Neutrosophic Fuzzy Soft Matrix Theory and Its Application in Group Decision Making

1. Introduction And Motivation

Classical or crisp notion gives the precise or deterministic value of an object to a set. But, sometimes due to the error in data collection, a wrong technique used for data collection, we don’t know the exact value of the missing data i.e there is an information gap. Such incomplete, indeterministic or imprecise information leads to uncertainty, vagueness,or ambiguity and it cannot be described by the classical set. This leads to a fuzzy set (FS) (Zadeh, 1965) introduced by Zadeh in 1965.The Fuzzy set uses the membership degree 𝜇_A(x)∈[0,1] to describe uncertainty. If 𝜇_A(x)=0 or 1, then there is no ambiguity arises and 𝜇_A(x) turned to a characteristic function. Due to complexity in some real-life problems, 𝜇_A(x) can’t be a crisp value, then we need to introduce an interval-valued fuzzy set (IVFS) (Gorzalczany, 1987) to remove such issue. In fuzzy set and interval-valued fuzzy set, there is no scope of considering the non-membership degree 𝛾_A(x) as we know there exist some fields of knowledge where we have to exhibit not only the membership value in support of evidence but also the non-membership value against the evidence. This leads to the introduction of the intuitionistic fuzzy set (IFS) (Atanassov, 1986). An IFS has been further extended by introducing an interval-valued intuitionistic fuzzy set (IVIFS) (Atanassov & Gargov, 1989). Some significant works related to fuzzy sets and their extensions are given in (Ashtiani,Haghighirad, Makui & Ali Montazer, 2009; De, Biswas & Roy,2001; Dengfeng & Chuntian, 2002; Garg, 2016; Mahmood, Ullah, Khan & Jan, 2019). In all types of fuzzy sets there exist difficulties to set the membership function in each particular case and it arises due to restriction given on membership function. Fuzzy set and its extensions are enabled to answer a question that is either true or false but it can’t able to answer a question that is neither true nor false (unknown) i.e., there is a contradiction. For example, a set of 10 true/false type questions are given to a student and the response of the student is recorded as 5 are true, 3 are false and the remaining 2 are unknown. Similarly, we give many instances in real-life where it is not possible to answer the evidence. Such situations are managed by a neutrosophic set (NS)(Smarandache, 2005). In NS, each element of the universe is characterized by three membership degrees such as truth-membership, indeterminacy-membership, and falsity-membership degree. In the neutrosophic set, each membership degree belongs to the non-standard unit interval ]-0,1⁺[. NS is a direct extension of the intuitionistic fuzzy set. After the introduction of NS, it has been progressed more rapidly as it provides a more general framework to explain uncertainty that contains indeterminacy and inconsistency. For the scientific and technical application we need standard unit interval and this problem is resolved by introducing a single-valued neutrosophic set(SVNS)(Wang, Smarandache, Zhang & Sunderraman, 2010) . So, SVNS can be treated as an instance of NS. For all practical application, we use SVNS instead of NS. NS has several applications among which some major contributions are proposed in (Huang, 2016; Patrascu, 2015; Ye, 2014; Ye, 2014). We find some instances in real life that contain both uncertainty and inconsistency and these can be handled by the combination of fuzzy set and neutrosophic set, which gives rise to the introduction of the neutrosophic fuzzy set(NFS)(Das, Roy, Kar, Kar & Pamučar, 2020). IN NFS, each membership degree is associated with neutrosophic components.