## 4 The Application of the Proposed Model
### Result Robustness and Sensitivity Analysis  
Based on the ranking results, this research discussed the ranking robustness and sensitivity of the results and how these influence the final decision-making. The result robustness has a critical impact on the result reliability and accuracy. In this research, one way to consider robustness in the ANP-fuzzy VIKOR method for PPE plan selection is to perform a sensitivity analysis. Sensitivity analysis focuses on texting the effects of changes in the value of parameter \(v\) on the ANP-fuzzy VIKOR model results, which helps to identify the most possible solution for optimal selection. As shown in Table 16, with the parameter \(v\) changing from 0.1 to 1.0, the \(Q_{i}\) values of different alternatives changed. For Alternative 1, it achieved a \(Q_{i}\) value of 0.363 when parameter \(v\) was assumed to be 0.1, while the \(Q_{i}\) value was 1.0 when \(v\) was set to be 1.0. Therefore, the \(Q_{i}\) value increased from 0.363 to 1.0 when parameter \(v\) increased. Alternative 2 achieved a \(Q_{i}\) value of 0.032 when \(v\) was 0.1, and the value increased to 0.320 with \(v\) increasing to 1.0. On the other hand, for Alternative 3, the value of \(Q_{i}\) was 0.9 when \(v\) was assumed to be 0.1; however, it decreased to 0 if the parameter \(v\) increased to 1.0. Therefore, when the value of parameter \(v\) was increasing, the \(Q_{i}\) value of Alternative 1 and Alternative 2 increased significantly, and the \(Q_{i}\) value of Alternative 3 decreased. As a result, these changes had a critical impact on the alternative ranking result.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\(v\)} & \multicolumn{4}{c}{**Value of \(Q_{i}\)**} \\ \cline{2-5}  & **Alternative 1 (A1)** & **Alternative 2 (A2)** & **Alternative A3 (A3)** & **Ranking** \\ \hline
0.1 & 0.3630 & 0.0320 & 0.9000 & A2 \(>\) A1 \(>\) A3 \\
0.2 & 0.4337 & 0.0641 & 0.8000 & A2 \(>\) A1 \(>\) A3 \\
0.3 & 0.5045 & 0.0961 & 0.7000 & A2 \(>\) A1 \(>\) A3 \\
0.4 & 0.5753 & 0.1281 & 0.6000 & A2 \(>\) A1 \(>\) A3 \\
0.5 & 0.6461 & 0.1602 & 0.5000 & A2 \(>\) A3 \(>\) A1 \\
0.6 & 0.7169 & 0.1922 & 0.4000 & A2 \(>\) A3 \(>\) A1 \\
0.7 & 0.7877 & 0.2242 & 0.3000 & A2 \(>\) A3 \(>\) A1 \\
0.8 & 0.8584 & 0.2562 & 0.2000 & A3 \(>\) A2 \(>\) A1 \\
0.9 & 0.9292 & 0.2883 & 0.1000 & A3 \(>\) A2 \(>\) A1 \\
1.0 & 1.0000 & 0.3203 & 0.0000 & A3 \(>\) A2 \(>\) A1 \\ \hline \hline \end{tabular}
\end{table}
Table 16: Result comparisons with different values of parameter \(v\) in the ANP-fuzzy VIKOR model.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**PPE Plan** & **ANP-Fuzzy VIKOR Scores (\(Q_{i}\))** & **Optimal Ranking** \\ \hline Alternative 1 & 0.646 & 3 \\ Alternative 2 & 0.160 & 1 \\ Alternative 3 & 0.500 & 2 \\ \hline \hline \end{tabular}
\end{table}
Table 15: The ANP-fuzzy VIKOR scores (\(Q_{i}\)) and optimal ranking of PPE plans.  
As illustrated in Figure 6, when the parameter \(v\) was lower than 0.5, the optimal solution among all alternatives was Alternative 2, and Alternative 1 was the second optimal solution, while Alternative 3 performed the worst. When the value of \(v\) was between the values of 0.5 and 0.3, Alternative 2 achieved the best among all alternatives, while Alternative 3 and Alternative 3 achieved worse than Alternative 1. When the value of \(v\) was set to be less than 0.2, Alternative 3 achieved the best. And Alternative 2 performed the second, while Alternative 1 performed the worst. Therefore, the results from the sensitivity analysis indicated that in most cases, Alternative 2 achieved the best among the three alternatives; however, Alternative 3 can be the optimal solution when parameter \(v\) is greater than 0.8.