## 3 Methodology
### Fuzzy-TOPSIS Method  
The selected alternative demonstrates the most significant deviation from the \"unfavorable ideal option\" and the nearest resemblance to the \"favorable ideal option\" [46]. The optimal selection is distinguished by incorporating the most favorable criteria, while the suboptimal choice is formed by amalgamating the least desirable performances. To address the inherent imprecision and subjectivity associated with experts' judgments, the use of fuzzy TOPSIS is employed. This approach involves expressing evaluations in linguistic words and transforming them into fuzzy terms [47]. The fuzzy-TOPSIS approach entails the following procedural stages:  
**Stage 1.**_The linguistic values \(\overset{\sim}{w}_{ji},j=1,2,\ldots m;i=1,2,\ldots n\) should be established for the options in relation to the criteria [48]. To omit the \"normalization stage, the fuzzy linguistic rating\" \(\overset{\sim}{w}_{ji}\) is defined using a range of \"normalized triangular fuzzy integers spanning\"between 0 and 10 as  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Fuzzy Score** & **Linguistic Scale** \\ \hline (0.0, 0.0, 0.1) & VL \\ (0.0, 0.1, 0.3) & L \\ (0.1, 0.3, 0.5) & ML \\ (0.3, 0.5, 0.7) & M \\ (0.5, 0.7, 0.9) & MH \\ (0.7, 0.9, 1.0) & H \\ (0.9, 1.0, 1.0) & VH \\ \hline \hline \end{tabular}
\end{table}
Table 2: Linguistic variables for each criterion’s significance value.  
displayed in Table 3 (\"Very Poor (VP), Poor (P), Medium Poor (MP), Medium (M), Medium Good (MG), Good (G), Very Good (VG)\") [49]._  
**Stage 2.**_The \"weighted normalized fuzzy decision\" matrix can be calculated using:_  
\[\widetilde{v}=\left[\widetilde{v}_{ji}\right]_{m\
eq j}j=1,2,\ldots m;i=1,2, \ldots n \tag{5}\]  
\[\widetilde{v}_{ji}=\widetilde{w}_{ji}*x_{i} \tag{6}\]  
_The measurement of \(x_{i}\) is conducted through the utilization of data obtained from expert surveys._  
**Stage 3.**_The favorable-ideal (\(B^{*}\) ) and unfavorable-ideal (\(B^{-}\)) possibilities can be evaluated using:_  
\[B^{*}=\left\{v_{1}^{*},\ldots\ldots\ldots,v_{j}^{*}\right\}=\left\{\left( \max_{i}v_{j},j\in\omega_{a}\right),\left(\min_{i}v_{i},j\in\omega_{d}\right)\right\} \tag{7}\]  
\[B^{-}=\left\{v_{1}^{-},\ldots\ldots\ldots,v_{j}^{-}\right\}=\left\{\left(\max _{i}v_{j},j\in\omega_{a}\right),\left(\min_{i}v_{i},j\in\omega_{d}\right)\right\} \tag{8}\]  
_The sets of benefit criteria are denoted as \(\omega_{a}\), whereas the sets of cost criteria are denoted as \(\omega_{d}\)._  
**Stage 4.**_Determine how far the choice is between the best possible outcome and the worst possible one._  
\[C_{j}^{*}=\sum_{i=1}^{n}c\left(\widetilde{V}_{i},\widetilde{V}_{j}\right),j=1,2,..m \tag{9}\]  
\[C_{j}^{-}=\sum_{i=1}^{n}c\left(\widetilde{V}_{i},\widetilde{V}_{j}\right),j=1,2,..n \tag{10}\]  
\[c\left(\widetilde{b},\widetilde{a}\right)=\sqrt{\left(\frac{1}{3}\right) \left[\left(b_{1}-a_{1}\right)^{2}+\left(b_{2}-a_{2}\right)^{2}+\left(b_{3}- a_{3}\right)^{2}\right]} \tag{11}\]  
_where \(b\) and a represent two triangular fuzzy integers denoted as \(\left(b_{1},\,b_{2},\,b_{3}\right)\) and \(\left(a_{1},\,a_{2},\,a_{3}\right)\), respectively._  
**Stage 5.**_Determine the degree of proximity of each alternative to the optimal choice. The proximity between alternative B8 and B9 can be described as follows:_  
\[ED_{j}=\frac{C_{j}^{-}}{c_{j}^{*}+C_{j}^{-}},j=1,2,\ldots n \tag{12}\]