Q-rung orthopair normal fuzzy aggregation operators and their application in multi-attribute decision-making

© 2019 by the authors. Q-rung orthopair fuzzy set (q-ROFS) is a powerful tool to describe uncertain information in the process of subjective decision-making, but not express vast objective phenomenons that obey normal distribution. For this situation, by combining the q-ROFS with the normal fuzzy number, we proposed a new concept of q-rung orthopair normal fuzzy (q-RONF) set. Firstly, we defined the conception, the operational laws, score function, and accuracy function of q-RONF set. Secondly, we presented some new aggregation operators to aggregate the q-RONF information, including the q-RONF weighted operators, the q-RONF ordered weighted operators, the q-RONF hybrid operator, and the generalized form of these operators. Furthermore, we discussed some desirable properties of the above operators, such as monotonicity, commutativity, and idempotency. Meanwhile, we applied the proposed operators to the multi-attribute decision-making (MADM) problem and established a novel MADM method. Finally, the proposed MADM method was applied in a numerical example on enterprise partner selection, the numerical result showed the proposed method can effectively handle the objective phenomena with obeying normal distribution and complicated fuzzy information, and has high practicality. The results of comparative and sensitive analysis indicated that our proposed method based on q-RONF aggregation operators over existing methods have stronger information aggregation ability, and are more suitable and flexible for MADM problems

Some q-rung orthopair fuzzy Muirhead means with their application to multi-attribute group decision making

Recently proposed q-rung orthopair fuzzy set (q-ROFS) is a powerful and effective tool to describe fuzziness, uncertainty and vagueness. The prominent feature of q-ROFS is that the sum and square sum of membership and non-membership degrees are allowed to be greater than one with the sum of qth power of the membership degree and qth power of the non-membership degree is less than or equal to one. This characteristic makes q-ROFS more powerful and useful than intuitionistic fuzzy set (IFS) and Pythagorean fuzzy set (PFS). The aim of this paper is to develop some aggregation operators for fusing q-rung orthopair fuzzy information. As the Muirhead mean (MM) is considered as a useful aggregation technology which can capture interrelationships among all aggregated arguments, we extend the MM to q-rung orthopair fuzzy environment and propose a family of q-rung orthopair fuzzy Muirhead mean operators. Moreover, we investigate some desirable properties and special cases of the proposed operators. Further, we apply the proposed operators to solve multi-attribute group decision making (MAGDM) problems. Finally, a numerical instance as well as some comparative analysis are provided to demonstrate the validity and superiorities of the proposed method

Complemental Fuzzy Sets: A Semantic Justification of q-Rung Orthopair Fuzzy Sets

This article introduces complemental fuzzy sets, explains their semantics, and presents a subclass of this model that generalizes intuitionistic fuzzy sets in a novel manner. It also provides practical results that will facilitate their implementation in real situations. At the theoretical level, we define a family of c-complemental fuzzy sets from each fuzzy negation c. We argue that this construction provides semantic justification for all subfamilies of complemental fuzzy sets, which include q-rung orthopair fuzzy sets (when c is a Yager’s fuzzy complement) and the new family of Sugeno intuitionistic fuzzy sets (when c belongs to the class of Sugeno’s fuzzy complements). We study fundamental operations and a general methodology for the aggregation of complemental fuzzy sets. Then, we give some specific examples of aggregation operators to illustrate their applicability. On a more practical level, constructive proofs demonstrate that all orthopair fuzzy sets on finite sets that satisfy a mild restriction are Sugeno intuitionistic fuzzy sets, and they are q-rung orthopair fuzzy sets for some q too. These contributions produce a new operational model that semantically justifies, and mathematically contains, “almost all” orthopair fuzzy sets on finite sets.Junta de Castilla y León y European Regional Development Fund (Fondo Europeo de Desarrollo Regional), CLU-2019-03, de apoyo a la unidad de investigación de excelencia “Economic Management for Sustainability” (GECOS)

Large-Scale Green Supplier Selection Approach under a Q-Rung Interval-Valued Orthopair Fuzzy Environment

As enterprises pay more and more attention to environmental issues, the green supply chain management (GSCM) mode has been extensively utilized to guarantee profit and sustainable development. Greensupplierselection(GSS),whichisakeysegmentofGSCM,hasbeeninvestigated to put forward plenty of GSS approaches

Extended Fuzzy Sets and Their Applications

This contribution deals with introducing the innovative concept of extended fuzzy set (E-FS), in which the S-norm function of membership and non-membership grades is less than or equal to one. The proposed concept not only encompasses the concept of the fuzzy set (FS), but it also includes the concepts of the intuitionistic fuzzy set (IFS), the Pythagorean fuzzy set (PFS) and the p-rung orthopair fuzzy set (p-ROFS). In order to explore the features of the E-FS concept, set and algebraic operations on E-FSs, average and geometric operations of E-FSs are studied and an E-FS score function is defined. The superiority of the E-FS concept is further confirmed with a score-based decision making technique in which the concepts of FS, IFS, PFS and p-ROFS do not make sense

Neutrosophic Set is a Generalization of Intuitionistic Fuzzy Set, Inconsistent Intuitionistic Fuzzy Set (Picture Fuzzy Set, Ternary Fuzzy Set), Pythagorean Fuzzy Set, q-Rung Orthopair Fuzzy Set, Spherical Fuzzy Set, and n-HyperSpherical Fuzzy Set, while Neutrosophication is a Generalization of Regret Theory, Grey System Theory, and Three-Ways Decision (revisited)

In this paper we prove that Neutrosophic Set (NS) is an extension of Intuitionistic Fuzzy Set (IFS) no matter if the sum of single-valued neutrosophic components is \u3c 1, or \u3e 1, or = 1. For the case when the sum of components is 1 (as in IFS), after applying the neutrosophic aggregation operators one gets a different result from that of applying the intuitionistic fuzzy operators, since the intuitionistic fuzzy operators ignore the indeterminacy, while the neutrosophic aggregation operators take into consideration the indeterminacy at the same level as truth-membership and falsehood-nonmembership are taken. NS is also more flexible and effective because it handles, besides independent components, also partially independent and partially dependent components, while IFS cannot deal with these. Since there are many types of indeterminacies in our world, we can construct different approaches to various neutrosophic concepts. Neutrosophic Set (NS) is also a generalization of Inconsistent Intuitionistic Fuzzy Set (IIFS) { which is equivalent to the Picture Fuzzy Set (PFS) and Ternary Fuzzy Set (TFS) }, Pythagorean Fuzzy Set (PyFS), Spherical Fuzzy Set (SFS), n-HyperSpherical Fuzzy Set (n-HSFS), and q-Rung Orthopair Fuzzy Set (q-ROFS). And all these sets are more general than Intuitionistic Fuzzy Set. We prove that Atanassov’s Intuitionistic Fuzzy Set of second type (IFS2), and Spherical Fuzzy Sets (SFS) do not have independent components. And we show that n-HyperSphericalFuzzy Set that we now introduce for the first time, Spherical Neutrosophic Set (SNS) and n-HyperSpherical Neutrosophic Set (n-HSNS) {the last one also introduced now for the first time} are generalizations of IFS2 and SFS. The main distinction between Neutrosophic Set (NS) and all previous set theories are: a) the independence of all three neutrosophic components {truth-membership (T), indeterminacy-membership (I), falsehood-nonmembership (F)} with respect to each other in NS – while in the previous set theories their components are dependent of each other; and b) the importance of indeterminacy in NS - while in previous set theories indeterminacy is completely or partially ignored. Also, Regret Theory, Grey System Theory, and Three-Ways Decision are particular cases of Neutrosophication and of Neutrosophic Probability. We have extended the Three-Ways Decision to n-Ways Decision