## 4 Approach
### Selection and Award of the New Owner  
Once the weights of each criterion have been obtained by the AHP method, the TOPSIS method will be used to determine the selection of the new owner. This point we define as: \(B\), the set of \(m\) potential buyers where each \(b\) is a buyer (citizen) \(B=\{b_{1},\ldots,b_{m}\}\) that has been associated with the set of criteria defined in the previous point \(b=\{c_{1},\ldots,c_{n}\}\), \(B=\{(c_{11},\ldots,c_{1n}),(c_{21},\ldots,c_{2n}),\ldots,(c_{m1},\ldots,c_{ mn})\}\).  
For this, we must define three simple but important things: Columns (number of criteria or attributes), Rows (number of buyers), Vector (weight for each attribute). As already mentioned in the determination of the weight with AHP, the attributes are important and are evaluated by smart government; these attributes can be: age, number of members in the family, profession, economic situation, etc. With this, we will form matrix M, and the first step (Equation (6)) will be carried out; we will then obtain a new matrix r (Table 7).  
Then, we apply Equation (7), where the vector of weights plays an important role. A third step is to find an ideal and negative-ideal solution, (Equations (8) and (9)) (Table 8).  
The fourth step is to calculate the separation measures (Equations (10) and (11)). The fifth step is to calculate relative closeness to the ideal solution, (Equation (12)) and, finally, we obtain the rank of the order of preference. This last step gives us the chosen citizen to be awarded the smart home (Table 9).  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Criteria** & **C1** & **C2** & **C3** & **C4** & **C5** & \multicolumn{4}{c}{**Normalized**} & \multicolumn{2}{c}{**Sum**} & **Eigenvector** \\ \hline C1 & 1 & 1/1 & 1/2 & 1/3 & 1/3 & 0.1000 & 0.1111 & 0.0714 & 0.0488 & 0.1538 & 0.4852 & 0.0970 \\ C2 & 1 & 1 & 1/2 & 1/2 & 1/3 & 0.1000 & 0.1111 & 0.0714 & 0.0732 & 0.1538 & 0.5096 & 0.1019 \\ C3 & 2 & 2 & 1 & 1/1 & 1/4 & 0.2000 & 0.2222 & 0.1429 & 0.1463 & 0.1154 & 0.8268 & 0.1654 \\ C4 & 3 & 2 & 1 & 1 & 1/4 & 0.3000 & 0.2222 & 0.1429 & 0.1463 & 0.1154 & 0.9268 & 0.1854 \\ C5 & 3 & 3 & 4 & 4 & 1 & 0.3000 & 0.3333 & 0.5714 & 0.5854 & 0.4615 & 2.2517 & 0.4503 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Example of matrix with five criteria, \(n=5\).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **C1** & **C2** & **C3** & **C4** & **C5** \\ \hline \(b_{1}\) & 7 & 9 & 9 & 6 & 8 \\ \(b_{2}\) & 8 & 7 & 8 & 5 & 7 \\ \(b_{3}\) & 9 & 6 & 8 & 5 & 9 \\ \(b_{4}\) & 6 & 7 & 8 & 8 & 6 \\ \(b_{5}\) & 8 & 7 & 7 & 8 & 6 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Matrix with five criteria (attributes) and five buyers.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Ideal and Negative-Ideal Solution** & \\ \hline \(A^{*}\) & 0.050914394 & 0.056443603 & 0.082956435 & 0.10138946 & 0.16565799 \\ \(A^{-}\) & 0.033942929 & 0.037629069 & 0.064521671 & 0.06336842 & 0.24848698 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Ideal and negative-ideal solution.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Separation Measures**} & **Relative Closeness to the Ideal Solution** & **Rank the Preference Order** \\ \hline \(\mathbf{D_{j}^{*}}\) & \(\mathbf{D_{j}^{-}}\) & \(\mathbf{C_{j}^{*}}\) & **Rank the Preference Order** \\ \hline
0.06180353 & 0.04060476 & 0.39649873 & \(b_{5}\) & 0.80006466 \\
0.04982153 & 0.05745851 & 0.53559366 & \(b_{4}\) & 0.79948347 \\
0.09351572 & 0.01931297 & 0.17117073 & \(b_{2}\) & 0.53559366 \\
0.02302864 & 0.09181794 & 0.79948347 & \(b_{1}\) & 0.39649873 \\
0.02300372 & 0.09205209 & 0.80006466 & \(b_{3}\) & 0.17117073 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Separation measures, relative closeness to the ideal solution, rank the preference order.