×
Grabisch, Michel

\(k\)-order additive discrete fuzzy measures and their representation. (English) Zbl 0927.28014

Fuzzy Sets Syst. 92, No. 2, 167-189 (1997).
Summary: In order to face with the complexity of discrete fuzzy measures, we propose the concept of \(k\)-order additive fuzzy measure, including usual additive measures and fuzzy measures. Every discrete fuzzy measure is a \(k\)-order additive fuzzy measure for a unique \(k\). A related topic of the paper is to introduce an alternative representation of fuzzy measures, called the interaction representation, which sets and extends in a common framework the Shaply value and the interaction index proposed by Murofushi and Soneda.

Cited in 1 Review
Cited in 204 Documents

MSC:

28E10 Fuzzy measure theory

Keywords:

pseudo-Boolean function; Möbius transform; discrete fuzzy measures; interaction representation; Shaply value; interaction index
Cite Review PDF
Full Text: DOI

References:

[1] Aumann, R. J.; Shapley, L. S., Values of Non-Atomic Games (1974), Princeton University Press: Princeton University Press Princeton · Zbl 0311.90084
[2] Chateauneuf, A.; Jaffray, J. Y., Some characterizations of lower probabilities and other monotone capacities through the use of Mobius inversion, Math. Social Sci., 17, 263-283 (1989) · Zbl 0669.90003
[3] Choquet, G., Theory of capacities, Ann. Institut Fourier, 5, 131-295 (1953) · Zbl 0064.35101
[4] Dempster, A. P., Upper and lower probabilities induced by a multivalued mapping, Ann. Math. Statist., 38, 325-339 (1967) · Zbl 0168.17501
[5] Denneberg, D., Non-Additive Measure and Integral (1994), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0826.28002
[6] Grabisch, M., The application of fuzzy integrals in multicriteria decision making, European J. Oper. Res., 89, 445-456 (1996) · Zbl 0916.90164
[7] Grabisch, M., Fuzzy measures and integrals: a survey of applications and recent issues, (Dubois, D.; Prade, H.; Yager, R., Fuzzy Sets Methods in Information Engineering: A Guided Tour of Applications (1996), Wiley: Wiley New York)
[8] Grabisch, M., \(k\)-order additive fuzzy measures, (Proc. 6th Intemat. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU). Proc. 6th Intemat. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU), Granada, Sapin (July 1996)), 1345-1350 · Zbl 1180.94055
[9] Grabisch, M.; Nguyen, H. T.; Walker, E. A., Fundamentals of Uncertainty Calculi, with Applications to Fuzzy Inference (1995), Kluwer Academic: Kluwer Academic Dordrecht
[10] Halmos, P. R., Measure Theory (1950), Springer: Springer Berlin · Zbl 0117.10502
[11] Hammer, P. L.; Holzman, R., On approximations of pseudo-Boolean functions, ZOR — Methods Models Oper. Res., 36, 3-21 (1992) · Zbl 0778.41009
[12] Murofushi, T.; Soneda, S., Techniques for reading fuzzy measures (III): interaction index, (Proc. 9th Fuzzy System Symposium. Proc. 9th Fuzzy System Symposium, Sapporo, Japan (May 1993)), 693-696, (in Japanese)
[13] Pap, E., Null-Additive Set Functions (1995), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0856.28001
[14] Rota, G. C., On the foundations of combinatorial theory I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie Verwandte Gebiete, 2, 340-368 (1964) · Zbl 0121.02406
[15] Schmeidler, D., Subjective probability and expected utility without additivity, Econometrica, 57, 3, 571-587 (1989) · Zbl 0672.90011
[16] Shafer, G., A. Mathematical Theory of Evidence (1976), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0359.62002
[17] Shapley, L. S., A value for n-person games, (Kuhn, H. W.; Tucker, A. W., Contributions to the Theory of Games. Contributions to the Theory of Games, Annals of Mathematics Studies, vol. II (1953), Princeton University Press: Princeton University Press Princeton), 307-317, vol. 28 · Zbl 0050.14404
[18] Shapley, L. S., Core of convex games, Int. J. Game Theory, 1, 11-26 (1971) · Zbl 0222.90054
[19] Smets, P., The combination of evidence in the transferable belief model, IEEE Trans. Pattern Analysis Machine Intelligence, 12, 5, 447-458 (1990)
[20] Sugeno, M., Theory of fuzzy integrals and its applications, (Ph.D. Thesis (1974), Tokyo Institute of Technology) · Zbl 0316.60005
[21] Zadeh, L. A., Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets Systems, 1, 3-28 (1978) · Zbl 0377.04002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.