# A VIKOR-Based Sequential Three-Way Classification Ranking Method
## 2 Preliminaries
### Sequential Three-Way VIKOR Model Based on Minimum Group Utility Value
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Different Methods** & **Ranking Results** & **The Optimal Candidate** & **The Inferior Candidate** \\ \hline ours & \(A_{1}\succ A_{4}\succ A_{5}\succ A_{6}\succ A_{2}\succ A_{3}\) & \(A_{1}\) & \(A_{3}\) \\ [PERSON] et al.’s method [29] & \(A_{1}\succ A_{4}\succ A_{5}\succ A_{6}\succ A_{2}\succ A_{3}\) & \(A_{1}\) & \(A_{3}\) \\ TOPSIS method [32] & \(A_{1}=A_{4}\succ A_{5}\succ A_{2}\succ A_{6}\succ A_{3}\) & \(A_{1},A_{4}\) & \(A_{3}\) \\ WAA operator method [33] & \(A_{1}=A_{4}\succ A_{5}\succ A_{6}\succ A_{2}\succ A_{3}\) & \(A_{1},A_{4}\) & \(A_{3}\) \\ EDAS method [34] & \(A_{1}=A_{4}\succ A_{5}\succ A_{6}\succ A_{2}\succ A_{3}\) & \(A_{1},A_{4}\) & \(A_{3}\) \\ [PERSON] et al.’s method [30] & \(A_{1}=A_{4}\succ A_{5}\succ A_{6}\succ A_{2}\succ A_{3}\) & \(A_{1},A_{4}\) & \(A_{3}\) \\ [PERSON] et al.’s method [6] & \(A_{5}\succ A_{6}\succ A_{1}=A_{4}\succ A_{2}\succ A_{3}\) & \(A_{5}\) & \(A_{3}\) \\ VIKOR method [28] & \(A_{1}\succ A_{4}\succ A_{5}\succ A_{6}\succ A_{2}\succ A_{3}\) & \(A_{1}\) & \(A_{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison results with other methods (Example 6). Figure 6: Visualization of comparison results with other methods of Example 6.

method. The consequences show that the proposed method is as effective as other methods in dealing with the above examples, and is more effective and simpler than other methods, which extends the application scope of sequential three-way decisions and VIKOR methods. To further test the proposed method, Example 5 is used to quantitatively analyze the proposed method. Firstly, the sensitivity analysis of the decision-making mechanism coefficient \(v\) in Equation (5) is carried out, and the values in the interval [0, 1] are taken in turn with a step of 0.1 to analyze the decision index value \(Q\) corresponding to the alternatives and the ranking order of the schemes. It can be observed from Figure 7 that with the continuous increase of \(v\), the proportion of the relative distance calculated by the group utility value becomes larger and larger. This means that the decision index value corresponding to the alternative with the smaller group utility value and the decision index value corresponding to the alternative with the larger single regret value and the group utility value corresponding to the alternative with the larger group utility value becomes larger and larger. However, the relative order of all schemes is not changed, and only when \(v=0\), the ranking result is \(\{A_{7}\succ A_{1}\succ A_{2}\succ A_{4}\succ A_{5}\succ A_{6}\succ A_{3}=A_ {8}\}\), and \(A_{3}\) and \(A_{8}\) are in the same priority. When \(v=0\), the group utility value is not considered in the calculation of the decision index value, and the individual regret values \(R_{3}\) and \(R_{8}\) of \(A_{3}\) and \(A_{8}\) are both 0.3 at this time. This possible decision error is caused by only referring to partial decision information, as mentioned in this paper. When \(V\
eq 0\), because the group utility values of \(A_{3}\) and \(A_{8}\) are not the same and the group utility value is considered in the calculation of the decision index value, the ordering consequence is \(\{A_{7}\succ A_{1}\succ A_{2}\succ A_{4}\succ A_{5}\succ A_{6}\succ A_{3} \succ A_{8}\}\), which is consistent with the ordering consequences of other cases. Therefore, the proposed method is stable. Next, the correlation between other methods and the ordering consequences of the proposed method is calculated by Example 5 as shown in Table 7 and Figure 8. It can be seen from Table 7 that the proposed method is significantly correlated with other methods and has high consistency in the ranking results. Figure 7: Influence of decision mechanism coefficient \(v\) on decision index value \(Q\). In addition, it can be observed that the correlation coefficients between the method proposed in this study and other methods are higher than or equal to the correlation coefficients between other methods.