Time-Division　Multiplexing　(TDM)　technology　is　widely　applied　to　solving　the　IO　limitation　problem　to　improve　the　routability　of　FPGA　system.　However,　the　increase　of　the　TDM　ratio　leads　to　a　significant　increase　in　system　delay.　Therefore,　a　Multi-Stage　Co-Optimization　FPGA　routing　(MSCOFRouting)　for　Time-Division　Multiplexing　is　proposed　in　this　paper　to　optimize　the　system　delay　and　the　routability　of　FPGA　system.　First,　an　adaptive　routing　algorithm　is　proposed　to　reduce　routing　congestion,　improve　the　routability,　solve　the　routing　optimization　problem　between　FPGAs,　and　provide　high-quality　routing　results　for　subsequent　TDM　ratio　assignment.　Second,　to　avoid　the　delay　degradation　caused　by　excessive　TDM　ratio　of　large-scale　net　groups,　a　TDM　ratio　assignment　algorithm　based　on　Lagrangian　relaxation　is　utilized　to　assign　the　initial　TDM　ratio　with　a　smaller　delay　to　the　edge　distribution　system　of　the　routing　graph.　In　addition,　a　multi-level　TDM　ratio　optimization　algorithm　is　used　to　reduce　the　TDM　ratios　of　the　net　group　with　maximum　TDM　ratios.　The　TDM　ratio　reduction　is　employed　for　the　net　group　and　the　FPGA　connection　pair.　Meanwhile,　a　multi-thread　parallelization　method　is　integrated　into　the　three　algorithms　above　to　improve　further　the　efficiency　of　MSCOFRouter.　Experiments　show　that　MSCOFRouting　can　obtain　the　results　satisfying　the　TDM　ratio　constraint,　and　achieve　the　best　routing　optimization　results　and　TDM　ratio　assignment　results.　©　2023　Science　Press.　All　rights　reserved.