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MIN and MAX operators for trapezoidal fuzzy intervals

MIN and MAX operators for trapezoidal fuzzy intervals Purpose – The purpose of this paper is to determine an extension of the MIN and MAX general analytical expression for triangular fuzzy intervals to trapezoidal ones when Zadeh's extension principle is considered. Design/methodology/approach – In order to determine the MIN and MAX analytical expressions, the paper exhibits the conventional interval relations and their extension in fuzzy case where trapezoidal fuzzy intervals are assumed. The formalization and the justification of the so‐built analytical expressions are then detailed where mathematical mappings are proposed. The potential use of these operators in the framework of uncertain aggregation operators and ranking fuzzy intervals is shown with illustrative examples. Findings – It is discovered that the MIN and MAX operations for fuzzy intervals can be formulated by a general analytical form. Practical implications – The proposed methodology can be directly applied for ranking fuzzy intervals and implementing a large class of uncertain aggregation operators, especially for two‐additive Choquet integral. Originality/value – The originality of the proposed technique resides in exploiting the interval relations between supports and kernels to express a general and compact analytical MIN and MAX expressions for fuzzy intervals. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png International Journal of Intelligent Computing and Cybernetics Emerald Publishing

MIN and MAX operators for trapezoidal fuzzy intervals

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Publisher
Emerald Publishing
Copyright
Copyright © 2010 Emerald Group Publishing Limited. All rights reserved.
ISSN
1756-378X
DOI
10.1108/17563781011028541
Publisher site
See Article on Publisher Site

Abstract

Purpose – The purpose of this paper is to determine an extension of the MIN and MAX general analytical expression for triangular fuzzy intervals to trapezoidal ones when Zadeh's extension principle is considered. Design/methodology/approach – In order to determine the MIN and MAX analytical expressions, the paper exhibits the conventional interval relations and their extension in fuzzy case where trapezoidal fuzzy intervals are assumed. The formalization and the justification of the so‐built analytical expressions are then detailed where mathematical mappings are proposed. The potential use of these operators in the framework of uncertain aggregation operators and ranking fuzzy intervals is shown with illustrative examples. Findings – It is discovered that the MIN and MAX operations for fuzzy intervals can be formulated by a general analytical form. Practical implications – The proposed methodology can be directly applied for ranking fuzzy intervals and implementing a large class of uncertain aggregation operators, especially for two‐additive Choquet integral. Originality/value – The originality of the proposed technique resides in exploiting the interval relations between supports and kernels to express a general and compact analytical MIN and MAX expressions for fuzzy intervals.

Journal

International Journal of Intelligent Computing and CyberneticsEmerald Publishing

Published: Mar 30, 2010

Keywords: Trapezoidal fuzzy intervals; MIN and MAX operators; Extension principle; Aggregation operators; Choquet integral; Fuzzy control; Fuzzy logic; Systems and control theory

References