Improving the anytime behavior of two-phase local search

30 Pages·2011·1.14 MB·English

AnnMathArtifIntell(2011)61:125–154
DOI10.1007/s10472-011-9235-0
Improving the anytime behavior of two-phase
local search
JérémieDubois-Lacoste·ManuelLópez-Ibáñez·
ThomasStützle
Publishedonline:1June2011
©SpringerScience+BusinessMediaB.V.2011
Abstract Algorithms based on the two-phase local search (TPLS) framework are
a powerful method to efficiently tackle multi-objective combinatorial optimization
problems.TPLSalgorithmssolveasequenceofscalarizations,thatis,weightedsum
aggregations, of the multi-objective problem. Each successive scalarization uses a
differentweightfromapredefinedsequenceofweights.TPLSrequiresdefiningthe
stopping criterion (the number of weights) a priori, and it does not produce satis-
factory results if stopped before completion. Therefore, TPLS has poor “anytime”
behavior.ThisarticleexaminesvariantsofTPLSthatimproveits“anytime”behavior
by adaptively generating the sequence of weights while solving the problem. The
aimistofillthe“largestgap”inthecurrentapproximationtotheParetofront.The
results presented here show that the best adaptive TPLS variants are superior to
the “classical” TPLS strategies in terms of anytime behavior, matching, and often
surpassing,themintermsoffinalquality,evenifthelatterrununtilcompletion.
Keywords Multi-objective·Anytimealgorithms·Two-phaselocalsearch
MathematicsSubjectClassification(2010) 68T20
J.Dubois-Lacoste(B)·M.López-Ibáñez·T.Stützle
IRIDIA,UniversitéLibredeBruxelles,Brussels,Belgium
e-mail:[email protected]
M.López-Ibáñez
e-mail:[email protected]
T.Stützle
e-mail:[email protected]
126 J.Dubois-Lacosteetal.
1Introduction
Two-phase local search (TPLS) [19] is a stochastic local search (SLS) method for
tackling multi-objective combinatorial optimization problems in terms of Pareto-
optimality[20].TPLSisanessentialcomponentofstate-of-the-artalgorithmsfora
numberofproblemssuchasthebi-objectivetravelingsalesmanproblem(bTSP)[17],
its variant with three objectives [21], and the bi-objective permutation flow-shop
scheduling problem (bPFSP) [5, 8]. In these algorithms, the solutions obtained by
TPLSarefurtherimprovedbyapplyingaParetoLocalSearch[22].
The central idea of TPLS is to start with a high quality solution for a single
objective (first phase) and then to solve a sequence of scalarizations of the multi-
objective problem (second phase). In TPLS, each successive scalarization uses a
slightly different weight and starts from the best solution found by the previous
scalarization.Originally,thesetofweightsand,hence,thenumberofscalarizations,
is defined before the execution of TPLS in order to equally distribute the compu-
tationaleffortalongtheParetofront[19].ThisstrategyimpliesthatstoppingTPLS
atanarbitrarytimebeforeithasperformedthepredefinednumberofscalarizations
would produce a poor approximation to the Pareto front in some regions. In this
sense,TPLSdoesnothaveagoodanytimebehavior.
Anytime algorithms [24] aim at producing an as high as possible performance
at any moment of their execution, without assuming a predefined termination
criterion. Recently, we proposed two variants of TPLS that improve its anytime
performance [6]. The first variant, Regular Anytime TPLS (RA-TPLS), uses a
regular distribution of the weight vectors, equally distributing the effort in all
directions of the objective space. We tested this variant on two bPFSPs, where the
objectivesresultinadifferentalgorithmbehaviorfortheunderlyingsingle-objective
algorithms.Fortheseproblems,wefoundthatanadaptiveTPLS,whichadaptsthe
weights in dependence of the shape of the Pareto front, performed better than a
regularstrategy.
Inthispaper,were-examineRA-TPLSandtheadaptiveTPLSalgorithmsforthe
bTSP and the bPFSP. In addition, we propose new design alternatives of adaptive
TPLS.First,westudytheeffectofperformingoneortwoscalarizationsperweight
(withdifferentseeds),andwegiveevidencewheneachapproachmaybebeneficial.
Second,weproposeanewmethodtochoosetheregionoftheobjectivespacewhere
thesearchshouldbeintensified,thatis,howtodefinethelargestgapinthecurrent
approximationoftheParetofront.InsteadofusingtheEuclideandistancebetween
solutions[6],thisnewmethodisbasedonanoptimisticestimateoftheimprovement
in the hypervolume indicator. Our experimental results show that the new method
further improves the results of the adaptive TPLS strategy in the bTSP, no matter
theshapeofthefront.
ThemainconclusionofourstudyisthattheadaptiveTPLSvariantsshowamuch
betteranytimebehaviorthantheoriginalTPLSalgorithms,andthebestperforming
adaptivevariantstypicallyreturnbetterqualityapproximationstotheParetofront,
asindicatedbythehypervolumeindicator.
The paper is structured as follows. In Section 2, we introduce some formal
definitionsonbi-objectiveoptimizationandwepresenttheoriginalTPLSalgorithms.
In Section 3, we present our first proposal to improve the anytime property of
TPLS.InSection4,weproposeanadaptiveTPLSframeworkthatnotonlysatisfies
Improvingtheanytimebehavioroftwo-phaselocalsearch 127
the anytime property, but that also adapts to the shape of the Pareto front to
maximizetheperformance.WeperforminSection5adetailedstatisticalanalysisto
compareallstrategies.InSection6,weproposetheuseofanoptimisticestimation
of the potential hypervolume contribution to direct the search of adaptive TPLS,
and we show that it leads to further improvements in the bTSP. In Section 7, we
graphically examine the differences in quality of the algorithms. We conclude in
Section8.
2Preliminaries
Inthispaper,wefocusonbi-objectivecombinatorialoptimizationproblemsandhere
wedescriberelevantbackgroundfortheremainderofthepaper.
2.1Bi-objectivecombinatorialoptimization
Inbi-objectivecombinatorialoptimizationproblems,candidatesolutionsareevalu-
atedaccordingtoanobjectivefunctionvectorf=(f , f ).Assuming,withoutlossof
1 2
generality,thatbothobjectivefunctionsmustbeminimized,thedominancecriterion
definesapartialorderamongobjectivevectorsasfollows.Giventwovectorsu,v∈
R2,wesaythatudominatesv(u≺v)iffu(cid:4)=vandu ≤v, i=1,2.Whenu⊀vand
i i
v⊀u, we say that u and v are mutually non-dominated. For simplicity, we extend
the dominance criteria to solutions, that is, a solution s dominates another one s(cid:6)
iff f(s)≺f(s(cid:6)). If no s(cid:6) exists such that f(s(cid:6))≺f(s), the solution s is called Pareto-
optimal.Inabi-objectiveoptimizationproblemwherenoaprioriassumptionsupon
the decision maker’s preferences are made, the problem becomes to find a set of
feasiblesolutionsthat“minimize”finthesenseofParetooptimality.Hence,thegoal
istodeterminethesetofallPareto-optimalsolutions,whoseimageintheobjective
space is called the Pareto front. Since this goal is in many cases computationally
intractable[9],inpracticethegoalbecomestofindthebestpossibleapproximation
totheParetofrontwithinaspecifictimelimit.
2.2Two-phaselocalsearch
Two-phaselocalsearch(TPLS)[19]isageneralalgorithmicframeworkthat,asthe
name suggests, is composed of two phases. In the first phase, a single-objective
algorithm generates a high-quality solution for one of the objectives. This high-
qualitysolutionservesasthestartingpointofthesecondphase,whereasequenceof
scalarizations,thatis,weightedsumaggregationsofthemultipleobjectivefunctions
into single scalar functions, are tackled. Each scalarization uses the best solution
found by the previous scalarization as the initial solution. TPLS will be successful
if effective single-objective algorithms are available, andsolutions that areclose to
eachotherinthesolutionspacearealsocloseintheobjectivespace.
The advantage of considering weighted sum scalarized problems when tackling
a multi-objective one is that an optimal solution for the former is also a Pareto-
optimalsolutionforthelatter.Differentscalarizingfunctionsmaybeused,suchas
Tchebychefffunctions,butthepropertymentionedabovedoesnotnecessarilyhold.
Inabi-objectiveproblem,anormalizedweightvectorisoftheformλ=(λ,1−λ),
128 J.Dubois-Lacosteetal.
Algorithm1Two-phaselocalsearch
1: π1 :=SLS1()
2: π2 :=SLS2()
3: Addπ1,π2toArchive
4: if1to2then
π(cid:6) :=π
5: 1
6: else
π(cid:6) :=π
7: 2
8: endif
9: foreachweightλdo
10: π(cid:6) :=SLS(cid:4)(π(cid:6),λ)
11: Addπ(cid:6)toArchive
12: endfor
13: RemoveDominated(Archive)
14: Output:Archive
λ∈[0,1]⊂R, and the scalar value of a solution s with objective function vector
f(s)=(f (s), f (s))iscomputedas
1 2
fλ(s)=λ· f1(s)+(1−λ)· f2(s). (1)
Depending on the sequence of weight vectors considered, there are two main
TPLSstrategiesintheliterature:
Single direction (1to2 or 2to1) The simplest way to define a sequence of scalar-
izationsistousearegularsequenceofweightvectorsfromthefirstobjectivetothe
secondorfromthesecondobjectivetothefirstone.Wecallthesealternatives1to2or
2to1,dependingonthedirectionfollowed.Forexample,thesuccessivescalarizations
in 1to2 are defined by the weights λ =1−(i−1)/(N −1), i=1,...,N ,
i scalar scalar
where N is the number of scalarizations. (For simplicity, we henceforth refer
scalar
toweightvectorsbytheirfirstcomponentonly,sincethesecondcomponentcanbe
easily derived from (1).) In 2to1 the sequence is reversed. Two drawbacks of this
simplestrategyarethat(i)thedirectionchosencangiveanadvantagetothestarting
objective,thatis,theParetofrontapproximationwillbebetteronthestartingside;
andthat(ii)oneneedstoknowinadvancethecomputationtimethatisavailablein
ordertodefineappropriatelythenumberofscalarizationsandthetimespentoneach
scalarization.DifferentfromtheoriginalTPLSproposal[19],inourimplementation
we first generate a very good solution for each single objective problem because
wehavehighperformingalgorithmsforthemandwewanttobeconsistentwithour
otherTPLSvariantsinSections3,4and6.However,weuseonlyoneofthesolutions
as a starting solution for further scalarizations. Algorithm 1 gives the pseudo-code
of the single direction TPLS. We denote by SLS and SLS the SLS algorithms to
1 2
minimize the first and the second single objectives, respectively. SLS(cid:4) is the SLS
algorithmtominimizethescalarizedproblem.
Doublestrategy WedenoteasDoubleTPLS(D-TPLS)[19]thestrategythatfirst
goessequentially fromoneobjective tothe other one,asin theusualTPLS. Then,
another sequence of scalarizations is generated starting from the second objective
Improvingtheanytimebehavioroftwo-phaselocalsearch 129
backtothefirstone.Thisis,infact,acombinationof1to2and2to1,wherehalfofthe
scalarizationsaredefinedsequentiallyfromoneobjectivetotheother,andtheother
halfintheoppositedirection.Thisapproachtriestoavoidthebiasofasinglestarting
objective.Tointroducemorevariability,inourD-TPLSimplementation,theweights
usedinthesecondTPLSpassarelocatedin-betweentheweightsusedforthefirst
TPLS pass. D-TPLS still requires to define the number of weights, and, hence, the
computationtime,inadvance.
3RegularanytimeTPLS
TheoriginalstrategyofTPLS,whichisbasedondefiningsuccessiveweightvectors
withminimalweightchanges,generatesverygoodapproximationstotheareasofthe
Paretofront“covered”bytheweightvectors[19,21].However,ifTPLSisstopped
prematurely,itleavesareasoftheParetofrontunexplored.Inthissection,wepresent
ourfirstproposaltoimprovetheanytimebehaviorofTPLS.
3.1Regularanytimestrategy
WehaveproposedaTPLS-likealgorithm,calledregularanytimeTPLS(RA-TPLS),
inwhichtheweightforeachnewscalarizationisdefinedinthemiddleoftheinterval
oftwopreviousconsecutiveweights[6].Thisstrategyprovidesafinerapproximation
totheParetofrontasthenumberofscalarizationsincreases,ensuringafairdistribu-
tionofthecomputationaleffortalongtheParetofrontandgraduallyintensifyingthe
search.Thesetofweightsisdefinedasasequenceofprogressivelyfiner“levels”of
2k−1 scalarizations(atlevelk)withmaximallydispersedweights(cid:5) inthefollowing
k
manner: (cid:5) ={0.5}, (cid:5) ={0.25,0.75}, (cid:5) ={0.125,0.375,0.625,0.875}, and so on.
1 2 3
Successivelevelsintensifytheexplorationoftheobjectivespace,fillingthegapsin
the Pareto front. The two initial solutions minimizing each objective could be seen
aslevel0:(cid:5) ={0,1}.OnceRA-TPLScompletesonelevel,thecomputationaleffort
0
has been equally distributed in all directions. However, if the search stops before
exploring all scalarizations at a certain level, the search would explore some areas
of the Pareto front more thoroughly than others. In order to minimize this effect,
RA-TPLSconsiderstheweightswithinonelevelinarandomorder.
In order to be an alternative to TPLS, RA-TPLS starts each new scalarization
from a solution obtained from a previous scalarization. In particular, the initial
solutionofthenewscalarization(usinganewweight)isoneofthetwosolutionsthat
wereobtainedusingthetwoweightvectorsclosesttothenewweight.Thealgorithm
computestheweightedsumscalarvaluesofthesetwosolutionsaccordingtothenew
weight, and selects the one with the better value as the initial solution of the new
scalarization.
The implementation of RA-TPLS requires three main data structures: L is the
i
set of pairs of weights used in previous scalarizations, wherei determines the level
ofthesearch;Sdisasetofpotentialinitialsolutions,eachsolutionbeingassociated
withthecorrespondingweightthatwasusedtogenerateit;Archiveisthearchiveof
non-dominatedsolutions.
Algorithm 2 describes RA-TPLS in detail. In the initialization phase, an initial
solutionisobtainedforeachobjectiveusingappropriatesingle-objectivealgorithms,
130 J.Dubois-Lacosteetal.
Algorithm2RA-TPLS
1: s1 :=SLS1()
2: s2 :=SLS2()
3: Adds1,s2toArchive
4: L0 :={(1,0)};Li :=∅ ∀i>0
5: Sd:={(s1,1),(s2,0)}
6: i:=0
7: whilenotstoppingcriterionmetdo
8: (λsup,λinf):=extractrandomlyfromLi
9: Li := Li\(λsup,λinf)
10: λ:=(λsup+λinf)/2
11: s:=ChooseSeed(Sd,λ)
12: s(cid:6) :=SLS(cid:4)(s,λ)
13: Adds(cid:6)toArchive
14: Sd:=Sd∪(s(cid:6),λ)
15: RemoveDominated(Sd)
16: Li+1 := Li+1∪(λsup,λ)∪(λ,λinf)
17: ifLi =∅theni:=i+1
18: endwhile
19: RemoveDominated(Archive)
20: Output:Archive
SLS ()andSLS ().Thesenewsolutionsandtheircorrespondingweights,λ=1and
1 2
λ=0,respectively,areusedtoinitializeL andSd.Inthenextphase,thewhileloop
0
is iterated until a stopping criterion is met. At each iteration, a pair of consecutive
weights (λ ,λ ) is subtracted randomly from L and used to calculate the new
sup inf i
weightλ=(λ +λ )/2.Then,procedureChooseSeedusesthisweightλtochoose
sup inf
a solution from the set of initial solutions Sd. To do so, first ChooseSeed finds
the two non-dominated solutions that were obtained from scalarizations using the
weightsclosesttoλ:
(cid:2) (cid:3)
s = s | max {λ :λ <λ}
inf i i i
(cid:2) (si,λi)∈Sd (cid:3) (2)
s = s | min {λ :λ >λ} .
sup i i i
(si,λi)∈Sd
Next, ChooseSeed calculates the scalar value of s and s according to the
sup inf
new weight λ following (1), and returns the solution with the smaller scalar value.
This solution is the initial solution for SLS(cid:4), the SLS algorithm used to tackle the
scalarizations. This algorithm produces a new solution s(cid:6), which is added to both
theglobalArchiveand(togetherwithitscorrespondingweight)tothesetofinitial
solutions Sd, from which any dominated solutions are removed. Finally, the set of
weightsforthenextlevel Li+1 isextendedwiththenewpairs(λsup,λ)and(λ,λinf).
This completes one iteration of the loop. If the current set of weights L is empty,
i
a level of the search is complete, and the algorithm starts using pairs of weights
from the next level Li+1. In principle, this procedure may continue indefinitely,
although larger number of scalarizations will lead to diminishing improvements in
theapproximationtotheParetofront.
Improvingtheanytimebehavioroftwo-phaselocalsearch 131
3.2Experimentalanalysis
Intheoriginalpublication,weappliedRA-TPLStotwobPFSPs[6].Inthispaper,we
performamorecomprehensivestudybyrepeatingtheexperimentsonalargersetof
instancesforthebPFSPsandincludingresultsonthebTSP.Thelatterallowsusto
showthattheshapeoftheParetofrontplaysafundamentalroleintheperformance
oftheRA-TPLSstrategy.
AllalgorithmsevaluatedinthispaperwereimplementedinC++,compiledwith
gcc4.4,andtheexperimentswererunonasinglecoreofIntelXeonE5410CPUs,
running at 2.33Ghz with 6MB of cache size under Cluster Rocks Linux version
4.2.1/CentOS4.
3.2.1Casestudy:bi-objectivetravelingsalesmanproblem(bTSP)
Given a complete graph G=(V,E) with n=|V| nodes {v ,...,v }, a set of edges
1 n
E,andacostassociatedtoeachedgec(v,v ),thegoalinthesingle-objectiveTSPis
i j
tofindaHamiltoniantour p=(p ,...,p )thatminimizesthetotaltourcost:
1 n
(cid:4) (cid:5) (cid:6)n−1 (cid:4) (cid:5)
minimize f(p)=c v ,v + c v ,v
pn p1 pi pi+1
i=1
ThesingleobjectiveTSPmaybedirectlyextendedtoamulti-objectiveformula-
tionbyassigningavectorofcoststoeachedge,whereeachcomponentcorresponds
tothecostofeachobjective.ThegoalthenbecomestofindthesetofHamiltonian
tours that “minimizes” a vector of objective functions, where each objective is
definedasabove.
Herewefocusonthebi-objectiveTSP(bTSP).Weassumethatthepreferencesof
thedecisionmakerarenotknownapriori.Hence,thegoalistofindasetoffeasible
solutions that “minimizes” the bTSP in the sense of Pareto optimality. The bTSP
is frequently used for testing algorithms and comparing their performance [9, 21].
Moreover,TPLSisamaincomponentofthecurrentstate-of-the-artalgorithm[17]
forthebTSP.
IsometricandanisometricbTSPinstances Wecreated10EuclideanbTSPinstances
by generating for each instance two sets of 1,000 points with integer coordinates
uniformlydistributedinasquareofside-length105.Wecalltheseinstancesisometric
becausebothdistancematriceshavesimilarrange.
In addition, we generated other bTSP instances, where the first distance matrix
(correspondingtothefirstobjective)isEuclideanwhereasthesecondmatrix(corre-
spondingtothesecondobjective)israndomlygeneratedwithdistancevaluesinthe
range [1,max ], with max ∈{5,10,25,100}. Given the different range of both
dist dist
distancematrices,wecalltheseinstancesanisometric.Wegenerated10instancesof
1,000nodesforeachvalueofmax ,thatis,40anisometricbTSPinstancesintotal.
dist
ExperimentalsetupforthebTSP Theunderlyingsingle-objectivealgorithmforthe
TSPisaniteratedlocalsearch(ILS)algorithmbasedonafirst-improvement3-opt
algorithm [13].1 In order to speed up the algorithm, we compute a new distance
1Thisalgorithmisavailableonlineathttp://www.sls-book.net/
132 J.Dubois-Lacosteetal.
matrixforeachscalarizationandwerecomputethecandidatesetsusedbythespeed-
up techniques of this ILS algorithm. For each scalarization, ILS runs for 1,000 ILS
iterations(equaltothenumberofnodesintheinstance).Withourimplementation
andcomputingenvironment,1,000iterationsrequirebetween0.5and1CPUseconds
dependingontheinstances.Eachofthetwoinitialsolutionsisgeneratedbyrunning
ILSfor2,000iterations.Finally,eachrunofthemulti-objectivealgorithmsperforms
30scalarizationsaftergeneratingthetwoinitialsolutions.Thenormalizationofthe
objectives, necessary when solving a scalarization or calculating a weighted sum
of the objectives, is performed by normalizing the two distance matrices to the
samerange.
We measure the quality of the results by means of the hypervolume unary
measure [10, 25]. In the bi-objective space, the hypervolume measures the area of
the objective space that is weakly dominated by the image of the solutions in a
non-dominated set. This area is bounded by a reference point that is worse in all
objectives than all points in all non-dominated sets measured. The larger is this
area,thebetterisanon-dominatedset.Tocomputethehypervolume,theobjective
valuesofallnon-dominatedsolutionsarenormalizedtotherange[1,2],thevalues
corresponding to the limits of the interval being the minimum and the maximum
values ever found for each objective. We use (2.1,2.1) as the reference point for
computingthehypervolume.
Experimental evaluation of RA-TPLS on bTSP instances We first study how the
differentTPLSstrategiessatisfytheanytimepropertybyexaminingthequalityofthe
Paretofrontas thenumber ofscalarizationsincreases.ForeachTPLSstrategy,we
plotthehypervolumevalueaftereachscalarizationaveragedacross15independent
runs.Figure1showsfourexemplaryplotscomparingRA-TPLS,1to2andD-TPLS
on two isometric bTSP instances, and two anisometric bTSP instances. We do not
show the strategy 2to1 for isometric instances because it performs almost identical
to 1to2 w.r.t. the hypervolume; however, for anisometric instances we include 2to1
due to its rather different behavior when compared to 1to2. These plots are repre-
sentativeofthegeneralresultsonotherinstances;thecompleteresultsaregivenas
supplementarymaterial[7].
For isometric bTSP instances, according to the top plots in Fig. 1, the three
strategies reach similar final quality. However, there are strong differences in the
development of the hypervolume during the execution of the algorithms. The
hypervolume of the Pareto front approximations generated by RA-TPLS shows a
quick initial increase, and for few scalarizations a much higher value than D-TPLS
and1to2.Hence,ifthealgorithmsareinterruptedbeforecompletingthepre-defined
numberofscalarizations,RA-TPLSwouldclearlyproducethebestresults.
ForanisometricbTSPinstances,wherethefirstobjectivecorrespondstoaEuclid-
eandistancematrixandthesecondobjectivecorrespondstoarandomlygenerated
matrixwithdistancevaluesintherange[1,max ],thebottomplotsinFig.1show
dist
acleardifferencebetweenstrategies1to2and2to1.Moreover,thesmallerthevalue
ofmax ,thelargeristhedifference.Thevalueofmax alsoaffectstheanytime
dist dist
behavior and final performance of RA-TPLS. For small max , both 1to2 and D-
dist
TPLS seem to outperform RA-TPLS at various times. This can be explained as
follows.Smallervaluesofmax resultinaverylargenumberofoptimalsolutions
dist
for the second objective. In such instances, the first scalarization of 2to1, which
Improvingtheanytimebehavioroftwo-phaselocalsearch 133
1 1
1. 1.
ume 0.9 ume 0.9
ol ol
v v
Hyper 0.7 Hyper 0.7
1to2 1to2
0.5 RDA−T−PTLPSLS 0.5 DRA−T−PTLPSLS
0 5 10 15 20 25 30 0 5 10 15 20 25 30
Number of scalarizations Number of scalarizations
(max = 5) (max = 100)
dist dist
2
1.
1
1.
1
ume 1. ume 1.0
ervol 1.0 ervol 0.9
p p
y y
H 9 1to2 H 0.8 1to2
0. 2to1 2to1
D−TPLS 7 D−TPLS
RA−TPLS 0. RA−TPLS
0 5 10 15 20 25 30 0 5 10 15 20 25 30
Number of scalarizations Number of scalarizations
Fig.1 Developmentofthehypervolumeoverthenumberofscalarizationsfor1to2,D-TPLSand
RA-TPLS for two isometric (top plots) and two anisometric (bottom plots) bTSP instances. For
anisometric instances we also plot the results of 2to1, since they differ strongly from 1to2. For
isometricinstances,thereisalmostnodifferencebetween1to2and2to1.Anisometricinstanceswith
intermediatevaluesofmaxdist(equalto10or25)showacompromisetrendbetweenthetwoextreme
values5and100(seesupplementarymaterial[7])
uses a nonzero weight for the first objective and a large weight for the second,
generates a solution with a value of the second objective being still optimal and
a good value of the first objective. This translates into a huge initial improvement
of the hypervolume. The underlying reason is that the initial solution minimizing
the second objective is weakly dominated by the solution returned for solving the
firstscalarization.Severalsubsequentscalarizationsof2to1improveonlyslightlythe
valueofthefirstobjective,whilekeepingtheoptimalvalueofthesecondobjective;
therefore,thehypervolumeimprovesveryslowly.Onlywhentheweightforthefirst
objective grows large enough, a solution with a non-optimal value of the second
objective is chosen, and the hypervolume starts improving in larger steps. On the
other hand, when starting from the first objective in 1to2, every scalarization finds
non-dominated solutions closer to each other, and the hypervolume grows initially
slower than what is observed for the first huge step in 2to1. However, as soon
as the weight of the second objective is large enough that only optimal values
of the second objective solutions are accepted, the hypervolume quickly reaches
its maximum. Finally, D-TPLS obtains better results because it progresses faster
towardsthesecondobjective.Allthesebehaviorsshowthatequallydistributingthe
computationaleffortinalldirectionsdoesnotpayoffintheseinstances.Itleadstoa
wasteofscalarizationswhenbeingclosetotheoptimumofthesecondobjective,and
134 J.Dubois-Lacosteetal.
veryslowprogresswhenbeingclosetotheoptimumofthefirstobjective.Thiseffect
willbeevenstrongerinthecaseofthebPFSP.
3.2.2Casestudy:bi-objectivepermutationflow-shopschedulingproblem
Theflow-shopschedulingproblem[14]modelsaverycommontypeofenvironment
in industry and it is therefore one of the most widely studied scheduling problems.
Inthe flow-shop schedulingproblem,asetof n jobs(J ,...,J ) is to beprocessed
1 n
onmmachines(M ,...,M ).Alljobsgothroughthemachinesinthesameorder,
1 m
i.e., all jobs have to be processed first on machine M , then on machine M , and
1 2
so on until machine M . A typical additional constraint is to forbid job passing,
m
and, as a result, the processing sequence of the jobs is the same on all machines.
Thus,anypermutationofthejobsisacandidatesolution.Thisformulationiscalled
permutationflow-shopschedulingproblem(PFSP).
InthePFSP,allprocessingtimes p forajob J onamachineM arefixed,known
ij i j
in advance, and non-negative. Furthermore, we assume that all jobs are available
at time 0. C denotes the completion time of a job J on the last machine M . The
i i m
makespan(C )isthecompletiontimeofthelastjobinthepermutation.ThePFSP
max
withmakespanminimizationformorethantwomachinesisNP-hardinthestrong
sense[11].
The other objectives studied in this paper are the minimization of the sum of
completiontimesa(cid:7)ndtheminimizationofthetotaltardiness.Thesumofcompletion
times is given by n C. The PFSP with sum of completion times minimization is
i=1 i
stronglyNP-hardalreadyfortwomachines[11].Eachjobmayhaveanadditional
associatedduedatedi.Thetar(cid:7)dinessofajobJiisdefinedasTi =max{Ci−di,0},and
thetotaltardinessisgivenby n T.ThePFSPwithtotaltardinessminimizationis
i=1 i
stronglyNP-hardevenforasinglemachine[4].
Asacasestudy,inthispaperwefocusontwobPFSPvariants:
(cid:7)
1. PFSP-(C , C) denotes the minimization of the makespan and the sum of
max i
completiontim(cid:7)es,and
2. PFSP-(C , T) denotes the minimization of the makespan and the total
max i
tardiness.
ExperimentalsetupforthebPFSP Recently,wedevelopedanew,hybridstate-of-
the-artSLSalgorithmforthesetwobPFSPs[5,8].Acrucialcomponentinthishybrid
algorithm is TPLS using effective iterated greedy (IG) algorithms [23] adapted to
each objective and combinations thereof. We use here the same IG algorithms to
implement the various TPLS variants. Concretely, each TPLS algorithm generates
twoinitialsolutionsforeachobjectivebyrunning1,000iterationsofthecorrespond-
ingIGalgorithm.Then,itperforms30scalarizations,eachscalarizationrunning500
iterationsoftheIGcorrespondingtothecombinationofobjectives.
Wegenerate10benchmarkinstanceswithn=50andm=20(50 × 20),and10
instanceswithn=100andm=20(100 × 20),followingtheproceduredescribed
by Minella et al. [18]. These instances are available as supplementary material [7].
Giventhelargediscrepanciesintherangeofthevariousobjectives,allobjectivesare
dynamicallynormalizedusingthemaximumandminimumvaluesfoundduringeach
run for each objective. We compute and plot the evolution of the hypervolume as
doneearlierforthebTSP.