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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.
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,SwainR.
,RevelleC.
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,(1988)"ThelocationofJournalofTheoreticalandAppliedInformationTechnology2005-2009JATIT.
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jatit.
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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.
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Chu,"Ageneticalgorithmforthesetcoveringproblem",EuropeanJournalofOperationalResearch,vol.
94,1996,pp.
392-404.
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Agreedyheuristicforthesetcoveringproblem.
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