Fuzzy Sets and Systems 118 (2001) 467–477 www.elsevier.com/locate/fss Entropy for intuitionistic fuzzy sets Eulalia Szmidt, Janusz Kacprzyk ∗ Systems Research Institute, Polish Academy of Sciences ul. Newelska 6, 01-447 Warsaw, Poland Received September 1997; received in revised form August 1998 Abstract A non-probabilistic-type entropy measure for intuitionistic fuzzy sets is proposed. It is a result of a geometric interpre- tation of intuitionistic fuzzy sets and uses a ratio of distances between them proposed in Szmidt and Kacprzyk (to appear). It is also shown that the proposed measure can be dened in terms of the ratio of intuitionistic fuzzy cardinalities: of F ∩ F c and F ∪ F c . c  2001 Elsevier Science B.V. All rights reserved. Keywords: Distance between intuitionistic fuzzy sets; Cardinality of intuitionistic fuzzy set; Entropy of intuitionistic fuzzy set 1. Introduction Fuzziness, a feature of imperfect information, results from the lack of crisp distinction between the elements belonging and not belonging to a set (i.e. the boundaries of the set under consideration are not sharply dened). A measure of fuzziness often used and cited in the literature is an entropy rst mentioned in 1965 by Zadeh [22]. The name entropy was chosen due to an intrinsic similarity of equations to the ones in the Shannon entropy [7]. However, the two functions measure fundamentally dierent types of uncertainty. Basically, the Shannon entropy measures the average uncertainty in bits associated with the prediction of outcomes in a random experiment. In 1972, De Luca and Termini [14] introduced some requirements which capture our intuitive comprehension of the degree of fuzziness. Kaufmann [9] proposed to measure the degree of fuzziness of any fuzzy set A by a metric distance between its membership function and the membership function (characteristic function) of its nearest crisp set. Another way given by Yager [20] was to view the degree of fuzziness in terms of a lack of distinction between the fuzzy set and its complement. Kosko [11 – 13] investigated the fuzzy entropy in relation to a measure of subsethood. In this paper, we propose a measure of fuzziness for intuitionistic fuzzy sets introduced by Atanassov [1 – 5]. The measure of entropy is a result of a geometric interpretation of intuitionistic fuzzy sets and basically uses * Corresponding author. E-mail address: kacprzyk@ibspan.waw.pl (J. Kacprzyk). 0165-0114/01/$ - see front matter c  2001 Elsevier Science B.V. All rights reserved. PII: S0165-0114(98)00402-3