Inclusion Measure and Similarity Measure of Intuitionistic and Interval-valued Fuzzy Sets

We begin the paper studying the axioms that the inclusion measure of an element in another element of a lattice and the similarity measure between two elements should satisfy. Next, we present a series of expressions of inclusion measures, first for intuitionistic values and then for intuitionistic fuzzy sets ( or interval-valued, since they have a strong connection), discussing in both cases their properties. Then we discuss similarity measure which can be transformed by the proposed inclusion measure based on their axiomatic definitions. The similarity measures defined by the proposed inclusion measures of IFSs are seem reasonable.

[1]  David L. Olson,et al.  Similarity measures between intuitionistic fuzzy (vague) sets: A comparative analysis , 2007, Pattern Recognit. Lett..

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

[3]  I. Turksen.  Interval valued fuzzy sets based on normal forms , 1986 .

[4]  Humberto Bustince,et al.  Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets , 1996, Fuzzy Sets Syst..

[5]  Chengyi Zhang,et al.  Similarity measures on three kinds of fuzzy sets , 2006, Pattern Recognit. Lett..

[6]  E. Dougherty,et al.  Fuzzification of set inclusion: theory and applications , 1993 .

[7]  I. Turksen,et al.  An approximate analogical reasoning schema based on similarity measures and interval-valued fuzzy sets , 1990 .

[8]  Weixin Xie,et al.  Subsethood measure: new definitions , 1999, Fuzzy Sets Syst..

[9]  Lotfi A. Zadeh,et al.  Outline of a New Approach to the Analysis of Complex Systems and Decision Processes , 1973, IEEE Trans. Syst. Man Cybern..

[10]  B. Baets,et al.  A comparative study of similarity measures , 1995 .

[11]  Humberto Bustince,et al.  Indicator of inclusion grade for interval-valued fuzzy sets. Application to approximate reasoning based on interval-valued fuzzy sets , 2000, Int. J. Approx. Reason..

[12]  Chris Cornelis,et al.  Sinha-Dougherty approach to the fuzzification of set inclusion revisited , 2003, Fuzzy Sets Syst..

[13]  Wen-Xiu Zhang,et al.  Theory of including degrees and its applications to uncertainty inferences , 1996, Soft Computing in Intelligent Systems and Information Processing. Proceedings of the 1996 Asian Fuzzy Systems Symposium.

[14]  Nikos Karacapilidis,et al.  Application of a similarity measure of fuzzy sets to fuzzy relational equations , 1995, Fuzzy Sets Syst..

[15]  C. Pappis.  Value approximation of fuzzy systems variables , 1991 .

[16]  Li Jun.  Theory of approximate reasoning in n-valued standard serial logic system , 2006 .

[17]  L. Kohout,et al.  FUZZY POWER SETS AND FUZZY IMPLICATION OPERATORS , 1980 .

[18]  Przemyslaw Grzegorzewski,et al.  Subsethood measure for intuitionistic fuzzy sets , 2004, 2004 IEEE International Conference on Fuzzy Systems (IEEE Cat. No.04CH37542).

[19]  Ranjit Biswas,et al.  An application of intuitionistic fuzzy sets in medical diagnosis , 2001, Fuzzy Sets Syst..

[20]  Chris Cornelis,et al.  Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application , 2004, Int. J. Approx. Reason..

[21]  Virginia R. Young,et al.  Fuzzy subsethood , 1996, Fuzzy Sets Syst..

[22]  Chris Cornelis,et al.  Inclusion Measures in Intuitionistic Fuzzy Set Theory , 2003, ECSQARU.

[23]  Weixin Xie,et al.  Some notes on similarity measure and proximity measure , 1999, Fuzzy Sets Syst..

[24]  Liu Xuecheng,et al.  Entropy, distance measure and similarity measure of fuzzy sets and their relations , 1992 .

[25]  R. Willmott.  Two fuzzier implication operators in the theory of fuzzy power sets , 1980 .

[26]  Jin-Han Park,et al.  On Fuzzy Inclusion in the Interval-Valued Sense , 2005, FSKD.

[27]  M. Gorzałczany.  A method for inference in approximate reasoning based on interval-valued fuzzy sets , 1987 .

[28]  C. Pappis,et al.  A comparative assessment of measures of similarity of fuzzy values , 1993 .