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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]

This paper is devoted to the research of full implication triple I method under Pythagorean fuzzy environment. We first propose the concepts of Pythagorean t-norm, Pythagorean t-conorm, residual Pythagorean fuzzy implication operator (RPFIO) and Pythagorean fuzzy biresiduum. The full implication triple I method for Pythagorean fuzzy modus ponens (PFMP) and Pythagorean fuzzy modus tollens (PFMT) are also established. In addition, the properties of full implication triple I method of PFMP and PFMT models including the robustness, continuity and reversibility are analyzed. Finally, a practical problem is discussed to demonstrate the effectiveness of the Pythagorean fuzzy full implication multiple I method in medical diagnosis. The advantages of the new method are also explained. Overall, compared with the existing methods, the proposed methods are based on logical reasoning rather than using aggregation operators, so they can more accurately and completely express decision information. [ABSTRACT FROM AUTHOR]

In this paper, by integrating the q-rung orthopair fuzzy set (q-ROFS) with the N-soft set, we first propose a q-rung orthopair fuzzy N-soft set (q-ROFNSS). Based on the q-ROFNSS, we explore the q-rung orthopair fuzzy N-soft weighted average (q-ROFNSWA) operator and q-rung orthopair fuzzy N-soft weighted geometric (q-ROFNSWG) operator, and investigate some properties of the q-ROFNSWG operator and q-ROFNSWG operator including idempotency, monotonicity and boundedness. Finally, two kinds of multiple-attribute group decision-making (MAGDM) methods based on q-rung orthopair fuzzy N-soft aggregation operators are established. In addition, a practical example is provided to illustrate the effectiveness and correctness of the new decision-making approaches. Through comparison with existing methods, the advantages of our method are also elaborated. [ABSTRACT FROM AUTHOR]

This paper develops a single-granulation hesitant fuzzy rough set (SGHFRS) model from the perspective of granular computing. In the multi-granulation framework, we propose two types of multi-granulation rough sets model called the optimistic multi-granulation hesitant fuzzy rough sets (OMGHFRSs) and pessimistic multi-granulation hesitant fuzzy rough sets (PMGHFRSs). In the models, the multi-granulation hesitant fuzzy lower and upper approximations are defined based on multiple hesitant fuzzy tolerance relations. The relationships among the SGHFRSs, OMGHFRSs and PMGHFRSs are also established. In order to further measure the uncertainty of multi-granulation hesitant fuzzy rough sets (MGHFRSs), the concepts of rough measure and rough measure about the parameters α and β are presented and some of their interesting properties are examined. Finally, we give a decision-making method based on the MGHFRSs, and the validity of this approach is illustrated by two practical applications. Compared with the existing results, we also expound its advantages. [ABSTRACT FROM AUTHOR]

Through a combination of hesitant fuzzy sets with rough sets, this study develops a single-granulation hesitant fuzzy rough set model from the perspective of granular computing. In the multi-granulation framework, we propose two types of multi-granulation rough set model, called the optimistic multi-granulation hesitant fuzzy rough sets and pessimistic multi-granulation hesitant fuzzy rough sets. In the models, the multi-granulation hesitant fuzzy lower and upper approximations are defined based on multiple hesitant fuzzy tolerance relations. The relationships among the single-granulation hesitant fuzzy rough sets, optimistic multi-granulation hesitant fuzzy rough sets and pessimistic multi-granulation hesitant fuzzy rough sets are also investigated. Finally, we develop an approximation reduction approach of multi-granulation hesitant fuzzy rough sets to eliminate redundant hesitant fuzzy granulations with a detailed example. [ABSTRACT FROM AUTHOR]

In this paper, we propose the theory of possibility Pythagorean fuzzy soft set and define some related the operations concerning this notion including the complement, union, intersection, AND and OR. Then a similarity measure is introduced to compare two possibility Pythagorean fuzzy soft sets for dealing with decision problems. Finally, a practical example is provided to illustrate the validity of this similarity measure in decision making problem. [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]