Numerical studies for the scheduling of continuous annealing lines

Höne loh, Hagen Alexande
A icle
Nume ical s udies o he scheduling o con inuous
annealing lines
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U.S. Unde Di e en Demand Scena ios
Hagen Alexande Höne loh, Nume ical S udies o he
Scheduling o Con inuous Annealing Lines
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ISSN: 2942-1861
Nume ical S udies o he Scheduling o Con inuous Annealing Lines
Hagen Alexande Höne loh
Leibniz Uni e si y Hanno e
Abs ac
The con inuous annealing o la s eel imp o es i s p ope ies o applica ions such as au omo i e manu ac u ing. Scheduling
hese p ocesses on Pa allel He e ogeneous Annealing Lines (PHALs) is complex due o di e se coil p ope ies, incompa ible
p ocess modes, and due da e cons ain s. In oducing s inge s o add ess incompa ibili ies be ween s eel shee s aises cos s,
ene gy use, and CO2 emissions, highligh ing he need o op imized scheduling. This hesis implemen s a ma hema ical model
in Py hon using he Gu obi sol e o op imize PHAL scheduling by minimizing s inge usage while mee ing a diness con-
s ain s. The model is ex ended o include coil-speci ic elease da es and expanded o add ess ade-o s be ween s inge use,
a diness, and due da e de ia ions, including ea liness. A compu a ional s udy e alua es he model unde a ious scena ios,
examining he e ec s o coil he e ogenei y, u gency, p ocess lexibili y, and s inge p ocessing imes. Resul s show ha op-
imized schedules educe s inge use and delays, pa icula ly unde high p ocess lexibili y. These indings demons a e he
po en ial o op imiza ion o imp o e e iciency and sus ainabili y in s eel p oduc ion while guiding u u e esea ch in dynamic
scheduling app oaches.
Keywo ds: con inuous annealing lines; Gu obi sol e ; scheduling op imiza ion; s eel indus y; s inge minimiza ion
1. In oduc ion
1.1. Subjec and mo i a ion
Ou mode n economy h i es on digi al ans o ma ion
and i s many ace s. One key aspec is he compu e iza ion o
p ocesses using ad anced digi al echnologies. Th ough com-
pu e iza ion, companies can c ea e and implemen op imal
schedules o hei p ocesses and he eby inc ease e iciency
and p oduc i i y, as companies always sough o imp o e
hei decision-making h ough new scheduling and planning
me hods.1Bu he po en ial impac o compu e iza ion and
digi aliza ion could be a g ea e han any hing be o e.
The s eel indus y is one o he many indus ies ha s and
o bene i signi ican ly om his de elopmen . By le e ag-
ing digi al echnologies, manu ac u e s a e able o minimize
p oduc ion ime, while also maximizing he use o esou ces
and hus hei p o i s.2The impac on his highly compe -
i i e indus y, which supplies us wi h ma e ials needed o
1Tang and Meng, 2021, p. 1
2Iannino e al., 2021, p. 620
e e yday appliances, ailways, o e en buildings, is as on-
ishing.3One p ocess in he s eel indus y, ha can bene i
g ea ly om digi aliza ion, is he con inuous annealing o
la s eel, a me hod o p ocessing s eel o change i s physi-
cal and mechanical p ope ies.4In his p ocess, coils o cold
olled la s eel a e p ocessed in u naces wi h di e en an-
nealing empe a u es and anspo speeds ha make up he
p ocess mode o an indi idual coil. Di e en p ocess modes
lead o di e en mechanical and physical p ope ies o he
s eel.4Hence, he mode is chosen acco ding o he desi ed
cha ac e is ics. Cold olled la s eel is essen ial o building
ca s, and household appliances and has many mo e a eas o
applica ion.5Scheduling he con inuous annealing p ocess
on con inuous annealing lines (CALs) is a di icul ask due
o he di e en p ope ies o he la s eel, p ocessing modes,
and o he aspec s like due da es. I he p ope ies o p o-
cessing modes o wo successi e coils a e oo di e en , sc ap
3Zhao and Yang, 2016, p. 3417 and Te ence, 2020
4Sa na, 2013
5Commodi y-Inside, 2019
DOI: h ps://doi.o g/10.5282/jums/ 10i3pp781-809
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H. A. Höne loh /Junio Managemen Science 10(3) (2025) 781-809782
coils, so-called s inge s, mus be added be ween he coils
o b idge hese di e ences, esul ing in addi ional ma e ial
cos s and a loss o e iciency.6
The impac o his loss o e iciency canno be o e s a ed.
No only does he manu ac u e lose aluable p oduc ion
ime, bu he also has o was e ene gy, inc easing he amoun
o CO2emi ed pe kg o s eel. Wi h a sha e o se en % o he
wo ld’s annual CO2emissions, he s eel indus y is al eady
one o he mos ene gy-in ensi e indus ies wo ldwide.7In
he meanwhile, in he Eu opean Union, he s eel sec o is ac-
ing an e e -inc easing cos o ene gy, as well as an inc ease
in CO2 p ice pe on o CO2 emi ed, which some expe s
p edic could each 50% by he end o he decade.8
Hence, manu ac u e s should ha e a se ious in e es in
op imizing hei p oduc ion schedules by minimizing he in-
oduc ion o s inge s and hus he cos s caused by ma e ial
and ene gy was age, loss o e iciency, and CO2emissions,
which can be achie ed h ough he use o digi al echnolo-
gies.
1.2. Resea ch ques ion and s uc u e o he wo k
This hesis is based on a pape by Wegel e al. (2024),
in which hey p opose a ma hema ical model o op imize he
scheduling o he con inuous annealing p ocess. The objec-
i e o his model is o minimize he in oduc ion o s inge s
in a schedule, while also conside ing a a diness cons ain
ha limi s he numbe o delays. I is designed o sho -
e m planning and can he e o e be used a he ope a ional
le el o ope a ions managemen .9So a , his model has only
been implemen ed in he Julia p og amming language and
used wi h a p op ie a y algo i hm. Thus, he model has no
ye been implemen ed in he popula Py hon p og amming
language. This implemen a ion o ms he basis o his hesis.
Du ing he cou se o his wo k, a nume ical s udy will
be conduc ed on his model and on i s expansions c ea ed
du ing he wo k. I will consis o di e en sec ions, which
s udy he impac o ce ain pa ame e s and scena ios on he
scheduling o CALs. The esul s will be ho oughly analyzed
and discussed o de i e eme ging ends and o mula e man-
age ial insigh s.
Fi s , he CALs o la s eel will be explained in e ms o
hei design and hei scheduling, which will be ollowed by
he cu en s a e o esea ch. A e wa ds, an explana ion o
he unde lying p oblem and he ma hema ical model i sel
will be gi en. This will be con inued by u he ex ensions
o he model wi h he aim o mimicking he eal-wo ld p o-
cess. The nume ical s udies conduc ed will be he co e o his
hesis. Di e en ends ha occu wi h inc easing ins ance
sizes will be explo ed i s , ollowed by sec ions in which he
impac o scena ios and he al e a ion o ce ain pa ame e s
on he scheduling p ocess will be in es iga ed. Fu he mo e,
6Besson, 1998, p. 29
7Join -Resea ch-Cen e, 2022
8Twidale e al., 2021 and K ukowska, 2021
9Ka akos as e al., 2020, p. 2 and Ka akos as e al., 2019, p. 2
he base model will be compa ed wi h he ex ended model
ega ding i s pe o mance and solu ions. The esul s o he
compu a ional s udy ca ied ou on he Py hon implemen-
a ion and he ins ances used a e p esen ed and discussed,
leading o a summa y o he wo k and an ou look o u u e
esea ch, u he ex ensions, and manage ial insigh s.
2. Con inuous Annealing Lines o Fla S eel
2.1. P ocess desc ip ion
The indus ial con inuous annealing p ocess is made pos-
sible by CALs, which consis o se e al sec ions. One such
CAL is depic ed in igu e 1. CALs a e no s andalone, hey a e
pa o a g ea e complex consis ing o di e en a eas dedi-
ca ed o di e en p ocessing me hods, like he cold olling
and gal aniza ion o s eel.10 This s udy will ocus on he an-
nealing o cold olled s eel, which is usually he bo leneck
o s eel p ocessing.11 In some cases, i may also be possible
and ad an ageous o anneal ho olled s eel.12 In compa -
ison o ho olling, cold olling happens a a lowe em-
pe a u es. Bu hese p ocesses a e complemen a y and no
subs i u ional, since he ho olling p ocess is an ups eam
p ocess o cold olling. While ho olling olls he s eel o
he desi ed wid h, cold olling educes i s hickness by up o
90 %, inc easing i s s eng h and ha dness bu also se e ely
diminishing i s duc ili y and inc easing i s b i leness.13 To
imp o e hese mechanical and physical p ope ies, he s eel
s ips a e annealed in CALs. Th ough ec ys alliza ion o mi-
c os uc u es and o he p ocesses, he s eel egains some o
i s los cha ac e is ics, especially i s duc ili y.14
The p ocess s a s a he en y sec ion wi h he coils o
cold olled la s eel and he so-called pay-o eel, as de-
pic ed in igu e 1. These coils s em om he ups eam cold
olling p ocess. The pay-o eels ixa e he coils o la s eel
wi h a mand el and o a e o con inuously unwind hem.16
Fu he downs eam, a welding machine au oma ically welds
he ail end o he head end o wo consecu i e coils oge he ,
hus p o iding con inuous s ip eeding o he succeeding
sec ions.17 The welding p ocess i sel is o he u mos im-
po ance since a weld b eak could esul in a comple e line
shu down.18 The e o e, compa ibili y be ween consecu i e
coils ega ding hei hickness and wid h has o be ensu ed.
I he wo consecu i e coils a e incompa ible wi h each o he ,
a s inge has o be added be ween hem o assu e a con in-
uous annealing p ocess, esul ing in a loss o e iciency and
highe ma e ial and ene gy cos s.19 S inge s can be eused,
bu only o a limi ed numbe o imes.20
10 Zhao and Yang, 2016, p. 3418
11 Li e al., 2023, p. 1
12 S eel-Wa ehouse, n.d.
13 Zhao and Yang, 2016, p. 1
14 Sa na, 2013
15 Taku ou e al. (2016), p. 113
16 Jiyuan-Shenzhou-Indus y, n.d.
17 Uni ed-En e p ises, n.d.
18 Williamson, n.d., p. 3
19 Besson, 1998, p. 29
20 Mujawa e al., 2004, p. 1
H. A. Höne loh /Junio Managemen Science 10(3) (2025) 781-809 783
Figu e 1: A Con inuous Annealing Line as implemen ed in a Japanese ac o y.15
A e deg easing by a deg easing uni , he s eel s ip en-
e s he p ocess sec ion, whe e i i s eeds in o an en y
loope .21 This loope coun e ac s in e up ions in he con-
inuous annealing p ocess, such as he welding p ocess, and
main ains a con inuous s ip speed h oughou he annealing
p ocess. I achie es his by mo ing i s olls apa om each
o he , hus inc easing he leng h o s ip s eel i can hold and
he dis ance he s eel has o a e se. The con inuous s ip
hen en e s se e al u naces, ha can each empe a u es o
up o 850 °C, while main aining a s ip speed o up o 800
m/min.22 The s eel s ip cycles h ough he u naces o se -
e al minu es, depending on he chosen p ocessing mode. As
p e iously desc ibed, a p ocessing mode is a combina ion o
annealing empe a u e and s ip speed, which leads o ce -
ain s eel p ope ies.23 Since he s ip speed and u nace
empe a u e can only be adjus ed in a ce ain ange om he
p eceding coil o he succeeding one, compa ibili y be ween
hei p ocessing modes is necessa y o else a s inge has o
be in oduced be ween hem.24 I should also be men ioned,
ha adjus ing he u nace empe a u e cos s a lo o ene gy
and should he e o e be minimized.25
A e he hea ea men , he s eel s ip is cooled down in
se e al s eps, un il i eeds in o he deli e y o exi loope .26
This loope wo ks like he en y loope and can he e o e
compensa e o in e up ions like main enance and he cu -
ing p ocess, which ollows downs eam.24 A e he cu ing,
he ension eel ecoils he s eel s ip in o he p e ious coils.27
Compa ed o he ba ch annealing p ocess, in which he s eel
is p ocessed as a coil, he con inuous annealing p ocess has
a highe e iciency and p oduc i i y, while also deli e ing a
mo e uni o m p oduc , ega ding he physical and mechan-
ical p ope ies o he s ip s eel. Some o i s disad an ages,
howe e , a e he la ge amoun s o space and capi al needed
o cons uc i .23
21 Uni ed-En e p ises, n.d.
22 Uni ed-En e p ises, n.d.
23 Sa na, 2013
24 Besson, 1998, p. 29
25 Zhao and Yang, 2016, p. 3417
26 Sa na, 2013
27 Jiyuan-Shenzhou-Indus y, n.d.
2.2. Di e en ia ion om li e a u e
The opic o scheduling has been a esea ch subjec since
he ea ly 20 h cen u y and is nowadays one o he mos e-
sea ched ields in ope a ions esea ch, wi h se e al hund ed
pape s published each yea .28 Scheduling in he s eel indus-
y pa icula ly is one o he mos di icul and complex p ob-
lems, due o he complexi y o he s eel indus y i sel and
he e o e, he e ha e been many a emp s o op imize ce ain
aspec s o i .29 This sec ion will ea u e di e en app oaches
conce ning CALs and con inuous gal anizing lines, a p ocess
u he downs eam o he annealing p ocess ha has many
simila i ies wi h CALs ega ding i s scheduling.30
Li e al. (2023) aimed o minimize ea liness and a -
diness cos s, as well as se up cos s, which occu h ough
changes in annealing empe a u e, on one p ocessing line.31
The au ho s no ed, ha hese objec i es con lic wi h each
o he since minimizing he se up cos s by cons uc ing la ge
ba ches o simila coils could inc ease he ea liness and a di-
ness cos s and ice e sa. S inge s we e no di ec ly consid-
e ed. To minimize he h ee objec i es, hey used an adap i e
mul i-objec i e di e en ial e olu iona y algo i hm (MODE)
based on deep ein o cemen lea ning (DRL). Se e al o he
pape s also include he use o MODEs o he op imiza ion o
CALs and o o he p ocesses as well, like o he ho olling
p ocess.32 The usage o MODEs can be highly e ec i e, as
can be seen in a s udy by Dong e al. (2021), in which hey
we e able o ind op imal schedules o he colo -coa ing
p ocess o up o 400 coils wi h h ee di e en objec i es.
Du ing he colo coa ing p ocess, p o ec i e o deco a i e
coa ings a e applied o he s eel.33
Zhao and Yang (2016) es ed a disc e e di e en ial e olu-
ion algo i hm (DDE), which uses disc e e job pe mu a ions
o ind op imal solu ions, o a e age he line capaci y and
minimize he changeo e cos s. These occu h ough anneal-
ing empe a u e changes in he u nace, and Zhao and Yang
(2016) concluded ha hei algo i hm pe o med well o up
o 90 coils on pa allel p ocessing lines.34 E en hough he di -
28 Po s and S use ich, 2009, p. 1
29 Ha junkoski and G ossmann, 2001, p. 1649
30 Zhao and Yang, 2016, p. 3418
31 Li e al., 2023, p. 1-2
32 Tang and Wang, 2010, p. 104-116 and Pan e al., 2019, p. 327-348
33 Sa na, 2014
34 Tasge i en e al., 2007, p. 271-273
H. A. Höne loh /Junio Managemen Science 10(3) (2025) 781-809784
e ences be ween annealing empe a u es o adjacen coils
we e conside ed, s inge s and a diness we e no . Since
DDEs a e di icul o apply o eal-li e p oblems, he algo-
i hms’ use ulness may be in ques ion.35
E olu iona y algo i hms also end o ocus on local op-
ima, which can be less op imal han a global op imum.36
The so-called Tabu sea ch sol es his p oblem and was used
o op imiza ion in con inuous gal anizing lines.37 Gao e al.
(2008) claimed hei espec i e app oaches o be e y e ec-
i e as hey we e able o sol e ins ances wi h up o 100 coils
in only some seconds. Bu i has o be no ed ha some as-
pec s like a diness we e no conside ed. A s udy by Pan e
al. (2017) aimed o minimize he o al weigh ed comple ion
imes on a pa allel CAL wi h up o 18 di e en lines and 120
coils and achie ed op imal solu ions in unde en minu es.38
One simila i y o he p e iously discussed s udies is he
de e minis ic cha ac e is ic o he used models. The e o e,
some au ho s used dynamic app oaches like a mul i-agen
sys em (MAS) o op imize dynamic scheduling p oblems,
which conside andomness.39 A MAS uses a i icial in elli-
gence and mul iple agen s o pe spec i es o sol e a p oblem
o p o ide lexible solu ions.40 Iannino e al. (2021) used
bo h de e minis ic and dynamic models o scheduling. The
au ho s used h ee di e en app oaches o op imize a day’s
schedule wi h a ound 2100 coils in se e al i e a ions, wi h
he objec i e o imp o e scheduling lexibili y. Fo sho - e m
planning wi h uns able ci cums ances, hey used a MAS. The
second app oach was a de e minis ic, mixed in ege linea
p og am (MILP), as is used in his s udy. The hi d and las
app oach u ilized a con inuous low model (CFM) o long-
e m scheduling, which uses a simpli ied model o schedule
he manu ac u ing p ocess o e a ious ime pe iods.41 Ian-
nino e al. (2021) summa ized ha all h ee app oaches ha e
hei ad an ages and disad an ages and ha hey should be
used complemen a ily a he han subs i u ionally o u ilize
each ad an age in he igh ci cums ances.
The second mos impo an s udy abou scheduling o
his hesis is by Mujawa e al. (2012). The au ho s p oposed
a MILP o he minimiza ion o bo h s inge s and a diness
in e ms o o al ime o e due pe coil. Due o echnical lim-
i a ions, hey we e only able o sol e he model wi h up o
15 coils, wi h a sol ing ime o close o h ee hou s. Because
o hese limi a ions, hey le e aged wo di e en heu is ics,
which we e able o yield easible solu ions o up o 150 coils,
bu could only minimize he numbe o s inge s in oduced
in he schedule and dis ega ded he a diness.42
None heless, he p oposed ma hema ical model mim-
icked he eal-wo ld p ocess o a highe deg ee han o he s.
Because o his, Wegel e al. (2024) decided o base hei
35 Zhao and Yang, 2016, p. 3418
36 Mi jalili and Gandomi, 2023, p. 393
37 Gao e al., 2008, p. 1829-1833
38 Zhang and Yang, 2014, p. 800-802
39 Cowling e al., 2004, p. 178-188 and Ouelhadj e al., 2004, p. 161-172
40 Balaji and S ini asan, 2010, p. 1-2
41 Iannino e al., 2021, p. 620-630
42 Mujawa e al., 2012, p. 440-444
s udy on he model p oposed by Mujawa e al. (2012) and
de elop i u he o be e pe o mance in he sol ing p o-
cess.
3. Op imiza ion model o Pa allel He e ogeneous An-
nealing Line Scheduling
3.1. P oblem desc ip ion
As men ioned in sec ion 2.1, he con inuous annealing
p ocess is e y complex, and many pa ame e s ha e o be
accoun ed o . One such pa ame e , ha was b ie ly men-
ioned be o e, is he indi idual due da e o each coil. These
a e necessa y, o schedule he u he downs eam p oduc-
ion s ages, like he gal aniza ion.43 These schedules only
ha e limi ed lexibili y and o no pu hem a isk, he max-
imum numbe o delays has o be bound. This limi a ion is
implemen ed h ough a se ice le el, which alue is ela i e
o he numbe o coils p ocessed. Thus, a g ea e ins ance is
g an ed wi h a highe se ice le el han a small o medium
one.
Un il now, only he speci ic cha ac e is ics o he coils and
hei p ocessing modes ha e been conside ed. Bu he lines
hey a e p ocessed on ha e di e en cha ac e is ics hem-
sel es. Some lines may only be able o p ocess coils wi h
ce ain hicknesses and wid hs, while o he s may be able o
p ocess all coils, ega dless o hei cha ac e is ics. When c e-
a ing a schedule o a con inuous annealing line, he manu-
ac u e hus also has o conside which coil can be p ocessed
on which o he pa allel he e ogeneous lines, he e o e de-
c easing he amoun o planning lexibili y.
This esul s in a schedule ha has o conside he di e -
en cha ac e is ics o he coils, hei desi ed speci ica ions in
e ms o s ip speed and annealing empe a u e, he compa -
ibili y o said cha ac e is ics and p ocessing modes o con-
secu i e coils, as well as hei compa ibili y wi h each p o-
cessing line and hei due da es, while i also has o comply
wi h he se ice le el and aims o minimize he numbe o
used s inge s. To be able o c ea e such a complex sched-
ule, Wegel e al. (2024) se ou assump ions, ega ding he
elease da es, in e nal se ice le el, cos s, and ope a ing con-
di ions.
To educe complexi y, i is i s assumed ha e e y coil
is a ailable and wai ing o p ocessing a he beginning o
he schedule. The e o e, each coil could be he i s o be
p ocessed in he schedule since hey ha e no elease da es.
The second assump ion conce ns he p e iously desc ibed
in e nal se ice le el. As al eady men ioned, he se ice le el
bounds he absolu e numbe o delayed coils o a ce ain
alue ha is ela i e o he ins ance size. Bu he in e nal
se ice le el also bounds he maximum delay ha can occu
in he schedule, since all coils ha e o be p ocessed du ing
i . This maximum delay depends on he maximum amoun
o ime needed o p ocess all coils in one schedule.
43 Zhao and Yang, 2016, p. 3418

H. A. Höne loh /Junio Managemen Science 10(3) (2025) 781-809 785
Table 1: No a ion o he ma hema ical model.
Symbol Meaning
Indices and index quan i ies
i,j∈1,...,NCoils
k∈1,...,KP ocessing lines
m∈1,...,Mik Feasible p ocessing modes o coil ion line k
n∈1,...,Mjk Feasible p ocessing modes o coil jon line k
Pa ame e s
diDue da e o coil i
pikm P ocessing ime o coil ion line kin mode m
αSe ice le el
MMaximum du a ion o he schedule
ci jkmn Cos s o adding a s inge be ween coils iin mode mand jin mode non line k
i jkmn Addi ional p ocessing ime o a s inge be ween coils iin mode mand jin mode non line k
Decision a iables
yi jkmn ∈ {0,1}Bina y a iable wi h alue 1, i coils iin mode mand jin mode na e p ocessed in sequence on
line k, else 0
δi jk ∈ {0,1}Bina y a iable wi h alue 1, i coils iand coil ja e p ocessed in sequence on line k, else 0
xikm ∈ {0, 1}Bina y a iable wi h alue 1, i coil iis p ocessed in mode mon line k, else 0
si≥0 S a da e o coil i
zi∈ {0,1}Bina y a iable wi h alue 1, i coil iis delayed, else 0
The hi d assump ion made se s he cos s o all coils o a
ixed alue. The easoning behind his is ha only he cos s
caused by he in oduc ion o s inge s should be conside ed
since we canno a oid he cos s caused by he p ocessing o
coils. Due o CALs being qui e s able and comple ely au o-
ma ed, Wegel e al. (2024) also assume ha no andomness
occu s du ing he p ocess and he e o e p opose a de e min-
is ic model, in which all pa ame e s a e known a p io i. This
is he ou h and las assump ion ega ding he ma hema ical
model.
Based on hese assump ions and he aspec s men ioned
p e iously, he au ho s build a de e minis ic op imiza ion
model ha aims o c ea e an op imal, cos minimizing, and
a diness-bound schedule o pa allel he e ogeneous anneal-
ing lines, which is based on he model by Mujawa e al.
(2012)44 In he ollowing sec ions, he no a ion o his ma h-
ema ical model and he model i sel will be explained, which
will be ollowed by u he ex ensions o i .
3.2. No a ion
This sec ion p esen s he no a ion o he ma hema ical
model p oposed by Wegel e al. (2024). The coils o la s eel
a e deno ed by N={1,..., N}and he pa allel con inuous
annealing lines hey a e p ocessed on by K={1,..., K}. E -
e y coil i∈ N has a speci ic due da e di>0. The maximum
44 Mujawa e al., 2012, p. 440
amoun o accep able delays, he se ice le el, is deno ed by
α∈N.
Each coil ialso has a se amoun o di e en p ocessing
modes Mik on each line k∈ K . The easible p ocess mode
mo coil ion line kincludes a speci ic annealing empe a u e
and s ip speed. I coil ican be p ocessed in a speci ic mode
on line kdepends on he cha ac e is ics o he coil iand o he
line k, as well as on he pa ame e s o he p ocessing mode
i sel . I is he e o e possible, ha no mode o he p ocessing
o coil ion line kexis s. I coil ican be p ocessed on line k
in mode m, he p ocessing ime pikm can be de i ed.
Fu he mo e, coil ion line kcould be succeeded by coil
j, wi h j∈ N , in he easible p ocessing mode n, wi h n∈
Mjk. In his case, he compa ibili y o coil iin mode mand
coil jin mode n ega ding hei wid h, hickness, annealing
empe a u es, and s ip speeds has o be e iewed. I he
coils and modes a e incompa ible wi h each o he , a s inge
has o be added be ween hem. The addi ional cos s caused
by in oducing a s inge in o he schedule be ween coil iin
mode mand coil jin mode non line ka e exp essed by ci jkmn,
while he addi ional p ocessing ime is con ained in i jkmn.
I he coils and hei espec i e modes a e compa ible, he
addi ional cos s and p ocessing ime will equal ze o, since
no s inge has o be in oduced.
The p ocessing schedule will be de ined by he ollowing
decision a iables. Each coil’s s a ime is ep esen ed by
si≥0, while i s p ocessing line and mode in he op imized
H. A. Höne loh /Junio Managemen Science 10(3) (2025) 781-809786
schedule a e indica ed by he bina y a iable xikm ∈ {0,1}
aking he alue o one i coil iis being p ocessed on line
kin mode mo ze o, i no . I he comple ion o coil iis
behind he scheduled due da e di, he bina y a iable zi∈
{0,1}will equal one o ze o, i coil i’s p ocessing was inished
in ime. The sequence o he schedule is indica ed by he
bina y a iable δi jk ∈ {0,1}, wi h he alue o one i coil i
p ecedes coil jon line ko ze o i his is no he case.
Addi ional in o ma ion ega ding hei espec i e modes
mand nis p o ided by he bina y a iable yi jkmn ∈ {0, 1}
wi h he alue o one i coil iis being p ocessed in mode m
and is succeeded by coil jin mode non line ko ze o, i no .
3.3. Ma hema ical model
min
z,s,x,δ,yZ=X
i∈N
X
j∈N
X
k∈K
X
m∈Mik
X
n∈Mjk
ci jkmn ·yi jkmn
s. .
si+X
k∈K
X
m∈Mik
pikm ·xikm ≤di+zi·M∀i∈ N , (1)
si+X
k∈K
X
m∈Mik
X
n∈Mjk
(pikm + i jkmn)·yi jkmn ≤
sj+M·(1−X
k∈K
δi jk)∀i,j∈ N , (2)
δi jk =X
m∈Mik
X
n∈Mjk
yi jkmn
∀i,j∈ N ,∀k∈ K , (3)
X
j∈N
X
n∈Mjk
yi jkmn ≤xikm
∀i∈ N ,∀k∈ K ,∀m∈ Mik, (4)
X
i∈N
X
m∈Mik
yi jkmn ≤xjkn
∀j∈ N ,∀k∈ K ,∀n∈ Mjk, (5)
X
k∈K
N+1
X
j=1
δi jk =1∀i∈ N , (6)
X
k∈K
N
X
i=0
δi jk =1∀j∈ N , (7)
N+1
X
j=1
δ0jk =1∀k∈ K , (8)
N
X
i=0
δi(N+1)k=1∀k∈ K , (9)
N
X
j=0
δjik =
N+1
X
j=1
δi jk ∀i∈ N ,∀k∈ K , (10)
X
i∈N
zi≤α, (11)
si≥0∀i∈ N , (12)
xikm,δi jk,yi jkmn,zi∈ {0, 1}
∀i,j∈ N ,∀k∈ K ,∀m∈ Mik,∀n∈ Mjk. (13)
The main objec i e o his ma hema ical model is he min-
imiza ion o cos s caused by he in oduc ion o s inge s.
The e o e, he objec i e unc ion minimizes he sum o p od-
uc s o ci jkmn and yi jkmn. By mul iplying ci jkmn and yi jkmn he
cos o a s inge is only ega ded i coil iin mode mand coil
jin mode non line ka e no compa ible wi h each o he and
a e p ocessed in sequence in he op imized schedule, hus
equi ing a s inge . I hey a e no p ocessed in sequence
o i hey a e compa ible wi h each o he o bo h, he alue
o he p oduc will equal ze o since yi jkmn o ci jkmn o bo h
will equal ze o, espec i ely. By aking he sum o e all coils,
lines and modes, all in oduced s inge s in he schedule a e
conside ed.
To mimic he eal-wo ld p ocess and main ain consis-
ency, se e al cons ain s a e necessa y and will be discussed
in he ollowing. Cons ain s (1) and (2) assu e ime con-
sis ency, while cons ain s (3)-(5) ensu e he consis ency o
decision a iables ha indica e he same in o ma ion. The
consis ency o he p ocessing sequence in he schedule is es-
ablished by he cons ain s (6)-(10). These a e ollowed by
h ee addi ional cons ain s, ha egula e gene al aspec s o
he ma hema ical model and he schedule.
The i s cons ain (1) akes he a diness o he schedule
in o accoun . The ime o comple ion o coil iis calcula ed
by adding i s p ocessing ime pikm in mode mon line k o
i s s a da e si. This comple ion da e has o be smalle o
equal o i s due da e di o he coil o be inished p ocess-
ing in ime. In his case, ziwould equal ze o due o he e-
s ic ion o he numbe o delayed coils by he se ice le el
and he possibili y ha he delay could be used somewhe e
else in he schedule o u he minimize he numbe o in o-
duced s inge s. Bu i he comple ion da e is a e he due
da e, zihas o equal one o no iola e he equa ion, since he
ime o comple ion o coil iwill be g ea e han i s due da e
di. Addi ionally, he p oduc o ziand Massu es ha coil i
has o be p ocessed some ime in he schedule since Mis de-
ined as maxk(Pi∈N ,Mik=;max j,m,n(pikm + i jkmn)), which is
he maximum amoun o ime necessa y o p ocess all coils
in he schedule. This cons ain has o be se o e e y coil,
as e e y coil has a due da e and could be delayed.
Cons ain (2) conside s he ime sequences o p ocessing
in he schedule. The le side ep esen s he ea lies ime a
succeeding coil jcould be p ocessed a e he p ocessing o
he p eceding coil iwas inished. This is achie ed by adding
he p ocessing ime pikm o coil iin mode mon line kand
he possible addi ional p ocessing ime o a s inge i jkmn, in
case o incompa ibili y o coil iin mode mand coil jin mode
n, o he s a da e o coil i. P ocessing o he succeeding
coil jcan only s a a e he p ocessing o coil io ha o
H. A. Höne loh /Junio Managemen Science 10(3) (2025) 781-809 787
he s inge is comple ed. Thus, sjhas o be g ea e o equal
o ha ime o comple ion. In he case ha coils iand ja e
p ocessed in sequence on line k,δi jk would equal one, and
hus, Mwould no be added o he igh -hand side. Bu as
his cons ain holds o all coils, i could compa e wo coils
wi h each o he , ha a e no p ocessed in sequence. To p e-
en his compa ison o ha e an e ec on he solu ion, Mis
added o sj, since in his case, δi jk will equal ze o. Thus, he
igh side will always be g ea e han o equal o he le side
since M≤sj+Mand si+(pikm + i jkmn)·yi jkmn ≤Min e e y
scena io, because o he na u e o Mdesc ibed p e iously.
Cons ain (3) assu es consis ency h oughou he a i-
ables δi jk and yi jkmn. Bo h a iables indica e he sequence
o coils iand j, as well as he line k hey a e p ocessed on,
and he e o e should be equal o each o he wi h he same
coils i,jand line k. Since yi jkmn also indica es he p ocessing
modes o bo h coils and canno ep esen a bina y alue wi h-
ou his in o ma ion, he sum o yi jkmn o e all modes has o
be aken o his cons ain o be e ec i e. This cons ain
also eassu es, ha he coils a e only p ocessed in one mode
and no in mul iple, hence in his case, wo yi jkmn would
equal one wi h he same coils and line. The e o e, δi jk would
ha e o equal wo, which is impossible due o he bina y na-
u e o δi jk.
Cons ain (4) con ibu es o he decision a iable con-
sis ency as well. Bo h a iables yi jkmn and xikm indica e he
p ocessing line kand he p ocessing mode mo coil i. Hence,
hey should be equal o each o he o e e y coil, mode, and
line. Bu in he case ha coil iis he las coil ha is being p o-
cessed on line k,yi jkmn would equal ze o since yi jkmn does
no ake he coils i∈ {0,N+1}in o accoun and could he e-
o e no be succeeded by ano he coil j. To no iola e his
es ic ion in his case, yi jkmn mus only be less o equal o
xikm. The e o e, wi h xikm equalling one, yi jkmn could be ei-
he one o ze o. Fu he mo e, by aking he sum o yi jkmn
o e all coils and modes, he cons ain p e en s coil i om
being succeeded by mul iple di e en coils in mul iple di -
e en modes and limi ing his numbe o he alue o xikm,
which is ei he one o ze o due o i s bina y na u e.
The e o e, coil ican only be succeeded by a single coil j
in a single mode non line k, i iis being p ocessed on line
kin mode mo by none i i is no . This holds o all coils,
lines, and modes.
The ollowing cons ain (5) is essen ially he same as (4)
bu o he succeeding coil j. Hence, i p e en s coil j om
succeeding mul iple coils in mul iple modes by limi ing his
numbe o xjkn. I has o be no ed, ha he indices o xjkn
in his cons ain di e om he indices o xikm in he o he
cons ain s since his cons ain should only es ain he suc-
ceeding coil j. Addi ionally, yi jkmn mus only be less o equal
o xjkn, since all yi jkmn’s would be ze o i jwould be he i s
coil o be p ocessed on line k, which would iola e he con-
s ain . This is because his cons ain only accoun s o coils
in Nand hus he i s coil on line kcanno be p eceded.
This es ic ion applies o all succeeding coils on e e y line
in e e y p ocessing mode.
Cons ain (6) es ic s he numbe o coils ha succeed
coil ion any line o one. This p e en s coil i om being
p ocessed and p eceded by se e al coils on di e en lines.
I his would be he case, he sum o δi jk o e all lines and
coils in [1, N+1]would be g ea e o coil i han one, hus
iola ing he es ain . I coil iis he las coil o be p ocessed
on line k, i could no be succeeded by any coil, esul ing in
δi jk equalling ze o. The e o e, a i ual coil N+1 has o be
in oduced so ha in he case men ioned abo e, coil iwould
no iola e his cons ain . This es ic ion applies o e e y
coil.
Cons ain (7) is simila o cons ain (6), as his con-
s ain es ic s he numbe o p eceding coils o coil jon
all lines o one and he e o e p e en s coil j om being p o-
cessed and p eceded mo e han once du ing he schedule.
The ex ension o Nby ze o can be explained wi h he con-
s ain (8). He e, iis se o he alue ze o. This i ual coil
ze o ma ks he beginning o he schedule on e e y line k.
Since his coil is no being p ocessed, i is no included in
mos o he cons ain s. Bu since coil ze o is he i s on e -
e y line k, i has o be succeeded by one coil j∈[1,N+1].
The e o e, cons ain (8) ensu es ha one coil jis he i s
one o be p ocessed on line k. No eal coils a e p ocessed on
line kdu ing he schedule i coil N+1 succeeds coil ze o.
This holds o e e y line because e e y line has o ha e a i -
ual i s coil ha is being succeeded by ano he coil, i ual
o eal.
Cons ain (9) on he o he hand ensu es ha one coil
i∈ {0,N } has o be he las eal coil o be p ocessed on line
k. To ma k his coil as he las coil o be p ocessed on line
k, i will be succeeded by ano he i ual coil N+1. As on
e e y line k he e has o be a las coil o be p ocessed, his
holds o e e y line.
The las sequence consis ency cons ain , cons ain (10),
assu es ha e e y eal coil ihas o ha e a p edecesso and
successo coil jon he line ki is being p ocessed on. Bu
i also accoun s o he ex eme case, ha a p ocessing line
kp ocesses no eal coil, which is why i conside s coil ze o
as a p edecesso on he le -hand side and coil N+1 as a
successo on he igh -hand side. Addi ionally, his conside s
ha one eal coil has o be he i s and ano he has o be he
las coil i eal coils a e p ocessed on he line, as desc ibed in
cons ain s (8) and (9). This es ic ion applies o e e y line.
Hence, he in oduc ion o he i ual coils ze o and N+1
p e en s δi jk om equalling ze o in he cases ha coil jis he
i s o coil iis he las coil o be p ocessed on line k, which
would iola e he sequence consis ency cons ain s.
The ele en h cons ain (11) es ic s he maximum
amoun o delayed coils o he se ice le el α. As his applies
o he en i e schedule, he sum o zio e all coils has o
be aken. Since p ocessing only s a s a he beginning o
he schedule, no s a da e sican ake a alue o less han
ze o. This is ensu ed by cons ain (12). As e e y coil needs
a s a da e, since e e y coil has o be p ocessed a some
poin du ing he schedule, his holds o e e y coil. The las
cons ain (13) ensu es ha he bina y a iables xikm,δi jk,
yi jkmn, and zionly equal he bina y alues o ze o o one and
also indica es which index is he elemen o wha pa ame e .
H. A. Höne loh /Junio Managemen Science 10(3) (2025) 781-809788
3.4. Expansions o he model
3.4.1. Release da es
The i s expansion ackles he assump ion o no coil-
speci ic elease da es made by Wegel e al. (2024), which was
explained in sec ion 3.1. To mimic a mo e ealis ic p ocess,
coil-speci ic elease da es we e added o he model wi h he
new pa ame e i. This pa ame e holds he speci ic elease
da e o each coil i, wi h he elease da es being dis ibu ed
wi h he same dis ibu ion as he u gency o he coils, bu
wi h ze o as he lowe and aas he uppe bound. The e o e,
in he case o Ve y high UC, he elease da es a e dis ibu ed
ia he uni o m dis ibu ion U(0, a). The UC and o he ac-
o s will be explained in sec ion 4.1 Fo he elease da es o
be conside ed in he model, he ollowing cons ain s had o
be added:
si≥ i∀i∈ N (14)
i≥0∀i∈ N (15)
Cons ain (14) es ic s he s a ime o coil i,si, om
being ea lie han i s elease da e i, while (15) p e en s he
elease da e om being ea lie han he s a o he schedule.
These hold o all coils.
This is only a mino ex ension o he model. The wo
majo ex ensions will be desc ibed in he ollowing sec ions.
3.4.2. Minimiza ion o s inge use and a diness
Since bo h a diness and cos s caused by s inge in o-
duc ion play a pi o al ole in he scheduling o pa allel he -
e ogeneous annealing lines (PHALs), Wegel e al. (2024) p o-
posed an expansion o he ma hema ical model. This ex-
pansion modi ies he objec i e unc ion in such a way, ha
he sum o in oduced s inge s and he absolu e numbe o
delays a e minimized, as depic ed below. Because o his,
he se ice le el αand cons ain (11) om he ma hema i-
cal model in sec ion 3.3 a e no necessa y anymo e and a e
emo ed. To accoun o di e en weigh s o a diness and
s inge cos s, he wo sub-objec i es a e mul iplied wi h new
pa ame e s. While _s se s he weigh o he s inge cos s,
_ a does he same o he a diness, exp essed by he sum
o delayed coils, wi h _s + _ a =1. To achie e his, an
addi ional es ic ion (16) was in oduced. This cons ain
ensu es ha he o al weigh o a diness and s inge cos s
does no di e om 100%.
min
z,s,x,δ,yZ=X
i∈N
X
j∈N
X
k∈K
X
m∈Mik
X
n∈Mjk
ci jkmn ·yi jkmn · _s
+X
i∈N
zi· _ a
_s + _ a =1 (16)
3.4.3. Minimiza ion o s inge use and due da e de ia ion
The second majo ex ension o he model is an enhance-
men o he i s ex ension by no only aking a diness in o
accoun bu also he ea liness o he schedule. I p oduc s
a e p oduced be o e he due da e, hey ha e o be s o ed in
wa ehouses, which leads o so-called in en o y holding cos s.
They a e no only caused by he s o age and handling o he
in en o y, bu also by insu ance and o he ac o s.45
Thus, he e is also an incen i e o minimize ea liness in
p oduc ion p ocesses. To minimize bo h he use o s inge s
and he de ia ion om he due da e, he objec i e unc ion
had o be al e ed acco dingly.
min
z,s,x,δ,yZ=X
i∈N
X
j∈N
X
k∈K
X
m∈Mik
X
n∈Mjk
ci jkmn ·yi jkmn · _s
+X
i∈N
( i· _ +ei· _e)· _dd
The i s pa o he objec i e unc ion is he same as in
he i s majo ex ension. Bu he second pa in oduces wo
new a iables and h ee new pa ame e s. The wo new con-
inuous a iables eiand ican be explained by desc ibing he
new i s cons ain :
si+X
k∈K
X
m∈Mik
pikm ·xikm +ei=di+ i∀i∈ N (17)
These new con inuous a iables, wi h a lowe bound o
ze o, wo k in such a way, ha only one o hem can ha e a
alue highe han ze o o coil i. I coil iin mode mon line k
was inished p ocessing be o e i s due da e di,si+pikm·xikm <
di. This would iola e he cons ain since bo h sides ha e o
be equal o each o he . The e o e, eiwill ake he alue o
di−(si+pikm ·xikm), which ep esen s he posi i e ime le
un il he due da e and he e o e he ea liness o coil i. In
his case, iwill equal ze o, since bo h eiand ia e o be
minimized. A ig ea e han ze o would lead o di+ i−
(si+pikm ·xikm)>di−(si+pikm ·xikm), he e o e inc easing
bo h iand eiand hus he objec i e alue.
The opposi e case, in which coil iis delayed, is simila .
Now, si+pikm ·xikm >di. To no inc ease his di e ence and
hence he objec i e alue, eiwill equal ze o. Because o he
necessa y equali y o bo h sides, iwill equal si+pikm ·xikm −
di, ep esen ing he posi i e ime ha coil iis delayed.
In he hi d and las case, in which he p ocessing o coil
iis inished p ocessing exac ly on ime, si+pikm ·xikm =di.
Since eiand ia e o be minimized, bo h a iables will equal
hei lowe bound ze o. In he al e ed objec i e unc ion, ei
and ia e mul iplied wi h he pa ame e s _ and _e, o
p o ide a possibili y o weighing he cos s o a diness and
ea liness. The hi d and las pa ame e _dd weighs he cos s
o due da e de ia ion agains he weigh o he s inge cos s
45 Du linge , 2015
H. A. Höne loh /Junio Managemen Science 10(3) (2025) 781-809 795
10 20 30 40 50 60 70
0
2
4
6
8
10
12
# coils
# s inge s
De aul sol e
Low
Medium
High
10 20 30 40 50 60 70
0
2
4
6
8
10
12
# coils
Tuned sol e
Low
Medium
High
Figu e 3: A e age numbe o in oduced s inge s pe Nand se ice le el in op imiza ions wi h he de aul and uned sol e .
he e o e i is no a esul o an ou lie as i was in he case
o 70 coils. Addi ionally, his sugges s ha his is no a an-
dom occu ence bu a end. Hence, i migh be bene icial o
obse e he ela i e a e age numbe o s inge s in oduced,
which a e depic ed in igu e 4.
10 20 30 40 50 60 70
10
20
30
40
# coils
Rela i e # s inge s [%]
Tuned sol e
Low
Medium
High
Figu e 4: A e age numbe o s inge s ela i e o he ins ance size
o op imiza ions wi h he uned sol e in %.
Figu e 4po ays he a e age numbe o s inge s ela-
i e o he ins ance size o he op imiza ions. To inc ease he
isualiza ion, only da a om he uned sol e is shown. The
igu e depic s, ha he absolu e numbe s o coils a e decei -
ing in his ega d and ha in ac , he esul s om he op i-
miza ion wi h 20 coils a e in line wi h he obse ed down-
wa d end. Now ha he exis ence o a cons an downwa d
end, excep o he case wi h seed 100 men ioned abo e, is
es ablished, a possible explana ion can be discussed.
To ecapi ula e, a s inge has o be in oduced be ween
adjacen coils i he di e ence be ween he physical o mode-
speci ic o bo h cha ac e is ics o hese coils is abo e he limi
speci ied by he p ocess lexibili y PF. The in oduc ion o a
s inge can be a oided i he schedule in oduces a hi d coil
be ween hese incompa ible coils, bu his is only possible i
he hi d coil is compa ible wi h he o he wo coils. An expla-
na ion o he high ela i e numbe o in oduced s inge s in
op imiza ions wi h small ins ance sizes could he e o e be he
absence o an insu icien numbe o hese hi d coils ha can
b idge he gap be ween wo incompa ible coils. By inc eas-
ing he numbe o coils, hese gaps could hen be closed o
na owed by he new coils. The likelihood o his inc eases
wi h he inc easing numbe o coils, which esul s in ewe
s inge s ha ha e o be in oduced due o di e ences in
cha ac e is ics.
To gi e a simpli ied example, in which only he anneal-
ing empe a u e o he coils is conside ed as compa ibili y
es ic ion: Th ee coils ha e o be scheduled in sequence,
wi h coil one ha ing an annealing empe a u e o 670 °C,
coil wo ha ing an annealing empe a u e o 740 °C and coil
h ee wi h an annealing empe a u e o 720 °C. The PF is 40
°C in a symme ic in e al and he annealing empe a u e is
only dis ibu ed in inc emen s o en °C, as explained in sec-
ion 4.1. Thus, coil wo and h ee a e compa ible wi h each
o he , bu no wi h coil one. Since he e is no o he coil o
b idge his gap, a s inge has o be added be ween hem.
Now, a ou h coil has been added o he schedule wi h an
annealing empe a u e o 700 °C. I is he e o e compa ible
wi h coil h ee, bu no wi h coils one and wo. Since no coil
is compa ible wi h coil one, a s inge s ill has o be added,
bu he ela i e numbe o s inge s dec eased om 33,3 % o
25 %. I ano he coil wi h an annealing empe a u e o 680
°C o 690 °C would be added, i would close he gap and lead
o a s inge - ee schedule. I i has a di e en annealing em-
pe a u e hough, i would s ill be compa ible wi h one o he
emaining coils and he e o e, s ill only one s inge would

H. A. Höne loh /Junio Managemen Science 10(3) (2025) 781-809796
be necessa y, dec easing he ela i e numbe o s inge s e en
u he . By adding mo e and mo e coils o his schedule, he
p obabili y ha one o he coils e en ually closes his gap in-
c eases.
The eason o he inc ease in he numbe o absolu e
s inge s om en o 20 coils could he e o e be ha mos
o he exis ing gaps could no be closed by he new coils and
ha some coils may ha e cha ac e is ics which we e incom-
pa ible wi h bo h ends o he gap, hus inc easing he numbe
o s inge s ha ha e o be in oduced. To summa ize, he
p obabili y o a single coil being compa ible wi h he o he
coils in he schedule is highe o a highe numbe o coils.
Thus, hese schedules wi h a highe numbe o coils need
ewe s inge s, since he e a e ewe gaps and mo e possi-
bili ies o sequence hese coils o a oid gaps. In he ac ual
op imiza ions pe o med in his s udy, he e a e mo e com-
pa ibili y es ain s as well as due da es o conside , which is
why in mos cases he e a e s ill s inge s in oduced in he
schedule, e en wi h highe numbe s o coils. The impac o
he es ic ion by due da es can be obse ed by compa ing
he a e age o he op imiza ions wi h a high and low se ice
le el because, in ac , mos op imiza ions do no u ilize he
en i e se ice le el. This is po ayed in igu e 5.
Since he se ice le el is a pa ame e o limi he o-
al amoun o delayed coils ela i e o i s ins ance size, he
h eshold o ela i e a diness o a high se ice le el is 20
% and so o h. As men ioned abo e, i can be obse ed ha
on a e age and wi h a high se ice le el, a no ins ance size
he ull se ice le el o 20 % is le e aged, wi h he highes a -
e age being 19,25 %. The da a sugges s, ha o mos cases a
se ice le el o 16 % o 17 % would su ice. This is suppo ed
by he ac , ha op imiza ions wi h a se ice le el o en %
always u ilize he en i e se ice le el o almos all o i . This
does no imply ha i he se ice le el would be inc eased o
o e 20 %, he sol e would no le e age i . Bu i s a es, ha
o ecei e he solu ions displayed in igu e 3, a se ice le el
unde 20 % would su ice, a leas in mos cases. As expec ed
wi h a low se ice le el, he sol e does no allow a delay o
coils.
The da a also sugges s a sligh dec ease in delays in op i-
miza ions wi h a highe numbe o coils h ough he uning
o he sol e . Bu , as p e iously desc ibed, his could be a -
ec ed by he selec i i y e ec . When compa ing he esul s o
igu e 3and igu e 5, i can be obse ed ha e en hough so-
lu ions ound wi h a se ice le el o 20 % always had a highe
numbe o delays, hey did no necessa ily yield mo e op i-
mal solu ions in e ms o s inge s han op imiza ions wi h
a se ice le el o only en %. In ac , he mos signi ican
di e ence in a e age s inge s in oduced be ween he wo
pa ame e iza ions was only 0,556 s inge s, which occu ed
in he case o an ins ance size o 60 coils wi h he de aul
sol e and hus could be in luenced by he selec i i y e ec .
The ques ion ha could a ise is, i no p o oundly posi-
i ely a ec ing he solu ions in e ms o in oduced s inge s,
wha is he bene i o using a highe se ice le el? This ques-
ion could be answe ed by aking a look a igu e 6.
The da a o highe ins ance sizes, especially 60 and 70
coils, seems o be s ongly a ec ed by he a o emen ioned
selec i i y e ec and will he e o e be dis ega ded in he
ollowing discussion conce ning he o al op imiza ion ime.
The op g aphs po ay, ha a highe se ice le el usually
leads o a lowe a e age o al op imiza ion ime, which could
sa e cos s caused by he ope a ion o he compu e , bu ha
in gene al, he o al op imiza ion ime inc eases p o oundly.
This end can be explained by he bo om g aphs, which dis-
play he ex ao dina y inc ease in a iables and cons ain s,
and hus he size o he MILP. The s a k inc ease in a iables
is mos ly d i en by bina y a iables, like yi jkmn and xikm.
Since he numbe o a iables and cons ain s a e no in lu-
enced by he pa ame e iza ion o he se ice le el o uning,
hey a e only displayed by one g aph each. When compa ing
he op wi h he bo om g aphs, a posi i e co ela ion can
clea ly be obse ed. The e o e, he p obable explana ion o
he inc ease in o al op imiza ion ime is he g ow h o he
model he sol e is ying o sol e.
The size o he model has wo di e en impac s on he
o al op imiza ion ime. The i s and mos ob ious is, ha
i has o conside a g owing numbe o cons ain s and a i-
ables when ying o ind a easible solu ion, which inc eases
he ime i akes o p o e he easibili y o i . Bu be o e he
model can be sol ed, i i s has o be buil , which is he sec-
ond impac o he model size. The ime i ook he sol e o
build he model ose om only 4,5 seconds wi h en coils o
o e 90 seconds wi h 50 coils, which does no seem o be a -
ec ed by he se ice le el o by uning. The e o e, he model
should be kep as small as possible o a oid was ing op i-
miza ion ime, which is especially impo an when ce ain
ci cums ances se e ely limi i . Wha seems o be a ec ed by
he se ice le el and by uning is he sol ing ime, which ac-
coun s o he majo i y o he o al op imiza ion ime. Tuning
can in ac inc ease he sol ing ime h ough di e en pa am-
e e iza ions o P ePasses and P esol e, as an example.54 In
he case o his s udy, he uning hus seems o inc ease he
success a e o op imiza ions, especially wi h g ea e ins ance
sizes, a he cos o sol ing ime. A highe se ice le el, on
he o he hand, seems o dec ease he sol ing ime, which
sugges s ha he sol e can ind op imal solu ions as e i i
is less es ic ed. The MIPGap also seems o bene i om an
inc eased se ice le el, as depic ed in igu e 7.
A posi i e end be ween he MIPGap and he ins ance
size can be obse ed, as well as he end ha op imiza ions
wi h mo e scheduling lexibili y end o ha e a highe g ade
o p o en op imali y, which can be explained by he lowe
sol ing imes. The lowe he se ice le el, he mo e op i-
miza ions each he sol ing ime limi o 720 seconds and
hus ha e o abo he sol ing p ocess. Hence, he sol e has
no o less ime o p o e he op imali y o he solu ion, esul -
ing in a highe MIPGap. This does no necessa ily esul in
a wo se solu ion, as he solu ions, in e ms o s inge s, be-
ween a se ice le el o en % and 20 % only di e ed sligh ly,
as po ayed in igu e 3. Thus, he gene al upwa ds end o
54 Gu obi, n.d.-b
H. A. Höne loh /Junio Managemen Science 10(3) (2025) 781-809 797
10 20 30 40 50 60 70
0
5
10
15
20
# coils
Rela i e a diness [%]
De aul sol e
Low
Medium
High
10 20 30 40 50 60 70
0
5
10
15
20
# coils
Tuned sol e
Low
Medium
High
Figu e 5: A e age a diness ela i e o he di e en ins ance sizes pe se ice le el in op imiza ions wi h he de aul and uned sol e .
10 20 30 40 50 60 70
200
400
600
800
# coils
To al ime [sec]
De aul sol e
Low
Medium
High
10 20 30 40 50 60 70
200
400
600
800
# coils
Tuned sol e
Low
Medium
High
10 20 30 40 50 60 70
1
2
3
·107
# coils
# o Va iables
10 20 30 40 50 60 70
0.5
1
1.5
2
·104
# coils
# o linea cons ain s
Figu e 6: A e age o al op imiza ion ime ( op) and numbe o a iables (bo om-le ), as well as a e age numbe o linea cons ain s
(bo om- igh ) pe se ice le el and ins an size in op imiza ions wi h he de aul and uned sol e .
he MIPGap could be a consequence o he inc easing sol -
ing imes o each seed. Hence, i could be ad an ageous
o choose a highe se ice le el i he a ailable op imiza ion
ime is se e ely limi ed o i a highe p o en g ade o op i-
H. A. Höne loh /Junio Managemen Science 10(3) (2025) 781-809798
10 20 30 40 50 60 70
0
20
40
# coils
MIPGap [%]
De aul sol e
Low
Medium
High
10 20 30 40 50 60 70
0
20
40
# coils
Tuned sol e
Low
Medium
High
Figu e 7: A e age MIPGap pe ins ance size and se ice le el wi h he de aul and uned sol e in %.
mali y o he easible solu ion is p e e ed. The da a om he
uned sol e seems o be a ec ed by he selec i i y e ec , bu
he MIPGaps o smalle ins ances seem o be simila o ha o
he de aul sol e . The men ioned inc ease in sol ing imes is
displayed in igu e 8wi h da a om op imiza ions sol ed by
he de aul sol e due o i s be e isualiza ion o his end.
All boxplo s we e c ea ed wi h he median and he highes
and lowes alue om he en op imiza ions, as well as he
25 % and 75 % qua ile.
Figu e 6al eady depic ed he inc easing a e age o al
ime. Bu his da a could sugges ha his applies o e e y
seed, when in ac i does no . The da a in igu e 8po ays
ha , s a ing om an ins ance size o 30, he di e ences in
sol ing imes inc ease d as ically i hey a e sol ed wi h he
wo lowe se ice le els. This inc ease can be obse ed om
40 coils onwa d, wi h he highes se ice le el. The highe
numbe o op imiza ions eaching he ime limi combined
wi h he highe a e age MIPGaps in op imiza ion bound by
he lowes se ice le el indica es, ha he sol e has no ime
o p o e he op imali y o he easible solu ion a e inding i .
Wi h a medium se ice le el, ewe op imiza ions ha e o be
abo ed due o he ime limi and hus ha e ime o p o e he
op imali y o he easible solu ion, which esul s in a lowe
a e age MIPGap. This is also ue o he highes se ice le el
ha displays he ewes numbe o ime limi e mina ions.
This concludes he i s pa o he nume ical s udies.
Based on he p esen ed da a, an ins ance size o 40 coils was
chosen o he ollowing sec ions since i is he highes num-
be o coils wi h a ole able amoun o ailed op imiza ions.
In addi ion, he op imiza ions we e conduc ed wi h he uned
sol e o inc ease he numbe o success ul op imiza ions.
4.2.3. Bes - and Wo s -case scena io
Fo he hi d pa , a Bes - and Wo s -case scena io was
de i ed om he di e en ac o s de ined by Wegel e al.
(2024), which we e desc ibed in sec ion 4.1. The esul s will
be compa ed wi h he esul s achie ed wi h he uned sol e
om he p e ious sec ion. In addi ion, he PF and HC we e
al e ed because i was suspec ed ha hese ac o s a e he
mos in luen ial among hem. A mo e de ailed analysis o
he in luence o each ac o and le el on he scheduling is
p o ided by Wegel e al. (2024).
Figu e 9compa es he numbe o s inge s in oduced
be ween he Bes -case scena io wi h a Ve y low PF and
he Basecase scena io. The da a om he Ve y high and
Basecase PF is no po ayed, as no s inge s a e in oduced
a any poin in hese scena ios. As p e iously obse ed, he
numbe o s inge s dec eases wi h highe se ice le els in
he Bes -case as well due o he highe scheduling lexibili y.
Compa ed wi h he esul s om he Basecase scena io, he
Bes -case yields be e solu ions, in e ms o s inge s, o
e e y le el o PF and o e e y se ice le el. E en wi h Ve y
low PF, a s inge educ ion o o e 50 % can be obse ed.
Thus, a lowe PF leads o an inc ease in s inge in oduc ion,
bu in his case, his e ec is p obably educed by he Ve y
low HC. Since hese e ec s could he e o e coun e ac each
o he , he s inge educ ion may be a esul o he lowe UC
and SPT. I ue, his could also esul in ewe delayed coils,
which a e depic ed in igu e 10.
By compa ing he a e age a diness in he Bes -case sce-
na io - Ve y low PF wi h he a diness om he Basecase
scena io, only a sligh dec ease can be obse ed. This di e -
ence inc eases wi h he B. PF, bu does no wi h he highes
PF. Thus, he Ve y low UC and SPT do no lead o se e ely
less a diness when HC and PF a e pa ame e ized as Ve y
low. Bu he inc eased scheduling lexibili y h ough mo e
elaxed ime cons ain s could s ill bene i he op imiza ion,
hus dec easing he numbe o in oduced s inge s.
Figu e 10 also displays he dis ibu ion sol ing imes. The
da a sugges s, ha he le el o PF, in he con ex o he Bes -
case scena io, has an immense impac on he sol ing ime.
The op imiza ions wi h he lowes PF ha e he highes sol -
ing imes, e en highe han in he Basecase scena io. Bu by
inc easing he PF o he Basecase le el, he median sol ing
H. A. Höne loh /Junio Managemen Science 10(3) (2025) 781-809 799
10 20 30 40 50 60 70
200
400
600
800
Time limi
# coils
Sol ing ime [sec]
Low
Medium
High
Figu e 8: Dis ibu ion o sol ing imes in seconds pe ins ance size and se ice le el wi h he uned sol e .
Low Medium High
2
4
6
Se ice le el
# s inge s
Bes -case - Vl. PF
Basecase
Figu e 9: A e age numbe o in oduced s inge s pe se ice le el
in he Bes -case scena io wi h Ve y low PF and Basecase scena io.
imes dec ease o a ou h. Fu he inc easing he PF o Ve y
high, howe e , does dec ease he sol ing imes only sligh ly.
Tha he sol ing imes o he wo highe in es iga ed PF le -
els a e ema kably lowe han in he Basecase scena io can be
explained by he pa ame e iza ion o he HC, UC and SPT. As
p e iously desc ibed, he Ve y low UC and SPT inc ease he
scheduling lexibili y by elaxing he ime cons ain s. The
Ve y low HC, on he o he hand, inc eases he gene al com-
pa ibili y be ween di e en coils, which leads o mo e ea-
sible, s inge - ee p ocessing sequences. All o hese ac o s
could esul in mo e solu ions being easible, hus dec eas-
ing he ime i akes he sol e o ind an op imal, easible
solu ion. This could be applied o he lowes PF as well bu
in e e se. I could sugges ha he Ve y low PF d as i-
cally limi s he numbe o easible, s inge - ee p ocessing
sequences o adjacen coils, which inc eases he sol ing ime
since he use o s inge s has o be minimized. In his ega d,
he o he e y bene icially pa ame e ized ac o s canno o -
se he Ve y low PF, which could indica e ha he PF is he
mos impac ul ac o o he ou . This s a emen could be
con i med by he da a om he Wo s -case scena io.
Figu e 11 po ays he impac o he HC on he Wo s -case
scena io in e ms o in oduced s inge s and ailed uns. As
p e iously desc ibed, Ve y low HC is bene icial o s inge -
ee sequences. In his case, he HC seems o impac he a -
e age numbe o s inge s mo e han he o he ac o s, dis-
played by he ac ha he lowes HC in combina ion wi h
he Wo s -case yields be e solu ions han he Basecase sce-
na io. This would con adic he s a emen , ha he PF is he
H. A. Höne loh /Junio Managemen Science 10(3) (2025) 781-809800
Low Medium High
0
2
4
6
Se ice le el
# delayed coils
Bes -case - Vl. PF
Bes -case - B. PF
Bes -case - Vh. PF
Basecase
Low Medium High 0
200
400
600
800
Time limi
Se ice le el
Sol ing ime [sec]
Bes -case Vl. PF
Bes -case B. PF
Bes -case Vh. PF
Basecase
Figu e 10: A e age numbe o delayed coils and dis ibu ion o sol ing imes in seconds pe se ice le el in he Bes -case ins ances and
Basecase scena io.
mos in luen ial ac o . Bu he da a om he wo o he HC
pa ame e iza ions suppo his claim. While he op imiza-
ions wi h Wo s -case and Basecase HC did s ill yield some
bu e y high solu ions, no op imiza ion was able o calcula e
a easible solu ion wi h he highes HC.
In gene al, he Wo s -case scena io seems o nega i ely
impac he numbe o success ul op imiza ions. As p e i-
ously desc ibed, he easons a e wo old. While all uns wi h
he lowes se ice le el ail due o in easibili y, op imiza ions
wi h a highe se ice le el ail because o he sol ing ime
limi , wi h wo excep ions wi h a medium se ice le el and
Ve y high HC. This demons a es he g ea impac o he
Wo s -case and especially o he HC on he easibili y o he
model. Be o e inalizing he second pa o his nume ical
s udy, one las es was conduc ed o analyze which o he
wo ac o s, HC and PF, impac he solu ion he mos . To ac-
complish his he case in able 10 was de ined.
Table 10: Case pa ame e iza ion o iden i y he mos impac ul
ac o .
Case HC UC PF SPT
Impac Ve y low Basecase Ve y low Basecase
The igu e 12 displays he di e ences in s inge s and de-
layed coils be ween he Basecase scena io and he Impac
case, by sub ac ing he a e age numbe o s inge s and de-
layed coils in he Impac case om he Basecase scena io.
I can be concluded, ha he PF has a g ea e impac
on hese aspec s o he solu ion han he HC. This obse a-
ion was con i med by a second Impac case, which used he
Basecase scena io and Ve y high PF and HC. Bu i has o be
no ed ha his only applies o he de ini ions o he di e en
le els o HC and PF by Wegel e al. (2024). Bo h ac o s indi-
ca e a s ong in luence on he cha ac e is ics o he solu ion
and should he e o e be op imized o achie e op imal solu-
ions in eal-wo ld applica ions. A e s udying he d as ic
e ec s o he Bes - and Wo s -case, di e en pa ame e iza-
ions o he model will be explo ed in he ollowing sec ion.
4.2.4. Al e a ion o p ocessing lines and due da es
Fi s , he numbe o p ocessing lines in he model was
al e ed. To ecapi ula e, in he i s case, an addi ional line
was added o he model, which can p ocess e e y coil. In
he second case, Remo al I, he p ocessing line ha can only
p ocess wide lines was emo ed, while he p ocessing line
ha can only p ocess na owe coils was emo ed in case
h ee. The line ha can p ocess e e y coil was no emo ed
because i would lead o in easibili y since some coils could
no be p ocessed.
Fo emos , none o he h ee cases displayed in igu e 13
seems o ha e a g ea impac on he numbe o in oduced
s inge s. As expec ed, he numbe o s inge s dec eased in
he addi ion case and inc eased in bo h emo al cases, bu
all hese e ec s we e only mino . The eason he addi ion o
one p ocessing line gene ally educes he numbe o s inge s
is ha an incompa ible coil o a sequence o coils, ha is in-
compa ible wi h ano he sequence o coils, can be p ocessed
on he addi ional line, hus educing he numbe o s inge s
by one.

H. A. Höne loh /Junio Managemen Science 10(3) (2025) 781-809 801
Low Medium High
0
5
10
15
20
25
Se ice le el
# s inge s
Wo s -case Vl. HC
Wo s -case B. HC
Basecase
Low Medium High 0
2
4
6
8
10
Se ice le el
# ailed uns
Wo s -case - Vl. HC
Wo s -case - B. HC
Wo s -case Vh. HC
Basecase
Figu e 11: A e age numbe o in oduced s inge s and ailed uns pe se ice le el in he Wo s -case ins ances and Basecase.
Low Medium High
0
1
2
3
Se ice le el
# s inge s
Low Medium High 0
1
2
3
Se ice le el
# delayed coils
Figu e 12: Di e ences in s inge s and delayed coils be ween he Impac case and Basecase pe se ice le el.
To gi e an example, ou sequences o coils ha e o be
scheduled, wi h each coil in a sequence being compa ible
wi h e e y o he coil in he same sequence. Howe e , he
sequences a e incompa ible wi h each o he . I hese ou se-
quences a e p ocessed on one p ocessing line, h ee s inge s
would ha e o be in oduced o ensu e compa ibili y. I a
new p ocessing line would be added, one o he sequences
could be p ocessed on he new line, educing he numbe o
s inge s o wo, bu no hing mo e could be done o educe
he numbe o s inge s u he .
The e o e, he maximum bene i o adding a p ocessing
line, in e ms o s inge use, is he educ ion o in oduced
s inge s by one. This bene i could inc ease i he schedule
in oduced s inge s o no iola e he se ice le el, bu his
was no he case o hese op imiza ions.
The eason he da a does no show a educ ion o he a -
e age in oduced s inge s by one is ha wo op imiza ions,
ha we e unsuccess ul in he Basecase scena io, we e suc-
cess ully sol ed in he addi ion case and ha e a s inge coun
o six, hus inc easing he a e age numbe o s inge s. In
ac , all h ee cases had only one ailed op imiza ion each
wi h a low se ice le el. The explana ion o his educ ion
could be he sol ing i sel since he sol e beha es di e en ly
e en i only one pa ame e changed.55 Because he Addi ion
case and he Remo al cases ha e opposi e e ec s, he addi-
ion o a p ocessing line inc eases he numbe o easible so-
lu ions and he emo al dec eases hem, i is he mos likely
explana ion.
The emo al o a p ocessing line in luences he numbe
o s inge s in he same way as an addi ion o a line does
bu in e e se. Thus, i no s inge s a e in oduced o no
iola e he se ice le el, he maximum inc ease in s inge s
is one. This is ue o e e y op imiza ion wi h a MIPGap
o ze o %. Howe e , some op imiza ions wi h a MIPGap o
ze o % did no su e any inc ease in s inge use, sugges ing
ha hey did no u ilize he line in he schedule. Bu some
op imiza ions wi h a MIPGap o o e ze o % had an inc ease
o up o h ee s inge s. Hence, he op imiza ions ei he had
55 Mil enbe ge , 2023b
H. A. Höne loh /Junio Managemen Science 10(3) (2025) 781-809802
Low Medium High
0
2
4
6
Se ice le el
# s inge s
Addi ion
Remo al I
Remo al II
Basecase
Figu e 13: A e age numbe o s inge s in he h ee line-al e ing
cases and Basecase scena io pe se ice le el.
o in oduce s inge s o comply wi h he se ice le el and he
sol e was unable o p o e he op imali y o he solu ion in
ime o he sol e simply could no ind he op imal easible
solu ion un il i had o e mina e he sol ing p ocess. This
e mina ion happened mo e o en wi h he addi ion o one
line and is p esen ed in igu e 14.
As po ayed, he majo i y o op imiza ions wi h an addi-
ional p ocessing line and a low se ice le el had o be e -
mina ed due o he ime limi . This esul s in a highe MIP-
Gap, as men ioned be o e. Highe sol ing imes and MIP-
Gaps han in he Basecase scena io a e also obse ed wi h
highe se ice le els bu a e no as se e e wi h he highes
one. This o e all end p obably esul s om he inc easing
model complexi y, which could also be obse ed in igu e 6
om he i s pa o his nume ical s udy. In his case, how-
e e , he inc ease esul s om he addi ional line and no
om addi ional coils. The ad e se e ec can be examined
in igu e 15.
The da a indica es a dec easing sol ing ime h ough
complexi y educ ion when applying a medium o high se -
ice le el, bu no wi h a low one. A possible explana ion
could be, ha he es ic ions caused by he emo al o a
p ocessing line in combina ion wi h he low se ice le el
esul in many o me solu ions being in easible and hus a
longe sea ch o a easible solu ion. Tha Remo al II dis-
plays lowe sol ing imes han Remo al I could sugges ha
he line which can only p ocess wide coils and was emo ed
in Remo al I, was u ilized o a highe deg ee han he line
ha was emo ed in Remo al II. The eason o his is he
pa ially iangula dis ibu ion o he coil wid h due o he
Basecase le el o he HC, which esul s in mo e coils being
wide han na owe . Hence, he e ec o he emo al o one
line depends on he HC se ing.
The las pa ame e o in es iga e is a diness, which indi-
ca es o no be se e ely a ec ed by he addi ion o emo al
o a p ocessing line. Tha he numbe o delays only com-
plies wi h he se ice le el and does no ha e o be mini-
mized could explain why a diness did no dec ease wi h an
addi ional line. Addi ionally, no inc ease in a diness sug-
ges s ha mos op imiza ions in Remo al I and Remo al II
had enough scheduling lexibili y o ind easible solu ions
wi h simila a diness, e en wi h only wo lines.
In con as , a diness was he mos impo an ac o in
he nex pa o he nume ical s udy, in which he ime ho i-
zon o he due da es was mo ed close o he beginning
o he schedule. These ea lie due da es a e expec ed o
ha e a ce ain e ec on he numbe o s inge s and delayed
coils. Wi h ea lie due da es and an imposed se ice le el,
he schedule has o implemen mo e s inge s o no exceed
he allowed a diness. This u he educes he scheduling
lexibili y due o he SPT and hus inc eases he di icul y o
inishing he p ocessing o o he coils be o e hei due da e.
The e o e, an inc eased numbe o ailed op imiza ions and
a highe s inge and a diness coun in success ul uns we e
expec ed.
The da a displayed in igu e 16 shows an inc easing num-
be o ailed op imiza ions, which aligns wi h he expec a-
ions. In case 100 %, mos op imiza ions a e e mina ed and
do no ail because o in easibili y, sugges ing ha hey could
be sol ed wi h a highe ime limi . Bu his changes wi h case
50 %, in which no op imiza ion wi h a low se ice le el is ea-
sible, while mos unsuccess ul op imiza ions wi h a highe
se ice le el a e e mina ed. By educing he alues o aand
bby 75 %, almos all op imiza ions wi h a low and medium
se ice le el a e in easible, bu only one op imiza ion wi h
a high se ice le el. The e o e, p o ided wi h a high se -
ice le el and a highe ime limi , mos op imiza ions could
s ill yield easible solu ions. Howe e , he necessa y sol ing
ime and he quali y o he esul ing solu ion canno be de-
e mined wi hou u he s udies.
The second expec a ion was a ise in he numbe o in o-
duced s inge s. Due o he e y low numbe o solu ions, he
ollowing in e p e a ions could be hea ily in luenced by he
selec i i y e ec . Because o his, only he mos impo an in-
o ma ion will be p esen ed in able 11 and b ie ly discussed.
I a case is no included in a se ice le el, hen no op imiza-
ions o ha case and se ice le el yield a easible solu ion.
As indica ed by he da a, he numbe o in oduced
s inge s inc eases h ough he shi o he due da es. In
addi ion, almos all op imiza ions comple ely u ilize he se -
ice le el. Bo h o hese obse a ions ep esen an inc ease
in compa ison o he Basecase scena io and align wi h he
expec ed esul s, hus concluding he hi d pa o he nu-
me ical s udy.
4.2.5. Nume ical s udies on he ex ended models
The las pa o he nume ical s udy consis s o se e al
es s conduc ed on he ex ended models desc ibed in sec-
ions 3.4.2 and 3.4.3. Bo h addi ionally inco po a e he
elease da es o coils om sec ion 3.4.1. The i s ex ended
model minimizes s inge s and absolu e a diness. Fi s ,
he a e age numbe o s inge s and delayed coils in he
H. A. Höne loh /Junio Managemen Science 10(3) (2025) 781-809 803
Low Medium High
0
200
400
600
800
Time limi
Se ice le el
Sol ing ime [sec]
Addi ion
Basecase
Low Medium High 0
5
10
15
20
Se ice le el
MIPGap [%]
Addi ion
Basecase
Figu e 14: Dis ibu ion o sol ing imes in seconds and a e age MIPGaps in % pe se ice le el in he Addi ion case and Basecase scena io.
Low Medium High
0
200
400
600
800
Time limi
Se ice le el
Sol ing ime [sec]
Remo al I
Remo al II
Basecase
Figu e 15: Dis ibu ion o sol ing imes in seconds pe se ice
le el in he Remo al I and II cases, as well as in he Basecase
scena io.
Basecase scena io wi h Ve y low and Ve y high UC will be
examined.
Figu e 17 depic s an expec ed end. The numbe o de-
layed coils is he highes i i has only a weigh o 20 % o he
objec i e alue, which has o be minimized. Addi ionally, he
numbe o s inge s is he lowes a his poin , since i has a
weigh o 80 %. Thus, he op imal solu ions seem o delay
some coils o minimize he numbe o s inge s, since i has
a much highe weigh . This end changes, wi h a diness
being educed a he cos o s inge s wi h highe weigh s o
a diness. As po ayed wi h Ve y low UC, he numbe o
delayed coils is, e en a i s highes poin , e y low. Thus,
he ade-o be ween s inge s and a diness canno be ob-
Low Medium High
0
2
4
6
8
10
Se ice le el
# ailed uns
100 %
75 %
50 %
25 %
Figu e 16: Numbe o ailed uns pe se ice le el in he ou
di e en due da e al e ing cases.
se ed as well as wi h he Ve y high UC on he igh side o
igu e 17.
In his scena io, an almos one- o-one ade-o be ween
he numbe o delayed coils and s inge s can be obse ed
wi h he inc easing weigh o he o me . The mechanism be-
hind he ade-o is simila o he one discussed in he hi d
pa o he s udy, ega ding he addi ion and emo al o p o-
cessing lines. I he a diness has almos negligible weigh ,
he coils a e sequenced in a way ha he numbe o s inge s
is minimized wi h no ega d o he numbe o delayed coils.
Bu i he a diness weigh inc eases, he objec i e alue may
bene i om in oducing s inge s o inish he p ocessing o
H. A. Höne loh /Junio Managemen Science 10(3) (2025) 781-809804
80:20 65:35 50:50 35:65 20:80
0
1
2
3
4
5
6
Weigh a io S inge :Ta diness
# s inge s
Ex ended model I - Vl. UC
80:20 65:35 50:50 35:65 20:80 0
0.1
0.2
0.3
0.4
0.5
0.6
# delayed coils
# delayed coils
# s inge s
80:20 65:35 50:50 35:65 20:80
0
2
4
6
8
Weigh a io S inge :Ta diness
# s inge s
Ex ended model I - Vh. UC
80:20 65:35 50:50 35:65 20:80 0
2
4
6
8
# delayed coils
# delayed coils
# s inge s
Figu e 17: A e age numbe s o delayed coils and s inge s wi h di e en UCs and weigh s in he i s ex ended model.
Table 11: S inge use and a diness in he cases 100 %, 75 % and
50 %.
Se ice
le el
Case # s inge s # delayed
coils
Low
100 % 6 0
Medium
100 % 6.667 4
75 % 7 4
50 % 7 4
High
100 % 5 7.83
75 % 5.75 8
50 % 7 8
coils be o e hei due da e, hus dec easing he numbe o
delayed coils a he cos o s inge s. By u he inc easing
he a diness weigh , mo e s inge s a e in oduced o com-
ply wi h due da es and he e o e o minimize he objec i e
alue. Wi h ea lie due da es, his end is be e o obse e
since i is ha de o comply wi h hem, which inc eases he
numbe o s inge s ha ha e o be in oduced o dec ease
he a diness by one uni and ice e sa.
Due o he di e en op imiza ion objec i es, he compa-
abili y be ween he ex ended model and he base model is
limi ed. Bu he esul s o he ex ended model wi h a a di-
ness weigh o 20 % a e simila o he esul s yielded wi h
a medium se ice le el in he base model, p esen ed in ig-
u e 3. The di e ence, howe e , is he UC, which is highe in
he ex ended model and hus highligh s i s pe o mance o
minimize s inge use and a diness a he same ime. Bu
his high pe o mance comes wi h he cos o inc eased sol -
ing ime, po ayed in igu e 18.
Gene ally, he e is a wide dispa i y in sol ing ime be-
ween di e en op imiza ions in he ex ended model, ega d-
less o he weigh a io, wi h mos median sol ing imes being
highe han ha o he Basecase scena io wi h a low se ice
le el.
The highes median sol ing imes and e mina ions in he
ex ended model can be obse ed wi h he wo highe s inge
weigh s o 80 % and 65 %. The median sol ing imes o he
o he ins ances a e signi ican ly lowe , which could sugges
ha he minimiza ion o s inge s is mo e complex han ha
o delayed coils. Bu his canno explain, why he median o
he sol ing ime inc eased by inc easing he a diness weigh
u he o 65 %. This e ec mos ly s ems om one op imiza-
ion, which su e ed an inc ease o 510 seconds in sol ing
ime om 210 seconds o 720 seconds. Wi hou his un,
he median would be 283 seconds, which is simila o he
ins ances wi h 50 % and 80 % a diness weigh and could
indica e ha his op imiza ion is an ou lie .
The sol ing imes o he ex ended model wi h a Ve y
high UC a e no depic ed, since all o hem had o be e -
mina ed due o he ime limi . None heless, e e y op imiza-
ion yielded a easible solu ion, which is a d as ic inc ease
o e he numbe o ailed uns in case 100 % o he hi d
pa o he nume ical s udy, displayed in igu e 16. Since
e e y esul wi h he ex ended model and a Ve y high UC
also had a low ela i e a diness and could he e o e com-
ply wi h he medium and high se ice le els implemen ed in
he base model a almos e e y weigh a io, i sugges s ha
i could be mo e pe o man o minimize bo h s inge s and
a diness han o implemen a se ice limi and only minimize