Backpropagation Neural Networks (BPNN) are a simple, effective type of feedforward network [66][67], made up of an input layer, one or more hidden layers, and an output layer. Each neuron computes a weighted sum of its inputs, adds a bias, and passes the result through an activation function, as shown in Equation (1) [68].

where the values at the nodes in the previous layer (i) and the layer currently active (j) are illustrated by x i and y j , respectively. The total nodal values obtained from the previous layer are depicted by n. The network’s biases and weights are denoted by b j and w i j .

For this study, a tangential hyperbolic function was utilized based on its ability to produce more precise outcomes. The function’s range is bounded by −1 and 1, and its mathematical expression is given in Equation (2) as:

The network’s final output ( y j ) is calculated by feeding the computed internal network value (net) through an activation function (f). During the training phase, weights and biases values are repeatedly adjusted to minimize the error between the predicted network values and the actual target values This mechanism is essentially a minimization problem [69]. For this study, the BPNN model was built and executed using MATLAB 2023a software. For a deeper look at how the BPNN works, comprehensive details on the BPNN itself are available in resources like the work presented by Jain et al. [70].

Support Vector Regression (SVR) is a variant of the Support Vector Machine adapted for regression tasks. As Zhang and O’Donnell [71] note, its structure is not fixed in advance—the input vectors that define it are determined during training. For a training set with input vector x and target f, the SVR estimator f(x) is given in Equation (3), where w is the weight vector, b the bias, and K(x) a non-linear basis function mapping the input into a higher-dimensional space.

where weight vector w = ( w 1 , w 2 , w 3 , ⋯ , w z ) , the bias term is b,and the basis function vector κ ( x ) = ( κ 1 ( x 1 ) , ⋯ , κ ( x z ) ) referring to a series of non-linear changes. The non-linear basis function is used to convert the R-dimensional input vector from a low-dimensional phase to a high-dimensional space κ ( x ) . Support Vector Regression (SVR) aims to identify the optimal separating hyperplane that enhances the model’s capacity to learn from data and effectively address a given problem [72]. Solving Equation (3) necessitates the use of Equation (4), while adhering to the constraints outlined in Equation (5). It is important to highlight that the structure of an SVR is not fixed in advance; rather, the input variables that contribute to the model are typically selected throughout the training process.

where the positive trade-off parameters, ɛ and M, regulate the measured deviation scale in maximization, while the slack factors ξ and ξ ∗ resolve the errors using the expression in Equation (5).

Support Vector Regression (SVR) relies on kernel functions, which function similarly to activation functions by implicitly projecting input data into a higher-dimensional space to enable better separation of classes. To construct accurate and efficient models, commonly used SVR kernels were evaluated, including the linear kernel, Gaussian kernel, and polynomial kernel functions [73]. These are represented by the following mathematical expressions, respectively.

where x, x ¯ represents the input feature vectors, x T ⋅ x ¯ is the product of the vectors γis the scaling factor, c is the coefficient term, and − x + x ¯ refers to the Euclidean distance between two vectors.

Metaheuristic algorithms draw on natural behaviors to solve complex optimization problems [52]. The Zebra Optimization Algorithm (ZOA) used here is a swarm-based method known for fast convergence, modelled on how zebra herds forage and evade predators together, balancing broad exploration with local refinement.

The Grey Wolf Optimizer (GWO) mimics the social hierarchy and hunting behavior of grey wolves [53] and is widely used across engineering and machine learning [74]. It identifies the three best solutions (α, β, δ) as leaders that guide the remaining wolves toward the global optimum, through three phases: encircling, hunting, and attacking the prey.

Bayesian Optimization (BO) uses a probabilistic surrogate model, typically a Gaussian Process, to approximate an expensive or complex objective function [75], rather than evaluating it directly many times. The surrogate gives a mean estimate and an uncertainty at each point, and BO uses this to choose where to sample next by maximizing an acquisition function, balancing exploration of new regions against exploitation of known good ones [75][76].

The complete dataset was randomly divided into a training dataset (80%) and a validation dataset (20%). The training dataset was used for model development and hyperparameter optimization. During model development, five-fold cross-validation was performed exclusively within the training dataset to identify the optimal model configurations and hyperparameters. In each fold, normalization parameters were computed (using Equation 9) only from the corresponding training subset and subsequently applied to the associated validation fold without refitting, thereby preventing information leakage. After model development, the final models were evaluated using the independent validation dataset, which was not involved in the training or hyperparameter optimization process [77].

where, the normalized data point of variable s i is represented by N i , the maximum scores are defined by s max , and the minimum scores of the measured samples are indicated by s min .

The baseline BPNN and SVR models were developed by evaluating multiple candidate learning strategies and selecting the best-performing configurations. For the BPNN, three training algorithms, namely Levenberg-Marquardt (LM), Scaled Conjugate Gradient (SCG), and Bayesian Regularization (BR), were investigated. The number of hidden neurons was varied from 1 to 150 to determine the optimal network architecture. Model performance was evaluated using the Mean Absolute Error (MAE), Mean Square Error (MSE), correlation coefficient (R), and coefficient of determination (R²). The training algorithm and network architecture that produced the lowest prediction errors and highest goodness-of-fit metrics on the validation dataset were selected as the baseline BPNN model.

For the SVR, three kernel functions, namely Linear, Polynomial, and Gaussian, were evaluated. The kernel yielding the best predictive performance was selected as the baseline model. The principal SVR hyperparameters, including the regularization parameter (C), kernel parameter (σ), and insensitive loss parameter (ε), were subsequently considered for optimization to further improve predictive performance.

The selected baseline BPNN and SVR models were subsequently integrated with the Zebra Optimization Algorithm (ZOA), Grey Wolf Optimizer (GWO), and Bayesian Optimization (BO) to improve predictive accuracy through optimal hyperparameter selection. For the BPNN, the optimization algorithms searched for the optimal network weights and biases, whereas for the SVR they optimized the regularization parameter (C), kernel parameter (σ), and insensitive loss parameter (ε). The objective function was defined as the mean squared error (MSE) computed over the training folds.

The optimization variables were normalized within the interval [−1, 1] to maintain a compact search space, improve computational efficiency, and achieve a balanced trade-off between exploration and exploitation. The preliminary experiments were conducted to identify ranges that consistently produced stable convergence while avoiding unnecessary computational cost. For ZOA, the population size varied from 10 to 120 zebras with iteration numbers ranging from 100 to 1500. The swapping probability (P_s) and exploration constant (C) were fixed at 0.5. For GWO, the population size ranged from 1 to 50 wolves, while the maximum number of iterations varied between 10 and 150 for the BPNN and between 10 and 200 for the SVR. Bayesian Optimization employed a Gaussian Process surrogate model with a Matérn kernel, using 5 to 50 initial sampling points and 50 to 400 optimization iterations. Expected Improvement and Upper Confidence Bound acquisition functions guided the search for the optimal solution.

Each optimization algorithm iteratively updated candidate solutions until the maximum number of iterations or the convergence criterion was satisfied. The optimal parameter set identified by each optimization algorithm was then used to retrain the corresponding BPNN or SVR model. The optimized models were subsequently evaluated using the validation dataset, while the training dataset was used for model development and optimization. Figure 4 presents the overall model development and optimization framework. All simulations were performed on an HP laptop equipped with an AMD Ryzen 5 processor and 12 GB of RAM.

To enhance the interpretability of the developed machine learning framework, SHapley Additive exPlanations (SHAP) [78] were employed to quantify the contribution of each input variable to model predictions. SHAP is a model-agnostic interpretability technique derived from cooperative game theory that assigns each feature a Shapley value, representing its average marginal contribution to the prediction across all possible combinations of input variables. Unlike conventional feature importance measures, SHAP provides both the magnitude and direction of each feature’s influence, thereby offering consistent and locally accurate explanations of model behavior.

Following model development and performance evaluation, SHAP analysis was performed on the model that demonstrated the highest predictive performance. Global feature importance was quantified using the mean absolute SHAP values computed from all observations in the validation dataset, allowing the relative influence of each blasting parameter on fragment size prediction to be ranked. In addition, SHAP summary plots were generated to illustrate how variations in individual input variables influenced the predicted fragmentation outcomes.

Figure 4.Conceptual framework of BPNN and SVR models integrated with optimization algorithms for blast-induced rock fragmentation prediction.

Model performance was assessed using MAE, MSE, correlation coefficient (R), and coefficient of determination (R²), which together capture prediction accuracy, error magnitude, and fit. Lower MAE and MSE indicate better accuracy, while higher R and R² indicate stronger agreement between predicted and observed values. These metrics are defined in Equations. (10)-(13) [79]-[81], where OF and PF denote the observed and predicted fragment sizes, their subscripted averages denote the dataset means, and n is the total number of samples.

In the statistical equations, OF_i and PF_idenote the observed and predicted fragment sizes for the i-th sample, respectively. The terms OF_av and PF_av represent the mean values of the observed and predicted fragment sizes across the entire dataset. The total number of samples used for model development is designated by n.

Table 3 shows that all three BPNN training algorithms exhibited good generalization, with training performance consistently outperforming validation performance by a small margin. This pattern indicates effective learning while maintaining good generalization to unseen data, with no evidence of overfitting. Among the evaluated algorithms, Bayesian Regularization achieved the best overall performance, recording the lowest training MAE (0.362) and MSE (0.781), together with the highest R (0.836) and R² (0.698). Its validation performance remained consistently strong (MAE = 0.389, MSE = 0.853, R = 0.812, and R² = 0.659), demonstrating superior predictive capability and robust generalization compared with the Levenberg-Marquardt and Scaled Conjugate Gradient algorithms. Consequently, Bayesian Regularization was selected as the baseline BPNN training algorithm for subsequent hybrid optimization.

Table 3. Statistical analysis results of the training algorithms modelled in the BPNN.

Table 4shows a similar pattern for the SVR models, with all three kernel functions exhibiting slightly better performance during training than during validation. The small differences between the two phases indicate good generalization ability while suggesting that the models did not overfit the training data. Among the evaluated kernels, the Gaussian kernel demonstrated the best overall performance, achieving the lowest training MAE (2.035) and MSE (4.612), together with the highest R (0.615) and R² (0.378). Its validation performance remained comparably strong (MAE = 2.127, MSE = 4.831, R = 0.592, and R² = 0.350), confirming its superior predictive capability and robustness compared with the Linear and Polynomial kernels.

Table 4. Statistical performance of candidate SVR kernel functions.

Based on these results, Bayesian Regularization and the Gaussian kernel were selected as the baseline BPNN training algorithm and SVR kernel, respectively. These baseline models were subsequently optimized using the Zebra Optimization Algorithm (ZOA), Grey Wolf Optimizer (GWO), and Bayesian Optimization (BO). The optimization variables were bounded within the interval [−1, 1] to maintain a compact search space, improve computational efficiency, and achieve a balanced trade-off between exploration and exploitation during the optimization process. The reported values correspond to the optimal solution returned by each optimization algorithm. Table 5 presents the baseline model architectures together with the optimal hyperparameter settings for each optimization algorithm.

Table 5.Baseline model architecture and optimal hyperparameter configurations for the optimization algorithms integrated with the BPNN and SVR models.

Table 6 compares the training and validation performance of the baseline models and their optimized hybrid counterparts.

Table 6.Training and validation performance of the baseline and optimized models.

Among all the evaluated models, ZOA-BPNN demonstrated the best predictive performance, achieving a validation MAE of 0.021, MSE of 0.046, R of 0.992, and R² of 0.984. These results indicate that the model explained approximately 98.4% of the variability in rock fragmentation, highlighting its excellent predictive accuracy and strong generalization capability. The superior performance of ZOA-BPNN can be attributed to the complementary strengths of the Zebra Optimization Algorithm and the BPNN. Conventional BPNN training relies on gradient-based optimization, which is susceptible to premature convergence and local optima. In contrast, ZOA combines effective global exploration with local exploitation, enabling a more comprehensive search of the network’s weight and bias space. By optimizing the initial weights and biases before training, ZOA provided a more favorable starting point for the learning process, thereby reducing convergence to poor local solutions, improving model stability, and enhancing prediction accuracy. The validation coefficient of determination (R²) results presented in Figure 5 further support these findings, with ZOA-BPNN exhibiting the highest R² value among all the developed models. GWO-BPNN and BO-BPNN also outperformed the standalone BPNN, increasing the coefficient of determination from 0.659 to 0.776 and 0.849, respectively. These improvements demonstrate the effectiveness of metaheuristic optimization in enhancing the predictive capability of neural network models, although neither algorithm matched the optimization efficiency achieved by ZOA. Among the support vector regression models, the standalone SVR produced the lowest predictive performance, with a validation MAE of 2.127 and an R² of 0.350. Nevertheless, integrating SVR with metaheuristic algorithms substantially improved its performance. ZOA-SVR emerged as the strongest SVR-based model, achieving a validation MAE of 0.033 and an R² of 0.837. This result further demonstrates the robustness and adaptability of ZOA, indicating that its optimization capability extends beyond neural networks to kernel-based learning algorithms.

Figure 5.Comparison of coefficient of determination results of the applied models.

To provide a comprehensive comparison of the developed models, the validation results were ranked using the four evaluation metrics, namely MAE, MSE, R, and R². For MAE and MSE, lower values indicate superior predictive performance and were therefore assigned higher rankings. Conversely, for R and R², higher values indicate stronger predictive capability and received higher rankings. Each model was ranked from 1 (best) to 8 (worst) for each metric, and the individual rankings were summed to obtain an overall score. Consequently, a lower total score represents better overall model performance. This ranking approach enables simultaneous consideration of prediction accuracy, goodness-of-fit, and model reliability, thereby facilitating an objective comparison of the standalone and hybrid intelligent models. Table 7 ranks all eight models across the four metrics.

Table 7. Overall ranking of the developed intelligent models based on validation performance.

The ranking confirms that ZOA-BPNN was the best-performing model, achieving first place across all four evaluation metrics with the lowest overall score of 4. ZOA-SVR ranked second, followed by BO-BPNN, demonstrating that the Zebra Optimization Algorithm consistently produced the greatest performance improvements irrespective of the underlying learning algorithm. GWO-BPNN also outperformed the standalone BPNN, while BO-SVR and GWO-SVR showed substantial improvements over the conventional SVR model. Among the baseline models, BPNN ranked fifth and considerably outperformed the standalone SVR, which placed last. These findings demonstrate that integrating metaheuristic optimization with machine learning models substantially enhances predictive performance, although the magnitude of improvement depends on the compatibility between the optimization algorithm and the underlying predictive model.

Following the comparative evaluation of all developed models, the ZOA-BPNN model demonstrated the highest predictive performance and was therefore selected for SHAP-based interpretation. Figure 6 presents the SHAP summary plot, which ranks the blasting parameters according to their mean absolute SHAP values and illustrates both the magnitude and direction of each variable’s influence on the predicted fragment size.

Powder factor emerged as the most influential predictor by a substantial margin, exhibiting the widest range of SHAP values (approximately −15 to +25). Higher powder factor values consistently shifted the predictions towards smaller fragment sizes, confirming its dominant role in controlling blast fragmentation. This finding is consistent with established blasting theory, where increasing the explosive energy per unit rock volume generally enhances rock breakage and produces finer fragmentation.

The SHAP feature importance plot (Figure 7) further corroborates these findings, ranking charge per hole and charge length as the second and third most influential variables, with mean absolute SHAP values of approximately 0.325 and 0.230, respectively. Bench height followed with a mean absolute SHAP value of approximately 0.194, while the remaining variables contributed comparatively less to the model predictions. Collectively, these results indicate that explosive energy distribution and blast geometry are the principal factors governing fragmentation performance.

It is important to note that, in this study, the blast design variables exhibit strong correlations, as demonstrated by the Pearson and Spearman analyses. In particular, powder factor, charge per hole, charge length, bench height, and the stemming variables are operationally related and jointly define the distribution of explosive energy within a blast. Consequently, their SHAP values should be interpreted within the context of these interdependent blasting parameters rather than as completely independent effects. Despite these correlations, the consistent identification of powder factor as the dominant predictor reinforces its critical influence on fragment size prediction and supports the robustness and engineering reliability of the developed ZOA-BPNN model.

Figure 6. SHAP summary plot showing global feature influence.

Figure 7.Mean absolute SHAP value bar chart for feature importance ranking.