Solving Irregular Strip Packing problems by hybridising simulated annealing and linear programming

19 Pages·2006·0.58 MB·English

EuropeanJournalofOperationalResearch171(2006)811–829
www.elsevier.com/locate/ejor
Solving Irregular Strip Packing problems by
q
hybridising simulated annealing and linear programming
A. Miguel Gomes a,b,*, Jose´ F. Oliveira a,b
aFaculdadedeEngenhariadaUniversidadedoPorto,DEECRuaDr.RobertoFriass/n,Porto4200-465,Portugal
bInstitutodeEngenhariadeSistemaseComputadoresdoPortoCampusdaFEUP,RuaDr.RobertoFrias378,Porto4200-465,Portugal
Availableonline14October2004
Abstract
InthispaperahybridalgorithmtosolveIrregularStripPackingproblemsispresented.Themetaheuristicsimulated
annealingisusedtoguidethesearchoverthesolutionspacewhilelinearprogrammingmodelsaresolvedtogenerate
neighbourhoods during the search process.These linear programming models, which are used to locally optimise the
layouts, derivefrom the applicationof compaction andseparationalgorithms.
Computational tests were run using instances that are commonly used as benchmarks in the literature. The best
results published sofarhavebeen improved bythisnew hybrid packingalgorithm.
(cid:1)2004Elsevier B.V.All rightsreserved.
Keywords:Cutting;Packing;Simulatedannealing;IrregularStripPacking;Nesting
1. Introduction minimisation,i.e.theportionsofthebigitemsthat
are not used to produce small pieces. In Cutting
TheIrregularStripPackingproblembelongsto andPackingproblems,on theonehand,decisions
the more general class of combinatorial optimisa- have to be taken on which pieces to produce and
tion problems—the Cutting and Packing prob- from which big item, which is a hard combinato-
lems. In these problems, one or more big items, rialproblem.Ontheotherhand,aspecificgeomet-
either material or space, must be divided into ric problem arises: how to cut the small pieces.
smaller pieces. Usually the objective is the waste These two problems are interlaced and solutions
must be feasible both from the ‘‘quantitative’’
viewpoint (i.e. minimum or maximum number of
small pieces to produce, availability of big items)
q Partially supported by FCT, POSI and FEDER (POSI/
andfromthegeometricviewpoint(i.e.nooverlap-
40908/SRI/2001(CPackMO)).
* Correspondingauthor. ping between the small pieces, containment of the
E-mailaddress:[email protected](A.M.Gomes). small pieces inside the big items).
0377-2217/$-seefrontmatter (cid:1)2004ElsevierB.V.Allrightsreserved.
doi:10.1016/j.ejor.2004.09.008
812 A.M.Gomes,J.F.Oliveira/EuropeanJournalofOperationalResearch171(2006)811–829
CuttingandPackingproblemsaremainlychar- (non-rectangular) shapes. Small pieces can be
acterised by: the number of relevant dimensions, placed with an arbitrary orientation.
the regularity or irregularity of the shapes of the Nesting problems in the garment industry have
small pieces and big items, the big items assort- several characteristics that are specific to this
ment and availability, the total number of small industry. These characteristics are inherent to the
pieces and the number of different types of small productionprocessandtoseveraltechnologicalis-
pieces. For a detailed classification of Cutting sues. In this type of industry, the plates are big
and Packing problems see Dyckhoff et al. (1988). rollsoffabricwithatotallengthuptoseveralkilo-
Additional information on Cutting and Packing metres. This leads to the consideration of a single
is available online on the ESICUP—EURO Spe- rectangle with a fixed width and an infinite length
cial Interest Group on Cutting and Packing web (a strip). The objective is the minimisation of the
site (http://www.apdio.pt/esicup). layoutlength.Inthegarmentindustryusuallyonly
The economic and ecologic impact of the Cut- two orientations are allowed for the small pieces:
ting and Packing problems is obvious in many the original and the one obtained by a 180(cid:2) rota-
industrial production processes and better solving tion. This restriction is due to the existence of
these problems contributes to a better usage of drawing patterns and to intrinsic characteristics
natural resources. Examples of industrial Cutting of the fabric(cid:1)s weave. Finally, in the garment
and Packing problems are: industry the total number of small pieces can rise
above 100, but usually from less than 20 different
• 1D Cutting-Stock problems in the Paper types, although with very different sizes. In this
industry; paper, an approach that takes advantage of these
• 2D Rectangular Cutting in the Furniture, characteristics is proposed. An example of a nest-
Home Textile and Glass industries; ing layout or cutting pattern is represented in
• 2D Irregular Packing in the Furniture, Shoe Fig. 1. A more detailed description of nesting
and Garment industries; problems and cutting and packing problems can
• 3D Container and Truck Loading. be found in Dowsland and Dowsland (1995).
In the industrial environment this problem is
The focus of this work is on the 2D Irregular usuallytackledbyexperiencedworkersthat‘‘man-
Strip Packing problem, also known as nesting ually’’ (with the help of CAD systems) build the
problem,andspeciallyonavariantoftheproblem layouts. The quality of the layouts produced by
thatarisesinthegarmentindustry.Initsmostgen- these specialised workers is high and, presently,
eral formulation, the nesting problem is a two automatic solutions can only barely match this
dimensional Cutting and Packing problem where level of quality. A list of (semi-)automatic com-
both the small and the big pieces have irregular mercial solutions is available in Hopper (2000).
Fig.1. Nestinglayoutexample.
A.M.Gomes,J.F.Oliveira/EuropeanJournalofOperationalResearch171(2006)811–829 813
In the literature we can find several algorithms tions of such algorithms must be efficiently
that follow different approaches to tackle the dif- performed. For the nesting problem, building
ferentissuespresentinnestingproblems.Forsome and evaluating layouts are the most important
of those issues there are techniques that clearly basic operations. Tasks such as avoiding overlap
out-perform all the others available. That is the between two pieces, layout compaction and plac-
case of the no-fit-polygon concept, used to ensure ing pieces inside the plate are included in these
the geometric feasibility of a layout, and the case operations.Theno-fit-polygonisusedtoefficiently
of the algorithms based on linear programming avoid overlapping among the pieces and to place
compaction models, used to obtain layouts that pieces inside the plate. The task of generating the
arelocaloptima(LiandMilenkovic,1995;Stoyan initiallayoutisperformedbyagreedy bottom-left
et al., 1996; Bennell and Dowsland, 2001). There placementheuristic. Thisheuristicusesasequence
arealsoafewsimplegreedyheuristicsthatarevery of pieces and places each one on the plate, with
effective in achieving a ‘‘good’’ layout rapidly the ability of filling holes. Linear Programming
(Danielsand Milenkovic, 1996; Gomes and Olive- models are used to compact layouts (Compaction
ira, 2002). In order to avoid local optima, meta- Model), improving their quality and removing in
heuristics have become very popular in this field feasibilities (Separation Model). These models
(OliveiraandFerreira, 1993; Błaz_ewicz andWalk- are intentionally kept linear and non-integer 1 so
owiak, 1995; Heckmann and Lengauer, 1995; that they can be easily solved by a standard linear
BennellandDowsland,1999;GomesandOliveira, programming package. These models are the core
1999; Hopper, 2000; Gomes and Oliveira, 2001; of the Compaction and Separation algorithms.
Bennell and Dowsland, 2001; Błaz_ewicz et al., Itisalsonecessarytohaveanadequategeomet-
2004).Averypromisingapproachistheuseofhy- ric representation and manipulation of the pieces.
brid algorithms that combine metaheuristics and Pieces are represented by convex and non-convex
linear programming compaction models (Bennell polygons, with oriented edges, and any curve is
and Dowsland, 2001). This is the approach fol- approximated by a set of exterior tangent edges.
lowed in this work, where a simulated annealing Theedgesareorientedinsuchawaythattheinte-
algorithm is used to guide the search through the riorofthepolygonisontheright-handsideofthe
solution space, being the neighbourhood structure edge. Holes inside the pieces are not allowed. In a
basedonlinearprogrammingcompactionmodels. pre-processing phase, all the pieces are simplified
Thispaperisstructuredasfollows.Thenextsec- to eliminate very narrow concavities and, there-
tionhasadescriptionofthebasicblocksofthepro- fore, avoid unnecessary computational effort.
posednestingalgorithm:theno-fit-polygonconcept,
the greedy bottom-left placement heuristic and the 2.1. The no-fit-polygon
compaction/separation algorithms. The following
sectionisusedtopresentthehybridalgorithm,with The no-fit-polygon concept is used to ensure
a detailed description of its main components: the feasible layouts, i.e. layouts where the pieces do
simulated annealing algorithm, the neighbourhood not overlap and fit inside the plate. This concept
structure—LOOCCAALLCOOMMPPAACCTT—and the multi-stage was first introduced by Art (1966) and later on
approach. Finally, computational results are pre- used by several authors. In other fields of knowl-
sentedandsomeconclusionsdrawn. edge this concept is also known as Minkowski
sums. Mahadevan (1984) presents a comprehen-
sive description of an algorithm for the no-fit-
2. Blocks to build nesting algorithms polygon generation based on a sliding scheme.
More recently, Bennell et al. (2001) published an
Algorithms based on a high level search algo-
rithm need to perform a fairly wide search over
the solution space to ensure the quality of the re- 1 Although,asaconsequence,insomesituationsthelayouts
sults obtained. This implies that the basic opera- obtainedarenotascompactasitwouldbepossible.
814 A.M.Gomes,J.F.Oliveira/EuropeanJournalofOperationalResearch171(2006)811–829
lem of finding the relative position of one point
and one polygon.
To achieve a feasible (without overlap) and
tight layout, each piece should have its reference
point on the boundary of at least one no-fit-poly-
gon (relative to another piece) and in the exterior
orintheboundaryofalltheotherno-fit-polygons
(relative to the remaining pieces).
(a) (b)
Theinner-fit-rectangleisaconceptderivedfrom
Fig.2. No-fit-polygonandinner-fit-rectangleexamples:(a)the the no-fit-polygon concept and represents the fea-
no-fit-polygon,NFPi,jand(b)theinner-fit-rectangle,IFRi,j. siblesetofpointsfortheplacementofonepolygon
inside a rectangle. This concept is used to ensure
alternative algorithm to compute the no-fit-poly- that all the pieces are placed inside the plate. The
gon based on Minkowski sums. In the present inner-fit-rectangle of piece j relative to rectangle
work a geometric library with an implementation i (IFR ) is obtained when the piece j (the orbital
i,j
of Mahadevan(cid:1)s algorithms, previously developed piece) slides along the internal contour of the rec-
and used by the authors Oliveira et al. (2000) tangle i (Fig. 2(b)). It is assumed that the plate is
and Gomes and Oliveira (2002), was used. larger than the biggest piece to place, i.e. that the
The no-fit-polygon of piece j relative to piece IFR always exists. The inner-fit-rectangle is also
i,j
i(NFP )isthelocus ofpointstracedbytherefer- representedbyanorientedpolygon,wherethefea-
i,j
ence point2 of piece j (R), when j slides along the sible positioning points are on the right-hand side
j
external contour of piece i. The relative orienta- oftheedges(theinterioroftheinner-fit-rectangle).
tionsofpiecesiandjarekeptfixedduringthisor-
bital movement. Piece j (the orbital piece) must 2.2. Greedy bottom-left placement heuristic
never intersect piece i (the stationary piece) and
they must always be in contact (Fig. 2(a)). The The greedy bottom-left heuristic is a placement
no-fit-polygons are represented by oriented poly- heuristic that converts a sequence of pieces into a
gons, where the feasible positioning points are on feasible layout. Pieces are placed inside the plate,
the left-hand side of the edges (the exterior of the onebyone,atthemostbottom-leftfeasibleplace-
polygon). mentpoint.Whenpositioningonepiecetheheuris-
From this definition it follows that: tictakesintoaccountthepreviouslyplacedpieces,
bothtoavoidoverlapandtofillholesleftemptyat
• if the reference point of piece j (R) is placed in earlier stages. In Gomes and Oliveira (2002), dif-
j
the interior of NFP then piece j overlaps piece ferent criteria to generate the sequence of pieces
i,j
i; are discussed. In this work, a new criterion is pro-
• if thereferencepoint ofpiece j(R)isplaced on posed: random weighted length. This criterion gen-
j
theboundaryofNFP thenpiecejtouchespiece eratesthesequenceofpiecesbyrandomlyselecting
i,j
i; the next piece to add to the sequence. The proba-
• if the reference point of piece j (R) is placed in bility of selecting one piece is proportional to that
j
the exterior of NFP then piece j does neither piece(cid:1)s length.
i,j
overlap nor touch piece i. To place each piece, the greedy heuristic builds
asetofplacementpointcandidates,whichincludes
The problem of finding the relative position of all the following points:
twopolygonsistransformedintothesimplerprob-
1. the vertices of the no-fit-polygons (NFP )
i,k
where k is the next piece to place and i stands
2 The reference point of a piece is the origin of the
for all the previously placed pieces (i=1,...,
coordinate system(0,0), overwhich the otherverticescoordi-
nates,ofthesamepiece,aredefined. k(cid:1)1);
A.M.Gomes,J.F.Oliveira/EuropeanJournalofOperationalResearch171(2006)811–829 815
2. the vertices of the inner-fit-rectangle (IFRp,k) X X
betweenthenextpiecetoplace(k)andtheplate
(p);
3. the intersections between any two edges of any Y Y
pair of no-fit-polygons described in item 1
(NFP and NFP ; i=1,..., k(cid:1)1; j=1,...,
i,k j,k
k(cid:1)1; i5j);
4. theintersectionsbetweenoneedgeofanyno-fit- Fig.3. Compactionexample.
polygon described in item 1 (NFP ; i=1,...,
i,k
k(cid:1)1) and one edge of the inner-fit-rectangle the motion of a piece must be continuous, i.e.
described in item 2 (IFRp,k). pieces can not jump over other pieces. Rotations
arenottackled.AproofoftheCompactionProb-
From this set, the points which lead to unfeasi- lemNP-hardnessisavailableinLiandMilenkovic
bleplacements,i.e.pointsthatareintheinteriorof (1993).
any no-fit-polygon or on the exterior of the inner- To use compaction models it is necessary to
fit-rectangle,areeliminated,sothatasetofadmis- haveaninitialfeasiblelayout. 4Thenthefirststep
sible placement points is obtained. It is trivial to is to analyse the layout and establish positioning
obtain the most bottom-left placement point of relationships between each pair of pieces. These
this set. This heuristic is described in detail in relationships are based on the corresponding no-
Gomes and Oliveira (2002). fit-polygons. As stated in Section 2.1, piece j does
Finally, it should also be mentioned that the not overlap piece i if the reference point of piece
no-fit-polygons and the inner-fit-rectangles can j is placed over or on the left-hand side of at least
be computed off-line, once they only depend on oneoftheorientededgesofNFP . 5Forapartic-
i,j
the shape of the pieces and do not depend on the ular feasible placement of pieces i and j (i.e. with-
actual places occupied by the pieces in the layout. out overlap) one or more edges of NFP will
i,j
Whendifferentorientationsarealloweditisneces- satisfy the previous condition. As, for each pair
sary to calculate no-fit-polygons for each pair of of pieces, only one constraint is needed, different
different orientations and an inner-fit-rectangle rulesmaybeusedtoselecttheedge/constraintthat
for each orientation. is added to the model (the so called active con-
straint). These different rules lead to different
2.3. Compaction algorithm implementations of the compaction model. In the
second step the model is solved by a standard lin-
One of the most powerful methods available to earprogrammingpackageandinthefinalstepthe
obtainhighqualitylayoutsistheuseoflinear/inte- layout is reanalysed in order to detect further
ger programming compaction models. Compac- compaction possibilities. This iterative process is
tion models can improve layouts by applying a needed because, after a successful compaction,
set of coordinated continuous motions to the pieces are positioned in different coordinates and
pieces, achieving layouts that are local optima theuseofnewsetsofactiveconstraintsmayallow
(Fig. 3).Ina compactedlayout themain structure further improvements. This iterative process stops
of the non-compacted layout is preserved, i.e. the when no improvement in the layout can be
relative positions of pairs of pieces are kept, 3
although the exact position of each piece may be
completely different. During this set of motions 4 Thisinitiallayoutcanbeobtainedinseveralways,e.g.a
the pieces are not allowed to overlap. Moreover, greedy heuristic, randomly placing the pieces on the plate or
manuallybyanexperiencedworker.Theinitiallayoutcaneven
betheresultofapreviouscompactionoperation.
3 Preservingrelativepositionsmeansthatifpieceiisonthe 5 This is only generally true for convex pieces. The gener-
left-hand side of piece j before the compaction, this relative alisation to non-convex pieces is presented in the end of this
positionstillholdsaftercompaction. section.
816 A.M.Gomes,J.F.Oliveira/EuropeanJournalofOperationalResearch171(2006)811–829
achieved. Usually very few compaction iterations In the present work, we use the strategy of
(lessthan10)arenecessarytocompletelycompact increasing the nesting efficiency by shortening the
a layout. layout length.
Different goals can instantiate the generic con- The general linear programming model for
cept of ‘‘layout quality’’. From the compaction compacting layouts is presented in Fig. 4. This
viewpointtheyallhaveincommonbeingmodelled general model has been used by other authors
as an application of an adequate set of forces to (Li and Milenkovic, 1995; Stoyan et al., 1996;
the pieces. Examples of these goals are: Bennell and Dowsland, 2001). Its actual imple-
mentations differ from author to author and are
• increasing nesting efficiency by shortening the explained at the end of this section. In this model,
layout length, which is achieved by applying a thedecisionvariablesx andy arethecoordinates
i i
right-left force to all the pieces; of the placement point of each piece, z is an aux-
• creating space, in the middle of the layout, to iliary variable that stands for the layout length
place new pieces without increasing the layout and N is the number of pieces in the layout. All
length.This canbe obtained byapplying forces the other constants in the model are described in
withdifferentdirectionstothepieces.Thedirec- Fig. 5. Eqs. (1) and (2) ensure the layout length
tion of the force for a particular piece depends minimisation. The set of constraints defined by
on the relative position of that piece regarding Eq. (3) limits the movement of the pieces, in each
thespacetobeopened(i.e. ifthepiece isabove one of the compaction iterations, to a maximum
the space, we should apply a bottom-up force value (DISTANCE) that depends on the dimen-
i
to the piece). sions of each piece, ensuring a smooth compac-
Fig.4. Linearprogrammingmodelforcompactinglayouts.
A.M.Gomes,J.F.Oliveira/EuropeanJournalofOperationalResearch171(2006)811–829 817
W0-Wi W0 In Eq. (7), d1 is the distance between the place-
ment point of piece j and the farthest edge of no-
(X0,Y0) X
fit-polygon NFP and d is the distance between
W* i,j 2
i theplacementpointofpiecejandthenearestedge
H*
i ofno-fit-polygonNFP .InFig.6theinstantiation
H0-Hi (xi,yi) Hi ofthismodelisillustrait,jed.Twopiecesiandj(Fig.
W 6(a)) do not overlap if and only if the placement
H i
0
pointofpiecej,R,isoverorontheleft-handside
j
Y
of at least one oriented edge of NFP (in the
i,j
Fig.5. Placingonepieceinsidetheplate. exampleofFig.6(c)thisconditionholdsforedges
d,eandf).Whenmorethanoneorientededgesat-
tion process. X(cid:2) and Y(cid:2) are constants that are ini- isfiestheabovecondition,theonethatismoredis-
i i
tialised with the coordinates of the placement tantfromR isselected.IntheexampleofFig.6(c)
j
point of piece i. Placing all the pieces inside the edge e is selected and the constraint associated
plate is guaranteed by the set of constraints de- with edge e is added to the model (Fig. 6(d)).
fined by Eq. (4). The set of constraints defined Two particular cases need special attention,
by Eq. (5) ensures that pieces do not overlap. since they correspond to saddle points. The first
When pieces i and j are placed at coordinates one is when the placement point, R, is over a ver-
j
(x,y) and (x,y), respectively, f(x (cid:1)x,y (cid:1)y) is tex of the no-fit-polygon NFP (e.g. the intersec-
i i j j j i j i i,j
defined as tion of edges a and b in Fig. 6(c)). In this case
we should add to the model a disjunction of the
fðxj(cid:1)xi;yj(cid:1)yiÞ constraints associated with the two edges (a,b).
8(cid:1)d2; if pieces i and j do not overlap; However, due to efficiency purposes (to avoid
>< 1 mixed-integer formulations) only one edge is
¼ 0; if pieces i touches j; ð7Þ
added:theonewiththesmallestslope.Thesecond
>:d22; if pieces i and j overlap: case is similar and occurs when the placement
(a) (b)
(c) (d)
Fig.6. Associatedconstraintsselection:(a)piecesiandj;(b)no-fit-polygon,NFP (c)edgesoftheno-fit-polygonand(d)associated
i,j
constraints.
818 A.M.Gomes,J.F.Oliveira/EuropeanJournalofOperationalResearch171(2006)811–829
point Rj is over a vertical edge of the no-fit-poly- NFPi,j c
gon NFPi,j(e.g.edge b inFig. 6(c)). Alike the pre- b
vious case, we should add to the model a
a
d
disjunctionoftheconstraintsassociatedwiththree
R
edges (e.g. edges a, b and c in Fig. 6(c)) to the j
model. However, only one edge is in fact added
i
to the model: the constraint associated with the e
vertical edge. These approximations can lead to
slightly worse layouts, but they are necessary to g
keep the global performance of the nesting algo- f
rithm at a good level.
Adifferentsituationoccurswhentheno-fit-pol-
h
ygon is non-convex and the edge associated with
the selected constraint belongs to a concavity. In Fig.7. Addingaconjunctionofconstraints.
these cases, instead of adding a single constraint,
it isnecessary toadd aconjunctionofconstraints:
all the constraints associated with the edges that tion of different initial layouts. Each one of these
boundtheconcavity. 6Thisisnotcritical,because initial layouts must be feasible and they are inde-
the addition of a conjunction of constraints does pendently compacted, in order to achieve the best
not imply the addition of any integer or binary layout.LiandMilenkovic(1995)proposesamod-
variable to the model. This situation is illustrated elwheredifferentforcesareappliedtothepiecesto
inFig.7.Inthisexample,theconstraintassociated enlarge the gaps between them, by using adequate
with edge c is satisfied. However, the conjunction coefficientsintheobjectivefunction.However,the
of constraints associated with edges d, e and f is pieces arerestricted tobe ‘‘star-shaped’’. Morere-
also satisfied (note that those are the edges that cently,MilenkovicandDaniels(1999)proposean-
bound the concavity). Since the distance between other model using a mixed-integer formulation.
edge c and the placement point R is less than the Finally, Bennell and Dowsland (2001) propose a
j
distancebetweenR andthenearestconcavitybor- modelbasedonLiandMilenkovic(1995),without
j
der(edged),theconjunctionofconstraintsassoci- any restriction on the shape of the pieces. The
atedwithedgesd,eandfischosentobeaddedto objectivefunctionistheminimisationofthelayout
the model. This results in an increase on the total length and redundant constraints are removed.
number of constraints, but without any relevant
impact in the overall performance of the nesting 2.4. Separation algorithm
algorithm.
Similarcompactionmodelshavealsobeenused Basedonthecompactionmodel,itispossibleto
byotherauthors.Differencesbetweenthedifferent develop another linear programming model, the
implementations can be found on the objective separation model. The idea behind the separation
function and on the rule used to select the con- model is to remove any infeasibility (i.e. overlap-
straintthatisaddedtothemodel,whenmorethan ping situations) by moving apart the pieces that
one fulfils the requirements. However, the major overlap. As in the compaction model this is done
differences are on the associated separation model by applying a set of coordinated continuous mo-
and on how the compaction model is used. Later tions to the pieces (Fig. 8).
on these differences are made clear. Stoyan et al. Inlayoutswithoverlappedpieces,theconstruc-
(1996)proposesanapproachbasedonthegenera- tionofthecompactionmodelfailsbecauseitisnot
possible to establish a valid relationship between
two pieces thatoverlap (Fig. 9). Ifpiece ioverlaps
6 This concept needs to be recursively extended if a piecej,theplacementpointofpiecej(Rj)isalways
‘‘concavityinsideanotherconcavity’’ispresent. inside the no-fit-polygon NFPi,j, i.e. Rj is on the
A.M.Gomes,J.F.Oliveira/EuropeanJournalofOperationalResearch171(2006)811–829 819
X X fðxj(cid:1)xi;yj(cid:1)yiÞ(cid:1)ai;j 60; ð8Þ
i6¼j
X
min a : ð9Þ
Y Y i;j
i;j¼1;...;N
Frequentlyitisnotpossibletoremovetheover-
lap in just one step and the separation model may
Fig.8. Separationexample.
have to be consecutively applied to obtain a feasi-
ble layout. In our computational experiments in
99% of the situations the overlap was removed in
a
less than 10 steps. In each step, the separation
d
e model needs to be rewritten accordingly to the
new positions of the pieces.
i NFP
i,j The main drawbacks of the separation model
are the sparsity of the layouts after the separation
R
i
process and the lack of guarantee that overlap is
always removed. The first one is easily overcome
by applying the compaction algorithm to the lay-
R
j
out that results from the separation algorithm.
j
The second drawback can only be overcome if it
b
is possible to increase both dimensions of the
c
plate.Unfortunately,thisisnotthecaseofnesting
problems where the width of the plate is fixed.
Fig.9. Activeedgeexamplewithoverlap. However,sincethelengthoftheplateis‘‘infinite’’,
in practice almost all layouts are effectively
separated.
right-hand side of all edges of NFP . To obtain a Based on the respective compaction models, Li
i,j
complete model of the problem, the constraint andMilenkovic(1995)andBennellandDowsland
associated with the edge closest to the placement (2001) also propose separation models. However,
point R should be added to the model. However, the separation model proposed in this work pre-
j
adding this constraint would naturally lead to an sents a major innovation: the use of a set of artifi-
infeasible model. To overcome this difficulty, an cial variables a and the associated set of
i,j
artificial variable a is introduced, turning the constraints defined by Eq. (8).With this modifica-
i,j
model feasible. In the example of Fig. 9, the con- tion,theseparationprocessisdonemoresmoothly
straint (Eq. (8)) associated with edge a is added and does not need to be achieved in just one step,
to the model, with the artificial variable a . The thus increasing the chances of success.
i,j
objective function of the separation model is the
minimisation of the sum of all artificial variables
(Eq. (9)). A feasible layout is obtained when all 3. Hybrid algorithm
theartificialvariableshaveavalueofzero.Unlike
other separation models, by using artificial varia- Thefocusofthisworkisonthehybridisationof
bles the elimination of overlap does not need to simulated annealing with linear programming
be achieved in just one step. This increases the models in the resolution of nesting problems.
chancesofsuccessfullyremovingoverlap.Thesep- The linear programming models arise in the com-
aration model has one additional difference when paction and separation algorithms presented
compared with the compaction model: the set of earlier. These algorithms can be used in a coordi-
constraintsusedtokeeptrackofthelayoutlength nated way, taking advantage of the strong points
(Eq. (2) in Fig. 4) is removed. of each one, to create an efficient neighbourhood
820 A.M.Gomes,J.F.Oliveira/EuropeanJournalofOperationalResearch171(2006)811–829
structure for nesting problems. The simulated Pure local search approaches frequently end up
annealing algorithm is used to guide the search entrapped in local optimal solutions, since they
over the solution space. Bennell and Dowsland canonlymovetobettersolutionsintheneighbour-
(2001) propose a similar approach where a differ- hood. Simulated annealing can be seen as an evo-
ent metaheuristic, a tabu search algorithm, is lution of those approaches, by allowing some
hybridised with compaction and separation mod- controlled uphill movements, in order to achieve
els. Besides the use of a different metaheuristic, globaloptimality.Acceptingamovementtoaworse
additional differences between the two hybrid ap- solution depends on a control parameter (the tem-
proachescanbefoundonthecompactionandsep- perature)andon the magnitudeof the variationof
aration linear programming models, as pointed theobjectivefunction,i.e.howworsethesolutionis.
outin theprevious section,and on theneighbour- Many real-life problems have been successfully
hood structure. Two additional differences can be tackledbyapproachesbasedonsimulatedanneal-
found on a more strategic level: the type of move- ing. The main reasons for this success are: the
ment used to move between neighbour solutions high quality of the solutions, the easy inclusion of
andtheacceptance,ornot, ofunfeasible solutions real-lifeconstraintsandtherobustnessoftheseap-
alongthesearch.InBennellandDowsland(2001), proaches. The main drawback is the large compu-
moving from one solution to another is done by tational effort needed.
moving one piece in the layout (insert movement) Inthiswork,weusethesimulatedannealingalgo-
and infeasible layouts (i.e. layouts with overlap) rithm described in Pirlot (1996). To apply a simu-
areallowed.Inthiswork,movingtoanothersolu- lated annealing algorithm to a particular problem,
tion is done by exchanging the positions of two several decisions must be taken. These decisions
piecesinthelayout(swapmovement)andunfeasi- canbedividedingenericdecisionsandspecificdeci-
ble layouts are not allowed along the search. 7 sions.The generic decisions include the initial tem-
Inthenextsubsections,themaincomponentsof perature (t ), the cooling scheme, the stopping
0
the hybrid algorithm are presented: the simulated conditionandtheplateaulength.Thespecificdeci-
annealingalgorithmandtheneighbourhoodstruc- sions are the evaluation function, the neighbour-
ture. The last subsection is dedicated to present a hoodstructureandtheinitialsolution.
multi-stageapproachthattriestoexplorethevari- Aftersomepreliminarytests,thefollowinggen-
ety, in terms of size, of the pieces that usually eric decisions were taken: an initial temperature t
0
arises in the garment industry. calculated so that the probability of accepting
worse solutions in the first plateau is greater than
3.1. Simulated annealing 50%; a geometric cooling scheme with a factor of
0.9; two plateaus without improvement as a stop-
The origins of simulated annealing can be ping condition; the plateau length was set equal
found in Metropolis et al. (1953), where an algo- to the number of combinations of pairs of pieces,
(cid:1) (cid:2)
rithm to simulate the cooling of material in a heat N
givenby ,whereNstandsforthetotalnum-
bath, a process known as annealing, is presented. 2
Later on, Kirkpatrick et al. (1983) proposed the ber of pieces.
use of simulated annealing as an approach to Problem specific decisions were taken based on
tacklecombinatorialoptimisationproblems.Since the blocks presented earlier: the initial solution is
then,therehasbeenanenormousquantityofwork obtainedbythegreedybottom-leftplacementheu-
where simulated annealing has been applied to ristic(Section2.2),beingthenextpiecetoplacese-
various combinatorial optimisation problems. lected according to the random weighted length
criterion. The neighbourhood structure used in
this work is based on the compaction and separa-
tion algorithms, presented in the previous section,
7 Layoutswithoverlapareallowedwithinaneighbourhood
andisdiscussedinthefollowingsubsection.There
movement,butthenewsolutionisonlyacceptediftheoverlap
iscompletelyremoved(seeSection3.2forfurtherdetails). isnoneedtoimplement anyadditionalevaluation