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

• Published in: IEEE transactions on fuzzy systems Vol. 28; no. 4; pp. 607 - 617
• Main Authors: Liu, Xianliang, Wan, Shuping
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.04.2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: 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
• DOI: 10.1109/TFUZZ.2019.2911918

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.