a note on linear models for two-group and three-group two-dimensional guillotine cutting problems

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A note on linear models for two-groupand three-group two-dimensionalguillotine cutting problemsH. H. Yanasse a & R. Morabito ba Instituto Nacional de Pesquisas Espaciais , Brazilb Universidade Federal de São Carlos , BrazilPublished online: 10 Oct 2008.

To cite this article: H. H. Yanasse & R. Morabito (2008) A note on linear models for two-groupand three-group two-dimensional guillotine cutting problems, International Journal of ProductionResearch, 46:21, 6189-6206, DOI: 10.1080/00207540601011543

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International Journal of Production Research,Vol. 46, No. 21, 1 November 2008, 6189–6206

A note on linear models for two-group and three-group

two-dimensional guillotine cutting problems

H. H. YANASSEy and R. MORABITO*z

yInstituto Nacional de Pesquisas Espaciais, Brazil

zUniversidade Federal de Sao Carlos, Brazil

(Revision received August 2006)

Due to the particular characteristics of certain cutting machines, special classes ofcutting patterns, such as 1-group, 2-group and 3-group, requiring shorterprocessing times appear in the furniture, hardboard and stone industries. In thisstudy we present integer linear models to generate 2-group and 3-groupconstrained and unconstrained two-dimensional guillotine cutting patterns. Themodels are derived from the linear models for 1-group guillotine cutting patternsproposed in Yanasse, H.H. and Morabito, R., Linear models for one-grouptwo-dimensional guillotine cutting problems. International Journal of ProductionResearch, in press. The performances of the models were evaluated by solving anumber of examples randomly generated and an actual example derived froma furniture company. These results were produced using GAMS modellinglanguage and the CPLEX solver, and they show that the models are effective tosolve problems of small and moderate size.

Keywords: Cutting and packing problems; 2-Group and 3-group guillotinecutting; Integer linear models; Furniture industry

1. Introduction

In different industrial cutting processes such as in the furniture and hardboardindustries, the cutting equipment is able to produce only guillotine cuts on the plates.A cut is of guillotine type if, when applied to a rectangle, it produces two newrectangles. In the first stage, parallel longitudinal guillotine cuts are produced on theplate (without moving it) to make a set of strips. In the second stage, these strips arepositioned, one by one, and the remaining parallel transversal guillotine cuts aremade on each strip. If there is no need for additional trimming (i.e. all pieces have thesame width in each strip), the pattern is called exact 2-stage guillotine, otherwise, it iscalled non-exact. If there are additional cutting stages on the pattern, for example,three, four, etc., the pattern is called 3-stage, 4-stage, etc., respectively. Other specialclasses of staged guillotine cutting patterns, named 1-group, 2-group, 3-group, etc.,also appear in the furniture and hardboard industries. Studies dealing with thecutting problem in these industries can be found in the following examples:Gilmore and Gomory (1965), Foronda and Carino (1991), Yanasse et al. (1991),

*Corresponding author. Email: [email protected]

International Journal of Production Research

ISSN 0020–7543 print/ISSN 1366–588X online � 2008 Taylor & Francis

http://www.tandf.co.uk/journals

DOI: 10.1080/00207540601011543

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Carnieri et al. (1994), Morabito and Garcia (1998), Morabito and Arenales (2000),

Gramani and Franca (in press) and Morabito and Belluzzo (2007). In this study we

present integer linear models to generate 2-group and 3-group constrained and

unconstrained two-dimensional guillotine cutting patterns.Gilmore and Gomory (1965) proposed a two-phase method based on the simplex

algorithm with a column generation procedure to generate 2-stage patterns. This

two-phase method works well if the ordered quantities of the pieces are ‘sufficiently’

large; in such cases the cutting problem is called unconstrained. For the constrained

case, authors have proposed alternatives for the rounding of relaxed solutions of the

simplex algorithm (e.g. Waescher and Gau 1996), or greedy heuristics combined withcolumn generation procedures (Hinxman 1980). Other studies dealing with staged

patterns can be found in, for example, Beasley (1985), Christofides and

Hadjiconstantinou (1995), Morabito and Arenales (1996), Riehme et al. (1996),

Hifi (1997), Hifi and Roucairol (2001), Vianna et al. (2002), Lodi and Monaci (2003)and Cui (2005). A special class of 2-stage patterns, called 1-group or checkboard

patterns, was introduced in Gilmore and Gomory (1965). In these patterns the

second stage cuts are performed simultaneously on the strips resulting from first

stage cuts. This implies that the second stage cuts can be produced together with the

first stage cuts, without moving the strips and, consequently, save processing time.The 1-group patterns can be non-homogeneous (a pattern is homogeneous if it

contains only pieces of the same type), and exact and non-exact depending on the

need for additional trimming, as illustrated in figure 1 (note that in the exact patterns

there is no need for additional trimming, since all pieces have the same width in eachhorizontal strip and all pieces have the same length in each vertical strip).

Gilmore and Gomory (1965) also discussed p-group cutting patterns, p41.

A p-group pattern is a guillotine pattern composed of p 1-group patterns. Examples

of p-group patterns with p¼ 2 and 3 are depicted in figure 2. The pattern in

figure 2(a) is called exact 2-group and the ones in figure 2(b) and (c) are exact3-group. Note that in figure 2(a) a vertical cut on the plate separates the pattern into

two 1-group patterns. Another way of obtaining a 2-group pattern would be by

(a)

(b)

(c)

(d)

Figure 1. 1-Group cutting pattern: (a) and (b) exact cases; (c) and (d) non-exact cases.

6190 H. H. Yanasse and R. Morabito

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a horizontal cut on the plate, separating the pattern into two 1-group patterns.Three-group patterns can be obtained in different ways, for example, by two verticalcuts on the plate separating the pattern into three 1-group patterns (figure 2b), herenamely 3i-group because of type-I cuts, or by a vertical cut and then a horizontal cuton the plate, separating the pattern into three 1-group patterns (figure 2c), herenamely 3t-group because of the type-T cut (see the bold lines in the figure).Obviously other ways of obtaining 3i-group and 3t-group patterns would be by twohorizontal cuts or by a horizontal cut and then a vertical cut on the plate. In otherwords, by type-I cuts we mean that the three parts of the 3-group pattern areobtained with one-stage cuts only; by type-T cuts we mean that the three parts of the3-group pattern are obtained with two-stage cuts.

Hundreds of studies are found in the literature dealing with cutting and packingproblems and their industrial applications, as reported in the surveys and specialissues in, for example, Dyckhoff and Waescher (1990), Dowsland and Dowsland(1992), Dyckhoff and Finke (1992), Lirov (1992), Sweeney and Paternoster (1992),Martello (1994a, b), Bischoff and Waescher (1995), Mukhacheva (1997), Dyckhoffet al. (1997), Arenales et al. (1999), Hifi (2002), Lodi et al. (2002), Wang andWaescher (2002), Waescher et al. (in press) and ESICUP (2006). However, only a few

(a) (b)

(c)

Figure 2. Other p-group cutting patterns: (a) exact 2-group; (b) exact 3i-group; (c) exact3t-group.

Two-dimensional guillotine cutting problems 6191

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studies presenting mathematical programming formulations for 1-group

two-dimensional guillotine cutting problems are found in the literature, for instance,

Morabito and Arenales (2000), Scheithauer (2002), Yanasse and Katsurayama(2005) and Yanasse and Morabito (in press). In particular, we are not aware of any

study proposing formulations for 2-group and 3-group cutting problems. Models for

such problems are useful for research and development of more effective solution

methods, exploring special features and particular structures, model decomposition,model relaxations, etc. These models are also helpful for evaluating the performance

of heuristic methods, since they allow (at least for problems of moderate size) an

estimation of the optimality gap of heuristic solutions.The present study proposes integer linear models to generate 2-group and

3-group (3i-group and 3t-group) constrained and unconstrained two-dimensional

guillotine cutting patterns, including exact and non-exact cases. These patterns

appear in different cutting processes as, for example, in the furniture, hardboard andstone industries. The models can be viewed as extensions of the linear models for

1-group guillotine cutting patterns proposed in Yanasse and Morabito (in press).

These models can be used in the column generation procedure of Gilmore and

Gomory’s approach aforementioned, or combined with repeated exhaustedreduction heuristics (Hinxman 1980), to solve cutting stock problems. It is important

to observe that the exact cases are of major interest in high production practical

settings like in the furniture and hardboard industries (Morabito and Arenales 2000).

The constrained exact case is also of particular interest in the stone industry(Scheithauer 2002). In practice, the non-exact pattern is less attractive than the exact

pattern for it requires trimmings that are machine expensive, therefore reducing the

process productivity.This text is organised as follows: in section 2 we present linear models for

2-group, 3i-group and 3t-group cutting problems based on the 1-group linear model

from Yanasse and Morabito (in press). In section 3, the performances of the

proposed models are evaluated and compared with the performance of the 1-group

model, by solving a number of examples randomly generated and an actual examplederived from a furniture company. These results were produced using the well-

known commercial modelling language GAMS and the mixed integer linear

programming solver CPLEX (Brooke et al. 1992). They show that the models are

effective to solve problems of small and moderate sizes. Finally, in section 4 wepresent concluding remarks and discuss perspectives for future research.

2. Linear models for 2-group and 3-group patterns

Without loss of generality, the models below assume that the orientation ofthe pieces is fixed, that is, the pieces cannot rotate. Consider the following

parameters:

L,W length and width of the plateli,wi length and width of piece type i (i¼ 1, . . . ,m)wmin minimum width of the piecesvi, bi value (e.g., the area of rectangle li�wi) and demand of piece type i.


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2.1 1-group model

For convenience, before deriving the models we briefly review the linear model forthe 1-group pattern presented in Yanasse and Morabito (in press). Let:

J,K number of different lengths li and widths wi, respectively.

vijk ¼ vi, if li � lj and wi � wk

0, otherwise:Variables:

�j number of times length lj is cut along L�k number of times width wk is cut along Waijk number of rectangles lj�wk containing a piece of type i (i.e. number of

type i pieces contained in all rectangles lj�wk)

maxXm

i¼1

XJ

j¼1

XK

k¼1

vijkaijk ð1Þ

XJ

j¼1

lj�j � L ð2Þ

XK

k¼1

wk�k � W ð3Þ

Xm

i¼1

aijk � �j�k, for all j, k ð4Þ

XJ

j¼1

XK

k¼1

aijk � bi, for all i ð5Þ

with

�j,�k, aijk � 0, integer, i ¼ 1, . . . ,m; j ¼ 1, . . . , J; k ¼ 1, . . . ,K: ð6Þ

Note that �j�k corresponds to the number of rectangles lj�wk in the pattern. Theobjective function (1) maximises the total value of the pieces cut in the pattern,constraints (2) and (3) guarantee that the piece lengths and widths do not exceed theplate length and width, respectively, constraints (4) limit the variables aijk to �j�k,constraints (5) refer to the availability of the pieces and constraints (6) refer to thenon-negativity and integrality of the variables. Models (1)–(6) can be adapted to dealwith the exact case by simply redefining vijk as:

vijk ¼vi, if li ¼ lj and wi ¼ wk for any i ¼ 1, . . . ,m

0, otherwise:

In this case, if two piece types i1 and i2 have the same size (li1,wi1)¼ (li2,wi2)¼ (lj,wk)and the same value vi1¼ vi2¼ vjk (e.g. the area of rectangle lj�wk), we can also reducethe number of model variables by defining ajk ¼

Pmi¼1 aijk, the number of rectangles

lj�wk containing a piece lj�wk (i.e. the number of pieces lj�wk).


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It is worth noting that for the case where the orientation of the pieces is not fixed,

it is sufficient to duplicate pieces in the model corresponding to the original ones

rotated. In this case constraint (5) on the limitation on the number of pieces should

be adjusted accordingly, that is, the sum of the pieces that are rotated, in addition to

the ones in their original orientation, is limited. This comment is also valid for the

models presented below.The non-linear models (1)–(6) can be linearised in the following way

(Harjunkoski et al. 1997, Yanasse and Morabito, in press): let �j ¼Psj

s¼1 2s�1�js,

where �js � {0, 1} and sj is so that: 2sj�1 � L=lj� �

< 2sj , that is, sj is the maximum

number of bits for a binary representation of �j (the same could be done choosing �k

instead of �j). The non-linear constraint (4) is rewritten as:

Xm

i¼1

aijk �Xsj

s¼1

2s�1�js�k, for all j, k

which can be replaced by the following set of linear constraints:

Xm

i¼1

aijk �Xsj

s¼1

2s�1fjks, for all j, k

fjks � �k, for all j, k, s

fjks � �k �Mð1� �jsÞ, for all j, k, s

fjks � M�js, for all j, k, s

�js 2 f0, 1g, for all j, s

where M is a sufficiently large number (e.g. W/wmin). Note that if �js¼ 1 then

fjks¼�k, on the other hand, if �js¼ 0 then fjks¼ 0. Models (1)–(6) is rewritten as:Model 1-group:

maxXm

i¼1

XJ

j¼1

XK

k¼1

vijkaijk ð7Þ

XJ

j¼1

ljXsj

s¼1

2s�1�js � L ð8Þ

XK

k¼1

wk�k � W ð9Þ

Xm

i¼1

aijk �Xsj

s¼1

2s�1fjks, for all j, k ð10Þ

fjks � �k, for all j, k, s ð11Þ

fjks � �k �Mð1� �jsÞ, for all j, k, s ð12Þ

fjks � M�js, for all j, k, s ð13Þ

XJ

j¼1

XK

k¼1

aijk � bi, for all i ð14Þ


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with �js 2 f0, 1g,�k, aijk � 0, integer, fjks � 0, i ¼ 1, . . . ,m; j ¼ 1, . . . , J;

k ¼ 1, . . . ,K, s ¼ 1, . . . , sj ð15Þ

Before moving to the next section, we point out that other possible 0–1transformations for the variables �j could be considered. For instance, thetransformation suggested in Johnston and Sadinlija (2004) is of interest in casethere are specific values of �j that are ruled out a priori and the number of values �jwe can assume is not large. This is so because the transformation is pseudopoly-nomial, hence, the number of 0–1 variables grows faster than the transformation wepresented previously.

2.2 2-group model

Let us consider the 2-group case in figure 2(a) where a vertical cut, say on L1,05L1�L, divides the plate (L,W) into two sub-plates (L1,W) and (L�L1,W). Themodel below considers L1 as a variable and imposes that the cutting patterns in thesesub-plates are limited to the 1-group. It can be viewed as a combination of two1-group models (7)–(15), one for each sub-plate h (h¼ 1 for sub-plate (L1,W) andh¼ 2 for sub-plate (L2¼L�L1,W)), in order to generate a 2-group pattern. Let:

Variables:

�kh number of times width wk is cut along W in sub-plate haijkh number of rectangles lj�wk containing a piece of type i in sub-plate h

(i.e. number of type i pieces contained in all rectangles lj�wk ofsub-plate h)

Lh length of sub-plate h.

Model 2-group (vertical):

maxX2

h¼1

Xm

i¼1

XJ

j¼1

XK

k¼1

vijkaijkh ð16Þ

XJ

j¼1

ljXsj

s¼1

2s�1�jsh � Lh, for all h ð17Þ

XK

k¼1

wk�kh � W, for all h ð18Þ

Xm

i¼1

aijkh �Xsj

s¼1

2s�1fjksh, for all j, k, h ð19Þ

fjksh � �kh, for all j, k, s, h ð20Þ

fjksh � �kh �Mð1� �jshÞ, for all j, k, s, h ð21Þ

fjksh � M�jsh, for all j, k, s, h ð22Þ

X2

h¼1

XJ

j¼1

XK

k¼1

aijkh � bi, for all i ð23Þ


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0 � L1 �L

2ð24Þ

with �jsh 2 f0, 1g,�kh, aijkh � 0, integer, fjksh � 0, i ¼ 1, . . . ,m; j ¼ 1, . . . , J;

k ¼ 1, . . . ,K, s ¼ 1, . . . , sj, h ¼ 1, 2: ð25Þ

Similarly to (1) and (7), the objective function (16) maximizes the total value of thepieces cut in the pattern. The term L1/L can be subtracted from the objectivefunction in order to avoid ties between 1-group and 2-group alternative optimalpatterns, once assuming without loss of generality that vi, i¼ 1, . . . ,m, are positiveinteger values. The term L1/L is in the interval [0, 1] and therefore, in case of a tie, itforces the model to choose a 1-group pattern (with L1¼ 0) rather than a 2-grouppattern (with L140). Constraints (17)–(23) correspond to constraints (8)–(14) foreach sub-plate h (note in (17) that Lh must be replaced by L�L1 when h¼ 2).Constraint (24) imposes that L1 is less than or equal to L/2, by symmetry. As in themodel 1-group, the exact case can also be treated by the model 2-group simplyredefining vijk by:

vijk ¼ vi, if li ¼ lj and wi ¼ wk for any i ¼ 1, . . . ,m0, otherwise:

(or redefining vijk and aijk by vjk and ajk, respectively, to reduce the number ofvariables, as discussed in section 2.1).

The vertical 2-group models (16)–(25) can be easily rewritten to deal with the casewhere a horizontal cut (instead of a vertical cut), for example, on W1, 05W1�W,divides the plate (L,W) into two sub-plates (L,W1) and (L,W1), resulting in thehorizontal 2-group model. Alternatively, this case can be solved by the vertical2-group model simply redefining a plate of size (W,L) and piece types of sizes (wi, li),i¼ 1, . . . ,m.

In table 1 models 1-group (7)–(15) and 2-group (16)–(25) are compared in termsof the number of variables and constraints.

2.3 3-group models

Let us consider initially the 3i-group case in figure 2(b) where two vertical cuts, sayon L1 and L1þL2, 05L1�L2�L, divide the plate (L,W) into three sub-plates(L1,W), (L2,W) and (L�L1�L2,W). Similarly to the model 2-group, the modelbelow considers L1 and L2 as variables and imposes that the cutting patterns in these

Table 1. Comparison of the number of variables and constraints of models 1-groupand 2-group.

Model Number of variables Number of constraints

1-group MJKþ (Kþ 1)PJ

j¼1

1þ log L=lj� ��

þ K 2þ JKþ 3KPJ

j¼1


þm

2-group 2mJKþ 2(Kþ 1)PJ

j¼1


þ 2Kþ 1 5þ 2JKþ 6KPJ

j¼1


þm


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sub-plates are limited to the 1-group. It can be viewed as a combination of three

1-group models (7)–(15), one for each sub-plate h (h¼ 1 for sub-plate (L1,W), h¼ 2

for sub-plate (L2,W) and h¼ 3 for sub-plate (L3¼L�L1�L2,W)), in order to

generate a 3i-group pattern. Let:

Variables:


(i.e. number of type i pieces contained in all rectangles lj�wk of

sub-plate h)Lh length of sub-plate h.

Model 3i-group (vertical):

maxX3

h¼1

Xm

i¼1

XJ

j¼1

XK

k¼1

vijkaijkh ð26Þ

XJ

j¼1

ljXsj

s¼1


XK

k¼1

wk�kh � W, for all h ð28Þ

Xm

i¼1

aijkh �Xsj

s¼1





X3

h¼1

XJ

j¼1

XK

k¼1


0 � L1 � L2 ð34Þ

L2 � L� L1 � L2 ð35Þ

with �jsh 2 f0, 1g, �kh, aijkh � 0, integer, fjksh � 0, i ¼ 1, . . . ,m; j ¼ 1, . . . , J;

k ¼ 1, . . . , K, s ¼ 1, . . . , sj, h ¼ 1, 2, 3: ð36Þ

Note in constraint (27) that Lh must be replaced by L�L1�L2 when h¼ 3.

Constraints (34)–(35) impose that L1 is less than or equal to L2, and that L2 is less

than or equal to L�L1�L2, by symmetry. Similarly to the model 2-group, the term

L1/Lþ (L1þL2)/L can be subtracted from the objective function (26) to avoid ties

between 1-group, 2-group and 3i-group alternative optimal patterns. Note that this

term is in the interval [0, 1] since 2L1þL2�L because of (34)–(35), and therefore, in

case of a tie, it forces the model to choose 1-group and 2-group patterns (with L1¼ 0

and/or L2¼ 0) rather than a 3i-group pattern (with L140 and L240). As in the


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models 1-group and 2-group, the exact case can also be treated by the model

3i-group simply redefining vijk by:

vijk ¼vi, if li ¼ lj and wi ¼ wk for any i ¼ 1, . . . ,m

0, otherwise:

(or redefining vijk and aijk by vjk and ajk, respectively, to reduce the number of

variables, as discussed in section 2.1).Similarly to the model 2-group, the vertical 3i-group models (26)–(36) can be

easily rewritten to deal with the case where two horizontal cuts (instead of two

vertical cuts) divide the plate into three sub-plates, resulting in the horizontal 3i-

group model. This case can also be dealt with using the vertical 3i-group model and

simply considering a plate of size (W,L) and piece types of sizes (wi, li), i¼ 1, . . . ,m.Now let us consider the 3t-group case in figure 2(c) where a vertical cut, for

example on L1, 05L1�L, divides the plate (L,W) into two sub-plates (L1,W) and

(L�L1,W), and then a horizontal cut, for example on W2, 05W2�W, divides the

plate (L�L1,W) into two sub-plates (L�L1,W2), (L�L1,W�W2). Similarly to the

model 3i-group, the model below considers L1 and W2 as variables and imposes that

the cutting patterns in these sub-plates are limited to the 1-group. It can be viewed as

a combination of three 1-group models (7)–(15), one for each sub-plate h (h¼ 1 for

sub-plate (L1,W1¼W), h¼ 2 for sub-plate (L2¼L�L1,W2) and h¼ 3 for sub-plate

(L3¼L�L1,W3¼W�W2)), in order to generate a 3t-group pattern. Let:

Variables:


(i.e. number of type i pieces contained in all rectangles lj�wk of

sub-plate h)Lh, Wh length and width of sub-plate h.

Model 3t-group (vertical):

maxX3

h¼1

Xm

i¼1

XJ

j¼1

XK

k¼1

vijkaijkh ð37Þ

XJ

j¼1

ljXsj

s¼1


XK

k¼1

wk�kh � Wh, for all h ð39Þ

Xm

i¼1

aijkh �Xsj

s¼1






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X3

h¼1

XJ

j¼1

XK

k¼1


0 � W2 �W

2ð45Þ

with �jsh 2 f0, 1g, �kh, aijkh � 0, integer, fjksh � 0, i ¼ 1, . . . ,m; j ¼ 1, . . . , J;

k ¼ 1, . . . , K, s ¼ 1, . . . , sj, h ¼ 1, 2, 3: ð46Þ

Similarly to the models 2-group and 3i-group, the term (L1þW2)/(LþW) can besubtracted from the objective function (37) to avoid ties between 1-group, 2-group

and 3t-group alternative optimal patterns; note that this term is in the interval [0, 1]and therefore, in case of a tie, it forces the model to choose 1-group and 2-grouppatterns (with L1¼ 0 and/or W2¼ 0) rather than a 3t-group pattern (with L140 andW240). Note in constraints (38) and (39) that Lh and Wh must be replaced by the

appropriate expressions when h¼ 2 or 3. Constraint (45) imposes that W2 is less thanor equal to W/2, by symmetry. As in the model 3i-group, the exact case can also bedealt with by the model 3t-group simply redefining vijk by:

vijk ¼vi, if li ¼ lj and wi ¼ wk for any i ¼ 1, . . . , m

0, otherwise:

(or redefining vijk and aijk by vjk and ajk, respectively, to reduce the number of

variables, as discussed in section 2.1). The horizontal 3t-group case can be dealt withanalogously as the horizontal 3i-group case.

3. Computational experiments

In this section, we present the computational results obtained by applying models2-group (models 16–25), 3i-group (models 26–36) and 3t-group (models 37–46) for

sets of examples solved by the 1-group models (7)–(15) in Yanasse and Morabito(in press). As aforementioned, since the exact cases are of major interest in practice,the computational results of this section refer only to exact 1-group, 2-group and

3-group models. These models were codified in the modelling language GAMS(Brooke et al. 1992) and solved by the integer linear programming solver CPLEX(version 7) in a microcomputer Pentium IV with 2.8GHZ, 512MBRAM. As in

Yanasse and Morabito (in press), we use the standard default settings of CPLEX,that is, we did not explore problem-oriented branching rules, initial heuristicsolutions, problem-specific valid inequalities, etc.

In the first set of examples the values of the pieces are equal to their respective

areas (i.e. vi¼ liwi); thus, a simple upper bound equal to the plate area LW might beimposed on their models. However, as pointed out in Yanasse and Morabito(in press) and also observed in our computational results, the execution times are, in

general, shorter when we do not impose this upper bound. Thus the values presentedin the tables below correspond to the results obtained without the upper bound. Thesolution values of the tables are in percentages of the plate area utilization. In some


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cases we also included the value of the objective function of the models (recall fromsection 2 that may be different because of the term to break ties).

Table 2 presents the input data of a simple constrained two-dimensionalguillotine cutting problem analysed in Vianna et al. (2002). We assume that theorientation of the pieces is fixed, that is, the pieces cannot rotate. Table 3 presents theresults obtained by the models with GAMS/CPLEX for such an example. The row‘Time (s)’ corresponds to the required computer runtime (in seconds) of GAMS/CPLEX. The optimal solutions were found in relatively small computer runtimes.The symbols (v) and (h) in the table indicate the vertical and horizontal p-groupsolutions, respectively. Note that the optimal values improve as we move from themodel 1-group to the model 3-group (as expected). Figure 3 depicts the p-groupcutting patterns for some of the solutions presented in table 3.

We run the same randomly generated constrained examples of Yanasse andMorabito (in press) with plate (L,W)¼ (100, 100) and m¼ 10, 20 and 50 piece types(li,wi) sampled from uniform distributions in the intervals [0.1L, 0.5L] and [0.1W,0.5W], respectively (after being sampled, li and wi were simply rounded). Quantitiesbi were also sampled (and then rounded) from uniform distributions in the intervals[1, L/liW/wi]. For each value of the parameter m, 10 instances were considered. Weassume that the orientation of the pieces is fixed, that is, the pieces cannot rotate.We imposed a time limit of 600 seconds for the computer runtime of each probleminstance. In table 4 the results obtained by the vertical p-group models with m¼ 10are presented. The ‘Solution (%)’ columns correspond to the plate area utilisation(in percentages).

Before moving to random instances with larger values of m, we illustrate theperformance of the models in a practical example of size m¼ 13 of a furniture factory

Table 2. Constrained example with (L, W)¼ (100, 100) andm¼ 5 piece types.

Pieces i li wi bi

1 33 14 82 20 38 43 15 43 44 40 43 35 31 44 3

Table 3. Results of the models for the example of table 2.

Example1-groupmodel

2-groupmodel

3i-groupmodel

3t-groupmodel

Objective function value� 64.50 79.64 (v) 85.41 (v) 90.04 (v)83.91 (h) 91.25 (h) 90.60 (h)

Plate area utilization (%) 64.50 79.64 (v) 85.42 (v) 90.04 (v)83.92 (h) 91.26 (h) 93.60 (h)

Time (s) 0.1 0.1 (v) 0.2 (v) 0.7 (v)0.2 (h) 0.2 (h) 0.4 (h)

*Value of the objective function with the term to break ties.


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(a) (b)

(c) (d)

3

4

3

4

4

3 3 3 3 4

5 5 5

3 3 3

3

4

4 4

1113 3 3 3 4

5 1 1

1

1

1

1

1

1

Figure 3. Some p-group cutting patterns for the example of table 2: (a) 1-group;(b) horizontal 2-group; (c) horizontal 3i-group; (d) horizontal 3t-group.

Table 4. Results of the models for random examples with m¼ 10 piece types.

Model 1-group Model 2-group Model 3i-group Model 3t-group

ExampleSolution(%)

Time(s)

Solution(%)

Time(s)

Solution(%)

Time(s)

Solution(%)

Time(s)

1 87.30 0.2 96.50 0.5 97.00 1.8 100.00 14.42 66.15 0.3 88.74 0.6 91.08 2.2 97.74 4.93 92.00 0.1 96.76 0.6 99.16 0.8 96.76 3.74 68.80 0.2 94.64 0.4 94.64 0.6 96.18 2.85 84.70 0.3 97.20 0.7 98.52 2.2 99.80 11.36 91.00 0.2 94.90 0.5 94.90 0.5 98.74 6.37 72.00 0.4 87.45 1.3 92.90 1.8 97.20 6.98 92.16 0.2 97.76 0.5 97.76 0.8 99.60 32.49 81.78 0.2 88.45 0.5 93.92 0.9 94.74 3.010 80.84 0.2 94.96 0.3 94.96 0.4 99.68 2.0

Average 81.67 0.2 93.74 0.6 95.48 1.2 98.04 8.8


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analysed in Morabito and Arenales (2000). Tables 5 and 6 present the input data and

the results obtained by the models for this example (the solution values in

the table correspond to the total areas of the pieces cut in the pattern). Note that the

optimal 1-group pattern is also optimal for the vertical and horizontal 2-group

patterns, but the optimal vertical 3i-group and 3t-group patterns are different.Following our experiments, we applied the models to solve the sets of 10

randomly generated examples with 20 and 50 piece types. Table 7 compares the

average results obtained by the vertical p-group models for the sets with m¼ 10, 20

and 50. Note that the mean optimal values improve as we move from m¼ 10 to

m¼ 50 (as expected), and as we move from the model 1-group to the model 3-group.

Table 5. Practical example of the furniture factory with (L,W)¼ (1850, 3670)and m¼ 13 piece types.

Pieces i li wi Pieces i Li wi

1 609 274 8 705 3002 380 274 9 465 3303 425 330 10 480 3304 650 361 11 674 3025 348 270 12 674 2706 705 270 13 636 2707 718 328

Table 6. Results of the models for the practical example in table 5.

Example1-groupmodel

2-groupmodel

3i-groupmodel

3t-groupmodel

Objective function value� 6 715 500 6 715 500 (v) 6 741 829 (v) 6 756 600 (v)6 715 500 (h) 6 715 500 (h) 6 756 599 (h)

Total area of the pieces cut 6 715 500 6 715 500 (v) 6 741 830 (v) 6 756 600 (v)6 715 500 (h) 6 715 500 (h) 6 756 600 (h)

Time (s) 0.2 0.9 (v) 1.3 (v) 39.1 (v)3.2 (h) 25.7 (h) 135.2 (h)

*Value of the objective function with the term to break ties.

Table 7. Average results of the models for random examples with m¼ 10, 20 and50 piece types.


Set of randomexamples

Solution(%)

Time(s)

Solution(%)

Time(s)

Solution(%)

Time(s)

Solution(%)

Time(s)

m¼ 10 81.67 0.2 93.74 0.6 95.48 1.2 98.04 8.8m¼ 20 91.42 1.1 97.23 4.5 98.14 13.1 98.93 133.0m¼ 50 98.15 12.3 99.81 215.5 99.90� 415.2� ��

*Average results of only 6 examples (out of 10) solved within the time limit.**No examples solved within the time limit.


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All models were able to solve all examples with 10 and 20 piece types within the timelimit of 600 seconds. Models 1-group and 2-group also solved all examples with50 piece types, while the model 3i-group solved only six of them (out of 10) andmodel 3t-group was unable to solve all of them within this time limit.

In the previous sets of random instances tested, the value of each piece wasproportional to its area. In the next set of instances, we take the same set of examplestested previously but with more arbitrary piece values. The value of each piece type iwas randomly generated in the interval [0.8 liwi, 1.2 liwi], where liwi is the piece area.We tested the vertical p-group models with instances with m¼ 10 piece types – theresults are presented in table 8 (note that since there may be pieces with valuesgreater than their areas, the solution values may be greater than the plate area).We observe no significant difference compared with the results in table 4.

Recall that on solving the cutting stock problem using the Gilmore and Gomory(1965) approach, a pattern generation problem must be solved at each iteration ofthe simplex method. Therefore, it is of great interest to have an efficient way to solveit. With the proposed approach we are able to solve instances up to 50 piece types inless than 600 seconds. For instances with 20 piece types, the average execution timesvaried from 1 up to 133 seconds, depending on the type of pattern (1-group, 2-groupor 3-group) generated. Even when we varied the values of the pieces (as we face whengenerating patterns during the overall optimisation of a cutting stock problem) theobserved average times did not change significantly (table 8).

4. Concluding remarks

In this study we propose linear models to generate 2-group and 3-group constrainedtwo-dimensional guillotine cutting patterns, including exact and non-exact cases, asextensions of the 1-group model presented in Yanasse and Morabito (in press). Tothe best of our knowledge, there are neither mathematical formulations nor exactalgorithms for the 2-group and 3-group pattern generation problems in the literature.

Table 8. Results of the models for random examples with m¼ 10 piece typeswith arbitrary values.


ExampleSolution(%)

Time(s)

Solution(%)

Time(s)

Solution(%)

Time(s)

Solution(%)

Time(s)

1 84.92 0.2 91.47 0.9 91.47 1.8 102.01 7.22 60.65 0.5 85.08 0.6 87.23 2.1 89.48 7.13 92.01 0.1 94.38 0.7 95.01 1.2 98.57 4.24 78.37 0.2 104.94 0.2 104.94 0.9 104.94 2.55 74.65 0.4 100.91 0.9 100.91 2.5 107.40 6.96 91.07 0.1 99.16 0.5 99.16 1.6 99.99 6.37 67.32 0.5 86.40 1.0 87.75 3.3 96.07 6.58 97.73 0.2 101.26 0.7 101.26 2.8 106.25 18.09 81.78 0.1 95.79 0.4 95.79 1.0 96.71 3.210 85.48 0.1 93.78 0.3 93.78 0.5 95.28 3.9

Average 81.40 0.2 95.32 0.6 95.73 1.8 99.67 6.6


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These models are useful for research and development of more efficient solutionmethods, exploring specific features and particular structures, model decomposition,model relaxations, etc. Such models are also helpful for the performance evaluationof heuristic methods, since they allow for (at least for problems of moderate size) theestimation of the optimality gap of heuristic solutions.

A comparison of the 1-group, 2-group and 3-group models performance wascarried out. We presented results of computational experiments using well-knowncommercial software such as the modelling language GAMS and the solver CPLEX(with standard default settings). These results showed that the models are effective tosolve problems of small and moderate sizes, and the computational efforts to solvethe models are very different. An interesting line of research is the study of effectiveupper and lower bounds, branching strategies and problem-specific valid inequalitiesin order to reduce the computer runtimes required to solve the models.

It is worth noting that the present approaches can be easily applied to deal withsimpler cases of p-group patterns where in each (or some) group(s) the patterns areconstrained to be homogeneous. Similarly, the approaches can be extended to copewith simpler n-stage patterns with particular configurations, for instance, theT-shaped 3-stage pattern recently studied in Cui (2005).

Acknowledgements

The authors thank the three anonymous referees for their useful comments andsuggestions. This research was partially supported by CNPq and FAPESP.

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a note on linear models for two-group and three-group two-dimensional guillotine cutting problems

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