The notion of a fuzzy subset was initially introduced by Zadeh[13] in 1965, for representing uncertainity.The idea of Intuitionistic fuzzy set was first published by Atanassov [2], as a generalization of the notion of the fuzzy set.In 2000, S.Sabu and T.V.Ramakrishnan [10] proposed the theory of multi-fuzzy sets in terms of multi-dimensional membership function and investigated some properties of multi –level fuzziness.Theory of multi-fuzzy set is an extension of theory of fuzzy sets. Y.Imai and K.Iseki introduced two classes of abstract algebras:BCK algebras and BCI-algebras[4].It is shown that the BCK-algebras is a proper subclass of the class of BCI-algebras.C.B.Kim and H.S.Kim[5] introduced the notion of a BGalgebra which is generalization of Balgebras.With these ideas, fuzzy subalgebras of BG-algebra were developed by S.S.Ahn and H.D.Lee[1].R.Muthuraj and S.Devi[6,7,8] introduced the concept of multi –fuzzy subalgebra and intuitionistic multi-fuzzy subalgebras in BG-algebra in 2016.Also in 2017 they also introduced intuitionistic multi-fuzzy ideals in BG-algebra.In this paper,we define a new algebraic structure of normal intuitionistic multi-fuzzy ideals in BGalgebra and discuss some of their related properties.Also we investigate the properties of Cartesian product of intuitionistic multi fuzzy BG-ideals