Abstract
This paper presents new axiomatic definitions of entropy measure using concept of probability and distance for interval valued intuitionistic fuzzy sets (IvIFSs) by considering degree of hesitancy which is consistent with the definition of entropy given by De Luca and Termini. Thereafter, we propose some entropy measures and also derived relation between distance, entropy and similarity measures for IvIFSs. Further, we checked the performance of proposed entropy and similarity measures on the basis of intuition and compared with the existing entropy and similarity measures using numerical examples. Lastly, proposed similarity measures are used to solve problems in the field of pattern recognition and medical diagnoses.
Highlights
Fuzzy set theory (Zadeh, 1965) is tool that can handle uncertainty and imprecision effortlessly
In this paper we have developed some of the distance, entropy and similarity measures by taking all the three degrees in account and applied it to pattern recognition and medical diagnoses under IvIFE
Liu (1992) defined distance and similarity measures for interval-valued intuitionistic fuzzy sets (IvIFSs) axiomatically which are given as follows: Definition 2: For any two IvIFSs A and B, a real valued function D: IvIFSs(Ω) × IvIFSs(Ω) ⟶ [0,1] is termed as a distance measure of IvIFSs on Ω, if it satisfies the below mentioned axioms: 1. For any crisp set A, we have D(A, A ) = 1
Summary
Fuzzy set theory (Zadeh, 1965) is tool that can handle uncertainty and imprecision effortlessly. Entropy and similarity measures are the central arenas that are investigated by various researchers under intuitionistic and interval-valued fuzzy environment (IFE and IvFE). Sun & Liu (2012), Hu & Li (2013), Zhang et al (2014) proposed entropy and similarity measure along with their relationship for IvIFSs. Applications of the aforesaid entropy, distance and similarity measures are for recognition of patterns, medical diagnoses, and decision making with multiple criteria and expert systems problems. Applications of the aforesaid entropy, distance and similarity measures are for recognition of patterns, medical diagnoses, and decision making with multiple criteria and expert systems problems Most of these distance, similarity or entropy measures do not consider hesitancy index between IvIFSs. Hesitance index play a very important role when membership and non-membership degree do not differ much for two IvIFSs but their hesitant index does.
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