Concep ual F amewo k o he Op imiza ion
o Capaci a ed Lo -Sizing and Scheduling
P oblem
Juan Pablo Fiesco, Ana Es eso(B), M. M. E. Alemany, and Raúl Pole
Resea ch Cen e on P oduc ion Managemen and Enginee ing (CIGIP), Uni e si a Poli ècnica
de València (UPV), Camino de Ve a s/n, 46022 Valencia, Spain
{j iesco,aes eso,ma e a, pole }@cigip.up .es
Abs ac . The complex na u e o he capaci a ed lo -sizing p oblem, pa icula ly
wi hin he scheduling, equi es a holis ic app oach ha conside s mul iple ac o s
and hei in ica e in e ac ions. In his con ex , a no el concep ual amewo k (CF)
o suppo he combined Capaci a ed Lo -Sizing and Scheduling P oblem (CLSSP)
h ough ma hema ical modelling is p oposed. The CF is de eloped h ough a ig-
o ous me hodology ha combines da a collec ion, da a analysis, li e a u e e iew
and concep ual amewo k de ini ion. I is composed by se en dimensions, wi h
di e en ca ego ies, in u n made up o elemen s ela ed o di e en aspec s o he
p oblem and i s modelling. The CF se es a dual pu pose: as a comp ehensi e ool
o he s uc u ed analysis o exis ing models acili a ing he gaps iden i ica ion
and as a guide o p oposing no el ma hema ical p og amming models o add ess
he combined complexi ies o he CLSSP including hose no ye co e ed.
Keywo ds: Concep ual F amewo k · Capaci a ed-Lo Sizing · Scheduling ·
Ma hema ical p og amming · Op imiza ion
1 In oduc ion
Op imiza ion o simul aneously lo -sizing and scheduling becomes c ucial as p oduc ion
quan i ies and capaci y u iliza ion hinge on he p ocess con igu a ions, hei du a ions,
and he sequencing employed (Ma ínez e al., 2019). This dual ocus is a ac ing a en-
ion om bo h academy and indus y (Copil e al. 2017) unde he name o Capaci a ed
Lo -Sizing and Scheduling P oblem (CLSSP) ha can be conside ed as an ex ension o
he Capaci a ed Lo -Sizing P oblem (CLSP). The CLSSP complexi y makes necessa y
a amewo k o cha ac e ize he p oblem unde s udy, o sys ema ically e ise and clas-
si y exis ing op imiza ion models and o iden i y gaps wi h he aim o de eloping no el
models o co e hem. The CF o iginali y lies in he ac ha i is s uc u ed o acili a e
he o mula ion o ma hema ical p og amming models, d awing a en ion o impo an
aspec s. Mos o hem can be iden i ied as a ely add essed suppo ing, he e o e, he c e-
a ion o inno a i e solu ions. The es o he pape is s uc u ed as ollows. In Sec . 2 he
igo ous esea ch me hodology applied o de i e he CF is p esen ed. Sec ion 3 desc ibes
in de ail he no el CF. Meanwhile, in Sec . 4 he main conclusions and u u e esea ch
lines a e ou lined.
© The Au ho (s), unde exclusi e license o Sp inge Na u e Swi ze land AG 2025
R. Ca asco-Gallego e al. (Eds.): CIO 2024, LNDECT 239, pp. 276–281, 2025.
h ps://doi.o g/10.1007/978-3-031-82334-3_47
Concep ual F amewo k o he Op imiza ion 277
2 Resea ch Me hodology
The esea ch me hodology p oposed by Lo en e-Ley a e al. (2024), ha combines he
me hods o Seu ing & Mülle (2008) and Es eso e al. (2018), is ollowed o de elop he
CF. This me hodology encompasses he ollowing s eps:
1. Ma e ial collec ion. I in ol es de ining and delimi ing he ma e ial o be collec ed. In
his pape , keywo ds such as “Capaci a ed Lo -Sizing P oblem”, “CLSP”, “Capaci-
a ed Lo -sizing and Scheduling P oblem”, “CLSSP”, “Ma hema ical P og amming”,
“MILP”, “Op imiza ion”, “Heu is ic”, “Me aheu is ic”, “Ma heu is ic” and “Hyb id”
we e used o conduc sea ches in he Web o Science and Scopus scien i ic da abases.
Only pape s including models o he CLSSP w i en in English we e conside ed.
2. Bibliog aphic da a p ocessing and isualisa ion. I in ol es analysing he o mal
aspec s o he ma e ial collec ed, such as publica ion da es and sou ces.
3. Ten a i e de ini ion o he CF. A d a o he CF is es ablished by iden i ying dimen-
sions, ca ego ies, and elemen s om p e ious li e a u e e iews such as Comelli e al.
(2008), Copile al. (2017), Ka imi e al. (2003), Quad & Kuhn (2008), and Robinson
e al. (2009). No el aspec s ha e been included based on he au ho ’s knowledge o
he p oblem and he esea ch objec i e o ien ed o o mula e op imiza ion models.
4. S uc u ed li e a u e e iew. I in ol es analysing and e alua ing ma e ials acco ding
o he ini ial d a o he CF. In his pape , 34 pape s we e analysed o en ich he ini ial
CF.
5. Re ined CF and Final p oposal. The ini ial d a is e ined by in eg a ing new dimen-
sions and ele an ca ego ies iden i ied du ing he e iew p ocess un il i s inal
p oposal.
3 Concep ual F amewo k o CLSSP
The applica ion o he p e ious me hodology esul s in he ollowing CF used o he
op imiza ion o he CLSSP h ough ma hema ical p og amming. This CF encompasses
se en dimensions, di ided in o ca ego ies and elemen s (Fig. 1).
The “SC physical cha ac e is ics” dimension comp ises h ee ca ego ies desc ibing
he physical a ibu es o he Supply Chain (SC). “Sec o ” de ines he scope in which
he model is applied (e.g. chemical, pape , ex ile, plas ic). “SC s ages” de ails key SC
ope a ions co e ed by he model: p ocu emen , p oduc ion, in en o y, dis ibu ion, and
sale. “Shop loo con igu a ion” speci ies he p oduc ion’s a ea se up: single-machine,
pa allel machines, lowshop, o jobshop.
The “Planning and scheduling cha ac e is ics” dimension is composed by i e ca e-
go ies. “Numbe o p oduc s” dis inguishes be ween single-p oduc and mul i-p oduc
scena ios. “P oduc cha ac e is ics” iden i ies dependencies o incompa ibili ies o p od-
uc s. “Sequence ypes” can be linea sequences, lacking a speci ic o de o p oduc s
du ing p oduc ion, o cyclic sequences, which acili a e simple and sa e changeo e s o
ope a o s (Tousain & Bosg a, 2006). Wi hin cyclic sequences, hey can be non-adap i e,
which adhe e o a ixed pa e n (Libe opoulos e al., 2013), and adap i e, which allow o
adjus men s in he pa e n. “Se up ype” comp ises sequence-dependen se ups, whe e
he p epa a ion ime o cos a ies wi h he p oduc ion sequence, sequence-independen
278 J. P. Fiesco e al.
Fig. 1. Concep ual amewo k o CLSSP
se ups una ec ed by he p oduc ion o de , quan i y-dependen se ups, whe e he p epa-
a ion ime o cos ises wi h ba ch size, and quan i y-independen se ups, no ied o
he quan i y o p oduc ion, and ca y-o e , whe e i is possible o con inue p oduc ion
om he p e ious pe iod in o he cu en pe iod wi hou he need o a new se up, hus
educing cos s and se up imes (Ka imi e al., 2003). “Tempo al aspec s” iden i ies he
planning ho izon and i s di ision in o a single o mul iple pe iods, and he g anula i y
o ime in e als ca ego ized as “Small bucke ” when p oducing one p oduc pe pe iod
and esou ce, o “Big bucke ” when p oducing mul iple p oduc s pe pe iod and esou ce
(Comelli e al., 2008; Ka imi e al., 2003).
In he “Decisional aspec s” dimension, decisions in luencing he CLSSP a e con-
side ed, ep esen ing he decision a iables o he model. Key decisions include he
lo sizing, he alloca ion o jobs o esou ces, hei sequencing and iming, he in en-
o y managemen , sale decisions, and wo k o ce- ela ed decisions such as o e ime and
shi s. Addi ional decisional aspec s such as ou sou cing, unme demand, and de e ed
demand a e included in “O he decisions”.
“Cons ain s” dimension includes ca ego ies ep esen ing po en ial limi a ions and
minimum equi emen s ac oss p oduc ion, in en o y, anspo a ion, and sales planning.
These include cons ain s ela ed o p oduc a ibu es, such as minimum ba ch sizes,
p oduc dependencies, incompa ibili ies, and sequencing ules. “Wo k o ce” cons ain s
ha e limi s on o e ime hou s, he maximum numbe o shi s, and egula ions go e ning
shi scheduling (Sahling e al., 2009). “In en o y” cons ain s in ol e limi a ions on
capaci y, alloca ion, and mini-mum s ock le els o e he planning ho izon (Popo ić
e al., 2023). “T anspo ” cons ain s a ec capaci y and minimum quan i y equi emen s
(Ma ínez e al., 2019). “Ma ke ” cons ain s may in ol e mee ing o al demand o a
minimum pe cen age he eo and he abili y o de e a maximum demand a e (La oche
e al., 2022). “Machines” cons ain s ela e o p oduc ion capaci y, alloca ion limi a ions,
and conside a ions ega ding main enance scheduling.
Concep ual F amewo k o he Op imiza ion 279
The “Objec i e” dimension is di ided in o h ee ca ego ies aligned wi h sus ainabil-
i y: economic, en i onmen al and social. Common CLSSP objec i es include minimising
in en o y and se up cos s, which all in o he economic ca ego y. En i onmen al objec-
i es ypically ocus on educing pollu ion and esou ce consump ion (Re el Helm ich
e al., 2015). In addi ion, a no el objec i e ha has no been widely add essed in CLSSP
models is cus ome sa is ac ion o employmen , which ep esen s he social dimension.
The “Modelling App oach” dimension p o ides a comp ehensi e amewo k o
unde s anding he ma hema ical p og amming models. The “Ma hema ical p og am-
ming model ype” ca ego y i ncludes Con inuous linea p og amming, In ege linea
p og amming, Bina y linea p og amming, Mixed in ege linea p og amming, and Non-
linea p og amming. The “Solu ion me hod” ca ego ises he app oaches employed in
sol ing he models such as exac solu ions, heu is ics, me aheu is ics, and hyb id me h-
ods. The “Model alida ion” iden i ies he alida ion p ocess used: Real case, Case s udy,
and Nume ical example.
Finally, in he “Unce ain y Modelling” dimension, unce ain y aspec s i n modelling
a e examined, conside ing whe he he modelling con ex is de e minis ic o unce ain.
In he unce ain con ex case, de ine which pa ame e s a e subjec o unce ain y, such
as demand o p ocessing ime, and speci y unce ain y modelling app oaches, such as
de e minis ic-based, p obabilis ic, possibilis ic, o scena io-based (Supi hak e al., 2010;
Cu cio e al., 2018).
4 Conclusions and Fu u e Resea ch Lines
A no el CF o he CLSSP has been p oposed ollowing a esea ch me hodology com-
p ised o i e s eps. The p oposed CF se es a dual pu pose. I can be employed as a
guide o p oposing new ma hema ical p og amming models and de i a i es o add ess
he combined p oblem o Capaci a ed Lo -Sizing and Scheduling. Al e na i ely, i se es
as a comp ehensi e amewo k o e iewing exis ing models in o de o iden i y exis ing
gaps and u u e esea ch lines.
As a p ospec i e esea ch di ec ion, a comp ehensi e s a e o he a o models
add essing he CLSSP h ough he p oposed CF should be conduc ed. In he li e a u e
e iew unde aken du ing he de elopmen o he CF me hodology, an ini ial explo a ion
o his s a e o he a has been ini ia ed, p o iding a basis o po en ial expansion and
subsequen publica ion.
In his ini ial analysis, p omising u u e esea ch lines ha e been iden i ied conce n-
ing he p oposi ion o models o suppo he CLSSP, such as he necessi y o modeling
cyclic p oduc ion sequences, pa icula ly emphasizing adap i e cyclic sequences, inco -
po a ing dependencies and incompa ibili ies among p oduc s, and add essing unce ain y
in he manu ac u ing sec o . Addi ionally, aligning he models wi h sus ainabili y is sug-
ges ed, wi h a heigh ened ocus on en i onmen al and social aspec s, which ha e ecei ed
limi ed a en ion.
Acknowledgemen s. This esea ch has been pa ially unded by Ho izon Eu ope Re . 101057294
“AI-D i en I ndus ial Equipmen P oduc Li e Cycle Boos ing Agili y, Sus ainabili y, and
Resilience (AIDEAS)” and Re . 101070076 “Op imizing P oduc ion And Logis ic Resou ces In
280 J. P. Fiesco e al.
The Time-C i ical Bio P oduc ion Indus ies In Eu ope (CLARUS)”; and he Regional Depa men
o Inno a ion, Uni e si ies, Science, and Digi al Socie y o he Gene ali a Valenciana “P og ama
In es igo” Re . INVEST/2022/330, which he Eu opean Union suppo ed - Nex Gene a ionEU
wi h Plan de Recupe ación, T ans o mación y Resiliencia.
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