Gislaine Mara Melega1*, Silvio Alexandre de Araujo1 and Raf Jans2

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©2016BrazilianOperationsResearchSociety
PrintedversionISSN0101-7438/OnlineversionISSN1678-5142
www.scielo.br/pope
doi:10.1590/0101-7438.2016.036.01.0167
COMPARISONOFMIPMODELSFORTHEINTEGRATEDLOT-SIZING
ANDONE-DIMENSIONALCUTTINGSTOCKPROBLEM
GislaineMara Melega1*, SilvioAlexandre deAraujo1 and RafJans2
ReceivedNovember30,2015/AcceptedApril1,2016
ABSTRACT.Productionprocessescomprisingboththelot-sizingproblemandthecuttingstockproblem
arefrequentinvariousindustrialsectors.Howevertheseproblemsareusuallytreatedseparately,whichcan
generatessuboptimaloverallsolutionandconsequentlycausesproductionlosses.Inthispaper,wepropose
different mathematicalmodelsfortheintegratedproblemcombiningalternativemodelsforthelot-sizing
andthecuttingstockproblem,inordertoevaluateandindicatetheimpactofthesechangesonthemodels’
performance. Anextensivecomputationalstudyisdoneusingrandomlygenerateddataandasasolution
strategyweusedacommercialoptimizationpackageandtheapplicationofacolumngenerationtechnique.
Keywords:CapacitatedLot-SizingProblems,One-DimensionalCuttingStockProblem,ColumnGenera-
tion.
1 INTRODUCTION
Technological advances and increasing competitioninindustrieshave become worldwidephe-
nomena. Some prominent aspects of these advances are the emphasis on knowledge and the
development of new technologies. Inserted on thiscontext, mathematical models thatdescribe
and improve the productionprocesses have been studiedand used as toolstosupportdecision
making in the business environment. Among the various decision processes, this work is part
ofthe tactical/operationalplanningofproductionwiththelot-sizingproblemandcuttingstock
problem. TheLot-SizingProblem(LSP)considersthetradeoffbetweenthesetupandinventory
costs todetermine atminimalcostthesize ofproductionlotstomeet thedemand ofeach final
product,whilethecuttingstockproblem(CSP)involvesthecuttingoflargeobjectsintosmaller
items,soastominimizethetotallossofmaterial.
The literature mostly deals with these two problems separately. However, in industrialsectors
such as furniture, paper and aluminum, the lot-sizingand cuttingstock problems are foundin
*Correspondingauthor.
1
DepartamentodeMatema´ticaAplicada,IBILCE, UNESP–UniversidadeEstadualPaulista,15054-000Sa˜oJose´do
RioPreto,SP,Brasil. E-mails:[email protected]; [email protected]
2HECMontre´alandCIRRELT,CanadaH3T2A7QC,Canada.E-mail:[email protected]
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MIPMODELSFORTHEINTEGRATEDLOT-SIZINGANDONE-DIMENSIONALCUTTINGSTOCKPROBLEM
consecutive phases. For these cases, a decoupled visionof the problems usuallyused by com-
panies, can provide good local solutionsbut, at the end of planningprocesses, these solutions
may conflictwiththeglobalobjectivesandfeasibilityofproduction.Thisfactwas observedin
Gramani et al. (2009) in the furniture industry. In this case, solutionsgenerated from the lot-
sizingproblemcan lead toinfeasiblesolutionsforthecuttingstockproblemwithregardtothe
productioncapacity of the cutting machine. If, on the other hand, for a given period, the pro-
ductionexceeds demand, thiscircumstance may generate cuttingpatternswithless loss, and a
decrease ofsetupcosts. However, thereisanincrease ofinventorycostsandphysicalspace for
allocationofitemstobestoredwillbeneeded,whichmaynotexistinpractice. Forthisreason,
therelevanceoftheseissuesintheindustrialsectorsandthebenefitsindealingwithproblemsin
anintegratedwaymakesthisanappropriateresearchtopictoday. Motivatedbythis,thepresent
paper proposesmathematical modelsandsolutionmethodsforsolvingtheintegratedlot-sizing
andcuttingstockproblem.
Inthiswork,the capacitated lot-sizingproblemismodeled usingthe mathematical modelpro-
posedbyTrigeiroetal. (1989),denotedherebyCL.Wealsoconsideredthevariableredefinition
strategy (Eppen & Martin, 1987) which reformulates the lot-sizingproblem as a shortest path
problem. This reformulationis called SP. The idea of this variable redefinition was originally
proposed foruncapacitated problemsand Jans & Degraeve (2004), Jans (2009), Fiorotto& de
Araujo(2014)andMelegaetal. (2013)extendedthistocaseswithcapacityconstraints,related
parallelmachines, unrelatedparallelmachines andvariousplants, respectively. Moreinforma-
tiononreformulationforlot-sizingproblemscanbefoundinDenizeletal. (2008),whichshows
severaltheoreticalandcomputationalresultsbetweensomereformulations.
To model the one-dimensional cuttingstock problem, we consider three mathematical models
from the literature. The first one is the model developed by Kantorovich (1960), here denoted
byKT.Thismodelisalsocalled thegeneralizedassignmentformulationforCSP(Degraeve &
Peeters, 2003)and determines thebest waytocutobjects tomeet the demand, minimizingthe
numberofobjectsused. Forthismodel,anupperboundonthenumberofobjectsisconsidered.
The second model dealt with,and perhaps the best knownamong the academic community, is
theoneproposedbyGilmore&Gomory(1961,1963)denotedherebyGG.Thismodelproduces
goodlowerboundswhencomparedtotheKTmodel,howeverithasanlargenumberofvariables,
whichindicatestheuseofcolumngenerationtodealwiththisdifficulty.
The thirdmodelwasproposedbyVale´riode Carvalho(1999,2002),here denotedVC. Theau-
thorsproposeanalternativemathematicalmodelfortheone-dimensionalcuttingstockproblem
basedonanarcflowproblem.Thus,findingavalidcuttingpatternforthecuttingstockproblemis
equivalenttofindingapathinadirectedacyclicgraph.Theauthorsalsoconsideradditionalcon-
straintstoguaranteethatdemandissatisfied.Thismodelisefficientinthesensethatitpresents
alinearrelaxationasgoodastheGilmoreandGomorymodel.
Furthermore, the cutting stock models described above were originally proposed for a single
object typeso theyare extended toconsider several types ofobjects instock(MOfrom multi-
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objects). In a second part ofthis study, these models are integrated intothe uncapacitated lot-
sizingproblemproposedbyWagner&Whitin(1958),herecalledWW.
Next, inSection2,there isa literaturereviewwiththemajorstudiesthataddress thelot-sizing
andthecuttingstockprobleminanintegratedway. TheSections3,4and5describetheproposed
integratedmodelswithcapacityconstraints,severalobjecttypesandareformulationforthelot-
sizing,respectively. InSection6,thesolutionstrategiesaredescribed. Thecomputationalresults
arepresentedinSection7. Andfinally,theconclusionsarepresentedinSection8.
2 LITERATUREREVIEW
The importance of addressing the problems in an integrated way has become increasingly ev-
ident in the literature and is closely related to industrial applications. Possibly Farley (1988)
had been the first paper that treats these problems in an integrated way. The author addressed
theprobleminanclothingindustry,wheretheprobleminvolvesirregulartwo-dimensionalcuts.
Hendryetal. (1996)conductedacasestudyonacoppercomponentsfactory.Theobjectivewas
toinvestigatealternativemethodsofgeneratingproductionplansforcastinginordertominimize
costs andmeet demand. Nona˚s& Thorstenson(2000,2008)studiedtheproblemcoupledwith
theadditionofsetupcostinaNorwegiancompanythatproducesoff-roadtrucks.Thecompany
needed aproductionplanthatminimizesthetotalcostinthecuttingprocessandintheproduc-
tionline. Therefore, the authorsproposed a model and different solutionmethods. Gramani &
Franc¸a(2006)formulatedamathematicalmodelfortheintegratedprobleminafurnitureindustry
and proposed a solutionmethodbased onan analogywiththeshortest pathproblem. Compu-
tational resultscompared the proposed method to the method used in the industry.Poltroniere
etal. (2008)studiedtheintegratedproblemappliedtothepaperindustry.Theauthorsanalyzed
the production process and proposed a mathematical model for machines operating in paral-
lel and presented twoheuristicsbased on Lagrangian relaxation consideringthe problems ina
decoupledway.
Ghidini & Arenales (2009) treated the coupled problem applied to the furniture industry. To
solvetheproblem,whichconsidersthefinalcompositionoftheproducts,theauthorsproposed
two heuristics based on the primal simplex method with column generation. A procedure for
obtaininganintegersolutionhasbeenproposed. Gramanietal.(2009)proposedamathematical
modelthatconsidersfinalproductsanda capacityconstraintonthesaw machineinafurniture
industry. Twosolutionmethodswereproposed. Oneisadecompositionheuristicandtheother
one is a Lagrangian based heuristic, in which the resulting problem is decomposed into two
subproblems. Asmoothingheuristictorecoverfeasibilityisused. Gramanietal. (2011)extend
theirmodelfrom2009toconsidercutitemsinventoryandthismodeldecomposedintotwo,one
forthelot-sizingproblemandoneforthecuttingstockprobleminordertoportraythesolution
methodthatindustriespracticed.Theintegratedproblemwassolvedusingthecolumngeneration
technique. The computational results showed that the column generation technique can obtain
gains of more than 12.7%, compared with the decompositionmodel used in practice. Also in
the context of a furniturefactory Santos et al. (2011) presented a mixed integer programming
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modelfortheintegratedprobleminthecontextofafurniturefactory.Theauthorsconsideredthe
cuttingpatternsusedbyindustryandn-groupscuttingpatterntosolvetheproblemwithasetof
realdataprovidedbythefactory. AnextensiontoGramanietal. (2011)modelwasproposedin
Vanzelaetal. (2013)inordertosimulatethepracticeofasmallfurniturefactory.
Alem & Morabito(2012)used robustoptimizationtoolstosolve theintegrated probleminthe
furniture industry, where the productioncosts and demand of products are uncertain parame-
ters. Computationaltestswereperformedusingrealandsimulateddata.Lea˜o&Arenales(2012)
studieda simplificationofthemodelstudiedinPoltroniereet al. (2008).Twoapproaches have
beenstudied:(i)forasingletypeofobjectand(ii)forvarioustypesofobjects.Threemathemat-
ical modelsforeach approach have been proposedandthey usedthe columngenerationas the
solutionmethod.Toobtainafeasiblesolution,thecolumnsobtainedwiththecolumngeneration
methodareusedandtheresultingintegerproblemwiththislimitedsetofcolumnsissolved.
Silva et al. (2014) presented two integer programming models for the integrated problem. In
thesemodels,boththebringingforwardofitemsforproductionandalsothestockingofreusable
leftoversforthecuttingprocessinlaterperiodswereallowed.Theauthorsproposedtwoheuris-
tics and evaluated the models through a computational study, checking the quality of the so-
lutionswhen compared to work described in the literature. In more recent papers, Poldi& de
Araujo(2016)studiedthemulti-periodcuttingstockproblemandMolinaetal.(2016)proposed
anintegratedlotsizingandpackingproblemthatisrelatedtotheproblemstudiedinthispaper.
Mostofthestudiesintheliteraturethataddresstheintegratedlot-sizingandcuttingstockprob-
lemstudyseveralcases oftheprobleminpractice. Inthesestudies,alternativeformulationsfor
theLSPandCSParenottestedinordertochoosetheformulationthatbestfitstheproblemand
thedataset. So,extendingtheideasofLonghietal.(2015)thispaper triestocover thegapby
proposingalternativemathematicalformulations,consideringdifferentfeaturesofthelot-sizing
and cutting stock problem such as capacity constraints and several object types. Furthermore,
throughtheuseofextensivecomputationaltests,itintendstoevaluateandpointouttheimpact
ofthesechangesintheperformanceofthemodels.
3 MATHEMATICALMODELSWITHCAPACITYCONSTRAINT
Inthissectionthelot-sizingproblemproposedbyTrigeiroetal. (1989)(CLmodel)isintegrated
withvariouscuttingstockformulationsasdescribedabove.Theintegratedlot-sizingandcutting
stockmodelisatwolevelproductionmodel,withproductionoffinalproductsatthefinallevel,
andthecuttingofthepiecesattheprecedinglevel.Theproposedmodelassumesthatfinalprod-
ucts can be kept in inventory, but pieces cannot be kept in inventory. Furthermore, the Billof
Materialindicatesthatonefinalproductrequiresexactlyonepiece,andeachtypeoffinalprod-
uctrequiresadifferentpiece.Therefore,thereisaone-to-onerelationshipbetweenfinalproducts
andpiecesinthemodel.
This might be a relevant model in various industrial applications. A first relevant setting is
present,forinstance,inthefurnitureindustry.Thereisademandforthefinalproducts(wardrobe,
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for example) which creates demand for subassemblies or components (a door, for example)
(Gramani&Franc¸a,2006).Afterthecuttingprocess,thecutitemspassthroughanotherproduc-
tiveprocess(e.g. drillingandpainting)beforebeingreadytobeassembledintoafinalproduct.
Usually these other productive processes are considered the bottlenecks of the complete pro-
duction process (Ghidiniet al., 2007; Santos et al., 2011; Alem & Morabito, 2013), and their
capacity is consideredinthemodel, whereas the finalassembly is notconsidered inthemodel
sinceitisnotabottleneckprocess. Ghidinietal. (2007)andAlem&Morabito(2013)considered
thedrillmachinecapacity intermsofitems. Santosetal. (2011)consideredaggregatedcapac-
ityconstraintsforalltheothermachines (exceptthecuttingmachine). Fortheone-dimensional
case, considered in this paper, the same type of capacity constraints appear. For instance, in
tubularfurnitureindustries,the bendingmachine is a bottleneckofthe productionprocess and
itscapacitymustbeconsideredinthemathematicalmodel.
A second potentialpractical backgroundappears in industrieswhere the cut items are directly
transformed in end items, i.e., there is no need to assemble several items to compose a final
product,butthereisa productionprocess betweenthecutitemsandtheenditems. Thecapac-
ity constraint in terms of end items means that the capacity constraints are related to the final
productionprocesstotransformthecutitemsintothefinalitems.
Considerthefollowingparametersanddecisionvariables:
Parameters:
T:numberoftimeperiods(indext);
I: numberofitems(indexi);
sc :setupcostofitemi inperiodt;
it
hc : unitholdingcostofitemi inperiodt;
it
st :setuptimeofitemi inperiodt;
it
vt : productiontimeofitemi inperiodt;
it
cap :capacity(inunitoftime)inperiodt;
t
d : demandofitemi inperiodt;
it
sd :sumofdemandofitemi fromperiodt untilperiodr;
itr
L:objectlength;
l :lengthofitemi;
i
co: fixedcostofanobject.
DecisionVariables:
X : productionquantityofitemi inperiodt;
it
S : inventoryforitemi attheendofperiodt;Y : binaryvariableindicatingtheproductionor
it it
notofitemi inperiodt.
The firstcuttingmodelusedtoformulatetheintegratedproblem,istheKT model,whichgives
themodelCLKTfromCL+KT.Forthis,considerthefollowingdataandvariablesspecifictothis
model:
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MIPMODELSFORTHEINTEGRATEDLOT-SIZINGANDONE-DIMENSIONALCUTTINGSTOCKPROBLEM
Q: numberofobjectsavailableinstock(indexq);
y :binaryvariablethatindicateswhetherobjectq isusedinperiodt;
qt
h :variablethatindicatestheamountofitemi cutfromobjectq inperiodt.
iqt
ModelCLKT
(cid:2)T (cid:2)I (cid:2)T (cid:2)Q
min (sc Y +hc S )+ coy (1)
it it it it qt
t=1 i=1 t=1q=1
Subjectto:
Xit +Si,t−1 =dit +Sit ∀i, ∀t (2)
X ≤sd Y ∀i, ∀t (3)
it itT it
(cid:2)I
(st Y +vt X )≤Cap ∀t (4)
it it it it t
i=1
(cid:2)I
l h ≤ Ly ∀q, ∀t (5)
i iqt qt
i=1
(cid:2)Q
h = X ∀i, ∀t (6)
iqt it
q=1
Y , y ∈{0,1} ∀i, ∀q, ∀t (7)
it qt
hiqt ∈Z+ ∀i, ∀q, ∀t (8)
Xit, Sit ∈R+ ∀i, ∀t (9)
The objectivefunction(1)minimizes the machine setup costs, inventoryholdingcosts and the
costofcuttingobjects. Constraints(2)are thedemandconstraints: inventorycarriedover from
thepreviousperiodandproductioninthecurrentperiodareavailabletobeusedtosatisfycurrent
demandandbuildupinventory. Constraints(3)forcethesetupvariabletooneifanyproduction
takesplaceinthatperiod.Next,thereisaconstraintontheavailablecapacityineachperiod(4).
Constraints(5)arefromthecuttingstockproblemandensurethatifobjectq isused, thecom-
binationofthelengthsofitemsthatwillbecutfromitshouldnotexceed itslength. Constraints
(6) are responsiblefor the integrationof the lot-sizingand cut stock problem decision, thatis,
wehavetocutasufficientamountofitemstomeettheplannedproductionamount. Finally,the
setofconstraints(7),(8)and(9)definethenon-negativityandintegralityconditions.Itisknown
thatthemodelKT forthecuttingstockproblemhasaweaklinearrelaxationandpresentsmany
symmetricsolution(Vale´riodeCarvalho,2002).
Observe that the variables X and constraints (6) could be eliminated by replacing X by
(cid:3) it it
Q
h intherestofthemodel.Howevertocomparethismodelwithotherproposedmodels,
q=1 iqt
thevariable X andtheconstraints(6)arekeptexplicitlyinthemodel. Thisremarkisalsovalid
it
fortheothermodelspresentedbelowintheSections3and4.
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The second integrated formulationpresented below (CLVC)uses the VC model for thecutting
stockproblem,whichismodeledasan arcflowproblem. Considerapathina directedacyclic
graph G = (V,A), with V = {0,1,...,L} and the set of arcs of the graph defined as A =
{(j,l);0 ≤ j < l < L and l− j = l forall i ≤ I}. Thelossesintheobjectgeneratedfrom
i
thecuttingprocessarerepresentedinthegraphbyadditionalarcsbetweenthevertices(j, j+1)
to j =0,...,L −1. AsdecisionvariablesfortheintegratedmodelCLVCwehave:
f : flowthroughthenetworkinperiodt;
t
z :numberofcuttingpatternswhichhaveanitemofsize(l− j)allocatedatadistance j from
jlt
thebeginningoftheobjectinperiodt.
ModelCLVC
(cid:2)T (cid:2)I (cid:2)T
min (sc Y +hc S )+ cof (10)
it it it it t
t=1i=1 t=1
Subjectto:
Xit +Si,t−1 =dit +Sit ∀i, ∀t (11)
X ≤sd Y ∀i, ∀t (12)
it itT it
(cid:2)I
(st Y +vt X )≤Cap ∀t (13)
it it it it t
i=1(cid:2)
− z =−f ∀t (14)
0lt t
(0,l)∈A
(cid:2) (cid:2)
z − z =0 l =1,...,L −1,∀t (15)
glt lht
(g,l)∈A (l,h)∈A
(cid:2)
z = f ∀t (16)
lLt t
(l,L)∈A
(cid:2)
zh,h+li,t = Xit ∀i, ∀t (17)
(h,h+li)∈A
Y ∈{0,1} ∀i, ∀t (18)
it
zlht, ft ∈Z+ ∀(l,h) ∈ A, ∀t (19)
Xit, Sit ∈R+ ∀i, ∀t (20)
Theobjectivefunction(10)minimizesthemachinesetupandinventorycostsoftheitems,aswell
asthecostoftheflowthroughthenetwork.Theflowsetforthisproblemrepresentsthenumber
of used objects (cutting patterns), since one flow unit defines a path, which in turn defines a
cuttingpattern.
Constraints(11),(12)and(13)refertothelot-sizingproblemandaredefinedasinCLKT.Theset
ofconstraints(14),(15)and(16)correspondtoflowconservationconstraintsandarecharacteris-
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MIPMODELSFORTHEINTEGRATEDLOT-SIZINGANDONE-DIMENSIONALCUTTINGSTOCKPROBLEM
ticoftheVale´riodeCarvalhomodel. Constraints(17)integratethetwoproblemsandconstraints
(18),(19)and(20)arenon-negativityandintegralityconstraints.
Thismodelalsopresentsmanysymmetricsolutions,oralternativesthatcorrespondtothesame
cuttingpatterns. Forthisreason,Vale´riodeCarvalho(1999)presentedsomereductioncriteriato
eliminatesomearcs, reducingthenumberofsymmetricsolutionswithouteliminatinganyvalid
cutting patterns from set A. This study applies the criterion consistingof allocating the items
in order of decreasing length in each cuttingpattern, that is, an item of length i can only be
1
placedafteranotheritemlengthi ifi ≤i ,oratthebeginningoftheobject. Anothercriterion
2 1 2
alsouseddoesnotallowstartingwithacuttingpatternwithloss. Thus,thefirstarc oflosswill
be insertedinthegraphata distancefromthebeginningoftheobjectrepresentingtheshortest
itemlength.
The third model CLGG combines the lot-sizing model CL with the cutting stock model GG
(Gilmore&Gomory,1961,1963),whichhasthefollowingparametersanddecisionvariables:
J:setofcuttingpatterns(index j);
a :parameterindicatingthequantityofitemi cutaccordingtothecuttingpattern j;
ij
x :variableindicatingthenumberofobjectscutaccordingtocuttingpattern j inperiodt.
jt
ModelCLGG
(cid:2)T (cid:2)I (cid:2)T (cid:2)
min (sc Y +hc S )+ cox (21)
it it it it jt
t=1 i=1 t=1 j∈J
Subjectto:
Xit +Si,t−1 =dit +Sit ∀i, ∀t (22)
X ≤sd Y ∀i, ∀t (23)
it itT it
(cid:2)I
(st Y +vt X )≤Cap ∀t (24)
it it it it t
(cid:2)i=1
a x = X ∀i, ∀t (25)
ij jt it
j∈J
Y ∈{0,1} ∀i, ∀t (26)
it
xjt ∈Z+ ∀j, ∀t (27)
Xit, Sit ∈R+ ∀i, ∀t (28)
The objective function(21)minimizes thesum ofsetup costs, inventorycosts and cost related
tothenumberofobjectsused inthecuttingprocess. Thesets ofconstraints(22),(23)and(24)
refer tothelot-sizingproblemandare defined as inthemodelCLKT. Constraints(25)linkthe
cuttingvariable withtheproductionvariableand finally the constraints(26), (27)and (28) are
non-negativityandintegralityconstraintsonthevariables. ThemodelGGisanextendedmodel
and can be obtained by applying Dantzig – Wolfe decomposition on the KT and VC models
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(Vale´riodeCarvalho,2002).Asaconsequencethismodelhasalargeamountofvariablesandis
generallysolvedbycolumngeneration.
4 MATHEMATICALMODELSWITHSEVERALTYPESOFOBJECTS
For the mathematical models presented in this section, we integrate the model proposed by
Wagner& Whitin(1958),here denotedbyWW withthecuttingstockmodelsextendedtocon-
sider several types of objects. The WW model describes the uncapacitated lot-sizingproblem.
Inthiscase,considerthefollowingparametersandadditionaldecisionvariables:
Parameters:
K: numberofobjecttypesavailableinstock(indexk);
e : plannedsupplyofobjecttypek atthebeginningofperiodt;
kt
L :objectlengthoftypek;
k
cw: costperunitofrawmaterialwaste.
DecisionVariable:
s : numberofobjectsoftypek stockedatendofperiodt.
kt
Themodeldescribedbelow(WWKTMO)modelsthecuttingstockproblembasedonmodelKT,
consideringseveraltypesofobjectsMO.Intheoriginalvariablesanewindexisinserted,which
referstothetypeofobjecttobecut.
Parameters:
(cid:4) (cid:3) (cid:5)
se : sumofplannedsupplyofobjectsoftypek fromperiod1toperiodt se = t e ;
kt kt a=1 ka
q ∈{1,...,se },indexfortheavailableobjects.
kt
DecisionVariables:
y :binaryvariablethatindicateswhetherobjectq oftypek isusedinperiodt;
qkt
h :quantityofitemi cutfromobjectq oftypek inperiodt.
iqkt
ModelWWKTMO
⎛ ⎞
(cid:2)T (cid:2)I (cid:2)T (cid:2)K (cid:2)sekt (cid:2)T (cid:2)K (cid:2)sekt (cid:2)I
min (sc Y +hc S )+cw⎝ L y − l h ⎠ (29)
it it it it k qkt i iqkt
t=1 i=1 t=1k=1q=1 t=1k=1q=1i=1
Subjectto:
Xit +Si,t−1 =dit +Sit ∀i, ∀t (30)
X ≤sd Y ∀i, ∀t (31)
it itT it
(cid:2)I
l h ≤ L y ∀q, ∀k, ∀t (32)
i iqkt k qkt
i=1
(cid:2)K (cid:2)sekt
h = X ∀i, ∀t (33)
iqkt it
k=1q=1
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MIPMODELSFORTHEINTEGRATEDLOT-SIZINGANDONE-DIMENSIONALCUTTINGSTOCKPROBLEM
(cid:2)sekt
ekt +sk,t−1 = yqkt +skt ∀k, ∀t (34)
q=1
Y , y ∈{0,1} ∀q, ∀i, ∀k, ∀t (35)
it qkt
hiqkt ∈Z+ ∀q, ∀i, ∀k, ∀t (36)
Xit, Sit,skt ∈R+ ∀i, ∀t (37)
The objectivefunction(29)minimizes machine setupcosts, inventoryholdingcosts and waste
cost in the cutting process. Constraints (30) and (31) refer to the lot-sizing problem and are
defined as in the CLKT model. The set of constraints (32) ensure that if an object q of type
k is used, the combination of the items that will be cut from it should not exceed its length.
Constraint(33) isresponsible forthe integrationof the decisionvariables of the lot-sizingand
cuttingstockproblems. Itensuresthatasufficientamountofitemsare cuttomeet thedemand.
Thesetofconstraints(34)ensurethatthenumberofobjectscutineach perioddoesnotexceed
theavailabilityinstock.
The integrated model presented below (WWVCMO), is based on the VC formulationto model
the cuttingstock problem. Since several object types are available in stock, we need to define
themaximum objectlength Lmax = max∀k{Lk}, correspondingtothenumbers ofnodesinthe
network.Thefollowingdecisionvariablesarealsoneeded:
f : flowthroughthenetworkintheperiodt fortheobjecttypek;
kt
z : numberofcuttingpatternswhichhaveaitemofsize(l − j)allocatedatadistance j from
jlt
thebeginningoftheobjectusedinperiodt.
ModelWWVCMO
⎛ ⎞
(cid:2)T (cid:2)I (cid:2)T (cid:2)K (cid:2)T (cid:2)I (cid:2)
min (scitYit +hcitSit)+cw⎝ Lkfkt − lizl,l+li,t⎠ (38)
t=1 i=1 t=1k=1 t=1 i=1(l,l+li)∈A
Subjectto:
Xit +Si,t−1 =dit +Sit ∀i, ∀t (39)
X ≤sd Y ∀i, ∀t (40)
it itT it
(cid:2) (cid:2)K
− z =− f ∀t (41)
0lt kt
(0,l)∈A k=1
(cid:2) (cid:2) (cid:10)K
z − z =0 l ={1,...,L −1}\ {L }, ∀t (42)
glt lht max a
(g,l)∈A (l,h)∈A a=1
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z = f ∀k, ∀t (43)
gLkt kt
(g,L(cid:2)k)∈A
zh,h+li,t = Xit ∀i, ∀t (44)
(h,h+li)∈A
PesquisaOperacional, Vol. 36(1),2016
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