Publication: On the structure of the k-additive fuzzy measures
Loading...
Full text at PDC
Publication Date
2009
Advisors (or tutors)
Journal Title
Journal ISSN
Volume Title
Publisher
International Fuzzy Systems Association (IFSA); European Society for Fuzzy Logic and Technology (EUSFLAT
Citations
Exportar
Abstract
The family of k-additive measures has been introduced as a midterm between probabilities and general fuzzy measures and finds a wide number of applications in practice. However, its structure is different from other families of fuzzy measures and is certainly more complex (for instance, its vertices are not always {0, 1}-valued), so it has not been yet fully studied. In this paper we present some results concerning the extreme points of the k-additive fuzzy measures. We give a characterization of these vertices as well as an algorithm to compute them. We show some examples of the results of this algorithm and provide lower bounds on the number of vertices of the n - 1-additive measures, proving that it grows much faster than the number of vertices of the general fuzzy measures. This suggests that k-additive measures might not be a good choice in modeling certain decision problems when the value of k is high but not equal to n.
Description
Proceedings of the Joint 2009 International Fuzzy Systems Association World Congress and 2009 European Society of Fuzzy Logic and Technology Conference, Lisbon, Portugal, July 20-24, 2009
UCM subjects
Unesco subjects
Keywords
Citation
M. Grabisch. Alternative representations of discrete fuzzy measures for decision making. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 5:587–607, 1997.
M. Grabisch. k-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems, (92):167–189, 1997.
L. S. Shapley. A value for n-person games. In H. W. Kuhn and A. W. Tucker, editors, Contributions to the theory of Games, volume II of Annals of Mathematics Studies, pages 307–317. Princeton University Press, 1953.
M. Grabisch. k-order additive discrete fuzzy measures. In Proceedings of 6th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU), pages 1345–1350, Granada (Spain), 1996.
P. L. Hammer and R. Holzman. On approximations of pseudoboolean functions. Zeitschrift fÜr Operations Research. Mathematical Methods of Operations Research, (36):3–21, 1992.
G. C. Rota. On the foundations of combinatorial theory I. Theory of MÖbius functions. Zeitschrift fÜr Wahrscheinlichkeitstheorie und Verwandte Gebiete, (2):340–368, 1964.
M. Sugeno and T. Terano. A model of learning based on fuzzy information. Kybernetes, (6):157–166, 1977.
J.-L. Marichal. k-intolerant capacities and Choquet integrals. In Proceedings of Tenth International Conference of Information Processing and Management of Uncertainty in Knowledgebased Systems (IPMU), pages 601–608, Perugia (Italy), July 2004.
J.-L. Marichal. Tolerant or intolerant character of interacting criteria in aggregation by the Choquet integral. European Journal of Operational Research, 155(3):771–791, 2004.
P. Miranda, M. Grabisch, and P. Gil. p-symmetric fuzzy measures. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 10 (Suppl.):105–123, 2002.
D. Dubois and H. Prade. A class of fuzzy measures based on triangular norms. Int. J. General Systems, 8:43–61, 1982.
G. Choquet. Theory of capacities. Annales de l’Institut Fourier, (5):131–295, 1953.
P. Miranda, M. Grabisch, and P. Gil. Axiomatic structure of k-additive capacities. Mathematical Social Sciences, 49:153–178, 2005.
C. A. Bana e Costa and J.-C. Vansnick. MACBETH- An interactive path towards the construction of cardinal value function. International Transactions in Operational Research, 1(4):489–500, 1994.
J.-L. Marichal, P. Meyer, and M. Roubens. Sorting multiattribute alternatives : the TOMASO method. Computer and Operations Resarch, 32(4):861–877, 2005.
M. Grabisch and J.-M. Nicolas. Classification by fuzzy integral-performance and tests. Fuzzy Sets and Systems, Special Issue on Pattern Recognition, (65):255–271, 1994.
J. Wang Z. Wang, K. Xu and G. Klir. Using genetic algorithms to determine nonnegative monotone set functions for information fusion in environments with random perturbation. International Journal of Intelligent Systems, 14:949–962, 1999.
E. F. Combarro and P. Miranda. Identification of fuzzy measures from sample data with genetic algorithms. Computers and Operations Research, 33(10):3046–3066, 2006.
J. H. Holland. Adaptation in natural and artificial systems. Ann Arbor: The University of Michigan Press, 1975.
D. Denneberg. Non-additive measures and integral. Kluwer Academic, 1994.
M. Sugeno. Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Institute of Technology, 1974.
D. Radojevic. The logical representation of the discrete Choquet integral. Belgian Journal of Operations Research, Statistics and Computer Science, 38(2–3):67–89, 1998.
P. Miranda, E.F. Combarro, and P. Gil. Extreme points of some families of non-additive measures. European Journal of Operational Research, 33(10):3046–3066, 2006.
J. H. van Lint and R. M. Wilson. A course in Combinatorics. Cambridge University Press, 1998.
A. Chateauneuf and J.-Y. Jaffray. Some characterizations of lower probabilities and other monotone capacities through the use of M¨obius inversion. Mathematical Social Sciences,(17):263–283, 1989.
G. Shafer. A Mathematical Theory of Evidence. Princeton University Press, Princeton, New Jersey, (USA), 1976.
P. Miranda and E.F. Combarro. On the structure of some families of fuzzy measures. IEEE Transactions on Fuzzy Systems, 15(6):1068–1081, 2007.
E. F. Combarro and P. Miranda. Adjacency on the order polytope with applications to the theory of fuzzy measures. Submitted.
E. F. Combarro and P. Miranda. On the polytope of non-additve measures. Fuzzy Sets and Systems, 159(16):2145–2162, 2008.
E. F. Combarro and P. Miranda. Characterizing the isometries on the order polytope with an application to the theory of fuzzy measures. Submitted.
K. G. Murty. Adjacency on convex polyhedra. SIAM Review, 13(3):377–386, 1971.
A. D. Korshunov. Monotone boolean functions. Russian Mathematical Survey, 58(5):929–1001, 2003.