Intuitionistic　fuzzy　set　(IFS),　a　classical　extension　of　fuzzy　set,　is　a　powerful　tool　to　describe　fuzzy　information　in　multi-criteria　decision-making　(MCDM)　because　of　its　membership　and　non-membership　functions.　Intuitionistic　fuzzy　aggregation　operators　are　the　standard　mathematical　tools　for　the　combination　of　several　inputs　with　respect　to　criteria　into　one　unique　output.　The　traditional　intuitionistic　fuzzy　aggregation　operator-based　methods　have　been　criticized　because　of　reasons　that　include　disregard　for　comprehensive　correlative　relationships　of　criteria　and　ignorance　for　combination　of　fuzzy　measures　of　criteria　and　weights　of　positions.　In　this　paper,　the　effectiveness　of　traditional　aggregation　operator-based　MCDM　techniques　is　improved　by　a　novel　method　to　select　the　best　alternative(s)　under　the　intuitionistic　fuzzy　environment.　We　begin　by　defining　the　Choquet　integral-based　intuitionistic　fuzzy　arithmetic　aggregation　(CIIFAA)　operator,　the　remarkable　properties　of　which　are　proved　in　details.　Further,　the　Choquet　integral　based　intuitionistic　fuzzy　hybrid　arithmetic　aggregation　(CIIFHAA)　operator　is　defined　as　well　as　the　proofs　of　its　properties.　The　CIIFHAA　operator　can　not　only　capture　the　comprehensive　correlative　relationships　of　criteria　in　a　simpler　manner,　but　also　combine　the　weights　of　positions.　Then,　we　put　forward　a　MCDM　method　based　on　CIIFHAA　operator　with　intuitionistic　fuzzy　evaluations.　An　illustrative　example　is　solved,　the　best　alternative　and　the　ranking　of　alternatives　are　obtained,　which　can　verify　the　feasibility　of　the　proposed　method.　Finally,　two　comparisons　are　conducted　to　illustrate　the　stability　and　advantages　of　the　proposed　method.　©　2021　Elsevier　Ltd