# A VIKOR-Based Sequential Three-Way Classification Ranking Method
## 2 Preliminaries
### Sequential Three-Way VIKOR Model Based on Minimum Group Utility Value
The alternatives for these two models to be separated into the positive area are consistent with the alternatives to be classified into the negative area, \(\{A_{3}\succ A_{4}\succ A_{1}\}\) and \(\{A_{8}\succ A_{2}\succ A_{7}\}\), respectively. It shows that these two models also follow the principle that the smaller the single regret, the larger the group utility, and the better the scheme, and the larger the single regret, the smaller group utility, and the worse the scheme. Through sequential three-way decisions and considering both single regret value and group utility value, decision makers can obtain more decision information to classify alternatives. The sequential three-way VIKOR model based on single regret value and group utility value reduces the range of the positive area and negative area, and divides the difficult solutions into the boundary area, which avoids the difference in ranking results due to misclassification to a certain extent. ## 5 Experiment and Analysis

In this chapter, the ranking results of the proposed two models in two examples are compared with other existing multi-attribute ranking methods to further illustrate the VIKOR-based sequential three-way model. The first example is from [PERSON] [6] and the second example is from [PERSON] [29]. The selected examples are from other papers on the combination of multiple attribute decision-making methods and sequential three-way decisions, and papers on the combination of fuzzy information systems and sequential three-way systems. These are representative in terms of multiple attribute decision-making methods and sequential three-way decisions, and the selected examples are also classic in terms of decision ranking and investment decision problems. Therefore, the selected examples are representative. **Example 5**. : _In order to the usefulness and feasibility of the newly proposed multiple attribute decision-making method related to TOPSIS, [PERSON] et al. [6] used the data related to an investment plan. The company needs to comprehensively consider the expected benefit, market saturation, environmental impact, social benefit, and energy saving degree of the eight investment plans proposed and select the best investment plan. The set of eight investment plans is \(A=\{A_{1},A_{2},A_{3},A_{4},A_{5},A_{6},A_{7},A_{8}\}\), the corresponding attribute set of investment plans is \(C=\{C_{1},C_{2},C_{3},C_{4},C_{5}\}\), the benefit attribute is \(\{C_{1},C_{4},C_{5}\}\), and the cost attribute is \(\{C_{2},C_{3}\}\). The set of weights corresponding to each attribute is \(W=\{0.3,0.1,0.15,0.25,0.2\}\), and the un-formized decision matrix corresponding to the eight investment alternatives is \(X=((X_{mn}))_{8\times 5}\)\((m=1,\cdots,8;n=1,2,3,4,5)\), as shown in Table 3. The visualization of Example 5 is presented in \[Q_{i}=v\frac{(S_{i}-S^{*})}{(S^{-}-S^{*})}+(1-\mathrm{v})\frac{(R_{i}-R^{*})} {(R^{-}-R^{*})} \tag{5}\]0_ \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(C_{1}\) & \(C_{2}\) & \(C_{3}\) & \(C_{4}\) & \(C_{5}\) \\ \hline \(A_{1}\) & 0.8 & 0.4 & 0.3 & 0.8 & 0.9 \\ \(A_{2}\) & 0.9 & 0.5 & 0.5 & 0.7 & 0.6 \\ \(A_{3}\) & 0.3 & 0.4 & 0.6 & 0.4 & 0.3 \\ \(A_{4}\) & 0.5 & 0.2 & 0.2 & 0.7 & 0.6 \\ \(A_{5}\) & 0.7 & 0.6 & 0.6 & 0.5 & 0.8 \\ \(A_{6}\) & 0.4 & 0.8 & 0.7 & 0.7 & 0.3 \\ \(A_{7}\) & 0.9 & 0.5 & 0.1 & 0.8 & 0.7 \\ \(A_{8}\) & 0.6 & 0.8 & 0.8 & 0.3 & 0.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Normalized decision matrix corresponding to investment alternatives. Method 1. Sequential three-way VIKOR model according to maximum individual regret value and minimum group utility value:

Step 1. Calculate the PIS and NIS of all investment plans according to Table 3, the larger value of benefit attribute. Step 2. Calculate the single regret value and group utility value of all schemes on account of (4) and (6), where \(R_{i}\) denotes the individual regret value of the \(i\)th scheme and \(S_{i}\) denotes the group utility value of the \(i\)th alternative.