# A VIKOR-Based Sequential Three-Way Classification Ranking Method
## 2 Preliminaries
### Sequential Three-Way VIKOR Model Based on Minimum Group Utility Value
At the same time, the diversity between the final ordering consequences of the two models also verifies this paper's idea that only considering the single regret value or group utility value to classify and judge the alternatives will lead to large errors, and a single individual regret value and group utility value can only afford incomplete decision information. Thus, to make the ordering consequences of all schemes more accurate, this paper will simultaneously consider the single regret value and group utility value to sort through the three-way sequential decision method. ## 4 Sequential Three-Way VIKOR Model Based on Group Utility Value and Individual Regret Value

The two models mentioned above mainly classify the alternatives based on the minimum single regret value or group utility value, and then rank the same and different regions by calculating the integral operator formula. Therefore, in the model proposed next, the single regret value and group utility value of alternatives will be considered simultaneously. The three-way sequential VIKOR model will be established on account of the maximum individual regret value and the minimum group utility value. Finally, the sequential three-way VIKOR model will be established on account of the minimum single regret value and the maximum group utility value. A good scheme should have a small single regret and a large group utility. The previous and the following models are classified on account of the three-way sequential decision method by calculating the single regret value and the group utility value of the scheme. Through the idea of three-way sequential decisions, all the schemes are reasonably and effectively separated into positive, boundary, and negative areas. The alternative is separated into the positive area, which signifies that it should be a good decision scheme and easy to be accepted by the decision maker. Dividing an alternative into a boundary region means that it has a large individual regret value and a small group utility value in the sequential three-way decision process, or on the contrary, that the decision maker needs to delay acceptance or rejection. The alternatives are classified into the negative region, which indicates that they should be inferior decision alternatives and easy to be rejected by decision makers. In the following part, all alternatives are separated into three areas--negative area, boundary area, and positive area--through three-way sequential decisions. Uncertain alternatives are put into the boundary region and the decision is delayed until more basis and support is obtained, to avoid misclassification and avoid the risk caused by directly accepting or rejecting alternatives. After that, the final sorting result is calculated according to the integral operator and the sorting rules of the different region and same regions. Next, the sequential three-way VIKOR model, based the maximum individual regret and the minimum group utility value, and the sequential three-way VIKOR model, based on the minimum individual regret value and the maximum group utility value, will be discussed. Sequential Three-Way VIKOR Model Based on Minimum Group Utility Value and Maximum Individual Regret Value

Unlike the previously proposed model according to a single value, the proposed model comprehensively considers the single regret value and the group utility value. In most cases, it is impossible to obtain an accurate decision scheme by only considering the partial decision information corresponding to a single value. The alternative with larger group utility does not necessarily have smaller individual regret, and the decision scheme with smaller individual regret does not necessarily have larger group utility. When dealing with these alternatives, a new method is needed to identify the classification results of the schemes by considering both the single regret value and the group utility value of the alternatives. **Theorem 2**.