We　propose　and　study　the　(MUFLP)-U-p　(universal　facility　location　problem　in　the　p-th　power　of　metric　space)　in　this　paper,　where　the　universal　facility　location　problem　(UFLP)　extends　several　classical　facility　location　problems　like　the　uncapacitated　facility　location,　hard-capacitated　facility　location,　soft-capacitated　facility　location,　incremental-cost　facility　location,　concave-cost　facility　location,　etc.　In　UFLP,　a　set　of　facilities,　a　set　of　clients,　as　well　as　the　distances　between　them　are　given.　Each　facility　has　its　specific　cost　function　w.r.t.　the　amount　of　clients　assigned　to　that　facility.　The　goal　is　to　assign　the　clients　to　facilities　such　that　the　sum　of　facility　cost　and　service　cost　is　minimized.　In　traditional　facility　location　problems,　the　unit　service　cost　is　proportional　to　the　distance　between　the　client　and　its　assigned　facility　and　thus　metric.　However,　in　our　work,　this　assumption　is　removed　and　a　generalized　version　of　universal　facility　location　problem　is　proposed,　which　is　the　so-called　(MUFLP)-U-n.　When　p　=　2,　it　is　also　known　as　l(2)(2)　measure　considered　by　Jain　and　Vazirani　[J.　ACM＇01　]　and　Fernandes　et　al.　[Math.　Program.＇15].　Particularly　in　this　case,　we　extend　their　work　to　include　the　aforementioned　variants　of　facility　location　and　a　local　search　based　(11.18　+　epsilon)-approximation　algorithm　is　proposed.　Furthermore,　the　reanalysis　of　the　proposed　algorithm　gives　a　p-related　performance　guarantee　for　general　p.　(C)　2020　Elsevier　B.V.　All　rights　reserved.