Minimal-costnetworkflowproblemswithvariablelowerboundsonarcflows

The minimal-costnetworkflowproblemwithfixedlowerandupperboundsonarcflowshasbeenwell studied.Thispaperinvestigatesanimportantextension,inwhichsomeorallarcshavevariablelower bounds.Inparticular,anarcwithavariablelowerboundisallowedtobeeitherclosed(i.e.,thenhaving zero flow)oropen(i.e.,thenhavingflowbetweenthegivenpositivelowerboundandanupperbound). This distinctivefeaturemakesthenewproblemNP-hard,althoughitsformulationbecomesmorebroadly applicable,sincetherearemanycaseswhereaflowdistributionchannelmaybeclosediftheflowonthe arc isnotenoughtojustifyitsoperation.Thispaperformulatesthenewmodel,referredtoasMCNF-VLB, as amixedintegerlinearprogramming,andshowsitsNP-hardcomplexity.Furthermore,anumerical exampleisusedtoillustratetheformulationanditsapplicability.Thispaperalsoshowsacomprehensive computationaltestingonusingCPLEXtosolvetheMCNF-VLBinstancesofuptomedium-to-largesize.