On the defect of complementarity of fuzzy measures

It is well-known that the fuzzy measures are non-additive. This means that the sum between the measure of a set and the measure of its complement is not equal with the measure of space. As a consequence, in this paper we introduce the concept of defect of complementarity. Also, for large classes of fuzzy measures this defect of complementarity is calculated or estimated. Finally, some applications and interpretations are presented.

[1]  Radko Mesiar,et al.  Generalizations of k-order additive discrete fuzzy measures , 1999, Fuzzy Sets Syst..

[2]  D. Ralescu.  Cardinality, quantifiers, and the aggregation of fuzzy criteria , 1995 .

[3]  Sorin G. Gal,et al.  On the defect of additivity of fuzzy measures , 2002, Fuzzy Sets Syst..

[4]  G. Choquet.  Theory of capacities , 1954 .

[5]  M. Sugeno,et al.  Non-monotonic fuzzy measures and the Choquet integral , 1994 .

[6]  Rudolf Kruse,et al.  A note on λ-additive fuzzy measures , 1982 .

[7]  M. J. Frank.  On the simultaneous associativity ofF(x,y) andx +y -F(x,y) , 1979 .

[8]  Keon Myung Lee,et al.  Identification of λ-fuzzy measure by genetic algorithms , 1995, Fuzzy Sets Syst..

[9]  On the null-additivity and the uniform autocontinuity of a fuzzy measure , 1993 .

[10]  David Cariolaro,et al.  On the measure of a fuzzy set based on continuous t-conorms , 1997, Fuzzy Sets Syst..

[11]  D. Denneberg.  Non-additive measure and integral , 1994 .

[12]  Jean-Luc Marichal,et al.  On Choquet and Sugeno integrals as aggregation functions , 2000 .

[13]  G. Klir,et al.  Fuzzy Measure Theory , 1993 .

[14]  M. Gupta,et al.  FUZZY INFORMATION AND DECISION PROCESSES , 1981 .

[15]  D. Dubois,et al.  Fuzzy cardinality and the modeling of imprecise quantification , 1985 .

[16]  Gregory T. Adams,et al.  The fuzzy integral , 1980 .

[17]  M. Sugeno,et al.  An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy , 1989 .

[18]  M. Sugeno,et al.  Fuzzy Measures and Integrals: Theory and Applications , 2000 .

[19]  S. T. Wierzcho.  An algorithm for identification of fuzzy measure , 1983 .

[20]  Didier Dubois,et al.  A class of fuzzy measures based on triangular norms , 1982 .

[21]  Settimo Termini,et al.  A Definition of a Nonprobabilistic Entropy in the Setting of Fuzzy Sets Theory , 1972, Inf. Control..

[22]  Lotfi A. Zadeh,et al.  A Theory of Approximate Reasoning , 1979 .

[23]  Maciej Wygralak.  Questions of cardinality of finite fuzzy sets , 1999, Fuzzy Sets Syst..

[24]  Michio Sugeno,et al.  Fuzzy Measure and Fuzzy Integral , 1972 .