Approximations of fuzzy numbers by trapezoidal fuzzy numbers preserving the ambiguity and value

Computers & Mathematics with Applications (Impact Factor: 1.7). 03/2011; 61(5):1379-1401. DOI: 10.1016/j.camwa.2011.01.005
Source: DBLP


Value and ambiguity are two parameters which were introduced to represent fuzzy numbers. In this paper, we find the nearest trapezoidal approximation and the nearest symmetric trapezoidal approximation to a given fuzzy number, with respect to the average Euclidean distance, preserving the value and ambiguity. To avoid the laborious calculus associated with the Karush–Kuhn–Tucker theorem, the working tool in some recent papers, a less sophisticated method is proposed. Algorithms for computing the approximations, many examples, proofs of continuity and two applications to ranking of fuzzy numbers and estimations of the defect of additivity for approximations are given.

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Available from: Octavia Bolojan, Oct 28, 2015
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    • "where a ≤ c ≤ d ≤ b ∈ R, u L : [a, c] −→ [0] [1] is a nondecreasing upper semicontinuous function, u L (a) = 0, u L (c) = 1, called the left side of the fuzzy number and u R : [d, b] −→ [0] [1] is a nonincreasing upper semicontinuous function, u R (d) = 1, u R (b) = 0, called the right side of the fuzzy number. "
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    • "Woxman [41] extends this canonical representation to discrete fuzzy numbers. A more recent paper that discusses such problems is given in [3]. They obtain the nearest trapezoidal approximation and nearest symmetric trapezoidal approximation to a given fuzzy number using the average Euclidean distance, preserving the value and ambiguity. "
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    • "In [12] some connections between trapezoidal approximation and aggregation were discussed. Other aspects of the trapezoidal approximation and its generalizations could be found in [1] [2] [6] [7] [10] [11] [26] [27]. "
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    ABSTRACT: The problem of the nearest approximation of fuzzy numbers by piecewise linear 1-knot fuzzy numbers is discussed. By using 1-knot fuzzy numbers one may obtain approximations which are simple enough and flexible to reconstruct the input fuzzy concepts under study. They might be also perceived as a generalization of the trapezoidal approximations. Moreover, these approximations possess some desirable properties. Apart from theoretical considerations approximation algorithms that can be applied in practice are also given.
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