On bipolar fuzzy soft topology with decision-making
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On bipolar fuzzy soft topology with decision-making Muhammad Riaz1
· Syeda Tayyba Tehrim1
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we bring out the idea of bipolar fuzzy soft topology (BFS-topology) based on bipolar fuzzy soft set (BFSset). BFS-topology is the generalization of the crisp topology. We discuss certain properties of BFS-topology including, BFS-closure, BFS-interior, BFS-exterior and BFS-frontier by utilizing BFS-points. We study the concept of BFS-subspace, BFS-neighbourhoods and BFS-base for BFS-topology with the help of detailed examples. Furthermore, we use BFS-topology in decision-making by applying an algorithm to deal with unpredictability. Keywords Bipolar fuzzy soft set · BFS-topology · BFS-base · Decision-making
1 Introduction Zadeh (1965) brought out the idea of fuzzy sets. The fuzzy set is the universality of classical set. In Chang (1968) interpreted fuzzy topology by using fuzzy set. In Molodtsov (1999) proposed the idea of soft sets in order to deal with unpredictability. Maji et al. (2002, 2003) discussed some important operations of soft sets and its implementation into the decision-making. Ali et al. (2009) proposed some modified operations of soft set theory. Ça˘gman et al. (2010), Shabir and Naz (2011) independently brought out the concept of soft topology. Kharal and Ahmad (2011) presented the idea of mappings of soft classes. Akram and Feng (2013), Akram and Adeel (2016) studied soft intersection Lie algebras and mpolar fuzzy graphs. Das and Samanta (2012, 2013) presented the abstraction of soft real set and soft metric spaces. Many authors have auspiciously adapted soft set theory in various fields [see Akram and Feng (2013), Aktas and Ça˘gman (2007), Ça˘gman et al. (2010), Feng et al. (2008), Kharal and Ahmad (2011), Maji et al. (2003), Majumdar and Samanta (2010), Shabir and Naz (2011)]. Maji et al. (2001) provided the impression of fuzzy soft sets. In Feng et al. (2010a, b, Communicated by A. Di Nola.
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Muhammad Riaz [email protected] Syeda Tayyba Tehrim [email protected]
1
Department of Mathematics, University of the Punjab, Lahore, Pakistan
2011) solved decision-making problems on fuzzy soft sets and provided amplification of soft set with rough set and fuzzy set. Ça˘gman et al. (2010, 2011b) studied fuzzy soft set and its utilization in decision-making. They also studied the fuzziness of parameters and combined it with fuzzy soft set. Varol and Aygun (2012) interpreted fuzzy soft topology. Zorlutuna and Atmaca (2016) brought out the abstraction of fuzzy parameterized fuzzy soft topology. Riaz and Naeem (2016a); Riaz and Naeem (2016b) established the novel ideas of measurable soft sets and measurable soft mappings. Riaz and Fatima (2017) presented certain properties of metric spaces. Riaz and Hashmi (2016a); Riaz and Hashmi (2016b) extended the idea of fuzzy parameterized fuzzy soft topology and proved important results which do hold in classical topology but do not hold in fuzzy parameterized fuzz
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