| Title:
|
$p$-symmetric bi-capacities (English) |
| Author:
|
Miranda, Pedro |
| Author:
|
Grabisch, Michel |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
40 |
| Issue:
|
4 |
| Year:
|
2004 |
| Pages:
|
[421]-440 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Bi-capacities have been recently introduced as a natural generalization of capacities (or fuzzy measures) when the underlying scale is bipolar. They allow to build more flexible models in decision making, although their complexity is of order $3^n$, instead of $2^n$ for fuzzy measures. In order to reduce the complexity, the paper proposes the notion of $p$-symmetric bi- capacities, in the same spirit as for $p$-symmetric fuzzy measures. The main idea is to partition the set of criteria (or states of nature, individuals,...) into subsets whose elements are all indifferent for the decision maker. (English) |
| Keyword:
|
bi-capacity |
| Keyword:
|
bipolar scales |
| Keyword:
|
$p$-symmetry |
| MSC:
|
03E72 |
| MSC:
|
03H05 |
| MSC:
|
28A12 |
| MSC:
|
28C05 |
| MSC:
|
28E05 |
| MSC:
|
28E10 |
| idZBL:
|
Zbl 1249.28021 |
| idMR:
|
MR2102362 |
| . |
| Date available:
|
2009-09-24T20:02:29Z |
| Last updated:
|
2015-03-23 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/135605 |
| . |
| Reference:
|
[1] Chateauneuf A., Jaffray J.-Y.: Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion.Mathematical Social Sciences 17(1989), 263–283 Zbl 0669.90003, MR 1006179, 10.1016/0165-4896(89)90056-5 |
| Reference:
|
[2] Choquet G.: Theory of capacities.Annales de l’Institut Fourier 5 (1953), 131–295 MR 0080760, 10.5802/aif.53 |
| Reference:
|
[3] Denneberg D.: Non-additive Measures and Integral.Kluwer, Dordrecht 1994 MR 1320048 |
| Reference:
|
[4] Dubois D., Prade H.: A class of fuzzy measures based on triangular norms.Internat. J. General Systems 8 (1982), 43–61 Zbl 0473.94023, MR 0671514, 10.1080/03081078208934833 |
| Reference:
|
[5] Grabisch M.: Pattern classification and feature extraction by fuzzy integral.In: 3d European Congress on Intelligent Techniques and Soft Computing (EUFIT, Aachen, Germany, August 1995), pp. 1465–1469 |
| Reference:
|
[6] Grabisch M.: Fuzzy measures and integrals: A survey of applications and recent issues.In: Fuzzy Sets Methods in Information Engineering: A Guide Tour of Applications (D. Dubois, H. Prade, and R. Yager, eds.), 1996 |
| Reference:
|
[8] Grabisch M.: $k$-order additive discrete fuzzy measures and their representation.Fuzzy Sets and Systems 92 (1997), 167–189 Zbl 0927.28014, MR 1486417, 10.1016/S0165-0114(97)00168-1 |
| Reference:
|
[9] Grabisch M., Labreuche C.: Bi-capacities.In: Proceedings of First int. Conf. on Soft Computing and Intelligent Systems (SCIC), Tsukuba (Japan), 2002 Zbl 1208.91029 |
| Reference:
|
[10] Grabisch M., Labreuche C.: Bi-capacities for decision making on bipolar scales.In Proceedings of the Seventh Meeting of the EURO Working Group on Fuzzy Sets (EUROFUSE) Varenna (Italy), September 2002, pp. 185–190 |
| Reference:
|
[11] Grabisch M., Labreuche, Ch.: Capacities on lattices and $k$-ary capacities.In: 3rd Int. Conf. of the European Soc. for Fuzzy Logic and Technology (EUSFLAT 2003), Zittau, Germany, September 2003, pp. 304–307 MR 1992695 |
| Reference:
|
[12] Hammer P. L., Holzman R.: On approximations of pseudo-boolean functions.Zeitschrift für Operations Research. Mathematical Methods of Operations Research 36 (1992), 3–21 MR 1141602, 10.1007/BF01541028 |
| Reference:
|
[13] Hungerford T. W.: Algebra.Springer-Verlag, Berlin 1980 Zbl 0442.00002, MR 0600654 |
| Reference:
|
[14] Mesiar R.: $k$-order additive measures.Internat. J. Uncertainty, Fuzziness and Knowledge-Based Systems 7 (1999), 423–428 10.1142/S0218488599000489 |
| Reference:
|
[15] Miranda P., Grabisch M.: $p$-symmetric fuzzy measures.In: Proceedings of Ninth International Conference of Information Processing and Management of Uncertainty in Knowledge-based Systems (IPMU), Annecy (France), July 2002, pp. 545–552 Zbl 1068.28013 |
| Reference:
|
[16] Miranda P., Grabisch, M., Gil P.: $p$-symmetric fuzzy measures.Internat. J. Uncertainty, Fuzziness and Knowledge-Based Systems 10 (2002), 105–123. Supplement Zbl 1068.28013, MR 1962672, 10.1142/S0218488502001867 |
| Reference:
|
[17] Rota G. C.: On the foundations of combinatorial theory I.Theory of Möbius functions. Z.r Wahrschein.und Verwandte Gebiete 2 (1964), 340–368 Zbl 0121.02406, MR 0174487, 10.1007/BF00531932 |
| Reference:
|
[18] Sugeno M.: Theory of Fuzzy Integrals and Iits Applications.PhD Thesis, Tokyo Institute of Technology, 1974 |
| Reference:
|
[19] Sugeno M., Fujimoto, K., Murofushi T.: A hierarchical decomposition of Choquet integral model.Internat. J. Uncertainty, Fuzziness and Knowledge-Based Systems 1 (1995), 1–15 Zbl 1232.93010, MR 1321933, 10.1142/S0218488595000025 |
| Reference:
|
[20] Sugeno M., Terano T.: A model of learning based on fuzzy information.Kybernetes 6 (1977), 157–166 10.1108/eb005448 |
| Reference:
|
[21] Tversky A., Kahneman D.: Advances in prospect theory: cumulative representation of uncertainty.J. Risk and Uncertainty 5 (1992), 297–323 Zbl 0775.90106, 10.1007/BF00122574 |
| Reference:
|
[22] Šipoš J.: Integral with respect to a pre-measure.Math. Slovaca 29 (1979), 141–155 MR 0578286 |
| Reference:
|
[23] Weber S.: $\perp $-decomposable measures and integrals for archimedean t-conorms $\perp $.J. Math. Anal. Appl. 101 (1984), 114–138 Zbl 0614.28019, MR 0746230, 10.1016/0022-247X(84)90061-1 |
| Reference:
|
[24] Yager R. R.: On ordered weighted averaging aggregation operators in multicriteria decision making.IEEE Trans. Systems, Man and Cybernetics 18 (1988), 183–190 MR 0931863, 10.1109/21.87068 |
| . |
