In　this　paper,　we　consider　the　lower-bounded　k-median　problem　(LB　k-median)　that　extends　the　classical　k-median　problem.　In　the　LB　k-median,　a　set　of　facilities,　a　set　of　clients　and　an　integer　k　are　given.　Every　facility　has　its　own　lower　bound　on　the　minimum　number　of　clients　that　must　be　connected　to　the　facility　if　it　is　opened.　Every　facility-client　pair　has　its　connection　cost.　We　want　to　open　at　most　k　facilities　and　connect　every　client　to　some　opened　facility,　such　that　the　total　connection　cost　is　minimized.　As　our　main　contribution,　we　study　the　LB　k-median　and　present　our　main　bi-criteria　approximation　algorithm,　which,　for　any　given　constant,　outputs　a　solution　that　satisfies　the　lower　bound　constraints　by　a　factor　of　and　has　an　approximation　ratio　of,　where　is　the　state-of-art　approximation　ratio　for　the　k-facility　location　problem　(k-FL).　Then,　by　extending　the　main　algorithm　to　several　general　versions　of　the　LB　k-median,　we　show　the　versatility　of　our　algorithm　for　the　LB　k-median.　Last,　through　providing　relationships　between　the　constant　and　the　approximation　ratios,　we　demonstrate　the　performances　of　all　the　algorithms　for　the　LB　k-median　and　its　generalizations.　©　2020,　Springer　Nature　Switzerland　AG.