Complutense University Library

The Polytope of Fuzzy Measures and Its Adjacency Graph

Combarro, Elías F. and Miranda Menéndez, Pedro (2008) The Polytope of Fuzzy Measures and Its Adjacency Graph. In Modeling Decisions for Artificial Intelligence: 5th International Conference, MDAI 2008 Sabadell, Spain, October 30-31, 2008. Proceedings. Lecture Notes in Computer Science (5285). Springer, Berlin, pp. 74-85. ISBN 978-3-540-88268-8

[img] PDF
Restricted to Repository staff only until 2020.

507kB

Official URL: http://www.springerlink.com/content/ekv20717683wl3gv/fulltext.pdf

View download statistics for this eprint

==>>> Export to other formats

Abstract

In this paper we deal with the problem of studying the structure of the polytope of fuzzy measure for finite referential sets. We prove that the diameter of tire polytope of fuzzy measures is 3 for referentials of 3 elements or more. We also show that the polytope is combinatorial, whence we deduce that the adjacency graph of fuzzy measures is Hamilton connected if the cardinality of the referential set is not 2. We also give some results about the facets and edges of this polytope. Finally, we treat the corresponding results for the polytope given by the convex hull of monotone boolean functions.


Item Type:Book Section
Uncontrolled Keywords:Fuzzy measures, monotone boolean functions, diameter, combinatorial polytopes
Subjects:Sciences > Mathematics > Topology
ID Code:17036
References:

Aigner, M.: Combinatorial Theory. Springer, Heidelberg (1979)

Allais, M.: Le comportement de l’homme rationnel devant le risque: critique des postulats de l’´ecole am´ericaine. Econometrica (21), 503–546 (1953) (in French)

Anscombe, F.J., Aumann, R.J.: A definition of subjective probability. The Annals of Mathematical Statistics (34),199–205 (1963)

Chateauneuf, A.: Modelling attitudes towards uncertainty and risk through the use of Choquet integral. Annals of Operations Research (52), 3–20 (1994)

Choquet, G.: Theory of capacities. Annales de l’Institut Fourier (5), 131–295 (1953)

Combarro, E.F., Miranda, P.: Identification of fuzzy measures from sample data with genetic algorithms. Computers and Operations Research 33(10), 3046–3066 (2006)

Combarro, E.F., Miranda, P.: On the polytope of non-additve measures. Fuzzy Sets and Systems 159(16), 2145–2162 (2008)

Dedekind, R.: Über Zerlegungen von Zahlen durch ihre grössten gemeinsamen Teiler. Festschrift Hoch Braunschweig Ges. Werke II, 103–148 (1897) (in German)

Dempster, A.P.: Upper and lower probabilities induced by a multivalued mapping. The Annals of Mathematical Statististics (38), 325–339 (1967)

Denneberg, D.: Non-additive measures and integral. Kluwer Academic, Dordrecht (1994)

Ellsberg, D.: Risk, ambiguity, and the Savage axioms.Quart. J. Econom. (75), 643–669 (1961)

Goldberg, D.E.: Genetic algorithms in search, optimization and machine learning. Addison-Wesley, Reading (1989)

Grabisch, M.: Alternative representations of discrete fuzzy measures for decision making. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 5, 587–607 (1997)

Grabisch, M.: k-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems (92), 167–189 (1997)

Grabisch, M.: The Möbius function on symmetric ordered structures and its application to capacities on finite sets. Discrete Mathematics 287(1-3), 17–34 (2004)

Hammer, P.L., Holzman, R.: On approximations of pseudo-boolean functions. Zeitschrift f¨ur Operations Research. Mathematical Methods of Operations Research (36), 3–21 (1992)

Kisielewicz, A.: A solution od Dedekind’s problem on the number of isotone Boolean functions. J. reine angew. Math. (386), 139–144 (1988)

Miranda, P., Combarro, E.F.: On the structure of some families of fuzzy measures. IEEE Transactions on Fuzzy Systems 15(6), 1068–1081 (2007)

Naddef, D., Pulleyblank, W.R.: Hamiltonicity and Combinatorial Polyhedra. Journal of Combinatorial Theory Series B 31, 297–312 (1981)

Radojevic, D.: The logical representation of the discrete Choquet integral. Belgian Journal of Operations Research, Statistics and Computer Science 38(2–3), 67–89 (1998)

Rota, G.C.: On the foundations of combinatorial theory I. Theory of Möbius functions. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete (2), 340–368 (1964)

Schmeidler, D.: Integral representation without additivity. Proceedings of the American Mathematical Society 97(2), 255–261 (1986)

Shapley, L.S.: A value for n-person games. In: Kuhn, H.W., Tucker, A.W. (eds.) Contributions to the theory of Games. Annals of Mathematics Studies, vol. II, pp. 307–317. Princeton University Press, Princeton (1953)

Shmulevich, I., Selke, T.M., Coyle, E.J.: Stack Filters and Free Distributive Lattices. In: Proceeding of the 1995 IEEE Workshop on Nonlinear Signal Processing, Halkidiki, Greece, June 1995, pp. 927–930 (1995)

Skornjakov, L.A.: Elements of lattice theory. Adam Hilger Ltd. (1977)

Sugeno, M.: Theory of fuzzy integrals and its applications. Ph.D thesis, Tokyo Institute of Technology (1974)

Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behaviour. Princeton University Press, New Jersey (1944)

Wendt, P., Coyle, E., Gallagher, N.J.: Stack filters. IEEE Transactions on Acoustics, Speech and Signal Processing, 898–911 (1986)

Wiedemann, D.: A computation of the eighth Dedekind number. Order 8, 5–6 (1991)

Deposited On:06 Nov 2012 12:33
Last Modified:07 Feb 2014 09:40

Repository Staff Only: item control page