Soft Computing

, Volume 22, Issue 9, pp 2809–2824 | Cite as

Hesitant intuitionistic fuzzy entropy/cross-entropy and their applications

  • Dengbao YaoEmail author
  • Cuicui Wang


In this paper, we introduce a new concept of hesitant intuitionistic fuzzy set (HIFS), which refines the dual hesitant fuzzy set and could be viewed as a more flexible tool to describe the uncertain information in reality. Since the uncertainty in HIFSs may be divided into three facets: fuzziness, intuitionism and hesitancy, we develop a fresh information-theoretic framework of uncertainty measures. We firstly propose the axiomatic principles of hesitant intuitionistic fuzzy entropy and give some distance-based entropy formulas. Then, a hesitant intuitionistic fuzzy cross-entropy is addressed to measure the discrimination of uncertain information between different HIFSs; the relationships between cross-entropy and entropy for HIFSs are also discussed. Moreover, some parameterized cross-entropy and entropy measures of HIFSs are investigated, and the decomposition formula suggests that hesitant intuitionistic fuzzy entropy may be expressed as the weighted average of fuzzy entropy, intuitionistic entropy and hesitant entropy. Finally, we demonstrate the efficiency of the proposed uncertainty measures for medical diagnosis and decision-making approach.


Hesitant intuitionistic fuzzy set Hesitant intuitionistic fuzzy entropy Hesitant intuitionistic fuzzy cross-entropy 



The authors are highly grateful to any anonymous referee for their careful reading and insightful comments, and the views and opinions expressed are those of the authors. The work is supported by the Talent Introduction Project of Anhui University (No. J01006134) and the Natural Science Key Project of Anhui Sanlian University (No. kjzd 2016001).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Economics SchoolAnhui UniversityHefeiPeople’s Republic of China
  2. 2.Foundation DepartmentAnhui Sanlian UniversityHefeiPeople’s Republic of China

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