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The interval-valued intuitionistic fuzzy multi-attribute decision making problems are investigated, in which the decision information is expressed as interval-valued intuitionistic fuzzy decision matrices, and the information about attribute weights is incomplete. The score matrix of the interval-valued intuitionistic fuzzy decision matrix is constructed. Based on the score matrix and the given attribute weight information, several optimization models are established to generate optimal weights for attributes. Furthermore, a procedure is developed to identify the best alternative(s). Finally, a global supplier selection example is used to illustrate the developed procedure.