The　efficiency　decomposition　and　frontier　projection　of　traditional　two-stage　network　data　envelopment　analysis　(DEA)　model　under　variable　returns　to　scale　(VRS)　are　often　not　equivalent;　which　not　only　contradicts　DEA　theory,　but　also　reduces　the　scientificity　of　the　model.　The　main　reason　for　this　inequivalence　is　that　there　is　a　synergistic　effect　of　variable　scale　return　in　two　different　stages.　Therefore,　this　paper　describes　the　production　frontier　of　two-stage　DEA　under　VRS　for　analyzing　this　synergistic　effect,　and　then　the　efficiency　evaluation　pitfalls　of　two-stage　DEA　under　VRS　are　identified.　From　the　input　orientation,　output　orientation,　non-orientation　perspectives,　different　two-stage　network　DEA　models　under　VRS　are　respectively　constructed　to　solve　these　evaluation　pitfalls,　and　the　equivalence　relationships　of　their　multiplier　model　and　envelopment　model　are　proved;　and　then　the　efficiency　decomposition　and　frontier　projection　with　equivalence　relationship　can　be　obtained　to　meet　the　different　needs　of　decision-makers.　Furthermore,　variable　intermediate　element　is　discussed　in　the　non-orientation　model　for　achieving　the　Pareto　optimality　of　two　stages　during　the　process　of　efficiency　decomposition　and　frontier　projection.　By　these　models,　the　theoretical　foundation　of　two-stage　network　DEA　under　VRS　has　been　further　improved.　Finally,　two　examples　are　provided　to　illustrate　the　effectiveness　of　the　new　models.　©　2024　Elsevier　B.V.