## 2 Methods
### Optimization Design Problem Determination
#### 2.2.3 Decision Variables and Constraints  
The building envelope has significant impacts on the building energy, daylight and cost performance [24]. Compared with residential buildings, the fenestration of public buildings has less limitations and the WWR values are relatively higher, which lead to more energy consumption. In this study, 10 building envelope decision variables were selected as decision variables, as shown in Table 1. Considering that the building is a public library, the range of the floor height variable was set to vary from 3.9 to 6.0 m. According to the original building facade design, windows on the facade were classified into different types according to their shape and size. The widths of five types of windows were selected as decision variables, and each was given a constraint based on the original facade design rules. The widths of the skylights of the building were also selected as decision variables and were distinguished into two types. The one on the flat roofs was constraint to 0.5-1.5m, and the other on the sloped roof was constraint to 1.0-7.5m. The variables related to windows and skylights have been marked in Figure 4. \begin{table}
\begin{tabular}{c l l} \hline \hline
**Variable name** & **Numeric type** & **Constraint** \\ \hline Floor height (H) & Continuous & 3.9–6.0 m \\ \hline Flat skylight width (W) & Continuous & 0.5–1.5 m \\ \hline Width of arched window A (\(\text{W}_{\text{a}}\)) & Continuous & 5.0–8.0 m \\ \hline Width of arched window B (\(\text{W}_{\text{b}}\)) & Continuous & 3.0–5.0 m \\ \hline Width of strip window 1 (\(\text{W}_{1}\)) & Continuous & 0.3–1.0 m \\ \hline Width of strip window 2 (\(\text{W}_{2}\)) & Continuous & 1.0–2.7 m \\ \hline Width of strip window 3 (\(\text{W}_{3}\)) & Continuous & 1.0–2.0 m \\ \hline Sloped roof skylight width (d) & Continuous & 1.0–7.5 m \\ \hline Wall construction type (\(\text{T}_{\text{wall}}\)) & Discrete & 0–8 (Integer) \\ \hline Window panel type (\(\text{T}_{\text{win}}\)) & Discrete & 0–3 (Integer) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Decision variables and constraints. Different types of building envelope constructions are varied in thermal, optical as well as cost properties, and the thermal properties often conflict with the cost property, which will lead to a conflict between energy and cost performance. In this research, wall construction type and window panel type of the building envelope were also selected as decision variables. As the reference building is located in the severe cold region of China, the U-value of the exterior wall was recommended to be between 0.45 and 0.5 according to the Design Standard for Energy Efficiency of Public Buildings (GB50189-2015) in China [34]. A typical composite wall construction was selected from the book of public building energy conservation structure charts (06J908-1) [35], and set to be the basic wall construction. From the exterior to the interior there were five layers including decorative brick layer, air space layer, insulation layer, 250 mm concrete block layer and 15 mm gypsum layer. The insulation layer was set to be varied on the material type and thickness, so that the different thermal and price properties can be assigned to the basic wall construction, as shown in Table 2. The average R-value in this table is a measurement of the wall thermal resistance and the average U-value refers to the heat transfer coefficient of the wall. The unit price of the wall was calculated based on the local engineering information supervision platform. Nine types of walls with different insulations were explored by the optimization algorithm, the variable values correspond to the integer of 0-8. In terms of window constructions, 4 types of window panels including double pane of clear glass, triple pane of clear glass, single silver low-e coating glass and double silver low-e coating glass were selected, which can be presented by the integer of 0-3.