The problem of multiple attribute making-decisions with interval-valued intuitionistic fuzzy information was analyzed.  Some operational laws of interval-valued intuitionistic fuzzy numbers, score function and accuracy function of interval-valued intuitionistic fuzzy numbers were introduced.  An optimization model based the maximizing deviation method was established, by which the attribute weights could be determined.  Intervalvalued intuitionistic fuzzy weighted arithmetic averaging (IIWAA) operator was used to aggregate the intervalvalued intuitionistic fuzzy information corresponding to each alternative.  According to the score function and accuracy function , the alternatives were ranked and the most desirable one(s) were selected.   Finally, a numerical example is used to illustrate the use of the proposed method.  The result shows the approach is simple, effective and easy to calculate.  An illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.