# Bipolar Pythagorean Fuzzy Sets and Their Application in Multi-attribute Decision Making Problems

Bipolar Pythagorean Fuzzy Sets and Their Application in Multi-attribute Decision Making Problems In this paper, the idea of the bipolar Pythagorean fuzzy sets (BPFSs) and its activities, which is a generalization of fuzzy sets, bipolar fuzzy sets (BFSs), intuitionistic fuzzy sets and bipolar intuitionistic fuzzy sets is proposed, with the goal that it can deal with dubious data all the more flexibly during the process of decision making. The key objective of this research paper has presented another variant of the Pythagorean fuzzy sets so called BPFSs. In bipolar Pythagorean fuzzy sets, membership degrees are satisfying the condition 0≤μp+x2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0 \le \left( {\mu_{p}^{ + } \left( x \right)} \right)^{2}$$\end{document} + vp+x2≤1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\left( {v_{p}^{ + } \left( x \right)} \right)^{2} \le 1$$\end{document} and 0≤μp-x2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0 \le \left( {\mu_{p}^{ - } \left( x \right)} \right)^{2}$$\end{document} + vp-x2≤1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\left( {v_{p}^{ - } \left( x \right)} \right)^{2} \le 1$$\end{document} instead of 0≤μpx2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0 \le \left( {\mu_{p} \left( x \right)} \right)^{2}$$\end{document} + vpx2≤1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\left( {v_{p} \left( x \right)} \right)^{2} \le 1$$\end{document} as is in Pythagorean fuzzy sets and 0≤μpx\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0 \le \mu_{p} \left( x \right)$$\end{document} + vpx≤1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$v_{p} \left( x \right) \le 1$$\end{document} as is in the intuitionistic fuzzy sets. Here, negative membership degree means the certain counter-property comparing to a bipolar Pythagorean fuzzy set. Also, the BPFSs weighted average operator and the BPFSs weighted geometric operator to aggregate the BPFSs is developed here. Further a multi attribute decision making technique is developed and the proposed aggregation operators are used. Finally, a numerical methodology for execution of the proposed system is introduced. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of Data Science Springer Journals

# Bipolar Pythagorean Fuzzy Sets and Their Application in Multi-attribute Decision Making Problems

, Volume 10 (3) – Jun 1, 2023
33 pages

/lp/springer-journals/bipolar-pythagorean-fuzzy-sets-and-their-application-in-multi-bNo6pTDHMz

# References (52)

Publisher
Springer Journals
ISSN
2198-5804
eISSN
2198-5812
DOI
10.1007/s40745-020-00315-8
Publisher site
See Article on Publisher Site

### Abstract

In this paper, the idea of the bipolar Pythagorean fuzzy sets (BPFSs) and its activities, which is a generalization of fuzzy sets, bipolar fuzzy sets (BFSs), intuitionistic fuzzy sets and bipolar intuitionistic fuzzy sets is proposed, with the goal that it can deal with dubious data all the more flexibly during the process of decision making. The key objective of this research paper has presented another variant of the Pythagorean fuzzy sets so called BPFSs. In bipolar Pythagorean fuzzy sets, membership degrees are satisfying the condition 0≤μp+x2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0 \le \left( {\mu_{p}^{ + } \left( x \right)} \right)^{2}$$\end{document} + vp+x2≤1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\left( {v_{p}^{ + } \left( x \right)} \right)^{2} \le 1$$\end{document} and 0≤μp-x2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0 \le \left( {\mu_{p}^{ - } \left( x \right)} \right)^{2}$$\end{document} + vp-x2≤1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\left( {v_{p}^{ - } \left( x \right)} \right)^{2} \le 1$$\end{document} instead of 0≤μpx2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0 \le \left( {\mu_{p} \left( x \right)} \right)^{2}$$\end{document} + vpx2≤1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\left( {v_{p} \left( x \right)} \right)^{2} \le 1$$\end{document} as is in Pythagorean fuzzy sets and 0≤μpx\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0 \le \mu_{p} \left( x \right)$$\end{document} + vpx≤1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$v_{p} \left( x \right) \le 1$$\end{document} as is in the intuitionistic fuzzy sets. Here, negative membership degree means the certain counter-property comparing to a bipolar Pythagorean fuzzy set. Also, the BPFSs weighted average operator and the BPFSs weighted geometric operator to aggregate the BPFSs is developed here. Further a multi attribute decision making technique is developed and the proposed aggregation operators are used. Finally, a numerical methodology for execution of the proposed system is introduced.

### Journal

Annals of Data ScienceSpringer Journals

Published: Jun 1, 2023

Keywords: FSs; IFSs; PFs; BFSs; BIFSs; BPFSs; 03E72; 93C42; 90B50