We　consider　a　network　operation　under　which　multiple　simultaneously　activated　routes　are　employed　to　distribute　multimedia　traffic　between　identified　source-destination　pairs.　We　aim　to　jointly　determine　the　path　selection　and　rate　allocation,　per　each　flow,　across　multi-path　routes,　to　ensure　the　utility　obtained　each　flow　is　as　high　as　feasibly　possible　while　striving　to　raise　the　utility　of　all　flows　in　a　max-min　fair　manner.　In　doing　so,　incorporate　the　communications　capacity　constraints　that　are　imposed　by　the　underlying　hybrid　of　directional　and/or　multiple-access　wireline　and　wireless　communications　media　employed　across　the　network　system.　We　develop　and　present　in　this　paper　an　optimal　algorithm　for　solving　such　complex　problem　that　involves　multiple　sequential　convex/linear　programs.　As　illustrative　examples,　we　demonstrate　the　effectiveness　of　our　solution　in　comparing　its　performance　with　that　obtained　under　the　use　of　a　traffic　regulation　scheme　that　is　not　overlaid　with　a　traffic　management　mechanism　that　serves　to　set-aside　resources　for　the　support　of　flows.