The　interval　neutrosophic　uncertain　linguistic　variables　(INULVs)　can　be　better　at　handling　the　uncertainty　of　the　decision-makers＇　cognition　in　multi-criteria　group　decision-making　(MCGDM)　problems.　Most　MCGDM　methods　with　INULVs　are　based　on　the　supposition　that　all　criteria　are　independent;　however,　they　may　be　correlative　in　real　decisions.　The　Choquet　integral　can　process　MCGDM　problems　with　correlated　criteria,　which　fail　to　aggregate　INULVs.　So,　it　is　necessary　and　meaningful　to　propose　the　MCGDM　method　with　INULVs　based　on　the　Choquet　integral　by　considering　the　correlations　between　the　attributes.　By　combining　INULVs　with　the　Choquet　integral,　we　propose　the　interval　neutrosophic　uncertain　linguistic　Choquet　averaging　(INULCA)　operator　and　the　interval　neutrosophic　uncertain　linguistic　Choquet　geometric　(INULCG)　operator.　These　two　operators　reflect　the　existing　correlation　between　two　adjacent　coalitions.　In　order　to　globally　consider　the　correlations　between　elements　or　their　ordered　positions,　the　generalized　Shapley　INULCA　(GS-INULCA)　operator　and　the　generalized　Shapley　INULCG　(GS-INULCG)　operator　are　further　proposed.　Furthermore,　some　models　based　on　the　grey　relational　analysis　(GRA)　method　for　determining　the　optimal　fuzzy　measures　on　the　expert　set　and　the　criteria　set　are　respectively　built.　Based　on　the　proposed　operators　and　built　models,　a　method　is　developed　to　cope　with　the　MCGDM　problems　with　INULVs,　and　the　validity　and　advantages　of　the　proposed　method　are　analyzed　by　comparison　with　some　existing　approaches.　The　method　proposed　in　this　paper　can　effectively　handle　the　MCGDM　problems　in　which　the　attribute　information　is　expressed　by　INULVs,　the　attributes＇　and　experts＇　weights　are　partly　known,　and　the　experts　and　attributes　are　interactive.