# A VIKOR-Based Sequential Three-Way Classification Ranking Method
## 2 Preliminaries
### Multiple Attribute Decision Making  
Multi-attribute decision making is very common in daily life, as shown in Table 1. The n alternatives are represented by the variable \(A=\{a_{1},a_{2},\cdots,a_{n}\}\), and the evaluation of each alternative involves \(m\) criteria. The set of \(m\) criteria is denoted by the variable \(V=\{v_{1},v_{2},\cdots,v_{m}\}\). The value of the object \(a_{i}\) under the criterion \(v_{j}(1\leq j\leq m)\) is represented by the variable \(e_{ij}(e_{ij}\in[0,1])\). The importance of each criterion is represented by the corresponding weight, which can reduce the impact of the differences of different criteria on the final decision. The variable \(W=\{w_{1},w_{2},\cdots,w_{m}\}\) denotes the set of \(m\) weights, where \(w_{j}\in[0,1]\), \(\sum\limits_{j=1}^{m}w_{j}=1\). The decision maker needs to rank each objective and combine the corresponding ranking information in the selection of the optimal scheme. The VIKOR method mainly includes five steps: calculating weight, calculating positive and negative ideal solutions, calculating individual regret value and group utility value, calculating decision index value, and sorting. The Lpmetric aggregation function is integrated into the calculation of \"negative ideal solution\" and \"positive ideal solution\" to obtain the compromise solution for each alternative. In the sorting process, the stability conditions and advantages are considered to obtain the ranking results according to the specific measurement between the ideal solution and each scheme. In recent years, many researchers [1; 2; 3; 4; 5] have carried out various types of expansion and improvement of the VIKOR method and combined it with other ideas and methods, showing better application effects compared with past methods. However, in addition to improving the decision results, these methods still have shortcomings in other aspects. In this paper, by introducing the VIKOR method into the three-way sequential decisions, a new VIKOR method combined with the sequential three-way decisions is proposed, which can classify and rank the alternatives. In order to easily compare and process the attribute information of the alternatives, it is necessary to normalize the attribute evaluation value of the multi-attribute information table. \[Y=\left(y_{ij}\right)_{m\times H} \tag{3}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(v_{1}\) & \(v_{2}\) & \(\cdots\) & \(v_{m}\) \\ \hline \(a_{1}\) & \(e_{11}\) & \(e_{12}\) & \(\cdots\) & \(e_{1m}\) \\ \(a_{2}\) & \(e_{21}\) & \(e_{22}\) & \(\cdots\) & \(e_{2m}\) \\ \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\cdots\) & \(\vdots\) \\ \(a_{n}\) & \(e_{n1}\) & \(e_{n2}\) & \(\cdots\) & \(e_{nm}\) \\ \(W\) & \(w_{1}\) & \(w_{2}\) & \(\cdots\) & \(w_{m}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Multi-criteria evaluation of information systems \(S=(A,V)\). where \(y_{ij}=\frac{x_{ij}}{\sqrt{\sum\limits_{j=1}^{n}x_{ij}^{2}}}\), \(x_{ij}\) is the element of set in the multi-attribute information table. VIKOR Model of Sequential Three-Way Decisions Based on Individual Regret Value or Group Utility Value  
Group utility value is a quantitative representation of decision-makers' views and attitudes towards gain and loss effects, which is used to measure decision-makers' subjective tendencies and preferences for certain things. The loss caused by the decision maker's mistake is represented by the individual regret value, which is a special measure of the optimal value compared to other values in the scheme. In the VIKOR method, the individual regret value and group utility value are obtained by calculating the \"specific measure\" between the ideal solution and each alternative. The smaller value of the group utility value means the greater group utility of the corresponding alternative, and the larger value of the individual regret value means the greater individual regret of the corresponding alternative.