Abstract
Complex fuzzy N-soft set (CFN-SS) is an important technique to manage awkward and unreliable information in realistic decision-making problems. CFN-SS is a blend of two separate theories, called N-soft sets (N-SSs) and complex fuzzy sets (CFSs), which are the modified versions of soft sets (SSs) and fuzzy sets (FSs) to depict vague and uncertain information in daily life problems. In this manuscript, the novel concept of CFN-SS is explored and their fundamental laws are discussed. CFN-SS contains the grade of truth in the form of a complex number whose real and imaginary parts are limited to the unit interval. Besides, we examine some algebraic properties for CFN-SS like union, intersections and justify these properties with the help of some numerical examples. To examine the superiority and effectiveness of the proposed approaches, the special cases of the investigated approaches are also discussed. A decision-making procedure is developed by using the investigated ideas based on CFN-SSs. Further, some numerical examples are also illustrated with the help of explored ideas to find the reliability and effectiveness of the proposed approaches. Finally, the comparative analysis of the investigated ideas with some existing ideas is also demonstrated to prove the quality of the proposed works. The graphical expressions of the obtained results are also discussed.
Highlights
With the development of the information age, the decisionmaking problems and decision-making environments are more and more complex
[7] A complex fuzzy set (CFS) is designated and defined by: S {(x, μ(x)) : x ∈ U }, This manuscript is settled as follows: In Section “Preliminaries” of this manuscript, we provide some fundamental definitions and their properties of fuzzy sets (FSs), complex fuzzy sets (CFSs), soft sets (SSs), Fuzzy SS (FSS), CFSs, N-soft sets (N-SSs), and fuzzy N-SS (FN-SS) for the readers
In Section “Complex Fuzzy N-Soft Sets” of this manuscript, we propose the notion of Complex fuzzy N-soft set (CFN-SS) and their functional representation
Summary
With the development of the information age, the decisionmaking problems and decision-making environments are more and more complex. Zhang and Zhan [22] described fuzzy soft β-covering based fuzzy rough sets and corresponding DM applications. The certain types of soft coverings based rough sets with applications were established by Zhan and Wang [23]. Jiang et al [24] defined the MADM approach to covering based variable precision fuzzy rough set. Zhan and Alcantud [29] presented a novel type of soft rough covering and tis application to multicriteria group decision-making. Zhan et al [31] presented covering based variable precision fuzzy rough sets with PROMETHEE-EDAS method. An application to rating problem by using TOPSIS-WAA method based on a covering-based fuzzy rough set is introduced in [32]. The notion of complex FSS was interpreted by Thirunavukarasu et al [34]
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