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By Bernd Reusch

This publication constitutes the refereed lawsuits of the ninth Dortmund Fuzzy Days, held in Dortmund, Germany, 2006. the bushy Days convention has validated itself as a global discussion board for the dialogue of recent leads to the sector of Computational Intelligence. all of the papers needed to endure a radical evaluation making certain a high-quality caliber of the programme. The papers are dedicated to foundational and useful concerns in fuzzy structures, neural networks, evolutionary algorithms, and computing device studying and hence conceal the complete diversity of computational intelligence.

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Extra info for Computational Intelligence, Theory and Applications: International Conference 9th Fuzzy Days in Dortmund, Germany, Sept. 18-20, 2006 Proceedings

Example text

N}. A Choquet measure [1] on the set N is a set function µ : P(N ) −→ [0, 1] satisfying (1) µ(∅) = 0, µ(N ) = 1, (2) S ⊆ T ⊆ N ⇒ µ(S) ≤ µ(T ). A. Marques Pereira et al. Given a Choquet measure µ we can define the Choquet integral [1–3] of a vector x = (x1 , . . , xn ) ∈ [0, 1]n with respect to µ as n Cµ (x) = [µ(A(i) ) − µ(A(i+1) )] x(i) , (2) i=1 where (·) indicates a permutation on N such that x(1) ≤ x(2) ≤ · · · ≤ x(n) . Also A(i) = {(i), . . , (n)} and A(n+1) = ∅. Notice that the Choquet integral with respect to an additive measure µ reduces to a weighted averaging operator, whose weights wi are given by the µ(i) values, µ(A(i) ) = µ((i)) + µ((i + 1)) + · · · + µ((n)), n n Cµ (x) = [µ(A(i) ) − µ(A(i+1) )] x(i) = i=1 n µ((i))x(i) = i=1 wi xi .

May be useful. Linguistic Summarization 23 A linguistic summary is meant as a natural language like sentence that subsumes the very essence (from a certain point of view) of a set of data. This set is assumed to be numeric and is usually large, not comprehensible in its original form by the human being. The original Yager’s approach to the linguistic summaries (cf. Yager [33], Kacprzyk and Yager [20], Kacprzyk et al. [21], and Kacprzyk and Zadro˙zny [28]) may be expressed as follows: – Y = {y1 , .

J. , Wiley, New York, 2004 Linguistic Summarization of Some Static and Dynamic Features of Consensus Reaching Janusz Kacprzyk, Slawomir Zadro˙zny, and Anna Wilbik Summary. Consensus reaching has been widely recognized as an important component of the decision-making process. In previous works Fedrizzi and Kacprzyk introduced a new concept of consensus referring to the idea of fuzzy majority and based on the Zadeh’s calculus of linguistically quantified propositions. , relevant) options. The approach was extended in further works by Fedrizzi, Kacprzyk, Nurmi, and Zadro˙zny.

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