Laboratoire d'informatique fondamentale de Lille
Thèse d'habilitation à diriger des re her hes
présentée par
François Clautiaux
spé ialité : informatique
New ollaborative approa hes
for bin-pa king problems
Philippe Baptiste Dire teur dere her he,E ole Polyte hnique (Rapporteur)
Ja ques Carlier Professeur, Université de Te hnologie de Compiègne (Examinateur)
Martine Labbé Professeur, Université Libre deBruxelles (Rapporteur)
Andrea Lodi Professeur, Université de Bologne (Rapporteur)
Aziz Moukrim Professeur, Université de Te hnologie de Compiègne (Examinateur)
El-Ghazali Talbi Professeur, Université de LilleI (Garant del'HDR)
Sophie Tison Professeur, Université de LilleI (Examinateur)
Je souhaite tout d'abord remer ier Philippe Baptiste, Martine Labbé et Andrea Lodi
d'avoira eptéd'êtrerapporteurspourmonhabilitationàdirigerdesre her hes. Mer i
aussi à SophieTison d'avoira epté de parti iper aujury.
Cette habilitation n'aurait pas de raison d'être sans les étudiants her heurs que
j'ai en adrés. Mer i tout d'abord à Ali Khanafer. Sa thèse représente une partie
importante de e do ument. Par son travail a harné, il a pla é la barre haute pour
ses su esseurs. Mer i égalementà NadiaDahmani etInes Bahri,qui ont toutes deux
ommen é leurtravailsous monen adrement, ainsi qu'àMatthieuGérard(un jour on
le onvain rade s'ins rireen thèse).
Au oursdemestravauxdere her he,j'aieuleprivilègede ollaborerave d'ex ellents
her heurs. Jemepermetsde distinguerAntoine Jouglet,SaïdHana, ainsique
Clau-dio Alves, ave qui j'ai le plus travaillé. J'ai beau oup appris auprès d'eux, tout en
appré iant leurs immenses qualités humaines. Mer i aussi à Ja ques Carlier et Aziz
Moukrim. En ommençant ma arrière de her heur sous leur dire tion, je n'avais
au une ex use en as d'é he .
Je remer ie également Rita Ma edo, Valério de Carvalho, Jürgen Rietz, Manuel
Iori, Mauro Dell'Ami o, Christophe Wilbaut et Raïd Mansi, pour les fru tueuses
ol-laborationsque nous menons etmènerons ensemble.
Entantqueresponsabledel'équipeDOLPHIN,El-GhazaliTalbiafaitlené essaire
pour me mettre dans les onditions qui m'ont permis de présenter ette habilitation.
Qu'il soitassuré de mon gratitude àson égard.
Un grand mer i aux ollègues de l'IUT A de Lille 1, qui auront été d'un grand
soutien depuis le début. Mer i également aux ollègues de l'équipe DOLPHIN qui
m'ontsoutenu. Jeremer ie en parti ulierNouredine Melab pour m'avoirfait partager
son expérien eet pour les bons moments passés en sa ompagnie.
A knowledgements iii
Introdu tion 1
1 De ompositionmethodsandstrategi os illationforbin-pa king
prob-lems with oni ts 5
1.1 Introdu tion . . . 5
1.2 A tree-de omposition based resolution s heme for the bin-pa kingwith
oni ts . . . 6
1.2.1 Tree-de omposition and graph triangulation . . . 8
1.2.2 A general s heme for applying a tree-de omposition to the
bin-pa kingproblem with oni ts . . . 9
1.2.3 The luster-separation problem . . . 10
1.2.4 Greedy heuristi s for the luster separation problem . . . 11
1.2.5 A tabu sear h based on the tree de omposition and strategi
os illation . . . 11
1.2.6 Time omplexity of the method . . . 13
1.2.7 Synthesis of the omputational experiments . . . 14
1.3 A olumn-generationalgorithmfor the min- oni t pa king problem . . 15
1.3.1 A simple ompa t model . . . 15
1.3.2 A set overing formulation . . . 16
1.3.3 Solvingthepri ingsubproblemusinglo alsear handlinear
pro-gramming . . . 17
1.3.4 Computingtheinitialsetof olumnsusingtabusear hand
strate-gi os illation . . . 19
1.3.5 Synthesis of the omputational experiments . . . 21
1.4 A olumngeneration algorithmfor the bin-pa king with fragileobje ts 22
1.4.1 Compa t models . . . 23
1.4.2 A set overing formulation . . . 25
1.4.3 SolvingthePri ingSubproblemusingdynami programmingand
linear programming . . . 27
1.4.4 Computingtheinitialsetof olumnsusingVariable-Neighborhood
2 Dual-feasible fun tions and extensions 35
2.1 Introdu tion . . . 35
2.2 Classi aldual-feasiblefun tions . . . 36
2.2.1 Denitions and properties . . . 36
2.2.2 Data-dependent DFF . . . 37
2.2.3 Appli ations of DFF . . . 38
2.2.4 Dis rete DFF and the modelof Gilmoreand Gomory . . . 38
2.3 A generalpointon viewonDFF . . . 40
2.4 Computationof CS-DFF . . . 41
2.4.1 Frameworks for reatingvalidCS-DFF . . . 41
2.4.2 A omparativeanalysis ofCS-DFF . . . 43
2.4.3 Summary of our literature study. . . 47
2.5 Extensions of DFF for various bin-pa kingproblems . . . 48
2.5.1 DDFF forthe bin-pa kingproblem (BP-DDFF) . . . 48
2.5.2 DDFF forthe bin-pa kingwith oni ts (BPC-DDFF) . . . 50
2.5.3 DFF for the bin-pa king problemwith fragileitems (BPFI-DFF) 53 2.5.4 DFF for two-dimensional bin-pa kingproblems (2BPP-DFF) . . 56
2.6 Con lusions,future works . . . 58
3 Mixing onstraint-programming and OR te hniques for solving re t-angle pla ement problems 61 3.1 Re tangle pla ementproblems . . . 61
3.2 Constraintprogramming . . . 63
3.3 The re tangle pla ementproblem (RPP) . . . 64
3.3.1 A onstraint-baseds heduling modelfor RPP . . . 65
3.3.2 Two-dimensional energeti reasoning . . . 67
3.3.3 UsingDFF in feasibility tests . . . 69
3.3.4 Knapsa k-based feasibilitytests . . . 71
3.3.5 Computationalexperiments: asynthesis . . . 72
3.4 The guillotine- uttingproblem (GCP) . . . 72
3.4.1 Guillotine- utting lasses . . . 73
3.4.2 A new graph-theoreti al model . . . 74
3.4.3 Cy le- ontra table graphs . . . 77
3.4.4 Computingthepatternsasso iatedwitha y le- ontra tablegraph 78 3.4.5 A onstraint-programmingapproa h . . . 81
3.4.6 Computationalexperiments: asynthesis . . . 83
3.5 Con lusions,future works . . . 83
In this do ument, we propose new models and resolution s hemes for problems that
belong tothe family of Cutting and Pa king (C&P) problems [90℄. The most "basi "
NP- ompleteproblemsintheC&Peldarethe bin-pa kingandthe knapsa k
prob-lems. In the former, the obje tive is to nd the minimum number of bins needed to
pa k allthe items.
Problem 1 (Bin-pa king Problem (BPP)) Given a set
I
of itemsi
of sizec
i
, what is the minimum number of bins of sizeC
needed to pa k all the items ofI
?The BPP is the main problem addressed here, but dierent knapsa k problems
also appear as subproblems throughout the do ument. In this se ond problem, all
items annot be pa ked in the ontainers, and the obje tive is to maximize the prot
asso iated with the input items.
Problem 2 (Knapsa k Problem (KP)) Givenaset
I
ofitemsi
ofsizec
i
andprotp
i
, and a knapsa k of sizeC
, nd the subset ofI
whose total size is smaller thanC
whi h maximizes the total protof the sele ted items.Therather simplestru ture of the lassi alBPP has made this problemone of the
most popular to test new methods, or to prove theoreti al results. For example, its
variantof utting-sto k has beenone oftherst problemstobesolved through olumn
generationmethods[50,51℄, andBPP isoneof thefavorite subje tsofapproximability
studies (see for example [38℄). These basi (yet hard) C&P problems an be seen
as "laboratories" in whi h new te hniques are tested. Therefore, many results rst
proposed for these problems lead toimprovements forthe resolution of many others.
Forsolvinghardmulti-dimensionalpa kingproblemse iently,theliteratureshows
that the best results are a hieved using meta-heuristi s, mathemati al programming
and onstraint programming (CP). The rst family of resolution method is ee tive
for large size instan es (see [72℄ for example), while CP an be more e ient than
heuristi s for problems involving one bin only (re tangle pla ement problems) [10℄.
For some parti ular utting problems, where there are many instan es of ea h item
type, mathemati al programming an even be faster than greedy algorithms. This
justies the fa t that most of our algorithms use dierent resolution methods in a
Inthisdo ument,weproposenewmodelsandmethodologiesthatweapplytothree
familiesofpa kingproblems. InChapter1,westudyde ompositionmethodsand
meta-heuristi sbasedonso- alledstrategi os illation. Weapplythesete hniquestopa king
problems with dierent kinds of oni ts. In Chapter 2, we deal with the on ept
of dual-feasible fun tions, whi h are used to derive polynomial-timelower bounds for
severalbin-pa kingproblems,andimprove utsforintegerlinearprograms. InChapter
3,weproposenewmodelsfortwodierentpa kingproblemsintwodimensions. Weuse
thesemodelsintomethodsthatusea ombinationofoperationsresear hand onstraint
programmingte hniques.
Throughout the do ument, we follow a onsistent methodology. A rst feature of
our workis touse te hniques fromdierent elds: most notablyintegerprogramming,
onstraintprogramming,meta-heuristi s, andgraphtheory. Although"hybrid" would
be too strong a word, we always use these dierent methods in a ollaborative way,
whi himprovesthe results that would beobtained by ea hmethodseparately.
Chapter 1 is devoted to dierent kinds of de omposition methods and so- alled
strategi os illation. We used these methods to design lower and upper bounding
strategies for pa king problems with oni ts. In parti ular, we show that these
de- ompositionmethods an lead toee tive ollaborativeresolution s hemes. Our
te h-niques relyontwotypesofde ompositions: Dantzig-Wolfe de omposition[39℄ oflinear
programs, andtree-de omposition [83℄of graphs. Werstpropose aresolutionmethod
for the bin-pa king problem with pairwise oni ts [49℄, based onthe de ompositionof
the oni t graph into several lusters. Ea h luster an be solved independently if a
partition based on the de omposition is omputed. This framework is exploited by a
tabu sear h, whi h assigns and remove items to/from lusters. The se ond problem
addressedisanewbin-pa kingproblemwith oni tsmetinamulti-obje tive ontext.
The numberof bins islimited, andwe needto minimizethe numberof oni ts inthe
bins. Forthisproblem,weproposeamethodbasedonlinear-programmingand olumn
generation. Ourmethodmakesagooduse ofheuristi andmeta-heuristi methodsfor
generating the initialbasis, and the olumns at ea h step of the pro ess. Finally, we
address thebin-pa kingproblemwith fragileitems [5℄, inwhi h oni ts aremodelled
byaleveloffragilityforea hitem. Weproposea olumngenerations hemeforsolving
this problem. We designed a dynami programming s heme for generating iteratively
the olumns, and a meta-heuristi method for initializing the master problem. The
three meta-heuristi s proposed in this hapter are based on the on ept of strategi
os illation, in whi h the sear h os illates from feasible to unfeasible, or omplete to
in omplete solutions.
Chapter 2 is learly onne ted tomathemati al programming and heuristi s. It is
dedi ated toso- alleddual-feasible fun tions (DFF)[75℄, relatedto olumn-generation
te hniquesforthe bin-pa kingproblemandduality. Thesefun tionsareused toderive
on ept of DFF, and hints based on experimentation to determine the problems for
whi hit anbegeneralized. Intherst partofthis hapter,wedes ribethe on ept of
DFFandshowhowitisrelatedtoother on epts intheliterature. Thenwesurveythe
literature in whi h dual-feasible fun tions are used (sometimes impli itly, sometimes
underadierentname),andstressthelinkwithsuperadditive fun tionsusedininteger
programming. Weshow that afew dierent te hniques are su ient togenerate most
fun tions of the literature. The se ondpart of Chapter 2 deals withextensions of this
on ept tovariousbin-pa kingproblems(mostnotablytwo-dimensionalproblems,and
the addition of oni t-based onstraints).
Chapter3 deals withanimportantissue intwo-dimensional utting/pa king
prob-lems: re tangle pla ement problems. This problem has been the subje t of a large
number of ontributions [9,10,4345,54,67,79,82℄. To our knowledge, methodsbased
on onstraintprogramming are the most e ient exa t algorithmsfor these problems
(see [10℄ for example). This an be explainedby the fa tthat linear relaxationsof the
integer models dedi atedtothese problems are generallyweak. Wepropose new
mod-elsfor two dierent re tanglepla ementproblems: the regular ase, and the guillotine
ase. We rst show that the regular ase is tightly linked with umulative s heduling
problems. This allows us to use powerful results from the s heduling eld (energeti
reasoning [41℄, bran hing s hemes, et .). The obtained method relies on an ee tive
ombinationof operationsresear h andCP te hniques. Wealsopropose therst
ee -tive graph-theoreti al modelfor the guillotine ase, whi h aptures the ombinatorial
stru tureof thepatterns,and helpsdesigningaCP-basedresolution method. Forboth
problems, the e ien ies of our methods, whi h are able to ompete with the best
methods of a large literature, relyon the strength of the new models, but alsoon the
De omposition methods and strategi
os illation for bin-pa king problems
with oni ts
The work des ribed in this hapter has been published in an international journal[65℄.
Two reports [31,64℄ are also submitted to international journals.
1.1 Introdu tion
In this hapter,we des ribeour lowerand upper boundsfor various bin-pa king
prob-lemswith oni tsusingde ompositionmethodsandmeta-heuristi sbasedonstrategi
os illation.
Handling oni tsisoneoftherstadditional onstraintstobedemandedin
indus-trialappli ations. In ompatibilities an bemodelledin many ways. Inthis do ument,
we address three variants: hard oni ts, soft oni ts, and fragilities. Given the
dif- ulty of these problems, and the time that would be entailed by an exa t resolution,
we fo us onheuristi and lower boundingmethods.
Pa kingproblems with oni ts are generallyhardertosolvethan the lassi albin
pa kingproblem (BP).Whereas asimple Integer Linear Programming(ILP)
formula-tion an nd good solutions for BP and in many ases good lower bounds in a small
amount of time, this is generally not the ase with oni ts. Sin e these problems are
di ult, a sensible way of addressing them is to de ompose them into subproblems
that will be hopefully easier to solve. We will fo us on two parti ular de omposition
methods: tree-de omposition ofgraphs and Dantzig-Wolfede omposition of ILP.
The methods that we des ribe in this hapter share some similarities. The rst,
as hinted above, is to rely on de omposition methods. The se ond is to generate
solutions using meta-heuristi s based on so- alled strategi os illation. The idea is
to os illate between two sets of solutions: omplete/in omplete for the rst problem,
over- onstrained/relaxed for the se ond, and feasible/non-feasible for the third. We
used this os illationstrategies be ause, given a neighborhood,the oni ts may forbid
non feasible or non omplete solutions are en ountered on the shortest path between
them).
The rst problem onsidered is the lassi al bin-pa king problem with oni ts
(BPC). In this problem, oni ts between two items are forbidden, and are modelled
with agraph. Weshowhowtree-de omposition anbeused tosolvethis problem.
Ap-plyingthisde ompositiontoBPCisnotstraightforward,sin endingapartitionofthe
itemsbasedonthetreede ompositionisahardproblem. Wepropose severalheuristi s
toaddressthisproblem,and atabusear hbasedona onstru tion/destru tions heme
where items are assignedto and de-assigned from lusters.
These ondproblem onsideredisanewproblem,whi hwenamemin- oni t
pa k-ing problem (MCBP). Wemet thisprobleminthe ontextofmulti-obje tive
optimiza-tion. The oni ts areofthe sametypeasthe rstproblem,butthistime, thenumber
of bins is limited, and the obje tive is to minimize the number of violated oni ts.
Weapply Dantzig-Wolfe de omposition toMCBP. Two di ulties arise: generating a
good initial basis and iteratively generating the olumns in an e ient manner. For
theinitialbasis,wedesignedatabu-sear hbasedonos illationbetweensolutionsusing
dierentnumbers of bins. Weproposed heuristi s, alo alsear hmethodand two ILP
models to generate the olumns iteratively. The former are improved by the means of
uts added tothe models.
Thethirdproblemfeaturesadierentvariantof oni ts. Ea hitemhasafragility,
andthe totalsizeoftheitemsinabin annotbelargerthanthesmallestfragilityofan
item inthe bin. Weused amethodology similartothe previousproblem( olumn
gen-eration). Initial olumns are generated by a meta-heuristi , a Variable Neighborhood
Sear h (VNS), based on an os illationbetween feasible and unfeasible solutions. The
subproblem is solved through a new dynami programming s heme. It outperforms
our two ILP models and it nds a solution within a small amount of time for all our
instan es.
1.2 A tree-de omposition based resolution s heme for
the bin-pa king with oni ts
In this se tion, we deal with the lassi al bin-pa king problem with oni ts. The
method we propose is generi and an be used for both one- and multi-dimensional
ases of the problem (the geometri onstraints are handled by sub-routines). In this
do ument, we fo us onthe two-dimensional ase.
Problem 3 (Two-dimensional Bin-pa king Problem with Coni ts (BPC)) Let
I = {1, . . . , n}
be a set of re tangular itemsi
of widthw
i
and heighth
i
, a binB
of widthW
and heightH
, andG = (I, E)
a oni t graph. Two itemsi
andj
are in oni t if(i, j) ∈ E
. What is the minimum number of binsneeded to pa k all items ofI
in binsof typeB
in su h away thattwo oni tingitems are notpa kedin the same bin, and no two items overlap?The problem is dened with a oni t graph. In the following, we mainly use the
ompatibilitygraph
G = (I, I × I \ E)
¯
. Ourde omposition method willbeapplied on this graph.Several heuristi s have been proposed forthe one-dimensional version of BPC [46,
49℄. The most ee tive are based on lassi al any-t algorithms originally designed
for BP, and on the sear h of liques in the graph. The rst-t de reasing algorithm
sort the items by de reasing size, and pa k the items one by one in this order in the
rst bin that an a ommodate it. Forthe two-dimensional ase, the same approa h
an be used. It leads to a larger omputing time sin e verifying that an item an
be pa ked into a bin is more di ult in two dimensions. Pra ti ally speaking, we
use the algorithm bottom-left of Coman [37℄. Other heuristi s an also be used (see
the methods des ribed in [71,73℄ for example). Unfortunately, in many ases, these
algorithms do not lead to interesting results, sin e they do not take into a ount the
stru ture of the graph. In the sequel, we show how a de omposition method an help
su h aalgorithmsto nd abetter solution.
Somemethodsdedi atedto the one-dimensional ase of BPC [49℄ relyonmaximal
liquesorstablesets inthegraph. Findingastablesetinthe ompatibilitygraphgives
a subset of items that have to be pa ked in dierent bins. On the ontrary, nding
a lique gives a subset of ompatible items that an be pa ked together. This notion
anbegeneralizedusingthe on ept oftree-de ompositionappliedtothe ompatibility
graph.
Inatree-de omposition,thegraphisde omposedinto lustersofverti es onne ted
in a tree. Ea h luster orresponds with a subproblem to solve. A property of this
de ompositionmethodis thatea h lique ofthe graph is ontainedentirely inatleast
one luster. Consequently, even if some oni ting items remain inside the lusters,
the ompatibility graph asso iated with ea h subproblem should be denser, and thus
algorithmsdesigned forthe lassi al BP should be more ee tive when appliedto the
dierent lusters.
On eade omposition is omputed, our methodsolvesthe problemrelatedto ea h
luster separately, and merges the solutions found. The most ru ial issue is that a
givenitem/vertex an belongtoseveral lustersofthe de omposition. In arstphase,
we assign ea hitem toa unique luster (and thusthis item is removed fromthe other
lusters). We show that nding the best partition of the items into the lusters is
NP-hard and we des ribe several heuristi s tond goodsolutions.
Finallyatabusear hbasedonour frameworkand strategi os illationisproposed.
The idea is to alternate onstru tion and destru tion phases in whi h items are
tiveness of our approa h.
1.2.1 Tree-de omposition and graph triangulation
Wenow denethe notion of tree-de omposition,whi hwillbeappliedto the
ompati-bility graph of the BPC inthe sequel.
A tree-de omposition is a spe ial mapping of a graph into a set of lusters linked
in atree.
Denition 1.2.1 (RobertsonandSeymour[83℄)Atree-de ompositionofagivengraph
G = (V, E)
is a pair(C, T )
whereT = (N , A)
is a tree with node setN
and edge setA
, andC = {C
i
: i ∈ N }
, is a family of subsets ofV
su hthat:1.
∪
i∈N
C
i
= V
,2.
∀(v, w) ∈ E
,∃C
i
∈ C
ontaining both itemsv
andw
,3.
∀i, j, k ∈ N
, ifj
is on the path fromi
tok
inT
, thenC
i
∩ C
k
⊆ C
j
.Figure1.1showsa graph
G
witheightverti es,and atreede ompositionofG
onto a tree with six nodes. The set of lusters isC = {C
1
= {0, 1}, C
2
= {1, 2, 5, 6}, C
3
=
{2, 3, 6}, C
4
= {3, 6, 7}, C
5
= {3, 4}}
.Figure 1.1: Agraph
G
andapossibletree-de ompositionforG
The width
w(C, T )
of a tree-de omposition is equal tomax
i∈N
(|C
i
| − 1)
. The treewidthtw(G)
ofa graphG
isdened asmin{w(C, T )}
wherethe minimumistaken overalltree-de ompositions(C, T )
ofG
. Whereasforsomegraphfamilies,su hastrees andseries-parallelgraphs,one an omputethetreewidthinlineartime, omputingthetreewidthofageneralgraphisa
N P
- ompleteproblem. Severalpapers aredevoted to heuristi s for this problem(see Koster et al. [68℄ or our paper [36℄. The most famous(and simple)methodisMaximal Cardinality Sear h (MCS) [85℄,whi hwillbe usedin
this do ument.
Thenotionoftree-de ompositionisstrongly onne tedwiththe lassoftriangulated
graphs.
Tarjanand Yannakakis[85℄showed thatany triangulatedgraph ontains atmost
n
maximal liques, and proposed analgorithmtoenumeratethese liques inlineartime.Computing a tree-de omposition for a graph is equivalent to nding a triangulation
of this graph, i.e. nding a suitable set of edges to add to the graph to obtain a
triangulatedgraph. Then, the lustersare obtained by enumerating inlinear time the
maximal liques of the triangulated graph.
1.2.2 A general s heme for applying a tree-de omposition to
the bin-pa king problem with oni ts
Inthis se tion,wepresentour frameworkforapplyingtree-de ompositiontotheBPC.
On e atree-de ompositionis obtained, whi h meansthat the set of lusterswas
iden-tied, ea h item has to be assigned to a spe i luster to prevent items belonging
to several lusters from being pa ked more than on e. We all su h an assignment a
luster-separation and show that nding the best luster-separation is
N P
- omplete. Then we propose arst familyof heuristi s to nd fast solutionsfor this problem.Given a oni t graph
G = (I, E)
, let us denote byG = (I, I × I \ E)
the orre-sponding ompatibility graph. Our method works as follows. The tree-de ompositionis rst applied tothe ompatibility graph
G
. Ea h luster is related to a set of items that indu esasmaller andhopefullyless densesubproblemthan the originalproblem.Then ea h luster is solved independently. If the density of the graph has been
sig-ni antly de reased, algorithmsdedi ated to the lassi al bin-pa kingshould be more
ee tive. Finally,the partial solutionsobtained are mergedinto aunique solution.
Now suppose the graph of Figure 1.1 is a graph of ompatibility. We an noti e
that a vertex may belong to several lusters. Forexample, vertex
6
belongs toC
2
,C
3
andC
4
. Ifthe orresponding item istreated as many times asthe vertex appears ina luster, the solutionobtained will be of weakquality.Algorithm 1 shows a step-by-step des ription of the new approa h. At line 1, the
graph of ompatibilityis tree-de omposed. A luster-separationis omputed atline2.
The separated lustersare then solved as subproblemsby the means of any resolution
method at lines 4-5. Finally, at line 6, an improving heuristi is applied to the nal
solution. Thislaststepisnotmandatory,butitallowsustopartially orre ttheee ts
of a bad luster-separation.
Whenthepartitionoftheitemsetisrealized,weuseaheuristi tosolvetheindu ed
subproblem. We used anadaptation of the Bottom-Left algorithm[37℄.
Notethatalthoughonlyheuristi sareusedinthisdo ument,ourframeworkallows
exa t methodstobeusedinea hstepofthe algorithm( omputingthe de omposition,
Algorithm 1: A Tree-De omposition based frameworkfor solving BPC.
input : Set
I
ofn
items andG = (I, E)
graph of oni ts. output: A Pa king of then
items ina setB
of bins.(C, T ) ←− T reeDecomposition(G)
; 1µ(C, T ) ←− ClusterSeparation(C, T )
; 2B ←− ∅
; 3 forea hC
i
∈ µ(C, T )
do 4B ←− B ∪ ResolutionMethod(C
i
)
; 5B ←− ImprovingHeuristic(B)
; 61.2.3 The luster-separation problem
An important issue in the new approa h isto nd a suitablepartitioning of the items
in the lusters. We all su h a partitioning a luster-separation.
Denition 1.2.3 Given a BPC instan e with a ompatibility graph
G = (I, E)
and its tree-de omposition(C, T )
, a luster-separationisa partitionof the set of itemsI
in the set of nodesC
su h that an item an be assigned to a nodeC
i
only if it belongs toC
i
in the tree-de omposition.Apossible luster-separationofthe de ompositionof Figure1.1is
C
1
= {0, 1}, C
2
=
{2, 5, 6}, C
3
= ∅, C
4
= {3, 7}, C
5
= {4}
.The hoi e of the luster-separation is the most ru ialpart of the algorithm. We
say thata lusterseparationis ompatible withagiven solutionif,inthissolution,two
items assigned to two dierent lusters are never pa ked in the same bin. We state
belowthattherealwaysexistsa luster-separationthat anleadtoanoptimalsolution.
Proposition 1.2.1 ForanyBPCinstan e
D
witha ompatibilitygraphG
andits tree-de omposition(C, T )
,thereexistsa luster-separationµ
of(C, T )
thatis ompatiblewith an optimal solution forD
.We alltheproblemofndingthebest luster-separationthebest- luster-separation
problem. We now state that this problem is
N P
- omplete for an arbitrary graph by redu ing the partitionproblem [48℄ to it.Denition 1.2.4 Let
D
be a BPC instan e with a ompatibility graphG = (I, E)
and(C, T )
its tree-de omposition andk
an integer value. The best- luster-separation problem onsists in nding a luster separation of(C, T )
ompatible with to a solution of valuek
, if it exists.1.2.4 Greedy heuristi s for the luster separation problem
Sin ethebest- luster-separationproblemis
N P
- omplete,areasonablewayofta kling this problemfordensegraphsistouse heuristi s. This phaseisthemost ru ialofourapproa h sin e hoosing a bad luster-separationwould lead tobad solutions.
Theheuristi sweproposebelongtothefamilyofgreedyalgorithmsbasedonan
ini-tialsortingof the lusters. Consider a tree-de omposition
(C, T )
. A luster-separation an be omputed asfollows: 1)numberthe lustersa ording toagiven ordering, and2)forea h lusterintheorder hosen,assignallremainingitemsofthe urrent luster
C
i
to a new setS
i
and remove these items fromthe subsequent lusters.Several riteriahave been proposedto explorethe luster treeasso iated toa
tree-de omposition (see e.g. [61℄). Two types of riteria were introdu ed in [61℄: lo al and
global. A lo al (resp. global) riterion evaluates the relevan e of a andidate luster
without(resp. by)takingintoa ounttheintera tionswithother lusters. Theauthors
also proposed two riteria, the luster size (lo al) and the luster neighborhood size
(global) orrespondingto the number of lusters onne ted to it.
In this do ument weintrodu e anew global riterion, the demand
D(i)
of anitemi
asto be the number of lusters that ontaini
. This riterion an begeneralized and applied to lusters as follows: the demand of a lusterC
k
is the sum of the demands of the items ofC
k
: D(C
k
) =
P
i∈C
k
D(i)
. A luster with a large demand shares many items with other lusters, and therefore this riterion identies the " entral" lustersof the de omposition. We also introdu e another lo al riterion that we all rand
orresponding torandomly sorting the lusters.
The hoi e of these simple heuristi s may bejustied by the fa t that they do not
entailalarge omputingtime,sin ethe useofany ompli atedheuristi for omputing
a luster-separationwould in rease the omputing time of algorithm1.
1.2.5 Atabu sear hbasedon thetree de ompositionand
strate-gi os illation
Lo alsear halgorithmsare widely a knowledged as powerful toolsforproviding
high-quality solutionsto awide variety of ombinatorialproblems. In the previous se tion,
we have stated that the luster separation problem was the ore of our resolution
approa h. Inordertoimprovetheresultsobtainedbythegreedyheuristi s,wedesigned
a tabu sear h that fo uses onthe luster separation phase.
In this se tion, we des ribe the tabu sear h algorithm, denoted as TS-TD in the
following. Tabu sear h [52℄ has already been used to solve pa king problems (see [56℄
1.2.5.1 Details of our tabu sear h
In our approa h, a solution
s
is represented by a ve torv
¯
of sizen
. Ea h element¯
v
i
of this ve tor re ordsthe urrent lusterto whi h itemi
isassigned. Wedenote byD
i
the set of possible lusters that an a ommodatei
. Forexample, a ording togure 1.1,thedomainofitem3
isD
3
= {3, 4, 5}
. Thesolutionspa eofTS-TDisdenedas the set of omplete and in omplete solutions. A solution is said to be omplete (resp.in omplete) when its variables are (resp. are not) all assigned. In our approa h, any
omplete solution has tobefeasible.
The initialization phase generates an initial non- omplete solution ve tor by
assigningea h item
i
su h that|D
i
| = 1
(see Figure1.2). The remainingitemsare set to−1
and willbeassigned to lusters duringthe sear h.Figure1.2: Ave torrepresentingasolutionanditsinitializationa ordingtogure1.1.
A move is the assignment of a variable
i
to a value in its domainD
i
(assign) or the value−1
(remove from the luster). The existen e of two types of movements is justied by the fa t that our TS involvestwo phases, onstru tion and destru tion.The onstru tionphaseguidestheTStowarda ompletesolutionwhilethedestru tion
phase desinstantiates some variables. Alternatingthese two phases plays the role of a
diversi ation pro ess in order to enable the TS to explore new regions of the sear h
spa e.
Ourobje tive fun tionuses two terms. Inabin pa king ontext, hoosing solely
therealobje tivefun tion,whi histominimizethenumberofbins,isratherpointless,
sin e many dierent solutionsstillingeneralhavethe samenumberofbins. Itis often
better to extend this oarser grained measure by the gap value, omputed from the
free area inea h bin.
Inour implementation,the tabu list (TL) isaset of moves lassiedtabuduring
some iterations (tabu tenure). The tabu tenure is a stati value equal to the total
number of possible moves. For example, the size of TL for the example of Figure 1.1
1.2.5.2 Strategi os illation
Our diversi ation strategy onsists inalternating the onstru tion and destru tion
phases following some dynami riteria based onthe number of iterations. Figure 1.3
shows thebehavior ofour TSthrough the sear hpro ess. Werst start a onstru tion
phase. We keep running our lo al sear h until all items of
s
are assigned. A destru -tion phase isthen appliedons
by de-assigning someitems inorder toenable the next onstru tion phase to explore new regions of the sear h spa e, and the pro edure isrepeated while no stopping riterion holds. This strategy may be ontrolled by two
parameters: the amplitude
a
and the frequen yf
. Theamplituderepresentsthe maxi-malnumberofba kward movestobeperformedduringadestru tionphase. Thevalueof the frequen y is equaltothe minimalnumberof omplete solutionstorea hduring
the sear h pro ess.
Figure 1.3: Asear htraje toryinthesear hspa e. Thefrequen y
f
isthenumber oftimesa ompletesolutionis obtained. Theamplitudea
representsthenumberofdesinstatiationstoperforminadestru tionphase.Our intensi ation phase onsists in applying an improving heuristi on ea h
omplete solution we nd during the sear h pro ess. It is based on the progressive
redu tion of the number of bins used by the urrent solution. The idea is to destroy
some bins and redistribute their ontentsto the remainingbins.
1.2.6 Time omplexity of the method
For graphs of bounded treewidth, the time omplexity of the algorithm is redu ed
ompared to that of anequivalent onstru tion heuristi .
The time omplexity of omputing a luster-separation a ording to an ordering
riterion depends on the number of items
n
, the number of lusters|C|
in the tree-de omposition and the widthw(C, T )
of the tree-de omposition. If a lo al sorting algorithmisused, the time omplexity of this phaseis inO(|C| × log(w(C, T )))
,whi h is inO(n × log(n))
for an arbitrary graph. For the global sorting strategies, the time omplexity isO(n
2
)
Forthe two-dimensional ase ofBPC, the time omplexityof BLC (asimple
adap-tation of the bottom-left heuristi ) is
O(n
3
)
. If the graph is su h that there is an
algorithmtondatree-de ompositionwhosewidthisbounded bya onstant,itwould
lead to a
O(1)
time omplexity in our framework (sin e the number of items in the subproblems would be a onstant).The time omplexity of the improving heuristi is
O(|I
B
| × n
2
)
where
I
B
is the
number of items to repa k. The time omplexity is entailedby the maximum number
of possible pla ements at any step of a pa king. A possibility for redu ing this time
omplexity isto onsider a onstantnumberof itemsin
I
B
and anumberof open bins
su h that the number of pa ked items is also bounded by a small onstant. In this
ase,thetime omplexityofthisphasewouldalsobe
O(1)
. Unfortunately,experiments showed that italsodramati allyweakens the ee tiveness of the improvingphase.Let us summarize these results by studying the ase of algorithm1 applied using
MCS to omputethe tree-de omposition,agreedyalgorithmbased onalo al riterion
to ompute the luster-separation, BLC for solving the resulting lusters and the
im-provingheuristi ,theoveralltime omplexitywouldbe
O(m+n×log(n)+n×w(C, T )
3
)
.
If
w(C, T )
isboundedbya onstant,thetime omplexitybe omesO(m+n×log(n))
, whereas the original onstru tivealgorithmsare atleast inO(n
3
)
. Forhuge instan es,
one an even avoid the sorting algorithm, whi h leads to a linear time omplexity
(although the qualityof the solutionobtained is expe ted tobeweak).
1.2.7 Synthesis of the omputational experiments
Wetestedourapproa honthetwo-dimensionalversionoftheBPC.Thetest aseswere
obtained from the lassi al two-dimensional bin-pa king ben hmarks. We generated
oni t graph randomly following the method used in [46℄. We used instan es of size
upto
120
items. WeusedtheframeworkParadisEO[16℄toimplementourtabusear h. For these instan es, the tree-de omposition framework improves signi antly theperforman e of thegreedy algorithmswith aslightlylarger omputing time. Thetabu
sear houtperforms the other methods, but needs alarger omputing time.
We also generated huge instan es (
2000
items) to validate the ee tiveness of the tree-de omposition. For this test, we used the greedy heuristi s based on thetree-de omposition and we swit hed o the improvement heuristi that is used after the
solutions are merged. For sparse oni t graphs, the method does not bring any
im-provement in term of omputing time, and deteriorates the quality of the solution,
sin e the width of the de omposition is lose to the number of items. When the
on-i t graph is dense, the omputing time is dramati allyredu ed, but there is a slight
1.3 A olumn-generation algorithm for the min- oni t
pa king problem
Inthe ontextofamulti-obje tivebin-pa kingproblem,westudiedasubproblemthat
we name the "min- oni t pa kingproblem".
Problem 4 (Min-Coni t Pa king Problem (MCPP)) Given a set
I
of items, a bin typeB
, a valueM
and a oni t graphG = (I, E)
where(i, j) ∈ E
ifi
andj
are not supposed to be pa ked in the same bin, ompute the minimum number of oni tsthat must o ur if the set
I
is pa ked inM
bins of typeB
.Thisproblemisjustied by the fa tthat resour es (bins) are notalwaysin innite
number, and thatthede isionmakermayfavorasolutionviolatingsome onstraintsif
its ost is small. The work we present isinserted in ageneral multi-obje tives heme,
and we use the knowledge of solutions related to dierent numbers of bins in our
resolution method.
We rst propose a ompa t formulation for this problem. This modelis weak and
thus we propose a reformulation based on a olumn-generation s heme. The model
itself is anadaptation of the set- overing modelof Gilmoreand Gomory [50,51℄. The
di ultyarisesinthepri ingsubproblem,abilinearknapsa kproblem,whi hisharder
to solve than the lassi al knapsa k problem. We reformulate this problem with two
dierent ILP models. These models demand too mu h time, so we designed a greedy
heuristi and a lo alsear h method based on swaps in order to nd good solutionsin
a faster manner. In our method, the ILP solver is run only if the heuristi s were not
able to nd a olumn of redu ed ost.
Atabusear hisalsoproposedtogenerateagoodinitialbasisfortheLP.Likeinthe
previousse tion,itisbasedonstrategi os illation. Forthismethod,wedonotos illate
from omplete toin omplete, but from a number of bins to another. This is justied
by the fa t that our methodis run inthe ontextof multi-obje tiveoptimization,and
thuswehavere orded some goodsolutionsrelatedtodierentnumbers ofbins. When
the number of bins is in reased, we will explore non-feasible solutions for our present
problem. When thenumberofbins isde reased, itleadstovalidyetmore onstrained
solutionthatwillmakeabetterusageofthespa einthebins(disregardingthenumber
of oni ts entailed).
1.3.1 A simple ompa t model
A simple linearization of a dire t quadrati model for the min- oni t problem leads
to the following ILP model. Variables
x
ik
are equal to1
ifi
is pa ked in bink
, and0
otherwise. Variablesy
ij
are equalto1
when itemsi
andj
are in the samebin. For an itemi
, letN(i)
be the set of itemsj
su h thati
andj
are in oni t. Let alsoN
be the items
j
ofN(i)
su h thatj > i
. This set is used to avoid ounting a oni t twi e in our models.min
X
i∈I
X
j∈N
+
(i)
y
ij
(1.1)X
k∈M
x
ik
≥ 1, i ∈ I
(1.2)n
X
i=1
c
i
x
ik
≤ C, k ∈ M
(1.3)y
ij
≥ x
ik
+ x
jk
− 1, i ∈ I, j ∈ N
+
(i), k ∈ M
(1.4)y
ij
∈ {0, 1}, i ∈ I, j ∈ N
+
(i)
(1.5)x
ik
∈ {0, 1}, i ∈ I, k ∈ M
(1.6)Constraints (1.2) ensurethat all items are pa ked, whereas onstraints(1.3) verify
the apa ity onstraint. Constraints (1.4) are su ient to ensure that if
i
andj
are pa ked together, theny
ij
willbe equaltoone.1.3.2 A set overing formulation
The linear relaxation of the ILP above is weak. Thus we used a formulation based
on a set overing model and the de omposition of Dantzig-Wolfe [39℄. The ILP is
de omposed into a restri ted master problem initialized with a set of olumns, and
optimized todetermine thevalueofthe dualvariables. Thedual informationispassed
to a subproblem that evaluates if there is a olumn that an be added to the master
problem and improve the urrent solution. If there is su h a olumn, the master
problemis reoptimized, otherwise the pro ess stops.
Let
P
bethe set ofpossible patterns,i.e. the set of possibleways of pa kingitems in a bin. Ea h possible patternis des ribed by a olumnp = (a
1p
, . . . , a
ip
, . . . , a
|I|p
)
T
,
where
a
ip
is equal to1
if itemi
is in the patternp
,0
otherwise. A new variableK
p
is introdu ed, whi h orresponds with the number of oni ts in ongurationp
. The de omposition is a simple adaptation of the model of Gilmore and Gomory [50,51℄dedi ated to the utting-sto k problem.
min
X
p∈P
x
p
K
p
(1.7)s.t.
X
p∈P
a
ip
x
p
≥ 1, ∀i ∈ I
(1.8)X
p∈P
x
p
≤ M
(1.9)x
p
∈ {0, 1}, ∀p ∈ P
(1.10)Twofamiliesof onstraintsare kept inthe master: onstraints(1.8) ensurethat all
items are ut, whereas (1.9) veries that the number of bins used is smaller than
M
. The apa ity onstraint is leftto the subproblem.The weakdual of (1.7)-(1.10)reads as follows.
max
X
i∈I
π
i
− θM
(1.11)s.t.
X
i∈I
a
ip
π
i
− θ ≤ K
p
, ∀p ∈ P
(1.12)π
i
≥ 0, ∀i ∈ I
(1.13)θ ≥ 0
(1.14)Dual variables
π
are related to the demand onstraints, whereas the dual variableθ
isrelated tothe numberof bins.1.3.3 Solving the pri ing subproblem using lo al sear h and
linear programming
In ordertogenerate the best olumnto addtothe urrent basis,the patternto addis
the onewiththe smallestredu ed ost,i.e. the pattern
p
forwhi hθ + K
p
−
P
i∈I
π
i
a
ip
isminimized. The only onstraintfora pattern
p
isthe apa ity onstraintof the bin. This isequivalent tosolving the followingbilinearproblem, alled pri ing subproblem.max
X
i∈I
(π
i
a
i
−
X
j∈N
+
(i)
a
i
a
j
) − θ
(1.15)s.t.
X
i∈I
a
i
c
i
≤ C
(1.16)a
i
∈ {0, 1}, ∀i ∈ I
(1.17)It is a generalization of the knapsa k problem, where ea h oni t between two
items redu es the value of the solution by one. The knapsa k problem with hard
oni ts has been studied by, among others, Hi and Mi hrafy [60℄. In our problem,
oni ts mayo ur, but they are penalizedin the obje tive fun tion.
Comparedtothe utting-sto kproblem,thepri ingsubproblemismoredi ultto
solve. Ittranspiredfromour rstexperimentsthatsolvingthe pri ingtooptimalityat
ea h step of the olumn generation algorithm leads to a large omputing time. Thus
wehavedeveloped amethodthat useslinear-programming,heuristi sandlo alsear h.
1.3.3.1 Two ILP models for the pri ing subproblem
Arstwayofsolvingthepri ingsubproblemisuseastraightforwardlinearizedversion
and
a
j
= 1
,and to0
otherwise.max
X
i∈I
(π
i
a
i
−
X
j∈N
+
(i)
b
ij
) − θ
(1.18)s.t.
X
i∈I
a
i
c
i
≤ C
(1.19)b
ij
≥ a
i
+ a
j
− 1, ∀i ∈ I, j ∈ N
+
(i)
(1.20)b
ij
∈ {0, 1}, ∀i ∈ I, j ∈ N
+
(i)
(1.21)a
i
∈ {0, 1}, ∀i ∈ I
(1.22)Abetter model an beproposed, usingadierenttypeof variables
b
i
forea hitem typei
. Ea hvariableb
i
isequaltothe numberof itemsofN
+
(i)
thatare pa ked with
i
. Note thata
i
= 0
impliesb
i
= 0
. In this model the number of variables remains linear, whereas it isquadrati inthe previous model.max
P
i∈I
(π
i
a
i
− b
i
) − θ
(1.23)s.t.
P
i∈I
a
i
c
i
≤ C
(1.24)P
j∈N
+
(i)
a
j
− |N
+
(i)| × (1 − a
i
) ≤ b
i
∀i ∈ I
(1.25)a
i
∈ {0, 1},
∀i ∈ I
(1.26)b
i
∈ N,
∀i ∈ I
(1.27)Bothmodels an beimproved by the means of uts. We des ribe these uts using
theformalismofmodel(1.23)-(1.27). Sin eoneofthe onstraintsisa lassi alknapsa k
onstraint, allte hniques related to this onstraint an be used. For example, if
N
max
is the maximum number of items that an be pa ked side by side, the following ut isvalid.
X
i∈I
a
i
≤ N
max
(1.28)Other hand-tailored te hniques an be used for this spe i problem. Note that
these uts are valid be ause we seek a solution of positive value. In other
words, these uts may ex lude the optimal solutionif this solution has not a positive
value.
First,notethatif
b
i
≥ π
i
,abettersolutionisobtainedby removingitemi
(be ause itleads tomore oni ts than itsvalue of prot). This leadstothe followingfamilyofuts.
b
i
≤ π
i
− 1, ∀i ∈ I
(1.29)Thisfamily of uts an begeneralized by onsidering sets of items. Let
I
f
bea set of items su hthatP
i∈I
f
π
i
−
P
i∈I
f
|N(i) ∩ I
f
| ≤ 0
an optimal solution. Consequently, if we are able to dete t su h a set
I
f
, we add the following ut.X
i∈I
f
a
i
≤ |I
f
| − 1
(1.30)Another ut an be added. First we ompute the minimum number
N
min
of items needed toobtain asum of protgreater than orequal toθ
.X
i∈I
a
i
≥ N
min
(1.31)An estimation of
N
min
an be omputed by sorting the items by de reasing value ofπ
i
and disregarding the oni ts.1.3.3.2 Heuristi solutions for the subproblem
Even with the uts above, and even if the sear h is stopped as soon as a solution of
positive ost isfound, our two ILPmodelsdonot lead tofastsolutionsin many ases.
Consequently, we developed heuristi sand alo alsear hmethodtofasten the olumn
generation pro ess.
Thegreedyheuristi sarebasedonaninitialorderingontheitems: de reasing
π
i
/c
i
, de reasing(π
i
+ |N(i)|)/c
i
, de reasing(π
i
+
P
j∈N (i)
π
j
)/c
i
We ompute the values of degrees in a dynami fashion (i.e. they are updated ea h time an item is sele ted orreje ted fromthe knapsa k).
Inthe ase wherethegreedy heuristi sare notable tond asolutionwith negative
redu ed ost, a lo al sear h phase is applied. It is based on two simple operators:
bit ip (sele t an unsele ted item or remove a sele ted item), and pairwise ex hange
(repla e an item by another item). At ea h step of the lo al sear h algorithm, all
possible bit-ipmoves aretested. Ifnoneimprovesthe valueof theobje tivefun tion,
the swapmovesare tested. Themethodstopswhennoimprovementhas been realized
in the lastiteration.
1.3.4 Computing the initial set of olumns using tabu sear h
and strategi os illation
Sin e the pri ing phase has a large omputational ost, nding a good initialbasis is
ru ial for our algorithm. Like in Se tion1.2, we use a strategi os illation in a tabu
sear h. This time, the os illation uses the fa t that the sear h is run in the ontext
1.3.4.1 Details of our tabu sear h
Given a set
I
ofn
items, a solution is a number of bin asso iated with ea h item ofI
. A solutionis a omplete pa kingin whi h the apa ity onstraintis not violatedin any bin (the sum of the sizes of the items is not larger than the size of the bin). Thenumber of bins is initiallyxed to
M
.The sear h spa e
S
is thus omposed of all possible ongurations meeting this onstraint. Aneighborofasolutions
isobtainedby hangingthebinasso iatedtoan item insu h a way that the apa ity onstraint remainssatised. A soft oni t(i, j)
is said to be violated ifi
andj
are pa ked in the same bin. Our obje tive fun tion isthe numberof violatedsoft oni ts. When anitemismoved fromabin toanother,the numberof oni ts isupdated only forthe two bins involved inthe move.
Weusedthreedierentfamiliesofheuristi algorithmsfortheinitializationphase
of the tabu sear h. The rst is the family of Any-Fit algorithms. The se ond isbased
on Soft Graph Coloring, and the third ona simple lo alsear h algorithm that tries to
improve iterativelyan initialsolution.
Ourtabu list(TL) onsistsofaset ofmoves lassiedtabuduringsomeiterations
(tabutenure). Thetabu tenureisastati value. On eTLisfull,itisresizedinasu h
way that the oldest halfis erased.
1.3.4.2 Strategi os illation
We developed a diversi ation strategy based on two types of modi ations
(top-down orbottom-up). Let
m
be the numberof bins used in the urrent solutions
m
. In a top-down (resp. bottom-up) strategy, a solutions
m
is repla ed by a solutions
m+d
(resp.s
m−d
) whered
is a possible value for the number of bins. These diversi ation strategies are ontrolledby aparameter alleddistan ed
. The diversi ation distan e represents the number of bins to add or remove from the urrent solution dependingon the hosen strategy.
In aseofatop-downstrategy,
d
binsareaddedtoasolutions
m
andthusasolutions
m+d
is obtained by distributing the ontents ofm
bins intom + d
bins by means of a lassi al bin pa king heuristi . In ase of a bottom-up strategy,d
bins are removed fromthe solutions,and the items inthesed
bins are distributed amongthe remaining bins. Thed
binsare hosenfollowingagiven riterion,likethebinswiththemaximum number of oni ts. Consequently, our method os illates between in rease phases, inwhi h itadds bins in order to redu ethe number of oni ts, and de rease phases,in
whi hbins are erased,and the algorithmsfo uses on the spa e used inthe bins.
Figure1.4illustratesthesetwodiversi ationstrategies. Theblue(resp. red)arrow
showsatop-down(bottom-up)diversi ationofasolution
s
l
(resp.s
m
)withadistan ed
l
(resp.d
m
).Figure1.4: Exampleillustratingthetwodiversi ationstrategiestop-downandbottom-up.
1.3.5 Synthesis of the omputational experiments
We implemented the methods des ribed above and tested them against ben hmarks
derived from the literature. We used instan es with up to
120
items, for whi h we generated bounds for the whole Pareto front (more than40
min oni t problems to solve in the worst ase).We were able tosolve alarge number ofinstan es tooptimality justby omputing
ourbounds. However, our omputationalexperiments onrmthe fa tthatthepri ing
subproblem to solve is mu h harder than the lassi al knapsa k problem, even using
our se ond ILP model. Even with this faster model, generating the olumns an take
time. The best results were obtained using the heuristi s and the lo al sear h, and
then the se ond ILP model. We improved the results by stopping the ILP as soonas
it gets an improving solution. Finding better algorithms for the pri ing subproblem
Figure 1.5: Aninstan eofBPP-FO,anon-optimalsolutionwiththreebinsandanoptimalsolutionwithtwobins.
Fragilitiesarerepresented bydottedre tangles,andsizeswithgreyre tangles.
1.4 A olumn generation algorithm for the bin-pa king
with fragile obje ts
In this se tion, we deal with a variant of pa king in whi h oni ts are modelled ina
dierentway. The problemisknownintheliterature astheBinPa king Problemwith
Fragile Obje ts (BPP-FO). The BPP-FO arises inthe tele ommuni ation eld and in
parti ular intheallo ationof ellularusers tofrequen y hannels(see Bansaletal.[5℄
and Chan et al.[24℄).
Problem 5 (Bin-pa king Problem with Fragile Obje ts (BPP-FO)) Givenaset
I
of itemsi
of sizec
i
and fragilityψ
i
, what is the minimum number of bins needed to pa k all the items ofI
in su h a way that in ea h bin, the sum of the sizes is smaller than the smallest fragility?An example of BPP-FO instan e, and two solutions is given in Figure 1.5. More
formally,let usdenote
I(k)
as the set of items assigned toa bink
, we need toensure thatX
i∈I(k)
c
i
≤ min
i∈I(k)
{ψ
i
}
(1.32)for allpossiblebins
k
.Theliteratureonthe BPP-FOisstillsmall. Bansaletal.[5℄presentapproximation
s hemes and probabilisti analysis. They onsider approximations both with respe t
tothenumberofbinsand tothefragilityofabin. Theypresent resultsforthe general
BPP-FO and for a spe ial ase, denoted the frequen y allo ation problem, in whi h
weight and fragility are stri tly orrelated one to the other. Chan et al. [24℄ onsider
instead the on-line version of the BPP-FO, in whi h an item arrives only after the
or unbounded, and present, for both ases, algorithms with asymptoti ompetitive
ratios.
Forthe BPP-FO,we rst propose simple ompa t models, whi h are able to solve
exa tly many random instan es in a fast manner. However, for some lasses of hard
instan es, these models are not su ient anymore. For this reason, we propose a
reformulationof the problemusingthe Dantzig-Wolfe de ompositionand olumn
gen-eration. The pri ing subproblem to solve is a knapsa k problem with fragile obje ts
des ribed above.
Problem 6 (Knapsa k Problem with Fragile Obje ts (KP01-FO)) Given
n
itemsi
with protp
i
, weightc
i
and fragilityψ
i
(i = 1, . . . , n
) and a single un apa itatedbin, nd the subset of items of largest total prot whose total weight is not larger than thefragility of any item in the bin.
Beingable tosolve this probleme iently isthe most ru ialissue in our
olumn-generationalgorithm. WeproposetwoILPmodelsandadynami -programmings heme
for KP01-FO. As we will see below, the dynami -programming s heme outperforms
both ILP models for allthe instan es weused.
Forgenerating a suitableinitialbasis, we designed a variable-neighborhoodsear h
(VNS)methodbasedonstrategi os illation. Weos illatefromfeasibletonon-feasible
solutions using dierent perturbation strategies. Our overall algorithm allows to nd
tight lowerand upper bounds ina fast mannerand lose the integrality gap for many
di ultinstan es thatwere notsolvedbythe ompa tmodels withintheallowedtime
limit.
1.4.1 Compa t models
Werst presenttwo ompa tformulationsforBPP-FOrequiringapolynomialnumber
of variablesand onstraints. We thendis uss athirdformulationusing anexponential
number of variables.
1.4.1.1 A simple formulation
Let us dene
ψ
max
= max
i=1,...,n
{ψ
i
}
to be the maximum fragility of an item. We deney
k
asabinaryvariabletaking value1
ifbink
isused, 0otherwise (k = 1, . . . , n
). We alsodenex
ik
as a binary variabletaking value1
if itemi
is assigned to bink
,0
otherwise (i, k = 1, . . . , n
). The BPP-FO an bemodeled asthe following ILP:min
n
X
k=1
y
k
(1.33)n
X
k=1
x
ik
= 1
i = 1, . . . , n
(1.34)n
X
j=1
c
j
x
jk
≤ ψ
max
+ x
ik
(ψ
i
− ψ
max
) i, k = 1, . . . , n
(1.35)x
ik
≤ y
k
i, k = 1, . . . , n
(1.36)y
k
∈ {0, 1}
k = 1, . . . , n
(1.37)x
ik
∈ {0, 1} i, k = 1, . . . , n.
(1.38)Constraints (1.34) impose that ea h item is assigned to a bin. Constraints (1.35)
require that the sum of the weights of the items pa ked in a bin does not ex eed the
fragilityof anyitem pa ked inthe same bin(if item
i
ispa ked inbink
therighthand sideofthe onstraintisequaltoψ
i
,otherwisethe onstraintisredundant). Constraints (1.36) are used to tighten the modellinear relaxation.Model(1.33)(1.38)isderivedfromthe lassi alBPP ompa tmodel. Ithasa lear
disadvantage that omesfromthe
O(n
2
)
Constraints(1.35),that modelthe non-linear
restri tion (1.32)by usingalargevalue(
ψ
max
). This value an worsen onsistently the model linear relaxation, and makes the formulation very dependent from the fragilityof the last item.
1.4.1.2 A better formulation
Were allthattheitemsaresortedbynon-de reasingvaluesof
ψ
i
,breakingtiesby non-in reasing values ofc
i
. Wedeney
i
asa binary variable takingvalue 1if itemi
isthe item with smallest fragility in the bin in whi h it is pa ked, 0 otherwise (i = 1, . . . , n
). We also denex
ji
as a binary variable taking value 1 if itemj
is assigned to the bin havingitemi
asitemwithsmallestfragility(bini
forshortinthefollowing),0otherwise (i = 1, . . . , n, j = i + 1, . . . , n
). The BPP-FO an bemodeledas the followingILP:min
n
X
i=1
y
i
(1.39)y
i
+
i−1
X
j=1
x
ij
= 1
i = 1, . . . , n
(1.40)n
X
j=i+1
c
j
x
ji
≤ (ψ
i
− c
i
)y
i
i = 1, . . . , n
(1.41)x
ji
≤ y
i
i = 1, . . . , n, j = i + 1, . . . , n
(1.42)y
i
∈ {0, 1}
i = 1, . . . , n
(1.43)x
ji
∈ {0, 1} i = 1, . . . , n, j = i + 1, . . . , n.
(1.44)Constraints(1.40) impose that eitheran itemis the smallestitem initsbin, either
it is assigned to a bin ontaining an item with smaller fragility. Constraints (1.41)
require that the sum of the weights of the items pa ked in a bin does not ex eed the
smallest fragility in the bin. Constraints (1.42) are again used to tighten the model
linear relaxation.
1.4.2 A set overing formulation
We present a modelthat builds upon the lassi al de omposition method by Gilmore
and Gomory [50,51℄. We dene a pattern as a feasible ombination of items. We
des ribe the pattern, say
p
, by a olumn(a
1p
, . . . , a
ip
,. . . , a
np
)
T
, where
a
ip
takes value1
if itemi
is inpatternp
,0
otherwise. LetP
be the set of all valid patterns, i.e., the set of patternsp
for whi hP
n
i=1
c
i
a
ip
≤ min
i=1,...,n
{ψ
i
a
ip
}
.Let also
z
p
be a binary variable taking value1
if patternp
is used,0
otherwise (p ∈ P
). The BPP-FO an be modeled asthe followingSet Covering problem:min
X
p∈P
z
p
(1.45)X
p∈P
a
ip
z
p
≥ 1 i = 1, . . . , n
(1.46)z
p
∈ {0, 1}
∀p ∈ P.
(1.47)Constraints (1.46)impose that ea h item
j
is pa ked inat least one bin.As the number of possible patterns may be very large, even solving the linear
relaxation of Model (1.45)(1.47) may be di ult. We approa hthis problem,as itis
usually done in the literature, by means of a olumn generation method.
Weinitializethemodelbyasubset
P ⊆ P
e
ofpatterns. Wethendroptheintegrality requirements (1.47)by repla ing them withz
p
≥ 0
,∀p ∈ e
P
. Notethat we are allowedtodropthe onditions
z
p
≤ 1
sin eredundant (indeedwe an alwaysrepla easolution in whi h there exists az
p
> 1
with a better one havingz
p
= 1
). We nallyasso iate dual variablesπ
i
(i = 1, . . . , n
) to onstraints(1.46).We operate inan iterativeway. We solve the linear modeljust outlinedand he k
if apattern(i.e., a olumn)with negativeredu ed ost exists. If itexists, thenwe add
it to the model and re-iterate, otherwise we proved the optimality of the (eventually
fra tional)solution obtained. The redu ed ost of apattern
p
is dened byc
p
= 1 −
n
X
i=1
π
i
a
ip
.
(1.48)A pattern is added to the model if it satises the fragility requirement and has a
negativeredu ed ost. Theexisten eofsu hpattern anthusbedeterminedbysolving
a KP01-FOwith obje tive fun tion
max
n
X
i=1
π
i
a
ip
(1.49) and subje t ton
X
i=1
c
i
a
ip
≤ min
i=1,...,n
{ψ
i
a
ip
}
(1.50)a
ip
∈ {0, 1}, ∀i ∈ I
(1.51)Note that one of the main issues in olumn generation is a long tail onvergen e,
during whi h the value of the optimum isonly marginallyimproved. Several methods
havebeen proposedtodeal withthis issue. One ofthe mostpromisingisthe notionof
dual uts introdu ed by Valerio de Carvalho [86℄. The idea is to add dual uts to the
master problem( olumns in the primal)to ex lude dual solutionsthat are dominated
by others. This notion an be extended to the BPP-FO asfollows.
Proposition 1.4.1 Fora givenitem
i
, ifthere existsa setS ⊂ I
su hthatP
j∈S
c
j
≤
c
i
andmin
j∈S
ψ
j
≥ ψ
i
thenX
j∈S
π
j
≤ π
i
(1.52)is a valid dual ut for (1.45)-(1.47).
Many uts of this type an be applied. Pra ti ally speaking, we use the of Type
I and II des ribed in [86℄. In the uts of type I, the set
S
above is of size one. The uts of type II onsider two items inS
. If the number of uts of type II is too large, only asubset of them an be applied. Pra ti allyspeaking, for the BPP-FO, this size1.4.3 Solving the Pri ing Subproblem using dynami
program-ming and linear programming
We solve the KP01-FO problem arising in the pri ing phase by means of two ILP
models and a dynami programmingalgorithm. Insights inthe omputationalresults
we obtained by using these three approa hes willbe given inSe tion1.4.5 below.
ILP Models
Wepresented two ompa t ILPmodelstosolvetheBPP-FO,thelatter(Model(1.39)
(1.44)) being better than the former (Model (1.33)(1.38)). A similar result an be
obtained for the KP01-FO.
A rst ILP model an be obtained by using a binary variable
x
i
taking value1
if itemi
is pa ked in the knapsa k,0
otherwise (i = 1, . . . , n
). The KP01-FO an be modeled as:max
n
X
i=1
π
i
x
i
(1.53)n
X
j=1
c
j
x
j
≤ ψ
max
+ x
i
(ψ
i
− ψ
max
), ∀i ∈ I
(1.54)x
i
∈ {0, 1} ∀i ∈ I
(1.55)The main dieren e with respe t to the lassi al ompa t model for the KP01 is
that hereweneed
n
onstraintstoimposethe maximum apa ity. Indeed, Constraints (1.54) impose that whenever anitemi
ispa ked in the knapsa k, then the sum of the weights of allthe items pa ked annot ex eed the valueψ
i
.A better ILP model an be obtained as follows. First, letus re all that items are
sorted by non-de reasing values of