Intuitionistic　fuzzy　sets　(IFSs)　and　interval-valued　intuitionistic　fuzzy　sets　(IVIFSs)　are　flexible　to　deal　with　the　vague　and/or　imprecise　information.　Thus,　many　multi-attribute　group　decision　making　(MAGDM)　problems　are　modeled　by　IFSs　or　IVIFSs.　The　comparison　of　two　IVIFSs　is　still　a　hot　topic,　and　thereby　this　paper　proposes　a　new　ranking　function　of　IVIFSs,　which　takes　into　the　amount　and　the　reliability　information　of　an　IVIFS　and　combines　the　advantages　of　TOPSIS.　Based　on　the　new　ranking　function,　we　establish　an　optimization　model　to　determine　the　attribute　weights　when　they　are　unknown　and　partially　known.　Moreover,　we　develop　an　effective　method　for　solving　MAGDM　problems　in　which　the　attribute　values　are　expressed　with　IVIFSs.　A　numerical　example　of　supplier　selection　problem　is　examined　to　demonstrate　applicability　and　feasibility　of　the　proposed　method.