Targeting　the　great　demand　for　adding　non-structural　elevators　to　old　residential　buildings,　this　article　proposes　an　updated　configuration　of　tuned　mass　damper　inerter　(U-TMDI)　applied　in　external　non-structural　elevators.　An　analog　2-DOF　system　is　established　to　describe　the　residential　building　controlled　by　an　inerter　elevator.　Then,　three　closed-form　optimal　solutions　for　designing　the　U-TMDI　are　derived　via　the　fixed-point　method,　equal　modal　damping　criterion,　and　the　infinity　damping　assumption.　Subsequently,　these　optimal　solutions　are　compared　and　discussed　involving　the　expressions　of　tuning　frequency　ratio　and　the　optimal　damping　parameters,　root　locus　diagram　and　supplemental　damping　ratios,　transfer　function,　and　robustness　to　frequency　variation,　respectively.　Finally,　a　residential　building　example　is　adopted　to　validate　the　feasibility　of　the　proposed　retrofitting　strategy　and　the　closed-form　optimal　design　solutions.　It　is　demonstrated　that　the　optimal　tuning　frequency　ratios　derived　by　the　fixed-point　method　and　equal　modal　damping　criterion　are　different　for　U-TMDI　due　to　the　influence　of　elevator　stiffness　ratio　eta$\eta　$,　while　its　degradation　forms　for　tuned　mass　damper　(TMD)　are　identical,　recognizing　the　importance　of　the　elevator　stiffness.　Moreover,　the　proposed　retrofitting　strategy　of　using　an　inerter　elevator　can　significantly　mitigate　the　main　structure　displacement　by　about　18%similar　to　23%　for　both　far-field　and　near-fault　earthquakes.