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Ranking of interval-valued intuitionistic fuzzy (IVIF) numbers is a most popular and elegant work in the area of decision-making of several real-world problems. Some limited methods have been presented concerning the ranking of IVIF sets in literature. In the present paper, we generalize the intuitionistic fuzzy (IF) number to interval-valued intuitionistic fuzzy number by defining interval membership and nonmembership functions instead of fixed-valued function and hence it will present uncertain situation better than IF numbers. It may also be applied in data analysis, industrial management, artificial intelligence, forecasting, time series and so on. In this paper, ranking methodology of IVIF numbers is presented, for this first we define the value and ambiguity of IVIF numbers. Proposed ranking method also is compared with existing ranking methods. Further, IVIF numbers are used to capture fuzziness and hesitation in transportation problem (TP), and we propose a new method to find optimal solutions of TP with IVIF number parameters and finally, a numerical example is given to demonstrate the proposed method.