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This article proposes an approach to multiple attribute decision making with partially known attribute weight information where individual assessments are provided as interval-valued intuitionistic fuzzy numbers (IVIFNs). By establishing an optimization model based on the deviation degree, the proposed approach derives a linear program for determining attribute weights. The weights are subsequently used to synthesize individual IVIFN assessments into an aggregated IVIFN value for each alternative. In order to rank alternatives based on their aggregated IVIFN values, a novel method is developed for comparing two IVIFNs by introducing the formula of possibility degree and the ranking vector of the possibility degree matrix. Finally, an illustrative outsourcing decision problem is employed to verify the proposed approach and to demonstrate its practicality and effectiveness.