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On the structure of the k-additive fuzzy measures

Combarro, Elías F. and Miranda Menéndez, Pedro (2009) On the structure of the k-additive fuzzy measures. In IFSA-EUSFLAT 2009 Proceedings. International Fuzzy Systems Association (IFSA); European Society for Fuzzy Logic and Technology (EUSFLAT, pp. 64-69. ISBN 978-989-95079-6-8.

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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.

Item Type:Book Section
Additional Information: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
Uncontrolled Keywords:Fuzzy measures, k-additive measures, polytope vertices
Subjects:Sciences > Mathematics > Operations research
ID Code:17277
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