## 5 Results
### Factor Analysis Based on Entropy-Weighted TOPSIS  
Building on this, the obtained Pareto solution set is analyzed to compare the heating carbon emissions and overall costs of various plans. Through the entropy weighting method, the data is normalized and weighted, resulting in the outcomes for seven key indicators, as shown in Table 5.  
The entropy weighting results highlight the insulation thickness of external walls as the most significant factor, followed by the south-facing window-to-wall ratio, roof insulation thickness, story height, north-facing window-to-wall ratio, and east-west facing window-to-wall ratio, in descending order of impact. This indicates that energy-saving optimizations for rural houses in the region should prioritize reducing the thermal conductivity of external walls and increasing the south-facing window-to-wall ratio. Subsequently, improving the performance of roof insulation materials and adjusting the design of window-to-wall ratios in other directions should be considered. Specifically, minimizing or avoiding windows in the east-west orientation can optimize the overall energy-saving effect. According to the calculation steps of the entropy weight TOPSIS method, after standardizing the values of each criterion, they are multiplied by the entropy weights previously determined to calculate the closeness coefficient C. This coefficient represents the overall performance of each scheme across all criteria, meaning that the higher the C value, the closer the scheme is to the ideal solution after considering all evaluation criteria. The results are shown in Figure 8.  
The graph clearly shows significant fluctuations in the C values, indicating considerable differences in the quality of the retrofitting schemes. About 25% of the schemes  
Figure 8: Relative closeness coefficient C value of the retrofitting schemes. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Indicator** & **Entropy Value** & **Difference Coefficient** & **Weight** & **Rank** \\ \hline Story height & 0.96 & 0.04 & 0.112 & 4 \\ Roof insulation & 0.95 & 0.05 & 0.139 & 3 \\ thickness & 0.92 & 0.08 & 0.217 & 1 \\ thickness & 0.91 & 0.09 & 0.241 & 6 \\  North-facing WWR & 0.97 & 0.03 & 0.092 & 5 \\  South-facing WWR & 0.95 & 0.05 & 0.140 & 2 \\  West-facing WWR & 0.98 & 0.02 & 0.059 & 7 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Results of entropy weight calculation. perform well, as these retrofitting strategies are closer to the ideal solution, with their relative closeness exceeding 0.7. Within the Pareto front solution, 20% of the plans are not economically viable, and the carbon reduction per unit of building area is less than 47.93 kg CO\({}_{2}\)/m\({}^{2}\). Compared to the baseline model, the maximum carbon reduction per unit of building area is achieved by Plan 43, with a value of 112.01 kg CO\({}_{2}\)/m\({}^{2}\), whereas Plan 62 has the highest economic feasibility, with a \(dC_{g}\) of \(-\)34,434.56 CNY, but the carbon reduction per unit of building area is 100.67 kg CO\({}_{2}\)/m\({}^{2}\). Hence, a higher investment in rural house optimization does not necessarily lead to greater carbon reduction. Balancing the carbon reduction per unit area and \(dC_{g}\) objectives, Plan 38 is identified as the optimal, with heating period carbon reduction and a global cost increase of 108.65 kg CO\({}_{2}\)/m\({}^{2}\) and \(-\)31,818.74 CNY, respectively, reducing building heating carbon emissions while also offering good economic benefits. The study utilizes roof-distributed \"self-consumption with surplus to grid\" for simulation, assuming that the degradation of the solar photovoltaic power generation system does not exceed 20% throughout its lifecycle [93], with a first-year degradation of 2% and no more than 0.5% per year thereafter. The calculated annual average photovoltaic power generation during the heating period is 3603.23 kWh, with 4583.77 kWh fed into the grid.