New Measures of Intuitionistic Inclusion and Similarity with Applications

Abstract

Similarity and inclusion are two most important conceptual ways for looking at any possible relationships between objects (sets). A similarity measure is used for estimating the degree of resemblance/similarity between two objects while an inclusion measure expresses the degree to which one of the two is covered by the other one. In this dissertation, we have presented two different yet equally important approaches of defining similarity and inclusion measures for intuitionistic fuzzy sets. The first approach can be regarded as a logical approach implanting intuitionistic fuzzy implication, bi-implications and t-norms while the second one is a purely set theoretic approach having cardinalities and sets operations involved in its construction. In the logical approach, the degree of inclusion/ similarity was obtained by composing the intuitionistic logical operators (implications and bi-implications) with fuzzy measures on intuitionistic fuzzy sets. For this purpose we initially defined some new normal fuzzy measures on intuitionistic fuzzy sets along with a class of scalar cardinality measure for intuitionistic fuzzy sets. Later, we introduced different classes of intuitionistic fuzzy bi-implication operators having axiomatic as well as constructive approaches. A study on the properties of these bi-implication operators by utilizing Lukasiewicz intuitionistic fuzzy implicator revealed some remarkable results. The intuitionistic fuzzy bi-implication operators along with new defined fuzzy measures gave rise to multiple classes of intuitionistic fuzzy bi-implicator based similarity measures for intuitionistic fuzzy sets. Also the same normal fuzzy measures were employed to obtain the degree of inclusion between two intuitionistic fuzzy sets when composed with intuitionistic fuzzy implication operators. The new implication based classes of inclusion and similarity measures fulfilled almost all of the universally accepted criteria’s. In the set theoretic approach, we employed one of the members of new introduced scalar cardinality of intuitionistic fuzzy sets to construct a four parametric family of cardinality based similarity and inclusion measures. Both of these parametric families, xi for different combinations of parameters, generated intuitionistic fuzzy versions of some of the famous crisp measures of time. Lastly, the utility of the new measures (similarity, inclusion, cardinality and others) of intuitionistic fuzzy sets introduced in this work is exhibited by utilizing them as a part of solution techniques to the problems arising in real life situations.