In the literature, there are two different approaches to define entropy of Atanassov intuitionistic fuzzy sets (AIFS, for short). The first approach, given by Szmidt and Kacprzyk, measures how far is an AIFS from its closest crisp set, while the second approach, given by Burrillo and Bustince, measures how far is an AIFS from its closest fuzzy set. On the other hand, divergence measures are functions that measure how different two AIFSs are. Our work generalizes both types of entropies using local measures of divergence. This results in at least two benefits: depending on the application, one may choose from a wide variety of entropy measures and the local nature provides a natural way of parallel computation of entropy, which is important for large data sets. In this context, we provide the necessary and sufficient conditions for defining entropy measures under both frameworks using divergence measures for AIFS. We show that the usual examples of entropy measures can be obtained as particular cases of our more general framework. Also, we investigate the connection between knowledge measures and divergence measures. Finally, we apply our results in a multi-attribute decision-making problem to obtain the weights of the experts.
This is a preview of subscription content, log in to check access.
A shorter version of this paper was published at EUSFLAT Conference (Montes et al. 2018). The current version has been updated with additional results, comments and proofs. We acknowledge the financial support by project TIN2014-59543-P.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
This article does not contain any studies with human participants or animals performed by any of the authors.
De Luca A, Termini S (1972) A definition of nonprobabilistic entropy in the setting of fuzzy theory. J Gen Syst 5:301–312MathSciNetzbMATHGoogle Scholar
Deng G, Jiang Y, Fu J (2015) Monotonic similarity measures between intuitionistic fuzzy sets and their relationship with entropy and inclusion measure. Inf Sci 316:348–369CrossRefzbMATHGoogle Scholar
Dubois D, Prade H (eds.) (2000) Fundamentals of Fuzzy Sets. SpingerGoogle Scholar
Farnoosh R, Rahimi M, Kumar P (2016) Removing noise in a digital image using a new entropy method based on intuitionistic fuzzy sets. In: 2016 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE)Google Scholar
Grzegorzewski P (2004) Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric. Fuzzy Sets Syst 148:319–328MathSciNetCrossRefzbMATHGoogle Scholar
Guo K (2016) Knowledge measure for Atanassov’s intuitionistic fuzzy sets. IEEE Trans Fuzzy Syst 24(5):1072–1078CrossRefGoogle Scholar
Guo K, Song Q (2014) On the entropy for Atanassov’s intuitionistic fuzzy sets: an interpretation from the perspective of amount of knowledge. Appl Soft Comput 24:328–340CrossRefGoogle Scholar
Pal N, Bustince H, Pagola M, Mukherjee U, Goswami D, Beliakov G (2013) Uncertainty with Atanassov intuitionistc fuzzy sets: fuzziness and lack of knowledge. Inf Sci 228:61–74CrossRefzbMATHGoogle Scholar
Szmidt E, Kacprzyk J (2006) An application of intuitionistic fuzzy set similarity measures to a multi-criteria decision making problem. In: Artificial Intelligence and Soft Computing ICAISC. Lecture Notes in Computer Science, vol. 4029, pp. 314–323. SpringerGoogle Scholar