## 4 Results and Discussion
### The Optimal Pareto Front Solution Sets  
The optimization procedure used to create the model and the run results are described below. Eight design variables and 1.44 \(\times 10^{7}\) combinatorial designs were evaluated in the case study. After 63 generations, the Pareto solutions successfully converged in less than 3 min. This study used the Intel Core i9-13950 HX processor for training. The machine learning surrogate model significantly shortened the required time, making the optimization process a viable approach in the context of engineering practice. Figure 8 shows the optimization results and Pareto front solutions, illustrating the Pareto front for minimizing energy consumption, carbon emissions, and life-cycle costs. Each solution represents an optimized building envelope design that balances the three objectives. As anticipated, trade-offs are evident between the objectives. The dispersion of solutions is influenced by the case study's characteristics and the notable differences in the market prices and embodied carbon emissions of building envelope materials. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Model** & **MAE** & **MSE** & **RMSE** & **R\({}^{2}\)** \\ \hline FRF & 1.32 & 2.67 & 1.63 & 0.86 \\ LightGBM & 0.39 & 0.29 & 0.54 & 0.98 \\ ANN & 1.24 & 2.33 & 1.53 & 0.88 \\ MVLR & 1.41 & 2.87 & 1.69 & 0.85 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Performance comparison of ML models. Figure 7: Optimization results and Pareto front solutions. As shown in Table 7, a comparative analysis was performed on the design solutions pertaining to the different energy efficiency standards, namely the case study building (CSB) and ultra-low-energy building (ULB), with the solutions corresponding to the three minimal objectives derived from the Pareto solutions. The CSB model represents the original design retrofitted based on the local energy efficiency design standard [51], and the ULB model is derived from the latest local energy efficiency design standard for ultra-low-energy residential buildings [71]. The values of all the design variables for the renovation strategies, as well as the corresponding EC, LCCE, and LCC values, are listed. Table 8 presents a comparison between the minimal single-objective optimization results, case study building, and ultra-low-energy building. Compared with the CSB model, the minimum energy consumption, life-cycle carbon emission, and cost schemes all \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Design Solutions** & **CSB** & **ULB** & **Minimal EC** & **Minimal LCCE** & **Minimal LCC** \\ \hline Insulation type & EPS & XPS & XPS & EPS & EPS \\ Thickness [mm] & Wall\({}_{N}\) & 100 & 300 & 300 & 220 & 300 \\  & Wall\({}_{S}\) & 100 & 300 & 300 & 160 & 300 \\  & Wall\({}_{E/W}\) & 100 & 300 & 260 & 280 & 260 \\  & Roof & 200 & 300 & 240 & 220 & 300 \\ Glazing type & Win\({}_{N}\) & Single-plane & W8 & W8 & W8 & W8 \\  & Win\({}_{S}\) & Single-plane & W8 & W7 & W4 & W7 \\ Shading [mm] & 0 & 1500 & 0 & 0 & 0 \\ EC [kWh/m\({}^{2}\)-a] & 131.60 & 51.20 & 52.59 & 52.63 & 56.53 \\ LCCE [kgCO\({}_{2}\)e/m\({}^{2}\)] & 1335.24 & 663.59 & 620.24 & 619.32 & 641.40 \\ LCC [\(\$/\)m\({}^{2}\)] & 165.85 & 149.25 & 139.07 & 139.39 & 97.24 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of single-objective solutions with the case study building and ultra-low-energy building. Figure 8: Optimization results and Pareto front solutions. achieved significant reductions. The three indicators were reduced to varying degrees, reflecting considerable energy savings, emission reductions, and cost advantages. Compared with the ULB model, the minimum energy consumption scheme in the Pareto solution also reduced energy consumption, carbon emissions, and life-cycle costs to varying degrees. The minimum carbon emission and life-cycle cost schemes significantly reduced carbon emissions and life-cycle costs by decreasing the use of building envelope materials, thereby achieving win-win economic and environmental benefits.