## 3 Methodology
### Formulation of the Multi-Objective Optimization Problem  
To identify optimal operational strategies that simultaneously minimize economic and environmental burdens associated with steel slag use in road base layers, a bi-objective mathematical optimization problem was formulated. The objective of this formulation was to search for feasible combinations of input parameters that jointly minimize the predicted total cost per unit volume and the associated CO2-equivalent emissions [47]. Formally, the optimization problem was expressed as a minimization over two scalar-valued objective functions f1(x) and f2(x), representing total unit cost (USD/m3) and total carbon footprint (kg CO2-e/m3), respectively. The vector of decision variables can be defined as x = [X1, X2, X3, X4, X5] \(\in\) R5. The two objective functions are expressed as follows in Equations (17) and (18). \[ \begin{array}{c}\min\\ \text{x}\end{array}  \tag{17}\]  
\[ \begin{array}{c}\min\\ \text{f2(x)}=\hat{\gamma}_{\text{CO2}}(\text{x})\cdot\left[\text{kg}\cdot\text{ CO2}-\text{e/m}^{3}\right]\end{array}  \tag{18}\]  
where \(\hat{\gamma}_{\text{cost}}(\text{x})\) and \(\hat{\gamma}_{\text{CO2}}(\text{x})\) are surrogate objective functions modeled using the Gradient Boosted Regressor (GBR), as described in Section 3.1. The decision variable vector x \(\in\) R5 consisted of five normalized, bounded, and independently controllable inputs, all of which had physical meaning in the logistics and processing chain. These variables included transport distance (in kilometers), energy intensity of mechanical processing (in kWh per ton of slag), initial moisture content at the source (in percent by weight), gradation adjustment level (modeled as an ordinal categorical parameter with three levels), and the regional electricity grid's CO2 emission factor (in kg CO2 per kWh). The inclusion of these variables was based on engineering relevance and SHAP-derived importance scores (see Section 3.2), and they collectively capture both spatial and technological variability in slag-based base layer construction. To ensure the realism and physical plausibility of the optimization outcomes, the search space was constrained based on both the statistical properties of the dataset and domain-specific knowledge. Each decision variable was restricted to lie within empirically observed bounds, as detailed in Table 1. For instance, transport distance was limited to a range of 5-250 km to reflect practical haulage conditions, and moisture content was confined to 2-15% to remain within the drying capacity of typical field operations. In addition to these range constraints, logical feasibility rules were embedded to preserve inter-variable consistency. For example, solutions involving high moisture content (\(X_{3}>10\%\)) were coupled with increased energy intensity (\(X_{2}\geq 1.8\)), reflecting the thermodynamic reality of additional drying requirements. Similarly, gradation adjustment levels (\(X_{4}=2\)) were associated with increased energy consumption due to crushing and screening operations. Moreover, all outputs were expressed per 1 m\({}^{3}\) of compacted material. This was achieved by scaling mass-based quantities (e.g., energy per ton, transport per ton-km) using an average dry bulk density of 2.1 ton/m\({}^{3}\), ensuring consistent comparison across configurations. The resulting problem is a continuous-discrete hybrid optimization model with a nonlinear, nonconvex objective landscape. Given the absence of closed-form expressions for the objective functions and the interdependencies introduced by feasibility rules, a metaheuristic evolutionary approach was required to navigate the solution space efficiently [48]. The adopted solution technique, described in Section 3.4, utilizes population-based search strategies to approximate the Pareto frontier of trade-offs between cost and emissions. The optimization problem exhibits a hybrid structure, incorporating both continuous (\(X_{1}\), \(X_{2}\), \(X_{3}\), \(X_{5}\)) and discrete (\(X_{4}\)) variables.