Three local search meta-heuristics for the Minimum Interference Frequency Assignment Problem (MI-FAP) in cellular networks,

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Interference Frequency Assignment Problem (MI-FAP)

in cellular networks,

Yasmine Lahsinat, Dalila Boughaci, Belaid Benhamou

To cite this version:

Yasmine Lahsinat, Dalila Boughaci, Belaid Benhamou. Three local search meta-heuristics for the Minimum Interference Frequency Assignment Problem (MI-FAP) in cellular networks,. Interna-tional Journal of Applied Metaheuristic Computing, IGI Global Publisher, 2019, 10 (3), pp.134-150. �10.4018/IJAMC.2019070107�. �hal-02098923�

DOI: 10.4018/IJAMC.2019070107



Copyright©2019,IGIGlobal.CopyingordistributinginprintorelectronicformswithoutwrittenpermissionofIGIGlobalisprohibited.

Three Local Search Meta-Heuristics

for the Minimum Interference

Frequency Assignment Problem

(MI-FAP) in Cellular Networks

Yasmine Lahsinat, LRIA-FEI- Computer Science Department, USTHB, BP 32 El-Alia Bab-Ezzouar, 16111, Algiers, Algeria Dalila Boughaci, LRIA-FEI- Computer Science Department, USTHB, BP 32 El-Alia Bab-Ezzouar, 16111, Algiers, Algeria Belaid Benhamou, AIX-Marseille University (AMU), LIS, Domaine Universitaire de Saint Jérôme Avenue Escadrille Normandie Niemen 13397 Marseille Cedex 20, Marseille, France

ABSTRACT Theminimuminterferencefrequencyassignmentproblem(MI-FAP)playsanimportantroleincellular networks.MI-FAPistheproblemoffindinganassignmentofasmallnumberoffrequenciestoalarge numberoftransceivers(TRXs)thatminimizestheinterferenceslevel.TheMI-FAPisknowntobe NP-Hard,thusitcannotbesolvedinpolynomialtime.Toremedythis,researchersusuallyusemeta-heuristictechniquestofindanapproximatesolutioninreasonabletime.Here,theauthorspropose threemeta-heuristicsfortheMI-FAP:avariableneighborhoodsearch(VNS)andastochasticlocal search(SLS)thatarecombinedtoobtainathirdandanewone,whichiscalledVNS-SLS.TheSLS methodisincorporatedintotheVNSprocessasasubroutineinordertoenhancethesolutionquality. Allthreeproposedmethodsareevaluatedonsomewell-knowndatasetstomeasuretheirperformance. TheempiricalexperimentsshowthattheproposedmethodVNS-SLSsucceedsinfindinggoodresults comparedtobothVNSandSLSconfirmingagoodbalancebetweenintensificationanddiversification. KeywoRdS

Assignment Problem, Local Search Methods, Minimum Interference Frequency, Radio Network, Stochastic Local Search (SLS), Variable Neighbourhood Search (VNS), VNS-SLS

1. INTRodUCTIoN Managingtheinterferencesinaradionetworkisarelevantissueconsideringthattheycanseverely damagethequalityofthecommunication.Inthispaper,weproposetostudythisissuewitharelated problemarisinginacellularradionetworknamelytheMinimumInterferenceFrequencyAssignment Problem.TheMI-FAPisawell-knownchallengingcombinatorialoptimizationproblem.Theissue istoassignafrequencytoeverytransceiverofthenetwork.Thedifficultiesappearduetothesetwo facts:theradioresourceisscarce,andthetremendousnumberoftransceiversdistributedoverthecell’s network.Therefore,itisunavoidabletoreusethefrequenciesmultipletimesbetweenthetransceivers

withinthenetwork.Butthiswillcauseinterferences.Indeed,whenstationsinclosegeographical areausethesameoradjacentfrequency,interferencesoccur.Toavoidthis,itisnecessarytoassign frequenciestoallthegeographicallycloseTRXsrespectingsomerequiredseparationconstraints. Inacellularradionetwork,theobjectiveisthentoassignfrequenciestoTRXswithinthedifferent cellsinsuchawaytominimizethetotalinterferences.Itisworthrecallingthesetwofacts:theTRXsin thesamecellscannotusethesamefrequencyandsomefrequenciesarenotoperationalinsomecells. Inthiswork,weconsiderthetwomajortypesofinterferences: • TheCo-channelinterferences:theseinterferencesappearwhentwogeographicallycloseTRXs usethesamefrequency(f); • TheAdjacent-channelinterferences:theseinterferencesappeariftwogeographicallycloseTRXs useneighboring(adjacent)frequencies(f+1,f-1). Intheeighties,HaledemonstratedthattheconsideredproblemisNP-hard.Heprovedthatthe MI-FAPisrelatedtothegraph-coloringproblem(Hale,1980).Therefore,thefrequencyassignment isanimportantandacomplextaskwhenplanningaradionetwork.Consideringthelimitedradio resource,thetremendousnumberofTRXs,aparticularattentionisgiventoachievethistaskavoiding oratleastminimizingtheinterferences. TheMI-FAPcannotbesolvedinpolynomialtime.Thus,usingmeta-heuristictechniquesoffers agoodcompromisetofindapproximatesolutionsfortheprobleminareasonabletime.Several meta-heuristicshavebeenproposedintheliterature.Thesemethodshaveshowntheirabilitytosolve variousproblemsthroughyears.Theyhadbeenusedtohandleschedulingproblem(Taghavifard, 2012),bioinformaticsproblem(Zemali,2016),bigdata(Benmounah,2017)andmanyotherproblems. Inthispaper,weproposethreemeta-heuristicsearchmethodsfortheMI-FAP.First,weadaptand studytheeffectivenessoftheVariableNeighborhoodSearch(VNS)fortheMI-FAP.Second,we proposeaStochasticLocalSearch(SLS)methodfortheMI-FAP.Finally,weproposetomergethese twolastmethodsinanewmethodwhichwecalledVNS-SLS.ThenewmethodVNS-SLS,includes theSLSmethodasalocalsearchsteptoattempttoenhancethesolutionquality.SinceVNShas gooddiversificationproperties,integratingSLSinit,willlogicallyresultinagoodbalancebetween intensificationanddiversification.WeexpectthenthatVNS-SLSwillhavethisfeaturewhichis essentialforproducinggoodresults.Wenotethatcollaborativeandhybridmethodsareworthwhile toensureasuccessofoptimizationtools.Therearetremendousworksthatusedhybridapproaches tohandleoptimizationproblems.Amongthem,wementionsomerecentworks(Wang,2017), (Boucheham,2017),(Abunaser,2015). Therestofthispaperisorganizedasfollows:section2givesabackgroundontheconsidered problem.Insection3,weproposeabriefliteraturereview.Section4explainsthekeycomponents ofthethreeproposedmethodsforsolvingtheMI-FAP.Section5presentssomeresultsofvarious testsandexperimentations.Finally,section6concludestheworkandgivessomefutureperspectives. 2. BACKGRoUNd WepresentinthissectiontheformaldescriptionoftheMI-FAPusedinthispaper.Letusconsidera radionetworkNcomposedofasetofcells:C = {C₁, C₂ …, C_i, …, C_nc},wherencisthenumberofcells. EachcellC_ihavingx_iTRXsrequiresx_idifferentfrequencies.Thenetworksmanageafixednumber

NFoffrequencies.Indeed,eachnetworkhasitsowninterval[F_min, F_max]ofoperationalfrequencies

with|F_min- F_max | = NF.Foreachcell,wehaveasetofoperationalfrequencies.Thismeansthatsome

cellscouldsufferfromsomeeventualrestrictionsandcannotusesomeofthefrequencies.Theco-channelandadjacentchannelconstraintsarerepresentedbytwomatricesdenotedIM_corespectively IM_adjthataredefinedasfollows:

• IM_co [nc][nc]:Containstheco-channelratebetweeneachpairofcellsofthenetwork.Anelement

IM_co[i][j]oftheIMcomatrixcorrespondstotherateofco-channelinterferencewhenaTRXof

thecelljusesthesamechannel(f)thanthatoneofaTRXtofthecellI;

• IM_adj [nc][nc]:Containstheadjacent-channelratebetweeneachpairofcellsofthenetwork.An

elementIMadj [i][j]oftheIMadjmatrixcorrespondstotherateoftheAdjacent-channelinterference

whenaTRXofthecelljusesanadjacentchanneltoaTRXtofthecelli.

ThesetwomatricesIMcoandIMadjcontaintheinteractionbetweencellswhentheTRXsusethe

sameoradjacentchannel.Theinterferencesareexpressedasrates(theirvaluesarebetween0and1). Wenotethat,area-basedandtraffic-basedaretwomainwaystorateinterferences.Inthispaper, weconsiderthearea-basedwaywheretheinterferencedescribestheaffectedareaandanelement

IM_co[i][j](orIMadj [i][j])describesthepercentageofpixelsintheservicearea(Kuurne,2001)of

celliaffectedbythecellj. TheMI-FAPhandlesdvariableswhereeachvariablecorrespondstoafrequencyassignedtoa TRXiwithinacellj.Thevalueofafrequencyispickedfromtherange1toNF. TheMI-FAPisanoptimizationproblemthatconsistsinfindingafrequencyassignmentforthe dTRXsofthenetworkthatminimizesthecostfunctionproposedin(Kuurne,2001)andgivenin thefollowingformula: i i nc j j nc ij co ij adj Y IM i j Z IM i j = = = =

∑∑

(

)

• Yij:isthenumberofdifferentpairsoftransceiversofthecellCiandCjthatoperatesonthe

samefrequency;

• Zij:isthenumberofdifferentpairsoftransceiversofthecellCiandCjthatoperatesonan

adjacentfrequency(f), (f+1/ f-1). 3. ReLATed woRKS Thefrequencyassignmentproblem(FAP)hasapracticalinterestinbothrealandtheoreticalaspects. Asthequalityoftheserviceofawirelesssystemdependshighlyontheoptimalmanagementofthe radioresource,theFAPisacoreprobleminthemodernwirelesssystems.Ithasrelevanceinmany fieldssuchasmilitaryandcivilapplications.TheFAPisachallengingcombinatorialoptimization problemthatdealswiththetuningofatremendousnumberoftransceivers,theirpositions,their frequenciesandtheinterferencesthatcouldoccur. ThepioneerwhointroducedthefrequencyassignmentproblemisMetzger.Hepublishedhis researchinthisfieldinthesixties(Metzger,1970).Sincethat,manyresearchersinvestigatedthis problem(Aardal,2007;Eisenblatter,2002;Leese,2002).SeveralformulationsoftheFAPhadbeen proposed.Thesurveypublishedin2007by(Aardal,2007)summarizedthedifferentformulations andobjectivesassociatedwiththeFAP.Hesaidthatingeneral,theobjectiveassociatedwiththe FAPistheminimizationofthenumberoffrequencies,therangeofthespectrum(span),orthetotal interferences.Therefore,dependingontheobjectivefunctiontooptimize,theFAPcanbeclassified asfollows(Aardal,2007):

• TheMinimumOrderFrequencyAssignment(MO-FAP)attempttominimizethenumberofthe frequenciesused; • TheMinimumSpanFrequencyAssignment(MS-FAP)attempttoreducetheintervalof frequenciesandgetassmallaspossiblethedifferencebetweenthehighestandthelowest frequency.Mostoftheearlyworksweredevotedtothisgoal(Mabed,2011); • TheMinimumBlockingFrequencyAssignment(MB-FAP)attempttoassignfrequenciesin suchawaythatnounacceptableinterferenceoccursandtheoverallblockingprobabilityofthe networkisminimized.Moreformallytheproblemisdefined(Eisenblätter,2000);

• The Minimum Interference Assignment Problem (MI-FAP) consists in minimizing the overallinterferences. Regardlessofitsformulation,theFAPrapidlygrewinpopularityandattractedmanyresearchers overthelastfiftyyears.WefocusinthisworkontheMI-FAPvariant.TheMI-FAPisaninteresting problemofoperationalresearcharisinginradionetworkswheretheissueistoassignfrequencies toeachTRXwiththeminimumlevelofinterferences.Hale(Hale,1980)modeledtheproblemasa graphcoloringproblemanddemonstratedthattheconsideredproblemisanNP-hard. Severalworksproposedthemeta-heuristicframeworktohandledifferentdatasetsoftheMI-FAP. AmongthemtheonesthatadoptedevolutionaryapproachessuchasEAsandAntColonyoptimization algorithms(ACO/EAs)proposedin2007(Luna,2007),ortheadaptationoftheculturalalgorithm (Alami,2008)publishedin2008.AnotherresearchstudiedtheparticleswarmtohandletheFAP (Bezuidenhout,2014).ArecentworkaddressedtheMI-FAPwithaHarmonySearch(Lahsinat,2017). Further,thereareworksthatadoptedsomesinglebasedsolutionmeta-heuristicmethodssuchasthe tabusearchalgorithmtosolvetheproblem(Dorne,1999),(MontemanniR.M.,2003),(Montemanni, 2010),(Lai,2015).Someotherworksthatdealtwiththeproblembyusingthesimulatedannealing principle(Duque-Anton,1993),(Beckmann,1999),(Colombo,2010).Otherworksadoptedhybrid methods(Luna,2011),(Mabed,2011)tosolvetheproblem.

4. THe PRoPoSed APPRoACHeS

Inthissection,wepresentthemainelementsoftheproposedapproaches.Wefirstdescribetheshared elements:thesolutionencoding,theobjectivefunctionmeasuringthequalityofthesolutionsandthe methodologyadoptedtoupdatethesolutionsmanipulatedbyallthetechniques.Thenwedescribe indetailthethreeproposedmethodsSLS,VNSandVNS-SLS.

4.1. The solution encoding

Animportantstepwhendesigningameta-heuristicisthesolutionencoding.Indeed,encodingrightly asolutioncaninfluencetheperformancesofthealgorithms.WeconsiderasasolutiontotheMI-FAPafrequencyplanthatminimizestheinterferences.AsolutionisexpressedbyavectorSolofd elementswithdthetotalnumberofTRXs.EachelementfromSolrepresentsthefrequencyassigned totheKth_{TRXofacelli.Eachfrequencyisrepresentedbyanumberfrom1toNF:} Sol←{f1;f2….fd} 4.2. Updating Strategy Typically,fortheMI-FAP,theupdatingschemeconsistsinchangingthefrequencyassignedtothe TRXs.Thestrategyadoptedmustconsiderthesetwoimportantelementaryconditions:donotassign thesamefrequencytoTRXswithinthesamecellanddonotassignanon-operationalfrequency. Thus,weadoptanupdatingstrategythatguaranteesbeforeperforminganychangesoffrequency

thatthenewfrequenciesarevalidandtherequiredseparationbetweentheassignedfrequenciesto TRXsinthesamecellisrespected.

4.3. The objective Function

Thepurposehereistoreachasolutionthatwouldminimizethelevelofinterferences.Inotherwords, wemustfindanassignmentofthefrequencythatminimizestheobjectivefunctionencodedbythe formulaalreadygiveninEquation(1).

4.4. The Variable Neighborhood Search for MI-FAP

ThevariableNeighborhoodSearch(VNS)algorithmisasinglesolutionorientedmeta-heuristic proposedin1997byMladenovicandHansen(Mladenovic,1997).Sincethenvariousvariantsof VNSwereproposed,butthefundamentalideaisthesystematicchangeofneighborhoodcombined withalocalsearch.Thismethodconfirmeditsperformancetosolveseveraloptimizationproblems (Mladenovic,1997). InordertouseVNSalgorithm,wemustdefinesomeelements: 1. AsolutionthattheVNSmanipulates; 2. AneighborhoodstructurethatshouldbedefinedasN_k(k=1…k=k_max); 3. Alocalsearchroutine; 4. Astoppingcondition,whichcouldbethemaximumnumberofiterationsorthemaximum CPUtime. InthespecificcaseofMI-FAP,theelementsoftheVNSaredefiningasfollows: • A Solution:Correspondstoafrequencyplan; • A Neighbor solution:Isasolutiongeneratedbyamovefromtheinitialsolution.Themoveis theoperatorthatmodifiestheassignmentofoneormoreTRXsintheinitialsolution,depending onthevalueofthek_max; • A Local Search:Weadoptedabasicone.Itstartswithaninitialsolution.Inthislocalsearch, wechoosetogeneratetheneighborbymodifyingtheconflictingelement:wefindthefrequency assignedtoaTRX,thatcausesthemostinterferenceswithitsneighborsandwechangeits frequencytothelessusedfrequency.Thenwetrytoimproveituntilastoppingcriterionismet. Thevariableneighborhoodsearch(VNS)processconsistsinthreemainphases:

• Initialization phase:Definesthesetofneighborhoodstructure Nkandgeneratestheinitial

solution x ;

• Main phase:(1)Setk :=1.(2)Untilk :=kmaxrepeatthefollowingsteps:a)Ashakingstep:

inthisstep,theprocessrandomlygenerates ′x ,with_x x_N k ′ ∈ .b)Alocalsearchstep:in thissteptheprocessappliesalocalsearchontheobtainedneighborx’toobtainalocally optimalsolution ′′x ; • Acceptance phase:Basedonthesolutionquality,decideseithertoaccepttheobtainedsolution ormovetothenextneighborhoodstructure.Formally:

◦ If ′′x isbetterthanthecurrentsolutionx set x := ′′x andrestarttheprocessfromthefirst neighborhoodstructureN1withx’’asaninitialsolution;

Fortheconsideredproblem,asolutioncorrespondstoafrequencyplan.Theinitialsolutionisgenerated randomly.Aneighborsolution x ' ofthesolution x ,isdefinedasasolutionwithkdecisionvariables havingdifferentvaluescomparingtox .FortheMI-FAP,thismeansthatbetweenthesolution x andits neighbor x ' wehavektransceiverswithdifferentfrequencies.Theneighboringsolutionisobtainedby applyingamovefromtheinitialsolution.Thismoveisanoperatorthatmodifiestheassignmentof frequenciestooneormoreTRXsintheinitialsolution x .Thenumberoffrequenciestomodifyinthe solutionx dependsonthecurrentvalueofNk.Thevalueofkmaxcorrespondstothemaximumnumber

ofdifferentfrequenciesassignedtoTRXsbetweenthesolutionx anditsneighborx ' . ThepseudocodeofVNSisgiveninAlgorithm1.

4.5. The Stochastic Local Search for MI-FAP

TheStochasticlocalsearchmethodsareefficientsearchtechniquestotacklealargerangeof combinatorialproblems.Theycoveredalargeclassofalgorithms.Wedistinguishbetweensimple iterativeimprovementmethodsandcomplexSLSmethods,suchasAntColonyOptimizationand EvolutionaryAlgorithms(HolgerH.Hoos,2004).HoosandStutzlementionthatmanyhigh-performanceSLSalgorithmsarebasedonacombinationofseveralsimplesearchstrategies, suchasIterativeBestImprovementandRandomWalk.TheconsideredSLSusedinthispaper probabilisticallymakesachoicebetweenarandomwalkandMin-Conflictsinspiredfromthe onedevelopedtohandletheoptimalwinnerdeterminationproblem(WDP)incombinatorial auctionspublishedin(Boughaci,2008). TheproposedSLStechniquefortheMI-FAPstartswithaninitialsolutionx andtriestofinda bettersolutioninthecurrentneighborhood.Theneighboringsolutionx ' ofthesolutionx isobtained bymodifyingthefrequencyassignedtoaselectedTRX.TheperturbedTRXisselectedaccording tooneofthetwofollowingcriteria:

Algorithm 1. The VNS for the MI-FAP

Require:aMI-FAPinstance,kmax,asolutionx,cost:theoverallvalueofinterferences.

Ensure:animprovedsolutionx

1:while(thestoppingcriterionisnotmet)do 2:k←1;

3:while(k≤kmax)do

4:x’←arandomneighborhoodsolution; 5:ApplyLocalSearchonx’toobtainx”; 6:ifcost(x”)<cost(x)then

7:x←x”;k←1; 8:else 9:k←k+1; 10:endif 11:endwhile 12:endwhile.

• ThefirstcriterionconsistsinchoosingtheTRXinarandomwaywithafixedprobabilitywp>0; • Thesecondcriterionconsistsinchoosingtheconflictingnode:wefindtheassignmentthat causesthemostinterferencewithitsneighborsandthenwemakeamoveonitbyassigning tothecorrespondingTRXthelessusedfrequency.Theprocessisrepeateduntilwemet stoppingcriterion. ThepseudocodeofSLSforMI-FAPisgiveninAlgorithm2.

4.6. A New Local Search Method VNS-SLS for MI-FAP

Inthissection,wepresentthemaincomponentsoftheproposednewapproachVNS-SLSthatcombines boththeVariableNeighborhoodSearch(VNS)andtheStochasticLocalSearch(SLS)methods.The VNS-SLSisaniterativeprocedure,whichstartsfromaninitialsolution x (i.eafrequencyassignment plan)generatedrandomly,andmovesstepbystepinthesearchspacewiththeaimtoimprovethe initialsolution.Ateachiteration,amovetoaneighborof x isperformed.Themoveconsistsin changingtheassignmentoffrequencytoaTRXselectedintheneighborhoodofx .Thepathbetween theinitialsolutionx andthegeneratedneighborisfixedwiththek_max.

TheSLSstepintheVNS-SLSmethodisalocalsearchprocessthatisusedtoimprovethe currentsolution.ThisSLSprocessusesaprobabilitytocontrolintensificationanddiversification. WeintegratedthisinVNSbecauseStochasticlocalsearch(SLS)algorithmsareamongthemost prominentandsuccessfultechniquesforsolvingseveralcomputationallydifficultproblemsinmany areasofcomputerscienceandoperationsresearch,includingpropositionalsatisfiability,constraint satisfaction,routing,andscheduling(Hoos,2004). Moreprecisely,inthiswork,weadoptedasimpleSLSasdescribedinsection4.5.Itallowstwo choices:(1)arandomwalkand(2)amin-conflictalgorithmusingaprobabilityparameter.Thesetwo choicesmakeabalancebetweenintensificationanddiversification.Indeed,thepuremin-conflict algorithmalonewillleadtoalocaloptimum.Thus,toavoidbeingstuckinalocaloptimum,weintroduce anoisethrougharandomwalk.Ittakesthegeneratedneighborof x andtriestoimproveit.

Algorithm 2. The SLS for MI-FAP

Require:AMI-FAPinstance,wp,asolutionx. Ensure:Animprovedsolutionx 1:while(Stoppingcriterionisnotmet)do 2:r←arandomnumberbetween0and1; 3:if(r<wp)then 4:pickarandomTRX 5:else 6:PickaconflictingTRX 7:endif 8:ModifyinxthefrequencyassignedtotheselectedTRXtoobtainx ' 9:x←x ' 10:endwhile

Thesolutionx '' obtainedattheendoftheSLSprocesswillbethecurrentsolutionintheVNS processwhenx '' isbetterthan x .Otherwise,theprocessrestartswiththenewneighborhood.The processisrepeateduntilastoppingcriterionismet. TheVNS-SLSmethodforMI-FAPissketchedinAlgorithm3. 5. CoMPUTATIoNAL eXPeRIMeNTS Inthissection,wepresentthecomputationalexperimentsachievedtocheckouttheperformances ofthethreemethods.TheexperimentsarecarriedonPChavinganIntelCorei5with4GBofRAM underWindows7.WedevelopedthemethodsusingtheJavalanguageprogramming.

5.1. The Considered datasets

Weevaluatedourmethodsonseveralpublicinstances.Duetothenon-deterministicnatureofthe threemethodsVNS,SLSandVNS-SLS,weconsider10runsbyinstanceandbyalgorithm.All instanceswerepickedupfromthecost259Benchmarksthatareavailableon(Eisenblätter,2000).We evaluatedVNS,SLS,andVNS-SLSonsevendifferentinstancesoftheconsideredproblemnamely: Tiny,K,Swisscom,Siemens1,Siemens2,Siemens3,andSiemens4.TheCOSTmeansEuropean UnionForumforcooperativescientificresearch(Eisenblatter,2002).TheCOST259containsseveral realisticradionetworkplanningscenarios. Wegivesomedetailsontheinstancestackled(Eisenblätter,2000): Siemens 01:Contains506cells,930TRXs,74differentfrequenciesavailablewithatotalnumberof 20417Co-channelinterferencesand10344adjacent-channelinterferencestobeavoided.

Algorithm 3. The VNS-SLS for MI-FAP

Require:anMI-FAPinstance,kmax,asolutionx,cost:istheoverallinterferences

Ensure:Animprovedsolutionx

1:while(Stoppingcriterionisnotmet)do 2:k←1;

3:while(k≤kmax)do

4:x '←arandomneighborhoodsolutionofx; 5:ApplySLSonx 'toobtain ’’x ;

6:ifcost(x '')<cost(x)then 7:x←x '';k←1; 8:else 9:k←k+1; 10:endif 11:endwhile 12:endwhile;

Siemens 02:Includes254cells,977TRXs,82differentavailablefrequencieswithatotalnumberof 30982Co-channelinterferencesand13970adjacent-channelinterferences. Siemens 03:Contains894cells,1623TRXs,55differentavailablefrequencieswithatotalnumber of63893Co-channelinterferencesand25105adjacentchannelinterferences. Siemens 04:Has760cells,2785TRXs,38differentavailablefrequencies,113658Co-channel interferences,and64341adjacentchannelinterferencestobeavoided. Swisscom:Composedby148cells,310TRXs,47availablefrequenciesandatotalnumberof535 adjacentchannelinterferences. K:264cells,267TRXs,49frequencies,27123Co-channelinterferences,and3199adjacentchannel interferencestobeavoided. Tiny:7cells,12TRXs,12frequencies,12Co-channelinterferencesconstraints,and9adjacent channelinterferencestobeavoided.

5.2. The Parameter Tuning

Forafaircomparisonamongthethreealgorithms,werunallthetechniquesusingthesameCPU time(1800seconds).Theadjustmentofthedifferentparametersoftheproposedapproachesisfixed byanempiricalstudy:

• TheVNSparameters:kmax=4,CPUtimeforthelocalsearchsub-routine(LS)=10seconds;

• TheSLSparameters:wp=0.25;

• TheVNS-SLSparameters:kmax=4,wp=0.25,CPUtimeforSLS=10seconds. 5.3. The effect of k_max

InordertoproposeanefficientVariableNeighborhoodSearchalgorithm,wededicateourfirstsets ofexperimentstofindtheadequatevalueofthek_maxparameterwhichisveryimportantintheVNS algorithm.Weexaminedfivepossiblevaluesofk_max=2,3,4,5,6,andachievesomeexperiments byusingtwoinstancesasshowninTable1.

TheresultspresentedinTable1indicatethatthebestsolutionsarefoundwhenusingak_max=4. Itisobviousthatincreasingthesizeoftheneighborhoodhasapositiveeffectontheperformances. Indeed,itbringsenoughdiversitythatavoidsbeingstuckinalocaloptimum.However,ifthesizeis toolargeitaffectsnegativelythequalityofthesolution.

5.4. The obtained Results

Table2givesthenumericalresultsofalloftheVNS,SLSandVNS-SLSmethodsonvariousinstances oftheMI-FAPproblem.Themean,thebest,theworstcostvalueandtheStandarddeviation(Std) resultsfoundbyeachmethodaregiven.Themeanvaluecorrespondstothesolutionqualityfound byeachalgorithmon10runs.Thebestandtheworstvaluesarerespectivelythebestandworst solutionsobtainedbyeachalgorithmon10runs.Thebestresults,intermofmeanandbestvalue, areinboldfont. TheresultsfromTable2showthatVNS-SLSandSLSmethodssucceedinfindinggoodresults foralmosttheconsideredinstances.TheyoutperformtheVNSalgorithmfortheSiemensandK

Table 1. Results obtained over two instances with different values of kmax

Instances VNS-1 (k_max = 2) VNS-2 (k_max = 3) VNS-3 (k_max = 4) VNS-4 (k_max = 5) VNS-5 (k_max = 6)

Siemens01 89.71 88.46 82.18 88.8 90

instances.WecanseethatthethreemethodsarecomparableontheTinyandSwisscominstances. Thisisduetothenatureofthesetwoproblemswhicharesmallinstances. FortheTinyinstance,weobservethatthethreemethodsfindtheoptimalsolution(i.ethebest=0). BothVNS-SLSandSLSfindtheoptimalsolutionforthe10performedrunsontheSwisscominstance. FortheKinstance,weobservethatSLSandVNS-SLSarecomparablewithaslightperformance infavoroftheSLSmethod.Forthisinstance,wedepictinFigure1theevolutionofthecostvalue overthe30minutesoftheexperimentforthethreemethods.ThecurveshowsthatVNS-SLSis fasterthanVNS.Wecanseethatforthesametime,theresultachievedbyVNS-SLSisbetterthan thestandardVNSwithalocalsearch.ThismeansthatthecombinationofSLSwithinVNSenhances theperformancesofVNSintermofboththetimeandthequalitypointofview.TheSLSallows escapingfromsomebadregionswhichleadstobetterresults. Forthedifficultinstances,theresultsobtainedbyVNS-SLSandSLSarecomparable.However, fortheSiemens1,Siemens3andSiemens4instances,wecanseeaslightperformanceinfavor ofVNS-SLS.Thestandarddeviationisrelativelysmall.Inordertoshowclearlytheperformance oftheproposedapproachesinsolvingMI-FAP,wedrawthebarplotsinFigure2.ThisFigure comparesthethreemethodsVNS,VNS-SLSandSLSintermofqualityofsolutionontheSiemens andKinstances.Wecanseethat,thetwomethodsSLSandVNS-SLSexploretheneighborsearch efficiently.ThisallowslocatinggoodqualitysolutioncomparingtotheVNS.WhenusingtheSLS

Table 2. The results obtained by the three proposed methods

Instances Statistical Features Method

VNS-SLS SLS VNS Siemens01 Best Mean Worst std 3.72 4.49 4.90 1.06 4.42 6.62 12.21 2.54 65.31 82.18 102.94 12.08 Siemens02 Best Mean Worst Std 2567.22 4988.30 8476.13 2774.46 2663.20 3000.81 4309.33 470.77 3681.11 5245.68 5643.34 525.57 Siemens03 Best Mean Worst Std 435.11 506.88 613.98 56.77 772.48 963.36 1189.63 120.78 970.49 1281.83 1939.33 420.59 Siemens04 Best Mean Worst Std 58204.47 62816.43 68462.09 3481.577 62820.30 67183.12 72804.13 2625.90 88035.89 91042.84 93194.87 1638.42 Tiny Best Mean Worst Std 0 0.14 1.46 0.46 0 0.08 0.48 0.15 0 0.08 0.48 0.18 Swisscom Best Mean Worst Std 0 0 0 0 0 0 0 0 0 0.34 1.33 0.46 K Best Mean Worst std 12.31 15.93 18.63 2.21 11.30 13.58 16.56 1.82 35.59 88.06 265.62 90.47

methodasalocalsearchintheVNSalgorithm,apositiveeffectandanenhancementisachievedin thequalitysolutioncomparedtothestandardVNSimplementedwiththelocalsearch.Indeed,the SLStechniquepermitstoenhancetheneighborsolutionsoftheVNSsearch.Moreprecisely,theSLS methodpermitstoensureagoodcompromisebetweenintensificationanddiversification.Thishelps themethodtoobtaingoodresults.Thediversificationphaseconsistsinselectingarandomneighbor solutionbyapplyingarandommove.Theintensificationphaseconsistsinselectinganeighborby changingtheconflictingnodeinthesolution. Further,wedrawFigure3toshowtheaverageresultsobtainedforalltheconsideredinstances. WecanseeinFigure3–athatbothSLSandVNS-SLSproducedresultsthatarebetterthanVNS foralltheSiemensinstances.WecanseeclearlythatthecombinationofSLSandVNSimprovethe resultsofVNSparticularlyforboththeinstancesSiemens01andSiemens03. FortheKinstance,wecanseeinFromFigure3-bthattheaveragesolutionsobtainedbybothSLS andVNS-SLSoutperformtheVNSmethod.TheSLScombinewellwiththeVNSitleadstoagood explorationofthesearchspace.ItenhancestheVNSsearchprocessallowingittogetbetterresults. FortheSwisscomandTinyinstances,thebehaviorofthethreemethodsisalmostthesame;they allsucceedtofindtheoptimalsolution. 5.5. Further Comparison Table3givesacomparisonwitharecentwork(LahsinatY.B.,2015)thatusedthesameformulation anddatasets.Theworkproposedthreehyper-heuristicstohandletheMI-FAP.Wenotethatthethree hyper-heuristicsmanagethesameheuristicsbutusedifferentmannertoselectthem.Thefirstone namelyHyper(1)isahyper-heuristicbasedonachoicefunctionwhichmeansthateachheuristichas aweightwhichevaluatesitsperformances.ThesecondnamelyHyper(2)usesarandomselectionof theheuristicsandthelastonenamelyHyper(3)isahybridonewhichusesaprobabilitytodecideto usethechoicefunctionorarandomchoice.

ThevaluesreportedinTable3forthedifferentapproachesaretheBestandaverageresults recordedfrom10runsforthesameoverallCPUtimeequalsto1800secondsforeachrun. WeobservethattheresultspresentedinTable3indicatethatourmethodsoutperformthose producedbythethreehyper-heuristicsforalltheconsideredinstances.Whencomparingtheresults recordedbytheVNS-SLSresultsandthehyper-heuristics,wecanobservethatfortheinstancesTiny andSwisscomthemethodsarecomparable. Toshowtheperformanceofourmethods,wegiveanothercomparisoninTable4. We compare our results with those of a Genetic algorithm and a Biogeography-based optimizationandtwohybridversionsoftheBiogeographyBasedoptimization:HBBO(1) andHBBO(2)(Lahsinat,2014).

ItisworthrecallingthatalltheresultsaregivenataCPUtimefixedto1800seconds.Itappears obviousfromtheresultspresentedinTable4thattheVNS-SLSandSLSproducedbetterresultsfor alltheSiemensandKinstances.FortheTinyandSwisscominstances,wecanobservethattheresults arethesame.WecanclaimthatSLScombinedwiththeVNSisagoodcombination.

Figure 3. The average results obtained on the considered instances

Table 3. A comparison with the hyper-heuristic approaches

Instances Results Methods

Our Better Achievement Hyper(1) Hyper(2) Hyper(3) Siemens01 Best_Average 3.724.49

(VNS-SLS)

47.70

88.78 21.2741.60 42.7949.04 Siemens02 Best_Average 2567.22(VNS-SLS)

3000.81(SLS)

4990,70

5815,79 5187.725422.59 4199.664672.32 Siemens03 Best_Average 435.11506.88

(VNS-SLS)

923,12

1409,34 884.641004.09 702.48787.99 Siemens04 Best_Average 58204.4762816.43

(VNS-SLS)

108232.4

110959.6 76968.7889047.31 87723.8391096.05

Swisscom Best_Average 0_{(SLS/VNS-SLS)} 0₀ 0₀ 0₀

K Best_Average 11.3013.58

(SLS)

80.22

99.27 22.3043.99 100.94115.29

The different comparisons provided above shows that the VNS-SLS is better than VNSandSLSstandaloneforsomeinstances.Indeed,itimprovesdramaticallytheresults obtainedbyVNSonalltheinstancesandimprovessomeothermethodspublishedpreviously. TheresultsobtainedbySLSandVNS-SLSontheexperimentedbenchmarkslookstobe comparable.Theresultsaretoocloseforsomeinstances.However,thehybridalgorithm VNS-SLSlooksstableandrobusttodealwithgreatsizeinstancesoftheMI-FAP.Ithasa goodbalancebetweenintensificationanddiversificationandallowsabetterexplorationof thesearchspace. 6. CoNCLUSIoN Thefrequencyassignmentisanimportanttaskinthedesignofradionetworkwheretheobjective istofindanoptimalfrequencyplanthatpermitstoensureagoodqualityofservice.Inthispaper,a VariableNeighborhoodSearch(VNS),aStochasticLocalSearchandanewhybridsearchmethod thatcombinestheVariableNeighborhoodSearchwiththeStochasticLocalSearchareproposedfor theMI-FAP.Theperformancesoftheproposedalgorithmsareevaluatedandcomparedusingwell-knowndatasets. TheexperimentsconductedshowthebenefitsoftheSLSasalocalSearchintheVNSprocess. Theresultsareveryencouragingandthenewhybridmethodsucceedsinfindinggoodresultsfor theconsideredbenchmarks.Thishybridmethodisstillinitscradleandmanypossibilitiesandideas arepossibletoenhanceitandavoidallthehindrancesthatlimititsperformance.Asaperspective, firstitwouldbeinterestingtoseeifwecanobtainbetterresultsbyincreasingthesearchtimefor theproposedapproaches.WealsoplantoexploreotherformsofSLSthatcouldbecombinedwith theVNS.Anotherpointthatcouldimproveourworkistoseeifthedifferentstepsintheproposed methodscouldbelaunchedinparallel.

Table 4. Comparison with BBO, HBBO(1), HBBO(2) and GA

Instances Results Our better achievement Method

BBO HBBO(1) HBBO(2) GA

Siemens01 Best_Average 3.724.49 (VNS-SLS)

223,62

250,18 32.1944.93 100.96104.40 140.68157.23 Siemens02 Best_Average 2567.22(VNS-SLS)

3000.81(SLS)

5941,29

6426,40 6325.195527.35 6325.196471.71 3778.615009.43 Siemens03 Best_Average 435.11506.88

(VNS-SLS)

1460,65

1544,94 1102.761202.81 1445.111498.99 1136.921276.527 Siemens04 Best_Average 58204.4762816.43

(VNS-SLS)

108401,08

110830,3 97204.81101117.4 110998.65112104.6 101010.6104288.8 Swisscom Best_Average 00

(VNS-SLS) 0 0 00 00 00.65 K Best_Average 11.3013.58 (SLS) 98.26 106.27 24.3033.99 108.94118.29 32.90040.788

ACKNowLedGMeNT

Theauthorswouldliketothanktheeditorandreviewersforcarefulreading,constructivecomments andvaluablesuggestionsthathelpedustoimprovethepaper.

AlsotheauthorswouldliketomentionthatthisworkwassupportedbythePHCTassili Project-17MDU987.

ReFeReNCeS

Aardal,K.,vanHoesel,S.P.M.,Koster,A.M.C.A.,Mannino,C.,&Sassano,A.(2007).Modelsandsolution techniquesforfrequencyassignmentproblems.Annals of Operations Research,153(1),79–129.doi:10.1007/ s10479-007-0178-0

Abunaser,A.M.,&Alshattnawi,S.(2015).Hybridizingartificialbeecolonyalgorithmwithmulti-parent crossoveroperator.International Journal of Applied Metaheuristic Computing,6(2),18–32.doi:10.4018/ IJAMC.2015040102

Alami,J.E.(2008).Usingculturalalgorithmforthefixed-spectrumfrequencyassignmentproblem.The Journal

of Mobile Communication,2(1),1–9.

Beckmann,D.K.(1999).Frequencyplanningwithrespecttointerferenceminimizationincellularradionetworks. Benmounah,Z.M.,Meshoul,S.,&Batouche,M.(2017).Scalabledifferentialevolutionaryclusteringalgorithm forBigDatausingmap-reduceparadigm.International Journal of Applied Metaheuristic Computing,8(1), 45–60.doi:10.4018/IJAMC.2017010103

Bezuidenhout,W.(2014).Optimisingthefrequencyassignmentproblemutilizingparticleswarmoptimisation [Doctoraldissertation].UniversityofJohannesburg.

Boucheham,A.,&Batouche,M.(2017).Hybridwrapper/filtergeneselectionusinganensembleofclassifiers andPSOalgorithm.International Journal of Applied Metaheuristic Computing,8(2),22–37.doi:10.4018/ IJAMC.2017040102

Boughaci,B.B.(2008).Stochastic Local Search for the Optimal Winner Determination Problem in Combinatorial

Auctions. In International Conference on Principles and Practice of Constraint Programming (pp. 593-597).

Springer. Colombo,G.A.,&Allen,S.M.(2010).AcomparisonofproblemdecompositiontechniquesfortheFAP.Journal of Heuristics,16(3),259–288.doi:10.1007/s10732-009-9116-4 Dorne,R.H.(1999).Tabusearchforgraphcoloring,T-coloringsandsetT-colorings.InMeta-heuristics(pp. 77-92).Boston,MA:Springer. Duque-Antón,M.,Kunz,D.,&Ruber,B.(1993).Channelassignmentforcellularradiousingsimulatedannealing.

IEEE Transactions on Vehicular Technology,42(1),14–21.

Eisenblatter,A.G.(2002).Frequencyplanningandramificationsofcoloring.InDiscussionsMathmatcaeGraph Theory(pp.51-88).

Eisenblätter,A.K.(2000,June9).COST259-WirelessFlexiblePersonalizedCommunications.Retrievedfrom http://fap.zib.de/problems/COST259/

Hale,W.(1980).Frequencyassignment:Theoryandapplications.InProceedings of the IEEE(pp.1497–1514). Hansen,P.,&Mladenović,N.(1997).Variableneighborhoodsearchforthep-median.Location Science,5(4), 207–226.

Holger,H.,&Hoos,T.S.(2004).Stochastic Local Search: Foundations and Applications.Elsevier.

Kuurne,A.(2001).Mobile Measurement based Frequency Planning in GSM Network.HelsinkiUniversityof Technology.

Lahsinat,Y.(2014).ImprovingBiogeographyBasedOptimizationbyUsingStochasticLocalSearchandSolving MI-FAPProbleminGSMNetworks.InBasedOptimization(ICSIBO’2014)(pp.112-119).Mulhouse. Lahsinat,Y.B.(2015).Trois hyper-heuristiques pour le problème d’affectation de fréquence dans un réseau

cellulaire. In 11th Journées Francophones de programmation par contraintes(pp.184–193).Bordeaux:JFPC.

Lahsinat,Y.B.(2017).HarmonySearchBasedAlgorithmsfortheMinimumInterferenceFrequencyAssignment Problem.InInternational Conference on Harmony Search Algorithm (ICHSA2017)(pp.179–189).Bilbao, Spain:Springer.doi:10.1007/978-981-10-3728-3_18

Yasmine Lahsinat is a PhD student in computer science. She currently works at the Department of Informatics, University of Science and Technology Houari Boumediene. Yasmine does research on combinatorial optimization, meta-heuristics and Artificial Intelligence. Yasmine is a member of the LRIA Artificial Intelligence Laboratory at the USTHB University.

Dalila Boughaci is a Full Professor in Computer Science at the University of Science and Technology USTHB (Algeria). She got her PhD from the University of Aix-Marseille (France) in 2008 and her “Habilitation” Post-doctoral diploma from USTHB University in 2009. Dalila Boughaci earned a Bachelor of Engineering degree in Computer Science from Algiers University, MS degree in Computer Science and her second PhD in programming systems from the University of Sciences and Technology, Beb-Ezzouar, Algiers in 1997, 2001 and 2008, respectively. Boughaci’s current research interests are in the areas of data mining, deep learning, evolutionary computation, artificial intelligence, meta-heuristics, multi-agent systems, network security, credit scoring and e-commerce. She has published several papers on these research topics in journals and conferences and directed several PhD, MS and BS students’ projects. Boughaci has taught Parallel computing, machine learning, Web service, Object Oriented Programming, algorithmic, software engineering, databases, Java, Agents and programming languages at Algiers, and served on several program committees. Boughaci is a member of the LRIA Artificial Intelligence Laboratory at the University of Algiers. She is the head of the research team: Optimization, Reasoning and Application of the LRIA Laboratory.

Belaid Benhamou is currently a Research Professor at Aix-Marseille University. He received his mathematical bachelor in 1983, his engineering degree in computer science in 1988 and his PhD in computer science in 1993 at the Aix-Marseille I University. He obtained an Assistant Professor job at the University of Aix-Marseille I in 1995 and in 2001, he obtained his Habilitation thesis after which he got a national qualification for Full University Professor. His research activities are in the field of artificial intelligence: constraint solving, logic and automated theorem proving, logic programming, non-monotonic reasoning, meta-heuristics and combinatorial optimization and data mining. He published over 120 articles and has supervised several doctoral theses on these areas.

Lai,X.H.,&Hao,J.-K.(2015).Pathrelinkingforthefixedspectrumfrequencyassignmentproblem.Expert

Systems with Applications,42(10),4755–4767.doi:10.1016/j.eswa.2015.01.025

Leese,R.H.(2002).Methods and Algorithms for Radio Channel Assignment.Oxford:OxfordUniversityPress. Luna,F.B.(2007).ACOvsEAsforsolvingareal-worldfrequencyassignmentprobleminGSMnetworks. In 9th Annual Conference on Genetic and Evolutionary Computation (pp. 94–101). London: ACM. doi:10.1145/1276958.1276972

Luna,F.E.-G.-R.,Estébanez,C.,León,C.,Chaves-González,J.M.,Nebro,A.J.,Aler,R.,&Gómez-Pulido, J.A.et al.(2011).Optimizationalgorithmsforlarge-scalereal-worldinstancesofthefrequencyassignment problem.Soft Computing,15(5),975–990.doi:10.1007/s00500-010-0653-4

Mabed,H.,Caminada,A.,&Hao,J.K.(2011).Genetictabusearchforrobustfixedchannelassignmentunder dynamictrafficdata.Computational Optimization and Applications,50(3),483–506.

Metzger,B.H.(1970).Spectrummanagementtechnique.InProceedings of the 38th National ORSA Meeting, Detroit,MI.

Montemanni,R.,Moon,J.N.,&Smith,D.H.(2003).Animprovedtabusearchalgorithmforthefixed-spectrum frequency-assignmentproblem.IEEE Transactions on Vehicular Technology,52(4),891–901.

Montemanni,R.S.,&Smith,D.H.(2010).Heuristicmanipulation,tabusearchandfrequencyassignment.

Computers & Operations Research,37(3),543–551.doi:10.1016/j.cor.2008.08.006

Taghavifard,M.T.(2012).SchedulingcellularmanufacturingsystemsusingACOandGA.International Journal

of Applied Metaheuristic Computing,3(1),48–64.doi:10.4018/jamc.2012010105

Wang,Y.,&Xu,N.(2017).Ahybridparticleswarmoptimizationmethodfortravelingsalesmanproblem.

International Journal of Applied Metaheuristic Computing,8(3),53–65.doi:10.4018/IJAMC.2017070104

Zemali,E.,&Boukra,A.(2016).Usingabio-inspiredalgorithmtoresolvethemultiplesequencealignment problem.International Journal of Applied Metaheuristic Computing,7(3),36–55.doi:10.4018/IJAMC.2016070103