Predictive Modeling for Carbon Footprint Optimization of Prestressed Road Flyovers

Abstract

This study addresses the challenge of minimizing carbon emissions in designing prestressed road flyovers by comparing advanced predictive modeling techniques for surrogate-based optimization. The research develops a two-stage optimization approach. First, a response surface is generated using Latin-hypercube sampling. Second, that response surface is optimized to identify design configurations with the lowest CO₂ emissions. The optimal configuration (deck #37)—base width 3.40 m, deck depth 1.10 m, and concrete grade C-35 MPa—achieved a carbon footprint of 386,515 kg CO₂, representing a reduction of 12% compared to the reference bridge. Among the models tested, the artificial neural network (ANN) achieved the highest predictive accuracy (RMSE = 8372 kg, MAE = 7356 kg), closely followed by the Kriging 1 model (RMSE = 9235 kg, MAE = 7236 kg). Results indicate that emissions remain minimal for deck depths between 1.10 and 1.30 m, base widths between 3.20 and 3.80 m, and concrete grades of C-35 to C-40 MPa. This study provides practical guidelines for reducing the carbon footprint of prestressed bridges and highlights the value of robust surrogate models in sustainable structural optimization.

1. Introduction

Minimizing carbon emissions remains a challenge in combating climate change, as it affects all aspects of human activity [1]. The construction sector holds a significant role, consuming 15% of the world’s freshwater and accounting for 50% of global material extraction, while also contributing 37% of CO₂ emissions related to energy use [2]. Among these, concrete production for construction accounts for over 5% of the total carbon footprint [3]; thus, emission reductions represent an essential pillar of research needed to create a more sustainable industry [4,5,6].

The sustainable construction of bridges is essential to support infrastructure development while maintaining long-term environmental, social, and economic balance. While bridges offer significant benefits, they also present significant challenges in these areas due to the CO₂ emissions generated throughout their life cycle. Embodied carbon focuses explicitly on the greenhouse gases generated during the production of a product, including extraction, transport, and manufacturing processes. A comprehensive review has been undertaken to summarize the values associated with embodied carbon and energy [7].

Recent work [8] has underlined how inventory models and carbon-pricing mechanisms interact with technology investment and operational decisions; these mechanisms can materially change the optimal engineering choices at the project level (e.g., green inventory under carbon tax, cap-and-trade, and offset regimes).

Since ancient times, designs have been improved. The right choice of materials or their optimal design can reduce emissions. Bridges are vital, and the performance of prestressed concrete (PC) bridges depends on the engineer’s experience. Although the design follows standard practices, its approval is obtained through trial and error, leading to variable costs despite code compliance. Structural optimization methods provide other avenues for design solutions that are not based solely on experience.

Structural optimization is typically carried out using two primary approaches [9]. The first involves exact methods, usually based on mathematical programming techniques. The second strategy relies on artificial intelligence-driven heuristic algorithms frequently inspired by nature. These algorithms have shown outstanding ability and adaptability in complex, large-scale optimization problems with non-linear characteristics typical in structural optimization problems [10]. Due to the nature of structural optimization problems, heuristic methods are frequently applied to solve them.

Additionally, studies from the supply chain and operations research communities demonstrate that integrated inventory and operational strategies can balance cost, product deterioration, and carbon emissions, and that multi-echelon supply chain design influences the environmental outcomes of infrastructure supply chains—insights that are conceptually relevant when inspecting material selection and logistics for bridge construction [11,12].

The significant disadvantage of structural design heuristics is the high cost of computation [13]. Approximations or surrogate models, generally known as metamodels, are used to solve this problem. A metamodel can be a replacement for a simulation model that can cut down the computational effort significantly. Some of the most common surrogate models used for structural design optimization are Kriging or Artificial Neural Networks (ANNs). Although applied to real-world structural design only to a limited extent, Kriging is nonetheless one of the most promising approaches. For example, Kriging has been used to settle articulated structure design, efficiently decoupling uncertainty quantification from optimizing [14]. This method has also been extended successfully to other domains, such as the design of prestressed bridges [15,16], the optimization of structural systems [17], the development of architectural models for residential structures [18], and the prediction of carbonation depths in concretes [19]. ANNs learn from training data and produce outputs by evaluating the non-linear relations among their inputs. Researchers utilize them to forecast structural performance [20] and aid in the multiobjective optimization of structures [21]. Radial basis functions (RBFs) have been employed for structural health monitoring of composite laminated beams [22] and frequency-domain structural damage identification in multi-member structures [23]. Gaussian process regression (GPR) has been applied to perform system fragility analysis of highway bridges using multi-output surrogate models [24] and to detect structural damage in bridges under varying temperature effects through hybrid data training approaches [25]. These examples illustrate the breadth of practical engineering problems where such surrogate models have been successfully integrated, reinforcing their relevance for the present research.

Overpasses are commonly constructed using continuous PC beams, suitable for spans between 10 m and 45 m. Slab solutions are competitive with precast beams due to their structural and construction advantages, such as adapting to complex layouts, simpler formwork, and eliminating pavement joints. Slab bridges are cost-effective for complex geometric conditions, like curves or low road heights. Traditional formwork systems are typically employed for spans beginning at 20 m and up to four spans (under 120–140 m). For longer spans, self-supporting formwork is required. Slab sections are concreted in situ for 30 m to 60 m spans, with cantilevering used for spans over 25 m.

Despite advancements, further research is needed to optimize embodied carbon in bridges. A hybrid firefly based heuristic was used to optimize a prefabricated trough girder bridge, displaying that a €1 cost reduction resulted in 1.75 kg of CO₂ [26]. Another investigation focused on optimizing a post-tensioned box-section pedestrian bridge, revealing that emissions can be minimized by increasing edge dimensions, adding more active cables, and reducing the concrete grade [27]. Multiobjective optimization was also conducted for a PC box girder bridge [28] and T-girder bridges [29]. A life-cycle assessment of a precast bridge revealed that the manufacturing use and maintenance phases have a higher environmental impact [30].

Studies comparing different metamodels have not determined a clear superior type [31,32]. Ryberg [33] highlights that performance is affected by parameter tuning, which varies depending on the software used. Radial basis functions (RBF) proved effective for scenarios with limited sample sizes, primarily due to their ease of implementation [34]. Kriging offers better accuracy with larger sample sizes when adequately calibrated. RBF is generally more reliable, though these studies have limitations, such as fixed parameters and few variables. Elwy and Hagishima [35] provide an up-to-date overview of optimization techniques supported by metamodels in building design.

This study analyzes the response surface generated by various metamodels to compare the predicted values from each model. It also examines how these models can identify the most advantageous features for predesigning PC slab bridges to moderate the CO₂ footprint in their construction process.

2. Materials and Methods

2.1. Case Study: Description of the Lightweight Prestressed Slab Bridge

The study aims to optimize the carbon footprint design of a deck for a post-tensioned road flyover prestressed and lightened slab with 24–34–28 m spans, typical of dual carriageway overpasses. As shown in Figure 1 and Figure 2, the slab has a rectilinear geometry with uniform depth and a deck width of 8.30 m, designed to carry two traffic lanes, each measuring 3.50 m. It features parapets of 0.65 m on both sides. Figure 1 and Figure 2 illustrate the geometry of the studied deck: Figure 1 shows the longitudinal view of the lightweight prestressed slab. In contrast, Figure 2 shows the cross-section, main dimensions, and hollow-core configuration used in the analysis.

This overpass, located at the 441st kilometer of the A-7 motorway near Cocentaina in the Alicante region, features the width of the base b = 4.00 m, a depth c = 1.35 m, and a cantilever length v = 1.75 m. Its sizes include c₁ = 0.30 m, c₂ = 0.20 m, and d = 0.40 m. The internal voids consist of four cylinders with a diameter ϕ = 0.60 m each, with internal voids measuring 0.14 m³/m² and the external voids of 0.51 m³/m².

The limit state design ensures structural safety by applying partial safety factors. Each design scenario is evaluated to confirm that the slab bridge does not exceed any limit states. The deck was modeled, analyzed, and designed using the CSiBridge@ Version 21.0.0 software. Evaluating each slab bridge’s structural response provided the acting stresses, represented as sectional forces. Additionally, the stress resultants for each element were calculated, reflecting the resistance sectional stresses. Stresses are determined independently based on the configuration of each structural component. Further information regarding the structural calculations can be found in [15,16].

2.2. Inputs and Carbon Inventory

The building of each deck generates CO₂ emissions, which are influenced by the concrete grade, the formwork, the amount of lighting, and the amount of steel. The emissions have been assessed, as shown in

Table 1. Deck unit costs.

Deck Unit	Unit	CO2 (kg)
C-30 concrete	m3	227.01
C-35 concrete	m3	263.96
C-40 concrete	m3	298.57
C-45 concrete	m3	330.25
C-50 concrete	m3	358.97
Steel reinforcement	kg	3.03
Steel prestressed	kg	5.64
Formwork	m2	2.24
Voids	m3	604.42

[36].

The construction uses Y-1860-S7 and B-500-S steel. The concrete grade ranges from 30 to 50 MPa. Additionally, using lightweight materials helps the overall weight, enhancing performance and cost-effectiveness. The slab formwork is critical in supporting and shaping the concrete, influencing the surface quality and the construction speed.

2.3. Sampling and Optimization Strategy

A previous study [15] proposed a two-phase methodology for optimizing the response surface, modeled using Kriging. This approach predicts embodied carbon using the MATLAB Kriging Toolbox, Version 2.0 (DACE) to construct the metamodel [37]. A set of 30 bridges was selected through Latin hypercube sampling (LHS), which generates random numbers with a uniform distribution to evaluate alternatives. LHS provides lower sample mean-variance compared to random sampling [38]. The approach involves selecting a random sample from each variable interval, ensuring that the entire range of each variable is explored while maintaining a structured, low-discrepancy sampling pattern. LHS improves the understanding of the design space, reduces systematic errors, ensures uniform sampling across the space, and is particularly suitable for computational testing. It offers flexibility, efficiency, and rapid result generation. A list of all the decks considered is provided in

Table 2. Data derived from calculated bridges.

Deck	Deck Depth (m)	Base Width (m)	Concrete Grade (MPa)	CO2 (kg)
1	1.45	4.35	35	439,416
2	1.55	4.10	35	460,393
3	1.45	4.75	35	455,722
4	1.70	3.80	45	484,897
5	1.20	3.85	40	407,988
6	1.55	3.60	45	456,668
7	1.20	4.85	50	472,401
8	1.15	4.50	50	471,362
9	1.35	3.95	30	406,654
10	1.30	4.45	30	436,703
11	1.35	4.25	45	455,374
12	1.50	4.55	30	434,674
13	1.60	4.20	40	503,797
14	1.25	4.70	40	462,915
15	1.50	4.05	45	482,659
16	1.30	4.90	40	477,491
17	1.65	3.65	35	444,714
18	1.65	3.45	45	464,051
19	1.25	3.50	45	420,514
20	1.40	3.30	40	443,840
21	1.45	3.90	45	464,536
22	1.35	3.60	35	416,584
23	1.50	3.35	45	455,442
24	1.50	4.50	45	490,669
25	1.55	3.20	30	403,972
26	1.25	3.00	50	423,112
27	1.40	3.45	45	470,008
28	1.50	3.55	35	418,839
29	1.70	3.85	40	468,898
30	1.15	3.70	40	394,616
31	1.15	3.40	35	411,077
32	1.25	3.35	35	398,614
33	1.15	3.65	45	422,934
34	1.15	3.35	40	395,465
35	1.15	3.25	40	397,154
36	1.15	3.55	40	391,370
37	1.10	3.40	35	386,515

.

The Kriging response surface was optimized with the first 30 data points in the diversification phase, while the intensification phase employs the subsequent five data points. In the first phase of diversification, the response surface was optimized using simulated annealing. Following this, a further sampling of five additional decks was conducted around the previously identified optimum, and the surface was re-optimized. This process led to discovering a new optimum in the intensification phase.

2.4. Surrogate Models

This research seeks to assess the performance of different predictive models to determine how much they can capture the information from the actual response surface. These techniques are chosen based on their proven ability to handle optimization problems with complex, non-linear response surfaces, such as those presented in this case. Various metrics will be defined to assess the error between the actual unknown data and the model’s predictions.

2.4.1. Radial Basis Functions

Radial basis functions (RBFs) [39] are among the methods used to construct complex systems’ surrogate models through function fitting and data-driven modeling. These functions are based on the principle of radial symmetry, which implies that their output depends solely on the distance between a target point and a central reference. RBFs exhibit strong performance when reconstructing values from irregularly distributed datasets, enabling the generation of continuous and differentiable approximations that remain robust in complex, multi-dimensional domains. An RBF, with input variables xi collectively represented as x_i, can be defined as:

$$f\left(\mathit{x}\right)=\sum _{i=1}^{N_{\Phi }}{\Phi }_i\left(\mathit{x}\right){\alpha }_i+b\left(\mathit{x}\right)$$

(1)

In this context, f = f(x) represents the output response, Φ_i = Φ_i(x) are the RBFs, N_Φ indicates the number of RBFs, α_i are the associated weights, and b = b(x) denotes the bias term. The number of RBFs selected corresponds to the number of input variables.

The particular type of RBF chosen—such as linear, Gaussian, or thin plate spline— has an impact on the model’s flexibility and smoothness. The selected RBF variants (linear, multiquadratic, inverse multiquadratic, and cubic) were chosen because they collectively span a range of smoothness and flexibility properties while remaining numerically stable for small-sample problems. Prior comparative studies in engineering metamodeling have shown that these kernels provide a practical trade-off between flexibility and overfitting in problems with moderate dimensionality and limited observations [34,39]. Thus, evaluating this set enables a representative comparison of the radial basis functions used in structural-design surrogate modeling. The most commonly applied types of RBFs include the following:

$$Linear: {\Phi }_i\left(r\right)=r$$

(2)

$$Multiquadratic: {\Phi }_i\left(r\right)=\sqrt{{\left({\displaystyle \frac{r}{{\epsilon }_i}}\right)}^2+1}$$

(3)

$$Inverse multiquadratic: {\Phi }_i\left(r\right)={\displaystyle \frac{1}{\sqrt{{\left(\frac{r}{{\epsilon }_i}\right)}^2+1}}}$$

(4)

$$Cubic: {\Phi }_i\left(r\right)=r^3$$

(5)

where r represents the Euclidean distance and (${\epsilon }_i$) denotes the shape parameter.

2.4.2. Kriging Metamodel

Kriging [40] predicts the value of a specific location u using a collection of n observed z-values (Figure 3). In this case, the focus is on estimating the embodied carbon of the bridge deck. This technique allows for response predictions without requiring comprehensive structural analysis. Kriging models produce consistent results as deterministic functions when applied to the same known data without random errors. The general expression for a Kriging model is given by:

$$y\left(x\right)=f\left(x\right)+Z\left(x\right)$$

(6)

In this formulation, Z(x) represents a stochastic residual with zero mean and spatial dependence, typically characterized through a correlation structure. Meanwhile, f(x) is the approximation function, similar to a regression model. This function captures the non-random tendency in the data—the part governed by known or predictable relationships—and is commonly expressed as follows:

$$f\left(x\right)=\sum _{i=1}^n{\beta }_i{f}_i\left(x\right)$$

(7)

In this context, f_i(x) denotes the predefined basis functions—such as low-order polynomials or other simple functional forms—while β_i are parameters to be estimated, which characterize the mean or trend component. This component can be represented in various forms: as an invariant mean in Ordinary Kriging, zero in Simple Kriging, or as a polynomial (e.g., linear or quadratic) in Universal Kriging. This study uses polynomial regressions of orders zero, one, and two to construct models labeled Kriging 1, Kriging 2, and Kriging 3, respectively. These models produce specified predictions, generating the same output for identical inputs when no random error exists.

In this study, the Kriging models were implemented using the MATLAB DACE toolbox [37], which estimates correlation parameters (including separate length scales for each input dimension and the nugget term) via maximum likelihood estimation. The nugget parameter, representing observation noise, was included with a small value to improve numerical stability, given the deterministic nature of the dataset. Anisotropy was addressed by independently estimating length scales for each design variable, allowing different correlation ranges per dimension. The Gaussian radial basis function (RBF) kernel was selected due to its proven robustness in structural optimization problems with small datasets [34,39]. Preliminary tests with alternative kernels, such as the Matérn and exponential functions, did not improve the cross-validated RMSE for the present dataset.

2.4.3. Artificial Neural Networks

An artificial neural network (ANN) comprises a structure that begins with an input node layer, followed by one or more intermediate hidden layers, and output layers, enabling the detection of intricate links among variables. The input layer receives the data, which is then processed by the hidden layer. The model undergoes training by iteratively adjusting the weights and backpropagating errors to enhance its accuracy. A detailed overview of ANN’s fundamental concepts, recent developments, and practical uses can be found in [41]. At the same time, additional discussions focus on their practical use and effectiveness, highlighting their broad applicability and proven impact [42].

A multilayer feedforward network is organized with an intermediate layer containing sigmoidal neurons, while the final layer consists of linear neurons. The hidden layer neurons are interconnected with the initial and final layers, with the hidden layer neuron number proportional to the initial and final parameter numbers.

The entry variables (x_i) are scaled by their associated weights (w_i,j) and subsequently offset by the addition of a bias term (b_j). Every node in the hidden layer calculates the following equation: ∑x_i − w_i,j + b_j. The output of each hidden unit is then generated by applying a sigmoid tangent function to this equation. In contrast, the final layer uses a linear function. This architecture, known as the multilayer perceptron (MLP), is widely utilized for approximating various functions, even when only a single hidden layer is utilized [43]. The backpropagation algorithm contributes to its success in modeling complex relationships by optimizing the weights and biases through error correction in training [44,45].

In a feed-forward network, data moves from the initial to the final layer, with learning tracked using data with predefined responses. The dataset is split into three subsets to assess overfitting: a training dataset for adjusting parameters, a validation dataset to detect overfitting, and a test set to assess the forecasting performance of the model on previously unseen data. The “early stopping” technique mitigates overfitting without explicitly partitioning the dataset into separate training and validation subsets. During the optimization process, performance metrics are monitored on both components to guide termination. If training errors continue to decrease while validation errors rise, this signals potential overfitting, prompting the termination of further training. In that case, the optimization progression is halted, preventing overlearning and ensuring the model can generalize effectively.

The neural network randomly chose 30 data points (24 training, 3 validation, and 3 test). The network architecture features a single hidden layer comprising five neurons. Simulation assesses model performance by generating predictions based on either training or previously unseen data, followed by an outcomes analysis. The initial stage involves cross-validation, a process that compares the output from the training phase with the results produced by the neural network simulation. This step is essential to assess the model’s accuracy and identify overfitting, which occurs when the model becomes excessively fitted to the training data. Cross-validation can be conducted across various partitions of the dataset—namely, training, validation, and test sets—to examine the risk of overfitting comprehensively.

The ANN models were implemented using MATLAB’s Neural Network Toolbox [46], employing the mean squared error (MSE) as the loss (performance) function during training. Network performance was monitored via MATLAB’s built-in performance plots, which display the MSE on the training, validation, and test sets across iterations, enabling visual detection of overfitting. Overfitting was controlled using the early stopping technique, where training was halted if the validation error did not improve for a predefined number of iterations. The architecture comprised a single hidden layer with five neurons, selected after preliminary trials to balance predictive accuracy and network simplicity for the small dataset. The 30 available samples were divided into 24 for training, 3 for validation, and 3 for testing. Given the limited dataset size, multiple independent runs with different random partitions were performed, and the predictions were averaged to improve robustness and reduce the influence of any single partition.

2.4.4. Gaussian Process Regression

The Gaussian Process Regression (GPR) model is a flexible, data-driven approach that does not rely on a fixed set of parameters offering a probabilistic regression framework [47]. It is advantageous when the relationship between entries and exits is complex and non-linear, as it can capture various underlying patterns. The Matern 5/2 kernel was used in this study [48], as it is well-suited for modeling smooth, non-linear relationships. Automatic hyperparameter optimization was employed to adjust the kernel parameters, enhancing the model and minimizing prediction errors. Both input and output variables were normalized using Z-score normalization to standardize the scale, which is crucial because GPR models are sensitive to data scaling.

Cross-validation with exclusion was employed to assess the model’s effectiveness and generalizability. The data were partitioned, with 80% assigned to training and 20% to testing. The model was developed using the training set, while the test dataset was used to evaluate its predictive accuracy. This technique reduces overfitting and ensures the model can predict new data.

To address the stochastic nature of the GPR model, 16 independent runs were conducted with different random seeds. The predictions were then averaged to minimize variability and yield more consistent results. This repetition, with distinct random seeds for each run, accounts for the model’s inherent variability due to random initialization, ensuring that the predictions are not overly influenced by the initial conditions of the optimization process and offering a more robust model performance evaluation.

The predictions obtained in the normalized domain were reverted to their original scale by applying the appropriate inverse transformation. The final output comprised the averaged predictions for the new data points denormalized to the original scale. Cross-validation and hyperparameter optimization were used to guarantee the reliability and robustness of the predictions.

A robust and dependable model was developed through the integration of Gaussian Process Regression with hyperparameter optimization, data normalization, and the averaging of predictions. By averaging across multiple runs, the model’s prediction stability was enhanced, effectively capturing the inherent stochasticity of the process and yielding consistent and reliable results.

2.5. Model Evaluation Metrics

Various metrics are employed to analyze the discrepancies between the actual results of novel solutions and their respective predicted values to evaluate the accuracy of the different models. Each model undergoes an evaluation process, assessing performance metrics to determine reliability and effectiveness [49]. Metrics include root mean square error (RMSE), mean absolute error (MAE), and the ratio of RMSE to the standard deviation of observations (RSR). These measurements give information on the model’s predictive ability under different conditions. Comparing estimations against error thresholds ensures robust performance assessment. The formulae for these metrics are given at (8)–(10).

$$RMSE=\sqrt{\sum _{i=1}^n{\displaystyle \frac{{\left({\widehat{y}}_i-y_i\right)}^2}{n}}}$$

(8)

$$RSR={\displaystyle \frac{RMSE}{SD}}$$

(9)

$$MAE=\sum _{i=1}^n{\displaystyle \frac{\left|{\widehat{y}}_i-y_i\right|}{n}}$$

(10)

where ${\widehat{y}}_i$ are the expected values, y_i denote the real values, and n corresponds to the total count of observations. SD refers to the observation’s standard deviation.

3. Results

In the initial diversification phase, the response surface was optimized using the simulated annealing algorithm, yielding deck #36 as the optimal solution. Subsequently, five additional decks were sampled near this optimum, and the surface was re-optimized. This second optimization, carried out during the intensification phase, identified a new optimum corresponding to deck #37. The local optima for each phase are represented by #36 and #37, respectively.

The observed data from Table 2 are represented in Figure 3, Figure 4, Figure 5 and Figure 6 with emissions as the output response, revealing multiple local optima and abruptness. A direct examination of Figure 3 shows that the lowest emissions are associated with the smallest base widths and deck depths within the specified ranges.

However, more detail regarding the location of the minimum emission values can be obtained by analyzing the contour plots representing the variables in pairs. Figure 4 demonstrates a two-dimensional representation of the response surface delineated in Figure 3, and a discernible trend is evident: a decline in CO₂ emissions is concomitant with a reduction in base widths and deck depths.

As demonstrated in Figure 5, the maximum and minimum emission values are concentrated around the C-40 concrete grade, with deck depth playing a critical role. Specifically, within a deck depth range of 1.10 to 1.30 m and concrete grades between 35 and 40 MPa, the carbon footprint reaches its minimum.

Figure 6 shows that the maximum and minimum emission values are associated with C-40 concrete grade. The lowest carbon footprint is observed within a base width range of 3.20 to 3.80 m. Therefore, an initial assessment of the contour plots suggests that the lowest emissions can be achieved using concrete grades between 35 and 40 MPa, a deck depth of 1.10 to 1.30 m, and the width of the base between 3.20 and 3.80 m. These preliminary recommendations align with the optimum identified by optimizing the response surface generated using Kriging. This optimum, represented by deck #37, features the width of the base at 3.40 m, a deck depth of 1.10 m, and a C-35 concrete grade.

4. Discussion

4.1. Comparison with Previous Studies

The results obtained in this study are compared with those reported in previous research on similar types of bridges and with recommendations provided by other authors.

Table 3. Measurements of the reference deck and the optimized solutions.

Objective Function	Slab Height (m)	Span/Deck Depth	Concrete (m3/m2)	Passive Steel (kg/m2)	Active Steel (kg/m2)
Reference	1.35	25.19	0.72	73.45	16.64
Cost	1.30	26.15	0.61	73.53	14.76
CO2	1.10	30.91	0.56	77.00	16.48
Energy	1.15	29.57	0.61	69.41	16.65

presents the key measurements of optimized bridges considering various objective functions such as cost [16] and embodied energy [15]. This study extends these optimization criteria by incorporating the reduction in CO₂ emissions, thereby offering a more comprehensive approach to bridge design optimization. Table 3 also includes the reference bridge—the real-world constructed structure described in Section 2.1. This inclusion enables a direct comparison between the optimized designs and the reference one, offering important context to evaluate the practical relevance and effectiveness of the optimization approaches used in this study.

The slab height decreases notably from 1.35 m in the reference to 1.10 m in the CO₂-optimized design, resulting in a higher span-to-depth ratio (30.91), which indicates a more slender and material-efficient structure. For prestressed slab bridges with cantilevers, recommended slenderness ratios provide benchmarks: The General Directorate of Public Roads of Spain [50] suggests 1/22 to 1/30, SETRA [51] 1/28, and Manterola [52] 1/25. The reference bridge’s slenderness (1/25.19) fits these ranges. The cost-optimized bridge is less slender than CO₂ and energy-optimized versions, implying cost-effectiveness at slenderness above 1/26 and sustainability goals near 1/30.

Concrete volume reduces accordingly, reaching 0.56 m³/m² in the CO₂ case, lowering the embodied carbon footprint. However, the reduced slab depth requires increased passive steel reinforcement (77.00 kg/m²) to maintain flexural strength, while active steel remains stable, indicating prestressing is driven mainly by structural needs. The cost-driven design balances material savings with feasibility, showing moderate reductions in slab height and concrete volume with minimal changes in steel. The energy-optimized solution lowers passive steel (69.41 kg/m²), likely reflecting steel’s higher embodied energy, favoring concrete within structural limits. These differences highlight trade-offs between efficiency, material properties, and sustainability, with CO₂ optimization favoring aggressive material cuts and increased steel, while cost and energy targets yield balanced solutions.

4.2. Surrogate Model Performance

Table 4. Real values and predictions corresponding to the optimal solutions identified during the diversification phase (#36) and the intensification phase (#37), along with their relative errors.

	Local Optima		Relative Error (%)
	#36	#37	#36	#37
Observed	391,370	386,515	0.00	0.00
Linear	402,802	376,571	2.92	−2.57
Linear RBF	405,515	410,217	3.61	6.13
Multiquadratic RBF	395,904	381,208	1.16	−1.37
Inverse multiquadratic RBF	397,924	373,258	1.67	−3.43
Cubic RBF	396,853	382,225	1.40	−1.11
Kriging 1	394,808	398,496	0.88	3.10
Kriging 2	389,612	373,195	−0.45	−3.45
Kriging 3	387,247	349,638	−1.05	−9.54
ANN average	400,619	399,219	2.36	3.29
GPR average	403,150	386,880	3.01	0.09

presents the real values alongside results from three Kriging surrogate models, a linear regression, and the local optimum obtained from the ANN’s diversification (#36) and intensification (#37) phases. While the Kriging models are deterministic, ANNs and GPR are inherently stochastic due to random training and validation splits. To account for this variability, each configuration was run 16 times with different random seeds, reducing the standard error of the mean and yielding more robust predictions. Mean values of each performance metric are reported, and the 95% confidence intervals for RMSE and MAE overlap, indicating no statistically significant differences between models. Early stopping on the validation set was applied to every run to limit overfitting.

The prediction of the 30 slabs used in the diversification phase was compared for all models (Table 1) to analyze the relative error. The linear model provides the lowest relative error concerning the optimum of the intensification phase. However, the scenario changes when examining the RMSE, RSR, and MAE for the prediction of slabs #31 to #37, as summarized in

Table 5. Used prediction model error metrics.

Predictive Models	RMSE	RSR	MAE	Rank (RMSE-RSR-MAE)
Linear	12,652	0.87	9012	4-1-4
Linear RBF	16,793	2.38	14,193	5-10-5
Multiquadratic RBF	34,182	1.50	23,144	9-7-9
Inverse multiquadratic RBF	34,439	1.45	26,686	10-6-10
Cubic RBF	21,773	1.62	17,433	7-9-7
Kriging 1	9235	1.59	7236	2-8-2
Kriging 2	19,189	1.37	15,428	6-5-6
Kriging 3	27,612	1.31	22,926	8-3-8
ANN average	8372	1.36	7356	1-4-3
GPR average	9246	0.90	5873	3-2-1

, which also includes an added ranking column (‘Rank (RMSE→MAE→RSR)’) to facilitate model selection. Models are ordered primarily by RMSE (lower is better), with MAE and RSR as tie-breakers when RMSE differences are slight. It can be observed that the neural network achieves the lowest RMSEs, closely followed by the Kriging 1 model.

Figure 7 illustrates the predicted-to-real values for points #31 to #37 across the metamodels. The prediction improves as the ratio approaches one. It can be observed that Kriging 2 and Kriging 3 tend to underestimate the predicted values. In contrast, Kriging 1, ANN, and GPR models achieve ratios closest to unity on average.

Figure 8 displays the metrics used to evaluate the effectiveness of each predictive model in this research. The values have been divided by the metrics of the linear model to allow for comparison of the normalized values. A lower metric value indicates improved predictive reliability of the model. The ANN and GPR models exhibit the best performance.

The ability of the ANN to identify optimal values is evaluated by analyzing the averages of 16 runs with the 37 observed values. Figure 9 predicts the minimum deck depth for a width of the base of 3.40 m and a concrete grade of C35. The optimal deck depth obtained with Kriging, 1.10 m, coincides with the neural network results, which indicate an increase in CO₂ emissions from this measure. This confirms the consistency between the two approaches to optimization.

With C35 concrete and a deck depth of 1.10 m, the ANN illustrates the variation in emissions as the deck width changes. Emissions are minimal between the 3.00 m and 3.40 m base, rising slowly thereafter and more steeply from 4.00 m onwards (Figure 10). This is consistent with the minimum base value found when optimizing the Kriging response surface, which was 3.40 m.

If the deck depth is now kept at 1.10 m and the base at 3.40 m, the neural network shows how CO₂ emissions evolve with the characteristic strength of the concrete (Figure 11). From 30 to 40 MPa, emissions remain at their lowest values, clearly increasing thereafter. This agrees with the value of 35 MPa obtained from the response surface optimization generated by the Kriging metamodel.

4.3. Practical Design Recommendations

The General Directorate of Public Roads of Spain [50] suggests a ratio between the depth of the deck and the main span length ranging from 1/22 and 1/30, whereas SETRA [51] advocates a value of 1/28 particularly for PC decks composed of three-span with wide lateral cantilever lengths. The ANN results guide the practical minimization of the carbon footprint in a PC deck with a 34 m main span. It is recommended to aim for the highest possible slenderness ratios, preferably within the range of 1/30, which would necessitate an increase in passive steel. Decreasing the concrete and active steel volume can compensate for this additional passive reinforcement. Additionally, the amount of lightening should be maximized to minimize concrete use further.

The following design guidelines are recommended for a PC road flyover with a main span of 34 m: a depth/span ratio exceeding 1/28, a concrete volume below 0.60 m³/m² of deck, and passive reinforcement above 120 kg/m³ of concrete. Active steel should remain under 17 kg/m² of the deck. The recommended concrete grade ranges between 35 and 40 MPa. The external voids should range from 0.40 to 0.50 m³/m², while the internal voids should be lower than 0.20 m³/m².

The ANN can identify a local optimum that closely aligns with the Kriging response surface local optimum. Nevertheless, to predict CO₂ emissions with higher accuracy, a heuristic optimization process must be applied to both Kriging models and ANN. While metamodels can effectively generate a response surface, minimizing it to locate a local minimum is crucial. Despite this, these models offer reliable project outcomes and significantly contribute to refining the definition of suitable value intervals for key design variables. The methodology is a valuable tool for designers who may not regularly employ heuristic optimization algorithms, thus enabling them to uncover potential opportunities for reducing economic costs and carbon footprint.

4.4. Limitations and Future Work

The present study provides a consistent and well-founded framework for optimizing the carbon footprint of prestressed lightweight slab bridges. Nonetheless, several aspects naturally suggest future research avenues beyond the current scope. The methodology was illustrated through a representative case study; its application to alternative bridge typologies would broaden the range of structural configurations addressed. Although the surrogate models and optimization strategies were carefully selected and calibrated, further enhancements could be achieved by incorporating larger datasets. Hybrid optimization schemes—combining heuristic search with surrogate-assisted refinement—can improve solution accuracy by exploiting the global exploration capacity of heuristics and the local precision of surrogate models, thereby reducing computational effort while increasing reliability.

Furthermore, integrating multi-objective formulations that balance CO₂ emissions with cost, structural performance, and constructability would provide a more comprehensive decision-making framework. The assessment of innovative materials, such as high-performance or low-carbon concretes, recycled aggregates, and advanced reinforcement systems, is expected to reduce embodied emissions and enhance durability and service life, amplifying long-term sustainability gains. Finally, incorporating life-cycle assessment to capture use-phase and end-of-life effects, together with validation through real construction projects, would strengthen the practical relevance and robustness of the methodology. These extensions represent natural continuations of the present work and hold significant potential to advance sustainable and efficient bridge engineering.

5. Concluding Remarks

This study compares Kriging and artificial neural networks (ANNs) for optimizing the carbon footprint in prestressed slab bridge design. The research uses Latin hypercube sampling (LHS) and two-phase optimization to identify key design parameters—deck depth, base width, and concrete grade—influencing CO₂ emissions. ANNs show superior predictive accuracy, outperforming other models in reducing RMSE and MAE. Verification indicates that neural networks effectively identify the local optimum, closely resembling values obtained by minimizing the Kriging response surface. Nevertheless, neither model accurately predicts the objective function and guides solutions toward promising areas. While these models closely define a response surface, refining the optimization to locate a more precise local minimum remains necessary. Optimal configurations include concrete grades between C-35 and C-40 MPa, deck depths from 1.10 to 1.30 m, and base widths from 3.20 to 3.80 m, highlighting the role of advanced surrogate models in sustainable bridge engineering.

This study provides practical recommendations, advocating for slenderness ratios above 1/28, concrete volumes below 0.60 m³/m², passive reinforcement over 120 kg/m³, and prestressing reinforcement under 17 kg/m². Enhanced lightening strategies, including external voids of 0.40 to 0.50 m³/m² and internal voids below 0.20 m³/m², further improve sustainability. ANN-based optimization enables eco-efficient designs while ensuring structural integrity. This methodology is a practical tool for identifying optimal configurations without excessive computational costs, bridging theoretical advancements with real-world applications.

Beyond its technical contributions, this study has potential impacts across research, economic, and sustainability domains. From a research perspective, it adds comparative evidence on the performance of surrogate models in structural optimization when working with small datasets. Economically, it supports identifying design configurations that simultaneously reduce costs and embodied CO₂. Societally and in terms of sustainability, it facilitates adopting environmentally friendly bridge design practices, contributing to greener and more responsible infrastructure development.