Complex Atanassov's Intuitionistic Fuzzy Relation

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.

[1]  Krassimir T. Atanassov,et al.  Intuitionistic fuzzy sets , 1986 .

[2]  Witold Pedrycz,et al.  Further relation equations with equality and difference composition operators , 1988 .

[3]  L. Kohout,et al.  Special properties, closures and interiors of crisp and fuzzy relations , 1988 .

[4]  J. Buckley.  Fuzzy complex numbers , 1989 .

[5]  W. Pedrycz,et al.  Fuzzy relation equations theory as a basis of fuzzy modelling: an overview , 1991 .

[6]  J. Mendel.  Fuzzy logic systems for engineering: a tutorial , 1995, Proc. IEEE.

[7]  Pedro Burillo López,et al.  On Intuitionistic Fuzzy Relations , 2017, Scientific Transactions in Environment and Technovation.

[8]  Humberto Bustince,et al.  Structures on intuitionistic fuzzy relations , 1996, Fuzzy Sets Syst..

[9]  Janusz Kacprzyk,et al.  Applications of intuitionistic Fuzzy sets in decision making , 1998 .

[10]  J. Kacprzyk,et al.  Group decision making under intuitionistic fuzzy preference relations , 1998 .

[11]  Vladik Kreinovich,et al.  Complex fuzzy sets: towards new foundations , 2000, Ninth IEEE International Conference on Fuzzy Systems. FUZZ- IEEE 2000 (Cat. No.00CH37063).

[12]  Humberto Bustince,et al.  Construction of intuitionistic fuzzy relations with predetermined properties , 2000, Fuzzy Sets Syst..

[13]  Janusz Kacprzyk,et al.  Distances between intuitionistic fuzzy sets , 2000, Fuzzy Sets Syst..

[14]  Abraham Kandel,et al.  On complex fuzzy sets , 2001, 10th IEEE International Conference on Fuzzy Systems. (Cat. No.01CH37297).

[15]  Janusz Kacprzyk,et al.  Entropy for intuitionistic fuzzy sets , 2001, Fuzzy Sets Syst..

[16]  Janusz Kacprzyk,et al.  Evaluation of agreement in a group of experts via distances between intuitionistic fuzzy preferences , 2002, Proceedings First International IEEE Symposium Intelligent Systems.

[17]  Lucien Duckstein,et al.  Comparison of fuzzy numbers using a fuzzy distance measure , 2002, Fuzzy Sets Syst..

[18]  J. Kacprzyk,et al.  An intuitionistic fuzzy set based approach to intelligent data analysis: an application to medical diagnosis , 2003 .

[19]  J. Baldwin,et al.  Entropy for Intuitionistic Fuzzy, Set Theory and Mass Assignment Theory , 2004 .

[20]  Eulalia Szmidt,et al.  Assigning the parameters for intuitionistic fuzzy sets , 2005 .

[21]  Qiang Zhang,et al.  Supplier Selection model based on distance measure between intuitionistic fuzzy sets , 2006, 2006 IEEE International Conference on Systems, Man and Cybernetics.

[22]  Tharam S. Dillon,et al.  δ-Equalities of Complex Fuzzy Relations , 2010 .

[23]  Scott Dick,et al.  ANCFIS: A Neurofuzzy Architecture Employing Complex Fuzzy Sets , 2011, IEEE Transactions on Fuzzy Systems.

[24]  Kai-Yuan Cai,et al.  Operation Properties and δ-Equalities of Complex Fuzzy Sets , 2011 .

[25]  Ting-Yu Chen,et al.  The ELECTRE multicriteria analysis approach based on Atanassov's intuitionistic fuzzy sets , 2011, Expert Syst. Appl..

[26]  Ming-yuan Chen,et al.  Induced generalized intuitionistic fuzzy OWA operator for multi-attribute group decision making , 2012, Expert Syst. Appl..

[27]  Jie Lu,et al.  A Method for Multiple Periodic Factor Prediction Problems Using Complex Fuzzy Sets , 2012, IEEE Transactions on Fuzzy Systems.

[28]  Krassimir T. Atanassov.  Intuitionistic Fuzzy Relations (IFRs) , 2012 .

[29]  Li Zheng,et al.  A novel approach to multi-attribute decision making based on intuitionistic fuzzy sets , 2012, Expert Syst. Appl..

[30]  Guiwu Wei,et al.  Some induced correlated aggregating operators with intuitionistic fuzzy information and their application to multiple attribute group decision making , 2012, Expert Syst. Appl..

[31]  Francisco Chiclana,et al.  Consistency of 2D and 3D distances of intuitionistic fuzzy sets , 2012, Expert Syst. Appl..

[32]  Abdul Razak Salleh,et al.  Complex intuitionistic fuzzy sets , 2012 .