# A VIKOR-Based Sequential Three-Way Classification Ranking Method
## 2 Preliminaries
### Sequential Three-Way VIKOR Model Based on Minimum Group Utility Value
: _The alternatives with the largest single regret value and group utility value or vice versa are classified as boundary regions in the model. Based on the concept of three-way decisions, a ranking rule considering the alternatives in different regions of the boundary domain is proposed, as shown in Figure 2._

Figure 2: Ranking rules for different decision regions. Proof of Theorem 2.: The individual regret of the alternatives divided into the positive region is relatively small and the group utility is relatively large, which is easy to be accepted by the decision maker. The schemes divided into the negative region have large single regret and relatively small group utility, which are easy for decision makers to reject. The single regret and group utility of the schemes separated into the boundary area are equivalent, so the decision maker needs to delay the decision. Therefore, the boundary region scheme is better than the negative region scheme and the boundary region scheme is inferior to the positive region scheme; this can be obtained by:\(\square\)

\[POS\{A\}\succ BND\{A\}\succ NEG\{A\}\]

The overall steps of the proposed model are the same as those of the model considering a single value, the main difference being that the proposed model considers both the single regret value and the group utility value. In the classification of the alternatives by each round of sequential three-way decisions, the scheme with the minimum group utility value is divided into the positive region, the alternative with the maximum individual regret value is separated into the negative region, and the alternative with the maximum individual regret value and the minimum group utility value is divided into the boundary region. The steps of sequential three-way VIKOR model according to maximum individual regret value and minimum group utility value are as follows (where Algorithm 2 is used in Step 3):

Step 1. Derive the NIS and PIS of schemes from the multi-attribute information table based on Definition 2. Step 2. Calculate the single regret value and group utility value for each alternative based on Definitions 3 and 4. Step 3. Algorithm 2 is used to classify the alternatives. Step 4. Calculate the decision index value for each alternative according to Equation (5). Step 5. Sort the schemes belonging to the same decision area in ascending rank of their decision index values. Step 6. According to the ordering regulations of different areas, the results of different areas are ordered to obtain the final ordering results. We illustrate the time sophistication of Algorithm 2 as follows: In Step 4, the extraction of the minimum value in \(S\) is denoted as \(A_{\max}\) and the extraction of the maximum value in \(R\) is denoted as \(A_{\min}\), and the time sophistication is \(O(|S|)\). Steps 6 to 11 add the alternatives to the corresponding domain, respectively, with time sophistication \(O(|k|)\). Steps 12 to 13 are sequential processing of the boundary region, \(S\) and \(R\), and the time complexity is \(O(|k|)\). In general, \(O(|k||S|)\) is the time sophistication of Algorithm 2. **Example 3**. : _The process of using this model to process the normalized attribute matrix \(X=((X_{mn}))_{S\times 3}\)\((m=1,\cdots 8;n=1,2,3)\) and the corresponding weight set \(W=\{0.3,0.5,0.2\}\) shown in Table 2 is as follows:_

_(1) derives the NIS and PIS of alternatives from the multiple message attribute information table on account of Definition 2._

_(2) calculates the individual regret value and group utility value of each scheme on account of Definitions 3 and 4, where \(R_{i}\) denotes the individual regret value of the \(i\) scheme and \(S_{i}\) represents the group utility value of the \(i\)th scheme._

\(S_{1}=0.3250\)_, \(S_{2}=0.9434\), \(S_{3}=0.0576\), \(S_{4}=0.4303\), \(S_{5}=0.5772\), \(S_{6}=0.6144\), \(S_{7}=0.8125\), \(S_{8}=0.9688\).