Generalized atanassov’s interval-valued intuitionistic Fuzzy index: construction of atanassov’s interval-valued Fuzzy entropy from interval-valued Fuzzy implication operators - Silva, Lidiane Costa da (2018)
Despite its widespread research in applied areas, recognized limitations of Fuzzy Logic (FL) justify the search for higher levels of abstraction, using extensions such as type-2 fuzzy logic (T2FL) for the representation of information in fuzzy reasoning systems. This proposal considers the two intersection areas of T2FL, the intuitionist fuzzy logic (A-IFL) and the interval-valued fuzzy logic (IvFL). The interval-valued intuitionist fuzzy logic introduced by Atanassov in 1969 (A-IvIFL) considers both the imprecision of data in the membership function and the hesitation in determining its complementary relation - the non-membership functions. In A-IvIFL approach, the principles of A-IFL are preserved and the forms of data representation are expanded, adding not only the hesitation information related to experts with respect to non necessarily complementary relations but also the imprecision information provided by the interval-valued intuitionistic fuzzy index. Firstly, we consider the study of main properties verified by the axiomatic concept of hesitation index names - the generalized Atanassov’s intuitionistic fuzzy index (A-GIFIx), and corresponding constructive methodology based on fuzzy implications. And, just in such context, this work introduces an extension of methodology which is able to preserve properties by making use of dual and conjugate operators. In addition, as the main contribution, the generalized Atanassov’s interval-valued intuitionist fuzzy index (A-GIvIFIx) is discussed, including its axiomatic concept and related constructive methodology characterized in terms of interval-valued fuzzy implications, preserving main properties of an A-GIFIx. This work also presents new ways of obtaining A-GIvIFIx via dual and conjugated constructions, by the action of negation operators and automorphisms, respectively. Among the several applications of A-GIvIFIx as similarity, correlation and distance measures, we introduce a methodology to obtain the entropy via A-GIvIFIx contributing with multi-attribute systems based on T2FL. The application of the concept of admissible linear orders makes the comparison between interval results possible.