A new concept of equivalence between intervals and the induced indistinguishability between interval-valued (IV) fuzzy sets are proposed and considered. A new notion of the degree of IV equivalence is presented where partial or linear orders and the width of intervals are involved reflecting uncertainty. Furthermore, construction methods of the considered equivalences are provided and the relation between them and other notions of IV equivalences are studied. Finally, a methodology to apply the proposed equivalences in the field of image processing is shown, along with an illustrative example.

Human beings represent and process imperfect information in a natural way, whereas effective approaches are still being looked for in the area of soft computing. Atanassov's intuitionistic fuzzy sets and especially Atanassov's intuitionistic fuzzy relations are are tools that make it possible to model in an effective way imperfect information. In this paper, we discuss the transitivity problem of Atanassov's intuitionistic fuzzy relations. The transitivity property reflects the consistency of a preference relation. Therefore, transitivity is important from the point of view of real problems appearing, e.g., in group decision making, choice, and utility theories widely making use of the preference procedures. We study here in detail transitivity property of Atanassov's intuitionistic fuzzy relations and propose a new algorithm for generating weakly transitivity among the alternatives (options) considered. Next, making use of the new algorithm we propose a procedure of ranking of the alternatives in a group decision-making problem.