Superior Performance of Artificial Neural Networks for Optimizing Concrete Mix Design Using Local Materials in Ayacucho, Peru (2024)

2.1 Experimental design and dataset

This study developed predictive models for concrete mix design using ANNs and MLRMs. The dataset comprised 806 experimental records, which is considered adequate for ANN training, validation, and testing when the input dimensionality (18 variables) and network complexity (18-11-12-4 architecture) are taken into account. This sample size ensures sufficient representation of variability in local material properties while reducing the risk of overfitting.

The concrete mixes covered three standard compressive strength classes widely used in Ayacucho’s construction practice: 140, 175, and 210 kg/cm². All materials were sourced from two representative local quarries La Moderna and Chillico chosen for their distinct mineralogical compositions and granulometric characteristics. Fine aggregates from La Moderna typically exhibited a fineness modulus between 2.86–3.53, whereas Chillico sands ranged from 2.58–3.89. Coarse aggregates from both quarries had nominal maximum sizes of ½, ¾″, and 1″ and bulk specific gravities between 2.35–2.88.

Data were collected from validated experimental procedures documented in undergraduate theses from the Universidad Nacional de San Cristóbal de Huamanga (UNSCH) and complementary academic reports. Each record included 18 input variables describing physical, granulometric, and mechanical properties of the constituent materials, as well as mix design parameters. Two strength variables were recorded: (1). Specified design strength (f′c): the target compressive strength established in the mix design process, typically at 28 days. (2). Measured compressive strength (fc): the experimentally obtained value from laboratory-cured specimens, which may differ from f′c due to material variability, curing conditions, and testing dispersion.

Table 1. Input and output variables used in the modeling of concrete mix design

The four output variables corresponded to the dosages, expressed per cubic meter, of cement, fine aggregate, coarse aggregate, and water.

Table 1 presents the input and output variables with descriptions, units, and basic statistics (mean ± standard deviation and observed ranges), allowing reproducibility and characterization of the dataset.

2.2 ANN model

A custom-built NeuroMAT Toolbox v1.0, developed by the research team in MATLAB R2020a, was used to implement the predictive ANN model, providing flexible configuration, real-time performance monitoring, and reproducible, scalable concrete mix design tailored to local conditions.

2.2.1 ANN Architecture and configuration

The optimal network architecture was selected after evaluating 25 MLP feedforward configurations. Considering multi-metric performance (MSE, RMSE, MAE, R, R²) and training efficiency, the final topology (18-11-12-4; Figure 1) comprised 18 input neurons (physical, granulometric, and mechanical parameters; Table 1), two hidden layers with 11 and 12 neurons, and 4 output neurons: Cement (C), Fine Aggregate (AF), Coarse Aggregate (AG), and Water (A).

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Figure 1. Architecture of the optimized ANN model (18-11-12-4)

All layers employed the hyperbolic tangent sigmoid activation (tansig), as shown in Eq. (1). Its bounded [−1, 1] range ensured compatibility with normalized data, avoided saturation beyond the scaling range, and enabled smoother convergence during regression of continuous targets, providing advantages over ReLU in stability and learning efficiency for this dataset [15-17].

$f(x)=\frac{2}{1+e^{-2 x}}-1$           (1)

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Figure 2. Training performance (MSE vs. epochs) and stopping at epoch 169

Compared to simpler architectures (e.g., 18-8-4 and 18-10-4), the 18-11-12-4 network (Figure 1) achieved lower errors and faster convergence, completing training in 4.577 seconds over 169 epochs (Figure 2), while avoiding overfitting.

2.2.2 Data preprocessing and normalization

Prior to training, data underwent two preprocessing steps:

(1). Outlier removal: A modified Z-score method was applied univariately to each of the 18 inputs and 4 outputs, using a threshold equivalent to a standard z-score > 3, to remove extreme values while preserving representative variability. Multivariate outlier detection was additionally performed using Mahalanobis distance.

(2). Normalization: All variables were scaled to the range [−1, 1] using MATLAB’s mapminmax function:

[X_norm, X_settings] = mapminmax(X, -1, 1);

[Y_norm, Y_settings] = mapminmax(Y, -1, 1);

(3). The [−1, 1] range was selected over [0, 1] because the tansig activation function outputs within this interval, enabling faster convergence, symmetric gradient propagation, and reduced bias shift during training.

New predictions were also automatically normalized and reversed:

Xnew_norm = mapminmax('apply', X_new, X_settings);

Ypred_norm = net(Xnew_norm);

Ypred = mapminmax('reverse', Ypred_norm, Y_settings);

Data were randomly split into mutually exclusive subsets:

•70% (564 samples) for training the model,

•15% (121 samples) for validation (early stopping and generalization assessment),

•15% (121 samples) for final testing of model performance.

All statistical analyses and modeling tasks were performed using MATLAB R2020a and RStudio 2024.12.0.

2.2.3 Training algorithm and parameters

The ANN was trained using the Levenberg-Marquardt (trainlm) backpropagation algorithm, ideal for small to medium-sized regression problems due to its fast convergence properties [18-20]. The learning process minimized the MSE between predicted and actual values using the following weight update rule:

$W_{\text {new }}=W_{\text {current }}-\left(J^T J+\mu I\right)^{-1} J^T E$             (2)

where, J: Jacobian matrix of partial derivatives, E: error vector, μ: damping parameter, I: identity matrix.

Training parameters included:

•Max. epochs: 1000,

•Performance goal: MSE < 0.001,

•Early stopping: Based on validation loss to avoid overfitting.

The final selected model converged in 169 epochs, reaching a minimum validation MSE of 6.7179×10⁻⁵, with training completed in 4.577 s.

2.2.4 Model evaluation and selection

Performance was assessed using MSE, RMSE, MAE, the correlation coefficient (R), and the coefficient of determination (R²). The selected network, saved as Mi_Red_23_30-Apr-2025_12-50-18, achieved the lowest prediction errors and the highest correlation among all candidates, with results summarized in Table 2.

Table 2. Selection of the RNA model with optimal number of neurons

Notes: The selection was based on a multi-metric evaluation framework. As summarized in Table 2, the chosen model achieved extremely low prediction errors, high correlation, and rapid convergence.

2.2.5 External validation with experimental mixes

The external validation of the artificial neural network (ANN) model Mi_Red_23_30-Apr-2025_12-50-18 was carried out using fifteen concrete mix designs that were not included in the training phase and were developed according to the ACI 221.1 procedure. These mixes were prepared with aggregates sourced from the La Moderna and Chillico quarries, with nominal compressive strengths of 140, 175, and 210 kg/cm². A total of 135 cylindrical specimens (6″×12″) were molded and cured following the guidelines of NTP 339.033:2021 and ASTM C31/C31M-2021. All specimens were stored in a curing chamber maintained at 20℃ ± 2℃ with a relative humidity above 95%, ensuring uniform curing conditions.

Table 3 provides a detailed description of the experimental datasets used in this validation stage. The data were subsequently processed by the neural network based on the previously learned weights, yielding predictions for the four main mixture components: cement, fine aggregate, coarse aggregate, and water. The validation procedure was implemented using the graphical interface of the NeuroMAT Toolbox v1.0, where the corresponding independent variables were entered. Finally, the model predictions are presented in Table 4, demonstrating the ANN’s predictive capacity when applied to experimental mixtures independent from the training dataset.

Table 3. Experimental data for the validation of the RNA model

Table 4. Mixture dosages predicted by the RNA model

Table 5. Percentage Error (PE) and Mean Squared Error (MSE) for ANN predictions compared to experimental dosages

Table 5 shows that the ANN model achieves a high level of accuracy, with both absolute and percentage errors generally low. The coarse aggregate registers the highest percentage error (4.02%), while the overall average error across all components remains below 1.5%. The higher deviation observed for coarse aggregate can be attributed to the intrinsic variability in particle packing and gradation within the experimental dataset, as well as the underrepresentation of certain high-strength mixes in which coarse aggregate proportions deviate more markedly from the mean. Regarding the MSE, the largest value also corresponds to coarse aggregate (220.10), followed by fine aggregate (92.01), although both remain within acceptable tolerance limits, confirming the model’s robustness despite these localized variations.

Figure 3 compares the experimental values (ACI 211.1) with those estimated by the ANN for cement, fine aggregate, coarse aggregate, and water. The blue line represents the experimental data, while the red dashed line corresponds to the model predictions. An adequate fit is observed between both curves, particularly for cement and water, with minor discrepancies in fine and coarse aggregates. These results confirm the accuracy and predictive capability of the ANN in the efficient design of concrete mixes.

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Figure 3. Experimental vs. predicted concrete mix components (ANN)

2.3 MLRM

2.3.1 Model specification and estimation

To benchmark the predictive capacity of the ANN, an MLRM was implemented using the same 18 input variables and 4 output variables. This method captures linear dependencies between predictors and dosage components. Recent studies have successfully applied similar models in concrete mix design and property prediction [21-23].

The multivariate regression model is formally expressed in matrix notation as:

$Y=X \cdot B+E$           (3)

where,

Y $\in$ ℝⁿˣ⁴: Matrix of dependent variables (dosage outputs: C, AF, AG, A)

X $\in$ ℝ^n×19: Matrix of independent variables, including an intercept and 18 predictors

B $\in$ ℝ^19×4: Matrix of regression coefficients

E $\in$ ℝ^n×4: Matrix of residual errors

The coefficient matrix B was estimated using the Ordinary Least Squares (OLS) method:

$\widehat{B}=\left(X^T \cdot X\right)^{-1} \cdot X^T \cdot Y$             (4)

This estimation minimizes the sum of squared residuals and provides the best linear unbiased estimators (BLUE), assuming the classical regression assumptions hold (linearity, homoscedasticity, independence, and normality of residuals).

(1) Implementation in R

The MLRM was implemented in RStudio 2024.12.0 using the built-in lm() function and the command summary(modelo) was used with the following structure:

modelo <- lm(cbind(y1, y2, y3, y4) ~ x1 + x2 + ... + x18, data = datos)

summary(modelo)

where, y1 to y4 denote the four output variables (C, AF, AG, A), and x1 to x18 represent the 18 input predictors.

(2) Regression equations

The resulting system produced four independent regression equations of the form:

$\hat{y}_k=\beta_{0 k}+\sum_{j=1}^{18} \beta_{j k} x_j+\varepsilon_k$            (5)

where,

$\hat{y}_k$: Predicted value of output variable k

$\beta_{0 k}$: Intercept for equation k

$\beta_{j k}$: Coefficient for predictor x_jin equation k

$\varepsilon_k$: Random error term associated with output k

Each equation captures the additive linear influence of the 18 input features on a specific output variable.

2.3.2 Statistical evaluation of fit

Residual diagnostics, statistical significance (p-values), and performance metrics were computed for each dependent variable. The complete data matrix included 806 samples with full multivariate observations.

(1) Goodness-of-fit metrics

All R² values were statistically significant (p < 0.001), indicating strong model performance, particularly for water and coarse aggregate prediction, as summarized in Table 6.

Table 6. Goodness-of-fit metrics for each dependent variable in the MLRM

(2) Model significance: ANOVA and MANOVA tests

Two complementary statistical tests were applied:

•Univariate ANOVA assessed each dependent variable individually, confirming that the predictors significantly influenced all four outputs (p < 0.001) [24].

•Multivariate MANOVA evaluated the predictors’ joint effect across all dependent variables, using Wilks’ Lambda, Pillai’s Trace, Hotelling-Lawley, and Roy’s Root, all of which confirmed strong multivariate significance (p < 0.001) as shown in Table 7 [25, 26].

Table 7. MANOVA statistics summary

Table 8. Computational demand comparison (MLRM vs. ANN)

The rationale for using MANOVA alongside ANOVA is that while ANOVA detects effects in each response variable independently, MANOVA accounts for correlations among responses, providing a more comprehensive assessment when dependent variables are related (as in concrete mix proportions) [24-26].

(3) Collinearity diagnostics

Variance Inflation Factor (VIF) analysis was conducted to assess multicollinearity among predictors. All VIF values were below the commonly accepted threshold of 5, indicating that collinearity was not a concern and that coefficient estimates are stable.

(4) Computational demand

Computational efficiency was compared between the MLRM and the ANN. As shown in Table 8, the MLRM computed solutions in less than one second via direct Ordinary Least Squares (OLS), while the ANN required 1–10 seconds due to iterative weight optimization over approximately 46–1000 epochs. This highlights a trade-off: the MLRM offers speed and simplicity for linear relations, whereas the ANN provides greater flexibility and non-linear mapping capabilities, albeit with higher computational cost.

2.3.3 Assumption diagnostics

Model assumptions were evaluated as follows (see Figures 4–6):

•Normality: Mahalanobis Q-Q plots (Figure 4) confirmed approximate multivariate normality.

•Homoscedasticity: Residual vs. fitted plots (Figure 5) showed no significant patterns, indicating constant variance.

•Independence: Serial correlation plots of MD² values (Figure 6) showed no significant autocorrelation.

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Figure 4. Chi-square Q-Q plot for multivariate residuals

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Figure 5. Residuals vs. fitted values plot

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Figure 6. Serial correlation plot of MD²

Following the validation protocol applied to the ANN model, a similar evaluation was performed for the MLRM. Table 9 presents the predicted mix proportions for 15 concrete mix designs obtained with the MLRM model. Table 10 summarizes the absolute (AE) and percentage errors between measured and predicted values. The average percentage errors for cement, fine aggregate, coarse aggregate, and water were 2.62%, 1.21%, 0.96%, and 0.87%, respectively, with a maximum PE of 6.24% observed in cement dosage.

Finally, Figure 7 shows the comparison between the actual experimental values (ACI 211.1) and those predicted by the RLMM model for the variables: cement, fine aggregate, coarse aggregate, and water. The blue lines represent the experimental data, while the dashed red lines represent the predictions. Overall, good agreement is observed, with a close fit for coarse aggregate and water. Cement and fine aggregate exhibit some variations but maintain the overall trend, demonstrating the model’s effectiveness.