Integrated procurement and reprocessing planning for reusable medical devices with a limited shelf life

Ricke s, S e en; Sahling, Flo ian
A icle — Published Ve sion
In eg a ed p ocu emen and ep ocessing planning o
eusable medical de ices wi h a limi ed shel li e
Heal h Ca e Managemen Science
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Ricke s, S e en; Sahling, Flo ian (2024) : In eg a ed p ocu emen and
ep ocessing planning o eusable medical de ices wi h a limi ed shel li e, Heal h Ca e
Managemen Science, ISSN 1572-9389, Sp inge US, New Yo k, NY, Vol. 27, Iss. 2, pp. 168-187,
h ps://doi.o g/10.1007/s10729-024-09664-9
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/315269
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Heal h Ca e Managemen Science (2024) 27:168–187
h ps://doi.o g/10.1007/s10729-024-09664-9
In eg a ed p ocu emen and ep ocessing planning o eusable
medical de ices wi h a limi ed shel li e
S effen Ricke s1·Flo ian Sahling2
Recei ed: 25 Ma ch 2022 / Accep ed: 3 Janua y 2024 / Published online: 25 Janua y 2024
© The Au ho (s) 2024
Abs ac
We p esen a new model o mula ion o a mul ip oduc dynamic o de quan i y p oblem wi h p oduc e u ns and a ep ocess-
ing op ion. The op imiza ion conside s he limi ed shel li e o s e ile medical de ices as well as he capaci y cons ain s o
ep ocessing and s e iliza ion esou ces. The ime- a ying demand is known in ad ance and mus be sa is ied by pu chasing
new medical de ices o by ep ocessing used and expi ed de ices. The objec i e is o de e mine a easible p ocu emen and
ep ocessing plan ha minimizes he incu ed cos s. The p oblem is sol ed in a heu is ic manne in wo s eps. Fi s , we use
a Dan zig-Wol e e o mula ion o he unde lying p oblem, and a column gene a ion app oach is applied o igh en he lowe
bound. In he nex s ep, he ob ained lowe bound is ans o med in o a easible solu ion using CPLEX. Ou nume ical esul s
illus a e he high solu ion quali y o his app oach. The compa ison wi h a simula ion based on he i s -come- i s -se ed
p inciple shows he ad an age o in eg a ed planning.
Keywo ds S e ile se ice depa men ·Ma e ial logis ics in hospi als ·Reusable medical de ices ·P ocu emen ·
Rep ocessing ·Limi ed shel li e ·Column gene a ion
Highligh s
•We p esen a no el model o mula ion o a mul ip oduc
dynamic o de quan i y p oblem wi h p oduc e u ns and
a ep ocessing op ion ha inco po a es he limi ed shel
li e o s e ile medical de ices and capaci y cons ain s on
ep ocessing and s e iliza ion esou ces.
•We p opose a wo-s age heu is ic app oach based on
Dan zig-Wol e e o mula ion and column gene a ion
echniques o gene a e high-quali y solu ions.
•Ou simula ion s udy shows a signi ican cos educ-
ion compa ed o he s anda d i s -come- i s -se ed
app oach. These esul s highligh he p ac ical ad an-
ages o he p oposed model.
BFlo ian Sahling
[email p o ec ed]
S e en Ricke s
s e [email p o ec ed] e .de
1Depa men o P oduc ion Managemen , Leibniz Uni e si ä
Hanno e , Königswo he Pla z 1, Hanno e 30167, Ge many
2RPTU School o Managemen and Economics, Chai o
P oduc ion Managemen , Uni e si y o
Kaise slau e n-Landau (RPTU), Go lieb-Daimle -S aße,
Kaise slau e n 67663, Ge many
1 In oduc ion
Cu en ly, hospi als a e unde conside able cos p essu e.
Ge many changed i s ca e ee sys em o a ee- o -ca e sys em
in 2004 which o ced Ge man hospi als o ins i u e cos -
sa ing measu es and p ocess op imiza ion. Un il he end o
2003, a daily a e was paid o each pa ien , which depended
on he ea men cos s ha we e ac ually incu ed. Due o
a lack o incen i es o educe cos s, a new billing p ocedu e
based on diagnosis- ela ed g oups (DRGs) was in oduced in
2004. Based on he p ima y diagnosis, pa ien s a e assigned
o a DRG. Fo each medical ea men , a uni o m la - a e
paymen is de ined, which ep esen s a ixed p ice o se -
ices ha do no depend on he indi idual pa ien . Thus, he
hospi al is only p o i able i he ac ual ea men cos s do no
exceed he speci ied ixed p ice. The DRG sys em he e o e
ans e s bo h he cos esponsibili y and he cos isk di ec ly
o hospi als.
Ra ionaliza ion ac i i ies o en ocus on he su gical a ea.
On he one hand, su ge ies a e he main sou ce o e enue. On
he o he hand i accoun s o a subs an ial po ion o he hos-
pi al’s o al cos s. Hence, in he li e a u e, many app oaches
ocus on he op imiza ion o his a ea. Howe e , li le a en-
ion has been gi en o he supply o s e ile goods. The supply
123
In eg a ed p ocu emen and ep ocessing... 169
o medical de ices is no p o i able i sel , e en hough cos
educ ions in his a ea would ha e a p o i -inc easing e ec .
Mo eo e , he supply o hese medical de ices is o eno mous
impo ance o he su gical a ea. Fo example, he hygiene
scandal a Mannheim Uni e si y Hospi al in Oc obe 2014
mean ha ac i i y in he en i e su gical a ea nea ly s opped
since s e ile goods we e no su icien ly ep ocessed. Fu -
he mo e, [21] poin ed ou ha he supply o s e ile goods
is o g ea impo ance o he su gical a ea, since su gical
eams assume ha he medical de ices equi ed o ope a-
ions a e no missing o con amina ed o ensu e he success
o he su ge y and a oid endange ing he heal h o he pa ien .
The ask o he s e ile se ice depa men (SSD) is o
supply he su gical a ea wi h he equi ed quan i y o s e -
ile medical de ices on ime o ensu e ha he su ge y uns
smoo hly. This wo k add esses he logis ical p ocesses ha
a e necessa y o ul ill his ask. The p ocesses in ol ed in
supplying s e ile goods g ea ly depend on whe he medical
de ices a e in ended o single o mul iple uses. The handling
o medical de ices is egula ed in he Eu opean Union in
Regula ion 2017/745 on medical de ices, c . [13]. Single-
use medical de ices a e in ended o be consumed when used
in he ope a ing oom. F om a logis ical poin o iew, p o-
cu emen and s o age p ocesses a e necessa y o single-use
medical de ices; c . [17]. Al hough eusable medical de ices
equi e addi ional ep ocessing, hey a e o en p e e able
o economic and ecological easons. Rep ocessing yields
an addi ional backwa d-o ien ed ma e ial low o medical
de ices. Thus, complexi y has inc eased due o his ep o-
cessing op ion. P ocu ing and ep ocessing a e wo op ions
o sa is ying he demand in he ope a ing oom. Fu he -
mo e, decisions abou he p ocu emen imes and quan i ies
di ec ly in luence he planning o ep ocessing, and ice
e sa. Hence, he p esen wo k concen a es on eusable
medical de ices. The ocus is on he de elopmen o an op i-
miza ion model and a sui able solu ion app oach ha can be
used o de e mine p ocu emen and ep ocessing plans ha
minimize he o al cos s. Howe e , hese plans mus ensu e
he imely p o ision o medical de ices. The wo k p esen ed
in his pape is based on conside a ions i s discussed in [31].
The emainde o his pape is s uc u ed as ollows:
Sec ion 2desc ibes he asks o he cen al s e ile supply
depa men . These asks include supplying he su gical a ea
wi h s e ile medical de ices and ep ocessing hese de ices
a e u iliza ion. Sec ion 3p o ides an o e iew o he ela ed
li e a u e.
In Sec ion 4, a new model o mula ion o in eg a ed
p ocu emen and ep ocessing planning o eusable medical
de ices wi h a limi ed s o age ime is p esen ed. A col-
umn gene a ion app oach is p oposed in Sec ion 5. Based
on he gene a ed es ins ances, he nume ical in es iga ions
in Sec ion 6e alua e he pe o mance o he p oposed solu-
ion app oach in e ms o compu a ional e o and solu ion
quali y. Finally, in Sec ion 7, he p esen ed esul s a e sum-
ma ized, and u he esea ch di ec ions a e desc ibed.
2 Rep ocessing and p ocu emen o eusable
medical de ices
This wo k ocuses on he p o ision, ep ocessing and p o-
cu emen o eusable s e ile goods in hospi als. Reusable
s e ile goods a e medical de ices ha a e in ended by he
manu ac u e o mul iple low-ge m and s e ile usage. They
mus be ep ocessed by he SSD a e use be o e hey can be
u ilized again.
The demand quan i ies o medical de ices can be de i ed
om he su ge y schedule as a esul o ope a ional su ge y
planning, which usually includes a planning ho izon o one
week, c . [15]. The aim o ope a ional su ge y planning is o
assign pa ien s o a speci ic day o he week wi h a s a ime
o he ope a ion as well as an ope a ing oom and eam. A
dis inc ion mus be made be ween elec i e and eme gency
pa ien s. Unlike eme gency pa ien s, elec i e pa ien s do no
ha e c i ical inju ies o illnesses ha equi e immedia e su -
gical ca e. Typically, elec i e pa ien s accoun o 80 o 90%
o he ope a ions in a hospi al. Thus, due o he sho plan-
ning ho izon o one week, ope a ions on elec i e pa ien s
can be planned unde almos de e minis ic condi ions. How-
e e , eme gency pa ien s canno be scheduled. Acco ding o
[4], hospi als ha e h ee op ions o dealing wi h medical
eme gencies in ope a ional su ge y planning. Fi s , a sepa a e
ope a ing oom can be ese ed exclusi ely o eme gencies.
Second, a po ion o he capaci y in each ope a ing oom
could be ese ed. Thi d, a combina ion could be conside ed.
Analogously, hese h ee op ions can be applied by he SSD
o cope wi h eme gency pa ien s. Addi ionally, sa e y s ocks
o pa icula medical de ices a e s ipula ed by law in some
coun ies. In Ge many, o example, he Fede al O ice o
Ci il P o ec ion and Disas e Assis ance p o ides an in en-
o y lis o s ocking speci ic medical de ices (see [5]). This
lis con ains, e.g., scisso s, scalpels, and o ceps.
Compliance wi h he su ge y schedule equi es he imely
a ailabili y and p o ision o essen ial medical de ices. Con-
sequen ly, he su gical a ea mus be closely coo dina ed wi h
he supply o s e ile goods o a oid delays. The co e ask
o he SSD is he imely supplying o ope a ing ooms wi h
s e ile medical de ices o he equi ed quan i y and qual-
i y. The main p ocess o supplying eusable medical de ices
can he e o e be di ided in o he ollowing subp ocesses:
p ocu emen , s o age, p o ision, anspo , ep ocessing and
disposal. Thus, hese p ocesses mus be coo dina ed e i-
cien ly.
Since hospi als usually do no p oduce medical de ices,
he SSD is also esponsible o he p ocu emen o medical
123
170 S. Ricke s and F. Sahling
de ices om ex e nal supplie s. Acco ding o egula ions,
medical consumables mus be disposed o a e use. Thus,
he ma e ial low o medical consumables h ough he hos-
pi al is s ic ly o wa d o ien ed and co esponds o a classic
supply chain. Reusable medical de ices, such as su gical
ins umen s, mus also be p ocu ed, bu hey can be used
se e al imes. Hence, om he pe spec i e o he SSD, he
su gical a ea is bo h a consume o s e ile medical de ices
and a supplie o hese p oduc s in a nons e ile condi ion.
Hence, a closed-loop supply chain mus be conside ed due
o he addi ional e e se ma e ial low om he ope a ing
oom o he SSD and he ep ocessing op ion. Wi h a po -
ion o app oxima ely 80% o wo king ime, ep ocessing
cons i u e he main asks o he SSD. The whole p ocess
can be desc ibed as a ep ocessing cycle ha is iden ical o
all eusable medical de ices. This ep ocessing cycle s a s
wi h he usage o medical de ices in ope a ing ooms. Med-
ical de ices a e ypically p o ided in su ge y-speci ic se s
ha mus be ex ac ed be o e usage. A e wa ds, he u ilized
medical de ices a e e u ned o he SSD. I is wo h men ion-
ing ha e en he medical de ices ha we e no used mus be
ep ocessed i hei packaging was opened o damaged.
A he beginning o a ep ocessing ope a ion, medical
de ices a e p ecleaned o emo e con amina ion. Addi ion-
ally, a p esc eening o medical de ices wi h une en su aces
is necessa y because con amina ion is di icul o emo e.
Damaged medical de ices a e disposed o and hus lea e he
ep ocessing cycle di ec ly. The decon amina ion o medical
de ices includes cleaning, disin ec ion, insing and d ying.
Fo his p ocess, medical de ices a e placed in sie es. These
sie es a e loaded in o washe -disin ec o s. The mal disin-
ec ion is based on he so-called A0concep . The A0 alue
desc ibes he ime du a ion equi ed o kill mic oo ganisms a
a gi en empe a u e. The equi ed A0 alue depends on he
isk classi ica ion o a medical de ice ha was in oduced
by [38] who classi ied medical de ices as nonc i ical, semi-
c i ical o c i ical based on he isk o in ec ion ela ed o
he usage on he pa ien . Theo e ically, each medical de ice
can be ep ocessed by any ime- empe a u e combina ion
in which he co esponding A0 alue is a leas as high as
he de ice-speci ic alue. Fo example, an A0 alue o 600
can be achie ed by “600 seconds a 80◦C” o by “60 sec-
onds a 90◦C”. Howe e , he empe a u e mus no exceed
he de ice-speci ic empe a u e ole ance s ipula ed by i s
manu ac u e . The inal insing and d ying gua an ee ha no
esidue o he chemicals used o cleaning emains on he
medical de ices.
A e wa ds, he cleaning esul s a e con olled, and he
unc ionali y is es ed. The medical de ices a e packed in o
su ge y-speci ic se s. Subsequen s e iliza ion is used o kill
he emaining mic oo ganisms. Va ious s e iliza ion ypes as
well as di e en ime- empe a u e combina ions a e a ail-
able. The s e iliza ion p ocedu e is selec ed based on he
equi emen s o he medical de ice. The mos able medical
de ices a e usually s e ilized by s eam o ho ai . Fo hea -
sensi i e medical de ices, di e en me hods a e a ailable
wi h lowe p ocess empe a u es. No ably, he comple e
ep ocessing ope a ion includes p ecleaning, decon amina-
ion and s e iliza ion.
I he medical de ices a e no p o ided di ec ly in ope -
a ing ooms, hey can be s o ed unp o ec ed on shel es o
p o ec ed in cabine s o d awe s. Howe e , he shel li e o
s e ilized medical de ices depends on he ype o packaging
and s o age condi ions. I hese de ices a e s o ed unp o-
ec ed, hey mus be used wi hin 48 hou s. I hey a e s o ed
p o ec ed, he shel li e can be up o 12 mon hs. The s o -
age space o s e ile medical de ices is usually limi ed since
he SSD is o en loca ed close o he ope a ing ooms; long
anspo a ion may inc ease he isk o con amina ion. Wi h-
d awal om s o age is based on he i s -in– i s -ou p inciple
o a oid exceeding he maximum s o age ime. I he maxi-
mum s o age ime is exceeded, he medical de ices mus be
ep ocessed again.
3 Rela ed wo k
The handling o medical de ices in hospi als has ecei ed
li le a en ion in li e a u e. An o e iew o he logis ics o
s e ile medical de ices can be ound in [45]. Mos scien i ic
publica ions examining he ep ocessing o medical de ices
desc ibe ules and legal equi emen s o ep ocessing and
s e iliza ion. Insigh s a e gi en, o example, by [22] and
[33].
In he li e a u e, he e a e a ew app oaches o ope a -
ing oom planning ha include he a ailabili y o medical
de ices o hei ep ocessing. Meskens e al. [24] p esen ed an
op imiza ion p oblem o he gene a ion o a su ge y sched-
ule, in which ep ocessable medical de ices a e conside ed
enewable esou ces. Each ype o ope a ion equi es a cha -
ac e is ic numbe o di e en medical de ices ha a e only
a ailable in limi ed quan i ies. Guine and Chaabane [14,44]
and [43] conside he esou ce limi a ions o medical de ices
when p epa ing su ge y schedules.
Ca doen e al. [6,7] include he necessa y ep ocessing
ime o medical de ices in ope a ing oom planning. A e
use, medical de ices a e no a ailable o a ixed numbe
o pe iods, so ope a ions o he same ype canno immedi-
a ely ollow one ano he . Al Hasan e al. [3] also c ea ed
a su ge y schedule aking in o accoun he a ailabili y o
medical de ices and he ep ocessing ime. To comply wi h
he su ge y schedule, a medical de ice can be p io i ized o
ep ocessing, leading o addi ional cos s.
Coban [8] o mula ed a mixed-in ege model o plan su g-
e ies and ep ocess medical de ices in an in eg a ed manne .
Howe e , only homogeneous medical de ices a e conside ed
123
In eg a ed p ocu emen and ep ocessing... 171
in his model. A ce ain numbe o s e ile medical de ices a e
p o ided o each ope a ion. I he numbe o s e ile medi-
cal de ices is no su icien , he ope a ion canno ake place
and mus be pos poned. I is no possible o o de missing
medical de ices o comply wi h he su gical plan. A e use,
he medical de ices can be ep ocessed and s o ed. Howe e ,
only one ype o s e iliza ion is conside ed.
Mos o he li e a u e on in en o y managemen o med-
ical de ices add esses medical consumables. Ahmadi e al.
[1] and [34] p o ide an o e iew. In he li e a u e on in en-
o y managemen o eusable medical de ices, ep ocessing
plays a a he seconda y ole. One eason is ha ep ocessing
is ou sou ced o an ex e nal se ice p o ide and is he e o e
no longe pa o he planning p oblem.
In [42], an ex e nal se ice p o ide conduc s he s e iliza-
ion o medical de ices. The au ho s conside an in eg a ed
lo sizing and anspo a ion p oblem wi h de e minis ic
demand o de e mine he op imal o de imes and quan i-
ies. Diaman e al. [11] also assume ha s e iliza ion is
ou sou ced. The au ho s de e mine he minimum quan i-
ies equi ed o ensu e a de ined se ice le el o s ochas ic
demand.
The majo i y o publica ions on ep ocessing medical
de ices deal wi h ei he he ules o be obse ed in ep o-
cessing o he eliabili y o he o e all p ocess. Se e al
publica ions ocus on decon amina ion esou ces, which,
acco ding o [10], a e he bo leneck o he en i e ep o-
cessing cycle. Oz u k e al. [28] de eloped a mixed-in ege
linea p og am o model he decon amina ion s ep as a ba ch
scheduling p oblem wi h mul iple iden ical machines. Be o e
decon amina ion, he incoming medical de ices a e g ouped
in o a ba ch and assigned o a machine. The ea lies possi-
ble planning ime is ob ained o each medical de ice. The
au ho s aim o minimize he o al ep ocessing ime. Oz u k e
al. [27] de elop a p oblem-speci ic b anch & bound heu is ic
o sol e la ge es ins ances. Xu and Wang [47] gene alize
he p oblem p esen ed by [28] o he case o noniden ical
machines wi h di e en capaci ies. Fu he mo e, [26] exam-
ines a special case in which an ex e nal se ice p o ide
conduc s s e iliza ion. The used medical de ices a e collec ed
a e he ope a ion and sen o he se ice p o ide .
Thewo ko [40,41] and [36] add esses he ques ion
o whe he medical de ices should be s e ilized in a cen-
alized o decen alized manne . Howe e , he ques ion o
cen alized o decen alized ep ocessing and he pu chasing
o ep ocessing esou ces is la gely a s a egic decision.
Lo sizing p oblems a e ela ed o o de quan i y plan-
ning. The e a e nume ous app oaches in he li e a u e ha
ake pe ishabili y o limi ed shel li e in o accoun , e.g., [19]
and [29]. In addi ion, he e a e app oaches in which limi ed
s o age capaci y is conside ed in planning. See [12] and [20]
o he one-p oduc case and [2,23,25] and [46] o he mul i-
p oduc case. Nume ous app oaches ake a emanu ac u ing
op ion in o accoun when planning lo sizes. See [39] and
[37] o he one-p oduc case wi hou capaci y es ic ions.
App oaches o capaci y- es ic ed lo sizing wi h emanu-
ac u ing a e conside ed, o example, by [30,35] and [9].
To ully map he planning si ua ion in he supply o
s e ile goods, p ocu emen and ep ocessing ac i i ies mus
be planned simul aneously by conside ing ep ocessing and
s o age capaci ies and he limi ed s o age ime o medical
de ices. Howe e , he publica ions p esen ed abo e co e
only a ew aspec s o hese equi emen s. To he bes o
ou knowledge, publica ions ha inco po a e p ocu emen ,
ep ocessing and he limi ed shel li e o medical de ices do
no exis .
4 The in eg a ed p ocu emen and
ep ocessing planning p oblem o
eusable medical de ices
4.1 Model assump ions
In he in eg a ed P ocu emen and Rep ocessing Planning
P oblem (PRPP), he planning ho izon is di ided in o Tdis-
c e e pe iods ( ∈T). Typically, he leng h o he planning
ho izon depends on he su ge y schedule. Consequen ly, a
planning ho izon o one week is o en assumed. We con-
side a se en-day week whe e each day consis s o wo shi s
wi h a leng h o 8 hou s, i.e., each pe iod equals one shi .
Kdi e en medical de ices (k∈K)can be p ocu ed o
ep ocessed. In Fig. 1,c .[31], he ma e ial low o eusable
medical de ices is desc ibed.
P ocu emen o medical de ices
Medical de ices can be p ocu ed om an ex e nal supplie .
Fo be e di e en ia ion, he no a ion ela ed o he p ocu e-
men p ocess has been ma ked wi h supe sc ip o. Va iable
ma e ial cos s pco
ka e incu ed o each o de ed quan i y Qo
k
o medical de ice kin pe iod . In addi ion, each o de
o medical de ice kincu s quan i y-independen o de ing
cos s oco
k. The bina y a iable γo
k equals 1 i medical de ice k
is o de ed in pe iod . O he wise, his a iable γo
k is equal
o 0. No ably, he p ocu ed medical de ices a e deli e ed in
s e ile and p o ec ed condi ions. Fu he mo e, hey can be
used o s o ed di ec ly wi hou delay.
Rep ocessing o medical de ices
The comple e ep ocessing ope a ion includes he s eps o
p ecleaning, decon amina ion and s e iliza ion, as desc ibed
in Sec ion 2. Howe e , [26] no ed ha decon amina ion
esou ces, i.e., he washe -disin ec o s o cleaning, disin ec-
ion, insing and d ying, a e o pa icula impo ance, as hey
123

172 S. Ricke s and F. Sahling
P ocu emen o
medical de ices
P o ec ed
s o age
Rep ocessing
and s e iliza ion
P epa a ion in
ope a ing ooms
Unp o ec ed
s o age
Uns e ile
s o age
Fig. 1 Ma e ial low o eusable medical de ices
o en ep esen a bo leneck in e ms o ime in he ep ocess-
ing cycle. Thus, i is su icien o ocus on decon amina ion
esou ces.
Di e en ep ocessing ypes (s∈S={1,...,S})can
be iden i ied om he exis ing ime- empe a u e combina-
ions (c . Sec ion 2). Howe e , due o di e en empe a u e
ole ances, no e e y ime- empe a u e combina ion is pe -
missible o each medical de ice, e.g., he molabile medical
de ices canno be ep ocessed a 90◦C. Thus, he se Ks
includes hose medical de ices ha can be ep ocessed by
ype s. The subse Sk, on he o he hand, includes hose
ypes h ough which medical de ice kcan be ep ocessed.
I is wo h men ioning ha a ep ocessing ope a ion does no
necessa ily equi e homogeneous medical de ices; a he ,
di e en medical de ices can be ep ocessed a he same ime.
The in ege decision a iable χ
s deno es he numbe o
ep ocessing ope a ions o ype sca ied ou in pe iod .
He e, he no a ion ela ed o ep ocessing has been ma ked
wi h supe sc ip . The du a ion o a ep ocessing ope a ion
o ype sis desc ibed by s
s, which depends nei he on he
assigned medical de ices no on he ep ocessing quan i y.
The capaci y c
limi s he numbe o ep ocessing ope a-
ions ha can be ca ied ou in pe iod can be dele ed. The
ixed cos s sc
sa e incu ed o each ep ocessing ope a ion o
ype s. No ably, because o he highe ene gy consump ion o
hea ing, high- empe a u e ep ocessing, al hough sho e in
du a ion, is mo e cos ly han ime- empe a u e combina ions
wi h lowe empe a u es bu longe du a ions. Fu he mo e,
he space olmax o a ep ocessing ope a ion ha is con-
sumed by medical de ices kwi h space equi emen s olkis
also limi ed.
The decision a iable Qp
ks speci ies he ep ocessed quan-
i y o medical de ice kin pe iod using ype s, which
is s o ed p o ec ed e.g. in cabine s o d awe s a e wa ds
(deno ed by supe sc ip p). The packaging o one uni o
medical de ice kincu s packaging cos s pcp
ki s o ed p o-
ec ed. Analogously, Qu
ks speci ies he ep ocessed quan i y
o medical de ice kin pe iod using ype s ha is s o ed
unp o ec ed e.g. on shel es a e wa ds (deno ed by supe -
sc ip u). Packaging cos s pcu
kalso apply o each uni i
s o ed unp o ec ed. Since p o ec ed s o age equi es addi-
ional packaging compa ed wi h ha o unp o ec ed s o age,
he packaging cos s o p o ec ed s o age a e highe (pcp
k>
pcu
k∀k∈K). The ep ocessed medical de ices can be used
o ul ill he demand a he end o a pe iod.
Shel li e and s o age o medical de ices
Fo p o ec ed s o ed medical de ices Ip
k , he maximum shel
li e clea ly exceeds he leng h o he planning ho izon, so
he shel li e can be neglec ed in his case. Howe e , o
unp o ec ed s o age, he s o age du a ion mus be moni o ed
explici ly. The index h∈H={0,...,hmax,hmax +1}
desc ibes he numbe o pe iods o which he medical
de ices ha e al eady been s o ed unp o ec ed. A e hmax +1
pe iods o unp o ec ed s o age, a medical de ice exceeds
he maximum shel li e and will lose i s s e ile condi ion.
The pa ame e αhdesc ibes he s a e o s e ili y in he s o -
age pe iod h. Fo s o age pe iod h≤hmax, he pa ame e
αhis equal o 1. O he wise, he pa ame e αhequals 0
(αh=0∀h>hmax).
The in ege decision a iable Iu
k h indica es he in en-
o y o he unp o ec ed s o ed medical de ice ka he end
o pe iod in s o age pe iod h. The decision a iable Ip
k
desc ibes he s ock o p o ec ed s o ed medical de ices ka
he end o pe iod . Fu he mo e, he in en o y o used and
uns e ile medical de ices ka he end o pe iod is deno ed
by I
k .
In each pe iod , he in en o y o unp o ec ed and p o-
ec ed s o ed s e ile medical de ices is limi ed by he capaci y
limi cIuo cIp. The pa ame e olkindica es he s o age
space equi emen o one uni o medical de ice k. Howe e ,
he s o age capaci y o used medical de ices is assumed o
be unlimi ed.
123
In eg a ed p ocu emen and ep ocessing... 173
Demand ul illmen and e u ns
The dynamic demand dk o medical de ice kin pe iod is
de i ed om he su ge y schedule. Thus, i is assumed ha he
pe iod-speci ic demand dk is known in ad ance and mus be
comple ely sa is ied. The equi emen s o medical de ice k
in pe iod can be co e ed by bo h p o ec ed and unp o ec ed
s o ed medical de ices. The espec i e wi hd awal quan i-
ies om s o age a e e e ed o as s aging quan i ies. The
decision a iable Au
k h co esponds o he s aging quan i y o
unp o ec ed s o ed medical de ice kin pe iod wi h s o age
du a ion h. Fu he mo e, Ap
k deno es he s aging quan i y o
p o ec ed s o ed medical de ice kin pe iod .
A e u iliza ion, a po ion 0 ≤βk≤1 o medical de ice k
e u ns o he depo o used medical de ices in pe iod .How-
e e , a ime delay o wo pe iods is assumed, so he e u ns k
o medical de ice kcan be de e mined by k =βk·dk, −2.
Due o damage o signs o aging, he non e u ning po -
ion (1−βk)o medical de ice kcanno be ep ocessed and
mus be disposed o .
The goal o he PRPP is o de e mine a easible p o-
cu emen and ep ocessing plan ha comple ely sa is ies he
de i ed demand and minimizes he p ocu emen and ep o-
cessing cos s.
4.2 Ma hema ical model o mula ion
Using he no a ion p esen ed in Table 1, he in eg a ed p o-
cu emen and ep ocessing planning p oblem o eusable
medical de ices can be ma hema ically modeled as ollows:
Model PRPP
min Z=
k∈K
∈Toco
k·γo
k +pco
k·Qo
k +
s∈S
∈T
sc
s·χ
s
+
k∈K
s∈Sk
∈Tpcu
k·Qu
ks +pcp
k·Qp
ks (1)
subjec o
Ap
k +
hmax

h=0
Au
k h =dk ∀k∈K, ∈T(2)

s∈Sk
Qu
ks −Au
k 0=Iu
k 0∀k∈K, ∈T(3)
αh·Iu
k, −1,h−1−Au
k h =Iu
k h ∀k∈K, ∈T,h∈H {0}
(4)
Table 1 No a ion used o he PRPP
Indices and index se s:
h∈Hse o s o age pe iods
(h∈{0,...,hmax,hmax +1})
k∈Kse o medical de ices (k∈{1,...,K})
s∈Sse o ypes (s∈{1,...,S})
∈Tse o pe iods ( ∈{1,...,T})
k∈Ks⊆Ksubse o medical de ices equi ing ype s
s∈Sk⊆Ssubse o ypes ha can ep ocess medical
de ice k
Pa ame e s:
αhshel li e indica o in s o age pe iod h
βkpo ion o ep ocessable medical de ice k
a e u iliza ion
bigMk su icien ly la ge numbe o medical
de ice kin pe iod
cIus o age capaci y o unp o ec ed s o ed
medical de ices
cIps o age capaci y o p o ec ed s o ed
medical de ices
c
capaci y o ep ocessing esou ces in
pe iod
dk demand o medical de ice kin pe iod
oco
k ixed p ocu emen cos s pe o de o
medical de ice k
pco
k a iable p ocu emen cos s pe uni o
medical de ice k
pcp
k a iable packaging cos s o one p o ec ed
s o ed uni o medical de ice k
pcu
k a iable packaging cos s o one
unp o ec ed s o ed uni o medical
de ice k
k e u ns o medical de ice kin pe iod
sc
s ixed cos s o a ep ocessing ope a ion o
ype s
s
sdu a ion o a ep ocessing ope a ion o
ype s
olkspace equi emen o ep ocessing o
s o ing one uni o medical de ice k
olmax space capaci y o ep ocessing
Decision a iables:
Ap
k ∈N0s aging quan i y o p o ec ed s o ed
medical de ice kin pe iod
Au
k h ∈N0s aging quan i y o unp o ec ed s o ed
medical de ice kin pe iod and s o age
pe iod h
Ip
k ∈N0p o ec ed end-o -pe iod in en o y o s e ile
medical de ice kin pe iod
I
k ∈N0end-o -pe iod in en o y o used medical
de ice kin pe iod
Iu
k h ∈N0unp o ec ed end-o -pe iod in en o y o
s e ile medical de ice kin pe iod and
s o age pe iod h
123
174 S. Ricke s and F. Sahling
Table 1 con inued
Qo
k ∈N0o de ed quan i y o medical de ice kin
pe iod
Qp
ks ∈N0 ep ocessing quan i y o medical de ice k
wi h ype sin pe iod wi h subsequen
p o ec ed s o age
Qu
ks ∈N0 ep ocessing quan i y o medical de ice k
wi h ype sin pe iod wi h subsequen
unp o ec ed s o age
γo
k ∈{0,1}bina y o de ing a iable o medical
de ice kin pe iod
χ
s ∈N0numbe o ep ocessing ope a ions o
ype sin pe iod
Ip
k, −1+Qo
k +
s∈Sk
Qp
ks −Ap
k =Ip
k ∀k∈K, ∈T
(5)
I
k, −1+ k +
hmax+1

h=1
(1−αh)·Iu
k, −1,h−1−
s∈SkQp
ks +Qu
ks =I
k
∀k∈K, ∈T(6)
Qo
k ≤bigMk ·γo
k ∀k∈K, ∈T(7)

k∈Ks
olk·Qp
ks +Qu
ks ≤ olmax ·χ
s ∀s∈S, ∈T(8)

s∈S
s
s·χ
s ≤c
∀ ∈T(9)

k∈K
hmax

h=0
olk·Iu
k h ≤cIu∀ ∈T(10)

k∈K
olk·Ip
k ≤cIp∀ ∈T(11)
Ap
k ,Au
k h ∈N0∀k∈K, ∈T,h∈H(12)
Ip
k ,I
k ,Iu
k h ∈N0∀k∈K, ∈T,h∈H(13)
Qo
k ,Qp
ks ,Qu
ks ∈N0∀k∈K,s∈Sk, ∈T(14)
χ
s ∈N0∀s∈S, ∈T(15)
γo
k ∈{0,1}∀k∈K, ∈T(16)
The in en o y balance cons ain s a e ep esen ed by (2)
o (6). Equa ions (2) ensu e ha he gi en demand dk is
ul illed comple ely by he cumula i e s aging quan i ies o
each medical de ice kin pe iod . Acco ding o cons ain s
(3), he in en o y o unp o ec ed medical de ices wi h s o -
age ime h=0 only consis s o he di ec ly ep ocessed
quan i ies in he conside ed pe iod , unless he de ices a e
di ec ly used o demand ul illmen . Equa ions (4) ep e-
sen he in en o y balance cons ain s o unp o ec ed s o ed
s e ile medical de ices wi h s o age ime h≥1. Howe e ,
hese es ic ions also ensu e ha medical de ices ha each
he maximum s o age ime h=hmax +1inpe iod will
lose hei s e ile condi ion. Equa ions (5) ep esen he in en-
o y balance cons ain s o p o ec ed s o ed medical de ices.
Cons ain s (6) desc ibe he in en o y balance equa ions o
nons e ile medical de ices, including medical de ices wi h
an expi ed s o age ime.
The cons ain s (7) link he in ege a iables o p ocu e-
men Qo
k wi h he bina y a iables γo
k . I medical de ice kis
o de ed in pe iod (Qo
k >0), an o de ing p ocess is needed.
This o ces he bina y o de a iable γo
k o he alue one. The
pa ame e bigMk ep esen s a su icien ly la ge numbe and
is de ined as ollows:
bigMk =
T

τ=
dkτ∀k∈K, ∈T.(17)
Cons ain s (8) combine he ep ocessing quan i ies Qp
ks
and Qu
ks wi h he numbe o ep ocessing ope a ions χ
s .
I a leas one medical de ice kis ep ocessed in pe iod
wi h ype s, i.e., kQp
ks +Qu
ks >0, he in ege a iable
χ
s equals he equi ed numbe o ep ocessing ope a ions.
The capaci y cons ain s (9) ensu e ha he gi en capaci y
o he ep ocessing esou ce is no exceeded; i.e., he maxi-
mum numbe o ep ocessing ope a ions ha can be ca ied
ou in pe iod is limi ed. Cons ain s (10) and (11) es ic
he s o age capaci ies o p o ec ed and unp o ec ed s o ed
s e ile medical de ices. Cons ain s (14) o(16) de ine he
dimensions o he decision a iables.
I he ep ocessing capaci y c
is se o ze o o all pe i-
ods, medical de ices canno be ep ocessed and he comple e
pe iod-speci ic demand dk o medical de ice kmus be sa -
is ied by p ocu emen . In his case, he PRPP co esponds o
an uncapaci a ed lo sizing p oblem wi h in en o y bounds.
Since his lo sizing p oblem is p o en o be NP-ha d (see
[2]), he PRPP is also NP-ha d. Due o he NP-ha dness o
he PRPP, he compu a ional e o o sol e his p oblem op i-
mally using a s anda d MILP sol e is usually p ohibi i ely
la ge o all bu iny p oblem ins ances. Thus, a heu is ic is
equi ed o de e mine an app op ia e solu ion wi hin a ea-
sonable ime ame.
123
In eg a ed p ocu emen and ep ocessing... 175
5 A solu ion app oach based on column
gene a ion
5.1 Idea o Dan zig-Wol e decomposi ion and
column gene a ion
The p oposed solu ion app oach is based on Dan zig-Wol e
decomposi ion, which is used o e o mula e he PRPP. The
PRPP is decomposed in o a mas e p oblem deno ed as MP-
PRPP and Kde ice-speci ic subp oblems deno ed as SP-
PRPPk. A column gene a ion (CG) app oach is applied o
sol e he mas e p oblem. The mas e p oblem is ini ialized
wi h a small numbe o columns. In an i e a i e p ocedu e,
he subp oblems a e sol ed o gene a e new columns o he
mas e p oblem. I a new column will lead o a educ ion
in he objec i e unc ion alue o he mas e p oblem, i is
inco po a ed in o he mas e p oblem. Howe e , he column
gene a ion app oach e mina es i no u he columns can be
gene a ed ha educe he cu en objec i e unc ion alue o
he mas e p oblem. The p ocess o he CG app oach o he
PRPP is desc ibed below. Fu he implemen a ion de ails o
he solu ion app oach can be ound in [31].
5.2 The mas e p oblem
F om a ma hema ical pe spec i e, he mas e p oblem co e-
sponds o a se pa i ioning e o mula ion o he PRPP. The
objec i e o he Se Pa i ioning P oblem (SPP) is o selec
exac ly one p ocu emen and ep ocessing plan o each med-
ical de ice ka a minimal o al cos . The selec ion o plans
mus mee he capaci y es ic ions o p o ec ed and unp o-
ec ed s o age as well as o ep ocessing.
Fi s , i is assumed ha all possible and easible p o-
cu emen and ep ocessing plans Nka e known o medical
de ice kin ad ance. A p ocu emen and ep ocessing plan
is easible i he demand is me in each pe iod. A p ocu e-
men and ep ocessing plan nis desc ibed by p ocu emen
quan i ies Qo(n)
k and decisions γo(n)
k . In addi ion, each plan n
p o ides in o ma ion ega ding he quan i ies o p o ec ed
and unp o ec ed ep ocessed medical de ices ko ype sin
pe iod , which a e desc ibed by pa ame e s Qp(n)
ks and Qu(n)
ks .
Fu he mo e, each plan ncon ains he end-o -pe iod in en-
o y o p o ec ed Ip(n)
k and unp o ec ed Iu(n)
k h s o ed medical
de ices kin pe iod , whe e he s o age du a ion his also
known.
The ixed and a iable p ocu emen cos s o medical
de ice kin plan ncan be de e mined wi h espec o γo(n)
k
and Qo(n)
k . Fo plan n, he a iable packaging cos s o med-
ical de ice kcan be calcula ed using he pa ame e s Qp(n)
ks
and Qu(n)
ks . Howe e , he ep ocessing cos s depend on he
selec ed plans and he numbe o ep ocessing ope a ions χ
s .
Thus, hese cos s mus be implici ly aken in o accoun in he
objec i e unc ion o he mas e p oblem.
The pa ame e s Qp(n)
ks and Qu(n)
ks allow o he de e mina-
ion o he capaci y equi emen s o ep ocessing medical
de ice kusing ype sin pe iod wi h plan n. In addi ion, he
equi ed s o age capaci y o p o ec ed o unp o ec ed s o -
age can be de i ed wi h espec o Ip(n)
k and Iu(n)
k h o medical
de ice kin pe iod wi h plan n.
Fo he selec ion o a plan n o medical de ice k, he
bina y a iable ϑkn is used, which is de ined as ollows:
ϑkn =1,i plan n ∈Nkis selec ed o medical de ice k
0,o he wise.
(18)
The objec i e o he mas e p oblem is o selec exac ly
one plan o each medical de ice kso ha he o al p ocu e-
men and ep ocessing cos s a e minimized and he capaci y
es ic ions a e me .
The model o mula ion o he mas e p oblem is in o-
duced using he addi ional no a ion in Table 2.
Model MP-PRPP
min Z=
k∈K
n∈Nk
∈Tpco
k·Qo(n)
k +oco
k·γo(n)
k ·ϑkn
+
k∈K
n∈Nk
s∈Sk
∈Tpcp
k·Qp(n)
ks +pcu
k·Qu(n)
ks ·ϑkn
+
s∈S
∈T
sc
s·χ
s (19)
subjec o dual a iables

k∈Ks
n∈Nk
olk·Qp(n)
ks +Qu(n)
ks ·ϑkn ≤ olmax ·χ
s
∀s∈S, ∈T→π
s (20)

s∈S
s
s·χ
s ≤c
∀ ∈T→π
s (21)

k∈K
n∈Nk
⎛
⎝
hmax

h=0
olk·Iu(n)
k h ⎞
⎠·ϑkn ≤cIu∀ ∈T→πIu
(22)

k∈K
n∈Nk olk·Ip(n)
k ·ϑkn ≤cIp∀ ∈T→πIp
(23)
123
182 S. Ricke s and F. Sahling
based on he ixed numbe o ep ocessing ope a ions. The
a e age compu a ional ime equi ed o de e mine his ini ial
solu ion o PRPP based on his ixa ion was less han one
second in he case o PC I, less han i e seconds o PC II
and less han 21 seconds o PC III.
When he ime limi TimLimUB is eached, on a e age,
he uppe bound de ia es by less han 1% om he CPLEX
e e ence solu ion o all PCs. This emphasizes he high solu-
ion quali y. This de ia ion ends o inc ease sligh ly in all
PCs i he ep ocessing capaci y c
is dec eased. The in e-
g ali y gap ∅In GapUB,LB is less han 2% e en in PC III and
is he e o e e y small.
In addi ion, o mo e han 10% o he ins ances, a solu ion
was ound ha is a leas as good o be e han he CPLEX
e e ence solu ion. The a e age compu a ional e o needed
o de e mine his solu ion amoun s o less han 10% o he
p o ided ime limi o CPLEX o gene a e a e e ence solu-
ion. Howe e , i is possible ha CPLEX will ind a good
solu ion a he e y beginning o he op imiza ion p ocess
and use he majo i y o he compu a ional e o o p o e op i-
mali y o o aise he lowe bound. Thus, he in es iga ions
below should allow a ai e compa ison be ween CPLEX and
he p oposed solu ion app oach.
Re e ence solu ions wi h a compa able compu a ional ime
To de ine an admissible ime limi o CPLEX o de e mine
a e e ence solu ion o each PC, he a e age compu a ional
ime TimCG o gene a ing he lowe bound (see Table 7)
was i s ounded up o he nex ull minu e. Then, he gi en
ime limi TimLimUB o gene a ing an uppe bound, acco d-
ing oTable8, was added. Since he compu a ional e o
o de e mining he lowe bound a ies depending on he
ins ance, he sum o he un imes was mul iplied by a ac o
o 1.5. The esul ing PC-speci ic ime limi is shown in col-
umn TimLimCPX
ed in Table 9. No ably, wi hin his new ime
limi , he solu ion app oach based on column gene a ion e -
mina es ea lie in mo e han 98% o he ins ances. This ime
limi co esponds o abou an eigh h o he p e ious ime
limi ha was gi en o CPLEX o de e mining he e e ence
solu ions in Sec ion 6.2.
Column FeasSolCPX
ed in he uppe pa o Table 9indi-
ca es he p opo ion o ins ances o which CPLEX ound
a easible solu ion wi hin he educed ime limi . Fo each
es ins ance, he in eg ali y gap In GapCPX
ed is de e mined.
Column ∅In GapCPX
ed shows he a e age in eg ali y gap o
each PC. In column In GapCPX
max, ed, he maximum in eg ali y
gap is gi en.
The esul s o he p oposed solu ion app oach a e desc ibed
in he lowe pa o Table 9. The s uc u e o his able is
simila o he s uc u e o Table 8. The addi ional column
De UB,CPX
max, ed indica es he maximum de ia ion o he uppe
bound om he new e e ence solu ion. Column Be SolUB
ed
again gi es he p opo ion o ins ances o which he objec-
i e unc ion alue is a leas as good o be e han he new
e e ence solu ion. No ably, o he compa ison, only es
ins ances o which CPLEX was able o ind a easible solu-
ion wi hin he speci ied new ime limi we e examined.
Wi hin his new ime limi , CPLEX was no able o ind a
easible solu ion o all TIs. The po ion o TIs o which no
easible solu ion was ound wi hin his ime limi inc eases
wi h he numbe o medical de ices. While CPLEX ound
a easible solu ion o mo e han 98% o ins ances in PC I
and II, CPLEX ailed o de e mine a easible solu ion o 28
o 216 es ins ances in PC III. Ou solu ion app oach, on he
o he hand, was able o gene a e a easible solu ion wi h less
compu a ional e o o all TIs.
Fo he TIs wi h a easible solu ion, he a e age in eg ali y
gap ∅In GapCPX
ed is ela i ely small o all PCs. In eg ali y
gaps highe han 25% a e only ound o indi idual ou lie s.
I is wo h men ioning ha he p oposed solu ion app oach
de e mines easible solu ions o high quali y ha de ia e
by less han 0.4% om he e e ence solu ion on a e age,
e en o he la ges PC. Fo app oxima ely 30% o he es
ins ances, he solu ion app oach e mina es wi h a easible
solu ion ha is a leas as good as o be e han he new
CPLEX e e ence solu ion.
Table 9 Compa ison o he
e e ence solu ions wi h a
educed ime limi
CPLEX sol e
FeasSolCPX
ed ∅In GapCPX
ed In GapCPX
max, ed TimLimCPX
ed
PC I 99.07% 0.52% 1.98% 480 s
PC II 98.61% 1.04% 2.70% 900 s
PC III 87.04% 1.22% 25.17% 1800 s
Solu ion app oach
FeasSolUB
ed ∅De UB,CPX
ed DecUB,CPX
max, ed Be SolUB
ed
PC I 100.00% 0.13% 1.37% 30.55%
PC II 100.00% 0.14% 3.78% 35.65%
PC III 100.00% 0.32% 16.15% 27.78%
123

In eg a ed p ocu emen and ep ocessing... 183
6.4 Compa ison o i s –come- i s -se ed
simula ion
To p o ide a baseline o u he compa ison, we ollow he
idea o [10] o use a i s -come– i s -se ed (FCFS) app oach
o ep ocessing medical de ices. Such a FCFS app oach is
qui e common in SSDs. The e o e, we implemen ed a simu-
la ion app oach which is guided by he FCFS p inciple.
The p ocedu e o each pe iod can be desc ibed as ol-
lows:
•A he beginning o each pe iod, he ne demand is de i ed
o each medical de ice kby aking he cu en in en-
o y in o accoun . No ably, he in en o y is wi hd awn
acco ding o he FIFO p inciple, whe e medical de ices
in unp o ec ed s o age a e p e e ed.
•Based on he pe iod-speci ic e u ns k o all medical
de ices, he e u ns a e andomly a anged uni by uni
and s o ed in se ial o de . Following his o de , we y o
assign all e u ned uni s o a ep ocessing ope a ion ia
he ollowing p ocedu e:
1. I a leas one ep ocessing ope a ion wi h su icien
capaci y is scheduled, he cu en medical de ice uni
is assigned o he ep ocessing ope a ion wi h he
as es ep ocessing ime,
i. i he e is unsa is ied demand o his medical
de ice, his uni is aken di ec ly o demand sa -
is ac ion;
ii. else i he ne demand has been ul illed and
enough s o age capaci y is le , his uni is s o ed
(unp o ec ed s o age p e e ed);
iii. o he wise he cu en uni is skipped.
•Rega dless o which o he abo e cases i. o iii. was
selec ed, in en o y le els a e se acco ding o in en o y
balance cons ain s (2) o(6).
2. else i he capaci y o he ep ocessing esou ce is
su icien and ei he ne demand o he cu en med-
ical de ice is unsa is ied o s o age capaci y is le ,
an addi ional ep ocessing ope a ion is scheduled
(p e e ably wi h he as es ep ocessing ime, ha
can ep ocess he cu en medial de ice uni ) and go
o 1;
3. o he wise he cu en uni is skipped.
•I he cu en medical de ice uni has no successo , he
assignmen o he cu en pe iod ends. Demand ha is
no me by nei he in en o y no ep ocessing mus be
p ocu ed. Medical de ice uni s ha ha e been skipped
a e conside ed i s in he ollowing pe iod +1.
We passed he ob ained simula ion esul s o he PRPP,
whe e we ixed he a iable alues, i.e., a iables (12)–
(16), o e i y easibili y o he p ocu emen and ep ocessing
plans.
Nex , we analyze he esul s o he simula ion om a man-
age ial pe spec i e; he e o e, six TI o PC 1 we e selec ed
o he simula ion ha di e in a ailable esou ce c (low,
medium, and high) and s o age capaci y cIu(low, high).
Fo each ins ance, 1000 eplica ions we e pe o med, and
he mean alues we e de e mined. These mean alues a e
compa ed wi h he co esponding solu ion o he monoli hic
PRPP. Ou e alua ion concen a es on he ob ained p ocu e-
men and ep ocessing plans.
An ins ance-speci ic cos compa ison is p o ided in
Table 10.
The esul s o he simula ion s udy show ha he mean
alues o he o al cos s a e on a e age 3.76 imes highe
han he objec i e unc ion alue o he monoli hic PRPP.
The 95% con idence in e al is a he small. In he case o
low esou ce capaci y, he o al cos is highe , bu he 95%
con idence in e al is smalle compa ed o he case o high
esou ce capaci y.
While he o al numbe o ep ocessing ope a ions sched-
uled du ing he simula ion is app oxima ely 17.5% lowe
han ha o he solu ion o he PRPP, he simula ed capaci y
consump ion is only sligh ly lowe wi h a de ia ion be ween
1.3% and 4.5%. This can be explained by he ac ha slow
ep ocessing ope a ions we e scheduled mo e o en han as
ep ocessing ope a ions. In con as , as ep ocessing ope -
a ions a e p io i ized du ing op imiza ion o inc ease he
numbe o ep ocessed de ices and hus a oid p ocu emen .
Du ing he simula ion, medical de ices ha can be ep o-
cessed by a as e ep ocessing ope a ion a e also in eg a ed
in o slow ep ocessing ope a ions. A e op imiza ion, his
case is a ely obse ed. Scheduling a la ge numbe o slow
ep ocessing ope a ions means ha he sca ce capaci y o
he decon amina ion esou ce is used up mo e quickly. As a
esul , o de s mus be placed myopically and he e o e much
mo e equen ly. Fo example, we ound ha he simula ed
solu ions exceeded bo h he o de s placed and he quan i ies
p ocu ed by a ac o o ou .
The simula ion esul s also show ha some medical
de ices exceeded hei maximum shel li e a e ep ocess-
ing. The e o e, ep ocessing o his de ice was unnecessa y.
This happens e y a ely o medical de ices wi h egula
high demand. Howe e , o p oduc s wi h spo adic demand,
we obse ed ha p oduc s exceeded shel -li e up o wice
a week. This esul s in unnecessa y ep ocessing cos s as
well as unnecessa y capaci y consump ion. In he op i-
mized solu ion, shel -li e exceedance does no occu a e
ep ocessing.
123
184 S. Ricke s and F. Sahling
Table 10 Cos compa ison
be ween PRPP and simula ion
s udy
Case (cIu/c ) Obj. PRPP Mean Value Mean De ia ion
incl. 95% con idence in e al
Case 1 (high/low) 95,793.5 370,727.3±190.6 387.01% ±0.20%
Case 2 (high/medium) 86,921.5 335,693.5±265.7 386.20% ±0.31%
Case 3 (high/high) 80,002.5 287,354.9±529.3 359.19% ±0.66%
Case 4 (low/low) 95,868.5 363,698.3±159.1 379.37% ±0.17%
Case 5 (low/medium) 87,482.5 332,738.4±203.7 380.35% ±0.23%
Case 6 (low/high) 80,937.5 294,942.2±424.5 364.41% ±0.52%
7 Conclusion and ou look
In his pape , we p esen ed a new model o mula ion o in e-
g a ed p ocu emen and ep ocessing planning o eusable
medical de ices wi h a limi ed shel li e. Based on a su ge y
schedule wi h a planning ho izon o one week, he objec-
i e o he PRPP is o de e mine a easible p ocu emen and
ep ocessing plan o he SSD in a hospi al. To sol e he
PRPP, a solu ion app oach based on he p inciple o Dan zig-
Wol e decomposi ion and column gene a ion was p esen ed.
As pa o an ex ensi e nume ical s udy, he solu ion quali y
o his app oach was examined. The in eg ali y gaps o he
lowe bound ob ained by column gene a ion a e e y close
o he op imal solu ion. The applied solu ion app oach yields
a high solu ion quali y, and he solu ions a e e y close o
he CPLEX e e ence alues. Howe e , he nume ical e o
is subs an ially educed. When he ime limi was educed,
unlike he p esen ed solu ion app oach, CPLEX was no able
o ind a easible solu ion o all TIs.
Fu u e esea ch will add ess di e en model ex ensions.
Rega ding he p ocu emen o medical de ices, minimum
o de quan i ies o quan i y discoun s as well as he selec-
ion o ex e nal supplie s can be in eg a ed in o he model
o mula ion. Rega ding ep ocessing, he model o mula-
ion can be ex ended o ake se e al pa allel esou ces in o
accoun o educe bo lenecks in ep ocessing ope a ions.
Howe e , i hese esou ces do no di e , his ex ension
in ol es many edundan o symme ic solu ions. Thus, addi-
ional cons ain s a e equi ed o a oid symme ies. Hence,
an adap a ion o he solu ion app oach is likely needed, since
a smalle numbe o in ege a iables can be expec ed o
appea in he inal solu ion o he elaxed mas e p oblem.
Addi ionally, he subp oblems can be sol ed heu is ically o
educe he nume ical e o . Howe e , o gua an ee a easible
lowe bound, he MILP sol e is only equi ed i he heu is ic
canno iden i y any u he plan wi h nega i e educed cos s.
This app oach can be ex ended o an exac b anch&p ice
app oach, whe e column gene a ion occu s in each node o
he sea ch ee.
Fu he mo e, he demand o medical de ices is de i ed
di ec ly om he su ge y schedule. Thus, he demand can
be assumed o be de e minis ic. I is also assumed ha he
su ge y schedule is execu ed wi hou any changes, such ha
he use o medical de ices always akes place in he spec-
i ied pe iod. Howe e , due o sho - e m s a absences o
u gen eme gency su ge ies, de ia ions om he schedule
may occu , leading o unce ain y in bo h demand and e u n
da a. To cope wi h hese unce ain ies, e.g. sample a e age
app oaches can be used, c . [16] and [18], o de e mine a
obus p ocu emen and ep ocessing plan. Depending on
he condi ion o he used medical de ice, i may be neces-
sa y o go h ough he ep ocessing cycle se e al imes. As a
esul , he a ec ed medical de ice may no be used immedi-
a ely a e ep ocessing. The e o e, u he esea ch should
also in es iga e he obus ness o he de e mined p ocu e-
men and ep ocessing plan. In addi ion, he simul aneous
conside a ion o s ochas ic demand and e u n da a should
be conside ed.
A De ailed desc ip ion o he es ins ances
Fo he gene a ion o es ins ances, see also [31], we assume
a hospi al in which su ge ies o elec i e pa ien s ake place in
wo shi s om Monday o F iday. The e o e, he demand o
medical de ices is posi i e in a mos en pe iods. He ea e ,
hese pe iods a e called su gical pe iods ∈T. In Table 11,
he su gical pe iods a e highligh ed in bold.
The SSD, on he o he hand, also ope a es in wo shi s
( ime slo s 1 and 2), bu hese shi s a e om Monday o Sun-
day. In an al e na i e scena io, he planning ho izon begins on
Thu sday o a y he posi ion o he su gical pe iods. Thus,
he hi d ime slo on Wednesday ep esen s he las pe iod
o he planning ho izon in he second scena io.
Table 11 Su gical pe iods (s a o planning: Monday)
Mon. Tue. Wed. Th . F i. Sa . Sun.
Time slo 1 147101316 19
Time slo 2 258111417 20
Time slo 3 3 6 9 12 15 18 21
123
In eg a ed p ocu emen and ep ocessing... 185
Using an ABC analysis, [32] examines bo h he demand
and he u no e a io o medical de ices. They poin ou ha
a ela i ely small po ion o medical de ices accoun o mos
o he demand. In con as , he majo i y o medical de ices
possess an i egula demand wi h small quan i ies. Following
his obse a ion, an XYZ analysis is used o assign medical
de ices acco ding o hei demand and u no e a io o one
o h ee classes κ∈{X,Y,Z}. The cha ac e is ics associa ed
wi h each class a e summa ized in Table 12.
Class Xincludes egula ly equi ed basic ins umen s.
Medical de ices in his class a e equi ed in each su gical
pe iod; i.e., he demand is compa a i ely high. Depa men -
speci ic ins umen s cons i u e Class Y, while Class Z
includes special su ge y-speci ic ins umen s. The medical
de ices in Class Za e he e o e equi ed less o en. Thus,
hese medical de ices a e only equi ed in one ou o en
su gical pe iods. Fo example, in PC I, en di e en medical
de ices belong o Class Z. Each o hese medical de ices is
equi ed in a di e en su gical pe iod. The demand o med-
ical de ices in Classes Yand Zis e enly dis ibu ed o e
he su gical pe iods.
The expec ed alue μκ
ko he demand o medical de ice k
is de e mined by a uni o m dis ibu ion in a gi en in e -
al depending on he assigned κ.μX
k∈[150,250],μY
k∈
[40,60]and μZ
k∈[4,6]apply o hese in ege in e als,
so he class-speci ic expec ed alues o he demand a e
μX=200, μY=50 and μZ=5. Using a ( unca ed)
no mal dis ibu ion, he demand dκ
k is hen gene a ed in he
su gical pe iods and ounded o he nex in ege . The s an-
da d de ia ion is σk=1
10 ·μκ
k, so he coe icien o a ia ion
cdis 1
10 . In an al e na i e scena io, he s anda d de ia ion is
σk=3
10 ·μκ
k; hus, he coe icien o a ia ion is 3
10 . No ably,
he no mal dis ibu ion heo e ically enables a nega i e eal-
iza ion o he demand. These ealiza ions a e se o 0.
Fo he p opo ion o e u ned medical de ices, h ee
di e en alues a e de ined in Classes Xand Y, namely,
βX
k,βY
k∈{98%,95%,90%}. Fo medical de ices o Class
Z, on he o he hand, i is assumed ha due o he low
u no e a io and he associa ed lowe le el o wea and
ea , no medical de ices need o be disposed o (βZ
k=1).
The e u n k o medical de ice kin pe iod is de e mined
by k =βκ
k·dk, −2. The e u n quan i ies in he i s and
second pe iods o he planning ho izon depend on he class,
and k1= k2=βκ
k·μdκ.
Table 12 Class-speci ic cha ac e is ics o medical de ices
su gical pe iods P opo ion Dis ibu ion o
wi h dk >0 demand
Class X10 o 10 X=20% dX
k ∼N(μX
k,σ2
k)
Class Y4o 10 Y=30% dY
k ∼N(μY
k,σ2
k)
Class Z1o 10 Z=50% dZ
k ∼N(μZ
k,σ2
k)
The ixed p ocu emen cos s oco
ka e no malized o oco
k=
250 o each medical de ice k. The a iable cos s pco
k o one
uni o medical de ice k ollow a uni o m dis ibu ion in he
eal- alued in e al pco
k∈[25,125]and a e hen ounded
o in s eps o 0.5.
The medical de ices a e assigned o he p ocessing ypes
based on he A0concep explained in Sec ion 2. Due o he
empe a u e sensi i i y o some medical de ices, i can be
expec ed ha only a compa a i ely small po ion o medical
de ices can be ep ocessed a e y high empe a u es. Since
compa ibili y es ic ions only exis o empe a u es ha a e
oo high, all medical de ices can be ep ocessed using he
ep ocessing ype wi h he lowes empe a u e. The diag am
in Fig. 4illus a es his ela ionship.
The medical de ices a e assigned o he subse Ks1wi h
p obabili y P[k∈Ks1]=1
3. I an assignmen is made, hese
medical de ices can also use ep ocessing ypes s2and s3.
O he wise, he emaining medical de ices a e assigned o he
subse Ks2wi h p obabili y 1
2; i.e., P[k∈Ks2]=1
3+2
3·1
2=
2
3. The s e iliza ion ype s3can be used by all medical de ices,
i.e., (P[k∈Ks3]=1).
The a iable packaging cos s o unp o ec ed and p o-
ec ed s o age a e pcu
k=1o pcp
k=2 pe uni o
medical de ice k. The ep ocessing cos s sc
spe ope a ion
o ype sand he associa ed du a ion s
sa e also based
on he A0concep . Consequen ly, he ep ocessing ime
dec eases wi h inc easing empe a u e. Fu he mo e, ep o-
cessing ypes wi h high p ocess empe a u es incu highe
cos s due o he highe ene gy equi emen . Table 13 shows
he ep ocessing cos s sc
sand he du a ion s
spe ope a ion
o ype s.
The space equi emen olk o ep ocessing one uni
o medical de ice kis uni o mly dis ibu ed in he in e al
olk∈[5,15]. The a e age space equi emen olκis de e -
mined o each class κ. The capaci y olmax pe ep ocessing
ope a ion depends on he PC. In PC I, olmax =250 capac-
i y uni s pe ope a ion a e a ailable, while in PC II o PC III,
500 o 1,000 capaci y uni s pe ope a ion a e a ailable. The
capaci y pe ep ocessing ope a ion is a ied in each p oblem
class by he ac o g∈{
4
/5,1,6
/5}.
Fig. 4 Assignmen o medical de ices o ep ocessing ypes Ks
123
186 S. Ricke s and F. Sahling
Table 13 Rep ocessing ype-speci ic cos s and du a ion pe ep ocess-
ing ope a ion
Cos s sc
sDu a ion s
s
Rep ocessing ype s1200 25
Rep ocessing ype s2100 40
Rep ocessing ype s350 60
The capaci y depends on he PC and he egula ep o-
cessing capaci y olmax. F om Monday o F iday, he e a e
c
=1−cem·|K|·κκ·μκ· olκ·s∈S s
s
2·|S|· olmax 
capaci y uni s a ailable in he i s and second ime slo s,
while his capaci y is hal ed on weekends. Fu he mo e,
he e is no capaci y a ailable in he hi d ime slo . Simila
o he su ge y schedule, he capaci y o ep ocessing medi-
cal de ices o eme gency pa ien s is ese ed in he SSD in
he amoun o cem ∈[0,1]. This ac o is examined in h ee
di e en o ms cem ∈{20%,10%,0%}.
The PC-speci ic s o age capaci y o p o ec ed medical
de ices is limi ed in each pe iod o
cIp=1
2·|K|·
κ
κ·μκ· olκ ∀ ∈T.
Depending on his size, wo o ms cIu∈1
/3·cIp,1
/2·
cIp a e analyzed. Iu
k01 =Iu
k02 =1
/10 ·μκapplies
o he ini ial in en o y o unp o ec ed and p o ec ed s o ed
medical de ices, and Ip
k0=1
3·μκapplies o each medical
de ice k. The ini ial in en o y o used medical de ices is
I
k0=1
/3·μκ.
Acknowledgemen s We would like o hank Di k B isko n and Jens
B unne o hei aluable suppo du ing he ini ial phase o his pape .
Fu he mo e, we would like o hank S e an Helbe o his help ul sup-
po du ing his esea ch p ojec . We hank he anonymous e iewe s
o hei ca e ul eading o ou manusc ip . Thei many insigh ul com-
men s and sugges ions helped o imp o e and cla i y his manusc ip .
The a icle has been de eloped om he PhD hesis o S e en Ricke s
(2022). The esul s p esen ed he e we e achie ed by compu a ions ca -
ied ou on he clus e sys em a he Leibniz Uni e si y o Hanno e ,
Ge many.
Funding Open Access unding enabled and o ganized by P ojek
DEAL.
Da a a ailabili y s a emen The es ins ances o his s udy a e a ailable
om he au ho s upon eques .
Decla a ions
Con lic s o in e es The au ho s decla e ha hey ha e no compe ing
in e es s.
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