## 2 Proposed Optimization Problem  
The problem in this study is to optimize two single objectives: the economic cost (C) and the embodied energy (E), considering material production, formwork, earth-fill, and excavation. Equations (1) and (2) allow us to evaluate the total cost and embodied energy of the construction. The unit price \(p_{i}\) and energy \(e_{i}\), which are the price and the energy linked with each construction unit (\(r\)), are multiplied by the unit's measurements (\(m_{i}\)) resulting from the optimization procedure. These equations must be minimized to satisfy the constraints problem by Equation (3). The embodied energy data take into account a cradle-to-gate analysis. This means that the energies consumed for every process consider the activities of extracting the raw materials, processing, manufacturing, and the transport of the materials to the construction site. Moreover, the cost takes into account the materials (concrete, reinforcing steel, and formwork) and other activities and elements required to evaluate the total cost of the construction. Data of prices and energy consumption in Table 1 have been collected by the BEDEC database of the Institute of Construction Technology of Catalonia [42]. Table 1 includes all prices and embodied energy for every construction unit. It is assumed that to produce reinforcing steel, the ratio of recycled scrap steel is approximately 40% and the manufacturing process is carried out by an electric arc furnace.  
\[C\left(\begin{matrix}\overrightarrow{x}\\ \end{matrix}\right)=\sum_{i=1}^{r}p_{i}\cdot m_{i}\left(\begin{matrix} \overrightarrow{x}\\ \end{matrix}\right), \tag{1}\]  
\[E\left(\begin{matrix}\overrightarrow{x}\\ \end{matrix}\right)=\sum_{i=1}^{r}e_{i}\cdot m_{i}\left(\begin{matrix} \overrightarrow{x}\\ \end{matrix}\right), \tag{2}\]  
\[G\left(\begin{matrix}\overrightarrow{x}\\ \end{matrix}\right)\leq 0, \tag{3}\]  
This study has developed a method to produce optimal structural solutions for buttressed earth-retaining walls for cost and energy requirements. The variables of the problem  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Unit** & **Energy (kW-h)** & **Cost (€)** \\ \hline \multicolumn{3}{c}{**Earth movement**} \\ m\({}^{3}\) of backfill & 78.32 & 14.12 \\ m\({}^{3}\) of backfill over the toe & 76.52 & 12.52 \\ m\({}^{3}\) of earth excavation & 44.65 & 10.56 \\ \multicolumn{3}{c}{**Foundation**} \\ kg of steel B400 & 10.39 & 1.11 \\ kg of steel B500 & 10.39 & 1.13 \\ m\({}^{3}\) of concrete C25/30 & 413.28 & 90.74 \\ m\({}^{3}\) of concrete C30/37 & 439.51 & 99.44 \\ m\({}^{3}\) of concrete C35/45 & 457.05 & 119.48 \\ m\({}^{3}\) of concrete C40/50 & 477.98 & 122.5 \\ m\({}^{3}\) of concrete C45/55 & 484.49 & 125.45 \\ m\({}^{3}\) of concrete C50/60 & 492 & 127.93 \\ m\({}^{2}\) of cleaning concrete & 24.8 & 9.28 \\ m\({}^{2}\) of formwork & 6.56 & 23.83 \\ \multicolumn{3}{c}{**Stem**} \\ kg of steel B400 & 10.4 & 1.29 \\ kg of steel B500 & 10.4 & 1.31 \\ m\({}^{3}\) of concrete C25/30 & 427.78 & 101.99 \\ m\({}^{3}\) of concrete C30/37 & 454.01 & 110.69 \\ m\({}^{3}\) of concrete C35/45 & 480.32 & 130.73 \\ m\({}^{3}\) of concrete C40/50 & 511.72 & 133.84 \\ m\({}^{3}\) of concrete C45/55 & 524.48 & 136.91 \\ m\({}^{3}\) of concrete C50/60 & 535.74 & 140.98 \\ m\({}^{2}\) of formwork & 86.57 & 53.26 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Prices and embodied energy values of the construction units [42].  
are discrete to adjust the procedure to reality, and vector x contains the design variables that describe the geometry of the wall and the reinforcing steel and concrete grade.