picksolved
solved 时间:2021-01-17 阅读:()
JournalofTheoreticalandAppliedInformationTechnology2005-2009JATIT.
Allrightsreserved.
www.
jatit.
org50A0-1MODELFORFIREANDEMERGENCYSERVICEFACILITYLOCATIONSELECTION:ACASESTUDYINNIGERIAAROGUNDADEO.
T.
,AKINWALEA.
T.
;ADEKOYAA.
F.
ANDAWEOLUDAREG.
DepartmentOfComputerScience,UniversityOfAgriculture,P.
M.
B2240,Abeokuta,OgunState,Nigeria.
E-mail:arogundade@acm.
org,atakinwale@yahoo.
com,lanlenge@yahoo.
comandaweolu@yahoo.
comABSTRACT:Facilitylocationselectionproblemisavariantofsetcoveringproblem.
Setcoveringproblemisaclassicalproblemincomputerscienceandcomplexitytheory.
Inthispapertwodifferenttechniquesareappliedtofacilitylocationproblems.
First,amathematicalmodeloffacilitylocationisintroducedandsolvedbyusingoptimizationsolver,TORA.
Secondly,thebalasadditivealgorithmofbranchandboundtechniquesisusedtosolvethefacilitylocationproblem.
TestsweremadeusingreallifedatafromacityinNigeria.
Wethenobservedthatbothalgorithmsindicatethesamenumberoffirestationsindifferentlocations.
Alsotheresultsobtainedbyapplyingandimplementingbalasadditiveweremoreexplanatorybyspecifyingthenamesofthelocationswherethefacilitiesaretobelocatedandthenamesofthelocationstobeservedbyeachofthefacilities.
Keywords:Setcoveringproblem,firestation,emergencyservice,branchandbound,integerlinearprogramming.
1INTRODUCTIONSetcoveringproblemisaclassicalproblemincomputerscienceandcomplexitytheory,andisoneofthemostimportantdiscreteoptimizationproblembecauseitservesasamodelforrealworldproblems.
Realworldproblemsthatcanbemodeledassetcoveringproblemincludeairlinecrewscheduling,nurseschedulingproblems,resourceallocation,assemblylinebalancing,vehiclerouting,facilitylocationproblemwhichisthemainfocusofthiswork.
Etc.
Setcoveringproblemisaproblemofcoveringtherowofanm-row/n-columnzero-onematrixwithasubsetofcolumnsatminimalcost[1].
ThesetcoverproblemisaclassicNP-hardproblemstudiedextensivelyinliterature,andthebestapproximationfactorachievableforitinpolynomialtimeis(logn)[2,3,4].
Arichliteraturehasbeendevelopedandseveralmodelshavebeenformulatedandappliedtothefacilitylocationproblemsoverthelastfewyears.
Thecomplexityoftheseproblemsisduetothemultitudesofquantitativeandqualitativefactorsinfluencinglocationchoices.
However,investigatorshavefocusedonbothalgorithmsandformulationindiversesettingintheprivatesector(e.
g.
industrialplants,retailfacilities,telecommunicationmastetc)andthepublicsectors(e.
g.
schools,healthcenters,ambulances,clinicsetc).
Inthiswork,ourinterestisononeofthepublicsectorfacilitylocationproblem,thefireandemergencyservicelocationproblem.
Infact,fireandemergencyserviceiscrucialinsavinglivesandvaluablepropertiesandthereforemustprovidehighlevelofqualityservicestoensurepublicsafety.
Butprovidingthesefacilitieseffectivelyisacomplexissuethatespeciallydependsonsomefactorsandmostespeciallyonthebestgeographicallocationofthefirefightingandemergenciesservicefacilities.
TheaimofthispaperthereforeistouseaSetCoveringmodeltoselecttheminimumfirestationsthatcouldserveallareasinabigcityinsuchawaythateachwardwillhaveequalbenefitsintermsofservicesfromthefirestationsandalsothefacilitywillbestrategicallyplaced.
TheprocessinvolvesgatheringdataaboutallthewardsinthecityusingtheGPS(GlobalPointSystem)soastogettheirdistancesfromeachotherusingGISsoftware(GeographicalInformationService).
WethendevelopedadecisionsupportsytemthatdeterminetheminimumnumberoffirestationsneededtoserveallthewardssuchthattheJournalofTheoreticalandAppliedInformationTechnology2005-2009JATIT.
Allrightsreserved.
www.
jatit.
org51distancebetweeneachwardandatleastonestationislessorequal10kilometersbysolvingthemathematicalmodelofthesetcoveringproblemusingtheBalasAdditivealgorithmaspecialcaseofbranchandboundthathandlesbinarylinearprogrammingproblem.
TheresultobtainedwascomparedtotheresultobtainedfromTORAsolver.
2LITERATUREREVIEWTheClassicalLocationSetCoveringProbleminvolvesfindingthesmallestnumberoffacilitiesandtheirlocationsothatdemandiscoveredbyatleastonefacility.
Itwasfirstintroducedby[12].
Theproblemrepresentseveraldifferentapplicationsettingincludingthelocationofemergencyserviceandtheapplicationsettingincludingthelocationofemergencyservicesandtheselectingofconservativesites.
Theproblemiscalledcoveringprobleminthatitrequiresthateachdemandbeservedor"covered"withinsomemaximumtimeanddistancestandards.
Ademandisdefinedascoveredifoneormorefacilitiesarelocatedwithinthemaximumdistanceortimestandardsofthatdemand.
ThesecondtypeofcoveringproblemiscalledtheMaximalCoveringLocationProblem[13].
Sincethedevelopmentofthesetwojuxtaposedproblemswereformed,therehavebeennumerousapplicationsandextensions.
SetCoveringProblemisoneofthemostprominentNP-completeproblem.
(Anexhaustivealgorithmmustsearchthroughall2msubsetsofStofindthosewhicharecoveringsubsetsandthenpicktheminimalfromamongthese[4]andcanformallybedefinedasfollow:Uistheuniversalset,SisacollectionofsubsetsofU,andc:S->Nisacostfunction.
ThegoalistofindacollectionS1,S2.
.
.
,SKofelementsofSsuchthatS1US2U.
.
.
USk=Uwithminimaltotalcost.
[16].
Significantresearchhasbeendirectedtowardstheproblemoflocatingandcoveringproblemsandseveralmethodshavebeenmadetoprovidesolutionsspecificallytothefacilitieslocationproblemandthesemethodsgenerallyinvolvetheuseofqueuingmodels[5],simulationandmathematicalprogramming,alsoacombinationofsimulationmodelandheuristicsearchroutines[6].
Alsoanextensivenumberofpapershavebeendedicatedtothesetcoveringproblem(SCP)andmanyexactalgorithms[7,10]whichcansolveinstanceswithuptofewhundredrowsandcolumns.
Acomparisonofsomeexactalgorithmscanbefoundin[9].
ApproximationalgorithmsplaysanimportantroleinsolvingSCP,giventhelimitationofexactmethodsandthelargelistofapplicationsusinglargesizeSCP[12].
Virtuallyeveryheuristicapproachforsolvinggeneralintegerproblemhasbeenappliedtosetcoveringproblems.
Thesetcoveringformulationnaturallylendsthemselvestogreedystart(i.
e.
anapproachthatateveryiterationmyopicallychoosesthenextbestsolutionwithoutregardsforitsimplicationonfuturemoves).
Interchangeapproacheshavealsobeenapplied;hereaswapofoneormorecolumnistakenwheneversuchaswapimprovestheobjectivefunctionvalue.
Newerheuristicapproachessuchasgeneticalgorithm,probabilisticsearch[8],simulatedannealing[11]andneuralnetworkhavealsobeentried.
Unfortunately,therehasnotbeenacomparativetestingacrosssuchmethodstodetermineunderwhatcircumstancesaspecificmethodmightperformbest.
Inaddition,onecanembedheuristicwithinanexactalgorithmsothatonecaniterativelytightentheupperboundandatthesametimeoneisattemptingtogetatightapproximationtothelowerboundforthisproblem.
Problemsarisinginpracticedonothoweverhaveperfectoridealmatrices.
Nevertheless,ithasbeenobservedincomputationalpracticethataslongastheproblemtobesolvedarerelativelyofmediumsize,linearprogrammingwithbranchandboundwillprovideintegersolutionquicklyandoptimally.
Howeverasthesubprogramsizeincreases,thenonintegralityofthelinearprogrammingsolutionincreasesdramaticallyanddoesthelengthandsizebranchingtree.
Itisforthislargeinstanceofproblemthatapproximationtechniques,reformulationandexactprocedureshavebeendevelopedthatexploittheunderlyingstructureoftheproblem.
IntegerLinearProgramming(ILPs)arelinearprogramsinwhichsomeorallofthevariablesarerestrictedtointeger(ordiscrete)values.
ILPhasimportantpracticalapplication.
Unfortunately,despitedecadesofextensiveresearch,computationalexperienceswithILPJournalofTheoreticalandAppliedInformationTechnology2005-2009JATIT.
Allrightsreserved.
www.
jatit.
org52havebeenlessthansatisfactory.
TodatetheredoesnotexistanILPcomputercodethatcansolveintegerlinearproblemsconsistently[15].
2.
1ProblemStatementConsiderafirestationlocationandallocationproblemhavingthefollowingfeatures:Afirestationlocatedinawardhastoserveasetofwards.
Eachwardtobeservedmustbelocatedatfixeddistancetothelocationofthefirestation.
Theminimumnumberoffirestationsthatcanserveallthewardsmustbedetermined.
Themathematicalmodelofthisproblemisformulatedasfollow.
MinZ=∑Cjxjj={1,2,…n}Subjectedto:∑aijxj≥1i={1,2,…m}xj={0,1}whereCjisthecostofinstallation,xirepresentsacoveringi.
xjwhichcantakethevalue0or1dependingonifwardiisincoveringxj.
3MethodologyThispaperaimstoobtainanoptimalsolutiontofireandemergencyfacilitieslocationproblem.
WeusetheGPS(GlobalPointSystem)equipmenttogetthecoordinatesofallthewardsinthecityunderconsideration.
Fromthescreenoftheequipment,wegottheNorth-axisandtheEast-axisofeveryparticularplacewevisited(37wards).
Afterthecollectionofthecoordinates,weinstalledtheGIS(GeographicalInformationSystem)softwareforanalysis.
WethensupplythecoordinatesofeachwardintotheGISwhichthenlocatethepositionofthewardsonthemapofOgunstate(seefigure1)andthereafterobtainedthedistancereadingsforeachwardtotheother.
Figure1.
WardslocationonthemapJournalofTheoreticalandAppliedInformationTechnology2005-2009JATIT.
Allrightsreserved.
www.
jatit.
org53Theresultobtainedfromthedistancereadingisa37by37matrixwhichwethentransformedintocoveringsaccordingtoaspecifieddistance(precisely10kmfromeachwards).
Forexample,thefirstcoverwhichis{1,2,3,4,5,6,7,8,9,10,15,21,25,26,27,28,29,30,31,32,33,34,35,36,37}indicatethosewardthatcanbecoveredwithintherangeof10kmfromward1.
Thepartoftheresultofthisprocessisshowninfigure2.
Accordingtoourfirstdefinitionofsetcoveringproblem,theuniversalsetUis{1,2…37}andF={C1,C2,…………….
C37},nowouraimistofindtheminimumSasubsetofFsuchthatitsunionwillgiveusU,andatthisstagethewardsareallcoveredwithequaldistancesandtheCipickedarethewardswherethefirestationshouldbelocated.
Thesedatawerethenslottedintothebalasadditiveandtorasolvertosolvethefacilitylocationproblem.
Theresultsfromthetwoalgorithmswerethencomparedtodeterminetheoptimalcase.
Figure2.
Wardscovering3.
1ModelsUsedToSolveFireAndEmergencyFacilityLocationProblem3.
1.
1.
BalasAdditiveAlgorithmTheadditivealgorithmwasoneoftheapproachesknownasbranchandboundandisusedtosolvelinearprogramsinn0-1variablesbysystematicallyenumeratingasubsetof2npossiblebinarynvectors,whileusingthelogicalimplicationofthe0-1propertytoensurethatthewholesetisimplicitlyexamined.
Thetechniqueemployedinthisalgorithmisbasedonsystematicallyassigningthevalue0and1tocertainsubsetofvariablesandexploringtheimplicationsoftheseassignmentsbyasequenceoflogicaltests.
Thesimplicityoftheprocedureanditseffectivenesswhendataarenottoolargemakesitabetterchoiceforthisresearchwork.
BalasAdditivealgorithmrequiredthattheproblembeputinstandardform:1.
{1,2,3,4,5,6,7,8,9,10,15,21,25,26,27,28,29,30,31,32,33,34,35,36,37}fromObantoko.
2.
{1,2,3,4,5,6,7,8,9,10,15,21,25,26,27,28,29,30,31,32,33,34,35,36,37}fromIkija3.
{1,2,3,4,5,6,7,8,9,10,15,21,25,26,27,28,29,30,31,32,33,34,35,36,37}fromAgoOko16{16,24}fromAlagbagba18{18,21,23}fromOsiele37{1,2,3,4,5,6,8,9,10,15,21,25,26,27,28,29,30,31,32,33,34,35,36,37}fromPansekeJournalofTheoreticalandAppliedInformationTechnology2005-2009JATIT.
Allrightsreserved.
www.
jatit.
org54TheobjectivefunctionisaformofminimizationThemconstraintsareallinequalitiesoftheform(≤)AllthevariablesxjarebinaryvariablesAllobjectivefunctioncoefficientsarenonnegativeAlgorithm:BalasAdditiveAlgorithm1Standardizetheproblemtotheform:MinZ=∑j∈NCjxjs.
t∑j∈Naijxj≤biforalli∈M.
whereM={1,2,…m}andN={1,2,…n}xj={0,1},forallj∈N2SetaninitialupperboundtoZ=+∞,seti=0,andJ={}.
3SelectthenextpartialsolutionJ,solvetheLPiofJandattempttofathomusingoneofthethreeconditionslistedbelow.
a.
Allcompletionviolatesoneormoreconstraints.
i.
ecomputei.
A={j:j∈N-J,aij≥0foralli∈MsuchthatSi≤0}ii.
N1=N–J–AIfNI={}thenfathomthepartialsolutionJb.
Allcompletionareinferiortotheincumbentz'i.
ecomputei.
B={j:j∈N1,Z+Cj≥Z'}ii.
N2=N1–BIfN2={}thenfathompartialsolutionJc.
IfconstraintiisviolatedbythezerocompletionofthepartialsolutionsothatSi{}thenfathomthepartialsolutionJIfallthefathomtestfail,Gotostep64.
Ifbettersolutionisfound,thenupdateZ5IfallelementsofJisfathomedi.
eunderlined,thenZisoptimalGotostep7ElsesetJJ,{-j}andrepeatfromstep36Performbranchingby:i)Selectfreevariableforforwardstepii)SetJJ,{+j}Seti=i+1andrepeatstep37Terminate3.
1.
2.
TORAOneofthepowerfulfeaturesofTORAisitsgraphicaluserinterface(GUI)whichenablesuserstoexpresstheirproblemsinanaturalwaythatisverysimilartostandardmathematicalnotation.
ThisfeatureofGUIallowsuserstochoosethenextactionbeingmenudriven.
Thisoffersflexibilitytouserstoincreaseordecreasethedatasizeortoremoveaparticularvariablecompletely.
TORAoptimizationsolverhasthefollowingattributes:a.
Sets,whichcompriseofobjectsinprogrammingmodelb.
Objectivefunctionoftheproblemc.
ConstraintsofProblemd.
inputdataJournalofTheoreticalandAppliedInformationTechnology2005-2009JATIT.
Allrightsreserved.
www.
jatit.
org554IMPLEMENTATIONANDRESULT4.
1FormatofinputdataInthispaper,theinputis38x380-1matrixwherecolumn2-38representseachcoveringandrow2-38representseachward.
Therefore,foreachcolumnandrow,theelementis1ifthewardiscoveredand0ifnotcovered.
E.
gthenameofthematrixisa,ifa[2][3]=1,itimpliesthatward3iscoveredbycovering2,otherwiseitisnotcoveredandthevaluewillbe0.
Thewholeinputfileformatforthisworkisshowninfigure4.
Theformatoftheoutputisinformofasolutionvectorcontainingonlyzerosandonesi.
e.
1ifacoveringisselectedand0ifnotselected.
Eachcoveringhasspecificnameofwardscoveringotherwardsthatarewithinthespecifieddistance.
(Thenameofeachwardandthenumberattachedtothemisshowninfigure3.
Figure3NamesofwardsandtheirnumberofidentificationTheinputmatrixshowninfigure4aandfigure4bwassavedastextfileandthebalasadditivealgorithmwasimplementedusingJavaprogramminglanguage.
Figure4aInputFileFormat1Obantoko,2Ikija,3Agooko,4ElegaHousing,5Iberekodo,6Agoika,7Ayetoro,8Okeago,9Totoro,10Itaosin,11Olorunda,12ImalaOrile,13IbaraOrile,14Ilewo/isaga,15Itaota,16Alagbagba,17Alabata,18Osiele,19Olodo,20Ilugun,21Agoodo,22Opeji,23Odeda,24Itesi,25Lafenwa,26Saje,27Itoko,28Ake,29Lantoro,30Ijemo,31Iporosodeke,32Irunbe,33Ijaye,34Okeitoku,35IjehunTitun,36Sabo,37Panseke.
JournalofTheoreticalandAppliedInformationTechnology2005-2009JATIT.
Allrightsreserved.
www.
jatit.
org56Figure4bInputFileFormatOncetheinputfilehasbeenselected,andthentheprogramcanberuntogeneratetheoutputrequired.
Theresultaftertheclickofthe"run"buttonisshowninfigure5below.
Figure5SetCoveringoutput.
JournalofTheoreticalandAppliedInformationTechnology2005-2009JATIT.
Allrightsreserved.
www.
jatit.
org57Thebalasadditiveresultaboveshowedthatcoveringisfoundandalsodisplayedthesolutionvector.
Itindicatedthatsixfirestationsareneededtoserveallthewardsandthelocationsofthosestationsareclearlystated.
ThesameresultwasobtainedfromTORAsoftwareintermsofoptimality,butthelocationsaredifferentandnotclearlystatedthoughitcanbetracedout.
Figure6ResultfromTORAsolverFigure6showstheresultoftheTORAsolver,theresultsissuchthatsixfirestationsarealsoneededtoserveallthewardseffectivelybutthelocationsofthefireservicestationarequitedifferentfromthatofBALASalgorithmthatwasimplemented.
Thelocationsindicatedbythesolverarelocations11,14,17,19,21,24whichcorrespondstothenamesofthefollowingwards(Olorunda,IlewoIsaga,alabata,Olodo,Ago-OdoandItesi)asshowninfigure3.
Torasolverdoesnotlistthenamesornumbersofthevillagesinitscovering.
5DISCUSSIONThenecessityofthedevelopmentoffacilitieslocationsoftwareforenhancingthedecisionmakingprocessandeventuallyproductivitycannotbeover-emphasized.
Theresultsobtainedinthisworkshowedthatsixfirestationsareneededtoserveeverywardsuchthatthemaximumdistancethatafirestationservicecangois10kilometers.
ItalsoshowedthatthelocationofthestationshouldbeObantoko,Olorunda,Ibaraorile,Olodo,OpejiandOdeda.
ThefirestationsatObantokowillrenderservicestotwentyfivewardswhichare:Obantoko,Ikija,Ago-oko,Elega,iberekodo,Ago-Ika,Ayetoro,Oke-ago,totoro,Ita-osin,Ita-ota,Ago-odo,lafenwa,Saje,Itoko,Ake,Lantoro,Ijemo,Iporo-sodeke,irunbe,Ijaye,Oke-itoku,Ijeun-titun,SaboandPanseke.
Olorundaservicestationwillservetwowardswhichare:OlorundaandImala-Orile.
Ibara-OrileservicestationwillserveIbara-orileandIlewo-Isagarespectively.
Olodoservicestationwillservethreewards.
Theyare:alagbagba,OlodoandIlugun.
Opejiwillservefourwardswhichare:alabata,OsieleandOpeji.
FinallyodedastationwillserveOdedaandItesi.
ThoughtheresultfromTORAsolveralsoindicatedthatsixfirestationsareneeded,itdidnotspecifytheactuallocationswherethestationsshouldbelocated.
ThisshortcominginTORAmakesouroutputandimplementationabetterone.
Theseresultsarepresentedinthetablebelowformoreclarity.
Itshowsthelocationswherethefacilitiesaretobeinstalledandalsothevillagestobecoveredbyeachofthefacility(coverings)onlyforbalasadditivealgorithm.
Theoutputfrombalasadditivealgorithmdoesnotshowfairdistribution.
InthecaseofthefacilityinlocationObantokowhichistoserve25locationswhileothersserveminimumoftwolocationsandmaximumofthree.
ThefacilityinObantokowillbeoverused.
JournalofTheoreticalandAppliedInformationTechnology2005-2009JATIT.
Allrightsreserved.
www.
jatit.
org58Table1:OUTPUTOFBALASADDITIVEALGORITHMLocationidentificationnumberLocationnameCoverings1Obantoko1,2,3,4,5,6,7,8,9,10,15,21,25,26,27,28,29,30,31,32,33,34,35,36,3711Olorunda11,1213Ibaraorile13,1419Olodo16,19,2022Opeji17,18,2223Odeda23,24TABLE2:OUTPUTOFTORASOLVERLocationIdentificationNumberLocationNameCoverings11OlorundaNotindicated14Ilewo-IsagaNotindicated17AlabataNotindicated19OlodoNotindicated21Ago-odoNotindicated24ItesiNotindicated6CONCLUSIONSANDRECOMMENDATIONSThisresultthereforeshouldraiseawarenessandcontributetotheaimofourgovernmenttoadoptthistoolwhichwilldefinitelyimprovethefunctionalityoffirestationsinNigeriabysavingalotofcitizen'slivesandproperties.
Itshouldalsobenotedthattheuseofthissystemisnotlimitedonlytofirestationsallocationalone,butalsotootherpublicfacilitieslikeschools,policestationsoastoincreaseresponsetimeandthereforereducecrime.
Itcanalsobeusedbyprivateestablishments.
REFERENCES[1]J.
EBeasleyandP.
C.
Chu,"Ageneticalgorithmforthesetcoveringproblem",EuropeanJournalofOperationalResearch,vol.
94,1996,pp.
392-404.
[2]V.
Chvatal.
Agreedyheuristicforthesetcoveringproblem.
MathematicsofOperationResearch,4(3):233-235,1979.
[3]U.
Fiege.
Athresholdforlnnforapproximatingsetcover.
JournaloftheACM,45(4):634-652,July1998.
[4]D.
S.
Johnson.
Approximationalgorithmsforcombinatorialproblems.
J.
Compute.
SystemScience9:256-278,1974.
[5]LarsonRichardC.
,"AHypercubeQueuingModelforfacilitylocationandredistrictinginurbanemergencyservices",ComputersandOperationsResearch1(1974)67-95.
[6]SavasES,"Simulationandcost-effectivenessanalysisofNewYork'semergencyambulanceservice",ManagementScience14(1969)608-627.
[7]M.
L.
FisherandP.
Kedia.
OptimalSolutionofsetcoveringproblemsusingdualheuristics.
ManagementScience,36:674-688,1990.
[8]FeoA.
andG.
C.
MauricioandA.
Resende,(2002)"AProbabilisticHeuristicforaComputationallyDifficultSetCoveringProblem",OperationsResearchLetters,8,67-71.
[9]A.
Caparara,M.
Fischetti,andP.
Toth.
AlgorithmsforsetcoverinproblemstechnicalreportOR-98-3,DEIS,UniversityofBologna,Italy1998.
[10]N.
ChristofidesandJ.
PPaixao.
Algorithmsforlargescalesetcoveringproblems.
AnnalsofOperationResearch,43:261-277,1993.
[11]M.
JBrusco,L.
W.
Jacobs,andG.
MThompson.
Amorphingproceduretosupplementasimulatedannealingheuristicsforcostandcoverage-correlatedsetcoveringproblems.
AnnalsofOperationResearch,86:611-627,1999.
[12]ToregasC.
,SwainR.
,RevelleC.
,andBergmanL.
,(1988)"ThelocationofJournalofTheoreticalandAppliedInformationTechnology2005-2009JATIT.
Allrightsreserved.
www.
jatit.
org59emergencyservicefacilities",Operationresearch19,pg1363-1373.
[13]ChurchRichardL.
,GerardRossa.
,(2003)"Themultilevellocationsetcoveringmodel"GeographicalAnalysisPublication,pg76-79.
[14]GareyM.
R.
andJohnsonD.
S.
,(2005)"Computerandinteractability:aguidetothetheoryofNP-completeness",pg34-38.
[15]Handya.
Taha.
,(2005)'OperationResearch:Anintroduction",Pearsoneducation(Singapore)Pte.
Ltd.
,pg391-397.
[16]Danielgulotta,(2006)"ApplicationoflinearProgrammingtosetcoveringandrelatedproblems"OnlinejournalofOperationResearch,pg1-2.
Allrightsreserved.
www.
jatit.
org50A0-1MODELFORFIREANDEMERGENCYSERVICEFACILITYLOCATIONSELECTION:ACASESTUDYINNIGERIAAROGUNDADEO.
T.
,AKINWALEA.
T.
;ADEKOYAA.
F.
ANDAWEOLUDAREG.
DepartmentOfComputerScience,UniversityOfAgriculture,P.
M.
B2240,Abeokuta,OgunState,Nigeria.
E-mail:arogundade@acm.
org,atakinwale@yahoo.
com,lanlenge@yahoo.
comandaweolu@yahoo.
comABSTRACT:Facilitylocationselectionproblemisavariantofsetcoveringproblem.
Setcoveringproblemisaclassicalproblemincomputerscienceandcomplexitytheory.
Inthispapertwodifferenttechniquesareappliedtofacilitylocationproblems.
First,amathematicalmodeloffacilitylocationisintroducedandsolvedbyusingoptimizationsolver,TORA.
Secondly,thebalasadditivealgorithmofbranchandboundtechniquesisusedtosolvethefacilitylocationproblem.
TestsweremadeusingreallifedatafromacityinNigeria.
Wethenobservedthatbothalgorithmsindicatethesamenumberoffirestationsindifferentlocations.
Alsotheresultsobtainedbyapplyingandimplementingbalasadditiveweremoreexplanatorybyspecifyingthenamesofthelocationswherethefacilitiesaretobelocatedandthenamesofthelocationstobeservedbyeachofthefacilities.
Keywords:Setcoveringproblem,firestation,emergencyservice,branchandbound,integerlinearprogramming.
1INTRODUCTIONSetcoveringproblemisaclassicalproblemincomputerscienceandcomplexitytheory,andisoneofthemostimportantdiscreteoptimizationproblembecauseitservesasamodelforrealworldproblems.
Realworldproblemsthatcanbemodeledassetcoveringproblemincludeairlinecrewscheduling,nurseschedulingproblems,resourceallocation,assemblylinebalancing,vehiclerouting,facilitylocationproblemwhichisthemainfocusofthiswork.
Etc.
Setcoveringproblemisaproblemofcoveringtherowofanm-row/n-columnzero-onematrixwithasubsetofcolumnsatminimalcost[1].
ThesetcoverproblemisaclassicNP-hardproblemstudiedextensivelyinliterature,andthebestapproximationfactorachievableforitinpolynomialtimeis(logn)[2,3,4].
Arichliteraturehasbeendevelopedandseveralmodelshavebeenformulatedandappliedtothefacilitylocationproblemsoverthelastfewyears.
Thecomplexityoftheseproblemsisduetothemultitudesofquantitativeandqualitativefactorsinfluencinglocationchoices.
However,investigatorshavefocusedonbothalgorithmsandformulationindiversesettingintheprivatesector(e.
g.
industrialplants,retailfacilities,telecommunicationmastetc)andthepublicsectors(e.
g.
schools,healthcenters,ambulances,clinicsetc).
Inthiswork,ourinterestisononeofthepublicsectorfacilitylocationproblem,thefireandemergencyservicelocationproblem.
Infact,fireandemergencyserviceiscrucialinsavinglivesandvaluablepropertiesandthereforemustprovidehighlevelofqualityservicestoensurepublicsafety.
Butprovidingthesefacilitieseffectivelyisacomplexissuethatespeciallydependsonsomefactorsandmostespeciallyonthebestgeographicallocationofthefirefightingandemergenciesservicefacilities.
TheaimofthispaperthereforeistouseaSetCoveringmodeltoselecttheminimumfirestationsthatcouldserveallareasinabigcityinsuchawaythateachwardwillhaveequalbenefitsintermsofservicesfromthefirestationsandalsothefacilitywillbestrategicallyplaced.
TheprocessinvolvesgatheringdataaboutallthewardsinthecityusingtheGPS(GlobalPointSystem)soastogettheirdistancesfromeachotherusingGISsoftware(GeographicalInformationService).
WethendevelopedadecisionsupportsytemthatdeterminetheminimumnumberoffirestationsneededtoserveallthewardssuchthattheJournalofTheoreticalandAppliedInformationTechnology2005-2009JATIT.
Allrightsreserved.
www.
jatit.
org51distancebetweeneachwardandatleastonestationislessorequal10kilometersbysolvingthemathematicalmodelofthesetcoveringproblemusingtheBalasAdditivealgorithmaspecialcaseofbranchandboundthathandlesbinarylinearprogrammingproblem.
TheresultobtainedwascomparedtotheresultobtainedfromTORAsolver.
2LITERATUREREVIEWTheClassicalLocationSetCoveringProbleminvolvesfindingthesmallestnumberoffacilitiesandtheirlocationsothatdemandiscoveredbyatleastonefacility.
Itwasfirstintroducedby[12].
Theproblemrepresentseveraldifferentapplicationsettingincludingthelocationofemergencyserviceandtheapplicationsettingincludingthelocationofemergencyservicesandtheselectingofconservativesites.
Theproblemiscalledcoveringprobleminthatitrequiresthateachdemandbeservedor"covered"withinsomemaximumtimeanddistancestandards.
Ademandisdefinedascoveredifoneormorefacilitiesarelocatedwithinthemaximumdistanceortimestandardsofthatdemand.
ThesecondtypeofcoveringproblemiscalledtheMaximalCoveringLocationProblem[13].
Sincethedevelopmentofthesetwojuxtaposedproblemswereformed,therehavebeennumerousapplicationsandextensions.
SetCoveringProblemisoneofthemostprominentNP-completeproblem.
(Anexhaustivealgorithmmustsearchthroughall2msubsetsofStofindthosewhicharecoveringsubsetsandthenpicktheminimalfromamongthese[4]andcanformallybedefinedasfollow:Uistheuniversalset,SisacollectionofsubsetsofU,andc:S->Nisacostfunction.
ThegoalistofindacollectionS1,S2.
.
.
,SKofelementsofSsuchthatS1US2U.
.
.
USk=Uwithminimaltotalcost.
[16].
Significantresearchhasbeendirectedtowardstheproblemoflocatingandcoveringproblemsandseveralmethodshavebeenmadetoprovidesolutionsspecificallytothefacilitieslocationproblemandthesemethodsgenerallyinvolvetheuseofqueuingmodels[5],simulationandmathematicalprogramming,alsoacombinationofsimulationmodelandheuristicsearchroutines[6].
Alsoanextensivenumberofpapershavebeendedicatedtothesetcoveringproblem(SCP)andmanyexactalgorithms[7,10]whichcansolveinstanceswithuptofewhundredrowsandcolumns.
Acomparisonofsomeexactalgorithmscanbefoundin[9].
ApproximationalgorithmsplaysanimportantroleinsolvingSCP,giventhelimitationofexactmethodsandthelargelistofapplicationsusinglargesizeSCP[12].
Virtuallyeveryheuristicapproachforsolvinggeneralintegerproblemhasbeenappliedtosetcoveringproblems.
Thesetcoveringformulationnaturallylendsthemselvestogreedystart(i.
e.
anapproachthatateveryiterationmyopicallychoosesthenextbestsolutionwithoutregardsforitsimplicationonfuturemoves).
Interchangeapproacheshavealsobeenapplied;hereaswapofoneormorecolumnistakenwheneversuchaswapimprovestheobjectivefunctionvalue.
Newerheuristicapproachessuchasgeneticalgorithm,probabilisticsearch[8],simulatedannealing[11]andneuralnetworkhavealsobeentried.
Unfortunately,therehasnotbeenacomparativetestingacrosssuchmethodstodetermineunderwhatcircumstancesaspecificmethodmightperformbest.
Inaddition,onecanembedheuristicwithinanexactalgorithmsothatonecaniterativelytightentheupperboundandatthesametimeoneisattemptingtogetatightapproximationtothelowerboundforthisproblem.
Problemsarisinginpracticedonothoweverhaveperfectoridealmatrices.
Nevertheless,ithasbeenobservedincomputationalpracticethataslongastheproblemtobesolvedarerelativelyofmediumsize,linearprogrammingwithbranchandboundwillprovideintegersolutionquicklyandoptimally.
Howeverasthesubprogramsizeincreases,thenonintegralityofthelinearprogrammingsolutionincreasesdramaticallyanddoesthelengthandsizebranchingtree.
Itisforthislargeinstanceofproblemthatapproximationtechniques,reformulationandexactprocedureshavebeendevelopedthatexploittheunderlyingstructureoftheproblem.
IntegerLinearProgramming(ILPs)arelinearprogramsinwhichsomeorallofthevariablesarerestrictedtointeger(ordiscrete)values.
ILPhasimportantpracticalapplication.
Unfortunately,despitedecadesofextensiveresearch,computationalexperienceswithILPJournalofTheoreticalandAppliedInformationTechnology2005-2009JATIT.
Allrightsreserved.
www.
jatit.
org52havebeenlessthansatisfactory.
TodatetheredoesnotexistanILPcomputercodethatcansolveintegerlinearproblemsconsistently[15].
2.
1ProblemStatementConsiderafirestationlocationandallocationproblemhavingthefollowingfeatures:Afirestationlocatedinawardhastoserveasetofwards.
Eachwardtobeservedmustbelocatedatfixeddistancetothelocationofthefirestation.
Theminimumnumberoffirestationsthatcanserveallthewardsmustbedetermined.
Themathematicalmodelofthisproblemisformulatedasfollow.
MinZ=∑Cjxjj={1,2,…n}Subjectedto:∑aijxj≥1i={1,2,…m}xj={0,1}whereCjisthecostofinstallation,xirepresentsacoveringi.
xjwhichcantakethevalue0or1dependingonifwardiisincoveringxj.
3MethodologyThispaperaimstoobtainanoptimalsolutiontofireandemergencyfacilitieslocationproblem.
WeusetheGPS(GlobalPointSystem)equipmenttogetthecoordinatesofallthewardsinthecityunderconsideration.
Fromthescreenoftheequipment,wegottheNorth-axisandtheEast-axisofeveryparticularplacewevisited(37wards).
Afterthecollectionofthecoordinates,weinstalledtheGIS(GeographicalInformationSystem)softwareforanalysis.
WethensupplythecoordinatesofeachwardintotheGISwhichthenlocatethepositionofthewardsonthemapofOgunstate(seefigure1)andthereafterobtainedthedistancereadingsforeachwardtotheother.
Figure1.
WardslocationonthemapJournalofTheoreticalandAppliedInformationTechnology2005-2009JATIT.
Allrightsreserved.
www.
jatit.
org53Theresultobtainedfromthedistancereadingisa37by37matrixwhichwethentransformedintocoveringsaccordingtoaspecifieddistance(precisely10kmfromeachwards).
Forexample,thefirstcoverwhichis{1,2,3,4,5,6,7,8,9,10,15,21,25,26,27,28,29,30,31,32,33,34,35,36,37}indicatethosewardthatcanbecoveredwithintherangeof10kmfromward1.
Thepartoftheresultofthisprocessisshowninfigure2.
Accordingtoourfirstdefinitionofsetcoveringproblem,theuniversalsetUis{1,2…37}andF={C1,C2,…………….
C37},nowouraimistofindtheminimumSasubsetofFsuchthatitsunionwillgiveusU,andatthisstagethewardsareallcoveredwithequaldistancesandtheCipickedarethewardswherethefirestationshouldbelocated.
Thesedatawerethenslottedintothebalasadditiveandtorasolvertosolvethefacilitylocationproblem.
Theresultsfromthetwoalgorithmswerethencomparedtodeterminetheoptimalcase.
Figure2.
Wardscovering3.
1ModelsUsedToSolveFireAndEmergencyFacilityLocationProblem3.
1.
1.
BalasAdditiveAlgorithmTheadditivealgorithmwasoneoftheapproachesknownasbranchandboundandisusedtosolvelinearprogramsinn0-1variablesbysystematicallyenumeratingasubsetof2npossiblebinarynvectors,whileusingthelogicalimplicationofthe0-1propertytoensurethatthewholesetisimplicitlyexamined.
Thetechniqueemployedinthisalgorithmisbasedonsystematicallyassigningthevalue0and1tocertainsubsetofvariablesandexploringtheimplicationsoftheseassignmentsbyasequenceoflogicaltests.
Thesimplicityoftheprocedureanditseffectivenesswhendataarenottoolargemakesitabetterchoiceforthisresearchwork.
BalasAdditivealgorithmrequiredthattheproblembeputinstandardform:1.
{1,2,3,4,5,6,7,8,9,10,15,21,25,26,27,28,29,30,31,32,33,34,35,36,37}fromObantoko.
2.
{1,2,3,4,5,6,7,8,9,10,15,21,25,26,27,28,29,30,31,32,33,34,35,36,37}fromIkija3.
{1,2,3,4,5,6,7,8,9,10,15,21,25,26,27,28,29,30,31,32,33,34,35,36,37}fromAgoOko16{16,24}fromAlagbagba18{18,21,23}fromOsiele37{1,2,3,4,5,6,8,9,10,15,21,25,26,27,28,29,30,31,32,33,34,35,36,37}fromPansekeJournalofTheoreticalandAppliedInformationTechnology2005-2009JATIT.
Allrightsreserved.
www.
jatit.
org54TheobjectivefunctionisaformofminimizationThemconstraintsareallinequalitiesoftheform(≤)AllthevariablesxjarebinaryvariablesAllobjectivefunctioncoefficientsarenonnegativeAlgorithm:BalasAdditiveAlgorithm1Standardizetheproblemtotheform:MinZ=∑j∈NCjxjs.
t∑j∈Naijxj≤biforalli∈M.
whereM={1,2,…m}andN={1,2,…n}xj={0,1},forallj∈N2SetaninitialupperboundtoZ=+∞,seti=0,andJ={}.
3SelectthenextpartialsolutionJ,solvetheLPiofJandattempttofathomusingoneofthethreeconditionslistedbelow.
a.
Allcompletionviolatesoneormoreconstraints.
i.
ecomputei.
A={j:j∈N-J,aij≥0foralli∈MsuchthatSi≤0}ii.
N1=N–J–AIfNI={}thenfathomthepartialsolutionJb.
Allcompletionareinferiortotheincumbentz'i.
ecomputei.
B={j:j∈N1,Z+Cj≥Z'}ii.
N2=N1–BIfN2={}thenfathompartialsolutionJc.
IfconstraintiisviolatedbythezerocompletionofthepartialsolutionsothatSi{}thenfathomthepartialsolutionJIfallthefathomtestfail,Gotostep64.
Ifbettersolutionisfound,thenupdateZ5IfallelementsofJisfathomedi.
eunderlined,thenZisoptimalGotostep7ElsesetJJ,{-j}andrepeatfromstep36Performbranchingby:i)Selectfreevariableforforwardstepii)SetJJ,{+j}Seti=i+1andrepeatstep37Terminate3.
1.
2.
TORAOneofthepowerfulfeaturesofTORAisitsgraphicaluserinterface(GUI)whichenablesuserstoexpresstheirproblemsinanaturalwaythatisverysimilartostandardmathematicalnotation.
ThisfeatureofGUIallowsuserstochoosethenextactionbeingmenudriven.
Thisoffersflexibilitytouserstoincreaseordecreasethedatasizeortoremoveaparticularvariablecompletely.
TORAoptimizationsolverhasthefollowingattributes:a.
Sets,whichcompriseofobjectsinprogrammingmodelb.
Objectivefunctionoftheproblemc.
ConstraintsofProblemd.
inputdataJournalofTheoreticalandAppliedInformationTechnology2005-2009JATIT.
Allrightsreserved.
www.
jatit.
org554IMPLEMENTATIONANDRESULT4.
1FormatofinputdataInthispaper,theinputis38x380-1matrixwherecolumn2-38representseachcoveringandrow2-38representseachward.
Therefore,foreachcolumnandrow,theelementis1ifthewardiscoveredand0ifnotcovered.
E.
gthenameofthematrixisa,ifa[2][3]=1,itimpliesthatward3iscoveredbycovering2,otherwiseitisnotcoveredandthevaluewillbe0.
Thewholeinputfileformatforthisworkisshowninfigure4.
Theformatoftheoutputisinformofasolutionvectorcontainingonlyzerosandonesi.
e.
1ifacoveringisselectedand0ifnotselected.
Eachcoveringhasspecificnameofwardscoveringotherwardsthatarewithinthespecifieddistance.
(Thenameofeachwardandthenumberattachedtothemisshowninfigure3.
Figure3NamesofwardsandtheirnumberofidentificationTheinputmatrixshowninfigure4aandfigure4bwassavedastextfileandthebalasadditivealgorithmwasimplementedusingJavaprogramminglanguage.
Figure4aInputFileFormat1Obantoko,2Ikija,3Agooko,4ElegaHousing,5Iberekodo,6Agoika,7Ayetoro,8Okeago,9Totoro,10Itaosin,11Olorunda,12ImalaOrile,13IbaraOrile,14Ilewo/isaga,15Itaota,16Alagbagba,17Alabata,18Osiele,19Olodo,20Ilugun,21Agoodo,22Opeji,23Odeda,24Itesi,25Lafenwa,26Saje,27Itoko,28Ake,29Lantoro,30Ijemo,31Iporosodeke,32Irunbe,33Ijaye,34Okeitoku,35IjehunTitun,36Sabo,37Panseke.
JournalofTheoreticalandAppliedInformationTechnology2005-2009JATIT.
Allrightsreserved.
www.
jatit.
org56Figure4bInputFileFormatOncetheinputfilehasbeenselected,andthentheprogramcanberuntogeneratetheoutputrequired.
Theresultaftertheclickofthe"run"buttonisshowninfigure5below.
Figure5SetCoveringoutput.
JournalofTheoreticalandAppliedInformationTechnology2005-2009JATIT.
Allrightsreserved.
www.
jatit.
org57Thebalasadditiveresultaboveshowedthatcoveringisfoundandalsodisplayedthesolutionvector.
Itindicatedthatsixfirestationsareneededtoserveallthewardsandthelocationsofthosestationsareclearlystated.
ThesameresultwasobtainedfromTORAsoftwareintermsofoptimality,butthelocationsaredifferentandnotclearlystatedthoughitcanbetracedout.
Figure6ResultfromTORAsolverFigure6showstheresultoftheTORAsolver,theresultsissuchthatsixfirestationsarealsoneededtoserveallthewardseffectivelybutthelocationsofthefireservicestationarequitedifferentfromthatofBALASalgorithmthatwasimplemented.
Thelocationsindicatedbythesolverarelocations11,14,17,19,21,24whichcorrespondstothenamesofthefollowingwards(Olorunda,IlewoIsaga,alabata,Olodo,Ago-OdoandItesi)asshowninfigure3.
Torasolverdoesnotlistthenamesornumbersofthevillagesinitscovering.
5DISCUSSIONThenecessityofthedevelopmentoffacilitieslocationsoftwareforenhancingthedecisionmakingprocessandeventuallyproductivitycannotbeover-emphasized.
Theresultsobtainedinthisworkshowedthatsixfirestationsareneededtoserveeverywardsuchthatthemaximumdistancethatafirestationservicecangois10kilometers.
ItalsoshowedthatthelocationofthestationshouldbeObantoko,Olorunda,Ibaraorile,Olodo,OpejiandOdeda.
ThefirestationsatObantokowillrenderservicestotwentyfivewardswhichare:Obantoko,Ikija,Ago-oko,Elega,iberekodo,Ago-Ika,Ayetoro,Oke-ago,totoro,Ita-osin,Ita-ota,Ago-odo,lafenwa,Saje,Itoko,Ake,Lantoro,Ijemo,Iporo-sodeke,irunbe,Ijaye,Oke-itoku,Ijeun-titun,SaboandPanseke.
Olorundaservicestationwillservetwowardswhichare:OlorundaandImala-Orile.
Ibara-OrileservicestationwillserveIbara-orileandIlewo-Isagarespectively.
Olodoservicestationwillservethreewards.
Theyare:alagbagba,OlodoandIlugun.
Opejiwillservefourwardswhichare:alabata,OsieleandOpeji.
FinallyodedastationwillserveOdedaandItesi.
ThoughtheresultfromTORAsolveralsoindicatedthatsixfirestationsareneeded,itdidnotspecifytheactuallocationswherethestationsshouldbelocated.
ThisshortcominginTORAmakesouroutputandimplementationabetterone.
Theseresultsarepresentedinthetablebelowformoreclarity.
Itshowsthelocationswherethefacilitiesaretobeinstalledandalsothevillagestobecoveredbyeachofthefacility(coverings)onlyforbalasadditivealgorithm.
Theoutputfrombalasadditivealgorithmdoesnotshowfairdistribution.
InthecaseofthefacilityinlocationObantokowhichistoserve25locationswhileothersserveminimumoftwolocationsandmaximumofthree.
ThefacilityinObantokowillbeoverused.
JournalofTheoreticalandAppliedInformationTechnology2005-2009JATIT.
Allrightsreserved.
www.
jatit.
org58Table1:OUTPUTOFBALASADDITIVEALGORITHMLocationidentificationnumberLocationnameCoverings1Obantoko1,2,3,4,5,6,7,8,9,10,15,21,25,26,27,28,29,30,31,32,33,34,35,36,3711Olorunda11,1213Ibaraorile13,1419Olodo16,19,2022Opeji17,18,2223Odeda23,24TABLE2:OUTPUTOFTORASOLVERLocationIdentificationNumberLocationNameCoverings11OlorundaNotindicated14Ilewo-IsagaNotindicated17AlabataNotindicated19OlodoNotindicated21Ago-odoNotindicated24ItesiNotindicated6CONCLUSIONSANDRECOMMENDATIONSThisresultthereforeshouldraiseawarenessandcontributetotheaimofourgovernmenttoadoptthistoolwhichwilldefinitelyimprovethefunctionalityoffirestationsinNigeriabysavingalotofcitizen'slivesandproperties.
Itshouldalsobenotedthattheuseofthissystemisnotlimitedonlytofirestationsallocationalone,butalsotootherpublicfacilitieslikeschools,policestationsoastoincreaseresponsetimeandthereforereducecrime.
Itcanalsobeusedbyprivateestablishments.
REFERENCES[1]J.
EBeasleyandP.
C.
Chu,"Ageneticalgorithmforthesetcoveringproblem",EuropeanJournalofOperationalResearch,vol.
94,1996,pp.
392-404.
[2]V.
Chvatal.
Agreedyheuristicforthesetcoveringproblem.
MathematicsofOperationResearch,4(3):233-235,1979.
[3]U.
Fiege.
Athresholdforlnnforapproximatingsetcover.
JournaloftheACM,45(4):634-652,July1998.
[4]D.
S.
Johnson.
Approximationalgorithmsforcombinatorialproblems.
J.
Compute.
SystemScience9:256-278,1974.
[5]LarsonRichardC.
,"AHypercubeQueuingModelforfacilitylocationandredistrictinginurbanemergencyservices",ComputersandOperationsResearch1(1974)67-95.
[6]SavasES,"Simulationandcost-effectivenessanalysisofNewYork'semergencyambulanceservice",ManagementScience14(1969)608-627.
[7]M.
L.
FisherandP.
Kedia.
OptimalSolutionofsetcoveringproblemsusingdualheuristics.
ManagementScience,36:674-688,1990.
[8]FeoA.
andG.
C.
MauricioandA.
Resende,(2002)"AProbabilisticHeuristicforaComputationallyDifficultSetCoveringProblem",OperationsResearchLetters,8,67-71.
[9]A.
Caparara,M.
Fischetti,andP.
Toth.
AlgorithmsforsetcoverinproblemstechnicalreportOR-98-3,DEIS,UniversityofBologna,Italy1998.
[10]N.
ChristofidesandJ.
PPaixao.
Algorithmsforlargescalesetcoveringproblems.
AnnalsofOperationResearch,43:261-277,1993.
[11]M.
JBrusco,L.
W.
Jacobs,andG.
MThompson.
Amorphingproceduretosupplementasimulatedannealingheuristicsforcostandcoverage-correlatedsetcoveringproblems.
AnnalsofOperationResearch,86:611-627,1999.
[12]ToregasC.
,SwainR.
,RevelleC.
,andBergmanL.
,(1988)"ThelocationofJournalofTheoreticalandAppliedInformationTechnology2005-2009JATIT.
Allrightsreserved.
www.
jatit.
org59emergencyservicefacilities",Operationresearch19,pg1363-1373.
[13]ChurchRichardL.
,GerardRossa.
,(2003)"Themultilevellocationsetcoveringmodel"GeographicalAnalysisPublication,pg76-79.
[14]GareyM.
R.
andJohnsonD.
S.
,(2005)"Computerandinteractability:aguidetothetheoryofNP-completeness",pg34-38.
[15]Handya.
Taha.
,(2005)'OperationResearch:Anintroduction",Pearsoneducation(Singapore)Pte.
Ltd.
,pg391-397.
[16]Danielgulotta,(2006)"ApplicationoflinearProgrammingtosetcoveringandrelatedproblems"OnlinejournalofOperationResearch,pg1-2.
哪里购买香港云服务器便宜?易探云2核2G低至18元/月起;BGP线路年付低至6.8折
哪里购买香港云服务器便宜?众所周知，国内购买云服务器大多数用户会选择阿里云或腾讯云，但是阿里云香港云服务器不仅平时没有优惠，就连双十一、618、开年采购节这些活动也很少给出优惠。那么，腾讯云虽然海外云有优惠活动，但仅限新用户，购买过腾讯云服务器的用户就不会有优惠了。那么，我们如果想买香港云服务器，怎么样购买香港云服务器便宜和优惠呢？下面，云服务器网(yuntue.com)小编就介绍一下！我们都知道...
80VPS：香港服务器月付420元;美国CN2 GIA独服月付650元;香港/日本/韩国/美国多IP站群服务器750元/月
80vps怎么样？80vps最近新上了香港服务器、美国cn2服务器，以及香港/日本/韩国/美国多ip站群服务器。80vps之前推荐的都是VPS主机内容，其实80VPS也有独立服务器业务，分布在中国香港、欧美、韩国、日本、美国等地区，可选CN2或直连优化线路。如80VPS香港独立服务器最低月付420元，美国CN2 GIA独服月付650元起，中国香港、日本、韩国、美国洛杉矶多IP站群服务器750元/月...
PIGYun月付14.4元起，美国洛杉矶/韩国VPS七月6折
PIGYun是成立于2019年的国人商家，提供香港、韩国和美西CUVIP-9929等机房线路基于KVM架构的VPS主机，本月商家针对韩国首尔、美国洛杉矶CUVIP-AS29、GIA回程带防御等多条线路VPS提供6-8.5折优惠码，优惠后韩国首尔CN2混合BGP特惠型/美国洛杉矶GIA回程带10Gbps攻击防御VPS主机最低每月14.4元起。下面列出几款不同机房VPS主机配置信息，请留意不同优惠码。...
solved为你推荐
-
域名注册商最全的域名注册商服务器空间租用租用服务器一年我要交多少钱php虚拟主机免费的国内的php虚拟主机有吗中国互联网域名注册负责我国境内internet用户域名注册是什么机构虚拟主机租用虚拟主机服务器租用要怎么选择？域名服务域名服务器是什么？虚拟主机申请在哪里可以申请到虚拟主机呢免备案虚拟空间虚拟免费空间网站怎么备案重庆虚拟空间现在重庆那家主机空间最好？网站空间商哪有好一点的网站空间商？欢迎友友们给我推荐下，