Negações e implicações fuzzy multidimensionais
Multidimensional fuzzy sets are a new extension of fuzzy sets on which the membership values of an element in the universe of discourse are vectors whose components are real numbers in the interval [0, 1], increasingly ordered. In the multidimensional fuzzy sets, distinct elements of the universe...
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| Formato: | doctoralThesis |
| Idioma: | pt_BR |
| Publicado em: |
Universidade Federal do Rio Grande do Norte
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| Endereço do item: | https://repositorio.ufrn.br/handle/123456789/52202 |
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| Resumo: | Multidimensional fuzzy sets are a new extension of fuzzy sets on which the membership values of an element in the universe of discourse are vectors whose components
are real numbers in the interval [0, 1], increasingly ordered. In the multidimensional fuzzy
sets, distinct elements of the universe of discourse may have membership values with
different numbers of components. The main application of this type of set are the multicriteria group decision making problems, in which, in the n-dimensional case, we have a
set of criteria, which are always evaluated by a fixed number n of experts and able to
identify the corresponding assignments for each of these specialists. The multidimensional case is used when some of these experts refrain to evaluate some of these situations
and, therefore, may be suitable for solving multi-criteria group decision making problems
with incomplete information(generated by omission or exclusion of opinion of some of the
experts). This thesis aims to investigate fuzzy negations and fuzzy implications on the set
of increasingly ordered vectors on [0, 1] with respect to some partial order, that is, in the
partially ordered set ⟨L∞([0, 1]), ≤⟩. In this thesis we study partial orders, giving special
attention to admissible orders on L∞([0, 1]). In addition, some properties and methods
to construct and generate such operators from fuzzy negations and fuzzy implications,
respectively, are provided. In particular, a notion of ordinal sums of n-dimensional fuzzy
negations and ordinal sums of multidimensional fuzzy negations will be proposed with
respect to specific partial orders, including an action of the group of automorphisms on
fuzzy implications on L∞([0, 1]) that preserves several original properties of the implication. Using a specific type of representable multidimensional fuzzy implication, we are able to generate a class of multidimensional fuzzy negations called natural m-negations. In the end, concepts of inclusion and similarity measure in multidimensional fuzzy sets will be proposed and an application in decision-making problems is presented. |
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