%0 Journal Article
%T On Intuitionistic Fuzzy Entropy of Order-¦Á
%A Rajkumar Verma
%A BhuDev Sharma
%J Advances in Fuzzy Systems
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/789890
%X Using the idea of R¨¨nyi¡¯s entropy, intuitionistic fuzzy entropy of order-¦Á is proposed in the setting of intuitionistic fuzzy sets theory. This measure is a generalized version of fuzzy entropy of order-¦Á proposed by Bhandari and Pal and intuitionistic fuzzy entropy defined by Vlachos and Sergiadis. Our study of the four essential and some other properties of the proposed measure clearly establishes the validity of the measure as intuitionistic fuzzy entropy. Finally, a numerical example is given to show that the proposed entropy measure for intuitionistic fuzzy set is reasonable by comparing it with other existing entropies. 1. Introduction In 1965, Zadeh [1] proposed the notion of fuzzy set (FS) to model nonstatistical imprecise or vague phenomena. Since then, the theory of fuzzy set has become a vigorous area of research in different disciplines such as engineering, artificial intelligence, medical science, signal processing, and expert systems. Fuzziness, a feature of uncertainty, results from the lack of sharp distinction of being or not being a member of a set; that is, the boundaries of the set under consideration are not sharply defined. A measure of fuzziness used and cited in the literature is fuzzy entropy, also first mentioned in 1968 by Zadeh [2]. In 1972, De Luca and Termini [3] first provided axiomatic structure for the entropy of fuzzy sets and defined an entropy measure of a fuzzy set based on Shannon¡¯s entropy function [4]. Kaufmann [5] introduced a fuzzy entropy measure by a metric distance between the fuzzy set and that of its nearest crisp set. In addition, Yager [6] defined an entropy measure of a fuzzy set in terms of a lack of distinction between fuzzy set and its complement. In 1989, N. R. Pal and S. K. Pal [7] proposed fuzzy entropy based on exponential function to measure the fuzziness called exponential fuzzy entropy. Further, Bhandari and Pal [8] proposed fuzzy entropy of order- corresponding to R¨¨nyi entropy [9]. Recently, Verma and Sharma [10] have introduced a parametric generalized entropy measure for fuzzy sets called ¡°exponential fuzzy entropy of order- .¡± Atanassov [11, 12] introduced the concept of intuitionistic fuzzy set (IFS), which is a generalization of the notion of fuzzy set. The distinguishing characteristic of intuitionistic fuzzy set is that it assigns to each element a membership degree, a nonmembership degree, and the hesitation degree. Firstly, Burillo and Bustince [13] defined the entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets. Szmidt and Kacprzyk [14] used a different approach
%U http://www.hindawi.com/journals/afs/2014/789890/