# A VIKOR-Based Sequential Three-Way Classification Ranking Method
## 2 Preliminaries
### Sequential Three-Way VIKOR Model Based on Minimum Group Utility Value
\begin{table}
\begin{tabular}{c c c c} \hline  & \(C_{1}\) & \(C_{2}\) & \(C_{3}\) \\ \hline \(A_{1}\) & 0.9000 & 0.5000 & 0.4000 \\ \(A_{2}\) & 0.2000 & 0.1000 & 0.8255 \\ \(A_{3}\) & 0.7465 & 0.9000 & 0.1000 \\ \(A_{4}\) & 0.4525 & 0.6000 & 0.4000 \\ \(5) sort the alternatives in the same region according to the decision index value in ascending order, so that the ranking result in the _POS_ domain is \(\{A_{3}\succ A_{4}\succ A_{1}\succ A_{5}\}\) and the ranking result in \(NEG\) domain is \(\{A_{6}\succ A_{7}\succ A_{8}\succ A_{2}\}\). (6) through the ranking rules of different regional alternatives, \(POS\{A_{m}\}\succ BND\{A_{m}\}\succ NEG\{A_{m}\}\), \(\{A_{3}\succ A_{4}\succ A_{1}\succ A_{5}\succ A_{6}\succ A_{7}\succ A_{8} \succ A_{2}\}\) is the final sorting result. ### Sequential Three-Way VIKOR Model Based on Minimum Individual Regret Value

Like the three-way sequential VIKOR model according to the minimum group utility value, the key steps of the ordinal three-way VIKOR model according to the minimum single regret value are to calculate the individual regret value for all alternatives, classify the alternatives using the sequential three-way decision method, calculate the decision indicator value of each scheme, and then sort schemes in the identical area. The final ranking results are acquired through the alternative ranking rules of different regions. **Definition 4**. : _Let there be \(m\) alternatives \(A=\{A_{1},A_{2},\cdots,A_{m}\}\), \(PIS=\{e_{1}^{*},e_{2}^{*},\cdots,e_{n}^{*}\}\), \(NIS=\left\{e_{1}^{-},e_{2}^{-},\cdots,e_{n}^{-}\right\}\), and a weight set is \(W=\{w_{1},w_{2},\cdots,w_{n}\}\), then the individual regret value of alternative \(A_{i}\) is defined as follows:_

\[R_{i}=\max_{j}w_{j}\frac{e_{j}^{*}-e_{ij}}{e_{j}^{*}-e_{j}^{-}} \tag{6}\]

where \(R_{i}\) denotes the individual regret value of the \(i\)th alternative. The ordinal three-way VIKOR model based on minimum individual regret value is illustrated as follows:

Step 1. Derive the NIS and PIS of schemes from the multi-attribute information table based on Definition 2. Step 2. The individual regret value is calculated for each scheme according to Definition 4. Step 3. Classify the alternatives by a similar procedure as in Algorithm 1. Step 4. The decision index values of each scheme are calculated according to Equation (5). Step 5. The schemes belonging to the identical decision area are sorted by the decision index value in ascending order. Step 6. According to the ordering regulation of different areas, the schemes of different areas are ordered to obtain the final ranking results. **Example 2** (like Example 1). : _The alternatives in Table 2 are calculated according to the above steps, and the final ranking result of the three-way sequential VIKOR model on account of the minimum individual regret value is \(\{A_{3}\succ A_{4}\succ A_{1}\succ A_{6}\succ A_{5}\succ A_{7}\succ A_{8} \succ A_{2}\}\). It can be acquired through the ranking solutions of all alternatives in Table 2 by the two models, which are different. The final ranking result of the three-way sequential VIKOR model on account of the minimum group utility value is \(\{A_{3}\succ A_{4}\succ A_{1}\succ A_{5}\succ A_{6}\succ A_{7}\succ A_{8} \succ A_{2}\}\). However, the final ranking result of the three-way sequential VIKOR model on account of the minimum individual regret value is \(\{A_{3}\succ A_{4}\succ A_{1}\succ A_{6}\succ A_{5}\succ A_{7}\succ A_{8} \succ A_{2}\}\). In the three-way sequential VIKOR model on account of the minimum group utility value, \(A_{5}\) is divided into the POS and \(A_{6}\) is divided into the NEG, but in the three-way sequential VIKOR model on account of the minimum individual regret value, \(A_{5}\) is divided into the NEG and \(A_{6}\) is divided into the POS, which eventually leads to diversities in the ultimate classification of results for the two models.