## 3 Materials and Methods
### Methods
#### 3.2.2 Fuzzy TOPSIS
\[\text{S}_{\text{i}}=\sum_{\text{j=1}}^{\text{m}}\text{M}_{\text{ij}}\times[ \sum_{\text{i=1}}^{\text{n}}\sum_{\text{j=1}}^{\text{m}}\text{M}_{\text{ij}}] ^{-1} \tag{1}\]  
\[\text{s.t}\sum_{\text{j=1}}^{\text{m}}\text{M}_{\text{ij}}=(\sum_{\text{j=1}}^ {\text{m}}\text{l}_{\text{ij}},\sum_{\text{j=1}}^{\text{m}}\text{m}_{\text{ij} },\sum_{\text{j=1}}^{\text{m}}\text{u}_{\text{ij}}) \tag{2}\]  
\[\sum_{\text{i=1}}^{\text{n}}\sum_{\text{j=1}}^{\text{m}}\text{M}_{\text{ij}}= (\sum_{\text{i=1}}^{\text{n}}\sum_{\text{j=1}}^{\text{m}}\text{l}_{\text{ij}},\sum_{\text{i=1}}^{\text{n}}\sum_{\text{j=1}}^{\text{m}}\text{m}_{\text{ij} },\sum_{\text{i=1}}^{\text{n}}\sum_{\text{j=1}}^{\text{m}}\text{u}_{\text{ij}}) \tag{3}\]  
\[\left[\sum_{\text{i=1}}^{\text{n}}\sum_{\text{j=1}}^{\text{m}}\right]^{-1}=[ \frac{1}{\sum_{\text{i=1}}^{\text{n}}\sum_{\text{j=1}}^{\text{m}}\text{u}_{ \text{ij}}},\frac{1}{\sum_{\text{i=1}}^{\text{m}}\sum_{\text{j=1}}^{\text{m}} \text{m}_{\text{ij}}},\frac{1}{\sum_{\text{i=1}}^{\text{n}}\sum_{\text{j=1}}^ {\text{m}}\text{l}_{\text{ij}}}] \tag{4}\]  
For alternatives, the respondents were asked to rate 84 alternatives in total: 12 alternatives for each of the 7 criteria. Considering fatigue associated with filling out the questionnaire survey, this study used a 5-point Likert scale for alternatives. The fuzzification of alternatives by criterion collected from multiple evaluators is calculated as in Equation (5), and the fuzzy weight is calculated as in Equation (6) [61]. \[\text{a}_{\text{ij}}=\text{min}_{\text{k}}(\text{a}_{\text{ij}}{}^{\text{k}}) \text{, }\text{b}_{\text{ij}}=\frac{1}{\text{k}}\sum_{\text{k=1}}^{\text{k}}\text{b} _{\text{ij}}{}^{\text{k}}\text{, }\text{c}_{\text{ij}}=\text{max}_{\text{k}}(\text{c}_{\text{ij}}{}^{\text{k}}) \tag{5}\]  
Figure 2: TFN M. \begin{table}
\begin{tabular}{c c} \hline \hline
**Language Scale** & **TFN** \\ \hline Extremely low (e.l) & (0, 0, 0.1) \\ \hline Very low (v.l) & (0, 0.1, 0.3) \\ \hline Low (l) & (0.1, 0.3, 0.5) \\ \hline Medium (m) & (0.3, 0.5, 0.7) \\ \hline High (h) & (0.5, 0.7, 0.9) \\ \hline Very high (v.h) & (0.7, 0.9, 1.0) \\ \hline Extremely high (e.h) & (0.9, 1.0, 1.0) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Fuzzy scale for evaluation criteria. \[w_{j1}=\min_{k}(w_{j1}^{k})\,\ w_{j2}=\frac{1}{k}\sum_{k=1}^{k}w_{j2}^{k}\,\ w_{j3}=\max_{k}(w_{j3}^{k}) \tag{6}\]  
Then, fuzzy decision matrices \(\widetilde{V}=(\widetilde{v_{ij}})\) and \(\widetilde{v_{ij}}=\widetilde{r_{ij}}\times w_{j}\) are obtained with a normalized fuzzy decision matrix \(\widetilde{R}=[\widetilde{r_{ij}}]\) (Equations (7) and (8)) and the weight of evaluation criteria (TFN) from Equation (4) [61]. \[\widetilde{r_{ij}}=(\frac{a_{\bar{ij}}}{c_{j}^{*}},\frac{b_{\bar{ij}}}{c_{j}^{ *}},\frac{c_{\bar{ij}}}{c_{j}^{*}})\ \text{and}\ \text{c}_{j}^{*}=\max_{i}(c_{ij})\ \left(\text{benefit \ criteria}\right) \tag{7}\]  
\[\widetilde{r_{ij}}=(\frac{a_{j}^{-}}{c_{\bar{ij}}},\frac{a_{j}^{-}}{b_{\bar{ ij}}},\frac{a_{j}^{-}}{a_{\bar{ij}}})\ \text{and}\ \text{c}_{j}^{-}=\min_{i}(a_{\bar{ij}})\ \left(\text{cost \ criteria}\right) \tag{8}\]  
Afterwards, the fuzzy positive ideal solution (FPIS) and the fuzzy negative ideal solution (FNIS) are calculated as described in Equations (9) and (10) [61]. \[\text{A}^{*}=\left(\widetilde{v}_{1}^{*},\widetilde{v}_{2}^{*},\,\cdots, \,\widetilde{v}_{n}^{*}\right)\text{, where}\ \widetilde{v}_{j}^{*}\ =\ \max_{i} \bigl{(}v_{j\bar{ij}}\bigr{)} \tag{9}\]  
\[\text{A}^{-}=\left(\widetilde{v}_{1}^{-},\widetilde{v}_{2}^{-},\cdots, \,\widetilde{v}_{n}^{-}\right)\text{, where}\ \widetilde{v}_{j}^{-}\ =\min_{i} \bigl{(}v_{j\bar{j}1}\bigr{)} \tag{10}\]  
To measure the distance to the FPIS and FNIS for each of the alternatives, this study used the two-TFN distance calculation method proposed by [PERSON] [57].