G iege , Milena; Heide , S e en; McRae, Sebas ian; Kope na, Thomas; B unne ,
Jens O.
A icle — Published Ve sion
Managing he pa ien po olio using ma hema ical
p og amming: decision suppo guidelines using a eal-
wo ld use case a a uni e si y hospi al
Jou nal o Business Economics
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: G iege , Milena; Heide , S e en; McRae, Sebas ian; Kope na, Thomas; B unne ,
Jens O. (2024) : Managing he pa ien po olio using ma hema ical p og amming: decision suppo
guidelines using a eal-wo ld use case a a uni e si y hospi al, Jou nal o Business Economics, ISSN
1861-8928, Sp inge , Be lin, Heidelbe g, Vol. 94, Iss. 9, pp. 1245-1260,
h ps://doi.o g/10.1007/s11573-024-01201-y
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/315428
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ORIGINAL ARTICLE
Jou nal o Business Economics (2024) 94:1245–1260
h ps://doi.o g/10.1007/s11573-024-01201-y
Abs ac
Many hospi als in Ge many a e acing escala ing economic p essu es. A e se e al
yea s o s agna ion, he numbe o inpa ien hospi al ea men s d opped by
13%
in 2020 compa ed o he p e ious yea . This nega i e endency can also be seen
in ope a ing hea e s (OTs). S a egic managemen o he case mix in hospi al OTs
now necessi a es a solid da a ounda ion. The case mix and he case mix index
ha e become cen al economic indica o s in con empo a y hospi al ope a ions. In
his wo k, we de elop a ma hema ical model o case mix op imiza ion a Augs-
bu g Uni e si y Hospi al in Ge many, which is based on an ex ensi e da a analysis
wi h desc ip i e me hods. The op imiza ion model is subjec o igo ous es ing and
e alua ion h ough an ex ensi e se ies o scena io analyses. The p ima y objec i e
is o calcula e a e enue-maximizing pa ien mix while espec ing he a ailable
sca ce pe sonnel esou ces in he OT and in ensi e ca e uni . This esea ch ma ks a
pionee ing e o in delinea ing he p ac ical in eg a ion o case mix planning in o a
hospi al’s ou ine ope a ions using ma hema ical op imiza ion. The analyses e eal
a s ong co ela ion be ween an upsu ge in e enue and an inc eased numbe o
cases. Fu he mo e, he esul s demons a e ha s a egic planning o he pa ien
mix has he po en ial o enhance e enue wi h exis ing esou ces. E en hough he
op imal pa ien mix may no be di ec ly implemen able in p ac ice, he indings
yield aluable insigh s o manage ial decision-making. A c i ical examina ion o
hese esul s also os e s a nuanced discou se on he u iliza ion o op imiza ion
models as decision suppo ools wi hin hospi al managemen .
Keywo ds Case mix · Ma hema ical op imiza ion · OT planning · Hospi al ·
Decision suppo
Accep ed: 9 Augus 2024 / Published online: 28 Augus 2024
© The Au ho (s) 2024
Managing he pa ien po olio using ma hema ical
p og amming: decision suppo guidelines using a eal-
wo ld use case a a uni e si y hospi al
MilenaG iege 1· S e enHeide 2· Sebas ianMcRae3· ThomasKope na4·
Jens O.B unne 1,5,6
Ex ended au ho in o ma ion a ailable on he las page o he a icle
1 3
M. G iege e al.
1 In oduc ion
Nume ous hospi als in Ge many a e acing inc eased economic p essu e. A e yea s
o s agna ion, he numbe o inpa ien hospi al ea men s declined by
13%
in 2020
compa ed o he p e ious yea , p ima ily due o he COVID-19 pandemic. Simila ly,
he o al numbe o su ge ies pe o med d opped by
10%
o
6.4
million (S a is isches
Bundesam 2021). Fu he mo e, he g owing sho age o medical p o essionals poses
an inc easingly daun ing challenge o Ge man hospi al managemen (Os e loh
2018). Quan i a i e Ope a ions Resea ch/Managemen Science (OR/MS) me hods
o e a po en ial solu ion o suppo hospi al managemen in dealing wi h hese com-
plex issues.
Ac i e s a egic managemen o he se ice po olio, commonly e e ed o as
case mix, can help hospi als main ain a sus ainable and economically iable se ice
o e ing despi e s agnan case numbe s and limi ed (pe sonnel) esou ces. The key
challenge lies in iden i ying he op imal mix and olumes o pa ien ca ego ies, a
complex planning ask known as he case mix planning p oblem (CMPP) (Ho e al.
2017). The co e o he CMPP is he op imal alloca ion o sca ce esou ces (e.g., ope -
a ing hea e (OT) capaci y) o s a egically manage he case mix. The basic objec i e
o s a egic case mix planning is o ul ill he se ice manda e, inc ease he quali y o
ca e, and/o maximize (pe sonnel) e enue (Ho e al. 2017; Waeschle e al. 2016).
The speci ic planning objec i es a y depending on he managemen pe spec i e
guiding he s a egic con ol o he se ice po olio.
This pape examines he applica ion o quan i a i e OR/MS me hods o s a egic
se ice po olio planning. Alongside in-dep h analyses, he pape del es in o he
po en ial and limi a ions o applying hese me hods in eal hospi al p ac ice. The
insigh s gained a e illus a ed using he Augsbu g Uni e si y Hospi al (UKA) as a
p ac ical case s udy. OR/MS is an in e disciplina y ield conce ned wi h de eloping
and applying ma hema ical op imiza ion models o decision suppo (Hulsho e al.
2012; an Wassenho e and Besiou 2013). In heal hca e, OR/MS models and me hods
ind di e se applica ions ac oss a ious complex planning p oblems, including OT
planning, shi scheduling, o se ice ange planning (E ha d e al. 2018; Heide e
al. 2022; Ho e al. 2017).
Gi en he p edominan ly isola ed conside a ion o su gical hospi als, making a
uni e sally applicable s a emen abou esou ce alloca ion wi hin a hospi al is chal-
lenging. This s udy add esses his issue by sys ema ically compa ing pe sonnel
expenses and e enues o he en i e hospi al. Assuming ha OT and in ensi e ca e
uni (ICU) capaci ies emain ela i ely s able in he sho o medium e m, he op i-
mized inc ease o educ ion o speci ic su ge y ypes and he dis ibu ion o a ailable
OT esou ces p o ide a edis ibu ion o he su gical se ice po olio. F om a s a-
egic managemen pe spec i e, he esul s may signal he need o selec i ely educe
less p o i able su ge ies o e he medium o long e m. Howe e , i should be no ed
ha such educ ion measu es would equi e ex e nal coope a ion, such as ans e ing
pa ien s o o he hospi als as pa o a coope a ion model, while con olling pa ien
in low ia e e ing pa ies.
To implemen CMPP h ough ma hema ical op imiza ion models a he ope a ional
le el, e enues gene a ed om inpa ien su ge ies can be alloca ed o co esponding
1 3
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Managing he pa ien po olio using ma hema ical p og amming:…
cos ca ego ies, including pe sonnel, ma e ial cos s, and in as uc u e, based on he
Ins i u ü das En gel sys em im K ankenhaus (InEK) ma ix (Deu sche K anken-
hausgesellscha e al. 2016). Hospi al in o ma ion sys ems’ OT con ol ools, com-
monly used in p ac ice, acili a e he compa ison o di ec ly assignable pe sonnel
expenses wi h hese e enues. Th ough his da a p epa a ion and analysis, along wi h
he help o op imiza ion echniques, i becomes easible o quan i y he ex en o
which pe sonnel expenses a e co e ed by he e enues o a ious su ge y ypes.
These esul s can be calcula ed bo h o indi idual su gical depa men s and agg e-
ga ed ac oss he en i e hospi al.
The pu pose o his wo k is o in oduce a ma hema ical op imiza ion me hod ha
inco po a es case mix planning seamlessly in o he daily ope a ion o hospi als and
o de elop decision suppo guidelines based on a use case a a uni e si y hospi al.
Ou wo k is s uc u ed as ollows. In Sec . 2, we in oduce ou case s udy and da a,
while Sec . 3 ou lines ou case mix op imiza ion me hod. In Sec . 4, we p o ide he
compu a ional analysis esul s a he UKA. A discussion o he indings and decision
suppo guidelines is p o ided in Sec . 5. Sec ion 6 includes a summa y and an ou -
look o u u e esea ch.
2 In oducing he case s udy and da a
The model is buil on a da ase comp ising 22,657 cases encompassing 2,100 dis-
inc su ge y ypes. Va ious da a cleaning s eps we e implemen ed o enhance pa ien
low con ol. The g ea e objec i e o his e inemen p ocess is o ensu e g ea e
e iciency. An o e iew o he da a p epa a ion p ocess is p esen ed in Fig. 1. Gi en
he exis ence o nume ous su ge y ypes wi h minimal occu ences in he e e ence
yea , hese ypes unde wen ca ego iza ion using an ABC analysis. Fo he u he
cou se o he s udy, solely su ge y ypes alling wi hin he A g oup a e u ilized. This
g oup encompasses all su ge y ypes ha collec i ely ep esen
80%
o case num-
be s, o de ed in descending o de o equency. Consequen ly,
15,609
cases and
433
su ge y ypes a e included in he analysis. The de e mina ion o a ailable sca ce
esou ces is hen ca ied ou based on he conside ed cases, as ou lined in Sec . 2. A
o al o
12
di e en depa men s a e conside ed.
3 De eloping a simple ma hema ical model o mula ion o he
CMPP
The s a egic managemen o he pa ien mix p esen s i sel as a ma hema ical op i-
miza ion challenge, wi h a de ailed model p o ided below. The p ima y objec i e o
case mix op imiza ion is o ensu e an economical se ice po olio in he long e m
while inc easing he quali y o ca e. The amewo k o hese decisions is shaped by
he a ailable esou ces, demand, and he hospi al’s ca e manda e.
S a egic case mix managemen ope a es on an agg ega e annual basis, conside -
ing bo h esou ces and case numbe s. Al hough Diagnosis-Rela ed G oups (DRG)
se e as a sui able classi ica ion sys em o e ospec i e analysis o he case mix,
1 3
1247
M. G iege e al.
hey p o e unsui able o ac i ely managing he se ice po olio (Salge and Ve a
2012). This is because assigning a su ge y o a DRG g oup ypically occu s e o-
spec i ely, and p ac ical po olio managemen based on DRG g oups is challeng-
ing. Consequen ly, pa ien g oups a e cha ac e ized o op imiza ion pu poses using
he su ge y ypes commonly employed in in e nal su gical planning. These in-house
su ge y ypes, numbe ing mo e han
2,000
, e ec i ely cap u e he ea men in en .
Examples o he su ge y ypes a e eimplan a ion o a pacemake , lung esec ion and
pulmona y esec ion. The eimbu semen , esou ce demands, and cos s associa ed
Fig. 1 Da a p epa a ion o he model
1 3
1248
Managing he pa ien po olio using ma hema ical p og amming:…
wi h pa ien s in a su ge y ype a e es ima ed based on he ela i e dis ibu ion o
DRGs wi hin ha su ge y ype.
In ou s udy, we ocus on maximizing he e enue gene a ed om s a ing. F om
he hospi al’s pe spec i e, ma e ial and in as uc u e cos s, o en e e ed o as
h oughpu cos s, a e conside ed empo a y, and maximizing e enue o case mix
poin s does no necessa ily inc ease con ibu ion ma gin. The a ionale o his is
ha ma e ial and in as uc u e cos s can luc ua e in esponse o changes in pa ien
demand and ope a ional p ocedu es, and a e equen ly adjus ed o accommoda e
sho - e m equi emen s. Fu he mo e, ce ain medical p ocedu es a e associa ed
wi h excep ionally high ma e ial cos s (e.g. cochlea implan s), which can dis o he
o e all cos analysis i no ca e ully conside ed. In con as , pe sonnel cos s ha e a
signi ican impac on o e all expendi u e and a e mo e s able o e ime (S a is isches
Bundesam 2023). Pe sonnel e enue is de e mined based on he p opo iona e pe -
sonnel e enue acco ding o he InEK cos calcula ion as pu suan o Sec ion 17b
(5) KHG (Hospi al Ca e Ac ) and he s a e p ime a e. In he nex s ep, he pe sonnel
e enue pe su ge y ype is es ima ed based on he his o ical ela i e dis ibu ion
o DRGs wi hin each su ge y ype. The op imiza ion o he case mix, as well as
o he pe sonnel e enue model, ope a es wi hin he cons ain s de ined by sca ce
esou ces. These esou ces encompass OT nu sing, OT anes hesia nu sing, OT physi-
cian anes hesia, OT ime, and ICU s a ing capaci y. In his con ex , su geon ime is
no conside ed a sca ce esou ce because alloca ing pe sonnel esou ces o he OT
always akes p ecedence o su geons, wi h o he asks ou side he OT conside ed
lowe in p io i y. The capaci y alloca ed pe esou ce o a pa ien is app oxima ed by
he espec i e mean alue o he obse ed esou ce commi men s pe su ge y ype.
Fu he mo e, he a ailable capaci y pe esou ce du ing he obse a ion pe iod, in
his case, a calenda yea , is de e mined. The sum o case numbe s pe su ge y ype
and he p e iously de e mined commi ed capaci y pe su ge y ype is calcula ed o
each sca ce esou ce. This is based on he obse a ion ha sca ce esou ces ha e no
expe ienced signi ican idle imes du ing he obse a ion pe iod.
In addi ion o esou ce cons ain s, uppe and lowe limi s on he numbe o
pa ien s pe su ge y ype de ine he amewo k wi hin which he case mix can be
op imized. Uppe limi s a e ypically de e mined by demand, while lowe limi s a e
in luenced by bo h he hospi al’s se ice manda e and he need o quali y assu -
ance in se ices whe e ea men ou comes depend on se ice olume. Fo indi idual
se ices, he lowe limi s a e p ede ined based on he minimum quan i y egula ion
pu suan o Sec ion 136b (1) Sen ence 1 Numbe 2 Ge man Social Code V (SGB V).
The desc ibed si ua ion can be depic ed using a ma hema ical CMPP model.
Table 1 de ines he se s, indices, pa ame e , and decision a iables.
The de e minis ic ma hema ical model con ains an objec i e unc ion wi h h ee
cons ain s, and i can be o mula ed as ollows.
Ma
x
s∈S
ps·Xs (1)
1 3
1249
M. G iege e al.
s. .
s∈S
d s ·Xs≤c ∀ ∈R (2)
Xs≤us∀s∈S
(3)
Xs≥ls∀s∈S
(4)
xs≥0∀s∈S
(5)
The objec i e unc ion (1) maximizes he o al p oduc o (p opo iona e) pe sonnel
e enue and he numbe o cases pe su ge y ype. This op imiza ion is aimed a max-
imizing pe sonnel e enue ac oss all depa men s. Cons ain s (2) gua an ee compli-
ance wi h capaci y limi s pe esou ce ype, p e en ing any de ia ion in he case
mix om causing he de ined sca ce esou ces o exceed hei a ailable capaci ies.
Cons ain s (3) and (4) se uppe and lowe limi s on he esul ing numbe o cases o
each su ge y ype. I is also possible o de ine indi idual limi s o each su ge y ype
i he necessa y da a is accessible. Cons ain s (5) a e he non-nega i i y cons ain s.
4 Sol ing he CMPP a Augsbu g Uni e si y Hospi al
This sec ion discusses he esul s o he case s udy desc ibed in Sec . 2. I conside s
h ee scena ios in ol ing po en ial de ia ion o
10%,15%
, and
20%
om he case
numbe s o he e e ence yea . In ou model, hese pe cen ages o
10%,15%
, and
20%
a e ep esen ed by he uppe and lowe limi s
us
and
ls
. Fo example, in he
10%
scena io, he numbe o cases pe su ge y ype can a y be ween
90%
and
110%
o he numbe o cases in he e e ence yea . In o he wo ds, any numbe o cases
be ween hese wo limi s can be ealized i he hospi al plans o i . Fo example, i a
su ge y ype o iginally has a case numbe o
505
, he lowe limi in he
10%
scena io
is
454
, and he uppe limi is
556
. This means ha any case numbe be ween
454
and
556
can heo e ically be ealized o his su ge y ype. The model selec s he numbe
o cases ha is mos p o i able acco ding o he objec i e unc ion. A dis inc ion is
made be ween pe sonnel e enue, p opo iona e e enue, and case mix.
Se s and indices
s∈S
Se o su ge y ypes
∈R
Se o esou ce ypes
Pa ame e
ps
A e age pe sonnel e enue pe su ge y
wi h su ge y ype
s
d s
A e age esou ce equi emen pe su ge y
pe esou ce ype
and su ge y ype
s
us
Case uppe limi pe su ge y ype
s
ls
Case lowe limi pe su ge y ype
s
c
Capaci y limi pe esou ce
Decision a iable
Xs
Case numbe pe su ge y ype
s
Table 1 Se s, indices, pa ame e ,
and decision a iables
1 3
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Managing he pa ien po olio using ma hema ical p og amming:…
4.1 Pe sonnel e enue
Figu e 2 illus a es ha he applica ion o he op imiza ion model yields highe alues
o bo h he o e all numbe o cases o be ea ed and pe sonnel e enue in all h ee
scena ios when compa ed o he cu en s a us. Depending on he speci ic scena io,
he inc ease in o al pe sonnel e enue anges om
+2.3%
and
+4.5%
, while he o al
numbe o cases inc eases by a ma gin o
+3.4%
and
+6.7%
, ac oss he scena ios,
all while main aining he same o lowe capaci y u iliza ion when compa ed o he
e e ence yea ’s alues.
An inc ease in he o al numbe o cases and pe sonnel e enue does no necessa -
ily ansla e o an inc ease o each depa men . Figu e 3 shows he ela i e changes
in he numbe o cases and pe sonnel e enue pe depa men in he
10%
scena io.
While mos depa men s expe ience g ow h in bo h case numbe s and pe sonnel
e enue, he e a e also depa men s whe e, om an o e all op imiza ion pe spec-
i e, i would be mo e economically sound o educe case numbe s and pe sonnel
e enue. One o he p ima y ac o s con ibu ing o hese educ ions is he ele a ed
ma e ial cos s associa ed wi h su ge ies a hese depa men s. Al hough ma e ial cos s
posi i ely impac case mix poin s pe depa men , hey ep esen solely empo a y
expenses no ac o ed in his model. Ano he con ibu ing ac o is an un a o able
a io o pe sonnel e enue o commi ed capaci y pe sca ce esou ce when compa ed
o o he ypes o su ge ies. In he de eloped op imiza ion model, his a io pe su ge y
ype is assessed ac oss depa men s and op imally employed wi h he objec i e o
maximizing pe sonnel e enue o he en i e hospi al.
None heless, changes in bo h key me ics a e no solely con ined o inc eases o
dec eases. Conside ing depa men 10, o example, i would be possible o dec ease
he case numbe s by adjus ing he case mix wi hou a ec ing he depa men ’s pe -
sonnel e enue. A de ailed o e iew o he e olu ion o pe sonnel e enues pe
depa men can be ound in he le pa o Table 2. To enable he moni o ing o he
p oposed changes a he depa men le el, he esul s ha e been o ganized o each
depa men and scena io
(10%,15%,20%)
.
Fig. 2 To al ela i e change in numbe o cases and pe sonnel e enue pe scena io
1 3
1251
M. G iege e al.
Figu e 4 p esen s an exce p o he ou comes p o ided o he espec i e depa -
men , using one depa men as an illus a i e example. In his ep esen a ion, bo h
he a ious scena ios and he di e en su ge y ypes a e displayed. Fo each ype o
su ge y, i is e iden whe he he e should be an inc ease o dec ease in he numbe o
cases wi hin he depa men . Fo example, pe sonnel e enue is mo e ad an ageous
o ha e a highe numbe o pacemake s implan ed in he depa men han in he e e -
ence yea . Thus, i is appa en ha o each scena io and each depa men , a cus om-
ized decision is made ega ding he ideal case mix. The model op imally adjus s he
numbe o cases pe su ge y ype o ei he he uppe o lowe limi (i.e. ei he exac ly
Table 2 De elopmen o pe sonnel e enue pe depa men and scena io (pe sonnel e enue model)
Pe sonnel e enue P opo iona e pe sonnel e enue
±10% ±15% ±20% ±10% ±15% ±20%
Dep. 1 +8.78% +13.17% +17.56% +8.45% +12.67% +16.90%
Dep. 2 +0.14% +0.22% +0.29% −0.68% −1.01% −1.35%
Dep. 3 +7.85% +11.77% +15.69% +7.78% +11.68% +15.57%
Dep. 4 −4.85% −7.27% −9.70% −5.47% −8.20% −10.94%
Dep. 5 +7.67% +11.50% +15.34% +6.57% +9.85% +13.13%
Dep. 6 +2.95% +4.42% +5.90% +0.61% +0.92% +1.22%
Dep. 7 −2.18% −3.27% −4.36% −3.73% −5.60% −7.47%
Dep. 8 +4.91% +7.37% +9.83% +3.51% +5.27% +7.03%
Dep. 9 +10.00% +15.00% +20.00% +10.00% +15.00% +20.00%
Dep. 10 −0.07% −0.10% −0.14% −0.70% −1.04% −1.39%
Dep. 11 −0.71% −1.07% −1.42% −1.52% −2.27% −3.03%
Dep. 12 +7.71% +11.56% +15.41% +5.09% +7.63% +10.17%
To al +2.26% +3.39% +4.52% +0.32% +0.47% +0.63%
Fig. 3 Rela i e change in case numbe s and pe sonnel e enue pe depa men in
10%
scena io (Dep.:
Depa men )
1 3
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Managing he pa ien po olio using ma hema ical p og amming:…
Funding The au ho s decla e ha no unds, g an s, o o he suppo we e ecei ed du ing he p epa a ion
o his manusc ip .
Open Access unding enabled and o ganized by P ojek DEAL.
Da a a ailabili y The da a ha suppo he indings o his s udy a e no openly a ailable due o easons o
sensi i i y and a e a ailable om he co esponding au ho upon easonable eques .
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Publishe ’s no e Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in published maps
and ins i u ional a ilia ions.
Au ho s and A ilia ions
MilenaG iege 1· S e enHeide 2· Sebas ianMcRae3· ThomasKope na4·
Jens O.B unne 1,5,6
Milena G iege
[email p o ec ed]
1 Heal h Ca e Ope a ions/Heal h In o ma ion Managemen , Facul y o Business and
Economics, Uni e si y o Augsbu g, Uni e si ä ss aße 16, 86159 Augsbu g, Ge many
2 Digi aliza ion and Business Analy ics Depa men , Augsbu g Uni e si y Hospi al,
S englins aße 2, 86156 Augsbu g, Ge many
3 Klinikum ech s de Isa , Ismaninge S aße 22, 81675 Munich, Ge many
4 Ope a ing Thea e Managemen , Ludwig-Maximilian Uni e si y Hospi al,
Ma chioninis aße 15, 81377 Munich, Ge many
5 Depa men o Technology, Managemen , and Economics, Technical Uni e si y o
Denma k, Kongens Lyngby, Denma k
6 Nex Gene a ion Technology, Region Zealand, Denma k
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