Abstract
In this paper, the concept of complex linear Diophantine fuzzy set (CLDFS), which is obtained by integrating the phase term into the structure of the linear Diophantine fuzzy set (LDFS) and thus is an extension of LDFS, is introduced. In other words, the ranges of grades of membership, non-membership, and reference parameters in the structure of LDFS are extended from the interval [0, 1] to unit circle in the complex plane. Besides, this set approach is proposed to remove the conditions associated with the grades of complex-valued membership and complex-valued non-membership in the framework of complex intuitionistic fuzzy set (CIFS), complex Pythagorean fuzzy set (CPyFS), and complex q-rung orthopair fuzzy set (Cq-ROFS). It is proved that each of CIFS, CPyFS, and Cq-ROFS is a CLDFS, but not vice versa. In addition, some operations and relations on CLDFSs are derived and their fundamental properties are investigated. The intuitive definitions of cosine similarity measure (CSM) and cosine distance measure (CDM) between two CLDFSs are introduced and their characteristic principles are examined. An approach based on CSM is proposed to tackle medical diagnosis issues and its performance is tested by dealing with numerical examples. Finally, a comparative study of the proposed approach with several existing approaches is created and its advantages are discussed.
Highlights
In classical set theory, the membership of elements in a set is evaluated in binary terms according to a bivalent condition whether an element belongs to the set
In the section “Complex linear Diophantine fuzzy sets”, we describe the construction of complex linear Diophantine fuzzy set (CLDFS), and propose some operations, propositions, and the methods of comparing two complex linear Diophantine fuzzy numbers (CLDFNs)
In the section “Cosine similarity measures for complex linear Diophantine fuzzy sets”, we propose two new approaches to calculate the coefficients of similarity and dissimilarity between two CLDFSs
Summary
The membership of elements in a set is evaluated in binary terms according to a bivalent condition whether an element belongs to the set. The cosine similarity measure for two vectors satisfies the following properties: where Q(rk), Q(rk) ∈ [0, 1] represent the grades of membership and non-membership of rk ∈ into the set Q with the condition 0 ≤ ( P (rk))q + ( P (rk))q ≤ 1. In 2019, Riaz and Hashmi [65] defined the linear Diophantine fuzzy sets, which are the generalized forms of the IFSs, PyFSs, and q-ROFSs, by integrating the grades of reference parameters to the grades of membership and non-membership in the constructions of the IFS, PyFS, and q-ROFS. The CLDFS (with the grades of membership, nonmembership and reference parameters are complexvalued) contains information that covers more amplitude and phase terms than the FS (with the grade of membership is real-valued), CFS (with the grade of membership is complex-valued), IFS (or PyFS, q-ROFS) (with the grades of membership and non-membership are real-valued), CIFS (or CPyFS, Cq-ROFS) We assert that the emerging similarity and distance measures will give the various options to the decision-makers based on their optimistic and pessimistic behavior in the decision-making process
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