Componentwise decomposition of some lattice-valued fuzzy integrals

Lattice-valued fuzzy measures are lattice-valued set functions which assign the bottom element of the lattice to the empty set, the top element of the lattice to the entire universe and satisfy the property of monotonicity. If the lattice is complete then a lattice-valued fuzzy integral of Sugeno type, with similar properties such as the Sugeno integral in its original form, can be introduced in a natural way. The main result of the paper is a componentwise decomposition theorem of an L-valued fuzzy integral to its L-valued fuzzy integrals components, where L is a complete lattice with negation and L={(@a,@b);@a,@[email protected]?L,@a=<@[email protected]?} is organized as a complete lattice too. This result is useful to obtain the properties of L-valued fuzzy integrals from the properties of L-valued fuzzy integrals and to calculate in a simple way the values of some integrals from the values of the components. The important case L=[0,1], when L becomes the lattice of the intuitionistic fuzzy values is distinctly discussed. An idea of application to synthetic evaluation of objects is also suggested.

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

[2]  Iuan-Yuan Lu,et al.  The evaluation of service quality using generalized Choquet integral , 2006, Inf. Sci..

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

[4]  Deli Zhang,et al.  On set-valued fuzzy measures , 2004, Inf. Sci..

[5]  Gwo-Hshiung Tzeng,et al.  Hierarchical MADM with fuzzy integral for evaluating enterprise intranet web sites , 2005, Inf. Sci..

[6]  Hong Yan,et al.  A Kriging Fuzzy Integral , 1997, Inf. Sci..

[7]  Krassimir T. Atanassov,et al.  Intuitionistic Fuzzy Sets - Theory and Applications , 1999, Studies in Fuzziness and Soft Computing.

[8]  Michel Grabisch,et al.  Classification by fuzzy integral: performance and tests , 1994, CVPR 1994.

[9]  Radko Mesiar,et al.  Pseudo-arithmetical operations as a basis for the general measure and integration theory , 2004, Inf. Sci..

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

[11]  Bobby Schmidt,et al.  Fuzzy math , 2001 .

[12]  Jong-Hwan Kim,et al.  Multicriteria Fuzzy Control Using Evolutionary Programming , 1997, Inf. Sci..

[13]  Do Sang Kim,et al.  Fuzzy integrals for set-valued mappings , 2001, Fuzzy Sets Syst..

[14]  Salvatore Sessa,et al.  On the fuzziness measure and negation in totally ordered lattices , 1986 .

[15]  Witold Pedrycz,et al.  Face recognition: A study in information fusion using fuzzy integral , 2005, Pattern Recognit. Lett..

[16]  Shiji Song,et al.  Generalized fuzzy integrals, part 3: convergent theorems , 1995 .

[17]  Haruki Imaoka,et al.  On a Subjective Evaluation Model by a Generalized Fuzzy Integral , 1997, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[18]  Michel Grabisch,et al.  Fuzzy integral for classification and feature extraction , 2000 .

[19]  Michio Sugeno,et al.  Fuzzy t -conorm integral with respect to fuzzy measures: generalization of Sugeno integral and choquet integral , 1991 .

[20]  Radko Mesiar.  Fuzzy measures and integrals , 2005, Fuzzy Sets Syst..

[21]  Didier Dubois,et al.  Qualitative Decision Theory with Sugeno Integrals , 1998, UAI.

[22]  Michel Grabisch,et al.  Interactive determination of a utility function represented as a fuzzy integral , 1993, Inf. Sci..

[23]  Michel Grabisch,et al.  The symmetric Sugeno integral , 2003, Fuzzy Sets Syst..

[24]  M. Grabisch.  Fuzzy integral in multicriteria decision making , 1995 .

[25]  J. Šipoš,et al.  Integral with respect to a pre-measure , 1979 .

[26]  Congxin Wu,et al.  Generalized fuzzy integrals. Part I: fundamental concepts , 1993 .

[27]  F. García,et al.  Two families of fuzzy integrals , 1986 .

[28]  S. Weber.  Two integrals and some modified versions-critical remarks , 1986 .

[29]  菅野 道夫,et al.  Theory of fuzzy integrals and its applications , 1975 .

[30]  Yian-Kui Liu,et al.  Fuzzy random programming with equilibrium chance constraints , 2005, Inf. Sci..

[31]  Guojun Wang,et al.  Endographic approach on supremum and infimum of fuzzy numbers , 2004, Inf. Sci..

[32]  A.I. Ban.  Intuitionistic fuzzy-valued fuzzy measures , 2004, 2004 2nd International IEEE Conference on 'Intelligent Systems'. Proceedings (IEEE Cat. No.04EX791).

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

[34]  Kwong-Sak Leung,et al.  Fuzzy numbers and fuzzification of the Choquet integral , 2005, Fuzzy Sets Syst..

[35]  M. Sugeno,et al.  Pseudo-additive measures and integrals , 1987 .