This work constitutes classical Hutchinson-Barnsley theory on the product intuitionistic fuzzy fractal space with the aid of iterated function system, in which a finite number of intuitionistic fuzzy B-contractions and intuitionistic fuzzy Edelstein contractions are enclosed. A fixed point theorem is exhibited by proving that the Hutchinson-Barnsley operator is an intuitionistic fuzzy B-contraction and intuitionistic fuzzy Edelstein contraction. To show the primary goal of the article, the Hausdorff product intuitionistic fuzzy metric space is constructed, then the notion of product intuitionistic fuzzy metric space on complete and compact spaces in the sense of intuitionistic fuzzy B-contraction and the intuitionistic fuzzy Edelstein contraction are defined.