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This paper is devoted to discussing the reverse triple I method based on the Pythagorean fuzzy set (PFS). We first propose the concepts of Pythagorean t-norm, Pythagorean t-conorm, residual Pythagorean fuzzy implication operator (RPFIO) and Pythagorean fuzzy biresiduum. The reverse triple I methods for Pythagorean fuzzy modus ponens (PFMP) and Pythagorean fuzzy modus tollens (PFMT) are also established. In addition, some interesting properties of the reverse triple I method of PFMP and PFMT inference models are analysed, including the robustness, continuity and reversibility. Finally, a practical problem is provided to illustrate the effectiveness of the reverse triple I method for PFMP in decision-making problems. The advantages of the new method over existing methods are also expounded. Overall, compared with the existing methods, the proposed methods are based on logical reasoning rather than using aggregation operators, so the novel methods are more logical, can better deal with the uncertain problems in complex decision-making environments and can completely reflect the decision-making opinions of decision-makers. [ABSTRACT FROM AUTHOR]

The dual hesitant fuzzy soft set is a generalized soft set by combining the dual hesitant fuzzy set and the soft set. Rough set theory is a powerful mathematical tool for dealing with uncertainty. Using a rough set theory, this paper develops a rough set approach to dual hesitant fuzzy soft set based on decision making problems. Firstly, by employing a dual hesitant fuzzy compatible relation, a new rough set model, called dual hesitant fuzzy compatible rough set, is first proposed, and some fundamental properties of this model are further examined. Then the concepts of roughness measure and approximate precision of a crisp set in dual hesitant fuzzy compatible approximation space are introduced and their basic properties are discussed. Furthermore, we present a rough set approach to dual hesitant fuzzy soft set based on decision making. Meanwhile, an example is provided to show the validity of this approach in dual hesitant fuzzy soft set based on decision making problems. Finally, by comparing the novel method with the decision making method based on intuitionistic fuzzy soft sets, we point out some advantages of the novel method. [ABSTRACT FROM AUTHOR]

In this paper, we establish a systematic framework for the study of hesitant fuzzy compatible rough sets by a constructive approach. In the constructive approach, we introduce the concept of lower and upper hesitant fuzzy compatible rough approximation operators and also investigate some properties of this model. Then the concepts of roughness measure and approximate precision of a crisp set in hesitant fuzzy compatible approximation space are proposed and their basic properties are further discussed. Based on the hesitant fuzzy compatible rough sets, a novel decision-making approach to hesitant fuzzy soft sets is established. Meanwhile, a practical example is provided to demonstrate the effectiveness of this method. Finally, by comparing the novel method with the decision making method based on fuzzy soft sets, we point out some advantages of the novel method. [ABSTRACT FROM AUTHOR]