Abstract
In this manuscript, two generalizations of fuzzy sets, intuitionistic fuzzy sets and picture fuzzy sets, known as spherical fuzzy sets and T-spherical fuzzy sets, are discussed and a numerical and geometrical comparison among them is established. A T-spherical fuzzy set can model phenomena like voting using four characteristic functions denoting the degree of vote in favor, abstinence, vote in opposition, and refusal with an infinite domain, whereas an intuitionistic fuzzy set can model only phenomena of yes or no types. First, in this manuscript, some similarity measures in the frameworks of intuitionistic fuzzy sets and picture fuzzy sets are discussed. With the help of some numerical results, it is discussed that existing similarity measures have some limitations and could not be applied to problems where information is provided in T-spherical fuzzy environment. Therefore, some new similarity measures in the framework of spherical fuzzy sets and T-spherical fuzzy sets are proposed including cosine similarity measures, grey similarity measures, and set theoretic similarity measures. With the help of some results, it was proved that the proposed similarity measures are a generalization of existing similarity measures. The newly-defined similarity measures were subjected to a well-known problem of building material recognition and the results are discussed. A comparative study of new and existing similarity measures was established and some advantages of the proposed work are discussed.
Highlights
The study of fuzzy set (FS) theory was initiated in Reference [1] where the membership degree s of an element of a set was defined by a characteristic function on a unit interval [0, 1]and the non-membership degree could be obtained by subtracting membership grade form 1.Atanassov [2] extended Zadeh’s concept of FS to intuitionistic fuzzy set (IFS) where the membership and non-membership degrees, i.e., s and d, are defined independently but with a constraint that their sum must belong to the interval [0, 1], i.e., sum(s, d) ∈ [0, 1]
This article described the background of IFSs, Pythagorean FSs and picturethat fuzzy setsum (PFS) in detail for observing the limited nature of their structures
It is discussed how the shortcomings that exists in current structures are improved using the framework of spherical fuzzy set (SFS) and TSFSs with the help of diagrams and numerical examples
Summary
The study of fuzzy set (FS) theory was initiated in Reference [1] where the membership degree s of an element of a set was defined by a characteristic function on a unit interval [0, 1]. Atanassov [2] extended Zadeh’s concept of FS to intuitionistic fuzzy set (IFS) where the membership and non-membership degrees, i.e., s and d, are defined independently but with a constraint that their sum must belong to the interval [0, 1], i.e., sum(s, d) ∈ [0, 1]. Due to the restriction of Atanassov’s model of IFS, values cannot be assigned to its characteristic functions, as sometimes the sum (s, d) exceeds [0, 1] interval. For some developments in these areas we refer readers to References [5,6,7,8,9,10]
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