# A VIKOR-Based Sequential Three-Way Classification Ranking Method
## 2 Preliminaries
### Sequential Three-Way VIKOR Model Based on Minimum Group Utility Value  
The key step of this model is to derive the group utility value corresponding to each alternative through alternatives, NIS, and PIS, and then classify the alternatives through three-way sequential decisions. The decision index value is calculated according to the measurement formula, and the ultimately ranking result is obtained through the ranking regulations of different regions and the same region. Therefore, the next step is to give the definition of PIS and NIS, as well as the measurement formula and the corresponding scheme comparison process. **Definition 2**. : _Worst solution (NIS) and optimal solution (PIS) of the alternatives attribute matrix._  
Where \(j\) is the condition attribute, the cost attribute set is \(J\), the benefit attribute set is \(I\), and the alternative and attribute sets are \(i\) and \(j\). According to the calculation results of the above formula, the NIS and PIS of all alternatives can be obtained, and then the group utility value of each alternative can be obtained on the basis of NIS and PIS. A smaller group utility value means that the corresponding scheme has better group utility. **Definition 3**. : _Let there be \(m\) alternatives, \(PIS=\left\{e_{1}^{*},e_{2}^{*},\cdots,e_{n}^{*}\right\}\), \(NIS=\left\{e_{1}^{-},e_{2}^{-},\cdots,e_{n}^{-}\right\}\), and a weight set \(W=\left\{w_{1},w_{2},\cdots,w_{n}\right\}\), then the group utility value of alternative \(A_{i}\) is defined as follows:_  
\[S_{i}=\sum\limits_{j=1}^{n}w_{j}\frac{e_{j}^{*}-e_{ij}}{e_{j}^{*}-e_{j}^{-}} \tag{4}\]where \(S_{i}\) is the group utility value of the \(i\)th scheme. **Theorem 1**. : _The schemes in the negative region are inferior to the schemes in the positive region, that is, the ranking rules belonging to the schemes in different regions are derived from the group utility values of all the schemes calculated._  
Proof of Theorem 1.: The alternatives in the positive region have relatively small group utility values and relatively large group utility. The alternatives that are in the negative region have larger group utility values and smaller group utility values. Therefore, the scheme for the negative region is inferior to the scheme for the positive region. To sum up, \(POS\{A\}\succ NEG\{A\}\). \(\square\)  
Since three-way sequential decisions can only classify the alternatives, it is necessary to introduce an integral operator to rank the schemes in the different region and same regions. [PERSON] et al. (2018) used the decision index value combining individual regret value and group utility value to rank, and the decision-making mechanism coefficient could help realize the minimization of single regret and the maximization of group utility. The larger the mechanism coefficient of decision-making process is, the more attention should be paid to the group maximum utility rather than the single regret. A smaller value of the decision index indicates a better alternative. By maximizing the team utility and minimizing the single regret, the compromise solution of each attribute of the alternative is obtained. The decision index value is defined as follows:  
\[Q_{i}=v\frac{(S_{i}-S^{*})}{(S^{-}-S^{*})}+(1-\mathrm{v})\frac{(R_{i}-R^{*})} {(R^{-}-R^{*})} \tag{5}\]  
where, \(Q_{i}\) denotes the decision index value of the \(i\)th alternative, \(S^{*}=\underset{i}{\min}S_{i}\), \(S^{-}=\underset{i}{\max}S_{i}\), \(R^{*}=\underset{i}{\min}R_{i}\), the weights of maximum team utility and decision index coefficients are \(v\), and \(v=0.5\) is usually assumed. According to Equation (5), the magnitude of \(Q_{i}\) value is related to the difference between \(S_{i}\) and \(S^{*}\), \(R_{i}\) and \(R^{*}\). \(S_{i}\) is closer to \(S^{*}\) and \(R_{i}\) is closer to \(R^{*}\). That is, the single regret value of the scheme is closer to the minimum value in the set of single regret values, and the team utility value is also closer to the minimum value in the set of team utility values.