–64,64,70,72,72}. This database contains 112 data points each for FS and STS. Based on earlier literature research, seven separate input parameters for model creation were identified: fine aggregate (FA), water to cement ratio (W/C),), coarse aggregate (CA), chipped rubber (CHR), crumb rubber (CR), super plasticizer (SP), and age. These input parameters were chosen because of their well-established effect on rubberized concrete mechanical properties and their constant availability across experimental experiments. Despite being influential factors, rubber particle size gradation, rubber surface treatment methods, cement content variations, supplementary cementitious materials (SCMs), admixture type, density, and air content were not consistently reported in the literature database. Such values would have lowered dataset size and weakened model stability and forecast reliability. Thus, the selected variables are the most consistently reported and statistically reliable parameters in the experimental dataset. The absence of these variables may affect prediction generalization for combinations with advanced rubber treatments or specific cementitious systems, as stated in the Study Limitations section.

The developed prediction models aim to estimate mechanical strength characteristics at different curing ages, rather than being limited to a 28-day strength evaluation, with curing age included as an independent input variable to reflect strength development behavior. Rubberized concrete’s mechanical properties (FS and STS) are given as a function of Eqs. (1) and (2), which consider these input variables to be the most important factors determining the mechanical properties of the material.

1

2

Where FS denotes flexural strength, STS signifies split tensile strength, CA refers to coarse aggregate content, FA indicates fine aggregate content, CHR represents chipped rubber content, CR stands for crumb rubber content, SP denotes superplasticizer dosage, Age indicates curing age (days), and W/C refers to the water-to-cement ratio. To guarantee consistency and comparability of datasets derived from various literature sources, all quantities of mixture components were standardized before modeling. The doses of CA, FA, CHR, CR, and SP were reported in kg/m³ according to standard concrete mix design standards. The rubber contents initially documented as volumetric replacement ratios or percentage substitutions in the source literature were transformed into corresponding mass-based values to ensure consistent dataset representation. Likewise, superplasticizer dosages expressed as a percentage of cement or binder content were translated into kg/m³ utilizing the relevant quantities of cementitious materials.

The assembled dataset was randomly shuffled before model development to avoid ordering bias and balance input variables across training and testing groups. A predetermined random seed was utilized during dataset partitioning to ensure reproducible random splitting. Duplicate and inconsistent records were thoroughly eliminated during database preparation. The dataset was acquired from numerous distinct experimental experiments, however each data point reflects a unique concrete mixture and independent testing observation. Records were treated as statistically independent samples. Random data partitioning was done after preprocessing and validation to reduce data leakage between training and testing subsets.

The prediction model is influenced by the pattern and distribution of input data. Figures 5 and 6 show the frequency distributions of the input data for FS and STS. Histogram-based distribution analysis is employed to characterize dataset variability rather than to directly illustrate model generalization capacity. The histograms indicate that the datasets display a broad distribution of strength values without significant clustering in limited ranges. This variability is crucial for training machine learning models, since it allows algorithms to identify nonlinear correlations between mixture parameters and mechanical performance. Furthermore, the lack of excessive skewness or data clustering diminishes model bias and improves predictive stability across various strength ranges. Table 2 also contains a large number of statistical values for data, which provide further insights into the database. This table illustrates data distribution (standard deviation), means, extremes, and patterns (kurtosis and skewness), making the input data easier to grasp. The dataset’s statistical diversity facilitates effective model training, while the model’s generalization capacity was assessed using comparisons of independent training and testing datasets, external validation criteria, and assessments of statistical error distribution, which are detailed in subsequent sections of this paper. The statistical diversity in the dataset justifies the creation of generalized prediction models that can operate reliably across various mixture compositions. The FS ranges from 0.64 to 16.04 MPA, whereas the STS spans from 0.94 to 4.81 MPa, indicating a broad variation in the output parameter. The extensive output ranges enhance the learning capacity of machine learning algorithms by ensuring adequate representation of both low and high strength levels. Skewness was utilized to determine data symmetry. This symmetry indicates a balanced representation of the dataset, so mitigating the possibility of predictive bias towards prevailing strength levels. The analysis revealed that nearly all variables were within the required range of − 3 to + 3, indicating good distributional symmetry. Furthermore, kurtosis was utilized to evaluate if the distribution was peaked or flat⁷³. The kurtosis values for almost all model variables were within the intended range of − 10 to + 10, indicating an appropriate distribution with no significant deviations from a normal distribution^74,75. These statistical attributes validate the dataset’s appropriateness for training ensemble and evolutionary machine learning models, which often necessitate diverse and well-distributed data to attain substantial predictive efficacy.

Fig. 5.

FS dataset frequency histograms.

Fig. 6.

STS dataset frequency histograms.

Table 2.

Analysis of the collected datasets using statistical methods.

Variables	Mean	Median	Standard deviation	Kurtosis	Skewness	Minimum	Maximum
FS (MPa)
Coarse aggregate	821.43	799.90	226.63	3.73	−1.28	0.00	1296.00
Fine aggregate	607.78	606.05	219.81	2.92	0.24	0.00	1354.70
Chipped rubber	46.66	0.00	162.46	20.91	4.32	0.00	1096.00
Crumb rubber	54.34	37.32	58.16	6.36	2.29	0.00	337.60
Superplasticizer	96.58	0.00	185.39	1.94	1.88	0.00	546.00
Age	27.53	28.00	12.07	18.33	3.42	7.00	90.00
W/C	0.49	0.47	0.10	−0.66	0.42	0.31	0.68
FS	4.59	4.03	2.99	5.74	2.23	0.64	16.04
STS (MPa)
Coarse aggregate	995.31	782.69	96.14	48.50	6.96	646.80	808.5
Fine aggregate	639.27	636.09	16.59	5.95	1.60	234.00	1354.70
Chipped rubber	2.29	0.00	0.86	17.14	4.17	0.00	49.00
Crumb rubber	50.20	34.00	4.86	3.09	1.92	0.00	226.50
Superplasticizer	98.13	0.00	18.21	1.75	1.84	0.00	546.00
Age	24.79	28.00	0.70	1.83	− 1.91	7.00	28.00
W/C	0.48	0.47	0.01	− 0.33	0.87	0.37	0.68
STS	2.78	2.62	0.09	− 0.65	0.37	0.94	4.81

Furthermore, Pearson correlation heat maps were created to analyze the interrelationships between input factors for FS and STS, as illustrated in Fig. 7. Understanding these correlations is critical because large relationships between predictors can lead to multicollinearity, complicating the interpretation of regression and machine learning models^{2,76,78–92,92}. The research revealed that the majority of pairwise correlation coefficients (r-values) stayed below the required threshold of 0.8, indicating a low probability of multicollinearity and more reliable model estimation^93,95,95. These results demonstrate that the dataset’s variables have an appropriate level of independence, allowing for the building of robust and interpretable predictive models.

Fig. 7.

Pearson correlation heat map of a FS, b STS.

GEP model development

Understanding the significance of proper variables in developing a valid and universal model is critical. Several tests runs and ideas from previous research were used to define the fitting parameters for the GEP method. The population size determines the duration of the process. In this work, GEP modeling was performed on a population of 150. The head size influences the structure of the software-generated model, with the former determining the number of sub-ETs and the latter dictating the complexity of each term. In this study, the head size and number of genes were adjusted to 20 and 10, respectively. Table 3 shows the fitting parameters utilized in GEP modeling. The GeneX software package was used for GEP modeling. A multi-generation run yields the simulation model with the fewest errors. Several parameter combinations were tested in order to select the best model, which produced the output model with the fewest statistical errors. Overfitting is a major problem with machine learning models; the model performs well when applied to original data, but its performance drastically decreases when applied to previously unknown data. To prevent overfitting, it has been advised to evaluate the model on previously unseen data^96,97. Consequently, the data is separated into two categories proportionately. After then, a testing set that was not utilized during the model’s development is employed to test it. For testing and training, the data in the current study was split into 30% and 70% groups, respectively. On every dataset, the produced models exhibit remarkable performance.

Table 3.

Parameters configuration for the construction of the gene expression programming (GEP) model.

Parameter	Setting	Description
General setting
Head size	20	Regulates the intricacy of produced expression trees
Function set	+, −, ×, ÷, exp, x², x³, Inv, ln, Max2, Min2, Atanh, Avg2, Tanh, NOT	Operators employed to formulate model expressions
Chromosomes	150	It decides size of population
Genes	10	It defines number of sub expression trees
Function linking	Addition	It combines gene outputs
Numerical constants
Data type	Floating-point	It allows for continuous numerical values
Lower bound	−10	It allows for minimum allowable constant value
Upper bound	10	It allows for maximum allowable constant value
Genetic operators
Rate of mutation	0.00138	Introduces random genetic variation
Transposition rate	0.00546	Shifts gene segments
Random cloning	0.00102	Copies randomly selected individuals
Best cloning	0.00260	Preserves best-performing individuals
RNC mutation	0.00206
Permutation rate	0.00546
Inversion rate	0.00546	Reverses gene segments
Recombination rate	0.00277	Facilitates the exchange of genetic material between chromosomes
IS transposition rate	0.00546	Incorporates gene sequences into chromosomes
Gene transposition rate	0.00277	Facilitates the translocation of whole genes across chromosomes

Development of the RF model

The Random Forest (RF) model was created utilizing the WEKA machine learning software package. The RF hyperparameters were refined via an iterative tuning approach to enhance predicted accuracy. Throughout the tuning process, several combinations of essential WEKA RF parameters, such as the number of trees (iterations), bag size %, and tree growth configuration, were methodically evaluated. The evaluation of model performance during hyperparameter tuning included statistical performance metrics, including the correlation coefficient (R), root mean square error (RMSE), and mean absolute error (MAE). The ultimate hyperparameter configuration was chosen for its balanced prediction accuracy, lowered error magnitude, and consistent performance across training and testing datasets. The quantity of trees (Number of iterations) was determined to optimize computational complexity and predictive efficacy. The RF model in WEKA was configured with bootstrap aggregation, with each tree trained on randomly generated data subsets. The bag size percentage was established at 100%, guaranteeing that each tree was trained using a bootstrap sampling of the complete training dataset. The maximum tree depth was not constrained (maximum depth = 0), permitting trees to develop entirely and include nonlinear interactions. To enhance transparency and reproducibility, Table 4 presents comprehensive configuration options accompanied by functional descriptions of each parameter utilized in the WEKA RF model. While Random Forest algorithms typically leverage Out-of-Bag (OOB) error for internal validation, this study did not include OOB estimation. Section 3 (Random Forest (RF) programming review) presents the theoretical framework and operational idea of OOB error to elucidate the general operation of the RF method. In this study, model performance was assessed by isolated training-testing data split, supplemented by several statistical performance measures, external validation criteria, and a comprehensive investigation of error distribution. This multi-tier validation methodology facilitates a dependable evaluation of predictive accuracy, generalization capacity, and model resilience across various datasets. This is consistent with findings from Liu et al. (2020), where RF was successfully used to predict the compressive strength of high-performance concrete, yielding robust results⁹⁸. Additionally, it was demonstrated that RF is more effective in modeling complex mix designs by outperforming other regression techniques in predicting the compressive strength of self-compacting concrete⁹⁹. RF demonstrated high predictive skills in rice husk ash-infused eco-friendly concrete (with accuracy frequently surpassing Gaussian Process Regression and Decision Tree models)¹⁰⁰. Furthermore, while calculating the compressive strength of sustainable self-consolidating concrete that contains SCMs such fly ash and ground granulated blast furnace slag (GGBS), RF showed excellent reliability¹⁰¹. The appropriateness of RF and its adjustable hyperparameters for forecasting mechanical responses in concrete systems is supported by these investigations. The final optimized hyperparameters employed in this investigation are compiled in Table 4.

Table 4.

Configuration parameters for the RF model developed using WEKA software.

Parameter	Setup	Function description
Bag size percentage	100	Bootstrap sample size
Debug	False	–
Run decimal places	2	–
Calc out of bag	False	–
Break ties randomly	False	–
Do not check capability	False	–
Num execution slot	1	–
Maximum depth	0	Unlimited tree depth
Batch size	100	–
Compute attribute importance	False	–
Num features	0	–
Num iterations	100	Number of decision trees
Output out of bag complexity statistics	False	–
Print classifier	False	–
Seed	1	–
Store out of bag predictions	False	–

GEP and RF models performance and external evaluation criteria

A range of statistical performance metrics, including the correlation coefficient (R), mean absolute error (MAE), root mean square error (RMSE), relative root square error (RRSE), and relative absolute error (RAE), as shown in Eqs. (3–7), were used to evaluate the validity, predictive performance, and accuracy of the developed models. Together, these metrics quantify the degree to which the expected values match the experimental data. Because they can estimate the size of prediction mistakes, RMSE and MAE are the two statistical metrics that are most frequently used in machine learning-based concrete modeling^30,102. By squaring the deviations, the RMSE highlights bigger errors, but the MAE offers a more balanced perspective of accuracy by providing a linear measure of the average difference between projected and observed values. Higher precision and dependability of the predictive model are indicated by lower RMSE and MAE values. The strength and direction of the linear relationship between expected and actual outputs are evaluated by the correlation coefficient (R), where values around unity indicate a perfect fit. Similar methods have been used in earlier research on the prediction of concrete strength using GEP¹⁰³, MEP¹⁰⁴, and RF^105,106, demonstrating that these statistical characteristics are trustworthy markers for verifying model performance during both training and testing phases. These metrics’ mathematical expressions are shown below. A lower RMSE, and MAE number indicates that the model is more precise and accurate. The mathematical equations for calculating these metrics are provided below.

3

4

5

6

7

Where n represents the number of points, and ei and mi represent the corresponding real and model values for the ith dataset. In contrast, the means of the actual and model values are represented by and , respectively. Higher model accuracy is indicated by an R-value near one, which indicates a strong connection between the variables. Additionally, it has been suggested in previous research that model performance is generally deemed satisfactory when the R-value exceeds 0.8^107,108. This threshold is not universal and may fluctuate based on dataset complexity, noise levels, and application domains. Therefore, the selected threshold serves as a general performance guideline rather than a strict acceptance criterion. The value of R shows the linearity between input and output data¹⁰⁹. R should not be used exclusively to assess the model’s performance since it is insensitive to the multiplication and division of output by a constant. Therefore, depending simply on R may lead to inaccurate performance interpretation. Rather, the mean error value is indicated by MAE and RMSE, where values close to 0 imply great model performance. Each error metric gives supplementary information regarding model accuracy. A large number of projected values with significant mistakes that exceed the expected range and require reduction are indicated by a high RMSE score. In contrast to RMSE, MAE typically produces smaller results and gives less weight to significant errors. Additionally, Table 6 presents an overview of the ranges of these statistical metrics values. Furthermore, external validation expressions were used to verify the model’s validity using the parameters specified in the pertinent literature, as shown in Table 5.

Table 6.

Best-fit values of performance matrices.

S. no	Metrics	Range	Best-fit value	Remarks
1	R	0, 1	1	Ideal fit
2	MAE	0, ∞	0	Ideal model value
3	RMSE	0, ∞	0	Ideal model value
4	RRSE	0, ∞	0	Ideal model value
5	RAE	0, ∞	0	Ideal model value

Table 5.

External Validation Criteria for GEP and RF.

S. no	Equation	Conditions	References
1		0.85 < k < 1.15	110
2		0.85 < < 1.15	110
3	) Where		111

To guarantee clarity in performance interpretation, it is essential to differentiate between the correlation coefficient (R) and the coefficient of determination (R²). The correlation coefficient denotes the strength of the linear relationship between predicted and experimental results, whilst R² signifies the fraction of variance elucidated by the model. External validation metrics (k, k′, R₀², and Rm) offer supplementary assessment of predictive calibration and dependability. Consequently, discrepancies between statistical and external validation outcomes signify an assessment of different aspects of model performance rather than numerical discrepancy.

The current work utilized a singular randomized 70/30 training-testing dataset partition for model validation. This technique facilitates independent assessment of predictive capabilities; however, model performance may be sensitive to the particular data split utilized in model building. To mitigate this constraint, various complementary statistical error measurements, external validation requirements, and comprehensive absolute error distribution studies were employed collectively to evaluate model dependability and prediction consistency. It is recognized that repeated resampling or cross-partition validation methods could offer further statistical validation of model stability across different dataset partitions and are thus advised for further research (Table 6).

Shapely additive explanation (SHAP)

A technique for assessing machine learning models called SHAP (Shapely Additive Explanations) was presented by Lundberg and Lee¹¹². All efforts to develop reliable estimating tools have been captured by machine learning models’ ability to learn from observed data and predict results in new data. However, the majority of ML modeling techniques have poor interpretability and excessive complexity. The explainability of prediction models is enhanced by the SHAP approach. SHAP is used in this study to determine the significance of the input parameters and how they affect the response parameter. Although a lot of machine learning research on building materials has achieved remarkable accuracy in predicting their goals, the interpretation of the ML models is not given enough thought. Several studies use the decision path and intuitive methods to calculate characteristic significance in machine learning models¹¹³. A Shapley additive (SHAP) based method was created to assess each input variable’s relative importance to the output and determine whether input variables improve the output in a positive or negative way. A thorough explanation of the SHAP approach can be found in references^114,115. Each independent variable’s significance to the response characteristics is shown by the Shapley value. This method is similar to parametric analysis, which evaluates how changes in one input parameter affect the result by changing one variable while keeping other factors constant. In this work, the SHAP is used to provide interpretations of each input variable as well as parameter relevance. It also makes it easier to analyze how input factors either favorably or negatively impact the model’s output^{112,116,118–123,123}. The average proportional contribution of an attribute value over all possible combinations is represented by SHAP, which is based on cooperative game theory. The average anticipated value of a variable with this attribute without the mean forecasted value of observations is called the SHAP number of a character. In order to facilitate interpretability, the output of an ML model is considered to be a linear sum of its input features scaled by matching SHAP values. The SHAP analysis is graphically depicted in Fig. 8.

Fig. 8.

SHAP attributes¹²⁴.

Results and discussion

GEP model regression analysis

Figures 9, 10, 11 and 12 show how well the GEP model predicts the mechanical characteristics of rubberized concrete. The experimental and predicted values of FS for the training and testing datasets are compared in Figs. 9 and 10, respectively. With coefficients of determination (R²) of 0.9887 for training and 0.9183 for testing, the regression lines show a high alignment between the two sets of values. These findings demonstrate good consistency between predicted and experimental data, indicating that the GEP model successfully captured the nonlinear relationship between input parameters and FS. The regression lines’ near-unity slopes attest to the model’s ability to generalize effectively to fresh data and its lack of underfitting or overfitting. These findings demonstrate substantial agreement between predicted and experimental data, indicating that the GEP model successfully captured the nonlinear relationship between input parameters and FS.

Fig. 9.

Experimental vs. GEP-predicted FS for training data.

Fig. 10.

Experimental vs. GEP-predicted FS for testing data.

Fig. 11.

Experimental vs. GEP-predicted STS for training data.

Fig. 12.

Experimental vs. GEP-predicted STS for testing data.

Gholampour et al.⁴³ argue that the amount of data and quality of the dataset utilized have a significant impact on machine learning model accuracy. The robustness and prediction accuracy of the model were enhanced in this work by a comparatively large dataset consisting of 112 records for both FS and STS. Similar findings were noted by Iqbal et al.¹⁰³ and Nafees et al.¹⁰⁶ who noted that GEP models outperform traditional regression techniques at capturing complex nonlinear interactions in concrete mixtures.

The STS regression charts are shown in Figs. 11 and 12. With R² values of 0.8671 for training and 0.8965 for testing datasets, the GEP model generated robust correlations once more. These values show that the model can accurately predict the experimental behavior of STS, even though they are marginally lower than those for FS. The model’s accuracy and flexibility for new data are confirmed by the strong agreement between measured and anticipated values. Overall, the relationship between mixing parameters and rubberized concrete’s mechanical characteristics was effectively represented by the GEP model. The created model may be a reliable prediction tool for material design and performance evaluation in sustainable concrete applications, as indicated by the strong correlation values, consistent regression slopes, and low error levels.

GEP model error analysis

The absolute error distribution between the experimental and GEP-predicted FS values for the testing and training datasets is shown in Figs. 13 and 14, respectively. For the FS testing data, the maximum, minimum, and average error values were 2.771 MPa, 0.00063 MPa, and 0.6855 MPa; for the training data, the corresponding values were 2.097 MPa, 0.0005 MPa, and 0.4323 MPa. The GEP model’s capacity to accurately predict unknown data based on the patterns discovered during training is demonstrated by these low error magnitudes. The stability and dependability of the GEP model for FS prediction are further confirmed by the close agreement between the anticipated and experimental findings.

Fig. 13.

Error distribution of FS predictions for the testing dataset.

Fig. 14.

Error distribution of FS predictions for the training dataset.

Similarly, the splitting tensile strength (STS) absolute error distribution is shown in Figs. 15 and 16. The training dataset’s maximum, minimum, and average errors were 0.7838 MPa, 0.0011 MPa, and 0.2172 MPa, respectively, while the testing dataset’s were 1.567 MPa, 0.00302 MPa, and 0.295 MPa. These findings demonstrate that the GEP model consistently outperforms the experimental results on both datasets.

Fig. 15.

Error distribution of STS predictions for the testing dataset.

Fig. 16.

Error distribution of STS predictions for the training dataset.

Large deviations occur very infrequently, according to the distribution of absolute errors. Significant disparities are uncommon for FS projections, with 91–94% of total errors falling between 0.2 MPa and 1.5 MPa and with 6–9% exceeding 1.5 MPa. For some data points, about 3% of errors were determined to be less than 0.01 MPa, demonstrating the accuracy of the model. Similar to this, 92% of testing errors in STS prediction fell between 0.1 and 0.6 MPa, with 8% exceeding 1.0 MPa; during the training phase, 94% of errors fell between 0.05 and 0.5 MPa, with just a small percentage exceeding 0.7 MPa.

Overall, the error distribution pattern demonstrates that the GEP model produces little mistakes across the majority of data points while maintaining strong predictive stability and generalization capabilities. The durability and effectiveness of the GEP model for calculating the mechanical characteristics of rubberized concrete are confirmed by the infrequency of large deviations and the concentration of errors within a small range.

RF model regression analysis

Figures 17, 18, 19 and 20 present regression plots that compare experimental values with RF-predicted values of FS and STS for both training and testing datasets. The regression equation for FS is y = 0.799x + 0.9975 (R² = 0.9799) in the testing dataset, and y = 0.777x + 0.5000 (R² = 0.9261) in the training dataset. In STS, the testing and training datasets yield regression equations of y = 0.9159x + 0.2094 (R² = 0.9817) and y = 0.8848x + 0.3345 (R² = 0.9703), respectively.

Fig. 17.

FS testing data comparison with RF model prediction.

Fig. 18.

FS training data comparison with RF model prediction.

Fig. 19.

STS testing data comparison with RF model prediction.

Fig. 20.

STS training data comparison with RF model prediction.

The model’s capacity to accurately represent the intricate, nonlinear interactions controlling the mechanical behavior of rubberized concrete is demonstrated by the consistently high R² values (above 0.92) across all datasets, which validate the excellent correlation between experimental and predicted values. Although the regression analyses show a significant correlation between predicted and experimental values, slope values slightly less than unity indicate the presence of small calibration bias, implying a slight propensity to underestimate strength values in specific data ranges. The clustering of data points near this reference line demonstrates the RF model’s acceptable prediction capabilities.

RF model error analysis

Figures 21, 22, 23 and 24 show the absolute error distribution of experimental findings vs. RF model predictions for FS and STS in both training and testing datasets. For FS, the testing dataset (Fig. 21) shows smaller dispersion, with maximum, minimum, and average errors of 2.0855 MPa, 0.008464 MPa, and 0.546 MPa, respectively. In comparison, the training dataset (Fig. 22) has a significantly wider range of errors, with a maximum error of 4.45 MPa, a minimum of 0.002638 MPa, and an average error of 0.576 MPa. This implies that, while the RF model maintains consistent predictive performance, there are occasional greater deviations, notably in the training data, which are most likely caused by the existence of complicated or extreme data points.

Fig. 21.

Error representation in FS testing data prediction model.

Fig. 22.

Error representation in FS training data prediction model.

Fig. 23.

Error representation in STS testing data prediction model.

Fig. 24.

Error representation in STS training data prediction model.

A similar pattern is seen in STS, where the training dataset (Fig. 24) has a maximum error of 0.6514 MPa, a minimum error of 0.00048 MPa, and an average error of 0.12367 MPa, while the testing dataset (Fig. 23) has slightly lower values of 0.4602 MPa, 0.000276 MPa, and 0.1244 MPa, respectively. The close agreement between training and testing errors demonstrates the RF model’s remarkable generalization ability, particularly for STS predictions.

Overall, the error distributions show that very small errors (< 0.01 MPa) are comparatively rare for both FS and STS, suggesting that near-perfect predictions are possible but uncommon. Only a small percentage of data points (usually fewer than 8–10%) surpass 2 MPa, indicating localized over- or under-estimation. The majority of FS mistakes are centered between 0.2 and 1.8 MPa. With over 90% of errors falling between 0.05 and 0.5 MPa and just a small percentage above 0.6 MPa, STS, on the other hand, shows a more condensed error range. These findings verify that the RF model predicts both mechanical properties with accuracy, stability, and dependability, with variances mostly falling within tolerable and moderate ranges.

Subsequent analysis of forecast discrepancies reveals that the most significant errors typically arise in mixtures with elevated rubber replacement levels or extreme quantities of components. These mixes frequently demonstrate intricate nonlinear mechanical behavior that poses greater challenges for data-driven algorithms to accurately capture. Nonetheless, the error distribution plots illustrate that such high deviations are rare and constitute a minor fraction of total predictions, suggesting that the general performance of the model is not much affected by single outlier data.

GEP and RF models performance assessment

A model’s validity is impacted by the number of data points needed to build it. For a model to be considered satisfactory, the dataset proportion to the number of inputs should be three^43,125. As was previously mentioned, different statistical measures (R, MAE, RMSE, RAE, and RRSE) are used to evaluate each of the three models’ performance.

Both models exhibit good predictive ability, as Table 7 and Figs. 25, 26, 27 and 28 illustrate, with R values near unity for all datasets. The RF model produced even higher correlations, reaching 0.998 for training and 0.990 for testing, while the GEP model produced R values of 0.990 and 0.991 for the training and testing phases, respectively, for the STS subset. Similarly, for the FS subset, the RF model continuously maintained strong correlations of 0.990 for both phases, while the GEP model generated R values of 0.970 (training) and 0.989 (testing). Strong agreement between the expected and experimental results is indicated by these high R values.

Table 7.

Various statistical calculations of the GEP and RF models.

	Model	Subset	R	MAE	RMSE	RAE	RRSE
GEP	STS	Testing	0.991	0.170	0.190	0.150	0.180
		Training	0.990	0.180	0.190	0.190	0.200
	FS	Testing	0.989	0.546	0.778	0.244	0.231
		Training	0.970	0.748	0.810	0.228	0.324
RF	STS	Testing	0.990	0.111	0.125	0.101	0.130
		Training	0.998	0.110	0.125	0.110	0.140
	FS	Testing	0.990	0.440	0.650	0.160	0.200
		Training	0.990	0.540	0.700	0.160	0.220

Fig. 25.

GEP-FS radar plot of performance metrics.

Fig. 26.

GEP-STS radar plot of performance metrics.

Fig. 27.

RF-FS radar plot of performance metrics.

Fig. 28.

RF-STS radar plot of performance metrics.

The great accuracy of both models is further supported by error-based measures. The RF model showed fewer errors, with MAE = 0.111 and RMSE = 0.125, for the STS testing dataset, whereas the GEP model generated MAE and RMSE values of 0.170 and 0.190, respectively. The training phase shows a similar pattern, with RF outperforming GEP once more (MAE = 0.110 and RMSE = 0.125) in contrast to GEP (MAE = 0.180 and RMSE = 0.190). Both models retain respectable predictive performance in the FS subset, despite the fact that the error values rise as a result of the dataset’s increased complexity. For example, the RF model lowered RMSE = 0.778 from the GEP testing dataset to 0.650, demonstrating RF’s superior learning capability.

Additionally, the models’ stability and capacity for generalization are strengthened by the relative error measures (RAE and RRSE). GEP obtained RAE = 0.150 and RRSE = 0.180 for the STS testing data, but RF obtained lower values of 0.101 and 0.130, respectively. Additionally, RF displayed lower RAE (0.160) and RRSE (0.200) in the FS testing phase than GEP (RAE = 0.244 and RRSE = 0.231). RF’s consistently lower RAE and RRSE scores show better generalization and less variation from the observed data.

Overall, the RF model shows marginally better accuracy and resilience across both STS and FS datasets, as shown by its higher correlation coefficients and lower error indices, even though both GEP and RF models show good predictive ability and dependable generalization.

External validation of GEP and RF models

External validation outcomes enhance statistical performance metrics by assessing calibration precision and predictive dependability beyond correlation-driven model fitting. The numbers of the extra criterion utilized for model validations are shown in Table 8. Regression line slopes have been suggested to be near 1¹¹⁰. Both k and K’ must be near 1 for a model to be considered correct. For an appropriate model, the squared coefficient R² and should likewise be near to 1¹²⁶. Another R_m criterion was established by Roy et al.¹¹¹ to assess the model’s external dependability. This criterion is met when R_m is greater than 0.5.

Table 8.

Various external validation values of the GEP and RF models.

S. No	GEP			RF
	Parameter	FS	STS	FS	STS
1	K	0.97	1.00	0.84	0.99
2	K ’	1.00	0.99	1.11	1.00
3	R 2	0.97	0.99	0.91	0.98
4		0.99	0.99	0.76	0.98
5	R m	0.77	0.89	0.55	0.95

For both datasets, the GEP model meets all of these criteria. The values of K = 0.97 and K′ = 1.00 for the FS subgroup show a good linear relationship between the measured and predicted values. Similarly, K = 1.00 and K′ = 0.99 are nearly perfect for the STS dataset. With R² = 0.97 and = 0.99 for FS and R² = 0.99 and = 0.99 for STS, the correlation indicators further support the GEP model’s excellent predictive ability. Furthermore, the GEP model’s strong external predictability is demonstrated by the modified correlation coefficient R_m, which is substantially above the acceptable level, reaching 0.77 for FS and 0.89 for STS.

On the other hand, especially for the FS dataset, the RF model exhibits relatively poorer external validation performance. The FS dataset does not quite meet the validation criteria, even though the STS subset still shows acceptable values (K = 0.99 K′ = 1.00 R² = 0.98, =0.98, and R_m = 0.95). The RF model produces K = 0.84 and K′ = 1.11 for FS, which are significantly different from unity. Additionally, its correlation measurements are much lower, with R² = 0.91 and =0.76, while the R_m value falls to 0.55, just above the permissible limit.

The comparison of GEP and RF models indicates that different machine learning algorithms may demonstrate varying strengths depending on the assessment measures employed. While RF frequently provides good statistical fitting accuracy and effective ranking between projected and experimental values, GEP may improve external prediction stability due to its symbolic regression structure. The external validation results suggest that, especially for the flexural strength dataset, the RF model may have slight calibration sensitivity in absolute prediction values, while still exhibiting robust correlation performance. This behavior aligns with ensemble-based models, which are adept at capturing nonlinear interactions but may occasionally yield minor systematic deviations from unity-based calibration standards. Thus, both models should be regarded as complimentary forecasting tools rather than solely competitive alternatives, as each model offers dependable predictive capabilities from distinct validation viewpoints.

The current study utilized a multi-level validation framework, incorporating training-testing dataset comparison, external validation criteria, absolute error distribution analysis, and SHAP-based interpretability evaluation, despite the prevalent use of cross-validation for assessing model stability. This integrated method offers a thorough evaluation of model resilience, predictive uncertainty, and generalization capacity.

The assembled rubberized concrete dataset is somewhat small, and frequent cross-validation may diminish the supply of training data, thus impacting model stability. Consequently, the chosen validation technique guarantees equitable model training and dependable evaluation.

Comparing GEP and RF models with related work

As shown in Table 9, the current work’s GEP and RF models are contrasted with comparable models that were previously created in the literature for assessing the characteristics of rubberized concrete. Using an ANN model with six input parameters, David Sinkhonde and Destine Mashava et al.¹²⁷ calculated the compressive strength of rubberized concrete including brick powder. With an R² value of 0.83, the ANN model’s performance was determined to be substantially worse than that of the current study. In order to predict the 28-day compressive, split tensile, and flexural strength of rubberized concrete, Senthil Vadivel et al.²⁰ substituted fine aggregate with fine rubber (FR) and coarse aggregate with coarse rubber (CR) using XG boost, CatBoost, Extra trees regressor, and Bagging regressor algorithms. The best R² value of 0.98 was obtained using ML models with five input parameters. Additionally, Musa Adamu et al.³³ obtained an R² value of 0.99 by using the ANN algorithm to predict the CS, STS, FS, and E characteristics of rubberized concrete that included fly ash and nano-silica. Furthermore, Jingkui Zhang et al.¹⁹ created a predictive model based on ELM with six input parameters. When compared to the results of this investigation, the ELM model’s performance was determined to be lower. Additionally, when predicting the FS and STS of rubberized concrete with more input factors, the GEP and RF models in this study performed better than all models in the literature. Therefore, it can be said that different mechanical properties of rubberized concrete may be predicted with confidence using GEP and RF ML algorithms.

Table 9.

Summary of prior predictive modeling techniques for rubberized and other concrete.

S. no	Technique	Property	Best R2 value	Reference
1	Artificial Neural Network (ANN),	CS of rubbercrete	0.83	127
2	ANN, Nonlinear Regression (NLR), and Linear Regression (LR)	CS of carbon nano tube modified mortar	0.99 (ANN)	128
3	Gaussian Process Regression (GPR), Ensemble Learning (EL), Support Vector Machine Regression (SVMR), and optimized GPR, SVMR, and EL	CS of LWC	(Optimized GPR, 0.9803) (SVMR, 0.9777) (GPR, 0.9740)	129
4	Multilayer Perceptron (MLP), SVM, and Decision Tree (DT)	CS of lightweight pumice concrete	0.914	130
5	ANN, LR, NLR	CS of self-compacting concrete (SCC) increased by palm oil fuel ash (POFA)	0.996–0.999 (ANN)	131
6	ANN, SVM, Ensemble boosting tree (EBT)	UHPC CS using pozzolanic materials	0.98	132
7	ANN, LR, NLR	CS of geopolymer mortar treated with POFA	0.99 (ANN)	133
8	Gradient-Boosted Trees (GBT), RF, Tree Ensemble (TE), Extreme Gradient Boosting (XGB), Keras Neural Network (KNN), Simple Regression (SR), Probabilistic Neural Network (PNN), Multilayer Perceptron (MLP), and Linear Regression (LR)	CS of LWC	0.90	134
9	Support Vector Machine (SVM), Artificial Neural Network (ANN), DT, GPR, and XGBoost.	Lightweight aggregate concrete (LWAC)	0.99	135
10	RF and GEP	CS of HSC	0.96	106
11	XG boost, Catboat, Extra trees regressor, Bagging regressor	CS, STS, FS	0.98	20
12	ANN	CS, STS, FS, E	0.99	33
13	ANN	CS, STS, FS, E	0.99	33
14	ELM	CS, STS, FS	0.96	19
15	REG, CART, CHAID, ANN, SVM	Slump, CS	0.85	136
16	SVM, GRP	CS	0.99	35
17	LR, XGBOOST	CS, STS	0.99	137
18	ELM	CS, STS, FS	0.96	19
19	REG, CART, CHAID, ANN, SVM	Slump, CS	0.85	136
20	SVM, GRP	CS	0.99	35
21	LR, XGBOOST	CS, STS	0.99	137
22	GEP	FS, STS	0.97	Present work
23	RF	FS, STS	0.99	Present work

SHAP analysis

A post-hoc explanation is required when models get more complicated. We used SHAP explanations to comprehend the fundamental logic of forecasts and how each variable affects the outcome. The relevance of the input parameters and their influence on the intended output are provided by SHAP interpretation, as will be covered in the following section.

This work utilized SHAP analysis through the model-agnostic KernelSHAP method, which is appropriate for understanding nonlinear and symbolic regression models, including GEP, as well as ensemble models like RF. KernelSHAP assesses the marginal contribution of each input variable by creating perturbed samples around observed data points and calculating weighted feature contributions to model predictions. The background dataset utilized for SHAP computation was derived from representative samples of the training dataset to maintain the statistical distribution and correlation structure of the input variables. This methodology guarantees that SHAP explanations accurately represent plausible combinations of mixture parameters. To assess the reliability of SHAP interpretations, SHAP values were analyzed over multiple sampling iterations. The relative ranking of feature relevance was similar across many evaluations, affirming the stability and dependability of the SHAP-based interpretability findings.

Importance of parameters

Figure 29a displays the mean absolute SHAP values for the FS GEP and RF models. The fine aggregate has a larger mean SHAP value (about 2.54) than the coarse aggregate and crumb rubber, which have mean SHAP values of roughly 2.15 and 0.5, respectively, as seen in Fig. 29a. The SHAP values of the remaining input variables are lower, suggesting that they have less of an impact on flexural strength. Figure 29b displays the mean SHAP values for the GEP and RF models that predict split tensile strength (STS). The STS of rubberized concrete is mostly determined by the superplasticizer and crumb rubber, as Fig. 29b illustrates, with mean SHAP values about 0.4 and 0.3, respectively. With a SHAP value of about 0.24, W/C became the dominant component. The STS of rubberized concrete is largely predicted by age, chipped rubber, and fine aggregate.

Fig. 29.

Input parameters importance: a FS, b STS.

Influence of parameter

Examining the SHAP values of those parameters during prediction is necessary to gain a better understanding of how the significant variable affected model outputs. The mean of the parameter SHAP values illustrates how the input variable affects the model’s output. In Fig. 30a, b, which shows the summary plots used to examine the effects of the input feature characteristics, the dot represents the Shapley value of a variable and a unique feature in the dataset. It shows the distribution of SHAP numbers for a particular feature along with the pattern of the related parameter. The individual SHAP value is displayed on the x-axis of the summary graphic, and the parameters utilized as research inputs and importance are listed in descending order on the y-axis. Database samples are represented by the dots, which range in size from minor (blue) to more substantial (red). The x-axis displays the value of prediction with respect to SHAP values for each parameter as the input parameter magnitude changes (from blue to red). Every parameter that has a high feature score is beneficial to the finished product. But as the input parameter has a more negative effect on the output, the feature value drops.

Fig. 30.

Influence plot of SHAP input parameters: a FS, b STS.

The FS SHAP summary plot, which illustrates how different input variables impact FS, is shown in Fig. 30a. Fine aggregate has the greatest impact, with SHAP values ranging from about − 2.5 to + 27.5. It also significantly improves FS when present in larger quantities. Furthermore, coarse aggregate is crucial; its SHAP values can reach + 7, meaning that adding more enhances FS. However, most of the chipped and crumb rubber have negative SHAP values, usually less than 0, indicating that their rise reduces FS. The W/C ratio shows an inverse tendency, with higher W/C having a negative impact on FS, with SHAP values primarily ranging between − 2.5 and 2.5. Superplasticizer and age have relatively little influence; SHAP values remain close to zero, indicating somewhat positive effects. These findings indicate that increasing aggregate content, decreasing rubber inclusion, and controlling the W/C ratio are necessary to improve the flexural performance of rubberized concrete.

Additionally, a comparable STS summary figure is shown in Fig. 30b. The red dots moving in the direction of positive SHAP values (+ 0.75) make it evident that a superplasticizer raises STS. The split tensile strength of rubberized concrete is more negatively impacted by the W/C ratio; a higher W/C results in a lower tensile strength, as shown by the red dots stretching towards negative SHAP values (− 0.75). Furthermore, with SHAP values ranging from − 0.50 to + 0.50, coarse aggregate (CA) and fine aggregate (FA) exhibit both positive and negative contributions, indicating that their influence varies according to their percentage in the mix. With SHAP values as low as −0.75, crumb rubber has a significant detrimental impact, suggesting that too much rubber lowers tensile strength. Age is important; negative SHAP values up to −0.25 indicate a more significant unfavorable effect on STS. Interestingly, STS is only slightly impacted by chipped rubber; SHAP values are mostly in the zero range, indicating that the model’s prediction is not greatly impacted. These findings highlight how important it is to maximize W/C and rubber content in order to enhance rubberized concrete’s tensile performance.

Conclusion

This work demonstrates the use of GEP and RF approaches to develop models for predicting the FS and STS of rubberized concrete and to conduct a comparative study of both the models. The offered models are empirical and derived from a large database containing numerous parameters gathered from experimental research described in the literature. A thorough database was created using published literature, with 112 outcomes for each FS and STS of rubberized concrete. This study looks at the seven most important input variables: CA, FA, CHR, CR, SP, W/C, and age. Various statistical metrics, including R, MAE, RMSE, RAE, and RRSE, were used to assess the efficacy of the provided models, which were also validated using external standards. In addition, SHAP analysis was used to determine the importance and impact of input parameters on response characteristics. Finally, the created models were compared to earlier findings. The main results can be stated here.

• Superior predictive accuracy of RF over GEP: For both FS and STS, the RF models consistently produced greater accuracy metrics than the GEP models. For instance, RF testing data achieved coefficients of determination in the range of 0.97–0.98, while GEP was usually between 0.89 and 0.91.

• Detailed statistical and error analysis: Statistical indicators (R, MAE, RMSE, RAE, and RRSE) evaluation confirmed the superiority of RF, which produced much lower errors) across all strength parameters compared to GEP. While GEP occasionally showed reasonable accuracy, its error margins were several times higher than those of RF, reinforcing the reliability of RF in predicting concrete strength properties.

• External validation of models: Both GEP and RF models exhibited satisfactory external prediction efficacy. The GEP model met all external validation criteria for both FS and STS datasets, whereas the RF model demonstrated robust external validation performance for STS and acceptable, though comparatively poorer external validation metrics for FS. The findings affirm that both models exhibit dependable generalization capabilities, while their prediction accuracy differ based on the mechanical feature assessed.

• Comparative validation: GEP and RF models outperform other ML algorithms in terms of prediction accuracy. In contrast to earlier research employing other ML algorithms like ANN, SVM, DT, KNN, PNN, MLP, GPR or hybrid tree-based models, the current GEP and RF techniques, especially RF provided equivalent or superior accuracy.

• SHAP Analysis: The SHAP analysis showed that a small number of governing mix parameters control the mechanical properties of rubberized concrete. Higher W/C and CR and CHR have a negative impact on FS, but fine and coarse aggregate contents have a beneficial impact. On the other hand, STS is primarily determined by the W/C, SP dosage, and CR; greater W/C ratios and excessive rubber content lower tensile performance. Overall, the SHAP results show that improving the mechanical performance of rubberized concrete requires careful control over aggregate proportions, rubber inclusion, and W/C.

• Engineering and practical implications: RF’s exceptional performance demonstrates its capacity to comprehend rubberized concrete mechanical behavior while retaining high prediction precision, which is especially useful for optimizing sustainable design. By measuring the impact of CR and CHR combinations on mechanical performance, the developed GEP and especially RF models can assist mix proportioning decisions and minimize cement use and experimental effort. The suggested models can accurately forecast the strength of concrete containing WTR, making them useful in practical applications. These solutions will reduce construction costs and time by offering early forecasts of rubberized concrete strength properties.

This research aims to repurpose WTR to create green concrete, promoting sustainable construction and environmental safety while reducing disposal costs, greenhouse gas emissions, and construction charges. Furthermore, this study is novel in creating a unified multi-source dataset and systematically comparing GEP and RF models under identical settings, offering a systematic alternative to time-consuming and resource-intensive laboratory testing. Additionally, combining SHAP-based interpretability with multi-level validation improves knowledge of parameter influence and predictive reliability. The predictive framework developed herein can be directly applied in practice to optimize mix design, contributing to the development of sustainable, high-performance concretes that align with global efforts to reduce carbon emissions in the construction industry.

Study limitations and future research direction

• Heterogeneity of Datasets: Despite the robust predictive efficacy of the established Gene Expression Programming (GEP) and Random Forest (RF) models, specific limitations must be recognized. The assembled dataset was derived from previously published experimental research, which inherently includes diversity in specimen preparation, curing conditions, source of materials, and testing methodologies. Despite meticulous data screening and standardization to reduce inconsistencies, a degree of dataset variability may still affect model generalization.

• Scope of Model Applicability: The prediction models are constrained by the range of input parameters reflected in the assembled experimental database. The models exhibit optimal reliability when utilized within the investigated ranges of mixture constituents, encompassing CA, FA, CHR, CR, SP, W/C, and curing age. The utilization of the models beyond these parameter ranges should be undertaken with caution, as prediction efficacy can decrease beyond the training data area.

• Exclusion of Supplementary Influential Variables: Several potentially significant parameters, including rubber particle gradation, rubber surface treatment methods, air content, density variations, and characteristics of supplementary cementitious materials, were not consistently documented in the source literature and thus could not be incorporated into the modeling framework. Integrating these characteristics in subsequent investigations may enhance predictive accuracy and model generalization.

• Restricted Dataset Dimensions: Despite the dataset being assembled from several experimental investigations and meticulously vetted, the overall amount of data points is still quite constrained due to the limited availability of experimentally reported mechanical properties of rubberized concrete. Augmenting the dataset by forthcoming experimental inquiries would further strengthen model robustness and forecast accuracy.

• Constraints of the Validation Strategy: The constructed models underwent validation by independent training-testing data segmentation, external validation standards, and error distribution assessment. A single randomized 70/30 dataset partition was used as the basis for the validation outcomes in this study, which could introduce sensitivity to the particular data split employed for developing the model. Nonetheless, resampling strategies based on cross-validation were not utilized. While many complementary validation metrics were utilized to assess model performance and predicted reliability, repetitive data partitioning or resampling validation methods could offer further statistical validation of model stability over diverse dataset splits. Subsequent research utilizing k-fold cross-validation or repeated resampling techniques may yield further understanding of model stability across various dataset partitions.

• Interpretability of Models and Predictive Representation: The GEP model offers symbolic predictive linkages, but the Random Forest model functions as an ensemble learning technique that does not yield explicit closed-form predictive equations. Thus, the RF model offers elevated predictive accuracy instead of streamlined analytical design formulations.

• Uncertainty Quantification and Robustness Evaluation: This study did not use bootstrap confidence intervals or repeated resampling to quantify uncertainty. Instead, independent testing datasets, external validation criteria, and statistical error distribution analysis were used to evaluate predictive reliability. The consistency between training and testing performances and external validation indicators suggests stable model behavior, but future studies should use bootstrapped confidence intervals, repeated cross-validation, and hyperparameter configuration sensitivity analysis to provide probabilistic performance bounds and improve model reliability assessment.

• Future studies ought to incorporate more extensive experimental datasets to enhance model robustness and generality. Furthermore, it is advisable to conduct extensive comparative studies that include various machine learning algorithms such as Multi-Expression Programming (MEP), artificial neural networks, fuzzy logic systems, advanced ensemble learning techniques, and hybrid modeling approaches to facilitate thorough benchmarking of the predictive performance of rubberized concrete mechanical properties. Subsequent investigations should also assess model robustness using repeated random data partitioning or resampling-based validation methods to further validate prediction stability. Furthermore, future research should integrate advanced statistical techniques for influence and uncertainty assessment, including leverage diagnostics, Cook’s distance, and hat matrix-based residual influence analysis, to evaluate the sensitivity of model predictions to influential data points and improve predictive reliability.

Supplementary Information

Below is the link to the electronic supplementary material.

Abbreviations

GEP

Gene expression programming

RF

Random forest

STS

Split tensile strength

FS

Flexural strength

CA

Coarse aggregate

FA

Fine aggregate

CHR

Chipped rubber

CR

Crumb rubber

SP

Superplasticizer

R

Correlation coefficient

ML

Machine learning

RMSE

Root mean squared error

RSE

Relative standard error

RAE

Relative absolute error

MAE

Mean absolute error

MSE

Mean squared error

SHAP

SHapley additive explanations

Author contributions

Muhammad Sheraz: Supervision, Conceptualization, Visualization, Methodology, Validation, Writing - original draft, Writing - review & editing, Project administration. Muhammad Talha: Software, Data curation, Formal analysis. Muhammad Alam: Resources, Data curation, Software. Muhammad Farhan: Writing - review & editing. Zia Ul Haq and Muhammad Rashid: Visualization and formal analysis. Hisham Alabduljabbar and Furqan Ahmad: Funding, and Data curation.

Funding

This research received no specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data availability

All data generated or analysed during this study are included in this published article [and its supplementary information files].

Declarations

Competing interests

The authors declare no competing interests.