Aldo G. S. Ventre

On some chains of fuzzy sets

Abstract A partial order relation σ is defined in the set F ( X ) of the fuzzy sets in X . If this ordering is induced in the subset F ( X ) of the measurable fuzzy sets in the set X with totally finite positive measure, then fσg implies that the entropy of the fuzyy set f is not less than the entropyof g. By means of this ordering a lattice L on F ( X ) is defined and a lattice structure is induced in the set of infinite chains in L . Furthermore the set F ′( X ) of the fuzzy sets of F ( X ) which assume value in a finite subset of the real interval [0,1] is considered and the following properties are stated: any chain of elements of F ′( X ) is an infinite sequence of functions convergent in the mean to an integrable function, and the entropy is a valuation of bounded variation on the sublattice of L whose elements are in F ′( X ). The chains on L can offer a model of a cognitive process in a fuzzy environment when their elements are determined by a sequence of decisions. The limit property traduces the determinism of a such procedure.

AGGREGATION AND CONSENSUS IN MULTIOBJECTIVE AND MULTIPERSON DECISION MAKING

The subject of the paper is related with the decision making problems with several objectives and a committee of experts. Models for aggregating opinions of experts and consensus reaching on the alternatives with respect to different objectives are presented. A modeling in a metric space is studied and a game theoretic interpretation is introduced. In this framework a procedure for monitoring and enhancing consensus is considered.

A MODEL FOR C-CALCULUS

Abstract C-calculus was introduced by the first author as a technique for the analysis of complex hierarchical systems.2 It has been further applied as a versatile tool in pattern recognition.1,3 Our present aim is to exhibit a model for C-calculus, and deal with its convergence and filtering properties.

On Fuzzy Integral inequalities and Fuzzy Expectation

and (s) I, Ydl.l>B does not imply Indeed let us consider the following example. Let X and Y be inexact variables satisfying the conditions: Q = [0, 11, 0 X(w)= o7 1. if oGO.3 if oBO.3 if w d 0.7 if w > 0.7. It is easy to check that if ~1 is Lebesgue measure then

Multicriteria and Multiagent Decision Making with Applications to Economics and Social Sciences

The book provides a comprehensive and timely report on the topic of decision making and decision analysis in economics and the social sciences. The various contributions included in the book, selected using a peer review process, present important studies and research conducted in various countries around the globe. The majority of these studies are concerned with the analysis, modeling and formalization of the behavior of groups or committees that are in charge of making decisions of social and economic importance. Decisions in these contexts have to meet precise coherence standards and achieve a significant degree of sharing, consensus and acceptance, even in uncertain and fuzzy environments. This necessitates the confluence of several research fields, such as foundations of social choice and decision making, mathematics, complexity, psychology, sociology and economics. A large spectrum of problems that may be encountered during decision making and decision analysis in the areas of economics and the social sciences, together with a broad range of tools and techniques that may be used to solve those problems, are presented in detail in this book, making it an ideal reference work for all those interested in analyzing and implementing mathematical tools for application to relevant issues involving the economy and society.

Consistency for Nonadditive Measures: Analytical and Algebraic Methods

We consider the problem of choosing an alternative in a set A = {A1, A2, ..., Am} of alternatives, given a set D = {d1, d2, ..., dh} of decision makers and a set Ω = {O1, O2, ..., On} of objectives. We assume that any decision maker dk assigns to any pair (alternative Ai, objective Oj) a number aijk that measures to what extent Ai satisfies Oj . We assume that Ω is a subset of a universal set U and, for every alternative Ai and decision maker dk, the function mik that associates aijk to Oj is a fuzzy measure. We propose to aggregate the scores aijk by means of a t-conorm⊕λ of a family Φλ of t-conorms such that every mik is a ⊕λ-decomposable measure. We consider also some algebraic and geometric representations of the Archimedean fuzzy unions and their additive generators in terms of the theory of hypergroups. By considering the Oj as events, we propose also to assign the scores aijk in such a way that for some λ the assessment is consistent and to aggregate such evaluations with the correspondent t-conorm ⊕λ. Finally we generalize the previous procedure by considering fuzzy measures of type 2, having as a range a set of fuzzy numbers with the interval [0, 1] as support. 2000 MSC: 03E72, 08A72, 20N20, 91B06, 91B14.

Coherence for Fuzzy Measures and Applications to Decision Making

Coherence is a central issue in probability (de Finetti, 1970). The studies on non-additive models in decision making, e. g., non-expected utility models (Fishburn, 1988), lead to an extension of the coherence principle in nonadditive settings, such as fuzzy or ambiguous contexts. We consider coherence in a class of measures that are decomposable with respect to Archimedean t-conorms (Weber, 1984), in order to interpret the lack of coherence in probability. Coherent fuzzy measures are utilized for the aggregations of scores in multiperson and multiobjective decision making. Furthermore, a geometrical representation of fuzzy and probabilistic uncertainty is considered here in the framework of join spaces (Prenowitz and Jantosciak, 1979) and, more generally, algebraic hyperstructures (Corsini and Leoreanu, 2003); indeed coherent probability assessments and fuzzy sets are join spaces (Corsini and Leoreanu, 2003; Maturo et al., 2006a, 2006b).

On fuzzy implication in De Morgan algebras

The behaviour of some fuzzy implications with respect to measure of fuzziness and their deductive power is studied in a lattice framework. The point of view of Bandler and Kohout [1] and Willmott [7] is followed.

Representations of the fuzzy integral

Abstract Several representation formulas of the fuzzy integral have been found, both for general integrand [2,3] and for integrand having suitable continuity properties [2]. In this paper some representations of the fuzzy integral are considered. The case of a topological structure added is treated. Furthermore an expression of the fuzzy integral of a fuzzy variable in terms of a decreasing rearrangement of f is given.

Fuzzy measures and convergence

Abstract Several theorems have been proved, e.g. by Ralescu, Ralescu and Adams, Puri and Ralescu, and Klement, in order to unify the approach to uncertainty of both statistical and fuzzy origin. The present paper is motivated by the same instances and deals with some extensions of the measure-theoretical bases of fuzzy measures and convergence properties, e.g. a B. Levi-like theorem.