Abstract
This paper presents distance measure between two complex Atanassov's intuitionistic fuzzy sets (CAIFSs). This distance measure is used to illustrate an application of CAIFSs in solving one of the most core application areas of fuzzy set theory, which is multiattributes decision-making (MADM) problems, in complex Atanassov's intuitionistic fuzzy realm. A new structure of relation between two CAIFSs, called complex Atanassov's intuitionistic fuzzy relation (CAIFR), is obtained. This relation is formally generalised from a conventional Atanassov's intuitionistic fuzzy relation, based on complex Atanassov's intuitionistic fuzzy sets, in which the ranges of values of CAIFR are extended to the unit circle in complex plane for both membership and nonmembership functions instead of [0, 1] as in the conventional Atanassov's intuitionistic fuzzy functions. Definition and some mathematical concepts of CAIFS, which serve as a foundation for the creation of complex Atanassov's intuitionistic fuzzy relation, are recalled. We also introduce the Cartesian product of CAIFSs and derive two properties of the product space. The concept of projection and cylindric extension of CAIFRs are also introduced. An example of CAIFR in real-life situation is illustrated in this paper. Finally, we introduce the concept of composition of CAIFRs.
Highlights
The idea of the concept of Atanassov’s intuitionistic fuzzy set (AIFS) was introduced by Atanassov [1], where he achieved his concept by adding the nonmembership term to the definition of fuzzy set (FS) that was given by Zadeh [2], while the fuzzy set has only one component, a membership function
In 2011, Jun et al [5] employed complex fuzzy sets to represent the information with uncertainty and periodicity, where they introduced a product-sum aggregation operator(PSAO-) based prediction (PSAOP) method to generate a solution of multiple periodic factor prediction (MPFP) problems
The phase terms that present production date for first attribute of ideal car can be given as follows: if the team of analysts thought that at least 80% of them believe that the ideal production date of car is suitable at the first attribute; and not more than 15% of them believe that the ideal production date of car is poor
Summary
The idea of the concept of Atanassov’s intuitionistic fuzzy set (AIFS) was introduced by Atanassov [1], where he achieved his concept by adding the nonmembership term to the definition of fuzzy set (FS) that was given by Zadeh [2], while the fuzzy set has only one component, a membership function. The CFS [4] has only one additional phase term, but in CAIFS [11], we have two additional phase terms This confers more range values to represent the uncertainty and periodicity semantics simultaneously, and to define the values of belongingness and nonbelongingness for any object in these complex-valued functions. The difference between CAIFS and AIFS is that in our concept, CAIFS has an ability to represent the problems with Atanassov’s intuitionistic uncertainty and periodicity simultaneously. In 2011, Jun et al [5] employed complex fuzzy sets to represent the information with uncertainty and periodicity, where they introduced a product-sum aggregation operator(PSAO-) based prediction (PSAOP) method to generate a solution of multiple periodic factor prediction (MPFP) problems. The main benefit expected to be gained from the introduction of the concept of CAIFR is the ability to denote both the presence and absence of association, interaction, or interconnectedness in one set of CAIFS instead of two sets as in CFS, where CAIFS has membership and nonmembership functions in the same set, whilst CFS contains only the membership functions and denotes the presence or absence of association, interaction, or interconnectedness
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