Interval-valued　intuitionistic　fuzzy　(IVIF)　sets　are　a　useful　tool　to　deal　with　fuzziness　inherent　in　decision　data　and　decision　making　process.　The　aim　of　this　paper　is　to　develop　a　methodology　for　solving　multiattribute　decision　making　(MADM)　with　both　ratings　of　alternatives　on　attributes　and　weights　being　expressed　with　IVIF　sets.　In　this　methodology,　a　weighted　Euclidean　distance　between　IF　sets　is　defined　using　weights　of　IF　sets.　A　pair　of　nonlinear　programming　models　is　constructed　based　on　the　concept　of　the　relative　closeness　coefficients　and　the　distance　defined.　Two　simpler　auxiliary　nonlinear　programming　models　are　further　derived　to　calculate　the　relative　closeness　coefficient　intervals　of　alternatives　to　the　IVIF　positive　ideal　solution,　which　can　be　used　to　generate　ranking　order　of　alternatives　based　on　the　concept　of　likelihood　of　interval　numbers.　The　method　proposed　in　this　paper　is　illustrated　with　a　real　example.