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Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations
- Publisher
- Springer Journals
- Copyright
- Copyright © Springer-Verlag GmbH Germany, part of Springer Nature 2021
- 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 Science – Springer Journals
Published: Jun 1, 2023
Keywords: FSs; IFSs; PFs; BFSs; BIFSs; BPFSs; 03E72; 93C42; 90B50