The notions of the bipolar complex fuzzy set (BCFS) and complex bipolar fuzzy set (CBFS) have been already given, but these notions of BCFS and CBFS have the problem that they contradict the basic ...definition of the complex numbers which we discussed in this article, and then we defined the new definition of BCFS. Our defined notion of BCFS is more closed to bipolarity as compared with already existing BCFS and CBFS, and more accurate. BCFS is the fusion of bipolar fuzzy set (BFS) which a decision analyst needs to describe the positive and negative aspects of an object and complex fuzzy set (CFS) which a decision analyst needs to handle two‐dimensional (two variables) information. When there is information of two variables with positive and negative aspects then a decision analyst needs BCFS to handle this information. In this article, we also interpreted some basic operations on BCFS like a complement, intersection, and union and explained them with the help of examples. Additionally, we defined the concept of type‐1 partially BCFS and type‐2 partially BCFS. Further, we interpreted some generalized trigonometric similarity measures such as generalized cosine similarity measure, generalized tangent similarity measure, generalized cotangent similarity measure, and generalized hybrid trigonometric similarity measure for BCFS. The weighted generalized trigonometric similarity measures are also presented in this article. After that, we applied these similarity measures (SMs) in two real‐life applications (pattern recognition and medical diagnosis) to show the benefits and advantages of our proposed SMs. Finally, we did a comparison of our demonstrated SMs with some existing SMs to show the superiority, usefulness, and effectiveness of our proposed SMs.
Complex fuzzy sets and complex intuitionistic fuzzy sets cannot handle imprecise, indeterminate, inconsistent, and incomplete information of periodic nature. To overcome this difficulty, we introduce ...complex neutrosophic set. A complex neutrosophic set is a neutrosophic set whose complex-valued truth membership function, complex-valued indeterminacy membership function, and complex-valued falsehood membership functions are the combination of real-valued truth amplitude term in association with phase term, real-valued indeterminate amplitude term with phase term, and real-valued false amplitude term with phase term, respectively. Complex neutrosophic set is an extension of the neutrosophic set. Further set theoretic operations such as complement, union, intersection, complex neutrosophic product, Cartesian product, distance measure, and
δ
-equalities of complex neutrosophic sets are studied here. A possible application of complex neutrosophic set is presented in this paper. Drawbacks and failure of the current methods are shown, and we also give a comparison of complex neutrosophic set to all such methods in this paper. We also showed in this paper the dominancy of complex neutrosophic set to all current methods through the graph.
The linguistic intuitionistic fuzzy sets (LIFSs) and linguistic Pythagorean fuzzy sets (LPFSs) are two linguistic orthopair fuzzy sets whose membership grades are pairs of linguistic terms from the ...predefined linguistic term sets (LTSs). One linguistic term indicates the membership degree (MD), while the other one gives the nonmembership degree (NMD). In each LIFS, the sum of the subscripts of MD and NMD is less than the cardinality of LTS. In the LPFSs, the sum of the squares of the subscripts of MD and NMD is less than the square of the cardinality of LTS. In this paper, we propose a general form of these two linguistic orthopair fuzzy sets, which can be named linguistic q‐rung orthopair fuzzy sets. We devise the operational laws, based on which, the linguistic q‐rung orthopair fuzzy weighted averaging (LqROFWA) operator and linguistic q‐rung orthopair fuzzy weighted geometric (LqROFWG) operator are developed to aggregate the linguistic q‐rung orthopair fuzzy numbers (LqROFNs). Then, the novel interactional operational laws that consider the interactions between the MD and NMD from different LqROFNs are given. The partitioned geometric Heronian mean (PGHM) operator can effectively solve the decision‐making problems in which the attributes grouped into the same clusters have interrelationships and the attributes belonging to different clusters have no interrelationship. Based on these novel operational laws and PGHM operator, the linguistic q‐rung orthopair fuzzy interactional PGHM (LqROFIPGHM) operator and linguistic q‐rung orthopair fuzzy interactional weighted PGHM (LqROFIWPGHM) operator are proposed and their properties are discussed. Based on the LqROFIWPGHM operator, an efficient multiattribute group decision‐making model is given to deal with the linguistic q‐rung orthopair fuzzy information. Finally, the superiorities of the interactional operational laws and LqROFIWPGHM operator are tested using some illustrative examples.
In this paper, we review the definition and basic properties of the different types of fuzzy sets that have appeared up to now in the literature. We also analyze the relationships between them and ...enumerate some of the applications in which they have been used.
Pythagorean fuzzy set, generalized by Yager, is a new tool to deal with vagueness considering the membership grade
μ
and non-membership
ν
satisfying the condition
μ
2
+
ν
2
≤
1
. It can be used to ...characterize the uncertain information more sufficiently and accurately than intuitionistic fuzzy set. Pythagorean fuzzy set has attracted great attention of many scholars that have been extended to new types and these extensions have been used in many areas such as decision making, aggregation operators, and information measures. Because of such a growth, we present an overview on Pythagorean fuzzy set with aim of offering a clear perspective on the different concepts, tools and trends related to their extension. In particular, we provide two novel algorithms in decision making problems under Pythagorean fuzzy environment. It may be served as a foundation for developing more algorithms in decision making.
In this article, a new linguistic Pythagorean fuzzy set (LPFS) is presented by combining the concepts of a Pythagorean fuzzy set and linguistic fuzzy set. LPFS is a better way to deal with the ...uncertain and imprecise information in decision making, which is characterized by linguistic membership and nonmembership degrees. Some of the basic operational laws, score, and accuracy functions are defined to compare the two or more linguistic Pythagorean fuzzy numbers and their properties are investigated in detail. Based on the norm operations, some series of the linguistic Pythagorean weighted averaging and geometric aggregation operators, named as linguistic Pythagorean fuzzy weighted average and geometric, ordered weighted average and geometric with linguistic Pythagorean fuzzy information are proposed. Furthermore, a multiattribute decision‐making method is established based on these operators. Finally, an illustrative example is used to illustrate the applicability and validity of the proposed approach and compare the results with the existing methods to show the effectiveness of it.
Recently, many researchers have studied some types of sets which are extension of fuzzy sets widely. Some of them are interval‐valued fuzzy set, intuitionistic fuzzy sets, type‐2 fuzzy sets, type‐n ...fuzzy sets, hesitant fuzzy sets (HFSs), dual fuzzy sets, and neutrosophic sets. In solving decision‐making problems, these sets have more advantages than classical sets. In this paper, we introduce a new concept called dual type‐2 hesitant fuzzy sets (DT2HFSs) by combining concepts of the dual HFS and the type‐2 fuzzy set. Then we give correlation coefficient formulas and weighted correlation coefficient formulas between two DT2HFSs and obtain some results related to the proposed correlation coefficient formulas. On the basis of the proposed correlation coefficient formulas, we develop the clustering method under DT2HF environment. Finally, we present an application of the proposed method for a problem to illustrate the process and validate the proposed method.
Pythagorean fuzzy set (PFS) is a significant soft computing tool for tackling embedded fuzziness in decision-making. Many computing methods have been studied to facilitate the application of PFS in ...modeling practical problems, among which the concept of correlation coefficient is very important. This article proposes some novel methods of computing correlation between PFSs via the three characteristic parameters of PFS by incorporating the ideas of Pythagorean fuzzy deviation, variance, and covariance. These novel methods evaluate the magnitude of relationship, show the potency of correlation between the PFSs, and also indicate whether the PFSs are related in either negative or positive sense. The proposed techniques are substantiated together with some theoretical results and numerically validated to be superior in terms of reliability and accuracy compared to some similar existing techniques. Decision-making processes involving pattern recognition and career placement problems are determined using the proposed techniques.
Fuzzy sets have a great progress in every scientific research area. It found many application areas in both theoretical and practical studies from engineering area to arts and humanities, from ...computer science to health sciences, and from life sciences to physical sciences. In this paper, a comprehensive literature review on the fuzzy set theory is realized. In the recent years, ordinary fuzzy sets have been extended to new types and these extensions have been used in many areas such as energy, medicine, material, economics and pharmacology sciences. This literature review also analyzes the chronological development of these extensions. In the last section of the paper, we present our interpretations on the future of fuzzy sets.
Intuitionistic fuzzy sets K.T. Atanassov, Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia (deposed in Central Science-Technical Library of Bulgarian Academy of Science, 1697/84), 1983 (in ...Bulgarian) are an extension of fuzzy set theory in which not only a membership degree is given, but also a non-membership degree, which is more or less independent. Considering the increasing interest in intuitionistic fuzzy sets, it is useful to determine the position of intuitionistic fuzzy set theory in the framework of the different theories modelling imprecision. In this paper we discuss the mathematical relationship between intuitionistic fuzzy sets and other models of imprecision.
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