An Interactive Method for 0-1 Multiobjective Problems Using Simulated Annealing and Tabu Search

19 Pages·2000·0.19 MB·English

JournalofHeuristics,6:385–403(2000)
(cid:176)c 2000KluwerAcademicPublishers
An Interactive Method for 0-1 Multiobjective
Problems Using Simulated Annealing
and Tabu Search
M.JOAQOALVESANDJOAQOCL´IMACO
FaculdadedeEconomia,UniversidadedeCoimbra/INESC,Av.DiasdaSilva,165,3000Coimbra,Portugal
e-mail:[email protected]
Abstract
Thispaperpresentsaninteractivemethodforsolvinggeneral0-1multiobjectivelinearprogramsusingSimulated
AnnealingandTabuSearch. Theinteractiveprotocolwiththedecisionmakerisbasedonthespecificationof
reservationlevelsfortheobjectivefunctionvalues. Thesereservationlevelsnarrowthescopeofthesearchin
eachinteractioninordertoidentifyregionsofmajorinteresttothedecisionmaker.Metaheuristicapproachesare
usedtogeneratepotentiallynondominatedsolutionsinthecomputationalphases.GenericversionsofSimulated
AnnealingandTabuSearchfor0-1singleobjectivelinearproblemsweredevelopedwhichincludeageneralroutine
forrepairingunfeasiblesolutions.Thisroutineimprovessignificantlytheresultsofsingleobjectiveproblemsand,
consequently,thequalityofthepotentiallynondominatedsolutionsgeneratedforthemultiobjectiveproblems.
Computationalresultsandexamplesarepresented.
Key Words: interactive multiple objective programming, zero-one programming, metaheuristics, simulated
annealing,tabusearch
Thecomputationalcomplexityofsingleobjectivecombinatorialproblemshasprovideda
basisforanincreasinginterestinthedevelopmentofheuristicapproachesaimedatdealing
withthoseproblems. However,mostoftheseapproachesareproblem-specific. Therefore
onethatappliestooneproblemusuallydoesnottoadifferentone. Overthelastdecade,
severaltechniquesfarmoregenerallyapplicablehavebeendevelopedsuchasthefollowing
metaheuristic ones: Simulated Annealing (Kirkpatrick, Gellat, and Vecchi, 1983), Tabu
Search(Glover,1986), andGeneticAlgorithms(Goldberg,1989)amongothers. Inspite
ofbeinggenericheuristicapproaches,inmostapplicationsreporteduntilnowtheirusehas
been problem-specific and different implementations have been developed for each new
combinatorialproblemtackled. Themaindifficultyofageneralpurposealgorithmliesin
theincorporationofageneralmechanismintendedtorestoretheprimalfeasibilityofthe
system. Infact,ithasbeenshownthattheideaofkeepingfeasibilityateverystageofthe
searchismoreadvantageousthanrelaxingtheconstraintsandusingapenaltyfunctionin
thecost(Abramson,Dang,andKrishnamoorthy,1996).
We believe that problem-specific implementations of any metaheuristics should out-
perform generic ones but, in our opinion, good general-purpose algorithms are be very
useful when dealing with problems with no particular structure. This fact motivated our
interest in metaheuristics and we started by developing Simulated Annealing and Tabu
386 ALVESANDCL´IMACO
Searchversionsforgenericsingleobjective 0-1linearprograms. Untilnow, fewgeneric
versionsofmetaheuristicmethodshavebeenreported. Examplesofsuchapproachesarethe
developmentsofAboudiandJo¨rnsten(1994)andLokketangen,Jo¨rnsten,andStoroy(1994)
for Tabu Search and the Simulated Annealing codes of Connolly (1992) and Abramson,
Dang,andKrishnamoorthy(1996).
It is now generally accepted that many real-world problems should consider multiple
criteria. Besides,the0-1multiobjectiveproblemisastimulatingareaofresearchduetoits
applications,namelyinareasconcerningcapitalbudgeting,projectselection,deliveryand
routing,facilitylocationproblemsandsoon. Interactivemethodsallowthecontributionof
thedecisionmaker(DM)duringthesolutionsearchprocessbyinputtinginformationthat
mayleadtosolutionsmoreinconsonancewithhis/herpreferences. Metaheuristicscanplay
animportantroleinprovidingeffectiveinteractiveprocessessincethecomputationaleffort
inherentinexactmethodsmay,dependingonthecircumstances,jeopardizetheinteractive
process.
Theadaptationofmetaheuristicstomultiobjectiveproblems,evenforparticularinstances,
has been little explored so far. However, we should mention the recent work of some
researchersinthisarea: Serafini(1994), Fortemps, Teghem, andUlungu(1994), Ulungu
(1993),CzyzakandJaszkiewicz(1996)andHansen(1997).
Inthispaperweproposeaninteractivemethodtosolvemultiobjective0-1linearproblems
whichisbasedontwometaheuristicapproaches: SimulatedAnnealingandTabuSearch.
Thebasicideaunderlyingthemethodistheprogressivesearchofnondominatedor“good”
approximations of nondominated (potentially nondominated) solutions that belong to re-
gionsofinteresttotheDM.Inthefirstphaseofthedecisionprocess,theDMmayimpose
bounds on the objective function values (reservation levels) that are used to narrow the
scopeofthesearchineachinteraction. Thesearchofpotentiallynondominatedsolutions
ismadeintheareaaroundthesolutionsthatoptimizeindividuallyeachobjectivefunction
inthatrestrictedregion. Itprovidesanapproximationofthecorrespondingnondominated
subset that tends to be closer to exact vectors in the extreme areas of the region (where
the individual optima for that region lie) rather than in the middle of the region. If the
DM wishes to know “better” approximations in the middle, he/she can then specify new
reservationlevelsthatnarrowtheregiontobeexploredwithrespecttothepreviousone.
Afteraglobalsearchfollowingmetaheuristicroutines,theDMmayuseexactmultiobjec-
tivetechniquesthathelphim/hertocometoafinalcompromiseregardinganondominated
solution. ThesetechniquesenabletheDMtoeithercheckifasolutiongivenbyametaheuris-
ticsisreallyanondominatedsolutionortocomputethenondominatedsolution(s)closest
toacriterionreferencepoint(whichrepresentsaspirationlevelsfortheobjectivefunctions)
byoptimizinganachievementscalarizingfunction(inthesenseofWierzbicki(1980)). In
ordertogiveabetterperceptionofthemaximaerrorsinvolvedinapproximationsandthus
abetterevaluationofthequalityofthosesolutions,theDMisprovidedwithboundsgiven
byspecificnondominatedsolutionsforthelinearrelaxationofthemultiobjectiveproblem.
TheprotocolproposedhereintointeractwiththeDMisinaccordancewithLarichevand
Nikiforov (1987) who concluded that, in general, good methods use information relative
totheobjectivefunctionvaluesinthedialoguewiththeDMinordertocreateasystemof
preferences. AccordingtoNakayama(1985)thismayleadtotwodifferentwaysofacting:
fixingaspirationlevelsorreservationlevels.
ANINTERACTIVEMETHODFOR0-1MULTIOBJECTIVEPROBLEMS 387
InSection1ofthispaperwepresentaversionofSimulatedAnnealingandanotherofTabu
Searchforgeneric0-1linearprogramswithasingleobjective. InSection2theinteractive
methodformultiobjectiveproblemsisdiscussedanddescribed. Computationalexamples
arepresentedinSection3. ThepapercloseswithconcludingremarksinSection4.
1. Simulatedannealingandtabusearchforsingleobjective0-1linearprograms
1.1. Simulatedannealing
ConsiderthegeneralschemeoftheSimulatedAnnealingalgorithm:
LetSdenotethesolutionspace,ctheobjectivefunctiontobeminimized(cost)and N
theneighbourhoodstructure.
Selectaninitialsolutionxo 2 S;setT :DTo >0(initialtemperature)
Repeat
Repeat
Randomlyselect y 2 N(xo);– :Dc(y)¡c(xo);
if– <0thenxo :D y
elsegeneraterandomd uniformlyintherange[0,1];
ifd <e¡–=T thenxo :D y;
untiliteration countDNrep
setT :Dfi.T/;
untilstopping conditionDtrue;
x istheapproximationtotheoptimalsolution.
o
T isthecontrolparameterwhichplaystheroleofthetemperatureinthephysicalsystem.
Initially,withlargevaluesofT,largeincreasesincostwillbeaccepted;asT decreasesonly
smallerincreaseswillbeacceptedandfinally,asthevalueofT approaches0,noincreases
incostwillbeacceptedatall. Thewayandtherateatwhichthisparameterisreducedis
usuallyreferredtoascoolingschedule. DetaileddescriptionsofSimulatedAnnealingcan
befoundinKirkpatrick,Gellat,andVecchi(1983),AartsandKorst(1989)andDowsland
(1993).
The algorithm given above is a very general one and some decisions must be made in
order to implement it: generic decisions¡TO, the stopping condition and the cooling
schedule(Nrepandfi.T/);problemspecificdecisions¡S,cand N.
TheSimulatedAnnealingversionweproposehereisdesignedfor0-1linearprograms
andwillbedenotedby0-1SAinwhatfollows. Letusbeginbythespecificdecisionsfor
0-1SA.
Aneighboursolutionfromthecurrentonecanbeobtainedbyamovewhichconsistsin
changing the value of one variable x from 0 to 1 or from 1 to 0. However, the question
i
ofthebetterwaytodealwithconstraintsnaturallyarises. Concerningspecificproblems,
someauthorshaveproposedrelaxingthefeasibilityconditionsincludingapenaltytermin
thecostfunctiontodiscourageviolationsoftherelaxedconstraints. Weexperimentedthis
approachinseveraltypesofproblemsandtheresultswerenotattractive. Difficultiesarose
intheselectionofanadequatepenaltyterm. Besides,inhighlyconstrainedsolutionspaces,
388 ALVESANDCL´IMACO
it is possible that during the whole process no feasible solution is found. The p-median
location problem whose formulation is presented in the appendix (entitled P-MEDIAN)
is such an example. Although a small problem, no solution was primal feasible during
aSimulatedAnnealingrunof1800iterationsbyapplyingjustsimplemovesliketheone
statedabove. Werecallthatweareinterestedinageneralalgorithmfor0-1linearprograms,
sowecannotprofitfromanyspecialstructureoftheconstraints.
Anotherpossibilityistotrytorestorethefeasibilityofeachnewneighboursolutioncom-
puted. Inthiscase,thesolutionspaceisrestrictedtothefeasiblesolutionspace. Connolly
(1992)developedaroutinetorestoresolutionfeasibilitycalledRESTOREwhichisusedin
theprogramGPSIMAN(ageneralpurposeSimulatedAnnealing). Ourcomputationalex-
perimentswiththisroutineledustorealizethatalargenumberofvariablesmustbeflipped
whentheprimalfeasibilityishardlyachievedandthealgorithmoftenbacktracks(byundo-
ingthelastmove),thusrequiringhighcomputationaleffort.Wedevelopedanewroutineto
restoreprimalfeasibilitytryingtoovercomethesedrawbacks(AlvesandCl´ımaco,1996).
Inouropinion,mostdifficultieswithConnolly’sroutineareduetothecomputingprocess
ofthemosthelpfulvariabletobeflipped. Ahelp-scoredependsonthewayavariablehelps
violatedconstraintswithoutregardtohowmuchitcan“destroy”satisfiedconstraints. In
ourroutine,theselectionofthemosthelpfulvariableconsiderstwoaspects: howhelpful
itisforanycurrently-unfeasibleconstraintandtowhatextentitdestroysthefeasibilityof
anycurrentlysatisfiedconstraint.
TheroutineweproposereliesonpartsofBalas’szero-oneadditivealgorithm(Balas,1965)
andworksasfollows:
† Initially,allvariablesare“free”excepttheone(randomlydetermined)whichwasfirst
flippedtoobtaintheneighboursolution.
Whilethesolutionisnotfeasibleandthefeasibilitycanberestored:
† findthemosthelpfulfreevariablex (ifnoneexists,feasibilitycannotberestored)
j
† changex valuefrom0to1orthereverse;x becomesnotfree
j j
Themosthelpfulfreevariablex isdeterminedinthefollowingmanner:
j
Withoutlossofgeneralityweassumethatallconstraintsare“•”
† Calculatetheslacks foreachconstraint(satisfiedornot): s DRHS ¡LHS
i i i i
† (constrainti isnotsatisfiedwhenevers <0)
i
† Create a subset P of free variables in which one variable is helpful for at least one
violatedconstraint. Pisdefinedas
PDfj free:.a <0andx D0/or.a >0andx D1/fori suchthats <0g
ij j ij j i
wherea isthecoefficientofvariablex intheconstrainti.
ij j
† Foreveryconstrainti suchthats <0,define y as
i i
X X
y Ds C .¡a /C a
i i ij ij
fj2P:aij<0andxjD0g fj2P:aij>0andxjD1g
ANINTERACTIVEMETHODFOR0-1MULTIOBJECTIVEPROBLEMS 389
† IfP D;orthereisanyy <0thenthefeasibilityofthesolutioncannotbeattained—
i
inthisparticularcase,0-1SAwillevaluateanunfeasiblesolutionpenalizingitinthe
costfunctionwhichmeansthatthisprocessdoesnotbacktrack.
Otherwise calculate the score (•0) of each variable in P. This score measures the
weaknessofahelpfulvariable:
8 X
>>< .si ¡aij/ if xj D0
SCORE D fi:siX¡aij<0g
j >>: .si Caij/ if xj D1
fi:siCaij<0g
† ThevariablewiththehighestSCOREisselectedtobeflipped.
A zero-score means that the variable is helpful without weakness. Thus, changing the
valueofthisvariableyieldsafeasiblesolution.
This procedure has performed well in most of the problems tested. However, 0-1SA
givesbetterresultsiftheselectionofthevariabletobeflippedisrandomwithprobabilities
proportionaltothevariablescoresinsteadofselectingthevariablewiththehighestscore
all the time. This change, which avoids the process to be biased, is only recommended
inproblemswherethefeasibilityofthesolutionsiseasilyrestored. 0-1SAautomatically
switchesfromthestochastictothedeterministicchoiceofthemosthelpfulvariablewhen
thefirstiterationsshowthatthefeasibilityoftheproblemisnoteasilyrestored. Thisisthe
caseoftheP-MEDIANproblem(inappendix).
Genericdecisionsin0-1SA
Theinitialsolution: theinitialsolutionisobtainedbyroundingtheoptimalsolutionforthe
linearrelaxationoftheproblem. Ifthissolutionisunfeasible,theproceduredescribed
aboveisusedtoattainthefeasibility.
Thecoolingschedule: thetemperaturereductionrelation T :D fi:T hasbeenconsidered
(usually,0:8•fi •0:99). T isthecontrolparameter(temperature)andisgivenbyK:t.
TheinitialtemperatureisTo D K:to withto beingcloseto1andthefinaltemperature
isK:tf withtf beingcloseto0. K isautomaticallydeterminedforeachproblem. After
severalexperimentsusingdifferent K valuesindifferentproblemswehavedecidedto
considerK D0:5c (wherec denotestheaverageabsolutecoefficientintheobjective
avg avg
function). Theparametersfi,to,tf andNreparespecifiedbytheuser.
Computationalresultsof0-1SA
We selected a set of 10 test problems which includes 8 multiple-constraint knapsack
problems(MCKP),ap-medianproblem(inappendix)andasetcoveringproblem(SCP).
TheMCKP(from28to105variables)andtheSCP(192variables,240constraints)were
takenfromtheliteratureandareavailableintheOR-LibrarybyBeasley(1990).TheMCKP
arethefollowing: PET5,PET6,PET7whichareoriginallyduetoPetersen(1967),SENTO1,
SENTO2whichareduetoSenyuandToyoda(1967)andWEING6,WEING7,WEING8
which are due to Weingarter and Ness (1967). The SCP is called SCPCYC6 and is from
GrossmanandWool(1997).Despitetheirsmallsize,theseMCKPwerechosenbecausethey
390 ALVESANDCL´IMACO
havebeentestedbyothermetaheuristicapproaches,namelythosefromKhuri,Ba¨ch,and
Heitko¨tter(1994),AboudiandJo¨rnsten(1994),Lokketangen,Jo¨rnsten,andStoroy(1994)
andDrexl(1988). TheSCPwaschosenduetoitsdifficulty. Inordertoprovideabetter
analysisofthequalityofthesolutionversustimetoeachproblem,wetookthecomputational
timeoftheoptimalsolutiongivenbytheCPLEXcommercialsolverasthereferencetime
to fix the 0-1SA parameters. We considered to D 0:98, tf D 0:01 (for SCPCYC6, tf D
0:001),fi D 0:95andNrepwasadjustedbetween1and20sothata0-1SArunwouldnot
takelongerthan50%ofthecorrespondingCPLEXtime. Thisisoflittlesignificancefor
theMCKPsincethemaximumCPLEXtimeis13secsforSENTO1. However,concerning
the SCPCYC6, owing to the fact that the CPLEX could not find the optimal solution in
10hours—givinga“betterintegervalue”of64—weconsideredthen2mins40secsruns
of 0-1SA that, in fact, provided better solutions. Both CPLEX and 0-1SA were run in a
personalcomputerPentium166MHz.
Table 1 shows a summary of the computational results produced by 20 runs of 0-1SA
foreachproblem.Itincludesthegapbetweentheoptimumandthebestsolutiongivenby
0-1SA(100£joptimum-bestj/joptimumj). ConcerningthesolutionqualityfortheMCKP,
somecomparisonswithotherprocedurescanbemade:
(i) 0-1SAgaveabetteraveragesolutionthantheGENEsYsalgorithmfromKhuri,Ba¨ch
andHeitko¨tter(1994)forallbutone(PET6)ofthetestproblems;
(ii) the“bestsolution”givenby0-1SAisbetterthan(orequalto)the“bestsolution”provided
byDrexl(1988)orAboudiandJo¨rnsten(1994)orLokketangen,Jo¨rnsten,andStoroy
(1994), respectively, for all, for all but one (PET6) and for all but one (PET7) of the
testproblems.
Asstatedbefore,0-1SAtriestorestoretheprimalfeasibilityofeachneighboursolution
producedbyafirstrandommove. Noteherethattherestoringroutinewascalledatevery
iteration of the p-median problem (performing nearly 5 additional moves each time) and
from10%(inWEING7)to70%(inSENTO1andWEING8)oftheiterationsoftheother
problems. The number of additional moves for each solution varied from 1 to 3 within
theseproblems.
Table1. Computationalresultsof0-1SA.
Problem: P-MEDIAN PET5 PET6 PET7 SENTO1 SENTO2 WEING6 WEING7 WEING8 SCPCYC6
n⁄m 20⁄9 28⁄10 39⁄5 50⁄5 60⁄30 60⁄30 28⁄2 105⁄2 105⁄2 192⁄240
Nrep 1 10 10 6 10 5 5 10 20 20
optimum 3,700 12,400 10,618 16,537 7,772 8,722 130,623 1,095,445 624,319 60
bestsol. 3,700 12,400 10,604 16,524 7,772 8,722 130,623 1,095,445 624,319 60
avgsol. 3,700 12,386 10,524.4 16,463.8 7,692.8 8,722 130,235 1,095,352 616,287 65.0
avgtime 0.10sec 0.44sec 0.39sec 0.33sec 3.5sec 1.27sec 0.11sec 0.39sec 1.9sec 160sec
GAPopt-best 0% 0% 0.13% 0.08% 0% 0% 0% 0% 0% 0%
ANINTERACTIVEMETHODFOR0-1MULTIOBJECTIVEPROBLEMS 391
1.2. TabuSearch
TabuSearch, likeSimulatedAnnealing, isaneighbourhoodsearchheuristicsdesignedto
avoidbeingtrappedinlocaloptima. However,incontrastwithSimulatedAnnealing,the
randomizationisde-emphasizedinTabuSearchassumingthatintelligentsearchshouldbe
basedonmoresystematicformsofguidance. Thesearchisconstrainedbyclassifyingcer-
tainmovesasforbidden(i.e. tabu)inordertopreventthereversal,orsometimesrepetitions,
ofthemoves. FormoredetailsseeGlover(1986,1989,1990a,1990b),GloverandLaguna
(1993).
Initssimplestform,TabuSearchmaybedescribedasfollows:
LetSbethesolutionspaceandctheobjectivefunctiontobeminimized
†Selectaninitialsolutionxo 2S;Letx⁄ :D xo
Thetabulistisinitiallyempty: TL:D;
Repeat
†Createacandidatelistofnon-tabumoves—ifapplied,eachmovewould
generateanewsolutionfromthecurrentone. So,letCandidate N.xo/bethe
setofcandidateneighboursolutions.
†Choose y 2Candidate N.xo)thatminimizesthefunctionevaluation.y/over
thisset.
Ifc.y/<c.x⁄/thenx⁄ :D y
†xo :D y andupdateTL.
untilaspecifiednumberofiterationshavepassedwithoutupdatingthebestsolution,x⁄.
TheTabuSearchversionweproposehereinisdevotedto0-1linearprograms.O-1TS/.
A move leading to a neighbour solution is defined by changing the value of one variable
from0to1orfrom1to0. Theinitialsolutioniscomputedasin0-1SA.
0-1TSincludesthreephases: afirstphaseofsearchthatonlyusesalistoftabumoves,
asecondonewhichisadiversificationphaseandfinally,anintensificationphase. Besides
short-termtabumemory,whichisusedthroughouttheprocess,frequency-basedmemories
are also included in the diversification and intensification phases. Short-term memory is
implementedbyatabulist(last-in-last-out)whichrecordsthelast#TL(thelengthofthe
tabulist)variableschangedfrom0to1orthereverse. Thesevariablescannotchangetheir
values(preventingthereversalofcertainmoves)unlesstheaspirationcriterionisapplied.
Theaspirationcriterionconsistsinoverridingthetabustatusofamovewhenityieldsthe
bestsolutionobtaineduptothen.
Thediversificationphasetriestogeneratesolutionsthatembodydifferentfeatures(vari-
able values) from those previously found, driving the search into new regions. It uses a
frequencymemorywhosedatahavebeencollectedfromthebeginningofthefirstphase.
This memory vector records the number of times each variable took the value 1 in fea-
sible solutions. At the diversification phase, a variable is allowed to change from 0 to 1
(1to0)ifthismoveisnon-tabuandthevariablefrequencyof1’sissmaller(larger)thana
threshold—thethresholdisinitializedtobetheaveragefrequencyof1’sofallthevariables
andisincreased(decreased)wheneverthereisnopossiblemove.
392 ALVESANDCL´IMACO
Theintensificationphasetriestoreinforcesolutionfeatureshistoricallyfoundtobegood.
Itusestwofrequencymemoryvectorswhosedataarebasedonsolutionqualityandhave
beencollectedfromthebeginningofthesecondphase: letxbestIandxworstIbe,respectively,
the best and the worst feasible solution produced until the end of the first (I) phase; the
memoryvectorsrecord1s goodandrecord1s badregisterthenumberoftimeseachvariable
tookvalue1,respectively,in“good”and“bad”feasiblesolutionsduringthesecondphase;a
solutionhasbeenconsidered“good”ifc.x/<c.xbestI/C–or“bad”ifc.x/>c.xworstI/¡2–
with– D 0:25[c.xworstI/¡c.xbestI/]. Attheintensificationphase,avariable x isallowed
i
to change from 0 to 1 (1 to 0) if this move is non-tabu and record1s good ‚ g1 and
i
record1s bad • b1 .record1s good • g0 and record1s bad ‚ b0). Initially, g1 D
i i i
g0 and b1 D b0 are given by the average value over record1s good and record1s bad,
respectively. These thresholds are relaxed whenever there is no possible move. In the
current implementation of 0-1TS the intensification phase is performed 3 times, starting
withthethreebestsolutionsobtaineduntiltheendofthesecondphase.
0-1TSgoesfromonephasetothenextoneafterpassing1Niterations(anumberspecified
bytheuser)withoutupdatingthebestsolution.
Like in Simulated Annealing, our computational experience showed that repairing un-
feasiblesolutionsimprovetheresultsremarkably. Therefore,movesthatrestorefeasibility
have higher priority. However, we have adopted a different strategy from 0-1SA: only
one move is performed at each iteration yielding a solution, possibly unfeasible, but that
surelywillmovetowardsfeasibilityinthenextiteration. Takingintoaccountthetabulist,
thefrequency-basedmemories, higherprioritymovesandtheaspirationcriterion, theset
Candidate N.xo) is defined in different ways whether xo is feasible or not and depend-
ingonthesearchphase. Ifthecurrentsolution xo isfeasible, thetabulist, theaspiration
criterion and the frequency-based memories (in the last phases of the search) are used to
identifytheelementsincludedinCandidate N.xo/. Theneighboursolutionyselectedwill
be the one (not necessarily feasible) that minimizes the function evaluation.y/ over this
set. Ifxo isnotfeasible,Candidate N.xo/onlyincludessolutionsobtainedbyanon-tabu
move that reduces the unfeasibility of the solution. According to the restoring routine
described above for 0-1SA, y will be the solution obtained by flipping the highest-score
variable. Therefore,thefunctionevaluation.y/justevaluatescandidatesolutionsobtained
by a move from one feasible solution (the former case). Since these candidate solutions
maybeunfeasible,evaluation.y/mustincludenotonlythecostfunctionbutalsoapenalty
termthatpenalizesviolatedconstraints. Afterexperimentingdifferentweightstopenalize
eachviolatedconstraintwerealizedthatlowerweightsperformbetterinthefirstiterations
(enabling an oscillation that diversify the search) and higher weights fit quite well in the
last iterations (when an intensification is desired). Thus, evaluation.y/ D c.y/C W¢v
wherev isthenumberofviolatedconstraintsandW istheweightthatvariesfromc to
min
c ,theminimumandmaximumnon-zeroabsolutecoefficientsintheobjectivefunction,
max
respectively.
Computationalresultsof0-1TS
0-1TSwasappliedtothesetofproblemsusedfor0-1SA.Theparameter1Nwasspecified
for each problem so that the computational times were similar to those spent by 0-1SA.
Severallengthsofthetabulist,relatedtothesquarerootofthenumberofvariables,were
ANINTERACTIVEMETHODFOR0-1MULTIOBJECTIVEPROBLEMS 393
Table2. Computationalresultsof0-1TS.
Problem: P-MEDIAN PET5 PET6 PET7 SENTO1 SENTO2 WEING6 WEING7 WEING8 SCPCYC6
1N 20 70 30 20 50 10 25 20 100 500
#TL 2–6 2–6 2–6 3–7 3–7 3–7 2–6 7–11 7–11 7–13
bestsol. 3,700 12,400 10,618 16,499 7,772 8,722 130,623 1,095,445 615,110 62
avgsol. 3,700 12,388 10,588.4 16,499 7,772 8,722 130,389 1,095,409 610,170 62.43
#TL-best all 6 2;4 all all all 2;3 7;8;9 11 7–11
avgtime 0.06sec 0.44sec 0.38sec 0.22sec 1.37sec 0.22sec 0.15sec 0.5sec 2.0sec 158sec
GAPopt-best 0% 0% 0% 0.23% 0% 0% 0% 0% 1.47% 3.33%
experimented. A summary of the computational results is given in Table 2. This table
includestherangeoftabulistlengths(#TL)usedforeachproblemandalsothebestand
averagesolutionsamongtheresultsproducedforthedifferent#TL.
Theresultsof0-1TSwithinoursetoftestproblemsshowedthatvariationsof#TLdonot
cause large variations in the solution quality. There is a relative stability in the solutions
thatusuallyaregoodapproximationsoftheoptimalsolution. Incontrast,severalrunsof
0-1SAmayproducesolutionsofverydifferentqualityduetorandomization. Therefore,the
averagesolutionof0-1SAoverseveralrunsis,ingeneral,worsethantheaveragesolution
of0-1TSfordifferenttabulistlengths. Eventhough,thebest0-1SAresultisoftenbetter
thanorequaltothe0-1TSresult. Thus,wecannotconcludethatthereisametaheuristics
bettersuitedforallthecasesthananotherone.
2. Aninteractivemethodformultiobjective0-1problems
2.1. Notationandbasicconcepts
Let f .x/ be the objective function i to be maximized in the multiobjective 0-1 linear
i
programwithk objectivefunctions:
max f .x/Dcix i D1;:::;k
i
s.t.x 2 S
whereS DfxjAx Db;x 2f0;1gng
Letzobetheimageofthefeasiblesolutionxo,sothatzo D.zo;:::;zo/D.f .xo/;:::;
1 k 1
f .xo//:
k
Thesolutionza dominateszb ifandonlyifza ‚ zb forallobjectivesi D 1;:::;k and
i i
za >zb foratleastoneobjectivei.
i i
Asolutionzissaidtobenondominatedifftheredoesnotexistanotheronethatdominates
it.
Let us call z a potentially nondominated solution (terminology used by Czyzak and
Jaskiewicz, 1996) or an approximation of a nondominated solution iff the existence of
394 ALVESANDCL´IMACO
anothersolutionthatdominatesitisstillunknown. Inwhatfollows,weusetheabbreviation
p.n.dtodesignate“potentiallynondominated”.
2.1. Theinteractivemethodscheme
The basic idea underlying the method we propose is the progressive and selective search
of p.n.d solutions by focusing the search, in each interaction, on a subregion delimited
by reservation levels specified by the DM for the objective function values. The use of
reservationlevelsaimsatbringingthesearchprocessgraduallyclosertoregionsofgreater
interesttotheDM.Eachcomputingphaseproducesasetofp.n.d. solutionsmoreconcen-
tratedaroundtheindividualoptimafortheobjectivefunctionsintheregionbeingexplored.
Subparts of a region will be explored deeper if the DM specifies new reservation levels
further narrowing the previous region. This approach tries to avoid the main drawbacks
ofgeneratingmethods,namelytheexcessiveamountofcomputationalresourcesrequired,
bothintimeandstoragespace,andthelargeamountofinformationpresentedtotheDM
ineachinteraction. Thus,themethoddoesnotintendtoprovideagoodapproximationfor
thewholesetofnondominatedsolutions,butjustforthesubregionsofgreaterinterestto
theDM.
SimulatedAnnealingandTabuSearchworkastwoalternativeandcomplementarycom-
putingprocedures.AcomputingphaseisperformedwhenevertheDMchoosesa(sub)region
tobeexplored. Itconsistsinrunningametaheuristicroutine(SimulatedAnnealingorTabu
Search)k times(k beingthenumberofobjectives). Thedistinctversionsofmetaheuristic
approaches are adaptations of 0-1SA and 0-1TS (described in Section 1) for the multi-
objectivecase.Theseproceduresaresimilartosingleobjectiveonesdifferingonlyinthe
gatheringofp.n.d. solutions. Asetofp.n.d. solutions,Set pnd;iscreatedatthestartand
updatedwheneverafeasibleneighboursolution(acceptedornot)isnotdominatedbyany
solutionpreviouslyfound. TheneighboursolutionisaddedtotheSet pndremovingallthe
solutionsdominatedbythisnewp.n.dsolution.
Theithrun.i D1;:::;k/ofametaheuristicroutineprivilegestheobjectivefunctioniby
usingstandardselectionandacceptancecriteria. Forinstance,theprobabilityofaccepting
theneighbour y ofthecurrentsolution xo attheithrunofSimulatedAnnealingisgiven
byminf1;e.fi.y/¡fi.xo//=Tg(recallthatobjectivefunction fi wasdefinedtobemaximized).
Thisimpliesthat,ifydominatesxoor,yisnondominatedinrelationtoxoandhasabetter
performancethanxo fortheobjectivei,then y isaccepted. Otherwiseithasaprobability
of being accepted less than 1. This depends on the magnitude of the decrease in the ith
objectivefunctionandthetemperatureT. However,inbothcasesthesetofp.n.d. solutions
canbeupdatedwith y.
The main reason for so doing—using a pre-defined objective function (i.e. that does
notchangeduringthewholerun)ineachrunofSimulatedAnnealingorTabuSearch—is
toavoiddamagingthe“convergence”characterofthesemetaheuristicroutinesthatcould
bringoutareallygoodsolutiontoonespecificfunction. Anacceptancecriterionjustbased
onadominance/nondominancerelationwouldleadtoamoredispersedsetofsolutionsbut
couldrisktobefarfromallpartsofthe“true”nondominatedfrontier. Therefore,sincethe
metaheuristicroutinerunsk timesforeachinteraction,privilegingeachobjectivefunction