## 3 Selection of Methods for Methodology
### Prommethee-IFS Method
If \(A_{j}>A_{k}\), the difference \(d_{i}\) between two IFNs is calculated as \(d_{i}\big{(}A_{j},\ A_{k}\big{)}=IFN_{ij}\ominus IFN_{ik}\), and if \(A_{j}<A_{k}\), the difference is calculated as \(IFN_{ik}\ominus IFN_{ij}\). (Equation (17)). \(PFN_{ij}\) corresponds to the value of the \(i\)-th criterion of the alternative \(A_{j}\), and \(PFN_{ik}\) corresponds to the value of \(k\)-th criterion of the alternative \(A_{k}\). \(P_{t}(d)=P_{t}\big{(}d_{i}\big{(}A_{j},\ A_{k}\big{)}\big{)}\) is the \(k\)-th preference function (Figure 7) for the \(i\)-th criterion selected from the six types of criteria functions proposed by the authors of the PROMETHEE method. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Linguistic Terms** & **Normalized Value \(r_{ij}\)** & **IFN** \\ \hline Extremely good (EG)/1.0 & 1.00 & (1.00, 0.00) \\ Very very good (VVG)/0.9 & 0.90 & (0.90, 0.10) \\ Very good (VG)/0.8 & 0.80 & (0.80, 0.20) \\ Good (G)/0.7 & 0.70 & (0.70, 0.30) \\ Medium good (MG)/0.6 & 0.60 & (0.60, 0.40) \\ Medium (M)/0.5 & 0.50 & (0.50, 0.50) \\ Medium bad (MB)/0.4 & 0.40 & (0.40, 0.60) \\ Bad (B)/0.3 & 0.30 & (0.30, 0.70) \\ Very bad (VB)/0.2 & 0.20 & (0.20, 0.80) \\ Very very bad (VVB)/0.1 & 0.10 & (0.10, 0.90) \\ Extremely bad (EB)/0.0 & 0.00 & (0.00, 1.00) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Fuzzification table for the criteria of the alternatives. \[P_{t}(d)=\left\{ \begin{aligned} 0,\ 0<d\left(A_{j},\ A_{k} \right)&\leq q\\ \frac{d\left(A_{j},\ A_{k}\right)-q}{p-q},\ q<d\left(A_{j},\ A_{k} \right)&\leq p\\ 1,\ d\left(A_{j},\ A_{k}\right)>p\end{aligned} \right. \tag{26}\]  
The preference function \(P_{t}(d)\) assigns values between 0 and 1 to quantify the degree of preference or indifference between two alternatives. Here, \(p\) and \(q\) are the preference function's thresholds, i.e., the maximum and minimum values of the selected criteria. (5) Calculation of the positive outranking flow:  
\[F_{j}^{+}=\sum_{k=1}^{n}\pi\big{(}A_{j},\ A_{k}\big{)},\ j=1,2,\ \ldots,\ n. \tag{27}\]  
Calculation of the negative outranking flow:  
\[F_{j}^{-}=\sum_{k=1}^{n}\pi\big{(}A_{k},\ A_{j}\big{)},\ j=1,2,\ \ldots,\ n. \tag{28}\]  
(6) Calculation of the positive net flow value \(F_{j}\) is a step that needs to be performed to determine the final rank of the alternative \(A_{j}\). This value is obtained by subtracting the negative flow values from the positive flow values:  
\[F_{j}=F_{j}^{+}-F_{j}^{-} \tag{29}\]  
_Note:_ To accommodate the characteristics of fuzzy sets, such as the non-negativity of fuzzy numbers, the calculation of negative net flow values is carried out using additional intuitionistic algebra, which can be represented as follows:  
\[F_{j}^{c}=F_{j}^{-}-F_{j}^{+} \tag{30}\]  
(7) Defuzzification is the reverse process of fuzzification, where fuzzy data is converted into crisp data. The process of defuzzification of the net flow value \(F_{j}\) involves calculating the score function \(S\big{(}F_{j}\big{)}\) for each alternative \(A_{j}\). Based on \(S\big{(}F_{j}\big{)}\) values, the ranking of the alternatives is determined as described in _Definition 4_. The sign of the final results of PROMETHEE-IFS is interpreted by directly ranking the alternatives with a positive score value \(S\big{(}F_{j}\big{)}\), where the best alternative would have the highest score value. However, for negative score values, the best alternative is the one with the smallest score value. Figure 7: Preference function of the PROMETHEE-IFS method.