By Sergei Ovchinnikov (auth.), Dr. Bernadette Bouchon-Meunier (eds.)

This ebook provides the most instruments for aggregation of knowledge given via a number of contributors of a gaggle or expressed in a number of standards, and for fusion of knowledge supplied via a number of assets. It specializes in the case the place the supply wisdom is imperfect, because of this uncertainty and/or imprecision needs to be taken under consideration. The publication includes either theoretical and utilized experiences of aggregation and fusion equipment in general frameworks: chance conception, facts concept, fuzzy set and probability conception. The latter is extra built since it permits to regulate either obscure and unsure wisdom. functions to decision-making, photograph processing, regulate and class are defined. The reader can discover a cutting-edge of the most methodologies, in addition to extra complicated effects and outlines of utilizations.

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**Example text**

We introduce now the concept of discrete fuzzy integrals, viewed as aggregation operators. For this reason, we will adopt a connective-like notation instead of the usual integral form, and the integrand will be a set of n values a,, ... , an in [0, 1]. Definition 2 Let p, be a fuzzy measure on X. The discrete Sugenointegral ofal, ... , an with respect to p, is defined by : n Sl'(a,, ... ,an) := V(a(i) 1\p,(A(i))), i=l where a(i)• i = 1, ... , n indicates a permutation of ai, i = 1, ... , n such that a(l) a(2) ~ ...

Fuzzy Sets and Systems 51 (1992), 295- 307. 35 [17] Yager, R. : Aggregation operators and fuzzy systems modeling. Fuzzy Sets and Systems 67 (1994),129 -146. [18] Yager, R. : Aggregating operators and fuzzy systems modelling. Techn. Report #MII-1401, Machine Intelligence Institute, lona College, New Rochelle, NY,1994. [19] Yager, R. R. : Uniform aggregation operators. Technical Report #MII-1501, Machine Intelligence Institute, lona College, New Rochelle, NY,1994. [20] Yager, R. R. : Full reinforcement operators in aggregation techniques.

That means that F is a min-max aggregation function. To prove the converse, suppose F is an associative T-S aggregation function for some t-norm T and t-conorm S. If k = 0 then F is at-norm, while k = 1 implies that F is a t-conorm. Suppose 0 < k < 1. First we prove that F(x, 0) = min(k, x). Indeed, F is associative, so we have for all x, y, z E [0, 1J that F(x, F(y, z)) = F(F(x, y), z). in (6) (6). Then, by definition of a T-S aggregation function, we get T(k, T(k, z)) = T(k, z) for all z E [0, 1].