Combinatorial Iterative Algorithms for Computing the Centroid of an Interval Type-2 Fuzzy Set

Computing the centroid of an interval type-2 fuzzy set (IT2 FS) is an important type-reduction method. The aim of this paper is to develop a new method to calculate the centroid of an IT2 FS when the problems of centroid computation of an IT2 FS are continuous. For the continuous centroid computatio...

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Veröffentlicht in:IEEE transactions on fuzzy systems Jg. 28; H. 4; S. 607 - 617
Hauptverfasser: Liu, Xianliang, Wan, Shuping
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.04.2020
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Algorithms
Centroid computation
Centroids
Combinatorial analysis
Computation
Computational complexity
Convergence
decision analysis
Frequency selective surfaces
Fuzzy sets
interval type-2 fuzzy sets (IT2 FSs)
Iterative algorithms
Iterative methods
Karnik–Mendel algorithms
Optimization
Uncertainty
ISSN:1063-6706, 1941-0034
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Zusammenfassung:Computing the centroid of an interval type-2 fuzzy set (IT2 FS) is an important type-reduction method. The aim of this paper is to develop a new method to calculate the centroid of an IT2 FS when the problems of centroid computation of an IT2 FS are continuous. For the continuous centroid computation problems, the structures of optimal solutions are strictly proven from mathematics for the first time in this paper. Furthermore, we also prove that the structures of the optimal solutions are unique in the sense of almost everywhere equal, i.e., if there are two optimal solutions f 1 (x) and f 2 (x), the Lebesgue measure of {x|f 1 (x) ≠ f 2 (x)} is equal to 0. Subsequently, a combinatorial iterative (CI) method is proposed to solve the roots of the sufficiently differentiable objective functions. It is proven that the convergence of the proposed iterative method is at least sixth order. Based on the proposed iterative method, two algorithms, called CI algorithms, are devised to compute the centroid of anIT2 FS. The efficiencies of CI algorithms are demonstrated by comparing the continuous Karnik-Mendel algorithms and the Hallye's methods with the CI algorithms through three numerical examples.
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ISSN:1063-6706
1941-0034
DOI:10.1109/TFUZZ.2019.2911918