Based on technique for order preference by similarity to an ideal solution(TOPSIS), by using the Euclidean distance of interval-valued intuitionistic fuzzy numbers(IVIFNs), the relative closeness degree of an IVIFN to the maximum IVIFN is proposed, and the excellent properties of the relative closeness degree are given. The properties show that the relative closeness degree serving as the ranking index is rational. Then, compared with some other ranking methods of IVIFN in literatures, the new ranking index shows higher discrimination capability. An interval-valued intuitionistic fuzzy multiple attribute decision making method is proposed by using the new ranking index, and an example is given to illustrate the effectiveness of the proposed method.