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In decision making problems, the increasing complexity of the world and the fuzziness of human thinking make it difficult to express a decision maker's preference information over alternatives and attributes with crisp data. As such, the interval-valued intuitionistic fuzzy sets (IVIFS) have become convenient and suitable to deal with the vagueness. This article investigates the multi-attribute decision making (MADM) problem under the IVIFS environment, where both the attribute weights and the assessments of alternatives with respect to attributes are given by the decision maker in terms of interval-valued intuitionistic fuzzy numbers. Several fractional programming models are derived from TOPSIS to determine the fuzzy relative closeness intervals of alternatives, and the corresponding decision-making method has also been developed. Feasibility and effectiveness of the proposed method are illustrated with an investment decision problem.