In this paper, we define some results on generalized interval-valued intuitionistic fuzzy sets (GIVIFSs). In fact, all interval-valued intuitionistic fuzzy sets (IVIFSs) are GIVIFSs but all GIVIFSs are not IVIFSs. We define two composite relations, four types of reflexivity and irreflexivity of GIVIFSs with some of their properties. We define generalized interval-valued intuitionistic fuzzy relation (GIVIFR) with some results. Also we define two operators C and I with some properties over GIVIFSs. Finally, an illustrative example is given to using GIVIFSs in a decision making problem.