A Pythagorean fuzzy set is one of the successful extensions of the intuitionistic fuzzy set to handle the uncertain and fuzzy information in a more wider way. In this paper, some new exponential similarity measures (SMs) for measuring the similarities between objects are proposed. For it, we used the exponential function for the membership and the non-membership degrees and hence defined some series of the SMs for PFSs. The various desirable properties and their relations are examined. Several counter-intuitive cases are given to show the effectiveness of the proposed measures with the existing SMs. Furthermore, examples to classify the pattern recognition and the decision-making problems are presented and compared with the existing approaches.

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Abstract-Fuzzy set theory was introduced by L.A. Zadeh in 1965. Immediately, it has many applications in practice and in building databases, one of which is the construction of a fuzzy relational database based on similar relationship. The study of cases of fuzzy relations in different environments will help us understand its applications. In this paper, the rough fuzzy relation on Cartesian product of two universe sets is defined, and then the algebraic properties of them, such as the max, min, and composition of two rough fuzzy relations are examined. Finally, reflexive, α-reflexive, symmetric and transitive rough fuzzy relations on two universe sets are also defined.

Today, soft computing is a field that is used a lot in solving real-world problems, such as problems in economics, finance, banking... With the aim to serve for solving the real problem, many new theories and/or tools which were proposed, improved to help soft computing used more efficiently. We can mention some theories as fuzzy sets theory (L. Zadeh, 1965), intuitionistic fuzzy set (K Atanasov, 1986). In this paper, we introduce a new notion of support-intuitionistic fuzzy (SIF) set, which is the combination a intuitionistic fuzzy set with a fuzzy set. So, SIF set is a directly extension of fuzzy set and intuitionistic fuzzy sets (Atanassov). Then, we define some operators on support-intuitionistic fuzzy sets, and investigate some properties of these operators.

Intuitionistic fuzzy sets (IFSs), including member and nonmember functions, have many applications in managing uncertain information. The similarity measures of IFSs proposed to represent the similarity between different types of sensitive fuzzy information. However, some existing similarity measures do not meet the axioms of similarity. Moreover, in some cases, they could not be applied appropriately. In this study, we proposed some novel similarity measures of IFSs constructed by combining the exponential function of membership functions and the negative function of non-membership functions. In this paper, we also proposed a new entropy measure as a stepping stone to calculate the weights of the criteria in the proposed multi-criteria decision making (MCDM) model. The similarity measures used to rank alternatives in the model. Finally, we used this MCDM model to evaluate the quality of software projects.

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