## 4 Application of IVT2F-TOPSIS for International Disputes Resolution
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multicolumn{11}{c}{**Alternatives**} & \multicolumn{11}{c}{**C\({}_{\mathbf{1}}\)**} \\ \hline \(A_{1}\) & 0.000 & 0.033 & 0.033 & 0.167 & 1.000 & 1.000 & 0.017 & 0.033 & 0.033 & 0.100 & 0.900 & 0.900 \\ \(A_{2}\) & 0.067 & 0.233 & 0.233 & 0.433 & 1.000 & 1.000 & 0.150 & 0.233 & 0.233 & 0.333 & 0.900 & 0.900 \\ \(A_{3}\) & 0.167 & 0.367 & 0.367 & 0.567 & 1.000 & 1.000 & 0.267 & 0.367 & 0.367 & 0.467 & 0.900 & 0.900 \\ \(A_{4}\) & 0.300 & 0.500 & 0.500 & 0.700 & 1.000 & 1.000 & 0.400 & 0.500 & 0.500 & 0.600 & 0.900 & 0.900 \\ \(A_{5}\) & 0.367 & 0.567 & 0.567 & 0.767 & 1.000 & 1.000 & 0.467 & 0.567 & 0.567 & 0.667 & 0.900 & 0.900 \\ \(A_{6}\) & 0.433 & 0.633 & 0.633 & 0.833 & 1.000 & 1.000 & 0.533 & 0.633 & 0.633 & 0.733 & 0.900 & 0.900 \\ \(A_{7}\) & 0.700 & 0.900 & 0.900 & 1.000 & 1.000 & 1.000 & 0.800 & 0.900 & 0.950 & 0.900 & 0.900 & 0.900 \\ \(A_{8}\) & 0.900 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 0.950 & 1.000 & 1.000 & 1.000 & 0.900 & 0.900 \\ \hline \multicolumn{11}{c}{**Alternatives**} & \multicolumn{11}{c}{**C\({}_{\mathbf{2}}\)**} \\ \hline \(A_{1}\) & 0.100 & 0.233 & 0.233 & 0.433 & 1.000 & 1.000 & 0.167 & 0.233 & 0.233 & 0.333 & 0.900 & 0.900 \\ \(A_{2}\) & 0.167 & 0.367 & 0.367 & 0.567 & 1.000 & 1.000 & 0.267 & 0.36 As shown in Step 3, the linguistic variables in Table 3 are used to calculate the weights of the importance matrix (\(Wp\)) for the attributes of the \(pth\) expert. The linguistic variables are converted to IVT2 FSs using Equation (3). Then, the average weighting matrix (\(\overline{W}\)) is created shown by applying Equation (4), as shown in Table 6. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline
**Criteria** & \multicolumn{10}{c}{**Average Weighting Matrix**} \\ \hline \(C_{1}\) & 0.233 & 0.433 & 0.433 & 0.633 & 1.000 & 1.000 & 0.333 & 0.433 & 0.433 & 0.533 & 0.900 & 0.900 \\ \(C_{2}\) & 0.433 & 0.633 & 0.633 & 0.833 & 1.000 & 1.000 & 0.533 & 0.633 & 0.633 & 0.733 & 0.900 & 0.900 \\ \(C_{3}\) & 0.567 & 0.767 & 0.767 & 0.933 & 1.000 & 1.000 & 0.667 & 0.767 & 0.767 & 0.850 & 0.900 & 0.900 \\ \(C_{4}\) & 0.767 & 0.933 & 0.933 & 1.000 & 1.000 & 1.000 & 0.850 & 0.933 & 0.933 & 0.967 & 0.900 & 0.900 \\ \(C_{5}\) & 0.633 & 0.767 & 0.767 & 0.833 & 1.000 & 1.000 & 0.700 & 0.767 & 0.767 & 0.800 & 0.900 & 0.900 \\ \(C_{6}\) & 0.333 & 0.500 & 0.500 & 0.667 & 1.000 & 1.000 & 0.417 & 0.500 & 0.500 & 0.583 & 0.900 & 0.900 \\ \hline \hline \end{tabular}  
A weighted decision matrix (\(\overline{WD}_{w}\)) is created using values from Table 6, as shown in Step 4. These matrix values are calculated with Equation (5), as shown in Table 7.  
\end{table} 
Table 6: Average weighting matrix. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline
**Criteria** & \multicolumn{10}{c}{**\(A_{1}\)**} \\ \hline \(C_{1}\) & 0.000 & 0.014 & 0.014 & 0.106 & 1.000 & 1.000 & 0.006 & 0.014 & 0.014 & 0.053 & 0.810 & 0.810 \\ \(C_{2}\) & 0.043 & 0.148 & 0.148 & 0.361 & 1.000 & 1.000 & 0.089 & 0.148 & 0.148 & 0.244 & 0.810 & 0.810 \\The negative ideal solution (NIS) for \(C^{-}=(z_{1}^{-},\ z_{2}^{-},\ \ldots,\ z_{m}^{-})\) and the positive ideal solution (PIS) for \(C^{+}=(z_{1}^{+},\ z_{2}^{+},\ \ldots,\ z_{m}^{+})\) are computed by applying Equations (11) and (12), respectively. The NIS and PIS results are given in Table 9.  
The distances from NIS and PIS are calculated by Equations (13) and (14), and the degree of closeness (\(DoC(x_{j})\)) is computed using Equation (15).