## 5 Case Study and Analysis
### Comparative Analysis
#### 5.3.2 Superiority of the Proposed Method  
To illustrate the strengths of the improved GBWM-PROMETHEE II method vividly, several widespread methods were applied in solving the above practical case with the q-ROF information, including VIKOR [45] and TOPSIS [18]. The comparison results are depicted by the form of ranking orders as Table 9. (Suppose \(q=3\)). As we can see from the Table 9, the rankings by utilizing three methods differ from the results by our method in this paper slightly. Meanwhile, for the sake of emphasizing the significance of the GBWM to derive each criterion weights, we further assigned the same importance degree directly to each criterion of the above-mentioned methods, then by means of \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\Pi(\alpha_{a},\alpha_{b})\) & \(\alpha_{1}\) & \(\alpha_{2}\) & \(\alpha_{3}\) & \(\alpha_{4}\) \\ \hline \(\alpha_{1}\) & & \(0.0034\) & \(-0.0206\) & \(0.1550\) \\ \(\alpha_{2}\) & \(-0.0034\) & & \(-0.0240\) & \(0.1516\) \\ \(\alpha_{3}\) & \(0.0206\) & \(0.0240\) & & \(0.1756\) \\ \(\alpha_{4}\) & \(-0.1550\) & \(-0.1516\) & \(-0.1756\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The global preference index \(\Pi(\alpha_{a},\alpha_{b})\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Methods** & **Ranking Values** & **Ranking Order** \\ \hline \(q-ROFBM\,s=1,t=0.5\)[44] & \(S(\alpha_{1})=0.2774,S(\alpha_{2})=0.2767,\) & \(\alpha_{3}>\alpha_{1}>\alpha_{2}>\alpha_{4}\) \\ The proposed method & \(\varphi(\alpha_{1})=0.0919,\varphi(\alpha_{2})=0.0827,\) & \(\alpha_{3}>\alpha_{1}>\alpha_{2}>\alpha_{4}\) \\  & \(\varphi(\alpha_{3})=0.1468,\varphi(\alpha_{4})=-0.3214.\) & \(\alpha_{3}>\alpha_{1}>\alpha_{2}>\alpha_{4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Ranking orders for different methods. comparing the final rankings as Table 10 to analyze the reliability of determining criterion weights by GBWM. In the following, the analysis concerning the comparison of different methods is described in detail. 1. Compare with the VIKOR method [45]. The VIKOR method [45] has the ability to select and rank from several alternatives in the face of the situation with conflicting criteria. However, from Table 9, we can conclude that this method cannot effectively distinguish the alternatives in our case. By means of calculating the values of \(S,R\), and \(Q\) with respect to the four alternatives as well as the sorting rules, the ranking result can be gained, as \(\alpha_{1},\alpha_{2},\alpha_{3}\) are all the compromise solutions. Nevertheless, the difference in the proposed method is providing a complete ranking order by comparing the values of the comprehensive net flow, as \(\alpha_{3}>\alpha_{1}>\alpha_{2}>\alpha_{4}\). In addition, when the weights were directly assigned to each criterion instead of obtained by means of the GBWM, the ranking result derived by the VIKOR also differed from our method, which is depicted in detail in Table 10. It can be easily concluded that \(\alpha_{1}\) and \(\alpha_{2}\) are the compromise solutions, so the VIKOR cannot select the optimal agricultural material supplier in the case. In other words, the ranking results obtained by the improved GBWM- PROMETHEE II in this paper are more forceful and reasonable. 1. Compare with the TOPSIS method [18]. The TOPSIS method considers the notion of similarity between each alternative and the ideal solution, as well as the negative ideal solution. From Table 9, the ranking based on the TOPSIS differed from the order obtained by the proposed method. This is mainly because the standard form of the TOPSIS is deterministic, while the improved GBWM-PROMETHEE II method in this paper considers the significance of pairwise comparisons, as well as the positive or negative evaluation information given by DMs. Moreover, from different orders shown in Table 10, by means of assigning the same