Geodesic
A general approach to decomposable bi-capacities
Kybernetika, Tome 39 (2003) no. 5, pp. 631-642 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

We propose a concept of decomposable bi-capacities based on an analogous property of decomposable capacities, namely the valuation property. We will show that our approach extends the already existing concepts of decomposable bi-capacities. We briefly discuss additive and $k$-additive bi-capacities based on our definition of decomposability. Finally we provide examples of decomposable bi-capacities in our sense in order to show how they can be constructed.
We propose a concept of decomposable bi-capacities based on an analogous property of decomposable capacities, namely the valuation property. We will show that our approach extends the already existing concepts of decomposable bi-capacities. We briefly discuss additive and $k$-additive bi-capacities based on our definition of decomposability. Finally we provide examples of decomposable bi-capacities in our sense in order to show how they can be constructed.
Classification : 03E72, 03H05, 28C99, 28E05, 68T37
Keywords: bi-capacity; cumulative prospect theory; decomposable capacity; uninorm 
@article{KYB_2003_39_5_a9,
     author = {Saminger, Susanne and Mesiar, Radko},
     title = {A general approach to decomposable bi-capacities},
     journal = {Kybernetika},
     pages = {631--642},
     year = {2003},
     volume = {39},
     number = {5},
     mrnumber = {2042345},
     zbl = {1249.28022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2003_39_5_a9/}
}
TY  - JOUR
AU  - Saminger, Susanne
AU  - Mesiar, Radko
TI  - A general approach to decomposable bi-capacities
JO  - Kybernetika
PY  - 2003
SP  - 631
EP  - 642
VL  - 39
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/KYB_2003_39_5_a9/
LA  - en
ID  - KYB_2003_39_5_a9
ER  - 
%0 Journal Article
%A Saminger, Susanne
%A Mesiar, Radko
%T A general approach to decomposable bi-capacities
%J Kybernetika
%D 2003
%P 631-642
%V 39
%N 5
%U http://geodesic.mathdoc.fr/item/KYB_2003_39_5_a9/
%G en
%F KYB_2003_39_5_a9
Saminger, Susanne; Mesiar, Radko. A general approach to decomposable bi-capacities. Kybernetika, Tome 39 (2003) no. 5, pp. 631-642. http://geodesic.mathdoc.fr/item/KYB_2003_39_5_a9/

[1] Calvo T., Kolesárová A., Komorníková, M., Mesiar R.: Aggregation operators: Properties, classes and construction methods. In: Aggregation Operators (T. Calvo, G. Mayor, and R. Mesiar, eds.), Physica–Verlag, Heidelberg 2002, pp. 3–104 | MR | Zbl

[2] Choquet G.: Theory of capacities. Ann. Inst. Fourier (Grenoble) 5 (1953–1954), 131–292 | DOI | MR

[3] Fodor J. C., Yager R. R., Rybalov A.: Structure of uninorms. Internat. J. Uncertain. Fuzziness Knowledge-based Systems 5 (1997), 411–427 | DOI | MR | Zbl

[4] Grabisch M.: $k$-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems 92 (1997), 167–189 | DOI | MR | Zbl

[5] Grabisch M., Baets, B. De, Fodor J.: On symmetric pseudo-additions and pseudo-multiplications: Is it possible to build a ring on [-1,1]? In: Proc. 9th Internat. Conference on Information Processing and Management of Uncertainty in Knowledge-based Systems (IPMU 2002), Volume III, Annecy (France), July 2002, pp. 1349–1363

[6] Grabisch M., Labreuche C.: Bi-capacities. In: Proc. First Internat. Conference on Soft Computing and Intelligent Systems (SCIC), Tsukuba (Japan), 2002 | Zbl

[7] Grabisch M., Labreuche C.: Bi-capacities for decision making on bipolar scales. In: Proc. Seventh Meeting of the EURO Working Group on Fuzzy Sets (EUROFUSE), Varenna (Italy), 2002, pp. 185–190

[8] Grabisch M., Labreuche C.: Capacities on lattices and $k$-ary capacities. In: Proc. 3rd Internat. Conference in Fuzzy Logic and Technology (EUSFLAT 2003), Zittau (Germany), 2003, pp. 304–307

[9] Greco S., Matarazzo, B., Slowinski R.: Bipolar Sugeno and Choquet integrals. In: Proc. Seventh Meeting of the EURO Working Group on Fuzzy Sets (EUROFUSE), Varenna (Italy), 2002, pp. 191–196

[10] Greco S., Matarazzo, B., Slowinski R.: The axiomatic basis of multicriteria noncompensatory preferences. In: Proc. Fourth International Workshop on Preferences and Decisions, Trento (Italy), 2003, pp. 81–86

[11] Klement E. P., Mesiar, R., Pap E.: Integration with respect to decomposable measures, based on a conditionally distributive semiring on the unit interval. Internat. J. Uncertain. Fuzziness Knowledge-based Systems 8 (2000), 701–717 | DOI | MR | Zbl

[12] Klement E. P., Mesiar, R., Pap E.: Triangular Norms. Kluwer, Dordrecht 2000 | MR | Zbl

[13] Mesiar R.: Generalization of $k$-order additive discrete fuzzy measures. Fuzzy Sets and Systems 102 (1999), 423–428 | DOI | MR

[14] Mesiar R.: $k$-order additive measures: Internat. J. Uncertain. Fuzziness Knowledge-based Systems 7 (1999), 6, 561–568 | DOI | MR

[15] Mesiar R.: $k$-order additivity and maxitivity. Atti Sem. Mat. Fis. Univ. Modena LI (2003), 179–189 | MR | Zbl

[16] Mesiar R., Saminger S.: Decomposable bi-capacities: In: Proc. Summer School on Aggregation Operators 2003 (AGOP 2003), University Alcala de Henares (Spain), 2003, pp. 155–158

[17] Miranda P., Grabisch, M., Gil P.: $p$-symmetric fuzzy measures. Internat. J. Uncertain. Fuzziness Knowledge-based Systems 10 (2002), 105–123 | DOI | MR | Zbl

[18] Pap E.: Null-Additive Set Functions. Kluwer, Dordrecht 1995 | MR | Zbl

[19] Sander W.: Associative aggregation operators. In: Aggregation Operators (T. Calvo, G. Mayor, and R. Mesiar, eds.), Physica–Verlag, Heidelberg, 2002, pp. 124–158 | MR | Zbl

[20] Wang Z., Klir G. J.: Fuzzy Measure Theory. Plenum Press, New York 1992 | MR | Zbl

[21] Weber S.: $\perp $-decomposable measures and integrals for Archimedean t-conorms $\perp $. J. Math. Anal. Appl. 101 (1984), 114–138 | DOI | MR | Zbl

[22] Yager R. R., Rybalov A.: Uninorm aggregation operators. Fuzzy Sets and Systems 80 (1996), 111–120 | DOI | MR | Zbl