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This article proposes a new axiomatic definition of entropy on interval-valued intuitionistic fuzzy sets (IVIFSs) and a method to construct different entropies on IVIFSs. Furthermore, a new multi-attribute decision making (MADM) method based on similarity measures using entropy-based attribute weights is proposed to deal with the decision making situations where the alternatives on attributes are expressed by IVIFSs and the attribute weights information is unknown. Instead of using traditional fuzzy entropy, which obtain attribute weights through the probabilistic discrimination of attributes, we utilize the interval-valued intuitionistic fuzzy (IVIF) entropy to assess attribute weights based on the credibility of the IVIF decision making matrix. Finally, two numerical examples are given to demonstrate the feasibility and validity of the newly proposed MADM method, by comparing it with other fuzzy MADM methods.