Let　be　a　given　directed　graph　in　which　every　edge　e　is　associated　with　two　nonnegative　costs:　a　weight　w(e)　and　a　length　l(e).　For　a　pair　of　specified　distinct　vertices　,　the　k-(edge)　disjoint　constrained　shortest　path　(kCSP)　problem　is　to　compute　k　(edge)　disjoint　paths　between　s　and　t,　such　that　the　total　length　of　the　paths　is　minimized　and　the　weight　is　bounded　by　a　given　weight　budget　.　The　problem　is　known　to　be　-hard,　even　when　(Garey　and　Johnson　in　Computers　and　intractability,　1979).　Approximation　algorithms　with　bifactor　ratio　and　have　been　developed　for　in　Orda　and　Sprintson　(IEEE　INFOCOM,　pp.　727-738,　2004)　and　Chao　and　Hong　(IEICE　Trans　Inf　Syst　90(2):465-472,　2007),　respectively.　For　general　k,　an　approximation　algorithm　with　ratio　has　been　developed　for　a　weaker　version　of　kCSP,　the　k　bi-constraint　path　problem　which　is　to　compute　k　disjoint　st-paths　satisfying　a　given　length　constraint　and　a　weight　constraint　simultaneously　(Guo　et　al.　in　COCOON,　pp.　325-336,　2013).　This　paper　first　gives　an　approximation　algorithm　with　bifactor　ratio　for　kCSP　using　the　LP-rounding　technique.　The　algorithm　is　then　improved　by　adopting　a　more　sophisticated　method　to　round　edges.　It　is　shown　that　for　any　solution　output　by　the　improved　algorithm,　there　exists　a　real　number　such　that　the　weight　and　the　length　of　the　solution　are　bounded　by　times　and　times　of　that　of　an　optimum　solution,　respectively.　The　key　observation　of　the　ratio　proof　is　to　show　that　the　fractional　edges,　in　a　basic　solution　against　the　proposed　linear　relaxation　of　kCSP,　exactly　compose　a　graph　in　which　the　degree　of　every　vertex　is　exactly　two.　At　last,　by　a　novel　enhancement　of　the　technique　in　Guo　et　al.　(COCOON,　pp.　325-336,　2013),　the　approximation　ratio　is　further　improved　to　.