Using the idea of Rènyi’s entropy, intuitionistic fuzzy entropy of order-α is proposed in the setting of intuitionistic fuzzy sets theory. This measure is a generalized version of fuzzy entropy of order-α 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 Rènyi 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
References
[1]
L. A. Zadeh, “Fuzzy sets,” Information and Computation, vol. 8, pp. 338–353, 1965.
[2]
L. A. Zadeh, “Probability measures of fuzzy events,” Journal of Mathematical Analysis and Applications, vol. 23, no. 2, pp. 421–427, 1968.
[3]
A. De Luca and S. Termini, “A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory,” Information and Computation, vol. 20, no. 4, pp. 301–312, 1972.
[4]
C. E. Shannon, “A mathematical theory of communication,” The Bell System Technical Journal, vol. 27, pp. 379–656, 1948.
[5]
A. Kaufmann, Introduction to the Theory of Fuzzy Subsets, Academic Press, New York, NY, USA, 1975.
[6]
R. R. Yager, “On the measure of fuzziness and negation part I: membership in the unit interval,” International Journal of General Systems, vol. 5, no. 4, pp. 221–229, 1979.
[7]
N. R. Pal and S. K. Pal, “Object-background segmentation using new definitions of entropy,” IEE Proceedings E: Computers and Digital Techniques, vol. 136, no. 4, pp. 284–295, 1989.
[8]
D. Bhandari and N. R. Pal, “Some new information measures for fuzzy sets,” Information Sciences, vol. 67, no. 3, pp. 209–228, 1993.
[9]
A. Rényi, “On measures of entropy and information,” in Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, pp. 547–561, University of California Press, Berkeley, Calif, USA, 1961.
[10]
R. Verma and B. D. Sharma, “On generalized exponential fuzzy entropy,” Engineering and Technology, vol. 5, pp. 956–959, 2011.
[11]
K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 20, no. 1, pp. 87–96, 1986.
[12]
K. T. Atanassov, Intuitionistic Fuzzy Sets, Physica, Heidelberg, Germany, 1999.
[13]
P. Burillo and H. Bustince, “Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets,” Fuzzy Sets and Systems, vol. 78, no. 3, pp. 305–316, 1996.
[14]
E. Szmidt and J. Kacprzyk, “Entropy for intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 118, no. 3, pp. 467–477, 2001.
[15]
W. Zeng and H. Li, “Relationship between similarity measure and entropy of interval valued fuzzy sets,” Fuzzy Sets and Systems, vol. 157, no. 11, pp. 1477–1484, 2006.
[16]
W. L. Hung and M. S. Yang, “Fuzzy entropy on intuitionistic fuzzy sets,” International Journal of Intelligent Systems, vol. 21, no. 4, pp. 443–451, 2006.
[17]
I. K. Vlachos and G. D. Sergiadis, “Intuitionistic fuzzy information—applications to pattern recognition,” Pattern Recognition Letters, vol. 28, no. 2, pp. 197–206, 2007.
[18]
Q. Zhang and S. Jiang, “A note on information entropy measures for vague sets and its applications,” Information Sciences, vol. 178, no. 21, pp. 4184–4191, 2008.
[19]
S. K. De, R. Biswas, and A. R. Roy, “Some operations on intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 114, no. 3, pp. 477–484, 2000.
