# A VIKOR-Based Sequential Three-Way Classification Ranking Method
## 2 Preliminaries
### Sequential Three-Way VIKOR Model Based on Minimum Group Utility Value
Currently, the team utility of the alternative is larger, and the single regret is smaller, and the value of \(Q_{i}\) is smaller. This indicates that the individual regret value and group utility value of the better decision alternative are always as small as possible. The steps of the three-way sequential VIKOR model according to minimum group utility value are as follows (where Algorithm 1 is used to Step 3):  
Step 1. According to Definition 2, the NIS and PIS of the schemes are deduced from the multiple property message table. Step 2. Calculate the group utility value for each alternative based on Definition 3. Step 3. Algorithm 1 is used to classify the decision regions of the scheme. Step 4. The decision index values of each scheme are calculated according to Equation (5). Step 5. Sort the schemes belonging to the identical decision area in ascending order of their decision index values. Step 6. The final ranking result is obtained by sorting the ranking rules about different regions shown in Figure 1.  
Figure 1: Connections between decisions regions, \(\rightarrow\) denotes \(\succ\). We will illustrate the time sophistication of Algorithm 1 as follows:  
\(A_{\max}\) and \(A_{\min}\) are obtained in step 4 with a corresponding time sophistication of \(O(|S|)\). Adding \(A_{\max}\) and \(A_{\min}\) to \(POS\) and \(NEG\) in steps 5 to 8, respectively, has a corresponding time sophistication of \(O(\frac{|S|}{2})\). Steps 9 and 10 are the sequential process of processing \(S\) and \(BND\). This time sophistication is \(O(|k|)\). Lastly, \(O(|k||S|)\) is the time complexity of Algorithm 1. Next, the working process of the proposed model is briefly explained by an example. **Example 1**. : _Suppose there are 8 alternatives \(A_{m}=\{A_{1},A_{2},A_{3},A_{4},A_{5},A_{6},A_{7},A_{8}\}\). The attribute set \(C=\{C_{1},C_{2},C_{3}\}\) of alternatives consists of two benefit attributes \(\{C_{1},C_{2}\}\) and one cost attribute \(\{C_{3}\}\), and the corresponding weight set is \(w=\{0.3,0.5,0.2\}\). The normalized attribute evaluation value table is shown in Table 2, and the uniformized determination matrix is expressed as \(X=((X_{\min})_{8\times 3})(m=1,\cdots 8;n=1,2,3)\)._  
(1) calculate PIS and NIS according to Definition 2 from the nature assessing values of all schemes in Table 2, which can be obtained by \(PIS=\{0.9,0.9,0.1\}\), \(NIS=\{0.1,0.1,0.9\}\). (2) obtain the team utility value \(S_{i}(i=1,2,3,4,5,6,7,8)\) of each alternative by PIS and NIS according to Definition 3. \(S_{1}=0.3250\), \(S_{2}=0.9434\), \(S_{3}=0.0576\), \(S_{4}=0.4303\), \(S_{5}=0.5772\), \(S_{6}=0.6144\), \(S_{7}=0.8125\), \(S_{8}=0.9688\). (3) classify all alternatives regionally according to Algorithm 1. \(POS^{4}=\{A_{3},A_{1},A_{4},A_{5}\}\), \(NGE^{4}=\{A_{8},A_{2},A_{7},A_{6}\}\). (4) calculate the decision index values of all schemes according to Equation (5) where \(Q_{i}\) represents the decision index value of the \(i\)th scheme, which can be obtained by \(Q_{1}=0.3642\), \(Q_{2}=0.9861\), \(Q_{3}=0\), \(Q_{4}=0.3513\), \(Q_{5}=0.6039\), \(Q_{6}=0.5936\), \(Q_{7}=0.8436\), and \(Q_{8}=0.9647\). ```
input:Group utility values for m alternatives \(k^{th}-\)level, \(k=0\). output:Three decision regions, \(POS^{k}\) and \(NEG^{k}\);
1 POS = \(\emptyset\), BND = {Am}, NEG = \(\emptyset\), k = 0, S = {Sm};
2 while\(BND^{k}\
eq\varnothing\)
3\(k++\), \(POS^{k}_{new}=NEG^{k}_{new}=\varnothing\);
4\(A_{\max}=\min(S)\), \(A_{\min}=\max(S)\);
5 Add\(A_{\max}\) to \(POS^{k}_{new}\)
6\(POS^{k}=POS^{k-1}\cup POS_{new}\);
7 Add\(A_{\min}\) to \(NEG^{k}_{new}\)
8\(NEG^{k}=NEG^{k-1}\cup NEG_{new}\);
9\(BND^{k}=BND^{k-1}\cup(POS_{new}\cup NEG_{new})\);
10\(S=S-(A_{\max}\cup A_{\min})\);
11 end while
12 Output\(POS^{k}(A_{m})\), \(NEG^{k}(A_{m})\).