This documen is he accep ed e sion o he esea ch pape :
•Resea ch a icle:
Paula Te ´an-Viade o, An onio Alonso-Ayuso, F. Ja ie Ma ´ın-Campo
(2023) A 2-dimensional guillo ine cu ing s ock p oblem wi h a iable-
sized s ock o he honeycomb ca dboa d indus y, In e na ional Jou nal
o P oduc ion Resea ch, DOI: 10.1080/00207543.2023.2279129
•Da a used in compu a ional expe imen s a e a ailable a :
Paula Te ´an-Viade o, An onio Alonso-Ayuso, F. Ja ie Ma ´ın-Campo
(2023) Da ase used in a icle “A 2-dimensional guillo ine cu ing s ock
p oblem wi h a iable-sized s ock o he honeycomb ca dboa d indus y”
( e sion 1.2), Zenodo, DOI: 10.5281/zenodo.10185247
No e: A copy o his documen is also a ailable he e.
A 2-dimensional guillo ine cu ing s ock p oblem wi h a iable-sized
s ock o he honeycomb ca dboa d indus y
Paula Te ´an-Viade oa, An onio Alonso-Ayusoband F. Ja ie Ma ´ın-Campoc
aFacul ad de CC. Ma em´a icas, Uni e sidad Complu ense de Mad id, Spain
bDSLAB - CETINIA, Uni e sidad Rey Juan Ca los, Spain
cDepa amen o de Es ad´ıs ica e In es igaci´on Ope a i a, Ins i u o de Ma em´a ica In e disciplina , Uni e -
sidad Complu ense de Mad id, Spain
ABSTRACT
This pape in oduces no el ma hema ical op imisa ion models o he 2-
Dimensional guillo ine Cu ing S ock P oblem wi h Va iable-Sized S ock ha ap-
pea s in a Spanish company in he honeycomb ca dboa d indus y. This p oblem
mainly di e s om he classical cu ing s ock p oblems in he s ock, which is consid-
e ed a iable-sized, i.e., we ha e o decide he panel dimensions, wid h, and leng h.
This app oach is help ul in indus ies whe e he s ock is p oduced simul aneously
wi h he cu ing p ocess. The s ock is hen cu in o smalle ec angula pieces ha
mus mee he cus ome s’ equi emen s, such as he ype o i em, dimensions, de-
mands, and echnical speci ica ions. Fu he mo e, in he p oblem ackled in his
pape , he cu s a e guillo ine, pe o med side o side. The p oposed ma hema ical
models a e alida ed using eal da a om he company, ob aining esul s ha d as-
ically educe he p oduced ma e ial and le o e s, educing ope a ion imes and
economic cos s.
KEYWORDS
Cu ing S ock P oblem; 2-Dimensional cu ing; Va iable-sized s ock; Mixed In ege
Linea Op imisa ion; Ca dboa d indus y.
1. In oduc ion
The indus ial wo ld is inc easingly acing complex p ocesses ha equi e expe sys-
ems o suppo decision-making he e o e i is necessa y o de elop use ul me hods
and echniques. This a ea o esea ch has been ex ensi ely s udied by Jean-Ma ie
P o h. See Dolgui and P o h (2010), which desc ibes se e al p oblems in supply chain
managemen , such as p icing, ou sou cing, in en o y, manu ac u ing, e c. On he con-
a y, P o h and Hillion (1990) p esen di e en ma hema ical ools o sol e hem such
1
as simula ion, linea o dynamic p og amming, and queueing heo y, among o he s.
Finally, Go il and P o h (2002), in oduce a gene al pe spec i e o he supply chain
design aking in o accoun he di e en le els o s a egies: s a egic, ac ical, and
ope a ional.
One o he p oblems ound in many indus ies is he Cu ing S ock P oblem (CSP),
in oduced by Kan o o ich (1960) and Gilmo e and Gomo y (1961), is one o he op i-
misa ion p oblems ha has been deeply explo ed in he li e a u e due o i s many ap-
plica ions in he indus y. The CSP is p esen in di e en sec o s such as glass (Pa e˜no
and Al a ez-Valdes (2021)), s one (Baykaso˘glu and ¨
Ozbel (2021)), wood (Kok en and
Sel (2020)), s eel (Sie a-Pa adinas e al. (2021) and An onio e al. (1999)), conc e e
(Signo ini, de A aujo, and Melega (2021)), cons uc ion (Lemos, Che i, and de A aujo
(2020)), pape (Kall a h e al. (2014)), and ex ile (Salem e al. (2023)). The cu -
ing p ocess is c ucial o companies p oducing pieces cu om a p e ious s ock. The
companies a e especially in e es ed in minimising he le o e s and p oduc ion imes,
al hough hey can conside o he goals, o example in An onio e al. (1999), educing
compu ing imes as much as possible while achie ing a easonable cos is used by he
sales depa men o espond in ‘ eal ime’ o he cus ome s’ demands.
Due o he di e en indus ial p ocesses and hei speci ic es ic ions, se e al e -
sions o he CSP ha e been de eloped based on speci ica ions such as dimensions,
pa e n cha ac e is ics, and cu ing es ic ions, among o he s. Bo h exac and non-
exac (heu is ics, me aheu is ics, and/o ma heu is ics) algo i hms ha e been explo ed
in e ms o p oblem esolu ion.
The p oblem s udied in his pape is mo i a ed by he collabo a ion wi h a medium-
sized Spanish company in he honeycomb ca dboa d sec o whose aim o he u u e
medium e m is o au oma ise he ope a ions o gain e iciency and e ec i eness. I is
pa o he 2-Dimensional Cu ing S ock P oblems (2DCSP). The 2DCSP is conce ned
wi h ob aining a se o di e en ec angula i ems cu om one o mo e ec angula
panels in s ock. In ou case, he size o panels is no p ede ined in ad ance and is de-
e mined by he op imisa ion model. The cu ing p ocess mus be de ined o p oduce
enough pieces o each i em o co e a p e iously known demand. Addi ionally, as de-
ailed below, di e en speci ica ions om he ca dboa d indus y mus be conside ed.
The main con ibu ions o his pape a e:
•A 0–1 linea op imisa ion model is in oduced o he Mul i-S ock 2-Dimensional
CSP. The p oposal imp o es he cu en ope a ion in he ac o y.
•A no el mixed 0–1 linea op imisa ion model o he 2DCSP wi h a iable-sized
s ock and guillo ine cu s, able o decide he dimensions o he panels p oduced,
is also in oduced.
•A eal p oblem om a Spanish company in he ca dboa d indus y is ackled.
Da a om he company a e used and ex ensi ely analysed, compa ing he cu en
ope a ion wi h he p oposals.
•As a esul , new, simple and s aigh o wa d s a egies a e p oposed o he
ope a ion o he company, p o iding up o a 50% le o e educ ion.
The emaining pa o his pape is o ganised as ollows: Sec ion 2 e iews he di e -
en a ian s o he Cu ing S ock P oblem, as well as he speci ica ions o mee in he
p oblem o s udy; Sec ion 3 desc ibes he p oblem o deal wi h and he cu en ope -
a ion managed by he company; Sec ion 4 in oduces wo ma hema ical op imisa ion
models o sol e he p oblem; Sec ion 5 p esen s an ex ensi e compu a ional expe i-
men based on eal-wo ld da a p o ided by he company; inally, Sec ion 6 concludes
and p esen s he u u e esea ch lines.
2
Panel Leng h
Panel Wid h
(a) 1DCSP
Panel Leng h
Panel Wid h
(b) 2DCSP
Figu e 1. Cap ion: 1DCSP s 2DCSP
Figu e 1. Al Tex : Two diag ams wi h a ec angula shape. On he le , cap ion (a) 1DCSP: Diag am wi h a
ec angula panel di ided in o di e en ho izon al s ips whe e he op s ip in dashed is le o e . On he igh ,
cap ion (b) 2DCSP: Diag am wi h a ec angula panel di ided in o h ee ho izon al s ips whe e he op s ip
in dashed is le o e . The o he wo s ips a e di ided in ec angula pieces.
2. The Cu ing S ock P oblem
The CSP was in oduced by Kan o o ich (1960) ( i s published in Russian in 1939),
ollowed by he seminal wo k p esen ed by Gilmo e and Gomo y (1965), which ex ends
he wo ks by he same au ho s o he 1-Dimensional CSP (1DCSP), Gilmo e and
Gomo y (1961) and Gilmo e and Gomo y (1963). In he la e , a column gene a ion
me hod is p esen ed o igh ening he lowe bound. Howe e , ha me hod does no
ob ain good esul s o he 2-Dimensional CSP (2DCSP).
Dyckho (1990) p esen s a consis en and sys ema ic app oach o a comp ehensi e
ypology in eg a ing he a ious kinds o p oblems o he cu ing and packing p ob-
lems. A deepe classi ica ion is p esen ed in Dyckho and Finke (1992). One o he
mos impo an cha ac e is ics is he numbe o dimensions o conside o he cu s.
Then, he CSP is classi ied in o 1-Dimensional Cu ing S ock P oblems (1DCSP),
whe e he s ock is cu in o s ips (pe o ming only leng hwise cu s on he s ock),
and 2-Dimensional Cu ing S ock P oblems (2DCSP), whe e he s ock is wid h and
leng hwise cu . Fig. 1 shows he di e ences be ween bo h p oblems. Fig. 1(a) shows a
panel cu leng hwise in o di e en s ips whe eas Fig. 1(b) shows a panel cu in o ec -
angula pieces. In bo h cases, he le o e s a e ep esen ed wi h he diagonal s iped
pa e n. Some o he dimensional p oblems ha e been s udied in he li e a u e, such
as he 1.5-Dimensional Cu ing S ock P oblem (1.5DCSP). In his a ia ion, he o -
de ed pieces a e gi en by hei leng hs and wid hs bu ix one dimension and lea e he
o he a iable depending on o he ac o s, such as weigh (see Sie a-Pa adinas e al.
(2021)). Haessle and Sweeney (1991) p esen s a good e iew o he 1DCSP, 2DCSP
and 1.5DCSP and hei main di e ences.
The 2DCSP is ela ed wi h wo classical packing p oblems: he Bin Packing P oblem
(BPP) and he S ip Packing P oblem (SPP). The 2-Dimensional BPP (2DBPP) is
a pa icula case o he 2DCSP whe e he demand o each i em is equal o one.
The SPP conside s a s ip o ix wid h and in ini e leng h and consis s in cu ing
all he i ems om he s ip by minimising he used s ip leng h. Lodi, Ma ello, and
Monaci (2002) e iew he ma hema ical models, lowe bounds, heu is ics, exac and
app oxima ion algo i hms o 2DSPP. Mos o he me hods p oposed a e heu is ics.
Oli ei a e al. (2016) e iew he heu is ics me hods and classi y hem acco ding o hei
ype: cons uc i e heu is ics, imp o emen heu is ics o e sequences, and imp o emen
heu is ics o e layou s. Lodi, Ma ello, and Vigo (2004) conside he 2DBPP and SPP,
whe e i ems mus be packed by le els. Fo u he in o ma ion on o he a ian s o he
CSP, oge he wi h he e alua ion o he di e en CSPs, see he ecen e iew Beze a
3
(a) Non-guillo ine cu s (b) Guillo ine cu s
Figu e 2. Cap ion: 2DCSP: Non-guillo ine s Guillo ine cu s
Figu e 2. Al ex : Two diag ams wi h a ec angula shape. On he le , cap ion (a) Non-guillo ine cu s: Diag am
wi h a ec angula panel di ided wi h non-guillo ine cu s o ming di e en pieces. On he igh , cap ion (b)
Guillo ine cu s: Diag am wi h a ec angula panel di ided in o h ee ho izon al s ips. Each s ip is di ided
in o di e en ec angula pieces o he same o less wid h han he s ip.
2 2
222
1
1
(a) Exac 2-s aged cu s
T imming
T imming
T imming
2 2
2 2 2 2 2
1
1
(b) Non-exac 2-s aged cu s
1
2 2
2
3
3
3
(c) Exac 3-s aged cu s
Figu e 3. Cap ion: Exac and non-exac s aged cu s
Figu e 3. Al ex : Th ee diag ams wi h a ec angula shape: cap ion (a) Exac 2-s aged cu s, cap ion (b)
Non-exac 2-s aged cu s and cap ion (c) Exac 3-s aged cu s.
Figu e 3. Long desc ip ion: On he le , cap ion (a) Exac 2-s aged cu s: Diag am wi h a ec angula panel
di ided in h ee ho izon al s ips. Each s ip is di ided in o ec angula pieces o he s ip wid h. The dashed
a ea a he end o he s ips is le o e . In he middle, cap ion (b) Non-exac 2-s aged cu s: Diag am wi h a
ec angula panel di ided in h ee ho izon al s ips. Each s ip is di ided in o ec angula pieces. In he i s
s ip some pieces’ wid hs a e smalle han he s ip gene a ing some le o e on he op o hem, imming
in dashed. On he ig h, cap ion (c) Exac 3-s aged cu s: Diag am wi h a ec angula panel di ided i s in o
wo ho izon al s ips. Each s ip, is di ided in o wo ec angula pieces and, hi d, he wo ec angula pieces
gene a ed ( igh hand side o he o iginal s ips) a e cu in o ho izon al s ips. The dashed a ea is le o e .
e al. (2019), ocused on he 2-Dimensional le el s ip packing p oblem.
An essen ial cha ac e is ic widesp ead in he indus ial sec o is he ype o cu s.
O en, he cu ing equipmen can p oduce only guillo ine cu s pe o med om side o
side pa allel o he edges p oducing wo new ec angles om a la ge one. In he exam-
ple shown in Fig. 1(a), all he cu s a e guillo ine. In he example shown in Fig. 1(b),
all he cu s a e also guillo ine bu in sequen ial o de . Fi s , leng hwise cu s a e pe -
o med, gene a ing wo ypes o s ips. Each one is hen guillo ine cu o p oduce he
smalle ec angles. Fig. 2 shows an example o non-guillo ine and guillo ine cu s.
The 2DCSP is classi ied as 2-s aged (2SCSP) o 3-s aged (3SCSP) i wo o h ee
sequen ial cu s a e needed, espec i ely. The i s s age consis s in pe o ming pa allel
leng hwise guillo ine cu s ha p oduce a se o ec angula s ips. Each s ip is indi id-
ually cu in he second s age wi h he emaining pa allel guillo ine c osscu s. I he e
is no need o an addi ional cu , i.e., all he pieces’ wid hs equal he o de ed dimen-
sions, he pa e n is called exac 2-s aged guillo ine. O he wise, i is called non-exac
since he pieces need a hi d s age, adding a cu o mo e away some sc ap o mee
he eques ed dimensions (see Salem e al. (2023) and And ade e al. (2014)). Fig. 3
shows h ee di e en cases: exac 2-s aged (Fig. 3(a)), non-exac 2-s aged (Fig. 3(b)),
and exac 3-s aged (Fig. 3(c)). Obse e in Fig. 3(b) ha a hi d cu is needed o he
smalle pieces in he i s s ip o mee he dimensions whe eas in he case showed
in Fig. 3(c), e en being exac , he hi d cu is needed o ob ain he inal pieces. See
4
Vande beck (2001), Al a ez-Valdes, Pa ajon, and Tama i (2002), Yanasse and Mo a-
bi o (2008) Sil a, Al elos, and de Ca alho (2010), Macedo, Al es, and de Ca alho
(2010), among o he s, o he 2SCSP and 3SCSP. Cin a e al. (2008) s udy he 2DCSP
wi h guillo ine cu s and hei a ian s. They s udy he SPP wi h wo, h ee, and ou
s aged pa e ns. Dola abadi, Lodi, and Monaci (2012) in oduce an exac ecu si e
p ocedu e ha cons uc s he se o guillo ine packing associa ed wi h a gi en se
o i ems as pa o he 2-Dimensional Knapsack P oblem (2DKP). Fu ini, Malagu i,
and Thomopulos (2016) p esen a amewo k o model gene al guillo ine cons ain s
in 2DCSP o mula ed as mixed-in ege linea p og amming, based on he o mula ion
p esen ed in Dyckho (1981). Fu he mo e, as a a ian o he 2SCSP, Ma in e al.
(2022) p opose p obably o he i s ime models o 2SCSP and 3SCSP wi h a limi ed
numbe o open s acks.
An elemen o conside in he Cu ing p oblems is he ea men o le o e s. They
can be classi ied as eusable le o e s o sc ap. The eusable le o e s a e conside ed
when he was e gene a ed can be used again i i mee s some speci ica ions ( ypically
minimum dimensions). Then, hose eusable le o e s a e conside ed s ock in u u e
p ocesses (see do Nascimen o, Che i, and Oli ei a (2022)). On he con a y, he sc ap
is di ec ly disca ded.
The s ock is also used o classi y he CSP. Typically, he CSP conside s a s ock
o med by an in ini e numbe o panels wi h dimensions W×L. Howe e , i is also
common o conside a he e ogeneous s ock, o med by panels wi h a ied dimensions.
This p oblem is known as he Mul i-S ock 2-Dimensional CSP (MS2DCSP). A pa ic-
ula case wi h he e ogeneous s ock is he p oblem wi h usable le o e s, whe e he e
is a high a iabili y o s ock dimensions and o en only one piece o s ock o each size.
Fu ini e al. (2012) in oduce a column gene a ion heu is ic o he MS2DCSP wi h
2-S aged guillo ine cu ing. La ely, Fu ini and Malagu i (2013) p esen some ma he-
ma ical models o MS2DCSP wi h 2-S aged guillo ine cu ing. P e iously, Pisinge
and Sigu d (2005) conside a closely ela ed p oblem, he 2DBPP wi h a iable bin
sizes.
In he li e a u e, he CSP conside s ha he s ock dimensions a e known in ad ance,
e en i i is he e ogeneous. In Mosallaeipou (2017) he p oblem o supplie ma e ial
selec ion and p oduc ion planning in he ca on box p oduc ion indus ies is p esen ed.
In his wo k, he panel sizes conside ed a e hose o e ed by he supplie s and he cu -
ing pa e ns a e gi en by a so wa e. Ano he p oblem ela ed o he ca dboa d sec o
is p esen ed in Sipahi (2022), whe e, in he i s s ep, he panel sizes a e de e mined
by a simula ed annealing algo i hm, and in he second s ep, he p oduc s a e assigned
using an in ege linea p og amming model. Ne e heless, he s ock dimensions could
be a decision as pa o he p oduc ion p ocess. This a ian , known as he 2DCSP
wi h Va iable-Sized s ock (2D-VSCSP), has been ecen ly p esen ed in Salem e al.
(2023). To he bes o ou knowledge, and as he au ho s men ion in he pape , his is
he only wo k dealing wi h he 2D-VSCSP. They p esen wo ma hema ical models.
The i s is based on he wo k by Lodi and Monaci (2003), while he second is based
on a Bin Packing P oblem (BPP). As he au ho s ema k in he pape , he p oblem
can be app oxima ed as a BPP since, in hei p ac ical applica ion ( ex ile sec o ),
he demand is no e y high, e en hough he demand o each i em is g ea e han
one uni . The p oblem we a e dealing wi h i s in o he 2D-VSCSP.
No e ha he 2D-VSCSP could be also ea ed as a 3-S aged S ip Packing P oblem
(3S-SPP). The i s s age p oduces he di e en panels, while he second and hi d
s ages p oduce he s ips and he inal i ems, espec i ely. I mo e han one wid h is
a ailable in s ock, he p oblem can be conside ed a Mul i-S ip (3S) Packing P oblem
5
(a) Honeycomb s uc u e (b) P oduced panels
Figu e 4. Cap ion: Honeycomb ca dboa d panels
Figu e 4. Al ex : Two pic u es. On he le , cap ion (a) Honeycomb s uc u e: Pic u e o he honeycomb
s uc u e, sandwich o wo laye s o g ey pape wi h he ne in he middle. Each cell is hexagonal. On he igh ,
cap ion (b) P oduced panels: Pic u e o a collec ion o honeycomb panels piled.
(MSPP).
3. P oblem desc ip ion
The p oblem conside ed akes place in he honeycomb ca dboa d indus y. The p od-
uc ’s name comes om he honeycomb s uc u e inside he ca on panel. The honey-
comb ca dboa d panels a e made o pape , wi h a sandwich s uc u e o wo pape
laye s and a pape ne inside ( he honeycomb s uc u e). Fig. 4 shows a g aphical
ep esen a ion o a honeycomb ca dboa d panel and some panels p oduced by he
company.
The success o he honeycomb panels is due o hei cha ac e is ics: g een, ligh ,
esis an , economical, and easy o manipula e. G een: i is made wi h ecycled and
ecyclable ma e ials. Ligh : compa ed wi h wood panels, commonly used in ans-
po a ion, i s weigh is a ound 1/6 he wood panels. Resis an : i esis s 5kg/cm2 o a
comp ession o 50 /m2. Economical: compa ed o wood, o o he Ex ended Polys y ene
(EPS) p oduc s, he p ice is signi ican ly lowe . Finally, i is easy o manipula e and
cu hanks o i s low weigh .
In i s beginnings, he honeycomb panels we e used o anspo a ion eplacing he
EPS. They a e e y esis an and ligh , making hem pe ec o p o ec ing he ans-
po ed p oduc s by abso bing he blows. Recen ly, his ma e ial has gained p esence
in he ad e ising and deco a ing sec o s as i is possible o p in on i .
The company ecei es e e y day di e en o de s ha mus be se ed. An o de
consis s o a lis o di e en i ems ( ec angula pieces) gi en by hei leng h (ℓi), wid h
(wi), and he numbe o uni s o se e (di). The pieces a e ob ained by p oducing
ca on panels and cu ing hem in o smalle i ems. The e o e, he company has o
decide: (1) how many ca on panels o p oduce, (2) hei dimensions, and (3) how o
cu hem aiming o educe he le o e s. I is wo h poin ing ou ha he pieces canno
be o a ed as he ma e ial is weake depending on how he pieces a e posi ioned.
The ac o y layou encloses wo di e en a eas, each de o ed o a phase in he
p oduc ion p ocess. The i s one con ains a machine, called line, o p oduce he hon-
eycomb ca dboa d panels (see Figs. 5(a) and 5(b)). The second a ea comp ises wo
cu ing machines ha cu he panels in o smalle pieces (see Fig. 5(c)).
The line un olls he pape olls used as co e s and in oduces in be ween hem
he honeycomb ne , ha is glued o he wo co e s. The pape oll wid h de ines he
panel’s wid h, while he leng h can be decided and adjus ed. The cu ing machine y
6
(a) Supply module o he line (b) Line machine (c) Cu ing Machine
Figu e 5. Cap ion: Line and cu ing machines
Figu e 5. Al ex : Th ee pic u es. On he le , cap ion (a) Supply module o he line: Pic u e o he machine
whe e he panels a e p oduced. The pape olls a e alloca ed o eed he machine in ou sha s. In he middle,
cap ion (b) Line machine: Pic u e o he machine p oducing a con inuous panel. On he igh , cap ion (c)
Cu ing Machine: Pic u e o a cu ing machine. The blades a e alloca ed along a sha o cu he ca on panels.
imposes minimum and maximum leng hs o he panels, say ℓand ℓ, espec i ely. A
wid h modi ica ion implies pape olls eplacemen , whe eas a leng h adjus men only
needs o se he new alue in he panel con ol. Bo h ac ions o ce he line o s op.
Once he panels a e p oduced, hey a e mo ed o he cu ing a ea. The cu ing
machines ha e a se o ci cula blades dis ibu ed along a sha ha pe o ms guillo ine
cu s. The blades can (1) ully c oss he panel ob aining sepa a ed pieces o (2) pa ially
c oss he panel keeping i oge he . In he la e , he pieces a e manually sepa a ed
by he cus ome . This op ion is equen ly chosen since packing on palle s is easie .
In addi ion, o ope a ion pu poses, many cus ome s p e e o ecei e palle s wi h
panels con aining only one e e ence, making hei manipula ion easie . The e o e,
each panel can only con ain one i em e e ence. The ope a ion p oduces le o e s bu
ne e imming as all he ob ained pieces ha e he same wid h as he s ip gene a ed.
The e o e, he p oblem leads o he exac 2-S aged 2DCSP.
Cu en ly, he company wo ks wi h panels o 1200×2400 mm2, no aking he ad-
an age o he lexibili y gi en by he line o be e adjus he panels’ dimensions
o he o de s. As he company se es o he p oduc s, hey also wo k wi h di e en
wid hs (pape olls) ha could be used o p oduce panels wi h di e en wid hs. In
an analogous way, hey could wo k wi h di e en leng hs ha a e easily con igu ed in
he line. The aim o his con ibu ion is o p opose ma hema ical models allowing he
company o explo e new s a egies and e en decide on a be e panel con igu a ion o
se e i s demand.
Then, he p oblem we a e ackling i s in o he 2DCSP amily wi h a iable-sized
s ock, 2D-VSCSP. As a as we know, 2D-VSCSP was ecen ly in oduced in Salem
e al. (2023). Models p esen ed in ha wo k only sol e cases wi h low o al demand
(up o 30 uni s) while ou company manages many di e en i ems (up o 90) wi h
high demands ( equen ly, housands o pieces pe i em). These models’ size depends
on he squa e o he o al numbe o pieces o de ed, making i impossible o apply
hem o ou p oblem.
Summa ising, he main hypo heses conside ed a e:
•Panels a e p oduced in he ac o y and ha e a ec angula shape (CSP).
•The panel dimensions (wid h and leng h) ha e o be decided (2D-VSCSP).
•O de s a e known be o e planning he panel p oduc ion.
•Pieces ha e a ec angula shape (2D-CSP) and canno be o a ed.
•Only guillo ine cu s a e allowed.
7
•Final pieces a e supplied in p e-cu panels ( hey a e no sepa a ed indi idually)
and each panel includes only one i em.
•The objec i e is o educe le o e s.
4. Ma hema ical op imisa ion models
In his sec ion, wo Mixed In ege Linea Op imisa ion Models o sol e he 2D-VSCSP
wi h guillo ine cons ain s a e p esen ed. Bo h models aim o de e mine he panels’
con igu a ion such ha he le o e s a e minimised. A con igu a ion is gi en by i s
leng h and wid h.
In he i s model, a se o po en ial con igu a ions (wid h and leng h) is p ede ined
by he company, and a subse o hem is selec ed. On he con a y, in he second one,
he con igu a ions a e p oposed by he model: he pape oll selec ed de ines he wid h,
and he leng h is a a iable in [ℓ, ℓ].
4.1. Model 1: Selec ion Model (SM)
The SM conside s a se o p ede ined con igu a ions, selec s a subse and assigns each
i em o one o hem. Each con igu a ion jis gi en by i s wid h Wjand leng h Lj. Uppe
bounds on he o al numbe o con igu a ions, wid hs and leng hs can be imposed, nc,
nwand nℓ, espec i ely. This model p o ides mo e op ions han he company’s cu en
ope a ion, which only uses he con igu a ion 1200×2400 mm2.
No e ha i only one con igu a ion is used, he e is no place o op imisa ion since
i is easy o calcula e he le o e gene a ed. I con igu a ion jis selec ed he numbe
o panels needed o se e he demand o i em iis:
npij =ldi
nij m,
whe e nij =jWj
wik×jLj
ℓikis he maximum numbe o pieces o i em i ha can be
p oduced wi h one panel o con igu a ion j.
Howe e , when mo e han one con igu a ion is selec ed, i is necessa y o decide he
assignmen o each i em o a con igu a ion and, hen, he e is a combina o ial numbe
o easible solu ions.
4.1.1. Se s
I={1, . . . , I},se o i ems, being I he o al numbe o di e en i ems.
J={1, . . . , J},se o a ailable con igu a ions, being J he o al numbe o con igu-
a ions conside ed.
W={ˆw1,ˆw2,..., ˆwn},se o a ailable con igu a ion wid hs.
L={ˆ
ℓ1,ˆ
ℓ2,...,ˆ
ℓm},se o a ailable con igu a ion leng hs.
Jˆw={j∈ J :Wj= ˆw} ⊂ J ,se o con igu a ions o wid h ˆw, o ˆw∈ W.
Jˆ
ℓ={j∈ J :Lj=ˆ
ℓ}⊂J,se o con igu a ions o leng h ˆ
ℓ, o ˆ
ℓ∈ L.
4.1.2. Pa ame e s
Wj,wid h o con igu a ion j, o j∈ J .
8
Lj,leng h o con igu a ion j, o j∈ J .
wi,wid h o i em i, o i∈ I.
ℓi,leng h o i em i, o i∈ I.
di,demand o i em i, o i∈ I.
sij,Le o e gene a ed i i em iis assigned o con igu a ion j, o i∈ I,j∈ J . I can
be calcula ed as
sij =npij ·Wj·Lj−di·wi·ℓi.
nc, nw, nℓ,maximum numbe o di e en con igu a ions, wid hs and leng hs ha can
be selec ed.
4.1.3. Decision a iables
xij = 1 i i em iis assigned o con igu a ion j, 0 o he wise, o i∈ I, j ∈ J .
yj= 1 i con igu a ion jis selec ed, 0 o he wise, o j∈ J .
uˆw= 1 i a leas one con igu a ion o Jˆwis selec ed, 0 o he wise, o ˆw∈ W.
ˆ
ℓ= 1 i a leas one con igu a ion o Jˆ
ℓis selec ed, 0 o he wise, o ˆ
ℓ∈ L.
4.1.4. SM ma hema ical o mula ion
min X
i∈I X
j∈J
sijxij (SM.1)
subjec o
X
j∈J
xij = 1 ∀i∈ I (SM.2)
xij ⩽yj⩽X
i∈I
xij ∀i∈ I, j ∈ J (SM.3)
yj⩽uˆw⩽X
j∈J ˆw
yj∀ˆw∈ W, j ∈ J ˆw(SM.4)
yj⩽ ˆ
ℓ⩽X
j∈J ˆ
ℓ
yj∀ˆ
ℓ∈ L, j ∈ J ˆ
ℓ(SM.5)
X
j∈J
yj⩽nc(SM.6)
X
ˆw∈W
uˆw⩽nw(SM.7)
X
ˆ
ℓ∈L
ˆ
ℓ⩽nℓ(SM.8)
xij, yj∈ {0,1} ∀i∈I, j ∈ J (SM.9)
uˆw∈ {0,1} ∀ ˆw∈ W (SM.10)
ˆ
ℓ∈ {0,1} ∀ˆ
ℓ∈ L (SM.11)
The objec i e unc ion (SM.1) minimises he o al le o e s gene a ed in he p o-
9
SM - |W| = 1 - nc= 1 VSM - |W| = 1 - nc= 1 VSM - |W| = 4 - nc= 1
Cons. Va s. 0–1 Va s. NonZe o Cons. Va s. 0–1 Va s. NonZe o Cons. Va s. 0–1 Va s. NonZe o
I1 56 53 52 217 783 507 259 2529 2655 1729 890 8633
I2 34 53 52 145 410 269 140 1319 1277 845 448 4133
I3 91 89 88 357 756 474 247 2373 2715 1727 907 8659
I4 167 185 184 685 1407 884 464 4417 4485 2847 1515 14235
I5 81 65 64 309 3641 2402 1208 12017 11396 7525 3794 37665
I6 80 77 76 305 521 317 167 1595 1907 1185 630 5977
I7 73 81 80 293 1985 1302 660 6501 7151 4711 2395 23531
I8 103 109 108 405 1847 1193 609 5973 6320 4107 2107 20579
I9 97 101 100 377 982 617 320 3096 3310 2101 1100 10561
I10 126 137 136 502 7417 4900 2466 24505 24838 16437 8286 82217
I11 89 89 88 349 4251 2804 1412 14023 14433 9539 4813 47719
I12 355 349 348 1429 8875 5789 2937 28993 30826 20199 10273 101199
I13 133 113 112 515 13427 8906 4466 44549 45035 29895 15003 149563
I14 218 197 196 863 14178 9377 4712 46915 47727 31609 15902 158177
I15 301 281 280 1203 18424 12178 6123 60931 62137 41135 20707 205851
I16 133 113 112 515 24287 16146 8086 80749 83591 55599 27855 278083
I17 301 281 280 1203 26062 17270 8669 86391 88135 58467 29373 292511
I18 148 133 132 576 20716 13757 6894 68809 71047 47213 23672 236173
I19 383 389 388 1552 22634 14953 7524 74808 77393 51223 25805 256299
I20 383 389 388 1552 30524 20213 10154 101108 104213 69103 34745 345699
Table 3. Models dimensions
16
nc
Sce. 1 2 3 4
1 2 21 26 26
2 7 32 39 42
3 11 33 37 43
4 – 49 55 62
Table 5. Le o e educ ion %
nc
Sce. 1 2 3 4
1 99 93 91 91
2 98 89 86 85
3 96 88 87 85
4 – 83 81 78
Table 6. A ea imp o emen %
1234
0
20
40
60
Maximum con igu a ions (nc)
Le o e educ ion (%)
Scena io 1
Scena io 2
Scena io 3
Scena io 4
Figu e 7. Cap ion: Le o e lines o each
scena io
Figu e 7. Al Tex : Lines cha o each sce-
na io showing he pe cen age o le o e e-
duc ion o nc= 1,2,3,4
1234
80
85
90
95
100
Maximum con igu a ions (nc)
A ea imp o emen (%)
Scena io 1
Scena io 2
Scena io 3
Scena io 4
Figu e 8. Cap ion: A ea lines o each sce-
na io
Figu e 8. Al Tex : Lines cha o each sce-
na io showing he pe cen age o a ea im-
p o emen o nc= 1,2,3,4
a iables (0–1 Va s.), and non-ze o elemen s (NonZe o). I is wo h poin ing ou ha
o nc>1, he numbe o a iables and cons ain s is app oxima ely p opo ional o
he alue o nc.
To e alua e he esul s, as he o al a ea o he demanded i ems is known, a key
elemen is he pe cen age o ma e ial p oduced ha is conside ed le o e . These pe -
cen ages a e epo ed in Table 4 o each scena io and each alue o nc. Fo each
ins ance, a colou scale is used, whe e he wo s pe cen age is da k, and he bes
pe cen age is ligh .
I can be obse ed ha he company ope a ion always epo s he wo s pe cen age
compa ed o he scena ios es ed. Howe e , o Scena io 1 (model SM) and nc= 1, in
mos o he ins ances, he selec ed con igu a ion is he one used by he company. As
expec ed, o each scena io, he highe he alue o nc, he be e he esul s ob ained.
Obse e ha he la ges imp o emen is om nc= 1 o nc= 2. In some speci ic
ins ances, in pa icula ins ances I5 and I14 (scena io 3) and I5, I10, and I13 (scena io
4), i can be obse ed ha he pe cen age o le o e s does no imp o e when he alue
o ¯ncis inc eased om 3 o 4. This is because, due o he inc eased complexi y o he
model, he op imise does no achie e he op imal solu ion and only p o ides a (good)
easible solu ion wi hin 1800 seconds.
Finally, he las ow o his able p o ides he a e age pe cen age ha is conside ed
le o e . Cu en ly, he company disca ds on a e age 36% o he ma e ial, while wi h
he ex eme Scena io 5, he le o e is educed o a hi d (13%).
To be e illus a e hese esul s, Tables 5 and 6 epo he a e age pe cen age o
le o e educ ion and ma e ial used imp o emen , espec i ely, using SM and VSM
agains he ac o y’s cu en ope a ion.
As men ioned, wi h Scena io 5, i is possible o educe he le o e o only 13%
17
Sce. 0 Scena io 1 Scena io 2 Scena io 3 Scena io 4 Scena io 5
company SM - W1VSM - W1VSM - W2-nw= 1 VSM - W2-nw= 2 VSM - W2
1.2×2.4 nc= 1 nc= 2 nc= 3 nc= 4 nc= 1 nc= 2 nc= 3 nc= 4 nc= 1 nc= 2 nc= 3 nc= 4 nc= 2 nc= 3 nc= 4 nw= 4-nc= 8
I1 25.2 19.4 18.0 17.6 17.6 15.8 13.7 12.8 12.6 14.3 13.7 12.8 12.6 10.3 9.1 8.7 6.3
I2 27.7 27.7 24.1 24.1 24.1 27.1 21.7 19.9 18.9 18.2 12.3 9.9 8.6 8.9 6.3 4.7 4.1
I3 32.9 31.7 23.1 21.4 21.3 21.7 16.8 14.2 12.9 21.7 16.8 14.2 12.9 16.8 13.5 11.0 7.4
I4 34.1 34.1 27.5 25.6 25.3 29.0 23.3 21.9 21.1 29.0 23.3 21.9 21.2 19.8 15.5 14.5 10.5
I5 33.3 19.2 18.7 18.6 18.6 19.2 18.1 17.5 17.3 7.1 4.1 3.7 5.5 4.6 2.8 3.2 1.4
I6 16.1 16.1 14.7 14.3 14.1 13.9 6.7 3.8 2.9 13.9 6.7 3.8 2.9 6.7 3.8 2.9 1.8
I7 17.6 17.6 15.7 15.3 15.2 17.2 8.5 5.4 4.1 17.2 8.5 5.4 4.1 8.5 5.8 4.3 3.0
I8 42.8 42.8 37.9 35.6 35.6 38.8 28.9 26.5 25.7 38.8 28.9 26.7 25.4 25.6 23.5 21.4 17.0
I9 28.5 28.5 19.5 19.3 19.3 23.1 18.7 16.1 14.5 23.1 18.7 16.1 14.5 15.1 10.9 9.5 6.3
I10 35.3 35.3 34.3 34.1 34.1 35.3 27.3 26.3 26.1 24.0 15.2 13.8 13.6 13.0 8.0 11.2 4.7
I11 35.5 35.5 33.6 33.6 33.6 29.1 23.0 22.0 21.5 29.1 23.0 22.0 21.4 20.3 19.0 18.1 16.8
I12 40.7 40.7 33.5 31.9 31.9 34.8 28.2 26.5 24.8 34.8 28.2 26.5 24.8 24.5 21.2 19.6 15.9
I13 39.7 39.7 37.6 37.2 37.2 36.8 29.7 29.1 28.3 32.2 26.9 23.9 23.9 22.1 19.0 23.6 11.8
I14 39.3 39.3 36.9 36.4 36.4 37.6 30.1 28.8 27.6 33.8 27.9 24.9 26.1 23.1 19.3 18.2 13.6
I15 38.0 38.0 36.0 35.4 35.4 36.0 28.4 26.9 25.9 36.0 28.4 27.1 26.3 22.8 20.6 17.4 15.6
I16 27.2 22.4 14.5 11.2 11.2 20.9 10.0 6.7 4.5 21.2 10.0 6.7 4.4 9.7 5.3 3.0 2.0
I17 38.3 38.3 36.2 35.7 35.7 36.4 28.8 27.1 27.7 35.9 28.8 31.1 27.0 22.8 20.6 19.7 16.1
I18 35.9 35.9 29.8 28.2 28.2 35.9 29.8 27.6 26.3 34.9 29.4 27.3 26.3 25.2 22.8 19.2 15.3
I19 35.4 35.4 29.5 27.7 27.7 35.4 29.6 26.9 26.2 34.8 32.0 29.2 26.4 24.5 22.6 19.5 13.7
I20 34.9 34.9 29.2 27.5 27.5 34.9 29.2 27.1 26.2 34.2 28.9 29.5 27.3 23.7 23.2 18.9 13.1
a . 35.5 35.1 30.4 29.1 29.1 34 27.2 25.2 24.3 32.8 27 25.8 24 22.1 19.9 17.4 12.9
W1={1200}
W2={1200,1400,1550,1600}
Table 4. Pe cen age o le o e
18
(73% educ ion wi h espec o he company’s ope a ion) esul ing in a o al ma e ial
used o 178,827 m2ins ead o 241,735 m2 o se e he 20 o de s. Howe e , his s a egy
is no an op ion o he company as i supposes o wo k wi h eigh di e en pape
olls ( wo pe wid h). In hese ables, i can be obse ed ha some s a egies using
no mo e han ou pape olls imp o e i s cu en ope a ion signi ican ly. F om he
analysis o hese esul s, i is possible o iden i y simple s a egies ha can e en hal e
he le o e amoun .
I only one con igu a ion can be used, he bes s a egy would be o conside Scena io
3 wi h nc= 1. Fo each ins ance, one wid h is selec ed om he ou a ailable in he
s ock, and only one leng h is ixed (de ined by he VSM model) o all he p oduced
panels. The le o e educ ion is 11% on a e age ( o ins ance I5, he educ ion is om
33% o 7%), ob aining in global o al ma e ial sa ing o 9,674 m2.
Ne e heless, conside ing he use o wo con igu a ions can lead o an impo an
p og ess. I ncis se o 2, Scena io 2 (only 1200 mm wid h is a ailable) and Scena io
3 ( ou wid hs a e a ailable and jus one can be selec ed), ob ain a big educ ion
in he pe cen age o le o e , 32% and 33%, espec i ely. I could be obse ed ha
bo h scena ios ga e e y simila imp o emen s, he e o e keeping he use o 1200 mm
wid h (Scena io 2) is mo e con enien o ope a ion pu poses. Wi h his ope a ion,
he company will ha e o adjus he machines only o ix he panels’ leng h o he wo
bes leng hs de ined by he model. In his case, he 32% in le o e educ ion u ns
in o 27,586 m2o ma e ial sa ings. No ice ha his sa ing is equi alen o 2×23 =
46 km o pape olls wi h 1200 mm wid h.
Finally, he bes esul s a e ob ained when he company conside s he op ion o
selec ing wo di e en wid hs pe o de (Scena io 4). Di e en o he p e ious ope -
a ion p esen ed, in his case, he line needs o s op o change he pape olls, which
is no oo ime-consuming as ou olls can be alloca ed. I nc= 2, he pe cen age o
le o e s is hal ed wi h espec o he amoun gene a ed wi h he cu en ope a i e
(49%). Wi h his ope a ion, he o al amoun o ma e ial used is educed by 42,244 m2.
No ice ha when nc= 4, he le o e educ ion is 62%; howe e , manipula ing ou
di e en con igu a ions makes he p oduc ion p ocess mo e complica ed.
Summa ising, i can be obse ed ha he e a e h ee s a egies ha can imp o e
d as ically he le o e educ ion wi hou implying signi ican changes in he company’s
cu en ope a ion:
•S a egy 1 (Scena io 3 - nc= 1): 11% le o e educ ion / 96% a ea.
•S a egy 2 (Scena io 2 - nc= 2): 32% le o e educ ion / 89% a ea.
•S a egy 3 (Scena io 4 - nc= 2): 49% le o e educ ion / 83% a ea.
Finally, Table 7 epo s some s a is ics on he compu a ional pe o mance o he
selec ed s a egies. The ollowing in o ma ion is p o ided: he alue o he incumben
solu ion ob ained (zIP ), he op imali y gap (in %), and he compu ing ime in sec-
onds (Time). Fo compa ison pu poses, he las column epo s he o al a ea o he
company’s cu en ope a ion.
No ice, ha he o al a ea o he demanded i ems is a lowe bound o he VSM
op imal solu ion (zIP ) and, he e o e, i can be conside ed o compu e he op imali y
gap. In Table 7, he op imali y gap epo ed is he minimum be ween he one ob ained
by Gu obi (GAPG) and he one calcula ed wi h he o al demanded a ea (a) as lowe
bound: GAP (%) = minGAPG,100zIP −a
zIP . Resul s using aas a lowe bound o zIP
in he VSM model a e no epo ed since hey a e wo se in e ms o compu ing ime
and/o incumben solu ion.
19
S a egy 1 S a egy 2 S a egy 3 Company
zIP Gap Time zIP Gap Time zIP Gap Time zIP
I1 196 0.0 7.6 195 0.0 35.9 187 10.3 1800∗225
I2 407 0.0 0.8 425 0.0 2.1 366 0.0 1.9 461
I3 526 0.0 1.5 495 0.0 4.8 495 0.0 44.7 613
I4 799 0.0 2.1 740 0.0 13.3 708 0.0 124.9 861
I5 864 7.1 1800∗980 18.1 1800∗841 4.6 1800∗1204
I6 994 0.0 0.9 917 0.0 1.2 917 0.0 15.0 1020
I7 1652 0.0 30.7 1494 0.0 96.5 1494 8.5 1800∗1659
I8 2297 0.0 1.9 1975 0.0 56.1 1889 5.9 1800∗2454
I9 1881 0.0 1.1 1779 0.0 6.5 1703 0.0 25.8 2022
I10 4795 0.0 8.5 5015 27.3 1800∗4190 13.0 1800∗5630
I11 5691 0.0 64.0 5245 0.0 733.7 5063 10.2 1800∗6261
I12 8663 0.0 781.8 7861 4.4 1800∗7473 4.4 1800∗9521
I13 9392 0.0 23.2 9059 13.4 1800∗8173 22.1 1800∗10555
I14 10242 0.0 6.2 9697 13.0 1800∗8818 23.1 1800∗11169
I15 16902 36.0 1800∗15111 12.2 1800∗14011 22.8 1800∗17430
I16 17800 21.2 1800∗15569 10.0 1800∗15523 9.7 1800∗19267
I17 22520 35.9 1800∗20281 21.2 1800∗18709 22.8 1800∗23394
I18 36138 26.9 1800∗33537 27.7 1800∗31480 25.2 1800∗36746
I19 38266 31.9 1800∗35441 27.6 1800∗33039 24.5 1800∗38627
I20 52036 31.7 1800∗48333 29.2 1800∗44864 23.7 1800∗52618
∗ ime limi exceeded
Table 7. Model esul s o he h ee bes ope a ion s a egies
I can be obse ed in Table 7 ha he a eas do no inc ease om S a egy 1 o
S a egy 3. No ice ha in gene al, he mo e complex he s a egy is, he highe he
compu ing imes. Fu he mo e, he e is a ela ion be ween he size o he ins ance and
he solu ion imes/GAPS. Ce ainly, he e a e o he ac o s ha inc ease he di icul y
o inding he op imal solu ion in a easonable ime, such as in ins ance I5, which
co esponds o an o de wi h small i ems (na ow and sho ) and a la ge a iabili y
in he quan i y demanded o each o hem (see Fig. 6). Rega dless, al hough some
o he GAPs a e high, he solu ion ob ained clea ly imp o es he cu en ope a ion o
he company.
6. Conclusions
The esul s e eal ha using expe sys ems based on ma hema ical op imisa ion o
suppo decision-making p ocesses p o ides signi ican ad an ages compa ed o he
ope a ional p ocesses designed based on he ope a o ’s expe ience, e en i hey ha e
a deep knowledge o he p oblem.
We ha e p esen ed wo no el linea op imisa ion models o a cu ing s ock p oblem
in he honeycomb ca dboa d indus y sugges ed by a Spanish company. This p oblem
belongs o he amily o 2-Dimensional Cu ing S ock P oblems wi h Va iable-Sized
s ock (2D-VSCSP) ecen ly in oduced by Salem e al. (2023). The models ha e been
alida ed using eal da a om he company. The esul s imp o e he cu en ope -
a ion in he ac o y by educing le o e s and panel p oduc ion. Fu he mo e, h ee
s aigh o wa d s a egies ha e been p oposed o he company ha sligh ly modi y i s
20
cu en ope a ion. Conce ning he numbe o con igu a ions, he ob ained esul s ha e
shown ha i is necessa y o conside wo con igu a ions a mos o a no able le o e
educ ion (un il 49%). Taking in o accoun he line speci ica ions, he impac o he
p oposed s a egies is no ele an in e ms o ime consump ion and ope a ion.
Finally, we p opose di e en esea ch lines: (1) igh he VSM model wi h new alid
cu s, ying o educe he GAP and he compu ing imes; (2) de elop (me a)heu is ics
o ob ain good easible solu ions in a sho ime, a oiding he use o comme cial
op imise s ha a e expensi e o he company; (3) as he company has wo cu ing
machines, conside job sequencing o minimise he p oduc ion imes.
Acknowledgemen (s)
The au ho s would like o hank he company manage s o p o iding us wi h eal
da a and o gi ing us insigh in o he company’s cu en ope a ion.
Disclosu e o in e es
The au ho s epo he e a e no compe ing in e es s o decla e.
Funding
This wo k has been suppo ed by g an PID2021-122640OB-I00 unded by
MCIN/AEI/10.13039/501100011033 and by ‘ERDF A way o making Eu ope’.
Da a a ailabili y s a emen
Due o he na u e o he esea ch, due o comme cial suppo ing no all da a is a ail-
able. We e e he eade s o Te ´an-Viade o, Alonso-Ayuso, and Ma ´ın-Campo (2023)
whe e o six ins ances, inpu da a and esul s ob ained a e epo ed.
Re e ences
Al a ez-Valdes, R., A. Pa ajon, and J.M. Tama i . 2002. “A compu a ional s udy o LP-based
heu is ic algo i hms o wo-dimensional guillo ine cu ing s ock p oblems.” OR Spec um
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Appendix A. Illus a i e small-scale case s udy
This sec ion p esen s a small-case s udy o illus a i e pu poses. The o de includes
he ollowing i ems:
I ems wi(mm)li(mm)di(mm)
i160 500 340
i2265 600 72
i3320 700 24
Fo he sake o cla i y, we assume ha only 1200 mm wide olls a e a ailable and ha
only one con igu a ion can be de ined. In addi ion, he company imposes a lowe and
uppe limi o ℓ= 1800 mm and ℓ= 3100 mm o he panels’ leng h.
Then, using he no a ion de ined abo e, we ha e: I={i1, i2, i3},J={j1},W=
{1200},J1200 ={j1},nc= 1, ℓ= 1800 and ℓ= 3100.
The i s s ep is o calcula e ij, he numbe o ows o each i em i ha i s in
con igu a ion j:
i1j1=jWj1
wi1k=j1200
60 k= 20, i2j1=j1200
265 k= 4, i3j1=j1200
320 k= 3
Then, i is possible o calcula e cij, he numbe o columns needed o mee he demand
o i em iusing con igu a ion j:
ci1j1=ldi1
i1j1m=l340
20 m= 17, ci2j1=l72
4m= 18, ci3j1=l24
3m= 8
The nex s ep is o calcula e a lowe and an uppe bound o he numbe o panels
needed o p oduce his numbe o columns o each i em. These bounds depend on
ℓ= 1800 mm and ℓ= 3100 mm, he minimum and maximum leng hs allowed o he
con igu a ion, espec i ely:
ki1j1=lci1j1
⌊ℓ
ℓi1
⌋m=l17
6m= 3, ki2j1=l18
5m= 4, ki3j1=l8
4m= 2,
ki1j1=lci1j1
⌊max{ℓi1,ℓ}
ℓi1
⌋m=l17
3m= 6, ki2j1=l18
3m= 6, ki3j1=l8
2m= 4.
The e o e, Ki1j1={3,4,5,6},Ki2j1={4,5,6}and Ki3j1={2,3,4}.
24
Finally, he alues o ℓijk a e calcula ed. Fo i em i1:
ℓi1j13= maxnℓ, ℓi1lci1j1
3mo = maxn1800,500l17
3mo = 3000,
ℓi1j14= maxnℓ, ℓi1lci1j1
4mo = maxn1800,500l17
4mo = 2500,
ℓi1j15= maxnℓ, ℓi1lci1j1
5mo = maxn1800,500l17
5mo = 2000,
ℓi1j16= maxnℓ, ℓi1lci1j1
6mo = maxn1800,500l17
6⌉o= 1800.
Analogously:
•I em i2:ℓi2j14= 3000, ℓi2j15= 2400 and ℓi2j16= 1800.
•I em i3:ℓi3j12= 2800, ℓi3j13= 2100 and ℓi3j14= 1800.
The objec i e unc ion o he VSM model esul s:
min 12003δi1j13+ 4δi1j14+ 5δi1j15+ 6δi1j16+ 4δi2j14+ 5δi2j15+ 6δi2j16+
2δi3j12+ 3δi3j13+ 4δi3j14
Cons ain s (VSM.2), all i ems mus be assigned o a con igu a ion:
xi1j1= 1, xi2j1= 1, xi3j1= 1.
Cons ain s (VSM.3), a con igu a ion is selec ed i and only i a leas one i em is
assigned o i :
xi1j1⩽yj1⩽xi1j1+xi2j1+xi3j1, xi2j1⩽yj1⩽xi1j1+xi2j1+xi3j1, xi3j1⩽yj1⩽xi1j1+xi2j1+xi3j1.
Cons ain s (VSM.4) assu e ha a wid h is used i and only i a con igu a ion o ha
wid h is selec ed.
yj1⩽u1200 ⩽yj1.
Cons ain s (VSM.5), a numbe o panels o be p oduced mus be selec ed o each
i em assigned o a con igu a ion:
zi1j13+zi1j14+zi1j15+zi1j16=xi1j1,
zi2j14+zi2j15+zi2j16=xi2j1,
zi3j12+zi3j13+zi3j14=xi3j1.
Cons ain s (VSM.6), he minimum con igu a ions’ leng h is imposed by he ℓijk cal-
cula ed:
3000zi1j13+ 2500zi1j14+ 2000zi1j15+ 1800zi1j16⩽Lj1,
3000zi2j14+ 2400zi2j15+ 1800zi2j16⩽Lj1,
2800zi3j12+ 2100zi3j13+ 1800zi3j14⩽Lj1.
25
