Abstract
Since the sufficient conditions for the maximum value of the intuitionistic fuzzy entropy are not unified and the hesitant fuzzy entropy cannot be compared when the lengths of the hesitation fuzzy elements are not equal, improved axiomatic definitions of intuitionistic fuzzy entropy and hesitant fuzzy entropy are proposed, and new intuitionistic fuzzy entropy and hesitant fuzzy entropy based on the improved axiomatic definitions are established. This paper defines the fuzzy entropy that satisfies the properties based on the axiomatized definition of fuzzy entropy and, based on the fuzzy entropy, defines new intuitionistic fuzzy entropy and hesitant fuzzy entropy, so that the three are unified in form. The validity and rationality of the proposed intuitionistic fuzzy entropy and hesitant fuzzy entropy are verified by analysis.
Highlights
Intuitionistic fuzzy sets [3,4,5,6] were proposed by Atanassov in 1983
Intuitionistic fuzzy sets are an important extension of fuzzy sets, which have a wide range of applications in many fields, such as multiattribute decisionmaking [7,8,9], pattern recognition [10], and image segmentation [11]
In 2009, Zhang [15] gave a new axiomatic definition of intuitionistic fuzzy entropy based on the distance between intuitionistic fuzzy sets. e axiomatic definition of intuitionistic fuzzy entropy in literature [12,13,14,15] fails to fully reflect the fuzziness of intuitionistic fuzzy sets, especially in terms of the necessary and sufficient conditions for Journal of Mathematics restraining the maximum value of intuitionistic fuzzy entropy, which does not meet reality, and fails to objectively describe the maximum or minimum fuzzy degree of the intuitionistic fuzzy set
Summary
Definition 1 (see [1]). Let X x1, x2, . . . , xn be the universe of discourse; a fuzzy set A in X is defined as. For any two fuzzy sets A and B in X, the following relations and operations can be defined:
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