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Journal articles on the topic 'Pythagorean Fuzzy'

Author: Grafiati

Published: 5 June 2025

Last updated: 2 August 2025

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1

Dughman, Hawwa M. S., Amnnah A. A. Rasheed, and Khadeejah A. A. Alqamoudi. "Pythagorean M-Fuzzy subgroups under a T-norm." International Science and Technology Journal 32, no. 1 (2023): 1–17. https://doi.org/10.62341/hakp0275.

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This study explores the structural properties of Pythagorean M- fuzzy groups, building upon prior works by several researchers. It introduces the Pythagorean M-Fuzzy subgroup as a generalization of intuitionistic fuzzy subgroups, along with Pythagorean M-Fuzzy cosets and Pythagorean M-Fuzz normal subgroups. The study also discusses the impact of group homomorphisms on Pythagorean M- Fuzzy subgroups under a T-norm. Keywords: Pythagorean fuzzy set, Pythagorean fuzzy subgroup, M-fuzzy subgroup, Pythagorean M-fuzzy subgroup

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BHUNIA, SUPRIYA, and GANESH GHORAI. "An Approach to Lagrange’s Theorem in Pythagorean Fuzzy Subgroups." Kragujevac Journal of Mathematics 48, no. 6 (2024): 893–906. http://dx.doi.org/10.46793/kgjmat2406.893b.

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The Pythagorean fuzzy environment is a modern way of depicting uncertainty. The concept of Pythagorean fuzzy semi-level subgroups of any group is described in this paper. The Pythagorean fuzzy order of an element in a Pythagorean fuzzy subgroup is introduced and established various algebraic attributes. The relation between the Pythagorean fuzzy order of an element of a group and the order of that group is established. The Pythagorean fuzzy normalizer and Pythagorean fuzzy centralizer of Pythagorean fuzzy subgroups are discussed. Further, the concept of Pythagorean fuzzy quotient group and the

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3

Preethi, N., and G. K. Revathi. "A view on Pythagorean fuzzy digital feeble topological spaces, nets and prefilters." Journal of Interdisciplinary Mathematics 28, no. 4 (2025): 1607–22. https://doi.org/10.47974/jim-2155.

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The conceptualization of Pythagorean fuzzy digital feeble topological space (PFDFTS) is acquainted by applying new operations # , * with illustrations and its properties are analyzed. Adjacent to this, the concepts of Pythagorean fuzzy digital Point, Pythagorean fuzzy digital feeble continuous, Pythagorean fuzzy digital feeble neighbourhood, Qu - Pythagorean fuzzy digital feeble neighbourhood, Q u - Pythagorean fuzzy digital feeble continuous, Pythagorean fuzzy digital feeble net, Pythagorean fuzzy digital feeble prefilters are also examined with its properties.

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4

Çıtak, Filiz. "Some Algebraic Properties of Pythagorean Fuzzy Bi-ideals." Sinop Üniversitesi Fen Bilimleri Dergisi 10, no. 1 (2025): 42–59. https://doi.org/10.33484/sinopfbd.1524118.

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Fuzzy sets have a significant place in solving problems involving uncertainty. There are many studies on fuzzy sets in decision-making, engineering, algebra, etc. In this study, we discuss the behavior of Pythagorean fuzzy sets, which are a kind of generalization of fuzzy sets in algebra. First, we define the Pythagorean fuzzy product and examine some of its properties. Then, we investigate the relationship between the Pythagorean fuzzy ideal and the Pythagorean fuzzy product. Then, we define Pythagorean fuzzy bi-ideal. We give the theorem that characterizes bi-ideals in terms of fuzzy bi-idea

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Bachadach, Idris, A. Talhaoui, S. Melliani, and Sofyane Achik. "Pythagorean fuzzy nil radical of Pythagorean fuzzy ideal." Boletim da Sociedade Paranaense de Matemática 42 (May 21, 2024): 1–14. http://dx.doi.org/10.5269/bspm.65957.

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In this work, we introduce the Pythagorean fuzzy nil radical of a Pythagorean fuzzy ideal of a commutative ring, we further provide the notion of Pythagorean fuzzy semiprime ideal, and we study some related properties. Finally, we give the relation between Pythagorean fuzzy semiprime ideals and the Pythagorean fuzzy nil radical of a commutative ring.

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Razaq, Abdul, Ghaliah Alhamzi, Asima Razzaque, and Harish Garg. "A Comprehensive Study on Pythagorean Fuzzy Normal Subgroups and Pythagorean Fuzzy Isomorphisms." Symmetry 14, no. 10 (2022): 2084. http://dx.doi.org/10.3390/sym14102084.

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The Pythagorean fuzzy set is an extension of the intuitionistic fuzzy set used to handle uncertain circumstances in various decisions making problems. Group theory is a mathematical technique for dealing with problems of symmetry. This study deals with Pythagorean fuzzy group theory. In this article, we characterize the notion of a Pythagorean fuzzy subgroup and examine various algebraic properties of this concept. An extensive study on Pythagorean fuzzy cosets of a Pythagorean fuzzy subgroup, Pythagorean fuzzy normal subgroups of a group and Pythagorean fuzzy normal subgroup of a Pythagorean

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Shareef, Ayesha, Uzma Ahmad, Saba Siddique, and Mohammed M. Ali Al-Shamiri. "Pythagorean fuzzy incidence graphs with application in illegal wildlife trade." AIMS Mathematics 8, no. 9 (2023): 21793–827. http://dx.doi.org/10.3934/math.20231112.

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<abstract><p>Chemical engineers can model numerous interactions in a process using incidence graphs. They are used to methodically map out a whole network of interconnected processes and controllers to describe each component's impact on the others. It makes it easier to visualize potential process paths or a series of impacts. A Pythagorean fuzzy set is an effective tool to overcome ambiguity and vagueness. In this paper, we introduce the concept of Pythagorean fuzzy incidence graphs. We discuss the incidence path and characterize the strongest incidence path in Pythagorean fuzzy

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8

A, Nasreen, and Latha S. Nair. "Quotient of ideals of a Pythagorean fuzzy lattice." International Journal of Engineering and Computer Science 14, no. 06 (2025): 27409–18. https://doi.org/10.18535/ijecs.v14i06.5172.

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In this paper, we defined operations on Pythagorean fuzzy ideals and the Pythagorean fuzzy ideal of a Pythagorean fuzzy lattice is introduced. Certain characterizations of these are provided. The residuals of Pythagorean fuzzy ideals are defined, and it is demonstrated that these residuals also form Pythagorean fuzzy ideals. Furthermore, it is shown that they are the largest Pythagorean fuzzy ideal of P .

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9

T. Anitha. "Characterization of Pythagorean Fuzzy Bi-Interior Ideal and Bi-Quasi-Ideal in Γ-Semirings". Communications on Applied Nonlinear Analysis 32, № 4s (2024): 150–67. https://doi.org/10.52783/cana.v32.2746.

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In this paper, we introduce the Pythagorean fuzzy bi-interior-ideals and Pythagorean fuzzy bi-quasi-ideals in Γ - semiring. More over we prove the every Pythagorean fuzzy left and right ideals are Pythagorean fuzzy bi-interior -ideal in Γ - semiring. Also we study the notion of Pythagorean fuzzy bi-quasi-ideal in Γ - semiring and characterize Pythagorean fuzzy bi-quasi-ideal in Γ - semiring.

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10

Hameed, M. Shazib, Salman Mukhtar, Haq Nawaz Khan, Shahbaz Ali, M. Haris Mateen, and Muhammad Gulzar. "Pythagorean Fuzzy N-Soft Groups." Indonesian Journal of Electrical Engineering and Computer Science 21, no. 2 (2021): 1030–38. https://doi.org/10.11591/ijeecs.v21i2.pp1030-1038.

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We elaborate in this paper a new structure Pythagorean fuzzy N-soft groups which is the generalization of intuitionistic fuzzy soft group initiated by Karaaslan in 2013. In Pythagorean fuzzy N-soft sets concepts of fuzzy sets, soft sets, N-soft sets, fuzzy soft sets, intuitionistic fuzzy sets, intuitionistic fuzzy soft sets, Pythagorean fuzzy sets, Pythagorean fuzzy soft sets are generalized. We also talk about some elementary basic concepts and operations on Pythagorean fuzzy N-soft sets with the assistance of illusions. We additionally define three different sorts of complements for Pythagor

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Azzam, A. A., Mohamed Aldawood, and Radwan Abu-Gdairi. "Pythagorean Fuzzy Soft Somewhat Continuous Functions." European Journal of Pure and Applied Mathematics 17, no. 4 (2024): 4147–63. http://dx.doi.org/10.29020/nybg.ejpam.v17i4.5430.

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In this work, we introduce the concept of Pythagorean fuzzy soft somewhat open sets utilizing the Pythagorean fuzzy soft interior operator, extending its application to Pythagorean fuzzy soft topological spaces. This study aims to enhance decision-making processes in future-assisted economies by addressing the limitations of existing fuzzy set theories. We investigate the distinctive properties of Pythagorean fuzzy soft somewhat open sets as a subclass of Pythagorean fuzzy soft somewhere dense sets. Additionally, we explore Pythagorean fuzzy soft somewhat metamorphism’s within the context of P

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12

Yang, Hai-Long, and Jia-Jia Zhou. "Interval-valued pythagorean fuzzy rough approximation operators and its application." Journal of Intelligent & Fuzzy Systems 39, no. 3 (2020): 3067–84. http://dx.doi.org/10.3233/jifs-191539.

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By combining interval-valued Pythagorean fuzzy sets with rough sets, the interval-valued Pythagorean fuzzy rough set model is first constructed in this paper. The connections between special interval-valued Pythagorean fuzzy relations and interval-valued Pythagorean fuzzy approximation operators are established subsequently. Then, we study the axiomatic characterizations of interval-valued Pythagorean fuzzy lower and upper approximation operators. Different axiom sets of interval-valued Pythagorean fuzzy set-theoretic operators ensure the existence of different types of interval-valued Pythago

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13

Soujanya, Gundeti, P. R. Kavyasree, and B. Surender Reddy. "Cubic Pythagorean Hesitant Fuzzy Linear Spaces and Its Relevance in Multi Criteria Decision Making." International Journal of Analysis and Applications 21 (November 28, 2023): 128. http://dx.doi.org/10.28924/2291-8639-21-2023-128.

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Pythagorean fuzzy sets and interval valued Pythagorean fuzzy sets have an important role in decision making techniques. Pythagorean hesitant fuzzy sets are time and again used in dealing with uncertain and vague data. The motive of this paper is to introduce the notion cubic Pythagorean hesitant fuzzy linear spaces. We also present the notion of P(R)-intersection, P(R)-union of cubic Pythagorean hesitant fuzzy linear spaces with examples. Secondly, a series of operators like cubic Pythagorean hesitant fuzzy weighted averaging aggregation operators, cubic Pythagorean hesitant fuzzy order weight

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14

Dhanan, Venkatesan. "Lukasiewicz Fuzzy Implication Operator on Pythagorean Fuzzy Tautological Matrices." International Journal on Recent and Innovation Trends in Computing and Communication 9, no. 6 (2021): 01–05. http://dx.doi.org/10.17762/ijritcc.v9i6.5473.

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In this paper, introduced Pythagorean fuzzy tautologial matrices and Pythagorean fuzzy cotautological matrices and some properties of Lukasiwicz implication operator over Pythagorean fuzzy tautologial matrices and Pythagorean fuzzy cotautological matrices are discussed. Also discussed the relation between implication with Lukasiewicz disjunction and conjunction operations of PFCMs and PFCTMs.

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15

Razaq, Abdul, and Ghaliah Alhamzi. "On Pythagorean fuzzy ideals of a classical ring." AIMS Mathematics 8, no. 2 (2023): 4280–303. http://dx.doi.org/10.3934/math.2023213.

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<abstract><p>The Pythagorean fuzzy set is an extension of the intuitionistic fuzzy set and is an effective approach of handling uncertain situations. Ring theory is a prominent branch of abstract algebra, vibrant in wide areas of current research in mathematics, computer science and mathematical/theoretical physics. In the theory of rings, the study of ideals is significant in many ways. Keeping in mind the importance of ring theory and Pythagorean fuzzy set, in the present article, we characterize the concept of Pythagorean fuzzy ideals in classical rings and study its numerous al

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16

Shakeel, M., K. Rahman, M. S. A. Khan, and Murad Ullah. "Induced Averaging Aggregation Operators with Interval Pythagorean Trapezoidal Fuzzy Numbers and their Application to Group Decision Making." Nucleus 54, no. 2 (2017): 140–53. https://doi.org/10.71330/thenucleus.2017.112.

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Pythagorean fuzzy number is a new tool for uncertainty and vagueness. It is a generalization of fuzzy numbers and intuitionistic fuzzy numbers. This paper deal with induced interval Pythagorean trapezoidal fuzzy numbers. In this paper we introduce induced interval Pythagorean trapezoidal fuzzy numbers and some operation on I-IPTFN, and we also define different types of operators for aggregating induced interval Pythagorean trapezoidal fuzzy numbers. We present induced interval Pythagorean trapezoidal fuzzy ordered weighted averaging (I-IPTFOWA) operator and induced interval Pythagorean trapezo

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17

Almuhaimeed, Areej. "Pythagorean Fuzzy HX-subgroups and Their Applications." European Journal of Pure and Applied Mathematics 18, no. 2 (2025): 5768. https://doi.org/10.29020/nybg.ejpam.v18i2.5768.

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In this paper, we introduce the notion of a pythagorean fuzzy HX-subgroup and a normal HX-subgroup. In addition, we prove various chracterisations for pythagorean fuzzy HX-subgroups and pythagorean normal HX-subgroups. Moreover, the notations of pythagorean fuzzy HX-subgroups homomorphisms and antihomomorphisms are introduced, and some related properties regarding the relationship between a pythagorean fuzzy set and its image are investigated. Characterisations of level pythagorean fuzzy HX-subgroups and normal HX-subgroups are proved. These results generalised some results regarding fuzzy HX-

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18

Xue, Yige, and Yong Deng. "Extending Set Measures to Orthopair Fuzzy Sets." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 30, no. 01 (2022): 63–91. http://dx.doi.org/10.1142/s0218488522500040.

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Yager have proposed the extending set measures to Pythagorean fuzzy sets, which is able to efficiently solve the problems of uncertain information representation in Pythagorean fuzzy environment. However, the Pythagorean fuzzy sets represent a limited range of fields, while the q-rung orthopair fuzzy sets can represent many fuzzy sets, including the Pythagorean fuzzy sets, Fermatean fuzzy sets, and intuitionistic fuzzy sets. In order to extend the extending set measures to Pythagorean fuzzy sets to a broader range, the paper proposes the extending set measures to orthopair fuzzy sets, which ca

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19

Akram, Muhammad, Inayat Ullah, Tofigh Allahviranloo, and S. A. Edalatpanah. "LR-type fully Pythagorean fuzzy linear programming problems with equality constraints." Journal of Intelligent & Fuzzy Systems 41, no. 1 (2021): 1975–92. http://dx.doi.org/10.3233/jifs-210655.

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A Pythagorean fuzzy set is a powerful model for depicting fuzziness and uncertainty. This model is more flexible and practical as compared to an intuitionistic fuzzy model. This research article presents a new model called LR-type fully Pythagorean fuzzy linear programming problem. We consider the notions of LR-type Pythagorean fuzzy number, ranking for LR-type Pythagorean fuzzy numbers and arithmetic operations for unrestricted LR-type Pythagorean fuzzy numbers. We propose a method to solve LR-type fully Pythagorean fuzzy linear programming problems with equality constraints. We describe our

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20

V.S., Subha and S. Sharmila. "Interval-valued Pythagorean fuzzy weak bi-hyperideals in hypersemigroups." Annals of Communications in Mathematics 4, no. 1 (2021): 35–44. https://doi.org/10.5281/zenodo.10048865.

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In this paper we introduce the concept of interval-valued Pythagorean fuzzy subsemihypergroup and interval-valued Pythagorean fuzzy weak bi-hyperideals in hypersemigroups. We show that the (˜α, β˜)−level set of interval-valued Pythagorean fuzzy weak bi-hyperideal is a weak bi-hyperideal in hypersemigroup. We characterize cartesian product of interval-valued Pythagorean fuzzy set and examine that the cartesian product of interval-valued Pythagorean fuzzy weak bi-hyperideals is also an interval-valued Pythagorean weak bi-hyperideal in hypersemigroups.

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21

Wei, Guiwu, and Mao Lu. "Pythagorean Hesitant Fuzzy Hamacher Aggregation Operators in Multiple-Attribute Decision Making." Journal of Intelligent Systems 28, no. 5 (2017): 759–76. http://dx.doi.org/10.1515/jisys-2017-0106.

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Abstract The Hamacher product is a t-norm and the Hamacher sum is a t-conorm. They are good alternatives to the algebraic product and the algebraic sum, respectively. Nevertheless, it seems that most of the existing hesitant fuzzy aggregation operators are based on algebraic operations. In this paper, we utilize Hamacher operations to develop some Pythagorean hesitant fuzzy aggregation operators: Pythagorean hesitant fuzzy Hamacher weighted average operator, Pythagorean hesitant fuzzy Hamacher weighted geometric operator, Pythagorean hesitant fuzzy Hamacher ordered weighted average operator, P

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Almuhaimeed, Areej. "Pythagorean Fuzzy Small Submodules." European Journal of Pure and Applied Mathematics 15, no. 1 (2022): 36–46. http://dx.doi.org/10.29020/nybg.ejpam.v15i1.4170.

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In this paper, we introduce the notion of a pythagorean fuzzy small submodule. We prove various characterisations for pythagorean fuzzy small submodules. We provide a relation between a pythagorean fuzzy small submodule and a basic small submodule. In addition, some important properties regarding pythagorean fuzzy small submodules are investigated.

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23

Mateen, MUHAMMAD Haris. "Pythagorean Fuzzy N-Soft Groups." Indonesian Journal of Electrical Engineering and Computer Science 21, no. 2 (2021): 1030. http://dx.doi.org/10.11591/ijeecs.v21.i2.pp1030-1038.

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<p>We elaborate in this paper a new structure Pythagorean fuzzy<br />$N$-soft groups which is the generalization of intuitionistic fuzzy<br />soft group initiated by Karaaslan in 2013. In Pythagorean fuzzy<br />N-soft sets concepts of fuzzy sets, soft sets, N-soft sets, fuzzy<br />soft sets, intuitionistic fuzzy sets, intuitionistic fuzzy soft<br />sets, Pythagorean fuzzy sets, Pythagorean fuzzy soft sets are<br />generalized. We also talk about some elementary basic concepts and<br />operations on Pythagorean fuzzy N-soft sets with the assistanc

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24

Zhang, Runtong, Jun Wang, Xiaomin Zhu, Meimei Xia, and Ming Yu. "Some Generalized Pythagorean Fuzzy Bonferroni Mean Aggregation Operators with Their Application to Multiattribute Group Decision-Making." Complexity 2017 (2017): 1–16. http://dx.doi.org/10.1155/2017/5937376.

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The Pythagorean fuzzy set as an extension of the intuitionistic fuzzy set characterized by membership and nonmembership degrees has been introduced recently. Accordingly, the square sum of the membership and nonmembership degrees is a maximum of one. The Pythagorean fuzzy set has been previously applied to multiattribute group decision-making. This study develops a few aggregation operators for fusing the Pythagorean fuzzy information, and a novel approach to decision-making is introduced based on the proposed operators. First, we extend the generalized Bonferroni mean to the Pythagorean fuzzy

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25

N. Preethi and G.K. Revathi. "A Pythagorean Fuzzy Digital Fine Space Approach for Selecting Menstrual Hygiene Products in Rural Areas." European Journal of Pure and Applied Mathematics 18, no. 3 (2025): 6598. https://doi.org/10.29020/nybg.ejpam.v18i3.6598.

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Pythagorean fuzzy digital fine b~-door space (PFD bf~-DS) is a novel notion of generalized Pythagorean fuzzy digital fine topological space that is presented in this paper. Additionally, a number of characterizations of Pythagorean fuzzy digital fine b~-door space are examined. Examples were presented to demonstrate the applicability of these ideas and also some characteristics and connections between other Pythagorean fuzzy digital fine topological spaces and Pythagorean fuzzy digital fine b~-door space were examined. Furthermore, an application which utilizes Pythagorean fuzzy digital fine t

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Shahzadi, Gulfam, Muhammad Akram, and Ahmad N. Al-Kenani. "Decision-Making Approach under Pythagorean Fuzzy Yager Weighted Operators." Mathematics 8, no. 1 (2020): 70. http://dx.doi.org/10.3390/math8010070.

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In fuzzy set theory, t-norms and t-conorms are fundamental binary operators. Yager proposed respective parametric families of both t-norms and t-conorms. In this paper, we apply these operators for the analysis of Pythagorean fuzzy sets. For this purpose, we introduce six families of aggregation operators named Pythagorean fuzzy Yager weighted averaging aggregation, Pythagorean fuzzy Yager ordered weighted averaging aggregation, Pythagorean fuzzy Yager hybrid weighted averaging aggregation, Pythagorean fuzzy Yager weighted geometric aggregation, Pythagorean fuzzy Yager ordered weighted geometr

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27

Hezam, Ibrahim M., Khaista Rahman, Ahmad Alshamrani, and Darko Božanić. "Geometric Aggregation Operators for Solving Multicriteria Group Decision-Making Problems Based on Complex Pythagorean Fuzzy Sets." Symmetry 15, no. 4 (2023): 826. http://dx.doi.org/10.3390/sym15040826.

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The Complex Pythagorean fuzzy set (CPyFS) is an efficient tool to handle two-dimensional periodic uncertain information, which has various applications in fuzzy modeling and decision making. It is known that the aggregation operators influence decision-making processes. Algebraic aggregation operators are the important and widely used operators in decision making techniques that deal with uncertain problems. This paper investigates some complex Pythagorean fuzzy geometric aggregation operators, such as complex Pythagorean fuzzy weighted geometric (CPyFWG), complex Pythagorean fuzzy ordered wei

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28

Jin, Feifei, Lidan Pei, Huayou Chen, Reza Langari, and Jinpei Liu. "A Novel Decision-Making Model with Pythagorean Fuzzy Linguistic Information Measures and Its Application to a Sustainable Blockchain Product Assessment Problem." Sustainability 11, no. 20 (2019): 5630. http://dx.doi.org/10.3390/su11205630.

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This study presents a novel multi-attribute decision-making (MADM) model on the basis of Pythagorean fuzzy linguistic information measures. To do so, we first present a new concept of Pythagorean fuzzy linguistic sets to describe fuzziness and inconsistent information, in which the Pythagorean fuzzy linguistic values (PFLVs) are represented by the linguistic membership degree and linguistic non-membership degree. Then, we introduce two axiomatic definitions of information measures for PFLVs, including Pythagorean fuzzy linguistic entropy and the Pythagorean fuzzy linguistic similarity measure,

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Nan, TaiBen, Haidong Zhang, and Yanping He. "Pythagorean fuzzy full implication multiple I method and corresponding applications." Journal of Intelligent & Fuzzy Systems 41, no. 1 (2021): 1741–55. http://dx.doi.org/10.3233/jifs-210527.

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The overwhelming majority of existing decision-making methods combined with the Pythagorean fuzzy set (PFS) are based on aggregation operators, and their logical foundation is imperfect. Therefore, we attempt to establish two decision-making methods based on the Pythagorean fuzzy multiple I method. This paper is devoted to the discussion of the full implication multiple I method based on the PFS. We first propose the concepts of Pythagorean t-norm, Pythagorean t-conorm, residual Pythagorean fuzzy implication operator (RPFIO), Pythagorean fuzzy biresiduum, and the degree of similarity between P

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Li, Zengxian, Guiwu Wei, and Mao Lu. "Pythagorean Fuzzy Hamy Mean Operators in Multiple Attribute Group Decision Making and Their Application to Supplier Selection." Symmetry 10, no. 10 (2018): 505. http://dx.doi.org/10.3390/sym10100505.

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In this paper, we extend the Hamy mean (HM) operator and dual Hamy mean (DHM) operator with Pythagorean fuzzy numbers (PFNs) to propose Pythagorean fuzzy Hamy mean (PFHM) operator, weighted Pythagorean fuzzy Hamy mean (WPFHM) operator, Pythagorean fuzzy dual Hamy mean (PFDHM) operator, weighted Pythagorean fuzzy dual Hamy mean (WPFDHM) operator. Then the multiple attribute group decision making (MAGDM) methods are proposed with these operators. In the end, we utilize an applicable example for supplier selection to prove the proposed methods.

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Zhang, Maoyin, Tingting Zheng, Wanrong Zheng, and Ligang Zhou. "Interval-Valued Pythagorean Hesitant Fuzzy Set and Its Application to Multiattribute Group Decision-Making." Complexity 2020 (February 13, 2020): 1–26. http://dx.doi.org/10.1155/2020/1724943.

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Pythagorean hesitant fuzzy sets are widely watched because of their excellent ability to deal with uncertainty, imprecise and vague information. This paper extends Pythagorean hesitant fuzzy environments to interval-valued Pythagorean hesitant fuzzy environments and proposes the concept of interval-valued Pythagorean hesitant fuzzy set (IVPHFS), which allows the membership of each object to be a set of several pairs of possible interval-valued Pythagorean fuzzy elements. Furthermore, we develop a series of aggregation operators for interval-valued Pythagorean hesitant fuzzy information and app

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Obbu, Ramesh, Tarakaramu Nainaru, G. YuvaRoopa Lakshmi, et al. "Planarity of Intuitionistic Pythagorean Fuzzy Graphs and its Application in Decision Making." European Journal of Pure and Applied Mathematics 18, no. 3 (2025): 6372. https://doi.org/10.29020/nybg.ejpam.v18i3.6372.

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This paper introduces the concept of Intuitionistic Pythagorean Fuzzy Graphs (IPFGs) and outlines their key characteristics as a means of addressing such uncertain scenarios by examining Pythagorean uncertain planarity values using weak, strong, and significant edges. An Intuitionistic Pythagorean Fuzzy Graph is a generalization of a traditional graph that can solve certain problems beyond the capabilities of both classical graph theory and fuzzy graph theory. The approach based on Pythagorean fuzzy graphs offers greater flexibility in handling human judgment data compared to other fuzzy model

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D, Ajay, Karthiga S, and Chellamani P. "A study on labelling of pythagorean neutrosophic fuzzy graphs." Journal of Computational Mathematica 5, no. 1 (2021): 105–16. http://dx.doi.org/10.26524/cm97.

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Pythagorean neutrosophic fuzzy set comprises elements with dependent membership (µ), non-membership (σ) and independent indeterminacy (β) functions with the flexibility 0 ≤ µ2 + β2 + σ2 ≤ 2. Pythagorean neutrosophic fuzzy graph is a new concept emerged by combining the concept of Pythagorean neutrosophic fuzzy set and fuzzy graph theory. In this paper, the authors present the labelling of Pythagorean neutrosophic fuzzy graphs and investigate their properties.

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Abdullah, Saleem, Muhammad Qiyas, Muhammad Naeem, Mamona, and Yi Liu. "Pythagorean Cubic fuzzy Hamacher aggregation operators and their application in green supply selection problem." AIMS Mathematics 7, no. 3 (2022): 4735–66. http://dx.doi.org/10.3934/math.2022263.

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<abstract><p>The green chain supplier selection process plays a major role in the environmental decision for the efficient and effective supply chain management. Therefore, the aim of this paper is to develop a mechanism for decision making on green chain supplier problem. First, we define the Hamacher operational law for Pythagorean cubic fuzzy numbers (PCFNs) and study their fundamental properties. Based on the Hamacher operation law of PCFNs, we defined Pythagorean cubic fuzzy aggregation operators by using Hamacher t-norm and t-conorm. Further, we develop a series of Pythagorea

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Li, Longmei, Tingting Zheng, Wenjing Yin, and Qiuyue Wu. "Novel Pythagorean fuzzy entropy and Pythagorean fuzzy cross-entropy measures and their applications1." Journal of Intelligent & Fuzzy Systems 41, no. 6 (2021): 6527–46. http://dx.doi.org/10.3233/jifs-210365.

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Entropy and cross-entropy are very vital for information discrimination under complicated Pythagorean fuzzy environment. Firstly, the novel score factors and indeterminacy factors of intuitionistic fuzzy sets (IFSs) are proposed, which are linear transformations of membership functions and non-membership functions. Based on them, the novel entropy measures and cross-entropy measures of an IFS are introduced using Jensen Shannon-divergence (J-divergence). They are more in line with actual fuzzy situations. Then the cases of Pythagorean fuzzy sets (PFSs) are extended. Pythagorean fuzzy entropy,

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B. Vijayalakshmi. "Contra M-continuous Maps in Pythagorean Fuzzy Topological Spaces." Communications on Applied Nonlinear Analysis 32, no. 6s (2025): 327–39. https://doi.org/10.52783/cana.v32.3299.

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In this paper, we introduce and investigate Pythagorean fuzzy contra -continuous maps in Pythagorean fuzzy topological spaces and also discuss about some properties and characterization of Pythagorean fuzzy contra -irresolute maps.

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Asif, Muhammad, Muhammad Akram, and Ghous Ali. "Pythagorean Fuzzy Matroids with Application." Symmetry 12, no. 3 (2020): 423. http://dx.doi.org/10.3390/sym12030423.

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The Pythagorean fuzzy models deal with graphical and algebraic structures in case of vague information related to membership and non-membership grades. Here, we use Pythagorean fuzzy sets to generalize the concept of vector spaces and discuss their basis and dimensions. We also highlight the concept of Pythagorean fuzzy matroids and examine some of their fundamental characteristics like circuits, basis, dimensions, and rank functions. Additionally, we explore the concept of Pythagorean fuzzy matroids in linear algebra, graph theory, and combinatorics. Finally, we demonstrate the use of Pythago

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Jin, Feifei, Danning Li, Shuyan Guo, Ligang Zhou, Yi Chen, and Jiaming Zhu. "Exponential information measures-driven Pythagorean fuzzy MADM method and its application to new energy battery supplier evaluation problem." Journal of Intelligent & Fuzzy Systems 44, no. 6 (2023): 9167–82. http://dx.doi.org/10.3233/jifs-223088.

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Under the Pythagorean fuzzy environment, this paper presents a multi-attribute decision-making (MADM) model based on exponential entropy measure and exponential similarity measure to evaluate new energy battery supplier’s performance. In this method, the notion of Pythagorean fuzzy linguistic sets (PFLSs) is first introduced by combining the linguistic fuzzy sets (LFSs) and the Pythagorean fuzzy sets (PFSs). Then, the axiomatic definitions of Pythagorean fuzzy entropy and Pythagorean fuzzy similarity measure are developed to measure the degree of uncertainty and similarity between two Pythagor

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Satirad, Akarachai, Ronnason Chinram, Pongpun Julatha, and Aiyared Iampan. "Rough Pythagorean Fuzzy Sets in UP-Algebras." European Journal of Pure and Applied Mathematics 15, no. 1 (2022): 169–98. http://dx.doi.org/10.29020/nybg.ejpam.v15i1.4254.

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This paper aims to apply the concept of rough sets to Pythagorean fuzzy sets in UP-algebras. Then we introduce fifteen types of rough Pythagorean fuzzy sets in UP-algebras and study their generalization. In addition, we will also discuss $t$-level subsets of rough Pythagorean fuzzy sets in UP-algebras to study the relationships between rough Pythagorean fuzzy sets and rough sets in UP-algebras.

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Meng, Lingyan, and Xiaoyan Wei. "Research on Evaluation of Sustainable Development of New Urbanization from the Perspective of Urban Agglomeration under the Pythagorean Fuzzy Sets." Discrete Dynamics in Nature and Society 2021 (August 16, 2021): 1–11. http://dx.doi.org/10.1155/2021/2445025.

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In this study, considering the traditional geometric operation laws and Pythagorean fuzzy information, we propose a variety of new distance measures of Pythagorean fuzzy sets such as generalized Pythagorean fuzzy geometric distance (GPFGD) measures and generalized Pythagorean fuzzy weighted geometric distance (GPFWGD) measures. Besides, some special issues including Hamming distance, Euclidean distance, and Hausdorff distance of these raised geometric distance measures are investigated. To testify the valid of these new presented distance measures, we build a decision-making model illustrated

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Yang, Wei, Jiarong Shi, Yong Liu, Yongfeng Pang, and Ruiyue Lin. "Pythagorean Fuzzy Interaction Partitioned Bonferroni Mean Operators and Their Application in Multiple-Attribute Decision-Making." Complexity 2018 (November 1, 2018): 1–25. http://dx.doi.org/10.1155/2018/3606245.

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The aim of this paper is to develop partitioned Pythagorean fuzzy interaction Bonferroni mean operators based on the Pythagorean fuzzy set, Bonferroni mean, and interaction between membership and nonmembership. Several new aggregation operators are developed including the Pythagorean fuzzy interaction partitioned Bonferroni mean (PFIPBM) operator, the Pythagorean fuzzy weighted interaction partitioned Bonferroni mean (PFWIPBM) operator, the Pythagorean fuzzy interaction partitioned geometric Bonferroni mean (PFIPGBM) operator, and the Pythagorean fuzzy weighted interaction partitioned geometri

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Basha, M. Asim, and M. Mohammed Jabarulla. "Pythagorean fuzzy shortest hyper path in a network." Journal of Interdisciplinary Mathematics 27, no. 5 (2024): 1163–73. http://dx.doi.org/10.47974/jim-1948.

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Objectives: The Pythagorean fuzzy shortest hyperpath problem balances distance minimization with network fuzziness and uncertainty, considering node lengths, membership and non-membership degrees, when accurate data is unavailable or traditional methods cannot handle uncertain information. Methods: Pythagorean Fuzzy Dijkstra’s Algorithm uses Pythagorean fuzzy numbers as edge weights, while the A* algorithm considers fuzziness and uncertainty. Evolutionary, metaheuristic, linear programming, and fuzzy decision-making techniques can be used to solve the Pythagorean fuzzy shortest hyperpath probl

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Khan, Muhammad Sajjad Ali, Saleem Abdullah, Asad Ali, and Khaista Rahman. "Pythagorean Hesitant Fuzzy Information Aggregation and Their Application to Multi-Attribute Group Decision-Making Problems." Journal of Intelligent Systems 29, no. 1 (2018): 154–71. http://dx.doi.org/10.1515/jisys-2017-0231.

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Abstract In this paper, we introduce the concept of the Pythagorean hesitant fuzzy set (PHFS), which is the generalization of the intuitionistic hesitant fuzzy set under the restriction that the square sum of its membership degrees is ≤1. In decision making with PHFSs, aggregation operators play a key role because they can be used to synthesize multidimensional evaluation values represented as Pythagorean hesitant fuzzy values into collective values. Under PHFS environments, Pythagorean hesitant fuzzy ordered weighted averaging and Pythagorean fuzzy ordered weighted geometric operators are use

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Wang, Jie, Guiwu Wei, and Hui Gao. "Approaches to Multiple Attribute Decision Making with Interval-Valued 2-Tuple Linguistic Pythagorean Fuzzy Information." Mathematics 6, no. 10 (2018): 201. http://dx.doi.org/10.3390/math6100201.

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The Maclaurin symmetric mean (MSM) operator is a classical mean type aggregation operator used in modern information fusion theory, which is suitable to aggregate numerical values. The prominent characteristic of the MSM operator is that it can capture the interrelationship among multi-input arguments. Motivated by the ideal characteristic of the MSM operator, in this paper, we expand the MSM operator, generalized MSM (GMSM), and dual MSM (DMSM) operator with interval-valued 2-tuple linguistic Pythagorean fuzzy numbers (IV2TLPFNs) to propose the interval-valued 2-tuple linguistic Pythagorean f

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Akram, Muhammad, Jawaria Dar, and Adeel Farooq. "Planar Graphs under Pythagorean Fuzzy Environment." Mathematics 6, no. 12 (2018): 278. http://dx.doi.org/10.3390/math6120278.

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Graph theory plays a substantial role in structuring and designing many problems. A number of structural designs with crossings can be found in real world scenarios. To model the vagueness and uncertainty in graphical network problems, many extensions of graph theoretical ideas are introduced. To deal with such uncertain situations, the present paper proposes the concept of Pythagorean fuzzy multigraphs and Pythagorean fuzzy planar graphs with some of their eminent characteristics by investigating Pythagorean fuzzy planarity value with strong, weak and considerable edges. A close association i

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Gnanachristy, N. B., та G. K. Revathi. "A View on Pythagorean Fuzzy Contra 𝓖^∗ Continuous Function". Journal of Physics: Conference Series 2115, № 1 (2021): 012041. http://dx.doi.org/10.1088/1742-6596/2115/1/012041.

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Abstract The new dimension of non-standard fuzzy sets called Pythagorean fuzzy sets which can handle the inaccurate data very strongly has been established in recent days. Even though intuitionistic fuzzy sets were generously used in decision making to handle the imprecise data the novelty and the voluminous of Pythagorean fuzzy environment gives motivation to use it in decision making process. The Pythagorean fuzzy topological spaces are the novel generalization of fuzzy topological spaces. Herein the concept of Pythagorean fuzzy contra 𝒢∗ continuous functions are explored. Interrelations hav

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Asif, Muhammad, Umar Ishtiaq, and Ioannis K. Argyros. "Hamacher Aggregation Operators for Pythagorean Fuzzy Set and its Application in Multi-Attribute Decision-Making Problem." Spectrum of Operational Research 2, no. 1 (2024): 27–40. http://dx.doi.org/10.31181/sor2120258.

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Pythagorean fuzzy set is a useful expansion of intuitionistic fuzzy set for dealing with ambiguities, which mostly occur in real-life problems. Hamacher t-norm also has important and compatible norms that incorporate a parameter that offers various options to decision-makers during the information fusion process, thereby enhancing their ability to model decision-making problems effectively compared to alternative methods. In this study, Hamacher operators are being used to introduce several Pythagorean fuzzy Hamacher interactive weighted averaging (PFHIWA), Pythagorean fuzzy Hamacher interacti

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Touqeer, Muhammad, Rimsha Umer, and Muhammad Irfan Ali. "A chance-constraint programming model with interval-valued pythagorean fuzzy constraints." Journal of Intelligent & Fuzzy Systems 40, no. 6 (2021): 11183–99. http://dx.doi.org/10.3233/jifs-202383.

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Pythagorean fuzzy sets and interval-valued Pythagorean fuzzy sets are more proficient in handling uncertain and imprecise information than intuitionistic fuzzy sets and fuzzy sets. In this article, we put forward a chance-constraint programming method to solve linear programming network problems with interval-valued Pythagorean fuzzy constraints. This practice is developed using score function and upper and lower membership functions of interval-valued Pythagorean fuzzy numbers. The feasibility of the anticipated approach is illustrated by solving an airway network application and shown to be

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Deng, Xiumei, Jie Wang, Guiwu Wei, and Mao Lu. "Models for Multiple Attribute Decision Making with Some 2-Tuple Linguistic Pythagorean Fuzzy Hamy Mean Operators." Mathematics 6, no. 11 (2018): 236. http://dx.doi.org/10.3390/math6110236.

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The Hamy mean (HM) operator, as a useful aggregation tool, can capture the correlation between multiple integration parameters, and the 2-tuple linguistic Pythagorean fuzzy numbers (2TLPFNs) are a special kind of Pythagorean fuzzy numbers (PFNs), which can easily describe the fuzziness in actual decision making by 2-tuple linguistic terms (2TLTs). In this paper, to consider both Hamy mean (HM) operator and 2TLPFNs, we combine the HM operator, weighted HM (WHM) operator, dual HM (DHM) operator, and dual WHM (DWHM) operator with 2TLPFNs to propose the 2-tuple linguistic Pythagorean fuzzy HM (2TL

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Akram, Muhammad, and Sumera Naz. "A Novel Decision-Making Approach under Complex Pythagorean Fuzzy Environment." Mathematical and Computational Applications 24, no. 3 (2019): 73. http://dx.doi.org/10.3390/mca24030073.

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A complex Pythagorean fuzzy set (CPFS) is an extension of a Pythagorean fuzzy set that is used to handle the vagueness with the degrees whose ranges are enlarged from real to complex subset with unit disc. In this research study, we propose the innovative concept of complex Pythagorean fuzzy graphs (CPFGs). Further, we present the concepts of regular and edge regular graphs in a complex Pythagorean fuzzy environment. Moreover, we develop a complex Pythagorean fuzzy graph based multi-attribute decision making an approach to handling the situations in which the graphic structure of attributes is

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