To appear in International Journal of Applied and Computational Mathematics (DOI: 10.1007/s40819-016-0286-0)
A Clustering Algorithm Based on Intuitionistic Fuzzy Relations for Tree Structure Evaluation ISMAT BEG Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore, Pakistan.
[email protected] and TABASAM RASHID School of Science, University of Management and Technology, Lahore-54770, Pakistan.
[email protected] Abstract: Intuitionistic fuzzy relations are used to construct hierarchical structures for the evaluation of vague complicated humanistic systems. A novel algorithm to develop partition trees at di¤erent levels according to di¤erent intuitionistic fuzzy triangular norm composition is presented. Examples are given to demonstrate the usefulness of the proposed algorithm. Keywords and phrases: Fuzzy cluster analysis; Intuitionistic fuzzy relation; Hierarchical evaluation structure.
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Introduction
Ordinary fuzzy set theory is used to show the vagueness in an information by Zadeh [36]. A fuzzy set A in the universe X is a mapping from X to [0; 1]: Bellman et al. [5] and Ruspini [20] started work in clustering with fuzzy sets. Nowa-days fuzzy clustering has been applied and studied in di¤erent areas (Bezdek [6], Dave [7], Trauwaert et al. [26]). Atanassov [1] gave the notion of intuitionistic fuzzy sets (IFS) which is an extension of Zadeh’s fuzzy set. An IFS A can be de…ned as a mapping from X to L = f(x1 ; x2 ) 2 [0; 1]2 jx1 + x2 1g ; where x1 and x2 are called membership and non-membership values for x 2 X; respectively. The class of all IFSs in X is denoted IF(X): IFS has proved to be a very suitable 1
tool to describe the uncertain or imprecise information. In [34], summation of any two numbers x = (x1 ; x2 ) and y = (y1 ; y2 ) from L is de…ned as: x + y = (x1 ; x2 ) + (y1 ; y2 ) = ((x1 + y1 )=2; (jx2 Here, we de…ne the ordering
L
y2 j)=2):
in L given as:
If x1 < y1 ; then (x1 ; x2 ) y2 then (x1 ; x2 )
