Aggregation operators: Properties, classes and construction methods.

*(English)*Zbl 1039.03015
Calvo, Tomasa (ed.) et al., Aggregation operators. New trends and applications. Heidelberg: Physica-Verlag (ISBN 3-7908-1468-7/hbk). Stud. Fuzziness Soft Comput. 97, 3-104 (2002).

One of the big issues in fuzzy logic concerns the extension of the classical logical operators such as disjunction and conjunction. More generally, the aggregation of a finite number of input values into a single output is a major problem in several sciences.

This long paper gives an extensive overview of \(n\)-ary aggregation operators on the unit interval, i.e. on \([0,1]^n-[0,1]\) mappings with \(n\) a natural number, and it may be seen as an excellent state-of-the-art report on aggregation operators.

In the second section, a list of potential properties of \(n\)-ary aggregation operators is given: boundary conditions, monotonicity, idempotency, (lower semi, upper semi) continuity, symmetry, associativity, bisymmetry, neutral element, annihilator, temporary breakdown property, self-identity property, strong self-identity property, shift-invariance, homogeneity, linearity, comonotone additivity and decomposability.

Section 3 treats the construction of aggregation operators satisfying some given properties.

In Section 4 some important aggregation operations based on the arithmetic mean are studied, among them: weighted arithmetic means, ordered weighted average operators introduced by Yager, quasi-arithmetic means, and ordered weighted quasi-arithmetic means.

Section 5 gives an overview of the aggregation operators based on integrals such as: Lebesgue integral-based aggregation, Choquet integral-based aggregation and Sugeno integral-based aggregation.

In Section 6, the authors discuss triangular norms and conorms – mostly used for the extension of binary conjunction and disjunction respectively – as well as their generalizations such as: uninorms, nullnorms and copulas.

The paper ends with aggregation operators based on the addition in the real line, i.e. the so-called generated aggregation operators.

This paper is warmly recommended to those readers looking for a comprehensive overview of recent developments on aggregation operators.

For the entire collection see [Zbl 0983.00020].

This long paper gives an extensive overview of \(n\)-ary aggregation operators on the unit interval, i.e. on \([0,1]^n-[0,1]\) mappings with \(n\) a natural number, and it may be seen as an excellent state-of-the-art report on aggregation operators.

In the second section, a list of potential properties of \(n\)-ary aggregation operators is given: boundary conditions, monotonicity, idempotency, (lower semi, upper semi) continuity, symmetry, associativity, bisymmetry, neutral element, annihilator, temporary breakdown property, self-identity property, strong self-identity property, shift-invariance, homogeneity, linearity, comonotone additivity and decomposability.

Section 3 treats the construction of aggregation operators satisfying some given properties.

In Section 4 some important aggregation operations based on the arithmetic mean are studied, among them: weighted arithmetic means, ordered weighted average operators introduced by Yager, quasi-arithmetic means, and ordered weighted quasi-arithmetic means.

Section 5 gives an overview of the aggregation operators based on integrals such as: Lebesgue integral-based aggregation, Choquet integral-based aggregation and Sugeno integral-based aggregation.

In Section 6, the authors discuss triangular norms and conorms – mostly used for the extension of binary conjunction and disjunction respectively – as well as their generalizations such as: uninorms, nullnorms and copulas.

The paper ends with aggregation operators based on the addition in the real line, i.e. the so-called generated aggregation operators.

This paper is warmly recommended to those readers looking for a comprehensive overview of recent developments on aggregation operators.

For the entire collection see [Zbl 0983.00020].

Reviewer: E. Kerre (Gent)