## 4 Results and Discussion
### The Optimal Pareto Front Solution Sets
Table 9 presents the top five solutions derived using the entropy-weighted TOPSIS method, evaluated against the criteria of minimizing EUI, LCCE, and LCC. With equal weights of 0.33 assigned to each criterion, the analysis identified solutions exhibiting the highest relative closeness values, signifying their proximity to the ideal solution while maintaining substantial separation from the negative ideal solution. The top-ranked solution achieved a relative closeness of 0.7964, followed sequentially by solutions with relative closeness values of 0.4767, 0.4475, 0.4131, and 0.4036. These rankings underscore the balanced integration of all criteria in the evaluation process, demonstrating the method's ability to prioritize Pareto-optimal solutions effectively. The findings highlight the utility of entropy-weighted TOPSIS as a robust tool for informed decision making in multi-criteria optimization scenarios. As shown in Figure 9 and Table 10, in comparison to the CSB model, all selected optimal solutions resulted in average energy savings of 56.62%, reductions in carbon emissions of 51.60%, and 24.27% decreases in life-cycle costs. In comparison to the ULB model, all selected optimal solutions resulted in a significant reduction in carbon emissions and life-cycle costs by 2.60% and 15.85%, respectively, despite slightly higher energy consumption. The majority of optimal solutions recommended EPS as the primary insulation material for the building envelope, which is attributed to its relatively low incremental cost and acceptable performance. Across all selected solutions, the insulation thickness on the south-facing wall was reduced to 160 mm, a noticeable decrease compared to the ULB model. This adjustment reflects the lower energy demand for the south-facing wall, given its higher solar exposure and heat-gain potential. Similarly, the east and west walls were recommended to have a reduced insulation thickness of 260 mm, as these orientations have a minimal overall impact on energy performance. In contrast, the north-facing wall was suggested to maintain \begin{table}
\begin{tabular}{l c c c c} \hline
**Design Solutions** & & **Minimal EC** & **Minimal LCCE** & **Minimal LCC** \\ \hline Compared with CSB & EC & \(-\)60.04\% & \(-\)60.01\% & \(-\)57.04\% \\  & LCCE & \(-\)53.55\% & \(-\)53.62\% & \(-\)51.96\% \\  & LCC & \(-\)16.15\% & \(-\)15.96\% & \(-\)41.36\% \\ Compared with ULB & EC & 2.71\% & 2.79\% & 10.41\% \\  & LCCE & \(-\)6.53\% & \(-\)6.67\% & \(-\)3.34\% \\  & LCC & \(-\)6.82\% & \(-\)6.61\% & \(-\)34.84\% \\ \hline \end{tabular}
\end{table} 
Table 8: Comparison between minimal single-objective optimization results, case study building, and ultra-low-energy building. \begin{table}
\begin{tabular}{l c c c c c} \hline
**Relative Closeness** & **A** & **B** & **C** & **D** & **E** \\ \hline \(D_{i}^{+}\) & 2.6837 & 6.6658 & 8.3124 & 8.3525 & 8.6695 \\ \(D_{i}^{-}\) & 10.4990 & 6.0726 & 6.7315 & 5.8794 & 5.8662 \\ \(C_{i}^{-}\) & 0.7964 & 0.4767 & 0.4475 & 0.4131 & 0.4036 \\ \hline \end{tabular}
\end{table} 
Table 9: The ranking of optimal solutions and relative closeness values. a 280-300 mm thickness for enhanced thermal performance. However, reducing the north wall insulation to 220 mm could significantly lower life-cycle carbon emissions (LCCE) at the expense of energy efficiency, leading to higher life-cycle costs (LCC). This trade-off arises because, in this case study, energy costs constitute a substantial portion of the total 30-year life-cycle costs. The optimization also highlights the importance of window selection. Solutions C, D, and E recommended using high-performance windows (W7 or W8) with triple-glazed, high-transmission Low-E glass. These windows help minimize life-cycle costs and carbon emissions while maintaining energy efficiency. Conversely, solution B recommended using lower-cost windows (W4 or W5, double-glazed), which increase annual energy consumption by 14.90% compared to the ULB benchmark. However, over a 30-year life