# A VIKOR-Based Sequential Three-Way Classification Ranking Method
## 2 Preliminaries
### Sequential Three-Way VIKOR Model Based on Minimum Group Utility Value
\(S_{1}=0.1262\), \(S_{2}=0.2857\), \(S_{3}=0.8404\), \(S_{4}=0.5642\), \(S_{5}=0.4571\), \(S_{6}=0.6911\), \(S_{7}=0.1167\), \(S_{8}=0.8167\). \(R_{1}=0.0500\), \(R_{2}=0.1000\), \(R_{3}=0.3000\), \(R_{4}=0.2000\), \(R_{5}=0.1500\), \(R_{6}=0.2500\), \(R_{7}=0.0667\), \(R_{8}=0.2500\). Step 3. Classify the alternatives based on Algorithm 2 by the maximum single regret value and the minimum group utility value, which can be obtained \(POS^{4}=\{A_{7},A_{1},A_{2},A_{5}\}\), \(NEG^{4}=\{A_{3},A_{6},A_{8},A_{4}\}\).The boundary region \(BND^{4}\) is empty because there is no alternative that has both the maximum single regret value and the minimum group utility value when classifying the alternatives by sequential three-way decisions. Step 4. Calculate the decision index values of all schemes according to Equation (5), and \(Q_{i}\) represents the decision index value of the \(i\)th scheme, which can be obtained by

\(Q_{1}=0.0066\), \(Q_{2}=0.2168\), \(Q_{3}=1\), \(Q_{4}=0.6092\), \(Q_{5}=0.4352\), \(Q_{6}=0.7969\), \(Q_{7}=0.0334\), \(Q_{8}=0.8836\). Step 5. On the basis of the ordering regulations of alternatives in the identical area, the alternatives in the identical area are sorted in ascending rank of decision index values, and the ranking result in the positive region is \(\{A_{1}\succ A_{7}\succ A_{2}\succ A_{5}\}\), and the ranking result in the negative region is \(\{A_{4}\succ A_{6}\succ A_{8}\succ A_{3}\}\). Step 6. The final ranking result \(\{A_{1}\succ A_{7}\succ A_{2}\succ A_{5}\succ A_{4}\succ A_{6}\succ A_{8} \succ A_{3}\}\) is obtained through the ordering regulations \(POS\{A\}\succ BND\{A\}\succ NEG\{A\}\) of different regions. Method 2. Sequential three-way VIKOR model based on minimum individual regret value and maximum group utility value:

Step 1. This step is the same as in Method 1. Figure 3: Visualization of Example 5. Table 6 shows that the ordering consequences of the proposed method are consistent with those of most methods. The ordering consequences of the proposed method are identical as those of [PERSON] [29] and VIKOR method [28]. The impurity between the proposed method and other methods is that \(A_{1}\) is used as the optimal solution in this paper. In addition to [PERSON] [6]'s method, other methods take \(A_{1}\) and \(A_{4}\) as optimal solutions at the same time. Table 6 shows that the proposed method has the same worst and optimal alternatives as most of the reference methods. According to Table 5, attributes \(\{C_{1},C_{3},C_{5}\}\) and \(\{C_{2},C_{4},C_{6}\}\) are evaluated by \(A_{1}\) and \(A_{4}\) with opposite values. Since the weights of \(\{C_{1},C_{2}\}\), \(\{C_{3},C_{4}\}\), and \(\{C_{5},C_{6}\}\) are the same, there is intuitively no difference between \(A_{1}\) and \(A_{4}\). Other methods mainly use TOPSIS for ranking, which is based on the set of distances from the ideal solution, and a compromise scheme with dominance ratio is improved by VIKOR, which is the basis of the TOPSIS method. The VIKOR method makes the decision of decision makers more conservative or radical by adding the decision mechanism coefficient since TOPSIS method. The process of TOPSIS method does not incorporate any subjective factors, so there are slight differences in the priority of \(A_{1}\) and \(A_{4}\). Besides that, most methods are consistent with the ranking results of the proposed