30 2015, XVIII, 3
Ekonomika a managemen
DOI: 10.15240/ ul/001/2015-3-003
In oduc ion
The quali y and e i ciency o an eme gency
medical sys em (EMS) depends mainly on he
numbe o ambulances ope a ing in a gi en
egion and he deploymen o s a ions whe e
he ambulances a e kep . Speci ying he p ope
numbe o ambulances is a sensi i e issue
balancing be ween wo opposing aspec s. On
one hand, he main ole o he EMS – o sa e
li es and educe human su e ing caused by
inju ies o illnesses – equi es a dense ne wo k
o eme gency s a ions. On he o he hand,
he e is a jus i i ed equi emen on he e i ciency
o public expendi u es. In he Slo ak Republic
he e is one ambulance a e e y s a ion. The
numbe and loca ions o s a ions in he whole
s a e e i o y a e de i ned by he Regula ions
o he Minis y o Heal h o he Slo ak
Republic No. 10548/2009-OL, 11378/2010-
OL and 14016/2010-OL. In acco dance wi h
he Regula ions, 273 s a ions a e cu en ly
deployed in he a ea o he Slo ak Republic.
The s a is ical compu a ions ha alloca e each
municipali y o he nea es s a ion sugges ha
he e a e signi i can di e ences in popula ion in
he egions se ed by indi idual ambulances,
he e o e some ambulances a e used less
equen ly han he o he s. The accessibili y
o he eme gency se ice in egions se ed by
o e loaded ambulances de e io a es, since he
p obabili y ha he nea es ambulance will be
busy a he momen o an eme gency call is
high. Assuming ha he numbe o ambulances
is gi en, we wan o in es iga e whe he
loca ing ambulances di e en ly migh esul in
a mo e e en dis ibu ion o hei wo kload and,
consequen ly, in a be e pe o mance o he
sys em.
To design a new deploymen o s a ions,
a ma hema ical p og amming model cons aining
he popula ion alloca ed o one s a ion can be
used. Ou o he abundan numbe o loca ion
models, a capaci a ed p-median model has
been chosen, since i maximises he e i ciency
o he sys em, bu i simul aneously inc eases
he ai ness o he deli e y o he EMS
se ice. The e i ciency c i e ion means ha
wi h a limi ed numbe o esou ces he bes
possible le el o he se ice is p o ided o
as many people as possible. In a p-median
model, he o al a el ime o ambulances o
po en ial pa ien s is a su oga e o e i ciency.
Besides e i ciency, equi y (o ai ness) is
a co e pe o mance dimension in a heal h ca e
sys em [17]. Fai ness is achie ed when each
cus ome ecei es he se ice o equi ed and/
o accep able quali y. This demand is ha d o
mee in a de e minis ic ma hema ical model
because o he s ochas ic na u e o he eal
sys em. De e minis ic models o he EMS
sys ems a e based on an implici assump ion
ha he e is always an ambulance a ailable o
espond o a call. Bu in he eal sys em his
may no be ue because he a i ing calls a e
s ochas ic e en s, and ea ing a pa ien is also
a andom a iable. Mo eo e , he a el ime
o an ambulance may be a ec ed by he a i c
and wea he condi ions. The e o e, he nea es
ambulance may happen o be busy when an
acciden occu s. Then ano he ambulance mus
be dispa ched o se e he call, o he se ice
mus be pos poned. Thus he eal se ice
le el becomes lowe han he compu ed one.
Howe e , we can limi he popula ion alloca ed
o one s a ion in he p oblem o mula ion and
so inc ease he p obabili y ha he nea es
ambulance will be a ailable a he momen o
an eme gency call.
The es o he pape is o ganized as
ollow: Sec ion 1 e iews he li e a u e on
loca ion models in public se ice sys ems.
A ma hema ical p og amming model o he
capaci a ed p-median p oblem is o mula ed
in Sec ion 2. Sec ion 3 desc ibes wo heu is ic
LOAD BALANCING LOCATION OF
EMERGENCY MEDICAL SERVICE STATIONS
Ľudmila Jánošíko á, Lýdia Gáb išo á, B uno Ježek
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31
3, XVIII, 2015
Business Adminis a ion and Managemen
me hods sol ing he p oblem and a new
heu is ic me hod based on p e iously de eloped
app oaches. The nume ical e alua ion o he
solu ions is p esen ed in Sec ion 4. The las
sec ion p o ides conclusions and p esen s
some pe spec i es o u u e esea ch.
1. Li e a u e Re iew
We ecommend he excellen pape by ReVelle
and Eisel [21] as an in oduc ion o he loca ion
analysis. The au ho s dis inguish wo basic
ypes o loca ion p oblems: con inuous loca ion
p oblems, which a e o he mos pa plana
p oblems and end o be non-linea op imiza ion
p oblems, and disc e e loca ion p oblems, which
a e mos o en ne wo k p oblems, in ol e ze o-
one a iables and esul in in ege p og amming
op imiza ion p oblems. Ano he classi i ca ion o
loca ion p oblems di e en ia es be ween he
p i a e and public sec o s. Rega ding disc e e
loca ion models in he public sec o , we e e
o [18] o he o e iew o basic models and
[4] o he case s udy in he Slo ak Republic.
Ambulance loca ion and eloca ion models
a e su eyed in [5]. In his e iew pape , he
models a e classi i ed in o wo main ca ego ies:
de e minis ic and p obabilis ic models.
De e minis ic models a e used a he
planning s age. They igno e s ochas ic
conside a ions ega ding he a ailabili y o
ambulances. They can be u he di ided in o
co e ing and alloca ion models. In co e ing
models, a maximum alue is p ese o ei he
dis ance o a el ime. I a se ice is p o ided by
a acili y loca ed wi hin his limi , hen he se ice
is conside ed accep able, and a cus ome is
conside ed co e ed by he se ice, i he has
a acili y si ed wi hin he p ese dis ance o ime.
The e a e wo ypes o objec i e: we may wan
o co e all cus ome s wi h minimum numbe o
acili ies o , gi en a limi ed numbe o acili ies,
o maximize co e age o he popula ion. In he
o me case, he p oblem is called a Loca ion
Se Co e ing P oblem (LSCP), in he la e case
we a e aced o a Maximal Co e ing Loca ion
P oblem (MCLP). The concep o co e age o
loca ion o ambulances is used o example by
A inghie i e al. [2]. The au ho s p opose he
Lowe -P io i y Calls Co e age model o p o ide
a lowe bound on he numbe o ambulances
needed o gua an ee a desi ed le el o he
eme gency se ice. In he alloca ion models, he
goal is o assign demand zones o ambulance
loca ions in o de o minimise he o al a el
ime om he ambulance loca ions o po en ial
pa ien s. One o possible models o his ype is
so-called p-median model ha is sol ed in [7]
and [22] o op imise ambulance loca ions in he
ci y o Niiga a (Japan).
P obabilis ic models e l ec he ac ha
ambulances ope a e as se e s in a queuing
sys em and hey canno always answe a call.
P obabilis ic models ha e been de eloped o
example by Chan a e al. [6] o Ingol sson e al.
[12]. The basic concep in hei app oach is so-
called busy ac ion o ambulances, which is he
p obabili y ha he ambulance will be occupied
a he momen o he call ecep ion, and i will
no be able o espond o he call. The p oblem
is ha he busy ac ion o an ambulance
depends on he loca ion o ambulances, mo e
speci i cally on he egion ha is se ed by each
ambulance, howe e , his is in ac he ou pu
om he loca ion model. So he models ha deal
wi h he busy ac ions as exogenous inpu s
canno gi e a ealis ic ou pu ei he , e en i hey
a e sol ed epea edly in an i e a i e p ocess,
whe e he busy ac ions o ambulances a e
adjus ed acco ding o he ou pu o he p e ious
i e a ion.
The way o cope wi h a empo al
una ailabili y o he nea es ambulance is o
in ol e a edundan ambulance. The concep o
backup co e age was i s in oduced by Hogan
and ReVelle [11]. Backup co e age means ha
a cus ome has a leas wo ambulances kep
a disposal in hei neighbou hood. Pi kul and
Schilling [20] and A az e al. [1] u he expand
he backup co e age o mula ion and conside
wo kload capaci ies o acili ies.
Mos s udies published so a deal wi h
ambulance loca ion in an u ban a ea like Milano,
I aly [2], Auckland Region, New Zealand [10], Belo
Ho izon e, B azil [23], Niiga a, Japan [7], [22].
Howe e , ou goal is o design a me hodology
applicable in a la ge-scale e i o y including
bo h u ban and u al a eas. The me hodology
comp ises a sui able ma hema ical p og amming
model and a simula ion model ha is used
o e alua e he pe o mance o he sys em in
a dynamic en i onmen . The compu e simula ion
pe o med wi h he solu ions o di e en
de e minis ic op imisa ion models [15] sugges s
ha he p-median model o ambulance loca ion
in a la ge-scale e i o y ou pe o ms backup
co e age models in e ms o expec ed a e age
a el ime, pe cen age o escue calls accessible
wi hin 15 minu es and he numbe o calls ha
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32 2015, XVIII, 3
Ekonomika a managemen
ha e o be pu on hold. Howe e , he p-median
model leads o signi i can di e ences in he
wo kload o indi idual ambulances. The e o e,
i seems easonable o limi he popula ion
alloca ed o one s a ion. This es ic ion esul s in
a ai dis ibu ion o ambulances’ wo kload and,
which is mo e impo an , in be e a ailabili y
o he ambulances. This way he p oblem o
s a ion loca ion becomes a weigh ed capaci a ed
p-median p oblem.
2. P oblem Fo mula ion
The goal in he weigh ed capaci a ed p-median
p oblem is o i nd he loca ion o a i xed numbe
o p s a ions in o de o minimise he o al a el
ime needed o each all po en ial pa ien s. The
model o he load balancing s a ion loca ion
p ese es he cu en numbe o s a ions and
looks o a new s a ion loca ion in he conce ned
egion. The demand zones a e indi idual
illages and ci ies. We suppose ha he
demand (i.e. he numbe o eme gency calls)
in a municipali y is p opo ional o he numbe
o i s inhabi an s. The e o e, in he objec i e
unc ion he a el ime o an ambulance
om i s base s a ion o a municipali y will be
mul iplied by he numbe o inhabi an s. We
assume de e minis ic a el imes (de i ned by
he dis ance and he a e age speed o a gi en
oad ype). The a e age speed is based on he
analysis by Ježek e al. [16].
The inpu s o he ma hema ical p og amming
model a e as ollow:
I he se o candida e loca ions,
J he se o municipali ies,
p he numbe o s a ions o be loca ed,
ij he sho es a el ime o an ambulance
om si e i  I o si e j  J,
bj he numbe o he inhabi an s o a municipali y
j  J,
Q he capaci y limi o an ambulance.
The decision on opening a s a ion (o mo e
s a ions) in a candida e loca ion i  I can be
modelled by he nonnega i e in ege a iable yi.
The alue o yi is he numbe o s a ions loca ed
in cen e i (o cou se, i may be ze o). The
assignmen o he municipali y j o he cen e i is
modelled by a bina y a iable xij. The a iable
xij akes alue o 1, i he municipali y j will be
se ed by an ambulance loca ed in he cen e i,
o he wise xij = 0.
A e hese p elimina ies, he model o he
weigh ed p-median p oblem can be w i en as:
Minimise

IiJj
jij b
ij
ij
x
ij (1)
Subjec o
1

Ii
ij
x
o j  J (2)
iij
yx 
o i  I, j  J (3)
Jj
ijj
xb 


Qyi o i  I (4)
py
Ii
i



(5)


0
Zy
i
o i  I (6)

1,0
ij
x
o i  I, j  J (7)
The basic sys em c i e ion (1) is he o al
a el ime o ambulances o all po en ial pa ien s
(o inhabi an s o municipali ies). Cons ain s (2)
ensu e ha e e y municipali y j will be assigned
o exac ly one cen e i. Cons ain s (3) ensu e
ha i he municipali y j is assigned o he cen e
i, hen a leas one s a ion mus be open in ha
cen e. Cons ain s (4) limi he o al numbe o
pe sons se ed by one cen e. Cons ain (5)
limi s he o al numbe o he s a ions ha can
be si ed. The emaining obliga o y cons ain s
(6) and (7) speci y he de i ni ion domains o he
a iables.
The capaci a ed p-median p oblem is known
o be NP-comple e. As a consequence, i canno
be sol ed o op imali y e en o mode a e-
sized p oblem ins ances [13]. Howe e , we
ace a la ge-scale p oblem ins ance consis ing
o all 2,916 municipali ies in Slo akia (by he
adminis a i e di ision alid in 2003). E e y
municipali y is ega ded as a candida e loca ion,
as well as a demand zone, i means I = J and
|I| = |J| = 2,916.
To ge a su i cien ly good solu ion in
a easonable ime, a decomposi ion echnique
can be used. In ou p e ious esea ch wo
decomposi ion heu is ic me hods we e
de eloped. Bo h o hem exploi a ma hema ical
p og amming app oach o sol e a subp oblem.
The p ima y esea ch aimed a he possibili y o
using he p oposed me hods o p ac ical la ge-
scale p oblem ins ances was p esen ed in [9]
and [14].
ij
ij
ij
ij
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33
3, XVIII, 2015
Business Adminis a ion and Managemen
3. Solu ion Me hods
– Decomposi ion Heu is ics Using
Ma hema ical P og amming
The p inciple o a decomposi ion echnique is
simple: i he p oblem canno be op imised as
a whole, op imise i in pa s. The app oach can
be used o e e y p oblem ha can be di ided
in o subp oblems. Then e e y imp o emen o
he subp oblem co esponds o an imp o emen
o he solu ion o he whole p oblem. Loca ion
p oblems mee his condi ion.
3.1 Lop Heu is ic
The i s decomposi ion heu is ic is based on he
app oach p oposed by Tailla d and published
o me ly unde he name o POPMUSIC [24],
and la e in [25] as local op imiza ion me hod
(LOPT). In his pape he la e no a ion is used.
To make he p oblem ac able, i s size is
educed be o e he op imiza ion. The goal
is o elimina e he a iables which a e less
likely o belong o a good o op imal solu ion.
The educ ion is pe o med in se e al s eps.
Fi s , he se I o 2,916 possible loca ions o
s a ions is educed. A new se
I
con ains
2,282 candida e loca ions and consis s o
all he municipali ies wi h he exis ing EMS
s a ions de i ned by he o i cial egula ions, and
all he o he municipali ies wi h a leas 300
inhabi an s. Second, one o mo e s a ions a e
placed a p io i in la ge ci ies wi h mo e han
25,000 inhabi an s, since 25,000 pe sons is
a capaci y limi o one ambulance (acco ding
o he analysis o he Slo ak EMS sys em [3]).
The numbe o s a ions kj ha mus be open
in he municipali y j  J we ge by di iding he
numbe o inhabi an s by he capaci y limi :
kj = [bj /25,000]. A he same ime, he demand
o he municipali y j is adjus ed o he new
alue ¯bj = bj – kj • 25,000. The o al numbe
o s a ions placed in la ge ci ies be o e he
op imiza ion is 
Jj j
kk (k = 50 in he case
s udy). The numbe o s a ions o be loca ed
(p = 273) is educed by his alue leading o
he new numbe p¯ = p – k = 223 . The se Ī and
pa ame e s ¯bj and p¯ a e he inpu s o he Lop
heu is ic. Since ¯bj < 25,000 o e e y j  J, he
heu is ic places one s a ion a he mos in he
candida e loca ion j.
Fu he elimina ion o a iables conce ns he
a iables x and is based on he assump ion ha
pa ien s will no be se ed by he ambulances
ha a e oo a away. Tha is why only hose
a iables xij emain in he model o which
he coe i cien ij is less han he p ede i ned
h eshold. The h eshold is de i ned by he alue
limi = α • max / √p¯ whe e max = max { ij : i  Ī, j  J}
and α is a pa ame e . In he Slo ak oad ne wo k
we ha e max = 279 minu es. Thus o α = 1.5
we ge he h eshold limi = 26 minu es. All hese
measu es educe he numbe o a iables by
94%. Howe e , we s ill ha e 431,569 bi alen
a iables, and he use o a decomposi ion
heu is ic is jus i i ed.
The Lop heu is ic s a s wi h he ini ial
loca ion o p¯ = 223 s a ions ha is compu ed
by an IP sol e unning in a limi ed ime. The
ini ial loca ions a e deno ed as empo a y
and inse ed in o he se C. Then a empo a y
s a ion is andomly selec ed. This selec ed
s a ion, oge he wi h a ew o i s closes
s a ions and municipali ies alloca ed o hem
in he cu en solu ion, o m a subp oblem wi h
s a ions, which is conside ably smalle han
he ini ial loca ion p oblem (see Fig. 1). The
loca ion o s a ions is op imized by using an
IP sol e . I a be e loca ion is ound, all hese
s a ions emain empo a y; o he wise he i s
s a ion is emo ed om C. Then a new s a ion
is andomly selec ed and he p ocess epea s
un il C is emp y.
3.2 Decomp Heu is ic
The second heu is ic deno ed as Decomp is
also based on he domain decomposi ion.
Bu in con as o he Lop heu is ic, he
decomposi ion is pe o med a he beginning
o he solu ion p ocess. The e i o y is
decomposed in o adminis a i e egions. I
means ha he se I ( emembe I = J) is di ided
in o a ew disjunc i e subse s. Each subse
includes all municipali ies o one o eigh Slo ak
adminis a i e egions. The size o he subse s
anges be ween 87 (B a isla a Region) and 664
municipali ies (P ešo Region). The subse s
de i ne eigh sepa a e p-median p oblems
(1) – (7) co esponding o he pa icula egions.
The cons an p in e e y p oblem is se in o de
o p ese e he cu en numbe o s a ions
in he gi en egion. The p oblems a e sol ed
sepa a ely. By he union o hei solu ions we
ob ain loca ions o all s a ions in he Slo ak
Republic.
The Decomp heu is ic consis s o he
ollowing ou phases:
1. The loca ion o p s a ions is compu ed
by sol ing he uncapaci a ed p-median
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34 2015, XVIII, 3
Ekonomika a managemen
p oblem (1) – (3), (5), (7) wi h he a iables
yi  {0,1} o i  I. Ad in e im he cons ain s
(4) a e elimina ed. Le 
Jj ijji xbB be
he numbe o inhabi an s o he nea es
municipali ies j which a e se ed by he
ambulance in he cen e i.
2. The alue Bi ep esen s he demand
alloca ed o he cen e i. I can be g ea e
o less han he capaci y limi Q o an
ambulance. To p o ide addi ional capaci y
in he cen es wi h high demand, s a ions
wi h small demand (Bi << Q) can be closed
and eloca ed o he cen es wi h high
demand (Bi >> Q). In his s ep we decide
on he numbe p1 o s a ions which can be
eloca ed.
3. The new loca ion o p – p1 s a ions is
compu ed ega dless o he capaci y
cons ain s. Le I* deno e he se o new
s a ion loca ions wi h he demand Bi
alloca ed o hem.
4. Now ee p1 s a ions can be eloca ed
among he cen es i  I* by sol ing a new
ma hema ical p og amming p oblem (8) –
(11). The a iable zi indica es how many
o eloca ed s a ions will be placed a he
cen e i  I*. We in oduce a new a iable
w ha ep esen s a lowe bound on he
numbe o s a ions in a cen e i ha sha e
he demand Bi. The ma hema ical model
maximizes he alue w and esul s in a new
numbe yi o s a ions ha consis s o one
s a ion loca ed in phase 3 and zi eloca ed
s a ions.
Maximise w (8)
Subjec o 1
*
pz
Ii
i


(9)
wBz ii  1 o i  I* (10)

0
Zzi o i  I* (11)
The abili y o he Decomp me hod o
i nd a be e s a ion loca ion is limi ed due
o he ac ha he numbe p o s a ions o
each subp oblem is bound by he exis ing
EMS s a ions in he gi en egion. The e o e
in he ollowing esea ch we ied o imp o e
ou solu ion by he adjus men o he inpu
pa ame e s o he Decomp heu is ic. We used
he esul s o he Lop me hod as inpu da a o
he Decomp me hod as ollow: om he Lop
solu ion we iden i i ed he numbe o s a ions
o each o he Slo ak egions and hen ound
an imp o ed loca ion in he egion by he
Decomp me hod. Fo he ime being, we call
his p ocedu e Lop -Decomp.
The Lop , Decomp and Lop -Decomp
p ocedu es we e implemen ed in he isual
de elopmen en i onmen Xp ess-IVE using
Fig. 1: A subp oblem wi h se e al cen es and municipali ies alloca ed o hem
Sou ce: own
ij
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35
3, XVIII, 2015
Business Adminis a ion and Managemen
he sol e Xp ess-Op imize 2.2.3 [8]. The
expe imen s we e pe o med on a pe sonal
compu e equipped wi h he In el Co e i7
p ocesso wi h 1.60 GHz and 8 GB o RAM. The
compu a ion ime did no exceed 25 minu es o
Decomp and 217 minu es o he Lop me hod.
4. E alua ion o he Al e na i e
Deploymen s
Each o h ee desc ibed me hods (Lop ,
Decomp and new Lop -Decomp) esul s in
a di e en loca ion o 273 s a ions. Each loca ion
is ep esen ed by he componen s o he ou pu
ec o y*. The alue yi > 0 indica es he numbe
o s a ions loca ed in he cen e i  I. The
ec o y* is he inpu o he e alua ion o he
al e na i e deploymen s and hei compa ison
mu ually and wi h he cu en si ua ion in he
Slo ak Republic.
The e alua ion is based on he assump ion
ha e e y municipali y is se ed by i s closes
ambulance. The assignmen o municipali ies o
cen es can be quickly compu ed by sol ing he
ollowing alloca ion p oblem:
Minimise 
IiJj
ij
ij
x (12)
Subjec o 1

Ii
ij
x o j  J (13)
iij
yx 
o i  I, j  J (14)

1,0
ij
x
o i  I, j  J (15)
He e yi a e no any mo e a iables bu
cons an s ha gi e he numbe o s a ions
loca ed in he cen e i  I. The op imal solu ion
o he p oblem (12) – (15) associa es each
cen e i wi h he subse o municipali ies

1,  ijixJjJ . Then he numbe o
pe sons (po en ial pa ien s) se ed by he
cen e i is 

i
Jj ji bB and he sha e o one
ambulance loca ed ain his cen e is Bi / yi.
Table 1 p esen s he basic cha ac e is ics
o he p oposed deploymen s o s a ions. The
column Numbe o cen es gi es how many
cen es we e chosen om 2,916 candida e
loca ions. The column Numbe o di e en ly
loca ed s a ions indica es how much he
solu ion o he model di e s om he cu en
deploymen o s a ions. We can see ha in
all solu ions abou 37% o 273 s a ions we e
loca ed di e en ly compa ed o he exis ing
design. The column To al a el ime o
ambulances o pa ien s ep esen s he e i ciency
o he sys em measu ed as he o al a el
ime o ambulances o all po en ial pa ien s.
The wo kload o ambulances exp essed
as he Numbe o people pe ambulance is
summa ised in he las h ee columns. The
a e age wo kload is iden ical in all designs
because all solu ions p ese e he numbe o
exis ing s a ions. Since ou goal was o p opose
an e en dis ibu ion o ambulances wo kload,
he mos impo an indica o was he ange
be ween he minimum and maximum wo kload.
The g ea es ange can be obse ed in he case
o he cu en deploymen o s a ions. The mos
e en dis ibu ion o wo kload was achie ed by
he combined Lop -Decomp heu is ic. These
ac s a e as well demons a ed in Fig. 2.
Me hod
o s a ion
loca ion
Numbe
o cen es
Numbe
o di e en ly
loca ed
s a ions
To al a el ime
o ambulances
o pa ien s
[million pe son*
minu es]
Numbe o people
pe ambulance
[in housands]
min a g max
cu en 209 0 14.06 1.3
19.8
70.9
Lop 233 104 11.86 5.3 41.8
Decomp 186 102 13.52 4.4 40.6
Lop -Decomp 186 101 13.43 5.8 33.6
Sou ce: own
Tab. 1: E alua ion o he deploymen s o s a ions
ij ij
ij
ij
ij
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36 2015, XVIII, 3
Ekonomika a managemen
Howe e , he uni o m dis ibu ion o
wo kload may be achie ed a he expense
o long a el imes o ambulances o some
illages. The e o e we in es iga e he impac
o he uni o m wo kload on he anspo a ion
accessibili y o municipali ies. Table 2 gi es
how many people a e accessible wi hin a gi en
ime limi . All alues abou he popula ion a e
gi en in % o he o al popula ion o he Slo ak
Republic. The column Ex eme con ains he
maximum a el ime and he popula ion o he
illage wi h he longes a el ime. As i can be
seen, he bes accessibili y was achie ed by
he Lop heu is ic.
The limi a ion o he applicabili y o he
p esen ed app oach consis s in he ac ha
we do no conside cos s o edeploymen
o s a ions. To achie e a mo e ai design,
he model changes subs an ially he cu en
deploymen o EMS s a ions (see he hi d
column in Table 1). To implemen p oposed
solu ions would hus equi e o es uc u e he
exis ing in as uc u e conside ably wha migh
be oo cos ly. I his issue is o impo ance, i can
be p ac ical o educe he numbe o eloca ed
s a ions, o o es ima e he cos o closing
and opening a s a ion a si e i and equi e
o al in es men cos s o be in an alloca ed
budge . To inco po a e such cons ain s in he
ma hema ical model, wo new a iables ui and
i mus be supplemen ed, which indica e he
numbe o closed and open s a ions a si e i,
espec i ely. They a e de i ned by cons ain s
(16) – (18), whe e si is he cu en numbe o
s a ions a si e i.
iii ysu  o i  I (16)
iii sy  o i  I (17)

0
Z u ii, o i  I (18)
Le n be he uppe bound o he accep able
numbe o eloca ed s a ions. Then he
cons ain ha limi s he numbe o changes
can be in he o m:



Ii
inu
(19)
This cons ain migh be eplaced by ano he
cons ain s equi ing ha he econs uc ion o
Fig. 2: Numbe o people assigned o one ambulance
Sou ce: own
EM_3_2015.indd 36EM_3_2015.indd 36 25.8.2015 10:51:1625.8.2015 10:51:16
37
3, XVIII, 2015
Business Adminis a ion and Managemen
he sys em will no exceed he o al budge q.
Le symbols i
c and i
o deno e he cos o closing
and opening a s a ion a si e i, espec i ely.
Then he co esponding cons ain is

q u
Ii
i
o
ii
c
i

 (20)
The p esen ed solu ion me hods would
be able o cope wi h such ex ension o he
model (1) – (7). Ano he way how o imp o e
he loca ion model is o eplace he capaci y
cons ain s (4) by an i egula i y measu e [19]
applied o ambulances’ wo kload.
Conclusion
In he pape , he p oblem o EMS s a ions
loca ion is o mula ed as a capaci a ed
p-median p oblem. A new decomposi ion
me hod is p oposed. The compa ison o he
new Lop -Decomp me hod wi h p e iously
de eloped heu is ics Lop and Decomp is
p esen ed wi h ega d o ambulances wo kload
and anspo a ion accessibili y. The bes esul s
in e ms o he dis ibu ion o wo kload we e
achie ed by he Lop -Decomp me hod, and in
e ms o he accessibili y by he Lop me hod.
Howe e , hese esul s a e jus he
es ima ions o he eal sys em pe o mance
because a de e minis ic ma hema ical
p og amming model igno es he s ochas ic
cha ac e o he modelled sys em. The bes way
how o es ima e pe o mance cha ac e is ics
be o e he implemen a ion o he solu ion in
he eal en i onmen is o use a compu e
simula ion model. Such a model o he EMS
sys em was buil wi hin he cu en esea ch
[15], ne e heless, we do no ha e ealis ic da a
needed o calib a e he model. In he u u e we
will endea ou o ob ain da a om esponsible
au ho i ies and o e i y he conclusions by
means o a compu e simula ion.
This esea ch was suppo ed by he Slo ak
Resea ch and De elopmen Agency unde
p ojec APVV-0760-11 “Designing o Fai
Se ice Sys ems on T anspo a ion Ne wo ks”.
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38 2015, XVIII, 3
Ekonomika a managemen
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