Integrated storage assignment for an E-grocery fulfilment centre: accounting for day-of-week demand patterns

Winkelmann, Da id; Tolkmi , F ede ik; Ul ich, Ma hias; Röme , Michael
A icle — Published Ve sion
In eg a ed s o age assignmen o an E-g oce y ul ilmen
cen e: accoun ing o day-o -week demand pa e ns
Flexible Se ices and Manu ac u ing Jou nal
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Winkelmann, Da id; Tolkmi , F ede ik; Ul ich, Ma hias; Röme , Michael (2024) :
In eg a ed s o age assignmen o an E-g oce y ul ilmen cen e: accoun ing o day-o -week
demand pa e ns, Flexible Se ices and Manu ac u ing Jou nal, ISSN 1936-6590, Sp inge US, New
Yo k, NY, Vol. 37, Iss. 2, pp. 558-598,
h ps://doi.o g/10.1007/s10696-024-09549-7
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/323317
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Flexible Se ices and Manu ac u ing Jou nal (2025) 37:558–598
h ps://doi.o g/10.1007/s10696-024-09549-7
In eg a ed s o age assignmen o anE‑g oce y ul ilmen
cen e: accoun ing o day‑o ‑week demand pa e ns
Da idWinkelmann1 · F ede ikTolkmi 1· Ma hiasUl ich1· MichaelRöme 1
Accep ed: 15 May 2024 / Published online: 20 June 2024
© The Au ho (s) 2024
Abs ac
In his pape , we add ess a s o age assignmen p oblem a ising in a ul ilmen cen-
e o a majo Eu opean e-g oce y e aile . The cen e can be cha ac e ised as a
hyb id wa ehouse, consis ing o a highly e icien and pa ially au oma ed as -pick-
ing a ea designed as a pick-and-pass sys em wi h mul iple s a ions, and a picke -
o-pa s a ea. The s o age assignmen p oblem in ol es he decisions o selec ing
p oduc s o be alloca ed o he as -picking a ea, assigning hese p oduc s o pick-
ing s a ions, and de e mining he speci ic shel es wi hin he designa ed s a ion. The
objec i e is o achie e high picking e iciency while main aining balanced wo k-
loads ac oss s a ions and espec ing p ecedence o de cons ain s. We o mula e his
h ee-le el p oblem using an in eg a ed mixed-in ege linea p og amming (MILP)
model. Compu a ional expe imen s wi h eal-wo ld da a demons a e ha ou in e-
g a ed app oach yields signi ican ly be e esul s han a sequen ial app oach, whe e
he selec ion o p oduc s o be included in he as -picking a ea is pe o med be o e
assigning s a ions and shel es. To enhance compu a ional e iciency, we p opose a
heu is ic solu ion app oach ha ixes SKUs o shel es, allowing us o ind be e
solu ions in sho e un imes compa ed o di ec ly sol ing he MILP model. Addi-
ionally, we ex end he in eg a ed s o age assignmen model o explici ly accoun o
wi hin-week demand a ia ion. In a se o expe imen s wi h day-o -week-dependen
demands, we show ha while a s o age assignmen based on a e age demand igu es
can lead o highly imbalanced wo kloads on ce ain days, he augmen ed model p o-
ides well-balanced s o age assignmen s o each day-o -week wi hou comp omis-
ing he solu ion quali y in e ms o picking e iciency. The bene i s o accoun ing
o demand a ia ion a e u he demons a ed h ough a simula ion-based analysis
using sampled weekly da a.
Keywo ds Re ailing· E-g oce y· S o age assignmen · Demand a ia ion·
Simula ion
Ex ended au ho in o ma ion a ailable on he las page o he a icle
559
In eg a ed s o age assignmen o anE‑g oce y ul ilmen …
1 In oduc ion
In e-g oce y e ailing, g oce y p oduc s a e o de ed online and deli e ed di ec ly a
a da e and ime chosen by he cus ome . In ecen yea s, he e-g oce y business has
expe ienced a g ow h a e in sales o 18.4% in he US in 2023 and is expec ed o
become he la ges ca ego y wi hin e-comme ce un il 2026 (D oesch 2024).
Many key playe s in he e-g oce y business a e omnichannel g oce s, ha ing hei
oo s in adi ional b ick-and-mo a e ailing. Wollenbu g e al. (2018) p o ide a
e iew o he ansi ion om b ick-and-mo a o a b ick-and-clicks g oce y e ailing
and he implica ions o unde lying logis ics ne wo ks. Ini ially, g oce s s a ed hei
e-g oce y business using in-s o e picking, whe e online o de s a e picked in exis -
ing b ick-and-mo a s o es close o he cus ome and hen deli e ed. Al hough his
pick s a egy is s ill used in u al egions, i is no sui able o handle he inc easing
olume o e-g oce y pu chases in la ge ci ies o me opoli an a eas. Consequen ly,
o inc ease he e iciency o he picking p ocess, which, acco ding o ou business
pa ne , accoun s o a subs an ial sha e o o al wa ehousing cos s, majo e-g oce y
e aile s ha e es ablished dedica ed wa ehouses, so-called ul ilmen cen es o da k
s o es, solely o picking e-g oce y o de s. While Hübne and Kuhn (2023) de elop
a model o shel space managemen in ligh o eal-wo ld eplenishmen p ocesses
in g oce y e ailing, esea ch on e-g oce y wa ehousing is s ill limi ed.
Wa ehousing is a key challenge o almos all e aile s (Gu e al. 2007). A sui able
wa ehouse con igu a ion depends on he asso men o he e aile , he cha ac e is-
ics o s ock keeping uni s (SKUs), as well as cus ome expec a ions, such as e y
high se ice le els o 97–99% (Ul ich e al. 2021) and sho deli e y imes, wi h
some e aile s e en o e ing same-day deli e y. In e-g oce y, mos e aile s o e an
asso men o abou 12,000 o 15,000 SKUs, some o which equi e special s o -
age condi ions like e ige a ion. Fo ou business pa ne , an a e age o de includes
abou 30 o 40 di e en SKUs (o de lines).1 While he need o sho deli e y imes
also a ises in classical (non-g oce y) e-comme ce, he numbe o o de lines in a
cus ome o de is qui e small— o example, Boysen e al. (2019) no e ha each
o de a Amazon Ge many comp ises 1.6 lines. This di e ence in lines pe o de
signi ican ly impac s wa ehouse design and ope a ion, as highligh ed by he ac ha
he ecen e iew on wa ehousing o e-comme ce by Boysen e al. (2019) explici ly
excludes he e-g oce y business.
In his pape , we add ess scien i ic decision suppo o a s o age assignmen
p oblem a ising in a ul ilmen cen e o a majo Eu opean e-g oce y e aile . This
e aile ope a es a ious ul ilmen cen es wi h di e en designs and deg ees o
au oma ion. The mos ecen ly es ablished cen e can be cha ac e ised as a hyb id
o pa allel wa ehouse, whe e a pa o he asso men is alloca ed o a adi ional
picke - o-pa s a ea, and he o he SKUs a e alloca ed o a pa ially au oma ed as -
pick a ea. In his a ea, boxes sequen ially mo e be ween s a ions wi hin a so-called
1 The la ge numbe o o de lines in g oce y e ailing is also emphasised by p e ious li e a u e, see e.g.
Fe nie e al. (2010). Addi ionally, Ul ich e al. (2021) s a e ha an a e age shopping baske in e-g oce y
e ailing has a alue o abou 80 o 90€.
560
D.Winkelmann e al.
picking loop2. A each s a ion, a picke pulls he SKUs o a gi en cus ome om a
shel and places hem in o he box. This con igu a ion can be classi ied as a pick-
and-pass sys em (Chia Jane 2000; Pan and Wu 2009).
Fo his ul ilmen cen e, we conside an in eg a ed s o age assignmen p oblem.
The e aile ’s goal is o assign SKUs o speci ic shel es wi hin designa ed s a ions o
he picking loop. Speci ically, his decision can be di ided in o h ee (hie a chically
ela ed) sub-decisions: The i s decision is o de e mine he subse o SKUs o be
handled in he picking loop, conside ing he limi ed s o age capaci y. The second
decision is o assign each SKU o a s a ion wi hin he picking loop o balance he
wo kload. The hi d decision is o assign SKUs o speci ic shel es wi hin he co -
esponding s a ion, wi h he aim o placing SKUs wi h a high numbe o picks close
o he picke . The o e all objec i e associa ed wi h hese decisions is o achie e a
high le el o ope a ional e iciency wi hin he wa ehouse. Addi ionally, as is ypical
o he e ail business (see e.g. T indade e al. 2022), he s o age assignmen mus
espec equi emen s such as main aining space be ween SKUs loca ed nex o each
o he and adhe ing o p ecedence o de cons ain s o ensu e ha hea y SKUs do no
damage agile ones in an o de box.
As obse ed by Boysen e al. (2019) and o he au ho s, demand a ia ion is a key
challenge in high-pe o mance e ail wa ehouses. I he demand o SKUs changes
due o seasonal impac s o long- e m ends, main aining a high le el o picking e i-
ciency ypically equi es adap ing he s o age assignmen by ea anging he s o age
loca ions o he SKUs. Howe e , o sho - e m demand a ia ions, such as day-o -
week-dependen demand o ce ain SKUs, ea anging SKUs is o en no possible
o p ac ical—as an example, his is he case in he e-g oce y ul ilmen cen e con-
side ed in his pape . To mi iga e he nega i e impac o such sho - e m demand
a ia ions, we p opose de e mining a a ia ion-awa e s o age assignmen , ha is,
a s o age assignmen ha pe o ms well ac oss mul iple demand scena ios, pa icu-
la ly o day-o -week-dependen SKU demands. The con ibu ions o his pape can
be s a ed as ollows:
We add ess a new h ee-le el s o age assignmen p oblem a ising in an e-g oce y
ul ilmen cen e wi h a pick-and-pass sys em o as o de picking. By doing so, we
con ibu e o he li e a u e on e-g oce y wa ehouse logis ics, which is ela i ely lim-
i ed compa ed o he ex ensi e body o esea ch dealing wi h non-g oce y e-com-
me ce and b ick-and-mo a wa ehousing (see Boysen e al. 2019, 2021).
We o mula e his h ee-le el p oblem as an in eg a ed Mixed-In ege Linea P o-
g amming (MILP) model and p o ide a heu is ic solu ion app oach based on i e a-
i e a iable ixing. In a se o expe imen s using eal-wo ld da a p o ided by a lead-
ing Eu opean e-g oce y e aile , we demons a e ha sol ing his in eg a ed model
is clea ly supe io o a s anda d sequen ial app oach, whe e he selec ion o SKUs
o be included in he as -pick a ea is made be o e aking zone/s a ion assignmen
decisions. Fu he mo e, we epo esul s om a sensi i i y analysis using simula ed
2 In he li e a u e, his ype o picking sys em is e e ed o as p og essi e zoning see e.g. he e iew on
wa ehouse o de picking by De Kos e e al. (2007).
561
In eg a ed s o age assignmen o anE‑g oce y ul ilmen …
da a o demons a e he e iciency and applicabili y o ou app oach o a b oade
ange o business cases.
Finally, o cope wi h day-o -week-dependen demand luc ua ions, we p opose
sol ing an augmen ed MILP model ha explici ly aims o ind a s o age assign-
men o he pick-and-pass sys em ha pe o ms well o each day o week. Using
eal-wo ld da a, we show ha while an assignmen based on a e age demands leads
o subs an ially imbalanced s a ion wo kloads on ce ain days, he a ia ion-awa e
solu ion main ains balance on each day o week, almos wi hou comp omising he
s o age assignmen objec i e. We u he demons a e he use ulness o he a ia-
ion-awa e app oach h ough a simula ion-based analysis wi h andomly sampled
demand da a o each week o he yea .
The emainde o he pape is s uc u ed as ollows: In he nex sec ion, we in o-
duce he business case and he eal-wo ld da a se conside ed in ou s udy. Sec ion3
p o ides a e iew o ela ed li e a u e, ollowed by Sec ion4, which co e s ou in e-
g a ed s o age assignmen model, compu a ional expe imen s, and he heu is ic solu-
ion app oach. In Sec ion5, we de elop a model ex ension ha accoun s o a ying
demand pa e ns wi h espec o days o week o he SKUs, which is e alua ed using
bo h eal-wo ld da a and a simula ion s udy. Finally, we summa ise ou majo ind-
ings in he Conclusion.
2 Desc ip ion o  helogis ic p ocesses anda ailable da a
The e-g oce y e aile analysed in his pape p ima ily ope a es wi h a wo-s age
dis ibu ion p ocess. In he i s s ep, SKUs a e supplied om na ional dis ibu ion
wa ehouses o local ul ilmen cen es. These supplies ypically occu on each wo k-
ing day om Monday o Sa u day. Upon a i al a a ul ilmen cen e, all SKUs a e
s o ed on hei alloca ed shel es. In he second s ep, cus ome pu chases a e ul illed
by hese local cen es based on he cus ome ’s loca ion. Mos o de s a e placed by
cus ome s a leas one day in ad ance. Al hough he e aile allows same-day o de s
o a limi ed ex en , he e is a ime lag be o e he deli e y om he wa ehouse occu s.
Addi ionally, he numbe o cus ome o de s is es ic ed by he a ailabili y o deli -
e y ime slo s. This allows he e aile o synch onise mos o de s o a ce ain day,
leading o he easonable assump ion o a cons an pick a e wi hin ou model. In he
ollowing, we desc ibe he logis ic p ocesses wi hin he ul ilmen cen e and de ail
he SKU da a. Finally, we p esen his o ical picking da a om he e aile , highligh -
ing ecu ing pa e ns depending on he day o week.
2.1 The p ocess o o de picking
In mos ul ilmen cen es, he e aile ope a es wi h a adi ional picke - o-pa s
sys em. To imp o e ope a ional e iciency, educe ope a ion imes, and inc ease
he numbe o pu chases se ed wi hin a day, he e aile has in oduced highe
le els o au oma ion in ce ain ul ilmen cen es. While a ully au oma ed picking
p ocess is cos -in ensi e, his pape conside s a pa ially au oma ed picking loop

562
D.Winkelmann e al.
wi hin a hyb id wa ehousing sys em es ablished in one o he e aile ’s ul ilmen
cen es. This hyb id sys em consis s o wo s o age a eas: (1) a pa ially
au oma ed picking loop and (2) a adi ional picke - o-pa s a ea. Al hough he
ope a ional e iciency is highe in he i s a ea, i s a ailable s o age space is
limi ed. Consequen ly, he e aile mus decide which SKUs should be included
in he picking loop and which should emain in he picke - o-pa s a ea. Gi en
ha an a e age cus ome o de comp ises abou 30 di e en SKUs, in gene al, no
o de can be comple ed by only one o he s o age a eas. Ins ead, assembling all
SKUs o a single pu chase usually equi es wo independen picking p ocesses in
bo h a eas. While he e is comp ehensi e li e a u e on adi ional picke - o-pa s
a eas (see e.g. Ca on e al. 1998; F anzke e al. 2017), he majo i y o picks in he
wa ehouse o he business pa ne unde conside a ion a e pe o med wi hin he
picking loop. A he same ime, op imising he picking loop is mo e complex due
o he exis ence o di e en s a ions. The e o e, his pape ocuses on op imising
he picking p ocess wi hin he mo e c ucial pick-and-pass a ea.
The picking loop consis s o eigh picking s a ions, wi h boxes sequen ially
isi ing he s a ions. Each box co esponds o one cus ome pu chase, and a
each s a ion, he picke e ie es he SKUs o ha pu chase om he shel es and
places hem in o he box. Once all SKUs o a speci ic cus ome pu chase a e
placed in o he box, i exi s he loop and he pu chase is loaded in o a ehicle o
deli e y. Figu e1 p o ides a schema ic ske ch o he picking loop.
Fig. 1 Rep esen a ion o he
picking loop
Fig. 2 Rep esen a ion o he
s uc u e o picking s a ions
563
In eg a ed s o age assignmen o anE‑g oce y ul ilmen …
Figu e2 illus a es he s uc u e o a ypical s a ion wi hin he picking loop. Each
s a ion consis s o six acks: ou in on o he picke ( wo ou e and wo inne ) and
wo behind he picke . The acks in on o he picke con ain ou shel es, each
wi h a heigh o 250mm ( ype 1), while he acks behind he picke con ain ou
shel es, each wi h a heigh o 450mm ( ype 2). In o al, he e a e 192 shel es wi hin
he en i e picking loop: 128 o ype 1 and 64 o ype 2. The s uc u e o shel es is
ep esen ed in Fig.13 in he Appendix. No e ha some SKUs can only be alloca ed
o ype 2 shel es due o hei indi idual heigh .
To a oid conges ion wi hin he picking loop and idle imes a some s a ions, he
e aile aims o balance he p ocessing ime and wo kload o picke s ac oss all s a-
ions. Fo a gi en o de , he ime spen a a s a ion depends on he numbe o picks
and he shel loca ions o he picked SKUs wi hin he acks. While i is easy o pull
SKUs om a shel a ace le el and in on o he picke , he picking p ocess is
mo e ime-in ensi e o SKUs loca ed on he op o bo om shel es o a ack, as
well as on shel es in he acks behind he picke . The e o e, in addi ion o deciding
which SKUs o alloca e o he picking loop, he e aile needs o assign each SKU
o a speci ic s a ion and shel , conside ing he goals men ioned abo e. In con as o
b ick-and-mo a e ailing, whe e SKU alloca ion o shel es also akes in o accoun
ma ke ing aspec s (c . Sigu dsson e al. 2009), he online e ailing se ing in da k
s o es allows he company he lexibili y o decide on SKU placemen based solely
on e iciency- ela ed objec i es. Howe e , he e aile mus conside addi ional con-
s ain s implici ly aken in o accoun by cus ome s in b ick-and-mo a e ailing,
such as placing la ge and hea y SKUs in o he box i s o mi iga e he isk o dam-
aging agile i ems. To a oid he e gonomic bu den o picking hea y i ems being
alloca ed o jus a ew picke s, he picke s o a e be ween s a ions h oughou he
day. This o a ion also educes he a iance in picking e iciency be ween s a ions
induced by human ouch and con ibu es o he plausibili y o he assump ion o a
cons an picking e iciency ac oss s a ions.
2.2 SKU da a
The da a se p o ided by he e-g oce y e aile co e s a o al o 4693 di e en SKUs.
I includes in o ma ion on he dimensions o each SKU, de e mining whe he he
SKU can be alloca ed o ype 2 shel es only o also o ype 1 shel es. Addi ionally,
each SKU is associa ed wi h a p ecedence o de ank, aking one o h ee alues: 1,
2, o 3. Rank 1 co esponds o hea y i ems which need o be alloca ed o an ea ly
s a ion, while ank 3 is used o agile i ems. All o he SKUs a e associa ed wi h
ank 2. Fu he mo e, each SKU has a a ge s ock based on expec ed cus ome
demand. This a ge , along wi h he size o he SKU, de e mines he amoun o
shel space ha needs o be alloca ed o he SKU. In ac , he a ge le el is a ec ed
by he eplenishmen cycle. Shel alloca ion mus also conside handling- ela ed
aspec s, such as he need o ese e space o a sepa a o i wo di e en SKUs a e
placed nex o each o he on a single shel . As men ioned p e iously, he a ailable
space wi hin he loop is insu icien o s o e all SKUs in he e aile ’s asso men .
564
D.Winkelmann e al.
The e o e, he decision on which SKUs a e included in he picking loop is based on
an impo ance sco e assigned o each SKU.
Relying on an impo ance sco e allows e aile s o include bo h quan i a i e and
quali a i e a iables in o he op imisa ion p ocess. While some ac o s a e ob ious,
such as he numbe o picks o space equi emen s de e mined by he olume o an
SKU, e aile s migh also wish o a ibu e highe impo ance o ce ain SKUs based
on his o ical da a o quali a i e human expe knowledge. Con e sely, SKUs wi h a
high alue and acing a highe isk o la ceny wi hin he picking loop compa ed o a
secu ed a ea ecei e a lowe sco e. The impo ance sco e enables a gene alisa ion o
ou app oach in e ms o a lexible, case-speci ic, o e en wa ehouse-speci ic op i-
misa ion o SKU alloca ion. In pa icula , his app oach can be gene alised o o he
e aile s, allowing hem o include addi ional in o ma ion based on ac o s such as
he shape o he wa ehouse (i.e. he picking loop and he picke - o-pa s a ea), he
le el o a ia ion in cus ome demand (e.g. due o seasonali y), o he weigh o an
SKU (e.g. i migh be mo e con enien o ca y hea y SKUs wi hin a box in he
picking loop a he han picking hem om he picke - o-pa s a ea). These con-
side a ions, while no di ec ly pa o he op imisa ion p oblem, can be implici ly
included h ough he impo ance sco e, enhancing he o e all e iciency and e ec-
i eness o he alloca ion p ocess.
In his pape , we conside he impo ance sco e as gi en and ely on he da a
p o ided by ou business pa ne . The e aile mainly bases he sco e alues on wo
dimensions: he space equi ed on he shel es, de e mined by he SKU olume,
and he a ge s ock le el. The olume o SKUs is no malised on a scale anging
om 0 o 1. Addi ionally, he numbe o o de lines o e ecen yea s ha include
his speci ic SKU is conside ed, wi h alues again no malised be ween 0 and 1.
Mul iplying bo h dimensions p o ides an impo ance sco e anging om 0 o 1,
whe e highe alues co espond o a highe impo ance o including his SKU in he
picking loop. The e migh be also some adjus men s o he impo ance sco e based
on conside a ions by human expe s ha a e no di ec ly quan i iable, as discussed in
he p e ious pa ag aph. No ably, we ind a high co ela ion be ween he impo ance
sco e and he numbe o picks o a speci ic SKU, wi h a co ela ion coe icien o
0
200
400
600
−15 −10 −5 0
Log Impo ance Sco e
F equency
(a) Log impo ance sco e
0
250
500
750
3579
Log Picks pe Yea
F equency
(b)Log numbe o picks
Fig. 3 His og ams o he log impo ance sco e and he log numbe o picks o he SKUs in he
asso men o he e aile app op ia e o he picking loop
565
In eg a ed s o age assignmen o anE‑g oce y ul ilmen …
0.718. The dis ibu ion o he impo ance sco e is s ongly posi i ely skewed, so
we illus a e he equency o he loga i hm o impo ance sco es o all SKUs in
Fig.3a. The log impo ance sco e is app oxima ely symme ic a ound i s mean o
−
7.81 wi h a s anda d de ia ion o 2.34. This implies ha only a small numbe o
SKUs ha e an impo ance sco e exceeding 0.1, while he sco e is ai ly small and
nea ly equal o he majo i y o SKUs. Due o he posi i e skewness o he o al
numbe o picks o he SKUs in he e aile ’s asso men app op ia e o inclusion in
he picking loop, we also show a his og am o he loga i hm o he numbe o picks
pe SKU in Fig.3b. This dis ibu ion is again oughly symme ic wi h a mean log
numbe o picks o abou 6.84. Fo mo e han 80% o he SKUs, he a e age numbe
o uni s pe o de line is a mos 2 (mean 1.70). This con i ms p io s a emen s by
Boysen e al. (2021) on he cha ac e is ics o e-g oce y pu chases. Fo some SKUs,
howe e , he a e age numbe o uni s pe o de line is la ge , wi h up o 13.92 uni s
(de ails can be seen in he boxplo in Fig.14 in he Appendix). The a ge s ock o
SKUs a ies ac oss he asso men . Mo e han 90% o he SKUs ha e a a ge s ock
o ewe han 20 uni s, wi h an a e age o 9.10 uni s, implying some lexibili y in
he assignmen due o he limi ed space needed o indi idual SKUs. Howe e , he
1% o SKUs wi h he highes a ge s ock ha e an a e age o 100.35 uni s, wi h a
maximum o 252 uni s. Addi ionally, he dimensions o 17.4% o he SKU equi e
alloca ion o ype 2 shel es.
2.3 His o ical picking da a
In addi ion o he cha ac e is ics o SKUs in oduced abo e, he da a se o he
e-g oce y e aile includes his o ical picking da a. This p o ides in o ma ion on he
a e age numbe o picks pe mon h o a speci ic day o week o SKUs wi hin he
e aile ’s asso men ha a e sui able o he picking loop, o he yea 2020. The
da a se includes he SKU ID, he day o week (wi h 1 co esponding o Monday and
6 o Sa u day), he mon h, and he co esponding a e age numbe o picks o each
Fig. 4 To al numbe o picks in
housands in he asso men o
he e aile app op ia e o he
picking loop depending on he
day o week (1 equals Monday,
6 Sa u day)
23.0
23.5
24.0
24.5
25.0
25.5
123456
Day o Week
To al Numbe o Picks in Thousands
572
D.Winkelmann e al.
a highe weigh should be placed on his sco e (i.e. choosing a la ge alue o
𝛼
).
Con e sely, i i is only some ule o humb o , e.g., highly co ela ed wi h he num-
be o picks, he impo ance sco e should no be o e alued (i.e. choosing a smalle
alue o
𝛼
). We will p o ide a discussion o he alue o
𝛼
in ou compu a ional
expe imen s.
Re aile s ypically aim o maximise p o i s. These p o i s a e in luenced by e e-
nues and cos s, such as hose o o de picking. While i is di icul o p ecisely quan-
i y he consequences o a ce ain s o age assignmen ega ding associa ed picking
cos s, ou model aims o educe hose cos s by enhancing ope a ional e iciency.
Nex , we p esen a MILP o mula ion o he in eg a ed p oblem. The p ima y
decision a iables in his o mula ion a e he bina y a iables
x ,
, which ake alue
1 i SKU is assigned o shel and 0 o he wise. The alues o hese a iables
de e mine he alues o he second se o a iables conside ed in he model: he
a iables
zk
, which ep esen he wo kload in e ms o he o al numbe o picks
assigned o s a ion k. Addi ionally, we in oduce he in ege a iable
yo
, which ep e-
sen s he las s a ion (i.e. he s a ion wi h he highes index k) o which an SKU wi h
p ecedence ank o is assigned. Gi en hese a iables and he pa ame e s in oduced
abo e, we can now p esen he MILP o mula ion o he in eg a ed p oblem6:
max 𝛼
𝛾1
∑
∈V
∑
∈R
s x ,
⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟
I
+
1−𝛼
𝛾2
∑
∈V
∑
∈R
1
d
p x ,
⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟
II
(1)
∑
∈R
x , ≤1∀ ∈V
(2)
k
x
,
≤y
o∀
o
∈
O
,
∈
V
o
,
∈
R
(3)
k x ,
≥
yo−1∀o∈O⧵{1}, ∈Vo, ∈R
(4)
z
k=
∑
∈V
∑
∈Rk
p x , ∀k∈
K
(5)
z
k≤(1+𝛿)⋅
1
|
K
|∑
l∈K
zl∀k∈K
(6)
z
k≥(1−𝛿)⋅
1
|
K
|∑
l∈K
zl∀k∈K
6 Fo a able con aining all no a ion used in he model, see Table6 in he Appendix.

573
In eg a ed s o age assignmen o anE‑g oce y ul ilmen …
The objec i e unc ion is a weigh ed combina ion o wo pa s: Pa I co esponds o
he maximisa ion o he o al impo ance sco e, while Pa II ep esen s he maximi-
sa ion o he a e age e iciency pe pick. To ensu e ha bo h pa s o he objec i e
unc ion, and consequen ly he o al objec i e alue, all wi hin he in e al [0,1], we
no malise he objec i e unc ion by di iding h ough
𝛾1
and
𝛾2
, espec i ely. He e,
𝛾1
co esponds o he objec i e alue when solely op imising he o al impo ance
sco e (Pa I o he objec i e unc ion), while
𝛾2
ep esen s he si ua ion whe e only
he picking e iciency is op imised. By adjus ing he pa ame e
𝛼
, he decision make
can con ol he ela i e impo ance o he wo objec i es.
Cons ain se (1) ensu es ha each SKU is assigned o a mos one shel in he
picking loop. Cons ain s (2) and (3) en o ce he p ecedence o de cons ain s: Con-
s ain se (2) equi es ha
yo
is a leas as la ge as he maximum s a ion index
k
o a
shel o which an SKU wi h o de ank o is assigned, and (3) ensu es ha all SKUs
wi h a p ecedence ank o o he han 1 a e assigned o a s a ion
k
≥
yo−1
. This means
hey a e ei he assigned o he las s a ion con aining an SKU wi h he nex smalle
ank o o a s a ion la e in he loop. Cons ain s (4)–(6) en o ce balanced wo kload
among he s a ions. Cons ain se (4) de e mines he alue o he auxilia y a iables
zk
, ep esen ing he o al numbe o picking ope a ions alloca ed o s a ion k. Using
his a iable, Cons ain s (5) and (6) ensu e ha he wo kload alloca ed o each s a-
ion espec s he maximum pe mi ed ela i e de ia ion om he a e age wo kload
among all s a ions. Cons ain s (7) ensu e ha he o al wid h o he SKUs assigned
o a shel plus he equi ed gaps be ween each pai o SKUs in a shel does no
exceed he wid h
w
o he shel . Finally, Cons ain s (8) and (9) en o ce he domains
o he a iables
x
,
and
yo
.
4.2 Compu a ional expe imen s
In his sec ion, we p esen he esul s o se e al expe imen s conduc ed wi h he
model desc ibed abo e, using eal-wo ld da a om he e-g oce y e aile conside ed
in his pape . In a i s se o expe imen s, we explo e he solu ion beha iou
conce ning he con e gence o he duali y gap, i.e. he ela i e di e ence be ween
a solu ion ound by he op imise and an uppe bound, o e ime. In addi ion, we
discuss he impac o he weigh ing ac o
𝛼
on he alues o he wo pa s o he
objec i e unc ion, conside ing a ixed ela i e de ia ion
𝛿
in he numbe o picks
be ween s a ions. This enables us o de e mine a ange o easonable alues o he
weigh ing ac o
𝛼
. Fu he mo e, we analyse he e ec o he allowed de ia ion
𝛿
be ween s a ions on he s uc u e o he solu ions. Finally, we compa e ou in eg a ed
h ee-le el s o age assignmen app oach o a sequen ial app oach, whe e we sol e
(7)
w
≥
∑
∈V
(w +g)⋅x , −g∀ ∈
R
(8)
x
,
∈{0, 1}∀ ∈V, ∈R
(9)
yo∈{1, …,|K|}∀o∈O
574
D.Winkelmann e al.
he a ea alloca ion p oblem (simila o he o wa d ese e p oblem), he assignmen
o s a ions, and he assignmen o selec ed SKUs o shel es consecu i ely. All
expe imen s we e conduc ed wi h he Gu obi op imise e sion 9.0.2 on a compu e
wi h 16 GB RAM and a AMD Ryzen™ 5 1600 3.2 GHz CPU.
4.2.1 Expe imen s on he un ime
In a i s analysis, we se an exempla y weigh ing ac o o
𝛼=0.5
and allow o
a de ia ion o picks be ween s a ions o
𝛿=1%
. Figu e6 depic s (a) he objec i e
alue and (b) he gap o he lowe bound o e a gi en un ime o up o 12h. We
highligh he esul ing alues a e one hou by he ed do ed lines. In his exempla y
se ing, a e he in ended un ime, a gap o 0.35% emains. Since we a e add essing
a ac ical p oblem o he e aile , ha is no egula ly sol ed, e en longe un imes
could be pe missible. Howe e , ou indings indica e slow p og ess in u he
educing he gap. Fo ins ance, e en a e an addi ional ou hou s o un ime, he
gap only diminishes by ano he 0.04 pe cen age poin s.
4.2.2 E ec o  heobjec i e weigh
˛
As p e iously in oduced, he objec i e unc ion comp ises wo pa s: he i s
(I) in ol es he sum o impo ance sco es o SKUs alloca ed o he picking loop
a ea, while he second (II) ela es o he picking e iciency in he picking loop. We
explo e he impac o di e en alues o he weigh ing ac o
𝛼
h ough a se ies o
expe imen s. Due o compu a ional cons ain s, we e mina e he op imisa ion when
ei he a gap o 0.5% o a p ede ined ime limi o 30min is eached, while limi ing
he ela i e de ia ion o picks be ween s a ions o
𝛿=1%
. Figu e7 p o ides an
o e iew o he alues o bo h pa s o he (no malised) objec i e unc ion ac oss
di e en alues o
𝛼
. Fo cla i y, we exclude he esul s o
𝛼=0
(sco e 0.912;
e iciency 0.997) and
𝛼=1
(sco e 0.999; e iciency 0.381). The le pa o he igu e
illus a es ha he no malised sum o impo ance sco es o SKUs assigned o he
98.6
98.8
99.0
99.2
99.4
99.6
99.8
0100 200 300 400 500 600 700
Run ime in Minu es
Objec i e Value
(a)Run ime s. objec i e alue
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0100 200300 400500 600700
Run ime in Minu es
Gap in Pe cen
(b)Run ime s. gap
Fig. 6 Rep esen a ion o he objec i e alue and gap o he lowe bound in pe cen depending on he
un ime in minu es o up o 12h using
𝛼=0.5
and
𝛿=1%
. The ed do ed line co esponds o a un ime
o 1h
575
In eg a ed s o age assignmen o anE‑g oce y ul ilmen …
0.990
0.993
0.996
0.999
0.00 0.25 0.50 0.75 1.00
Alpha
Rela i e Sco e
(a) Pa I(impo ance sco e)
0.984
0.988
0.992
0.996
0.00 0.25 0.50 0.75 1.00
Alpha
Rela i e Dis ance
(b)Pa II (pickinge iciency)
Fig. 7 No malised alues o Pa I (impo ance sco e) and Pa II (picking e iciency) o he objec i e
unc ion depending on he weigh ing ac o
𝛼
o
𝛿=1%
0
500
1000
1500
2000
2500
3000
010000 20000
Picks
Dis ance
alpha = 1
0
500
1000
1500
2000
2500
3000
010000 20000
Picks
Dis ance
alpha = 0.75
0
500
1000
1500
2000
2500
3000
010000 20000
Picks
Dis ance
alpha = 0.5
0
500
1000
1500
2000
2500
3000
010000 20000
Picks
Dis ance
alpha = 0.25
0
500
1000
1500
2000
2500
3000
0 10000 20000
Picks
Dis ance
alpha = 0
Fig. 8 Alloca ion o SKUs and co esponding picks o shel es wi h gi en dis ance o he picke o
di e en alues o
𝛼
and
𝛿=1%
. No e ha he igu e is limi ed o SKUs wi h a heigh o up o 250mm
576
D.Winkelmann e al.
picking loop achie es i s peak o
𝛼≥0.35
. Meanwhile, Pa II o he objec i e
unc ion emains ela i ely s able o
𝛼≤0.6
and declines o la ge alues o
𝛼
.7
Figu e8 p o ides u he insigh s in o he s uc u e o he solu ions by displaying
he numbe o picks o a gi en dis ance be ween he picke and he shel ac oss di -
e en alues
𝛼∈{0, 0.25, 0.5, 0.75, 1}
. SKUs wi h a heigh exceeding 250mm a e
excluded om hese plo s as hey can only be alloca ed o ype II shel es (see Fig.15
in he Appendix o hei alloca ion). Fo
𝛼=1
(i.e. a scena io whe e he dis ance
be ween he picke and he co esponding shel does no in luence he objec i e
alue), Fig.8 e eals a non-sys ema ic pa e n in he alloca ion o SKUs o shel es.
Con e sely, o
𝛼≤0.75
SKUs wi h a high numbe o picks end o be alloca ed o
shel es close o he picke , wi h only ma ginal changes obse ed o smalle al-
ues o
𝛼
. Howe e , he e emain some ou lie s in each scena io. Fo ins ance, in he
alloca ion o
𝛼=0.5
, ce ain SKUs wi h a high numbe o picks a e s ill alloca ed
o shel es wi h dis ances o 1900mm and 2850mm, espec i ely. These SKUs ypi-
cally ha e a la ge wid h, leading he model o p io i ise he alloca ion o mo e bu
smalle SKUs wi h a high numbe o picks o e hese SKUs o shel es close o he
picke .
Fo
𝛼=0
, whe e he ocus is solely on picking e iciency wi hou conside ing he
impo ance sco e, he e is a dec ease in he numbe o SKUs alloca ed o shel es
close o he picke , pa icula ly o SKUs wi h a small numbe o picks. Meanwhile,
he o al numbe o SKUs alloca ed o he picking loop inc eases by app oxima ely
20%, and he o al numbe o picks ises by 5–6%. Howe e , he o al impo ance
sco e dec eases by nea ly 10% compa ed o o he alues o
𝛼
conside ed. The
a e age wid h aken on he shel by SKUs alloca ed o he picking loop is a ound
20% smalle in his case. This con i ms ha he impo ance sco e conside s
addi ional ac o s, such as he olume o he SKU (co ela ion coe icien o 0.52
be ween he equi ed wid h and he a io o he sco e o he numbe o picks o
an SKU). O e all, he analyses unde line he con as ing beha iou o bo h pa s
7 No e: We encoun e an ou lie o he sco e when using
𝛼=0.45
, p ima ily due o he un ime
limi a ion, which would no ypically occu in p ac ice wi h la ge un imes.
Table 2 Summa y s a is ics on he numbe o SKUs included in he picking loop, he space u ilisa ion in
he picking loop, he objec i e alue, he maximum ela i e de ia ion in picks be ween s a ions, and he
esul ing gap a e a un ime o one hou o di e en alues o allowed de ia ion
𝛿
and a ixed weigh ing
ac o
𝛼=0.5
𝛿
# SKUs Space
u ilisa ion
(%)
No m. de . o
obj Pa I (%)
No m. de .
obj Pa II
(%)
De . o obj
alue (%)
Maximum
el. de .
(%)
Gap (%)
0.1% 1491 98.00 0.18 1.00 0.59 0.097 0.63
1.0% 1496 98.51 0.10 0.54 0.32 0.908 0.35
5.0% 1505 98.52 0.10 0.56 0.33 4.961 0.37
10.0% 1502 98.64 0.14 0.45 0.29 9.450 0.33
un es ic ed 1531 99.41 0.07 0.11 0.09 16.572 0.12
577
In eg a ed s o age assignmen o anE‑g oce y ul ilmen …
and he impo ance o a combina ion wi hin he objec i e unc ion. Consequen ly,
𝛼
should all wi hin he in e al [0.35, 0.6]. Fo ou ongoing analyses, we ix
𝛼
a 0.5.
4.2.3 E ec o  hewo kload balancing pa ame e
ı
In he ollowing analysis, we del e in o he e ec o he pe mi ed de ia ion o picks
be ween s a ions
𝛿
on he esul ing objec i e alue ob ained a e a un ime o one
hou . Using
𝛼=0.5
, we a y he pe mi ed ela i e de ia ion be ween s a ions ac oss
𝛿∈{0.1%, 1.0%, 5.0%, 10.0%}
, while also examining he esul s when balancing
cons ain s a e dis ega ded. Table2 p esen s he numbe o SKUs included in he
picking loop, along wi h he co esponding de ia ion o he (no malised) o al
impo ance sco e o hese SKUs (Pa I o he objec i e unc ion), picking e iciency
(Pa II o he objec i e unc ion), o al objec i e alue ela i e o he alues ob ained
by op imising bo h pa s indi idually wi hou espec ing balancing cons ain s (i.e.
𝛾1
and
𝛾2
), and he emaining gap a e one hou o un ime. Speci ically, he numbe s
p esen ed o bo h pa s, as well as he o al objec i e, co espond o one minus he
ac ual alue o he (pa o he) objec i e unc ion. When balancing cons ain s a e
no espec ed, he highes objec i e alue is achie ed wi h a emaining gap o 0.12%.
While his model is easy o sol e, he sugges ed assignmen is no ably imbalanced,
wi h a de ia ion be ween s a ions o up o 16.57%. Fo
𝛿∈{1%,5%, 10%}
, he
objec i e alues and gaps exhibi simila i ies, while he ac ual maximum ela i e
de ia ions a y conside ably, albei emaining sligh ly below he co esponding
pe mi ed de ia ion
𝛿
in each case. Rema kably, we a e e en able o balance he
assignmen on he le el
𝛿=0.1
. As his se ing is mo e complex, he objec i e alue
de ia ion om 1 is nea ly wice as high as o
1%≤𝛿≤10%
(0.59% compa ed o
0.32% o
𝛿=1%
) d i en by a ela i ely highe emaining gap which is also abou
wice as high a e a un ime o one hou . Ac oss all cases, we obse e a e y
high u ilisa ion o he picking loop o a leas 98%, wi h app oxima ely one- hi d
o he sui able SKUs alloca ed o he picking loop. No ably, educing he wo kload
de ia ion ma ginally dec eases he space u ilisa ion as i becomes mo e challenging
o ind an alloca ion mee ing his limi . F om a manage ial poin o iew, hese
indings sugges ocusing on limi ing he allowed de ia ion o
𝛿≤1%
. This app oach
ensu es wo kload balance be ween s a ions, minimising he isk o conges ion, while
main aining a high objec i e alue accoun ing o he impo ance o SKUs alloca ed
o he picking loop as well as picking e iciency.
4.2.4 In eg a ed s sequen ial s o age assignmen
Finally, we compa e ou in eg a ed model o a sequen ial h ee-s age app oach,
whe e we i s sol e he subp oblem akin o he o wa d ese e alloca ion p oblem,
i.e. he selec ion o SKUs o be alloca ed o he e icien picking loop a ea (Pa
I o ou objec i e unc ion), wi hou conside ing picking e iciency (Pa II o ou
objec i e unc ion) o espec ing balancing cons ain s. Rema kably, his model can
be sol ed wi h a gap o 0.11% a e a un ime o only 24s. I sugges s including
1533 SKUs in o he picking loop, leading o a de ia ion o he o al (no malised)
sco e om 1 o 0.1%. Compa ing hese esul s o Table2, we ind ha he sco e

578
D.Winkelmann e al.
imp o es only sligh ly, while we alloca e 37 SKUs mo e o he picking loop han
when also accoun ing o picking e iciency and limi ing he de ia ion in picks
be ween s a ions o
𝛿=1%
. Howe e , he un ime educes comp ehensi ely.
We hen assume he se o hese 1533 SKUs as gi en and alloca e hem o s a ions
wi h he aim o minimising he de ia ion in he numbe o picks be ween s a ions.
Wi hin a un ime o one hou , which is su icien o sol e he in eg a ed p oblem
e icien ly, we a e no able o ind a easible solu ion o an assignmen ha sa is ies
a maximum de ia ion o 1%. Ins ead, we ob ain an objec i e alue o he de ia ion
be ween s a ions o mo e han 21%. Thus, we emo e hose SKUs wi h he small-
es impo ance sco e un il we a e able o sol e he p oblem wi hin he 1% de ia-
ion cons ain . This holds o he 1516 SKUs wi h he highes impo ance sco e in
he se de e mined be o e, while he co esponding o al impo ance sco e dec eases
only sligh ly.
Finally, assuming he assignmen o SKUs o s a ions as gi en, we aim o max-
imise he picking e iciency wi hin he s a ions by de e mining he loca ion on he
shel es o each SKU. Since each s a ion can be op imised independen ly, his
decomposed p oblem can be sol ed o op imali y wi hin less han 4min o an indi-
idual s a ion. We ob ain a de ia ion o 2.2% o he (no malised) objec i e Pa II
and a o al de ia ion o he objec i e alue o 1.1% when using a weigh ing ac o
𝛼=0.5
again. As his objec i e alue is clea ly in e io compa ed o he in eg a ed
app oach (de ia ion o he objec i e alue 0.3% o
𝛿=1%
), he esul s unde sco e
he impo ance o an in eg a ed model compa ed o a sequen ial app oach o he
p oblem unde conside a ion.
4.3 Heu is ic solu ion app oach
Ou compu a ional expe imen s conduc ed in he p e ious sec ion e eal ha a e
12h o un ime, an op imal solu ion is no a ained; a small gap o less han 1%
emains (wi h he exac magni ude depending on he allowed wo kload de ia ion
𝛿
, see Table2). Figu e6 pa icula ly illus a es ha imp o emen s in he objec i e
alue diminish as un imes inc ease. Consequen ly, we in oduce a heu is ic solu ion
app oach o ackle he model, aiming o ind sa is ac o y solu ions wi hin sho e
un imes. Ou heu is ic ocuses on educing he solu ion space o enhance sea ch
e iciency. Fo his pu pose, we ix he s o age loca ions o SKUs in an i e a i e
scheme. S a ing wi h he basic model wi hou any ixa ions, he sol e is endowed
wi h a p ede ined un ime o ind a solu ion. Subsequen ly, we ix he shel loca ions
o a ce ain numbe o SKUs, based on he solu ion alues ob ained in he bes solu-
ion ound. The selec ion o SKUs o be ixed is accomplished acco ding o hei
impo ance sco e (highe impo ance sco e i s ). Subsequen i e a ions a e con-
duc ed on he model wi h an inc easing numbe o ixed SKUs, u ilising he solu ion
om he p e ious i e a ion.
Based on a se o ini ial expe imen s, we es di e en alues o he un ime,
he numbe o ixed SKUs in each i e a ion, and c i e ia o selec ing he SKUs o
ix. We ind ha he bes esul s o ou da a se , wi h
𝛿=1%
, a e achie ed when
limi ing he un ime o 5 min and ixing 100 SKUs in each i e a ion. A e 15
579
In eg a ed s o age assignmen o anE‑g oce y ul ilmen …
i e a ions, aking abou 75 min in o al, we ind a solu ion co e ing 1526 SKUs
wi h a de ia ion o he (no malised) objec i e o 0.07% (wi h an op imali y gap o
0.11%). Compa ing hese esul s o hose ob ained in Sec .4.2, we obse e ha he
solu ion ob ained wi h he heu is ic app oach is supe io e en o he scena io wi h
un es ic ed wo kload de ia ion (see Table2). The supe io i y also holds compa ed
o he de ia ion o he objec i e alue om he basic model a e a un ime o 12h
(de ia ion o 0.11%).
4.4 Sensi i i y analysis wi ha i icial SKU da a
Gi en ha ou compu a ional expe imen s ely solely on he da a se p o ided by ou
business pa ne , i aises ques ions abou he sensi i i y o he esul s wi h espec
o he se o SKUs. A he same ime, he heu is ic solu ion app oach p oposed in
he p e ious sec ion o e s us he oppo uni y o e icien ly sol e he model wi hin
easonable un imes. To assess he impac o he da a se on he solu ion beha iou ,
we p oceed as ollows: (1) we gene a e simula ed SKU da a se s based on he
s uc u e o he da a se p o ided by he e aile , and (2) we sol e he model o hese
se s. In he ollowing, we will desc ibe he da a-gene a ing p ocess be o e p esen ing
he solu ion esul s.
4.4.1 Gene a ing SKU da a
The da a se p o ided by he e aile includes i e a iables o each SKU ele an
o op imising he s o age assignmen : he impo ance sco e, he numbe o picks
accomplished, he wid h, he heigh , and he o de ank o he SKU. As depic ed in
Fig.3, we can app oxima e bo h he log numbe o picks and he log impo ance sco e
wi h no mal dis ibu ions. Addi ionally, we can simpli y by assuming ha he heigh
and wid h o SKUs also ollow no mal dis ibu ions. Calcula ing he co a iance
ma ix
Σ2
be ween he loga i hm o impo ance sco es and picks accomplished, as
well as heigh and wid h, enables us o andomly gene a e SKU da a based on a
0
100
200
300
0100 200300
Heigh in Basic Se
Heigh in Simula ed Se
(a)Heigh
0
100
200
300
400
500
0 100 200 300 40
05
00
Wid h in Basic Se
Wid h in Simula ed Se
(b)Wid h
Fig. 9 Rela ion be ween he so ed heigh (a) and wid h (b) o he SKUs o he basic se (x-axis) and he
gene a ed se 1 (y-axis)
580
D.Winkelmann e al.
mul i a ia e no mal dis ibu ion wi h he same means o he ma ginal dis ibu ions
and dependence s uc u e as in he basic da a se p o ided by he e aile . To ensu e
he gene a ed da a se aligns wi h he business case, we main ain he same numbe
o SKUs in each se . Figu e9, displaying he so ed heigh s (a) and wid hs (b) o he
basic da a se (x-axis) and he i s gene a ed da a se (y-axis), con i ms he ela i ely
good i o no mal dis ibu ions in his case.8 Finally, we de e mine he o de ank o
each SKU. Since he basic da a se co e s e y ew SKUs wi h ank 1, we simpli y
by conside ing he bina y case wi h ank 2 and 3 only. Gi en he posi i e co ela ion
be ween he heigh and he o de ank, as well as be ween he wid h and he o de
ank, we es ima e a logis ic eg ession model a emp ing o explain he o de ank by
he heigh and wid h o he co esponding SKU. U ilising he es ima ed coe icien s
o his eg ession model allows us o andomly selec he o de ank o gene a ed
SKUs based on he unde lying p obabili y de e mined by hei heigh and wid h. In
o al, we gene a e en di e en se s o SKUs wi h in o ma ion on he i e a iables
s a ed abo e.
4.4.2 Resul s
Table3 summa ises he esul s ob ained om he heu is ic solu ion app oach
applied o en di e en da a se s gene a ed acco ding o he da a s uc u e o he
SKU se p o ided by he e aile . We compa e ou indings o hose o he basic
se (see Sec .4.3 o de ailed esul s) in he bo om line o he able. Each se
equi es i s own
𝛾1
and
𝛾2
. Howe e , compu ing hese he same way as be o e o
all se s would be oo ime-consuming, so we simpli y by no using he solu ions
bu he uppe bounds o he op imisa ions men ioned in Sec . 4.1. Ac oss all
8 No e: The e is a small p obabili y o gene a ing SKUs wi h a nega i e heigh and/o wid h. In hese
cases, we exclude he co esponding SKU and d aw again om he unde lying dis ibu ion.
Table 3 Resul s on he
simula ion-based analysis
o 10 di e en se s o SKUs
s a ing he space u ilisa ion, he
de ia ion o he objec i e alue
om 1, he maximum ela i e
de ia ion in picks be ween
s a ions, and he emaining gap
o each se
Se Space
u ilisa ion
(%)
De ia ion o
objec i e alue
(%)
Maximum
ela i e de ia ion
(%)
Gap (%)
1 99.73 0.07 0.91 0.06
2 99.54 0.15 0.41 0.14
3 99.74 0.06 0.92 0.05
4 99.79 0.05 0.98 0.04
5 99.79 0.05 0.99 0.04
6 99.59 0.08 0.99 0.07
7 99.66 0.09 0.72 0.08
8 99.71 0.14 0.80 0.13
9 99.70 0.06 0.96 0.05
10 99.70 0.07 0.99 0.06
Basic 99.60 0.07 0.94 0.11
581
In eg a ed s o age assignmen o anE‑g oce y ul ilmen …
summa y s a is ics, i.e. he space u ilisa ion, he de ia ion o he objec i e alue,
he maximum ela i e de ia ion be ween s a ions, and he gap, we obse e
simila esul s, ha a e also close o hose ob ained in he basic se . In each
case, he space u ilisa ion is close o 100%, indica ing ha selec ing he mos
impo an SKUs o be alloca ed o he picking loop is a c ucial ask in each
se . Fu he mo e, we demons a e ha we a e able o sol e he model wi h he
heu is ic solu ion app oach in easonable un imes close o op imali y. This
is e iden om he de ia ions o he objec i e alue being close o 0, as well
as e y small gaps. Only ega ding he maximum ela i e wo kload de ia ion
be ween di e en s a ions, we do ind some a ia ion among he di e en se s.
While in se en ou o en se s he e is a de ia ion o mo e han 0.9%, which
also holds o he basic se , o se 2, his a ia ion is only 0.41%. Howe e , in
conclusion, we can s a e ha e en o di e en se s o SKUs gene a ed acco ding
o he same s uc u e as p esen in he SKU se p o ided by he e aile , ou
model allows o e icien solu ions. This sensi i i y analysis gene alises he
esul s ob ained be o e and also sugges s conside ing he business case o
o he e aile s by adjus ing he co a iance ma ix, e.g. o allow o a di e en
composi ion o he impo ance sco e wi h less co ela ion o he numbe o picks
in u u e wo k.
5 Coping wi hsho ‑ e m demand a ia ion
The gene al model de eloped in he p e ious sec ion enables he e aile o
add ess he h ee-le el s o age assignmen p oblem: deciding which SKUs om
he asso men should be alloca ed o he picking loop, as well as de e mining
he assignmen o SKUs o s a ions and shel es wi hin he wa ehouse. Howe e ,
as e idenced by he da a desc ibed in Sec .2.3, he demand o SKUs (and hence
he numbe o picks) a ies signi ican ly ac oss days, weeks, o e en mon hs o
he yea . This unde sco es he impo ance o balancing wo kload o each day o
week indi idually, enabling he e aile o op imise he assignmen o SKUs o
s a ions and shel es. In his sec ion, we del e in o he signi icance o accoun ing
o a ia ion in demand when assigning SKUs o shel es. We s a by analysing
he e icacy o he s o age assignmen gene a ed by he heu is ic in oduced in
he p e ious sec ion, pa icula ly ega ding he po en ial imbalance be ween
s a ions ac oss di e en days o week. Subsequen ly, we enhance ou model
o mula ion by cons aining he de ia ion o picks be ween s a ions on he le el
o days o week, and hen compa e bo h app oaches. Again, we employ he
heu is ic app oach in oduced in Sec .4.3 o sol e he model. This compa a i e
analysis allows us o quan i y he bene i s o explici ly conside ing a ia ion in
demand when making s o age assignmen decisions. Howe e , i is impo an
o acknowledge ha he e aile needs o spend e o on da a collec ion, da a
p ocessing and compu a ional powe o he de ailed analysis. The e o e, his
analysis se es as he ounda ion o de e mining whe he he bene i s o he
de ailed solu ion ou weigh he associa ed e o s o he e aile .
588
D.Winkelmann e al.
o 110. Thus, he e is a highe isk o an imbalanced alloca ion o SKUs o s a ions
in his scena io, e en i we limi he de ia ion a he le el o a e ages pe mon h
and day o week. Conside ing di e en alues o he coe icien o a ia ion enables
us o compa e a ious se ings and gene alise he esul s ob ained in his sec ion o
o he e aile s, o example.
Using he gene a ed alues o he numbe o picks accomplished on indi idual
wo king days, we compu e he wo kload o each s a ion and wo king day o e he
en i e yea . This enables us o de e mine he de ia ion om he a e age o e all
eigh s a ions o a o al o
52 ⋅6=312
days. Repea ing he da a gene a ion p ocess
acco ding o he unde lying dis ibu ion o each SKU and calcula ing he wo kload
de ia ion 100 imes ensu es he eliabili y o he analysis. Speci ically, we calcula e
bo h he a e age absolu e de ia ion be ween a single s a ion and he a e age ac oss
all s a ions o each wo king day and he maximum absolu e de ia ion. Compa ing
he a e age esul s ac oss all simula ion uns o he solu ion based on he alloca ion
unde he basic model (see Sec ion4) o hose ob ained when using he a ia ion-
awa e model (see Sec ion5), Fig.12a gi es he a e age o e 100 simula ion uns
o he mean absolu e de ia ion be ween a single s a ion and all s a ions o e 52
weeks and six days o week wi hin each week. Figu e12b shows he maximum
absolu e de ia ion o e all indi idual wo king days and s a ions, again a e aged
o e he same 100 simula ion uns. In bo h igu es, we compa e esul s ob ained
unde he alloca ion de e mined by he basic model wi h
𝛿=1%
( ed solid line) o
hose ob ained unde he a ia ion-awa e model wi h
𝛿=1%
(blue do ed line), bo h
sol ed by he heu is ic.
While we limi he de ia ion in he numbe o picks be ween di e en s a ions
o
𝛿=1%
( o he basic model) and
𝛿 =1%
( o he a ia ion-awa e model,
accoun ing o day-o -week a ia ion) in he op imisa ion model, hese de ia ions
a e based on he a e age numbe o picks o e a whole yea . Howe e , in p ac ice,
he ac ual numbe o picks migh a y om week o week o he same day o week.
Speci ically, ou esul s demons a e ha e en wi h a small coe icien o a ia ion
(
CV =0.05
), he maximum absolu e de ia ion o a single s a ion om he a e age
o e all s a ions esul s in a alue o mo e han 4%, which is ou imes highe han
he in ended le el o 1%. When elying on he a ia ion-awa e model, which akes
in o accoun a ia ion be ween di e en days o week, again on he le el o a e ages
o e he whole yea , we ind an ac ual de ia ion o 2.46% (compa ed o 1% in ended
by he model). E en hough he ac ual de ia ion is again la ge han in ended by he
model, we can show ha day-o -week-speci ic cons ain s allow o a conside able
educ ion in de ia ion, in his case by abou 43%. The same esul holds o he
mean absolu e de ia ion, which educes om 1.01% o 0.57%. Fo inc easing alues
o he coe icien o a ia ion, we ind highe mean absolu e de ia ions (inc easing
o 2.33% o he basic model and 2.16% o he a ia ion-awa e model, espec i ely,
o
CV =0.3
), as well as highe maximum absolu e de ia ions in his case (11.18%
o he basic model and 10.43% o he a ia ion-awa e model). A he same ime,
he bene i o he a ia ion-awa e model educes o 7.48% o he mean absolu e
de ia ion and 6.75% o he maximum absolu e de ia ion. In summa y, his analysis
demons a es he bene i o he a ia ion-awa e model in oduced in his sec ion in a

589
In eg a ed s o age assignmen o anE‑g oce y ul ilmen …
p ac ical se ing whe e he numbe o picks a ies be ween di e en weeks, wi h he
highes bene i obse ed when he a ia ion is ela i ely small.
6 Conclusion
In his pape , we p esen an in eg a ed app oach o ackle a h ee-le el s o age
assignmen p oblem encoun e ed in a ul ilmen cen e ope a ed by a leading
Eu opean e-g oce y e aile . The ul ilmen cen e is cha ac e ised as a hyb id
wa ehouse, combining a highly e icien , pa ially au oma ed picking loop wi h a
less e icien picke - o-pa s a ea. While he demand o e-g oce ies has inc eased
in ecen yea s, he ma ke has become mo e compe i i e, necessi a ing e-g o-
ce y e aile s o enhance hei ope a ional e iciency. A key challenge lies in he
assignmen o SKUs o shel es wi hin a ul ilmen cen e. We op imise a bi-
objec i e alue unc ion o he e aile , conside ing he impo ance o SKUs allo-
ca ed o he highly e icien picking loop, while also add essing picking e iciency
dependen on he dis ance be ween a picke and he shel es. To p e en conges-
ion wi hin he picking loop, we addi ionally impose cons ain s in ou p oposed
op imisa ion model, limi ing he pe mi ed ela i e de ia ion in he numbe o
picks be ween di e en s a ions.
Ou esul s indica e ha we can e icien ly sol e he model wi h a emaining
gap o less han 0.7% wi hin one hou in mos scena ios. Since we add ess a ac i-
cal p oblem o he e aile , which has no o be sol ed egula ly bu only in cases
o signi ican changes in he asso men o he e aile o cus ome p e e ences,
such un imes a e easonable and can e en be ex ended. Addi ionally, we p o-
pose a heu is ic solu ion app oach ha akes less han wo hou s o ob ain solu-
ions su passing hose ound by a s anda d sol e a e 12h. The ob ained esul s
clea ly demons a e he supe io i y o ou in eg a ed app oach compa ed o sol -
ing he alloca ion o he picking loop and he assignmen o s a ions and shel es
sequen ially. The indings emain consis en ac oss di e en se s o SKUs gene -
a ed wi hin a simula ion-based analysis.
This pape also add esses he challenge o day-o -week-dependen demand
a ia ion o speci ic SKUs. In ou business case, demand a ia ion is no ably
high a he beginning o a week and jus be o e he weekend. Th ough a se ies
o expe imen s, we demons a e ha a s o age assignmen based solely on day-
o -week-agnos ic a e age demand igu es ends o exhibi a highly imbalanced
wo kload on ce ain days o week. To mi iga e his issue, we ex end he a o e-
men ioned s o age assignmen model o conside day-o -week-dependen demand
a ia ion. Ou indings e eal ha his ex ended model p oduces s o age assign-
men s ha mee he wo kload balance equi emen s imposed o each day o week
wi hou comp omising he quali y o he solu ions in e ms o he (e iciency-o i-
en ed) objec i e alue. Fu he mo e, by gene a ing simula ed da a based on di -
e en coe icien s o a ia ions o accoun o a ia ion be ween di e en weeks,
we unde sco e he bene i s o he ex ended model o mula ion. This app oach also
o e s manage ial insigh s in o he ac ual de ia ion, going beyond eliance on he
a e age numbe o picks o e he en i e yea .
590
D.Winkelmann e al.
Fu u e wo k could include e inemen s such as indi idual le els o pe mi ed
de ia ion be ween s a ions based on he o al numbe o picks on a pa icula day
o week, ha is, a ying
𝛿
wi h espec o he day o week
∈T
. Since conges ion
is mo e c i ical on days o week wi h high wo kload, his model ex ension could
u he educe ope a ional ine iciencies o e-g oce y e aile s. The simula ion-
based analysis on he a ia ion in he numbe o picks ac oss weeks also o e s
he po en ial o de elop ad anced models. Fo ins ance, inco po a ing me hods
o lea n om weeks wi h a high le el o de ia ion could u he educe wo kload
imbalance. While andom demand a ia ion migh be add essed by employing
obus o s ochas ic op imisa ion app oaches, he a ailabili y o da a spanning
mul iple yea s would addi ionally enable he de ec ion o s uc u al long- e m
demand changes, he eby jus i ying a ea angemen o he s o age assignmen .
Fu he mo e, gi en ha ou pape ocuses solely on he de ailed s o age
assignmen in he as -picking a ea, a na u al ex ension would be o include
he s o age assignmen o he picke - o-pa s a ea in an in eg a ed app oach.
Fu he mo e, se ing up a de ailed simula ion o daily ope a ions could gene a e
insigh s in o p ocessing imes o he numbe o o de s ha could be accep ed on a
single day, ansla ing in o a measu e o ope a ional cos s. Finally, while we add ess
he op imisa ion o an exis ing ul ilmen cen e, u u e esea ch could also conside
he s a egic p oblem o designing wa ehouses. This would in ol e he decision on
he size o he picking loop, he numbe o s a ions, and he con igu a ion o shel es.
Appendix1: Rep esen a ion o shel es
See Fig.13.
Fig. 13 Rep esen a ion o he s uc u e o shel es
591
In eg a ed s o age assignmen o anE‑g oce y ul ilmen …
Appendix2: Boxplo o  henumbe o uni s pe o de line
See Fig.14.
Appendix3: MILP o mula ion o  hein eg a ed h ee‑le el s o age
assignmen
See Table6.
Fig. 14 Boxplo o he a e age
numbe o uni s wi hin one
o de line o he SKUs in
he asso men o he e aile
app op ia e o he picking loop
246810 12 14
A e age Numbe o Uni s pe SKU Wi hin an O de
592
D.Winkelmann e al.
max
𝛼
𝛾1
∑
∈V
∑
∈R
s x ,
⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟
I
+
1−
𝛼
𝛾2
∑
∈V
∑
∈R
1
d
p x ,
⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟
II
Table 6 Table o se , pa ame e s, and decision a iables o he MILP
Se s
VSe o SKUs
KSe o s a ions in he picking loop, o de ed and indexed
by in ege s (
K
={1, …,
|K|}
RSe o shel es in he picking loop wi h
Rk
deno ing he subse o shel es
a s a ion
k∈K
and
R
co esponding o he subse o shel es R
ha can i SKU
∈V
OSe o p ecedence o de anks wi h
O={1, …,|O|}
,
whe e
o ≤o ′
i has o be assigned o an ea lie s a ion han
′
Pa ame e s
s
Impo ance sco e o SKU
∈V
p
Numbe o picks o SKU
∈V
h
Heigh aken in he shel SKU
∈V
w
Wid h aken in he shel by he a ge s ock o SKU
∈V
k
S a ion
k∈K
shel
∈R
is loca ed in
zk
Wo kload a s a ion
k∈K
. The a e age o e all s a ions is deno ed by z.
𝛿
Th eshold deno ing he maximum pe mi ed ela i e de ia ion o
wo kload
zk
o a s a ion
k∈K
om he a e age z o e all s a ions
h
Heigh o shel e
∈R
w
Wid h o shel e
∈R
d
Dis ance be ween he picke and shel e
∈R
wi h picking e iciency
1
d
gMinimum dis ance be ween each wo SKUs s o ed nex o each o he
𝛼
Weigh ing ac o o he objec i e unc ion
𝛾1
Objec i e alue when op imising he o al impo ance sco e only
𝛾2
Objec i e alue when op imising he picking e iciency only
Decision a iables
x , ∈{0, 1}
1 i SKU
∈V
is assigned o shel e
∈R
yo
Las s a ion o which an SKU wi h o de ank
o∈O
can be assigned
593
In eg a ed s o age assignmen o anE‑g oce y ul ilmen …
Appendix4: Augmen ed MILP o mula ion accoun ing o demand
a ia ion in hes o age assignmen
See Table7.
∑
∈R
x , ≤1∀ ∈
V
k x , ≤yo∀o∈O, ∈Vo, ∈R
k x , ≥yo−1∀o∈O⧵{1}, ∈Vo, ∈R
zk=∑
∈V
∑
∈Rk
p x , ∀k∈
K
zk≤(1+𝛿)⋅
1
|K|∑
l∈K
zl∀k∈
K
zk≥(1−𝛿)⋅
1
|K|∑
l∈K
zl∀k∈
K
w ≥∑
∈V
(w +g)⋅x , −g∀ ∈
R
x , ∈{0, 1
}∀
∈V, ∈R
y
o
∈{1, …,
|
K
|}∀
o∈
O
max 𝛼
𝛾1
∑
∈V
∑
∈R
s x ,
⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟
I
+
1−𝛼
𝛾2
∑
∈V
∑
∈R
1
d
p x ,
⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟
II
∑
∈R
x
,
≤1∀ ∈V
k x , ≤yo∀o∈O, ∈Vo, ∈R
k x , ≥yo−1∀o∈O⧵{1}, ∈Vo, ∈R
z
k=∑
∈V
∑
∈Rk
p
x , ∀k∈K, ∈
T
z
k≤(1+𝛿 )⋅
1
|K|∑
k∈K
z
k∀k∈K, ∈
T
z
k≥(1−𝛿 )⋅
1
|K|∑
k∈K
z
k∀k∈K, ∈
T
w ≥∑
∈V
(w +g)⋅x , −g∀ ∈
R
x , ∈{0, 1
}∀
∈V, ∈R
y
o
∈{
1,
…
,|
K
|}∀
o∈
O

594
D.Winkelmann e al.
Table 7 Table o se , pa ame e s, and decision a iables o he MILP
Se s
VSe o SKUs
KSe o s a ions in he picking loop, o de ed and indexed
by in ege s (
K
={1, …,
|K|}
RSe o shel es in he picking loop wi h
Rk
deno ing he subse o shel es
a s a ion
k∈K
and
R
co esponding o he subse o shel es R
ha can i SKU
∈V
OSe o p ecedence o de anks wi h
O={1, …,|O|}
,
whe e
o ≤o ′
i has o be assigned o an ea lie s a ion han
′
TSe o days o week
Pa ame e s
s
Impo ance sco e o SKU
∈V
p
Numbe o picks o SKU
∈V
h
Heigh aken in he shel SKU
∈V
w
Wid h aken in he shel by he a ge s ock o SKU
∈V
k
S a ion
k∈K
shel
∈R
is loca ed in.
zk
Wo kload a s a ion
k∈K
. The a e age o e all s a ions is deno ed by z.
The wo kload a s a ion
k∈K
a day o week
∈T
is deno ed by
z
k
𝛿
Th eshold deno ing he maximum pe mi ed ela i e de ia ion o
wo kload
zk
o a s a ion
k∈K
om he a e age z o e all s a ions.
The h eshold a day o week
∈T
is deno ed by
𝛿
h
Heigh o shel e
∈R
w
Wid h o shel e
∈R
d
Dis ance be ween he picke and shel e
∈R
wi h picking e iciency
1
d
gMinimum dis ance be ween each wo SKUs s o ed nex o each o he
𝛼
Weigh ing ac o o he objec i e unc ion
𝛾1
Objec i e alue when op imising he o al impo ance sco e only
𝛾2
Objec i e alue when op imising he picking e iciency only
Decision a iables
x , ∈{0, 1}
1 i SKU
∈V
is assigned o shel e
∈R
yo
Las s a ion o which an SKU wi h o de ank
o∈O
can be assigned
Appendix5: Shel dis ance o SKUs wi hheigh exceeding 250mm
See Figs.15 and 16.
595
In eg a ed s o age assignmen o anE‑g oce y ul ilmen …
2000
2500
0 500010000 15000
Picks
Dis ance
alpha = 1
2000
2500
0500010000 15000
Picks
Dis ance
alpha = 0.75
2000
2500
05000 10000 15000
Picks
Dis ance
alpha = 0.5
2000
2500
0 500010000 15000
Picks
Dis ance
alpha = 0.25
2000
2500
0500010000 15000
Picks
Dis ance
alpha = 0
Fig. 15 Alloca ion o SKUs and co esponding picks o shel es wi h gi en dis ance o he picke o
di e en alues o
𝛼
and
𝛿=1%
. No e ha he igu e is limi ed o SKUs wi h a heigh exceeding 250mm
596
D.Winkelmann e al.
Funding Open Access unding enabled and o ganized by P ojek DEAL.
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Monday
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Tuesday
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