# A VIKOR-Based Sequential Three-Way Classification Ranking Method
## 2 Preliminaries
### Sequential Three-Way VIKOR Model Based on Minimum Group Utility Value
\(R_{1}=0.2500\), \(R_{2}=0.5000\), \(R_{3}=0.0576\), \(R_{4}=0.1875\), \(R_{5}=0.3397\), \(R_{6}=0.3125\), \(R_{7}=0.4375\), \(R_{8}=0.4688\)._

_(3) divides all the alternatives into corresponding regions according to Algorithm 2. \(POS^{3}=\{A_{3},A_{1},A_{4}\}\), \(BND^{3}=\{A_{5},A_{6}\}\), \(NEG^{3}=\{A_{2},A_{8},A_{7}\}\).__(4) calculates the decision indicatrix value of each scheme according to Equation (5), where \(Q_{i}\) denotes the decision index value of the \(i\)th alternative._

\(Q_{1}=0.3642\)_, \(Q_{2}=0.9861\), \(Q_{3}=0\), \(Q_{4}=0.3513\), \(Q_{5}=0.6039\), \(Q_{6}=0.5936\), \(Q_{7}=0.8346\), \(Q_{8}=0.9647\)._

_(5) sorts alternatives for the same region in ascending order of decision index values based on the ordering regulations of alternatives in the identical area. The ordering result in the positive region is \(\{A_{3}\succ A_{4}\succ A_{1}\}\), the ranking result in the boundary area is \(\{A_{6}\succ A_{5}\}\), and the ranking result in the negative area is \(\{A_{7}\succ A_{8}\succ A_{2}\}\)._

_(6) Finally, the ranking rules use different regions when ranking the alternatives are shown in Figure 2. The final sorting result is \(\{A_{3}\succ A_{4}\succ A_{1}\succ A_{6}\succ A_{5}\succ A_{7}\succ A_{8} \succ A_{2}\}\)._

Sequential Three-Way VIKOR Model Based on Maximum Group Utility Value and Minimum Individual Regret Value

In the sequential three-way decision classification, the scheme with the largest group utility value is separated into the negative area, and the scheme with the smallest individual regret value is divided into the positive region. If the alternative with the minimum single regret value and the maximum group utility value is also divided into the boundary region, the other steps of the proposed model are roughly like the sequential three-way VIKOR model according to the maximum individual regret value and the minimum group utility value. ```
Input: Group utility values and individual regret values for m alternatives \(k^{th}-level\), \(k=0\); output: Three decision regions, \(POS^{k}\)\(BND^{k}_{new}\) and \(NEG^{k}\);
1 POS = \(\emptyset\), BND = {Am}, NEG = \(\emptyset\), k = 0, S = [Sm], R = [Rm]
2 while\(BND^{k}\
eq\varnothing\)do
3\(k++\), \(POS_{new}=NEG_{new}=\varnothing\);
4\(A_{\max}=\min(S)\), \(A_{\min}=\max(R)\);
5 if\(A_{\max}\)\(=\)\(A_{\min}\)
6 Add \(A_{\max}\) to \(BND_{new}\)
7\(BND^{k}_{new}=BND^{k-1}_{new}\cup BND_{new}\);
8 else Add \(A_{\max}\) to \(POS_{new}\)
9\(POS^{k}_{new}=POS^{k-1}_{new}\cup POS_{new}\);
10 Add \(A_{\min}\) to \(NEG_{new}\)
11\(NEG^{k}=NEG^{k-1}\cup NEG_{new}\);
12\(BND^{k}=BND^{k-1}-(POS_{new}\cup BND_{new}\cup NEG_{new})\);
13\(S=S-(A_{\max}\cup A_{\min})\);
14
15 end while
16 Output \(POS^{k}(A_{\min})\),\(BND^{k}_{new}(A_{m})\),\(NEG^{k}(A_{m})\). ```

**Algorithm 2** Sequential three-way VIKOR model based on minimum group utility value and maximum individual regret value

**Example 4**. : (_Similar to Example 3), Using this model to process Table 2 can acquire the final ordering result of the schemes of the three-way sequential VIKOR model according to the minimum single regret value and the maximum group utility value as \(\{A_{3}\succ A_{4}\succ A_{1}\succ A_{6}\succ A_{5}\succ A_{7}\succ A_{8} \succ A_{2}\}\). It can be seen



From Example 1 and Example 2, it can be seen that there is \(A_{5}\) difference in the final ranking results of \(A_{5}\) and \(A_{6}\). Example 1 divides \(A_{5}\) into the positive area and \(A_{6}\) into the negative area, while Example 2 divides \(A_{6}\) into the positive area and \(A_{5}\) into the negative area, which eventually leads to the difference in the ranking results. Therefore, it is difficult to acquire an accurate ranking consequence by only considering the partial decision information of \(A_{5}\) single individual regret value or group utility value. By comprehensively considering the single regret value and group utility value, the result that \(A_{6}\) is better than \(A_{5}\) is consistent with the result obtained by VIKOR method.