## 4 The Application of the Proposed Model
### Case Study
#### 4.1.2 Application of the Fuzzy VIKOR Method
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Criteria** & **Alternative 1** & **Alternative 2** & **Alternative A3** \\ \hline C1 & (0.294, 0.533, 0.761) & (0.326, 0.565, 0.783) & (0.304, 0.533, 0.761) \\ C2 & (0.228, 0.457, 0.696) & (0.272, 0.489, 0.717) & (0.337, 0.576, 0.793) \\ C3 & (0.261, 0.478, 0.696) & (0.293, 0.511, 0.728) & (0.337, 0.587, 0.793) \\ C4 & (0.370, 0.609, 0.815) & (0.337, 0.565, 0.772) & (0.250, 0.467, 0.696) \\ C5 & (0.217, 0.424, 0.663) & (0.272, 0.489, 0.717) & (0.402, 0.641, 0.837) \\ C6 & (0.207, 0.413, 0.641) & (0.370, 0.620, 0.826) & (0.337, 0.576, 0.793) \\ C7 & (0.196, 0.413, 0.641) & (0.315, 0.543, 0.761) & (0.326, 0.554, 0.783) \\ C8 & (0.207, 0.413, 0.652) & (0.402, 0.641, 0.837) & (0.337, 0.554, 0.772) \\ C9 & (0.315, 0.543, 0.761) & (0.370, 0.587, 0.793) & (0.272, 0.511, 0.750) \\ C10 & (0.359, 0.576, 0.783) & (0.283, 0.500, 0.728) & (0.337, 0.576, 0.804) \\ C11 & (0.380, 0.609, 0.804) & (0.315, 0.543, 0.761) & (0.283, 0.500, 0.728) \\ C12 & (0.228, 0.424, 0.652) & (0.370, 0.598, 0.804) & (0.348, 0.587, 0.804) \\ C13 & (0.272, 0.489, 0.707) & (0.402, 0.641, 0.837) & (0.402, 0.652, 0.859) \\ C14 & (0.304, 0.543, 0.783) & (0.272, 0.489, 0.707) & (0.304, 0.543, 0.772) \\ C15 & (0.272, 0.489, 0.728) & (0.370, 0.598, 0.804) & (0.348, 0.576, 0.804) \\ \hline \hline \end{tabular} Note: The linguistic variables are represented as follows: VH (very high), H (high), F (fair), L (low), and VL (very low).  
\end{table} 
Table 9: Expert assessment results for Alternative 1. (3) Determination of the best and worst values. The best\(f_{j}^{*}\) and the worst\(f_{j}^{-}\) with respect to each criterion were identified based on the Step 3 described in Part 3.4. Accordingly, the values of the best\(f_{j}^{*}\) and the worst\(f_{j}^{-}\) are shown in Table 11.  
(4) Value calculation of Utility (S\({}_{i}\)) and Regret (G\({}_{i}\)). Therefore, the normalized score of the alternative \(i\) with criterion \(j\) can be calculated. Then, the normalized fuzzy decision matrix for the companies with respect to the criterion performance of each alternative is obtained, as shown in Table 12.  
The values of \(S_{i}\) and \(G_{i}\) were calculated as illustrated in Table 13, and the values of \(S^{*}\), \(S^{-}\), \(G^{*}\), and \(G^{-}\) were further determined as shown in Table 14. \begin{table}
\begin{tabular}{c c c} \hline
**PPE Plan** & \(S_{i}\) & \(G_{i}\) \\ \hline Alternative 1 & (0.622, 0.623, 0.649) & (0.116, 0.116, 0.116) \\ Alternative 2 & (0.417, 0.448, 0.464) & (0.080, 0.085, 0.090) \\ Alternative 3 & (0.365, 0.365, 0.364) & (0.193, 0.193, 0.193) \\ \hline \end{tabular}
\end{table} 
Table 13: The values of \(S_{i}\) and \(G_{i}\) for each alternative. \begin{table}
\begin{tabular}{c c c} \hline
**Criteria** & **Alternative 1** & **Alternative 2** & **Alternative A3** \\ \hline C1 & (0.067, 0.067, 0.067) & (0.000, 0.000, 0.000) & (0.044, 0.066, 0.065) \\ C2 & (0.116, 0.116, 0.116) & (0.070, 0.085, 0.091) & (0.000, 0.000, 0.000) \\ C3 & (0.086, 0.086, 0.086) & (0.049, 0.060, 0.057) & (0.000, 0.000, 0.000) \\ C4 & (0.000, 0.000, 0.000) & (0.053, 0.060, 0.070) & (0.193, 0.193, 0.193) \\ C5 & (0.114, 0.114, 0.114) & (0.081, 0.080, 0.078) & (0.000, 0.000, 0.000) \\ C6 & (0.048, 0.048, 0.048) & (0.000, 0.000, 0.000) & (0.010, 0.010, 0.009) \\ C7 & (0.039, 0.039, 0.039) & (0.003, 0.003, 0.006) & (0.000, 0.000, 0.000) \\ C8 & (0.080, 0.080, 0.080) & (0.000, 0.000, 0.000) & (0.027, 0.030, 0.028) \\ C9 & (0.010, 0.010, 0.014) & (0.000, 0.000, 0.000) & (0.018, 0.018, 0.018) \\ C10 & (0.000, 0.000, 0.023) & (0.080, 0.080, 0.080) & (0.023, 0.000, 0.000) \\ C11 & (0.000, 0.000, 0.000) & (0.029, 0.026, 0.025) & (0.043, 0.043, 0.043) \\ C12 & (0.029, 0.029, 0.029) & (0.000, 0.000, 0.000) & (0.004, 0.002, 0.000) \\ C13 & (0.024, 0.024, 0.024) & (0.000, 0.002, 0.003) & (0.000, 0.000, 0.000) \\ C14 & (0.000, 0.000, 0.000) & (0.053, 0.053, 0.053) & (0.000, 0.000, 0.008) \\ C15 & (0.011, 0.011, 0.011) & (0.000, 0.000, 0.000) & (0.002, 0.002, 0.000) \\ \hline \end{tabular}
\end{table}