Bipolar N -soft set theory with applications
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METHODOLOGIES AND APPLICATION
Bipolar N-soft set theory with applications Hüseyin Kamacı1 · Subramanian Petchimuthu2
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, the notion of bipolar N -soft set, which is the bipolar extension of N -soft set, and its fundamental properties are introduced. This new idea is illustrated with real-life examples. Moreover, some useful operations and products on the bipolar N -soft sets are derived. We thoroughly discuss the idempotent, commutative, associative, and distributive laws for these emerging operations and products. Also, we set forth two outstanding algorithms to handle the decision-making problems under bipolar N -soft set environments. We give potential applications and comparison analysis to demonstrate the efficiency and advantages of algorithms. Keywords Soft sets · N -soft sets · Bipolar N -soft sets · Operations of bipolar N -soft sets · Decision making
1 Introduction The real-world issues that are frequently encountered in many fields such as medical, engineering, economics, and environmental science are closely associated with uncertainty and impreciseness. To illuminate these issues, in 1965, Zadeh (1965) introduced the concept of fuzzy set and immediately afterward many of the algebraic properties in classical set theory were adapted for these sets. Many authors studied the combinations and relations of intuitive, hesitancy and bipolarity with the fuzzy set (Ali et al. 2019; Atanassov 1986, 2012; Lee 2000; Saikia et al. 2020; Torra 2010). In 1999, Molodtsov (1999) described the concept of a soft set based on the binary-valued logic. It is a convenient mathematical tool for classifying objects concerning the specified parameters. As a sophisticated version of the fuzzy set, the notion of the fuzzy soft set was depicted and studied in various directions (Majumdar and Samanta 2010; Roy and Maji 2007; Petchimuthu et al. 2020).
Communicated by V. Loia.
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Subramanian Petchimuthu [email protected]
1
Department of Mathematics, Yozgat Bozok University, 66100 Yozgat, Turkey
2
Department of Mathematics, University College of Engineering, Nagercoil, Tamilnadu 629004, India
In 2000, Lee (2000) put forward the idea of a bipolarvalued fuzzy set derived by extending the grade of membership of the fuzzy set from the interval [0, 1] to [−1, 1]. In Karaaslan and Karata¸s (2015) and Shabir and Naz (2013), the authors discussed the bipolarity of soft sets. By generalizing bipolar-valued fuzzy sets and bipolar soft sets, the concepts of bipolar fuzzy soft sets (Abdullah et al. 2014; Riaz and Tehrim 2019), fuzzy bipolar soft sets (Naz and Shabir 2014), bipolar multi-fuzzy soft sets (Yang et al. 2014), bipolar fuzzy soft expert sets (Al-Qudah and Hassan 2017), bipolar neutrosophic soft sets (Ali et al. 2017), bipolar fuzzy soft graphs (Akram et al. 2018b), rough fuzzy bipolar soft sets (Malik and Shabir 2019) were defined. Moreover, their algebraic structures were investigated and thereby applied to multi-criteria decision-making pr
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