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Top1. Introduction
Fuzzy sets and its generalizations such as intuitionistic fuzzy sets (IFSs) and interval valued intuitionistic fuzzy sets (IvIFSs) are meant to handle uncertain and imprecise situations. Uncertainty, ambiguity, vagueness and imprecision are common in human nature and real-world problems, fuzzy sets and its generalizations such as intuitionistic fuzzy sets (IFSs) and interval-valued intuitionistic fuzzy sets (IvIFSs) are meant to handle such situations. Various approaches have been developed to deal with uncertainty using these tools. Fuzzy sets were originally developed by Zadeh (1948). In fuzzy sets belongingness of each element is based on degree of association or membership, whereas IFSs which was developed by Atanassov (1986) and IvIFSs which was derived by Atanassov and Gargov (1989) are based on point value or interval for membership degree and non-membership degree respectively. Amalgamation of intervals for membership and non-membership in fuzzy sets formulate IvIFSs which are more flexible and widely applicable to deal with real world problems. The growing intricacies of the social-economic environment and deficiency of knowledge or data about the problem domains can be dealt with IvIFSs, which are characterized by membership and non-membership functions whose values are intervals, instead of real numbers. To identify a pattern in different fields, measure of similarity is one of the most suitable tool wherein optimal decision or optimality criteria is defined. Numerous researchers have worked on both measures and applied it to various real-world problems. A lot of work is done by many researchers on entropy and similarity measures under fuzzy and intuitionistic fuzzy environment, but the efforts in interval valued intuitionistic fuzzy environment is limited. Some of the related work is mentioned as follows:
Zang et al. (2010), Zhang et al. (2011; 2013), Chen et al. (2013), Hu and Li (2013), Zhang et al. (2014), Wei and Zhang (2015), Xie and Lv (2016), Zhao and Xu (2016) introduces different parametric/ trigonometric entropies measures for IvIFSs which satisfies the extended form of Deluca-Termini axioms for fuzzy entropy. These researchers also derive relationship between proposed entropy measures and various other information theoretic measures such as distance, similarity and cross entropy etc. Research work also applied suggested work on different types of problem involving pattern recognition, decision making, multi- attribute decision making (MADM), and medical diagnoses using hypothetical data. Wei et al. (2011), Jing and Min (2013), Sun and Liu (2013), Jin et al. (2014), Gupta et al. (2016a; 2016b), Zhao and Xu (2016), suggests generalized entropies, relation with information theoretic measures under fuzzy, intuitionistic fuzzy and interval valued intuitionistic fuzzy environment. The aforesaid papers applied the proposed generalization for recognizing patterns, medical diagnoses, MADM or multi-criteria decision making (MCDM). Wu et al. (2014) considered the hesitancy degree to define similarity measure for IvIFSs and presented certain examples to prove the efficiency of the proposed measure. Applications of measure are validated for pattern recognition and MADM problems. Yu et al. (2016) proposed an interactive MCDM approach which is applicable in intuitionistic fuzzy environment. Cao et al. (2016) developed an approach to deal with MCDM problems in interval valued intuitionistic settings. Zhu and Li (2016) proposed entropy for IFSs and compared it with some existing measures to check its validity and applicability. Jiang and Wang (2018) constructed a new intuitionistic fuzzy entropy and compared it with existing measures.
Peng et al. (2017) and Li et al. (2017) proposed fuzzy aggregation operators based on the Archimedean t-Norms and t-Conorms to aggregate hesitant intuitionistic fuzzy and hesitant fuzzy values. They also applied it to MCDM. Zhou et al. (2015; 2017) studied grey discrete and continuous numbers under uncertain situation with application in MCDM. Liu and Li (2018) proposed Nonlinear programming methodology based on TOPSIS for MADM under Interval valued intuitionistic fuzzy environment. Yu et al. (2018) derived some operators for intuitionistic uncertain 2-tuple linguistic (IU2TL) variables and to deal with heterogeneous relationships, the IU2TL continuous Bonferroni mean operator is developed. Lastly developed method is used for solving multi-attribute group decision making problems.