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Interval-valued intuitionistic fuzzy sets generalize the concept of intuitionistic fuzzy sets and are well suited to dealing with vagueness. This article investigates multi-attribute decision making problems in which all the information provided by the decision makers is expressed as interval-valued intuitionistic fuzzy decision matrices where each of the elements is characterized by interval-valued intuitionistic fuzzy number. Information about attribute weights is partially known and may be constructed by various forms. Several optimization models are presented to generate optimal weights for attributes, and the corresponding decision-making methods have also been developed. Feasibility and effectiveness of the proposed methods are illustrated with an example of investment decision problem.