# Study on Multi-Objective Optimization-Based Climate Responsive Design of Residential Building
## 4 Discussion of Optimization Results
### Optimization Results of Residential Buildings in Typical American Cities  
Figure 13 shows the optimization process with Duluth as an example. From the figure, it can be seen that the Pareto frontier of multi-objective optimization is more and more dense before the 9 th generation, and then gradually begins to converge until the 30 th generation. After 30 generations, the Pareto front has hardly changed. Figure 14 shows the optimization results of different climatic zones. In the three-dimensional space, all of Pareto non-dominated solutions with BED (building energy demand), GC (global cost), and DH (discomfort hour percentage) as objective functions are generated. These solutions represent trade-offs in design, because no other solution can improve (i.e., reduce) these three objectives at the same time. In order to better describe the optimization parameters, the three-dimensional solution is projected on the two-dimensional plane BED (horizontal axis)-GC (vertical axis) in Figure 15. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**N**} & \multirow{2}{*}{**Design Variable**} & **Climate** & **Climate** & **Climate** & **Climate** & **Climate** \\  & & **Zone 2** & **Zone 3** & **Zone 5** & **Zone 6** & **Zone 7** & **Zone 8** \\ \hline \multirow{2}{*}{1} & & **Houston** & **Los Angeles** & **Chicago** & **Helena** & **Duluth** & **Fairbanks** \\
2 & & 0 & 0 & 0 & 0 & 0 \\  & & 0.1 & 0.4 & 0.5 & 0.4 & 0.8 & 0.8 \\
3 & & 0.1 & 0.3 & 0.5 & 0.6 & 0.75 & 0.75 \\
4 & & 0.1 & 0.3 & 0.03 & 0.04 & 0.05 \\
5 & & 0.08 & 0.05 & 0.08 & 0.12 & 0.12 & 0.12 \\
6 & & 0.03 & 0.03 & 0.03 & 0.05 & 0.06 & 0.06 \\
7 & & 0.25 & 0.35 & 0.3 & 0.3 & 0.3 & 0.4 \\  & & 0.25 & 0.25 & 0.5 & 0.36 & 0.36 & 0.9 \\  & & 0.08 & 0.06 & 0.06 & 0.06 & 0.06 \\
8 & & 0.08 & 0.08 & 0.08 & 0.08 & 0.08 \\  & & 0.08 & 0.08 & 0.08 & 0.08 & 0.08 \\  & & 0.08 & 0.08 & 0.08 & 0.08 & 0.08 \\
10 & & 0.08 & 0.08 & 0.08 & 0.08 & 0.08 \\  & & 0.08 & 0.08 & 0.08 & 0.08 & 0.08 \\
11 & & 0.08 & 0.08 & 0.08 & 0.08 & 0.08 \\  & & 0.08 & 0.08 & 0.08 & 0.08 & 0.08 \\  & & 0.08 & 0.08 & 0.08 & 0.08 & 0.08 \\
12 & & 0.08 & 0.08 & 0.08 & 0.08 & 0.08 \\  & & 0.08 & 0.08 & 0.08 & 0.08 & 0.08 \\  & & 0.08 & 0.08 & 0.08 & 0.08 & 0.08 \\
13 & & 0.08 & 0.08 & 0.08 & 0.08 & 0.08 \\ \hline \hline \end{tabular}  
* refers to the unified change of block thermal conductivity and block density in parametric simulation. For example, the block thermal conductivity of roof are consistently change with block density of roof. Same as external wall and ground floor control.  
\end{table} 
Table 10: Design parameters of reference buildings in various cities of the United States. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Reference Building** & **Climate** & **Climate** & **Climate** & **Climate** & **Climate** & **Climate** \\
**Performance Indicator** & **Zone 2** & **Zone 3** & **Zone 5** & **Zone 6** & **Zone 7** & **Zone 8** \\ \hline  & **Houston** & **Los Angeles** & **Chicago** & **Helena** & **Duluth** & **Fairbanks** \\ \hline BED (kWh/m2): Building & 56.75 & 7.28 & 73.78 & 72.03 & 93.78 & 169.77 \\ Energy Demand & 323.52 & 322.63 & 378.58 & 335.89 & 389.09 & 475.73 \\ GC (§m2): Global Cost & 383.35 & 8.33 & 59.17 & 41.67 & 61.25 & 62.30 \\ DH (\%): Discomfort Hours & 329.32 & 318.63 & 338.08 & 297.90 & 337.62 & 382.54 \\ IC (§m2): Investment Cost & 329.32 & 318.63 & 338.08 & 329.79 & 337.62 & 382.54 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Performance indicators for residential building design in typical cities of the United States. Figure 14: Pareto optimization frontiers of typical cities in the United States. Figure 13: Multi-objective optimization iteration with Duluth as an example. The study provides two optimal solutions, namely the energy-saving optimal solution (nZEB optimal) that minimizes building energy demand and the cost optimal solution (C-O optimal solution) that minimizes global costs. These two optimal solutions correspond to two different goals of public demand and private demand.