## 3 Background
### Technique for Order of Preference by Similarity to Ideal Solution  
The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) is a multi-criteria decision analysis method developed by [PERSON] and [PERSON] in 1981. It is a decision method based on choosing between different alternatives that have the shortest geometric distance from the positive ideal solution and the longest geometric distance from the negative ideal solution. It should be mentioned that the positive ideal solution is the sum of all the best values for each attribute, and, conversely, the negative ideal solution is the sum of all the worst values of each attribute. Given a set of \(m\) alternatives and a set of \(n\) criteria with a set of \(n\) weights, we have a matrix \(\left(f_{ij}\right)_{m\ X\ n}\). The notation for the TOPSIS method is in the Table 3.  
\begin{table}
\begin{tabular}{c c c} \hline
**Number** & **Notations** & **Description** \\ \hline
1 & A & Matrix \\
2 & \(a_{ij}\) & Element of matrix (column \(i\), row \(j\)) \\
3 & \(n\) & Size of matrix (in this case a square matrix) \\
4 & \(\omega_{i}\) & weight \\
5 & \(\mathbf{w}\) & eigenvector \\ \hline \end{tabular}
\end{table}
Table 2: Notation for the AHP method.  
The TOPSIS procedure is performed in six stages, as follows [48]:  
1. Calculate the normalized decision matrix. The \(r_{ij}\) is calculated as \[r_{ij}=f_{ij}/\sqrt{\sum_{j=1}^{m}f_{ij}^{2}},\;\;j=1,\ldots,m\;;\;i=1,\ldots,n\] (5)
2. Calculate the weighted normalized decision matrix. The weighted normalized value \(v_{ij}\) is calculated as \[v_{ij}=w_{i}r_{ij},\;\;j=1,\ldots,m;i=1,\ldots,n\] (6) where \(w_{i}\) is the weight of the \(i\)th attribute or criterion, and \[\sum_{i=1}^{n}w_{i}=1\] (7)
3. Determine the ideal and negative-ideal solution. \[A^{*}=\{v_{1}^{*},\ldots,v_{n}^{*}\}=\left\{\binom{max}{j}v_{ij}|i\in I^{ \prime}\right),\binom{min}{j}v_{ij}|i\in I^{\prime}\right)\}\] (8) \[A^{-}=\left\{v_{1}^{-},\ldots,v_{n}^{-}\right\}=\left\{\binom{min}{j}v_{ij}|i \in I^{\prime}\right),\binom{max}{j}v_{ij}|i\in I^{\prime}\right)\}\] (9) where \(I^{\prime}\) is associated with benefit criteria, and \(I^{\prime\prime}\) is associated with cost criteria.
4. Calculate the separation measures, using the \(n\)-dimensional Euclidean distance. The separation of each alternative from the ideal solution is given as \[D_{j}^{*}=\sqrt{\sum_{i=1}^{n}(v_{ij}-v_{i}^{*})^{2}},\;\;\;\;j=1,\ldots,m\] (10) Similarly, the separation from the negative ideal solution is given as \[D_{j}^{-}=\sqrt{\sum_{i=1}^{n}(v_{ij}-v_{i}^{-})^{2}},\;\;\;\;j=1,\ldots,m\] (11)  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Number** & **Notations** & **Description** \\ \hline
1 & m & Number of alternative \\
2 & n & Number of criteria \\
3 & \(f_{ij}\) & Element of matrix (column \(i\), row \(j\)) \\
4 & \(r_{ij}\) & Element of normalized decision matrix \\
5 & \(w_{i}\) & Weight of the \(i\)th attribute or criterion \\
6 & \(v_{ij}\) & Element of weight normalized decision matrix \\
7 & \(A^{*}=\{v_{1}^{*},\ldots,v_{n}^{*}\}\) & Set ideal solution \\
8 & \(A^{-}=\{v_{1}^{*},\ldots,v_{n}^{*}\}\) & Set negative ideal solution \\
9 & \(I^{\prime}\) & Benefit criteria \\
10 & \(I^{*}\) & Cost criteria \\
11 & \(D_{j}^{*}\) & Separation from the ideal solution \\
12 & \(D_{j}^{-}\) & Separation from the negative ideal solution \\
13 & \(C_{j}^{*}\) & Relative closeness to the ideal solution \\ \hline \hline \end{tabular}
\end{table}
Table 3: Notation for the TOPSIS method.  
Calculate the relative closeness to the ideal solution. The relative closeness of the alternative \(a_{j}\) with respect to \(A^{*}\) is defined as \[C_{j}^{*}=D_{j}^{-}/\left(D_{j}^{*}+D_{j}^{-}\right),\ \ j=1,\ldots,m\] (12) Rank the preference order.