Bilateral fuzzy sets and their three-way decisions: a new perspective of fuzzy logic

Abstract

 Fuzzy sets provide an effective method for dealing with uncertain and imprecise problems. For data of intermediate fuzzy distribution, membership degrees of objects whose attribute values are larger or smaller than the normal value would be the same and carried out the same decision. However, objects with different values mean that the information they contain is different for the decision-making problem. The decision process of calculating membership degrees in fuzzy set will lose the information of data itself. Therefore, bilateral fuzzy sets and their three-way decisions are proposed. First, the deviation degree is proposed in order to distinguish these objects. Compared with the membership degree, the deviation degree extends the mapping range from [0, 1] to [- 1, 1]. For six typical membership functions, their corresponding deviation functions are discussed and deduced. Second, the concept of bilateral fuzzy sets is proposed and the corresponding operation rules are analyzed and proved. Then, three-way decisions and approximations based on bilateral fuzzy sets are constructed. Next, for the optimization of threshold, principle of least cost is extended to the three-way decisions model based on bilateral fuzzy sets, and theoretical derivation is carried out. Finally, based on probability statistics, the principle based on confidence interval is proposed, which provides a new perspective for threshold calculation.