Conjuntos fuzzy multidimensionais

Abstract

Since the emergence of fuzzy set theory, many extensions proposals have been given. One of them is the n-dimensional fuzzy set in which its elements are tuples of size n whose components are values in [0, 1], ordered in increasing form, called n-dimensional intervals. Generally, these sets are used to develop tools that aid in modeling situations involving decision-making where given a problem and an alternative, each n-dimensional interval represents the opinion of n specialists on the degree to which an alternative meets a given criterion or attribute for this problem. However, this approach is not able to deal with situations in which a particular expert can, for example, refrain from any decision-making criteria, and therefore, we would have in the same problem coexisting n-dimensional intervals with different values of n or where the set of specialists changes for each pair alternative/attribute. Thus, we need a new fuzzy set extension in which its elements (intervals) can have any dimensions. In this work, we present the concept of multidimensional fuzzy sets as a generalization of the ndimensional fuzzy sets in which the elements can have different dimensions. We also present a way to generate comparisons (ordering) of these elements of different dimensions, discuss conditions under which these sets have lattice structure and introduce the concepts of admissible orders, multidimensional aggregation functions and fuzzy negations on multidimensional fuzzy sets. In addition, we deepen studies on ordinal sums of fuzzy negations.