Facility location based on adjusted present value

Rezaei, Hamid eza; Bos el, Na halie; Ho elaque, Vincen ; Pé on, Oli ie ; Vi iani,
Jean-Lau en
A icle
Facili y loca ion based on adjus ed p esen alue
Ope a ions Resea ch Pe spec i es
P o ided in Coope a ion wi h:
Else ie
Sugges ed Ci a ion: Rezaei, Hamid eza; Bos el, Na halie; Ho elaque, Vincen ; Pé on, Oli ie ; Vi iani,
Jean-Lau en (2025) : Facili y loca ion based on adjus ed p esen alue, Ope a ions Resea ch
Pe spec i es, ISSN 2214-7160, Else ie , Ams e dam, Vol. 14, pp. 1-22,
h ps://doi.o g/10.1016/j.o p.2024.100319
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/325799
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Facili y loca ion based on Adjus ed P esen Value
Hamid eza Rezaei a,b, Na halie Bos el b,c, Vincen Ho elaque d, Oli ie Pé on a,b,∗,
Jean-Lau en Vi iani d
aIMT A lan ique, 4 ue Al ed Kas le , F-44307 Nan es Cedex, F ance
bLabo a oi e des Sciences du Numé ique de Nan es (LS2N, UMR CNRS 6004), Nan es, F ance
cNan es Uni e si é, Nan es, F ance
dUni e si é de Rennes, CNRS, CREM – UMR 6211, Rennes, F ance
ARTICLE INFO
Keywo ds:
Facili y loca ion
Supply chain
Finance
Adjus ed p esen alue
ABSTRACT
Supply chain ne wo k design aims o op imize s a egic decisions such as acili y loca ion decisions.
These decisions ha e a majo impac on he supply chain, bu also on he inancial alue o he company.
Howe e , inancial conside a ions a e o en omi ed om acili y loca ion ma hema ical models.
This pape add esses he challenge o iden i ying a ele an inancial indica o ha can be p ac ically
implemen ed in acili y loca ion models ac oss di e en indus ies.
This pape makes se e al con ibu ions: he Adjus ed P esen Value (APV) is iden i ied as such a inancial
indica o ; we p opose a ma hema ical o mula ion ha embeds he APV in a acili y loca ion model maximizing
i m alue; compu a ional expe imen s demons a e he ac abili y o he model. Finally, we compa e he
ma hema ical model wi h a sequen ial app oach ha i s op imizes logis ical decisions and hen inancial
decisions. The p oposed model imp o es he sequen ial app oach up o 5.5%, inc eases he ma ke co e age
and an icipa es acili y loca ion decisions.
1. In oduc ion
Op imizing s a egic decisions in supply chain ne wo k design in-
ol es key decisions such as loca ing acili ies, de e mining hei ca-
paci y, and deciding when o open hem. I also in ol es de e mining
he p oduc lows in he logis ics ne wo k unde conside a ion. Supply
chain ne wo k design has been he subjec o ex ensi e li e a u e and
many e iews (see, e.g. [1,2]). Mos o his li e a u e iden i ies he
acili y loca ion p oblem and i s ex ensions as he co e o supply chain
ne wo k design models. The p oblem is known o be NP-ha d.
The p ima y goals o acili y loca ion in he con ex o supply chain
managemen a e (i) o de ine he ne wo k i sel , by loca ing acili ies
and de ining he alloca ion o p oduc lows o hese acili ies and
(ii) o de e mine he op imal p oduc lows in his logis ics ne wo k.
Classically, his ne wo k is op imized ei he by minimizing o al logis-
ics cos s o by maximizing he p o i gene a ed by he dis ibu ion o
goods.
Ideally, he co esponding ma hema ical models should ake in o
accoun he in e ac ions be ween supply chain managemen and o he
company depa men s (e.g. ma ke ing, human esou ces, inance) as
well as ex e nal ac o s (e.g. compe i o s, consume beha io , inancial
ma ke s).
∗Co espondence o: IMT A lan ique, 4 ue Al ed Kas le , F-44307 Nan es, F ance.
E-mail add ess: [email p o ec ed] (O. Pé on).
Despi e hei impo ance, i is s iking ha inancial ins umen s
a ely appea as ull componen s o acili y loca ion models. Howe e ,
mos companies inance hei s a egic in es men s by eso ing o
deb . I he e o e becomes clea ha he s a egic planning o logis ical
and inancial decisions mus go hand in hand. This would enhance
cos managemen by enabling mo e e ec i e capi al alloca ion and
deb inancing, and educing ope a ional expenses. By analyzing i-
nancial implica ions, companies can s a egically alloca e esou ces o
ensu e ha acili y in es men s a e bo h cos -e ec i e and in line wi h
hei o e all inancial objec i es. This assessmen o loca ion decisions
u he enhances esou ce alloca ion, leading o in es men s ha a e
e icien and aligned wi h inancial goals. Such in eg a ion imp o es
in es men s a egies, secu es ad an ageous inancing e ms, and a-
cili a es be e nego ia ion oppo uni ies, esul ing in lowe bo owing
cos s and g ea e inancial s abili y.
In addi ion o in e nal inancial bene i s, well-in o med loca ion
decisions can ha e o he posi i e e ec s, like local economic de el-
opmen . Choosing si es ha con ibu e o job c ea ion and le e aging
a ailable economic incen i es can s imula e local economies, c ea -
ing addi ional alue beyond he company’s immedia e inancial in e -
es s [3]. Al hough hese ac o s con ibu e o he company’s epu a ion
and long- e m success, hey a e beyond he scope o his pape .
h ps://doi.o g/10.1016/j.o p.2024.100319
Recei ed 21 June 2024; Recei ed in e ised o m 2 Oc obe 2024; Accep ed 19 No embe 2024
Ope a ions Resea ch Pe spec i es 14 (2025) 100319
A ailable online 20 Decembe 2024
2214-7160/© 2024 The Au ho (s). Published by Else ie L d. This is an open access a icle unde he CC BY license (
h p://c ea i ecommons.o g/licenses/by/4.0/ ).
H. Rezaei e al.
Fig. 1. T ade-o heo y o capi al s uc u e.
[4] men ioned he s ong in e ac ion be ween he inancial ac o s
and he s a egic planning. In closed-loop supply chains, [5] no ed ew
s udies conside hese as decision a iables. T adi ional supply chain
managemen app oaches conside he cash lows gene a ed by ope -
a ional decisions, bu no hose esul ing om inancial decisions. [6]
obse e ha he majo i y o models only b oadly examine he cos s and
e enues associa ed wi h he design o supply chains and igno e o he
inancial elemen s ha a e ela ed o i s u u e pe o mance. The e o e,
a s ong mo i a ion o his pape is o join ly conside he impac o
inancial and ope a ional decisions on he loca ion s a egy.
Financial heo y seeks o maximize he alue o he company and
s a es ha he alue o any asse is equal o he p esen alue o he
u u e cash lows discoun ed a an app op ia e discoun a e called he
cos o capi al. The discoun ed cash lows alua ion models can be ound
in any inance ex book o eade s ha would like o explo e he
model mo e in de ail, o example [7,8].
In co po a e inance, key decisions abou sou ces o inance a e
based on bo h equi y and deb . The heo e ical and p ac ical ques ion
is wha should be he mix o equi y and deb ha maximizes he
company’s alue? To answe his ques ion, he ade-o heo y o
capi al s uc u e [9] explo es wo opposi e consequences o he deb :
(i) he ax ad an ages associa ed wi h he ac ha paying in e es on
deb educes he co po a e axes and (ii) he dis ess disad an ages
associa ed wi h he ac ha , as he i m’s le e age inc eases, so does
he p obabili y o de aul and hence he Expec ed Bank up cy Cos s
(EBC). The ade-o heo y s a es ha he ad an ages o deb ( ax shield
bene i s) a e balanced by i s disad an ages (bank up cy cos s).
Acco ding o his heo y, he e is an op imal deb le el ha maxi-
mizes a i m’s alue (Fig. 1). As deb inc eases (ho izon al axis), bo h
he ax shield bene i and he bank up cy cos o deb ise. The ho izon-
al solid line ep esen s he unle e ed alue o he i m, i.e., he alue
o he i m wi hou aking in o accoun he cash low consequences o
inancial decisions. The ax shield, ep esen ed by he solid diagonal
line, is linea ly inc easing, while he cos o bank up cy is con ex
non-linea . The alue o he i m (blue cu e) is he e o e a conca e
non-linea unc ion. The peak o his cu e indica es he op imal deb
alue. This pape explici ly in oduces he bene i s and disad an ages
o deb in o de o de e mine he op imal mix o deb and equi y a
company should se le o und i s acili y loca ion s a egy.
The s a ing poin o mos discussions on he impac o inancing
on i m alua ion is he seminal wo k o Modigliani and Mille [10].
Since hen, se e al me hods ha e been p oposed o inco po a ing
he e ec s o deb in o cash low discoun ing alua ion models. The
Fig. 2. Calcula ion o he APV.
wo main app oaches a e ei he (i) o inco po a e he consequences
o inancial decisions in o he discoun ing a e: F ee Cash Flow (FCF)
discoun ed a he Weigh ed A e age Cos o Capi al (WACC), Capi al
Cash Flows (CCF) discoun ed a he p e- ax WACC, Economic Value
Added (EVA) discoun ed a he WACC (also known as he Ma ke Value
Added (MVA)) [11]; o (ii) o include hem in he Adjus ed P esen
Value (APV).
Acco ding o [12], all o he me hods p oposed in he li e a u e lead
o he same alua ion o he company when p ope ly applied. Howe e ,
some me hods a e mo e app op ia e han o he s in speci ic cases.
This pape ocuses on he APV [13], which ep esen s he alue o
a le e aged i m ( aking in o accoun he cash low consequences o
inancial decisions). I is a widely used measu e o alue c ea ion in
he con ex o discoun ed cash low alua ion models. The APV has
se e al main ad an ages: (i) I o e s de ailed in o ma ion on he ac o s
con ibu ing o he i m’s alue [14], dis inguishing he unle e ed i m
alue om he alue added by inancial decisions. (ii) I allows a
de ailed analysis o he alue de i ed om he choice o a pa icula
inancial s uc u e by isola ing he con ibu ion o ax bene i s o he
co po a e alue c ea ion [14]. (iii) As explained by [15], he change in
le e age equi es a pe iodic complex eassessmen o he WACC. On
he con a y, he APV wo ks unde bo h cons an and a iable deb
a ios o e he o ecas pe iod. Thus, maximizing APV sa is ies ou goal
o inco po a ing inancial conside a ions in o a acili y loca ion model.
Mo e speci ically, APV allows he ad an ages and disad an ages o deb
o be inco po a ed in o he company’s cash lows. I he company does
no use deb , APV is educed o he classic NPV model.
The unle e ed i m alue, also known as he base case Ne P esen
Value (NPV) [16], is he alue gene a ed by he ope a ional decisions
alone. This is why we call i Ope a ionally Gene a ed Value (OGV), as
opposed o he Financially Gene a ed Value (FGV). OGV is calcula ed
by discoun ing u u e ope a ional cash lows a he unle e ed cos o
equi y (o deb - ee cos o capi al). FGV is he p esen alue o he
ad an ages and disad an ages o deb .
As shown in Fig. 2, he APV is de e mined by wo ypes o decisions.
Fi s , logis ical decisions consis o selec ing acili ies om a se o
candida e loca ions o deli e goods o a se o cus ome s. This a ec s
only he OGV. Second, inancial decisions de e mine he deb le el
needed o inance logis ics in es men s, s agge ed o e ime.
This only a ec s he FGV. This led us o compa e wo app oaches,
which consis o sol ing he p oposed MILP model sequen ially (OGV
hen FGV) o all a once. These wo app oaches a e desc ibed in de ail
in Sec ion 6.
The main con ibu ions o his pape a e: (i) o p opose he Adjus ed
P esen Value (APV) as a inancial indica o used o op imize he
u u e impac o s a egic supply chain decisions on he u u e alue
o he i m (ins ead o he classical cos unc ion), (ii) o p opose a
mixed in ege linea p og amming (MILP) model ha in eg a es acili y
loca ion decisions and hei inancial consequences o e a s a egic
ho izon, (iii) o assess he ac abili y o his MILP by s a e-o - he-a
sol e s, and (i ) o e alua e he po en ial bene i s o in eg a ing inan-
cial conside a ions in o a acili y loca ion model, h ough a compa ison
Ope a ions Resea ch Pe spec i es 14 (2025) 100319
2
H. Rezaei e al.
wi h a sequen ial app oach ha op imizes logis ical decisions i s and
he inancial decisions second.
The s uc u e o he pape is as ollows. Sec ion 2posi ions his wo k
in he ela ed li e a u e. Sec ion 3desc ibes he assump ions and he
global s uc u e o ou p oblem. Sec ion 4p esen s he ma hema ical
o mula ion. Sec ion 5desc ibes he da a gene a ion p inciples used o
c ea e new ins ances. Sec ion 6p esen s he esul s o ou compu a-
ional expe imen s. Sec ion 7p esen s some manage ial insigh s and
concludes wi h u u e esea ch di ec ions.
2. S a e o he a
In hei e iew on acili y loca ion and supply chain manage-
men , [1] classi y inancial ac o s in h ee ca ego ies: (i) in e na ional
ac o s, including axes, du ies, a i s, exchange a es, ans e p ices,
and local con en ules, (ii) inancing and axa ion incen i es o e ed
by go e nmen s and (iii) in es men expendi u es, usually limi ed by
he o al a ailable budge . In he i s ca ego y, [17] pleads o he
conside a ion o di e en ax egimes and du ies, exchange a es,
ans e p ices and di e ences in ope a ing cos s. [18] p opose a acili y
loca ion model in o sho ing con ex , wi h bo h ac ical and s a egic
decision le els. The inancial decisions a e he ans e p icing and wo
a iables alloca ing logis ics cos s o a ious s akeholde s o he supply
chain.
Budge cons ain s can be ound in nume ous acili y loca ion mod-
els. Fo example, [19] add essed a acili y loca ion p oblem conside ing
budge cons ain s wi h he aim o minimizing he o e all dis ance
a eled. Simila ly, [20] p oposed a mul i-pe iod ma hema ical model
minimizing o al business cos s. The a ailable budge limi a ion is he
inco po a ed inancial ac o . [21] add essed a mul i-pe iod s ochas ic
acili y loca ion p oblem o maximize he bene i s o he i m, wi h a
budge cons ain on in es men s and he possibili y o se ing a a ge
o he e u n on in es men . See also, e.g., [22–26]
In many pape s, he only inancial conside a ion is o op imize
NPV. Al hough NPV is an app op ia e measu e o he p o i abili y o
an in es men , i does no cap u e he alue con ibu ed by inancing
decisions [16,27].
In he ollowing pa ag aphs, we e iew publica ions acco ding o
hei app oach measu ing inancial alues.
F ee cash low (FCF) discoun ed a WACC. [27], p opose an in eg a ed
s a egic- ac ical model whose objec i e unc ion, called
Co po a e Value, is he di e ence be ween he FCF discoun ed a WACC
and ne deb . To cope wi h he di icul y o a a iable WACC a e,
hey conside i as a ixed pa ame e o e he planning ho izon. This
app oach a o ably compa es wi h he adi ional NPV-o ien ed model.
[28] ex end he wo k o [27] by in oducing he unce ain y o demand,
p ice, and in e es a es o he model. The p esen ed s ochas ic model
is hen compa ed wi h a de e minis ic model, indica ing a signi ican
pe o mance imp o emen .
Economic alue added (EVA). EVA is an absolu e key igu e based
on ea nings ha ocuses on he pe o mance o a single pe iod [29],
while all he ollowing publica ions deal wi h a mul i-pe iod planning
ho izon. In hese cases, he objec i e unc ion is he sum o o ecas ed
EVAs.
[30] de elop a ma hema ical model o designing a ou -echelon
supply chain unde demand unce ain y, op imizing ne c ea ed alue,
measu ed by EVA. They assume a cons an WACC and compa e hei
inancial model wi h a non- inancial one ha igno es inancial anal-
ysis. While he inancial model c ea es mo e sha eholde alue, he
non- inancial model shows highe e u n on equi y.
The same au ho s p opose a bi-objec i e MILP model ha cap u es
ade-o alues be ween inancial pe o mance, measu ed wi h EVA,
as he i s objec i e and c edi sol ency, using Al man Z-sco e [31], as
he second objec i e unc ion [32] . Ou APV model elies on a simila
ade-o be ween he opposi e e ec s o in e es ax shield bene i and
he p esen alue o he deb . [33] add ess a h ee-echelon, mul i-
pe iod, mul i-i em closed-loop, s a egic and ac ical acili y loca ion
p oblem. EVA is maximized while he WACC is de ined as a cons an
pa ame e . [34] p opose a mul i-p oduc , mul i-pe iod, ou -echelon
model add essing inancial decisions like cash and isk managemen ,
capi al s uc u e, and e enue/cos managemen . They e alua e inan-
cial dimensions (co po a e alue, Change in Equi y, and EVA) agains
p o i maximiza ion.
[35] p opose a uzzy MILP model o design a global supply chain
ne wo k ha conside s logis ical and inancial lows simul aneously. To
assess he inancial pe o mance, EVA is maximized. [36] p opose a
MINLP model analyzing he impac o economic unce ain y on supply
chain inancial heal h. The model designs a mul i-pe iod closed-loop
supply chain o maximize EVA unde demand unce ain y. Besides EVA,
he au ho s also use NPV and WACC as objec i e unc ions, wi h a ixed
WACC a e.
EVA is also used as he objec i e unc ion in he mul i-pe iod model
p oposed by [37], which conside s deb epaymen s and new capi al
en ies as decision a iables. S ill, he WACC is assumed o be a ixed
pa ame e . Thei model leads o highe EVA in compa ison o he model
o [30]. [38] add ess a s ochas ic supply chain ne wo k design model
which aims a maximizing EVA. In e es a es (sho - e m and long-
e m), expec ed e u n on s ock ma ke , and isk- ee a e o in e es
a e subjec o unce ain y.
Ma ke alue added (MVA). MVA is a ool o ans o m EVA o a
mul i-pe iod basis. I measu es he p esen alue o u u e EVAs by
discoun ing hem a he WACC a e [29]. Un o una ely, he MVA loses
he EVA’s p ope y o being compa ible wi h luc ua ing deb a ios.
[39] add ess a mul i-pe iod, mul i-p oduc sus ainable supply chain
ne wo k design p oblem maximizing he MVA o he i m. Ye , he
au ho s conside he WACC a e as a ixed pa ame e .
The e iewed li e a u e p o ides a ange o me hodologies o in-
eg a ing inancial conside a ions in o acili y loca ion models, each
wi h i s s eng hs and limi a ions. F ee Cash Flow (FCF) and Eco-
nomic Value Added (EVA) a e commonly used bu ha e cons ain s
ela ed o ixed inancial pa ame e s. Ma ke Value Added (MVA) ad-
d esses mul i-pe iod analysis bu s uggles wi h luc ua ing deb a ios.
No ably, Adjus ed P esen Value (APV) is iden i ied as a p omising ap-
p oach o handling bo h mul i-pe iod models and a iable deb a ios.
Table 1summa izes hese app oaches and highligh s he po en ial o
APV o o e a mo e adap able and comp ehensi e inancial alua ion
amewo k o acili y loca ion decisions.
To ou knowledge, Capi al Cash Flow (CCF) and Adjus ed P esen
Value (APV) ha e no been employed as alua ion me hods in acili y
loca ion models and emain ela i ely unexplo ed in his con ex . This
obse a ion is co obo a ed by he ecen e iew conduc ed by [6].
3. P oblem desc ip ion and assump ions
In his sec ion, we desc ibe he main se ings and assump ions o he
mul i-pe iod capaci a ed acili y loca ion p oblem ha will be o mally
p esen ed in Sec ion 4. The no a ions a e summa ized in Appendix A.
3.1. Gene al se ing
The main goal o his pape is o p opose a ma hema ical model o
he join op imiza ion o acili y loca ion and inancing decisions. We
wan o ensu e ha : (i) he di e ences be ween he p oposed model
and adi ional models based on cos minimiza ion mus be angible and
in e p e able, and (ii) he logis ical assump ions mus be as ‘‘simple’’ as
possible, in o de o highligh he inancial in e p e a ions.
The i s ea u e mo i a ed us o conside a company s a ing om
a blank page. The e is no ope a ing acili y a he beginning o he ime
ho izon, no cu en loans. The e o e, ul illing all cus ome demands
is no manda o y. This assump ion suppo s a g adual expansion o
Ope a ions Resea ch Pe spec i es 14 (2025) 100319
3
H. Rezaei e al.
Table 1
Financial alua ion app oaches in acili y loca ions models.
Valua ion app oach Compa ibili y wi h Re e ences
Mul iple Fluc ua ing
pe iods deb a io
F ee Cash Flow (FCF) ✓✕[27] [28] [34]
Economic Value Added (EVA) ✕✓[30] [32] [34] [33] [35] [36] [37] [38]
Ma ke Value Added (MVA) ✓✕[39]
Capi al Cash Flow (CCF) ✓✕–
Adjus ed P esen Value (APV) ✓ ✓ his pape
Fig. 3. Time scale o supply chain decisions and hei inancial impac .
he logis ics ne wo k, allowing he company o a oid he obliga ion o
opening nume ous candida e acili ies in pe iod 1 wi h no addi ional
openings a e wa d.
Based on he second ea u e, we conside a logis ics ne wo k wi h
wo laye s: a se o candida e p oduc ion acili ies and a se o
cus ome s o be se ed om he selec ed acili ies.
The model is based on a s a egic ime ho izon ( ime pe iods a e
ypically yea s). All da a and pa ame e s a e assumed de e minis ic.
The main goal o he model is o de e mine which candida e acili-
ies o open, when o open hem o e he planning ho izon, which cus-
ome s will be se ed by he chosen acili ies and how o inance hese
decisions wi h an app op ia e mix o deb and equi y. The objec i e
unc ion o be maximized is he company’s APV.
3.2. Time ho izon
Maximizing APV equi es compu ing he p esen alue o he u u e
ope a ional and inancial cash lows o e mul iple pe iods. The ime
ho izon = {1,…, 𝑇} ep esen s he se o pe iods in which he
loca ion decisions a e applicable, i.e., he company can bo ow and
in es money a any pe iod 𝑡∈. Pe iod 0 ep esen s he ini ial s a e
o he supply chain. The ma hema ical model p esen ed in Sec ion 4is
assumed o be sol ed a pe iod 0, and conce ns decisions ha apply in
any pe iod 𝑡∈.
E en so, he logis ical and inancial decisions aken in his ime
ho izon will ha e a much longe impac on he company’s cash lows.
Assume ha a new acili y s a s ope a ing a some pe iod 𝑡∈and
ha some money will be bo owed o inance his decision (possibly in
addi ion o he use o in e nal and/o ex e nal equi y). Gi en a payback
pe iod o 𝑁 ime pe iods and a acili y li e ime 𝐿, he impac o his
decision on he deb will be obse ed un il pe iod 𝑡+𝑁while he
associa ed cash low will be obse ed un il pe iod 𝑡+𝐿− 1.
Time ho izon needs o be ex ended. To do so, we de ine he ime
ho izon ex ension ′= {𝑇+ 1,…, 𝑇+ max(𝐿, 𝑁)} as he se o pe iods
du ing which he selec ed acili ies a e s ill ope a ing and he inancial
impac o logis ical decisions can be obse ed a e he ime ho izon
.Fig. 3illus a es he case whe e a new acili y is selec ed a some
pe iod 𝑡∈and gene a es cash low un il pe iod 𝑇+𝐿− 1.
3.3. Logis ics and ma ke ea u es
Each candida e acili y 𝑗∈𝐽has a known capaci y 𝐶𝑗. Each acili y
has a ixed opening cos 𝑂𝑗paid once i he acili y 𝑗∈𝐽is selec ed.
I also has a ixed yea ly unning cos 𝐹𝑗paid e e y yea p o ided he
acili y is ope a ing. The uni a y p ocessing cos 𝜇𝑗is paid o each uni
o p oduc manu ac u ed in he acili y.
Since he model is ini ialized by a blank page, we conside ha a
new opened acili y will no be closed du ing he ho izon . The lis o
candida e acili ies in acili y loca ion models is gene ally much la ge
han he o al numbe o acili ies ac ually selec ed. Thus, we conside
an uppe bound 𝐽𝑚𝑎𝑥 on he o al numbe o selec ed acili ies. The isk
associa ed wi h he selec ion and all logis ical ope a ions is assumed o
be he same in each acili y.
Ou mul i-pe iod model add esses ma ke s wi h ime- a iable and
de e minis ic cus ome s’ demands. We assume ha eliable demand
o ecas is a ailable o he whole ime ho izon . The demand o
cus ome 𝑖∈in pe iod 𝑡∈is deno ed 𝐷𝑖𝑡. The selling p ice o each
cus ome can depend on many ac o s, including he cus ome ’s ma ke ,
i s demand le el and i s nego ia ion skills. Fo hese easons, we assume
a selling p ice 𝑃𝑖 o each cus ome 𝑖∈. We do no o ce he company
o se e all cus ome s a e e y pe iod. Unsa is ied cus ome s’ demands
a e simply los and back o de s a e o bidden. Following he all-o -
no hing p inciple, a cus ome ’s demand is ei he en i ely ul illed o no
a all. We apply he idea o inc emen al se ice [40]: cus ome s ha a e
being se ed in a pe iod mus be se ed in all subsequen pe iods ( he
alloca ion o cus ome s o acili ies migh change in di e en pe iods).
3.4. Financial ea u es
We assume all acili ies ope a e in a homogeneous inancial en i on-
men wi h a single ax a e 𝜂and no exchange a es. Consequen ly, he
equi y cos o capi al 𝐾𝐸is loca ion-independen and emains cons an
o e ime, as all in es men s a e subjec o he same business isk
wi hin he indus y and ma ke . This homogenei y means all candida e
acili ies and logis ical ope a ions sha e simila sys ema ic isks. Addi-
ionally, we assume ha in es men s ela ed o loca ion decisions ca y
he same isk as he company’s ‘‘business as usual.’’
Fo a new acili y 𝑗∈opened in pe iod 𝑡∈, he i m has o
inance i s ixed opening cos 𝑂𝑗by an app op ia e mix o deb and
equi y.
3.4.1. Deb inancing
Deb inancing amoun s o bo owing money om a bank. The e a e
|𝑇|disc e e pe iods in which bo owing is possible. In ligh o his, we
de ine |𝑇|sepa a e loans, each o which is de ined a a pe iod 𝑡∈𝑇
Ope a ions Resea ch Pe spec i es 14 (2025) 100319
4

H. Rezaei e al.
and may o may no be ac i a ed. We will e e o he loan ha was
igge ed in pe iod 𝑡as 𝑙𝑡 h oughou his s udy.
We assume ha he loans ha e he same eimbu semen du a ion
𝑁, and a cons an annui y epaymen me hod (homogeneous ma ke s).
Howe e , he cos o loan 𝑙𝑡, deno ed by 𝐾 𝐷𝑡, depends on he accu-
mula ed deb a io o pe iod 𝑡, such ha he highe he deb a io, he
highe he cos o loan. The inc ease in le e age leads o a highe isk
o bank up cy o he company, p omp ing deb holde s o demand a
highe isk p emium.
Fo each loan 𝑙𝑡, he annui y amoun 𝐴𝑡associa ed wi h a loan
amoun 𝐵𝑡, a cos o loan 𝐾 𝐷𝑡and a numbe 𝑁o equally sized
paymen s is gi en by:
𝐴𝑡=𝐵𝑡
𝐾 𝐷𝑡× (1 +𝐾 𝐷𝑡)𝑁
(1 +𝐾 𝐷𝑡)𝑁− 1.
Assuming ha he company will only use loans wi h he same
ma u i y 𝑁,𝑁≥𝑇, and cons an annui y epaymen me hod, he
amoun bo owed in pe iod 𝑡will be epaid om pe iod 𝑡+ 1 o pe iod
o 𝑡+𝑁.
3.4.2. Equi y inancing
In equi y inancing, money is suppo ed by sha eholde s in he o m
o in e nal inancing (company’s cash holdings) ep esen ed by he
a iables 𝐼 𝐸𝑡and ex e nal equi y ep esen ed by he a iables 𝐸 𝐸𝑡. The
company will i s seek in e nal inancing based on a ailable cash and
hen ex e nal inancing. The amoun o cash a ailable depends on he
le el o e ained ea nings and he amo iza ion policy. The pa ame e 𝛿
de ines he payou a io, i.e., he ac ion o ea nings paid as di idends.
The emaining amoun will be e ained by he company. I will be
added o exis ing cash holdings.
In an accoun ing app oach, dep ecia ion ep esen s a yea ly de-
c ease in angible asse s’ alue o e hei li e ime. Among di e en
me hods o dep ecia e he asse s (see, e.g., [27]) we conside a s aigh -
line dep ecia ion scheme, be ween he ini ial alue 𝑂𝑗and he sal age
alue 𝑆 𝑉𝑗o he selec ed asse s 𝑗∈. The amoun o dep ecia ion, as a
non-cash accoun ing expense, enables he company o build up ese es
ha can be used o u u e in es men s.
As men ioned abo e, he change om NPV o APV is made by
adding wo complemen a y elemen s: Tax Shield Bene i s and Expec ed
Bank up cy Cos (see Fig. 2). EBC is he p esen alue o di ec (legal
and accoun ing cos s) and indi ec bank up cy cos s, a con inuum o
cos s ha inc ease a an accele a ing a e as exposu e o bank up cy
inc eases, e.g., inc eased in e es expenses, los c edi , los sales, in-
e icien ope a ions. Di ec cos s ake he o m o adminis a i e ex-
penses ( us ee’s ees, legal ees, e e ee’s ees), and in he ime los
by execu i es in liquida ion [41].
[8] calcula es EBC as he p oduc o he p obabili y o bank up cy,
deno ed 𝑝, by he bank up cy cos s 𝛾×𝑂 𝐺 𝑉, whe e 0< 𝛾 <1is a
known ixed pa ame e . We choose o exp ess bank up cy cos s as a
ac ion o he i m alue be o e bank up cy as usually done bo h in
heo e ical [42] and empi ical s udies [43,44].
The a ious app oaches only di e on he measu es o he i m alue
(book e sus ma ke alue, o al e sus equi y alue), in he model, we
choose he ma ke ope a ional alue o he company. A company will
go bank up in pe iod 𝑡∈i he ma ke alue o i s asse s alls below
he alue o deb a his pe iod. The highe he deb h eshold ela i e
o he company’s asse s, he mo e di icul i will be o he asse alue
o each i .
4. Ma hema ical o mula ion
This sec ion de ails he ma hema ical model maximizing APV sub-
jec o logis ical and inancial cons ain s. Sec ions 4.1 and 4.2 enume -
a e he model cons ain s ela ed o he Ope a ionally Gene a ed Value
(OGV) and he Financially Gene a ed Value (FGV), espec i ely.
4.1. OGV: Ope a ionally gene a ed alue
Fo each candida e acili y 𝑗∈, he opening cos 𝑂𝑗is paid once
i he acili y is selec ed, and yea ly ope a ing ixed cos s 𝐹𝑗paid a
e e y pe iod when he acili y is ope a ing. In addi ion, each ope a ing
acili y has a p ocessing cos 𝜇𝑗 o each uni o p oduc p ocessed by
his acili y. We assume a li e ime 𝐿 > 𝑇 o all candida e acili ies
which is consis en in a s a egic poin o iew.
The dis ance be ween a cus ome 𝑖∈and a acili y 𝑗∈is
deno ed as 𝐷 𝑖𝑠𝑡𝑖𝑗 . We assume ha he anspo a ion cos be ween wo
loca ions is p opo ional o he dis ance a eled and he load ca ied,
wi h a uni anspo a ion cos 𝜔o e he whole ne wo k. Finally, we
conside a selling p ice 𝑃𝑖 o cus ome 𝑖∈.
Due o ade ules be ween geog aphical a eas as well as a ious
logis ical cons ain s, some cus ome s migh no be deli e ed by some
acili ies. Thus, we in oduce an accessibili y bina y pa ame e 𝑉𝑖𝑗
which akes he alue 1 i he cus ome 𝑖∈is accessible om acili y
𝑗∈.
We conside wo amilies o bina y decision a iables and one
amily o con inuous decision a iables. The a iable 𝑦𝑗 𝑡, akes he
alue 1 i he acili y 𝑗∈is ope a ing in pe iod 𝑡∈, and 0
o he wise. The a iable 𝑥𝑖𝑡 akes alue 1 i cus ome 𝑖∈is se ed
in pe iod 𝑡∈, and 0 o he wise. The a iable 𝑞𝑖𝑗 𝑡deno es he quan i y
deli e ed by acili y 𝑗∈ o cus ome 𝑖∈in pe iod 𝑡∈.
𝑦𝑗0= 0 ∀𝑗∈(1)
𝑦𝑗 ,𝑡−1 ≤𝑦𝑗 𝑡∀𝑗∈, 𝑡∈(2)
∑
𝑗∈
𝑦𝑗 ,𝑇 ≤𝐽𝑚𝑎𝑥.(3)
𝑥𝑖,𝑡−1 ≤𝑥𝑖𝑡 ∀𝑖∈, 𝑡∈(4)
𝑞𝑖𝑗 𝑡≤𝑉𝑖𝑗 𝐷𝑖𝑡 𝑦𝑗 𝑡∀𝑖∈, 𝑗∈, 𝑡∈(5)
∑
𝑗∈
𝑞𝑖𝑗 𝑡=𝐷𝑖𝑡 𝑥𝑖𝑡 ∀𝑖∈, 𝑡∈(6)
∑
𝑖∈
𝑞𝑖𝑗 𝑡≤𝐶𝑗𝑦𝑗 𝑡∀𝑗∈, 𝑡∈(7)
𝑒𝑗 𝑡=𝐹𝑗𝑦𝑗 𝑡+𝜇𝑗∑
𝑖∈
𝑞𝑖𝑗 𝑡+𝜔∑
𝑖∈
(𝐷 𝑖𝑠𝑡𝑖𝑗 𝑞𝑖𝑗 𝑡) ∀𝑗∈, 𝑡∈(8)
𝑟𝑗 𝑡=∑
𝑖∈
(𝑃𝑖𝑞𝑖𝑗 𝑡) ∀𝑗∈, 𝑡∈(9)
Cons ain s (1) es ablish ini ial condi ions: in pe iod 0, none o
he candida e acili ies is selec ed. Cons ain s (2) s a e ha selec ed
acili ies canno be closed du ing he ime ho izon (nei he du ing
he complemen a y ime ho izon ′whe eas 𝑦𝑗 ,𝑡 is ozen when 𝑡∈′).
Cons ain s (3) de ine an uppe bound 𝐽𝑚𝑎𝑥 on he numbe o selec ed
acili ies. Cons ain s (4) impose o se e in pe iod 𝑡a cus ome who
was se ed in pe iod 𝑡− 1.
Cons ain s (5) s a e ha he a iable 𝑞𝑖𝑗 𝑡is s ic ly posi i e only
i acili y 𝑗is opened and accessible (𝑦𝑗 𝑡= 1) and 𝑉𝑖𝑗 = 1. This
quan i y canno exceed he demand 𝐷𝑖𝑡. Cons ain s (6) calcula e he
o al quan i y deli e ed o each cus ome , which is ei he 0 when 𝑥𝑖𝑡 = 0
o he o al demand 𝐷𝑖𝑡 when 𝑥𝑖𝑡 = 1(no pa ial sa is ac ion o a
pa icula cus ome ’s demand). Some cus ome s may be deli e ed om
se e al di e en acili ies, a a gi en pe iod. The capaci y cons ain s
(7) en o ce he o al quan i y shipped by a selec ed acili y 𝑗∈ o be
a mos equal o i s capaci y 𝐶𝑗.
Cons ain s (8) calcula e he o al amoun o logis ics expenses 𝑒𝑗 𝑡
ela ed o acili y 𝑗∈in pe iod 𝑡∈𝑇. This amoun is he sum
o he yea ly ixed cos , he p ocessing cos and he anspo a ion
cos . Cons ain s (9) calcula e he o al e enue 𝑟𝑗 𝑡gene a ed by acili y
𝑗∈in pe iod 𝑡∈.
Ope a ions Resea ch Pe spec i es 14 (2025) 100319
5
H. Rezaei e al.
4.1.1. Calcula ion o OGV
We conside a ax a e 𝜂and he equi y cos o capi al o he
unle e ed company 𝐾𝐸. Fo each candida e acili y 𝑗∈, he ini ial
alue and he sal age alue and deno ed 𝑂𝑗and 𝑆 𝑉𝑗, espec i ely.
A acili y 𝑗∈opened in pe iod 𝜏∈has a li e ime 𝐿 > 𝑇. I
gene a es a iable cash lows be ween pe iod 𝜏and 𝑇and a cons an
cash low 𝐶 𝐹𝑗 𝑇a e pe iod 𝑇, as long as is i ac i e.
The calcula ion o OGV equi es de e mining which acili ies a e
ope a ing a each pe iod 𝑡∈∪′. We ex end he de ini ion o bina y
a iables 𝑦𝑗 𝑡 o ′as ollows: o any 𝑗∈, 𝑡∈′,𝑦𝑗 𝑡is equal o
1 i acili y 𝑗is ope a ing in pe iod 𝑡, and 0o he wise. I 𝑦𝑗 𝑇= 1
and he acili y was opened in pe iod 𝜏∈, hen 𝑦𝑗 𝑡= 1 o all
𝑇+ 1≤𝑡≤𝜏+𝐿− 1, and 0 o he wise. Then, o all 𝑡≥𝜏+𝐿,𝑦𝑡𝑗 = 0.
I 𝑦𝑗 𝑇= 0, all a iables 𝑦𝑗 𝑡a e equal o 0.
𝐷 𝑒𝑝𝑗=𝑂𝑗−𝑆 𝑉𝑗
𝐿∀𝑗∈(10)
𝐸 𝐵 𝐼 𝑇𝑗 𝑡=𝑟𝑗 𝑡−𝑒𝑗 𝑡−𝐷 𝑒𝑝𝑗𝑦𝑗 ,𝑡−1 ∀𝑗∈, 𝑡∈(11)
𝐶 𝐹𝑗 𝑡= (1 −𝜂)𝐸 𝐵 𝐼 𝑇𝑗 𝑡+𝐷 𝑒𝑝𝑗𝑦𝑗 ,𝑡−1 ∀𝑗∈, 𝑡∈(12)
𝐹 𝐶 𝐹𝑗 𝑡= (1 −𝜂)𝐸 𝐵 𝐼 𝑇𝑗 𝑡− (𝑂𝑗(𝑦𝑗 𝑡−𝑦𝑗 ,𝑡−1)
−𝐷 𝑒𝑝𝑗𝑦𝑗 ,𝑡−1) ∀𝑗∈, 𝑡∈(13)
𝑦𝑗 ,𝑡−1 ≥𝑦𝑗 𝑡∀𝑡∈′∖{𝑇+ 1} (14)
∑
𝑡∈
𝑦𝑗 𝑡+∑
𝑡∈′
𝑦𝑗 𝑡=𝐿 𝑦𝑗 𝑇∀𝑗∈(15)
Cons ain s (10) calcula e he s aigh -line dep ecia ion 𝐷 𝑒𝑝𝑗o
acili y 𝑗∈as a linea unc ion o i s ini ial alue (opening cos )
𝑂𝑗, sal age alue 𝑆 𝑉𝑗, and li e ime 𝐿.
Cons ain s (11) calcula e he Ea ning Be o e In e es and Taxes
𝐸 𝐵 𝐼 𝑇𝑡associa ed wi h acili y 𝑗∈in pe iod 𝑡∈. I is he di e ence
be ween he e enues and expenses o acili y 𝑗∈(be o e in e es
and ax) in pe iod 𝑡and he dep ecia ion ac o 𝐷 𝑒𝑝𝑗occu ing i he
acili y 𝑗ope a es in pe iod 𝑡− 1.
Cons ain s (12) calcula e he ope a ing cash low associa ed wi h
acili y 𝑗∈in pe iod 𝑡∈. The i s e m is he accoun ing esul
a e ax. The second e m ein oduces he dep ecia ion because i is
no a cash ou low.
In cons ain s (13), he ee cash lows 𝐹 𝐶 𝐹𝑗 𝑡measu es he abili y
o he i m o gene a e cash. I is he di e ence be ween ope a ing and
non-ope a ing cash in lows and ou lows associa ed wi h each acili y.
Cons ain s (14) and (15) model he ex ension o a iables 𝑦𝑗 𝑡 o he
pe iods 𝑡∈′.
The Ope a ionally Gene a ed Value (OGV) is de ined by:
𝑂 𝐺 𝑉=∑
𝑗∈(∑
𝑡∈
𝐹 𝐶 𝐹𝑗 𝑡
(1 +𝐾𝐸)𝑡+∑
𝑡∈′
𝐶 𝐹𝑗 𝑇𝑦𝑗 𝑡
(1 +𝐾𝐸)𝑡).(16)
This exp ession is non-linea due o he p oduc o a iables 𝐶 𝐹𝑗 𝑇
and 𝑦𝑗 𝑡. The linea iza ion p ocess is explained in Appendix C.2.
4.2. FGV: Financially gene a ed alue
I a acili y 𝑗∈is opened in pe iod 𝑡∈, he i m has
o decide how o inance i s ini ial alue 𝑂𝑗. The company can mix
deb inancing, de ailed in Sec ion 4.2.1 and equi y inancing, de ailed
in Sec ion 4.2.2. We conside h ee amilies o con inuous decision
a iables: 𝑏𝑜𝑟𝑟𝑜𝑤𝑡 ep esen he amoun o money bo owed in pe iod
𝑡∈;𝐼 𝐸𝑡and 𝐸 𝐸𝑡 ep esen he in e nal equi y he ex e nal equi y in
pe iod 𝑡∈, espec i ely.
Eq. (17) s a es ha acili y sunk cos s a e unded by he deb ,
in e nal unding o by ex e nal equi y.
∑
𝑗∈
𝑂𝑗(𝑦𝑗 𝑡−𝑦𝑗 ,𝑡−1) =𝑏𝑜𝑟𝑟𝑜𝑤𝑡+𝐼 𝐸𝑡+𝐸 𝐸𝑡∀𝑡∈.(17)
4.2.1. Deb inancing cons ain
Fo loan 𝑙
𝑡, we de ine 𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒
𝑡𝑡,𝑟𝑒𝑝𝑎𝑦
𝑡𝑡 and 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑡𝑡 espec i ely as
he o al amoun s ill ali e, he annui y amoun epaid, and he alue
o he in e es associa ed wi h ha loan in pe iod 𝑡∈∪′.
𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒
𝑡𝑡 =⎧
⎪
⎨
⎪
⎩
0𝑡 <
𝑡
𝑏𝑜𝑟𝑟𝑜𝑤
𝑡𝑡=
𝑡
𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒
𝑡,𝑡−1 −𝑟𝑒𝑝𝑎𝑦
𝑡𝑡 +𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑡𝑡 𝑡 >
𝑡
(18)
𝑟𝑒𝑝𝑎𝑦
𝑡𝑡 =⎧
⎪
⎨
⎪
⎩
0𝑡≤
𝑡
𝑏𝑜𝑟𝑟𝑜𝑤
𝑡
𝐾 𝐷
𝑡(1+𝐾 𝐷
𝑡)𝑁
(1+𝐾 𝐷
𝑡)𝑁−1 
𝑡 < 𝑡≤
𝑡+𝑁
0𝑡 >
𝑡+𝑁
(19)
𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑡𝑡 =𝐾 𝐷
𝑡𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒
𝑡,𝑡−1 ∀𝑡∈∪′(20)
Cons ain s (18) calcula e he alue o he loan balance associa ed
wi h loan 𝑙
𝑡, in pe iod 𝑡∈∪′. I s alue is 0un il i s ac i a ion
in pe iod 𝑡=
𝑡. As i is ac i a ed (𝑡=
𝑡), he balance equals he
amoun bo owed (=𝑏𝑜𝑟𝑟𝑜𝑤
𝑡), hen o 𝑡 >
𝑡, i is dec eased by he deb
amo iza ion du ing he pe iod, which is equal o 𝑟𝑒𝑝𝑎𝑦
𝑡𝑡 -𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑡𝑡.
The epaymen o mula is gi en by (19). A e loan 𝑙
𝑡is ac i a ed
in pe iod 
𝑡∈, i s epaymen s a s a he beginning o he ollowing
pe iod (𝑡=
𝑡+ 1) un il i is ully epaid (𝑡=
𝑡+𝑁).
Cons ain s (20) calcula e 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑡𝑡. Bo h cons ain s (19) and (20)
a e non-linea . Thei linea iza ion is explained in Appendices C.3, and
C.4, espec i ely.
4.2.2. Equi y inancing cons ain
𝑁 𝑂 𝑃 𝐴𝑇𝑡= (1 −𝜂)(∑
𝑗∈
𝐸 𝐵 𝐼 𝑇𝑗 𝑡−∑

𝑡∈
𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑡𝑡) ∀𝑡∈(21)
𝑐 𝑎𝑠ℎ𝑡=𝑐 𝑎𝑠ℎ𝑡−1 + (1 −𝛿)𝑁 𝑂 𝑃 𝐴𝑇𝑡+𝐸 𝐸𝑡
+∑

𝑡∈
(𝑏𝑜𝑟𝑟𝑜𝑤
𝑡𝑡 −𝑟𝑒𝑝𝑎𝑦
𝑡𝑡 +𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑡𝑡)
−∑
𝑗∈
(𝑂𝑗(𝑦𝑗 𝑡−𝑦𝑗 ,𝑡−1) −𝐷 𝑒𝑝𝑗𝑦𝑗 ,𝑡−1) ∀𝑡∈(22)
𝐼 𝐸𝑡≤max(0, 𝑐 𝑎𝑠ℎ𝑡−1) ∀𝑡∈(23)
To ease he p esen a ion calcula ion o in e nal equi y a iables 𝐼 𝐸𝑡,
we in oduce wo in e media e a iables: he Ne Ope a ing P o i A e
Taxes o pe iod 𝑡∈, deno ed 𝑁 𝑂 𝑃 𝐴𝑇𝑡, is ob ained by emo ing he
in e es expenses om he EBIT (see cons ain s (11)) in pe iod 𝑡∈,
and hen mul iplying he esul by he e m (1 −𝜂), whe e 𝜂is he
i m ax a e. The cash lows 𝑐 𝑎𝑠ℎ𝑡gene a ed in pe iod 𝑡and owned by
sha eholde s (cons ain s (22)) a e equal o he esidual cash lows once
all s akeholde s, including lende s and S a e, ha e been emune a ed.
Finally, cons ain s (23) s a e ha 𝐼 𝐸𝑡canno exceed he cash
a ailable a he end o he p eceding pe iod, 𝑐 𝑎𝑠ℎ𝑡−1. The linea iza ion
o his equa ion is explained in Appendix C.5.
4.2.3. Calcula ion o FGV and APV
The p obabili y o de aul is deno ed by 𝑝. The a iable 𝑒𝑞 𝑢𝑖𝑡𝑦𝑡
ep esen s he alue o he equi y in pe iod 𝑡∈. The pa ame e 𝛽 >1
models he ola ili y o he company’s asse s. The pa ame e 0< 𝜁 <1
is he maximum pe cen age o he o al asse s ha can be used o
deb s ela ed o new acili ies. We ecall ha he pa ame e 0< 𝛾 <1,
de ined in Sec ion 3.4 is used o calcula e he bank up cy cos . Since
cash lows and ax shield bene i s sha e he same sys ema ic isk, hey
a e discoun ed a he same a e 𝐾𝐸.
𝑒𝑞 𝑢𝑖𝑡𝑦𝑡=𝑒𝑞 𝑢𝑖𝑡𝑦𝑡−1 + (1 −𝛿)𝑁 𝑂 𝑃 𝐴𝑇𝑡+𝐸 𝐸𝑡∀𝑡∈(24)
𝑝=(∑
𝑡∈𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒
𝑡𝑇
∑
𝑡∈𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒
𝑡𝑇 +𝑒𝑞 𝑢𝑖𝑡𝑦𝑇)𝛽
(25)
∑

𝑡∈
𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒
𝑡𝑡 ≤𝜁(𝑐 𝑎𝑠ℎ𝑡+∑
𝑗∈∑
𝑡′≤𝑡
(𝑂𝑗(𝑦𝑗 𝑡′−𝑦𝑗 ,𝑡′−1)
−𝐷 𝑒𝑝𝑗𝑦𝑗 ,𝑡′−1)) ∀𝑡∈.(26)
Ope a ions Resea ch Pe spec i es 14 (2025) 100319
6
H. Rezaei e al.
𝐹 𝐺 𝑉= (1 −𝑝)𝜂∑
𝑡∈∪′∑
𝑡∈𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑡𝑡
(1 +𝐾𝐸)𝑡−𝑝 𝛾 𝑂 𝐺 𝑉(27)
In cons ain s (24), he alue o equi y in pe iod 𝑡∈is de ined as
i s alue a ime 𝑡− 1plus he e ained ea nings a ime 𝑡plus he new
cash p o ided by sha eholde s a ime 𝑡. This alue is used o calcula e
he p obabili y o de aul in cons ain (25). This p obabili y is an
inc easing unc ion o he deb a io ∑
𝑡∈𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒
𝑡𝑇 ∕(∑
𝑡∈𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒
𝑡𝑇 +
𝑒𝑞 𝑢𝑖𝑡𝑦𝑇)in pe iod 𝑇.
The p obabili y o de aul is in luenced by he company’s capaci y
o gene a e enough cash low o mee i s deb obliga ions. The deb
a io quan i ies he ela i e signi icance o he deb in ela ion o cash
low. Since we do no di ec ly model cash low isk, we app oxima e i
indi ec ly using he coe icien 𝛽. The linea iza ion o his cons ain is
explained in Appendix C.8.
To a oid inancial dis ess in ea lie pe iods, we conside an uppe
bound on he le el o deb , se by Cons ain s (26). The igh -hand side
ep esen s he ne alue o he o al asse s in pe iod 𝑡∈𝑇, mul iplied
by pa ame e 𝜁. Cons ain (27) calcula es he alue o FGV. The i s
e m co esponds o he ax shield bene i (TSB) and he second one
o he expec ed bank up cy cos (EBC) as illus a ed in Fig. 2. As deb
le els inc ease, so do in e es paymen s and he esul ing ax shield
bene i s. Howe e , as shown in Eq. (25), a highe deb le el also aises
he p obabili y o de aul and he co esponding bank up cy cos s.
Finally, he objec i e unc ion o be maximized is he APV, which
is he sum o OGV and FGV, as w i en in cons ain (28):
𝐴𝑃 𝑉= (1 −𝑝𝛾)𝑂 𝐺 𝑉+ (1 −𝑝)𝜂(∑
𝑡∈∪′∑
𝑡∈𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑡𝑡
(1 +𝐾𝐸)𝑡).(28)
The p esen alue o ax shields is compu ed by discoun ing he
annual in e es amoun ∑
𝑡∈𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑡𝑡 a he a e 𝐾𝐸, mul iplied by he
i m ax a e 𝜂.
The company will ecei e he in e es ax shields wi h a p obabili y
1 −𝑝(i.e., i i is no bank up ed) and pay a bank up cy cos 𝛾 𝑂 𝐺 𝑉
wi h p obabili y 𝑝. F om his equa ion, i is clea ha he le e age has
a mixed impac on APV. On he one hand, i has a nega i e impac
h ough he bank up cy isk (𝑝) and, on he o he hand, a posi i e one
h ough he ax shield bene i (las e m o he APV unc ion).
5. Da a gene a ion
As explained in Sec ion 2, we did no ind a ela ed benchma k
ins ance ha could be used o alida e and expe imen wi h his model.
The e o e, new ins ances we e gene a ed ollowing he gene a ion ules
used by di e en au ho s. As men ioned abo e, APV is pa icula ly
ele an o e alua ing s a egic decisions in la ge-scale ac i i ies. Thus,
he gene a ed ins ances mimic a supply chain wi h mul iple ma ke s,
each ma ke ha ing i s own cos s, p oduc p ice, e c.
We gene a ed ins ances wi h 60 o 270 cus ome s. Following [45],
he numbe ||o candida e acili ies is de ined as 10% o he numbe
o cus ome s. The maximum numbe o open acili ies (𝐽𝑚𝑎𝑥) is de ined
as ⌈0.5||⌉. In all ins ances, ||= 5and |′|= 10 pe iods we e
conside ed. The nex subsec ions de ail he s eps ollowed by he da a
gene a ion.
5.1. Logis ics da a and pa ame e s
All loca ions we e gene a ed in a 1000 ×1000 g id. Bo h axes o he
g id a e decomposed in o 5 in e als o size 200, de ining 25 squa ed
a eas called egions.
Ma ke s. The g id is pa i ioned in o wo o i e ma ke s, ha is a
connec ed se o egions. Each ma ke ecei es an economic index ha is
used as a p oxy o indica e i s economic si ua ion and gene a e he cos s
and p ices in acco dance. The a ec ed cos s and p ices a e, speci ically
he p ocessing cos (𝜇), he opening cos pa ame e (𝜑), and he selling
p ice (𝑃) as shown in Table 2. The economic indices a e gene a ed wi h
a uni o m dis ibu ion be ween 50 and 150. The highe he economic
index, he highe he ma ke ’s cos s and p ices.
Cus ome s and candida e acili ies. To gene a e he loca ion o cus ome s
and candida e acili ies, we used wo app oaches: he coo dina es o
he cus ome s and candida e acili ies we e ei he andomly gene a ed
wi h a uni o m dis ibu ion o e he en i e g id o we used a clus e ed
pa e n. In he la e case, we loca ed a ound 60% o cus ome s ( esp.
candida e acili ies) in a subse o 4 o 5 egions, hen he emaining
40% we e andomly loca ed wi h a uni o m dis ibu ion o e he g id.
Cus ome s’ demand. We gene a ed he cus ome s’ demand acco ding
o wo di e en demand p o iles: Fi s , ollowing [46], he cus ome s’
demands a each pe iod a e gene a ed wi h a uni o m dis ibu ion in
he in e al [100, 300]. Then, each cus ome ’s demand s ill lies in he
in e al [100, 300], wi h he addi ional p ope y ha he o al demand
g ows by a ac o in he in e al [1.05, 1.25] be ween wo successi e
pe iods. No e ha al hough he sum o cus ome s’ demands g ows o e
ime, he indi idual demand o some cus ome s may dec ease be ween
wo successi e pe iods due o he une en dis ibu ion o g ow h."
Capaci y o he acili ies. Each candida e loca ion has a gi en capaci y.
We ha e gene a ed h ee sizes o acili ies, named small,medium and
la ge, ep esen ing 80%, 100% and 130% o he a io 𝐷∕𝐽𝑚𝑎𝑥 espec-
i ely, whe e 𝐷is he a e age demand pe pe iod. The capaci y o each
acili y is andomly chosen such ha abou 1/3 o acili ies a e small,
1/3 a e medium and 1/3 a e la ge.
Logis ics cos s. Se e al cos s in he ma hema ical model a e a ec ed by
a ia ions in he cos o li ing ac oss di e en ma ke a eas. To accoun
o his, we in oduced an in e media e pa ame e , e e ed o as he
economic index, o gene a e hese cos s. Fo each ma ke , he economic
index is gene a ed wi h a uni o m dis ibu ion be ween alues 50 and
150. The ange [50,150] is decomposed in o 5 in e als, and o each
in e al, he p ocessing cos , he opening cos and he selling p ice a e
gene a ed acco ding o a uni o m dis ibu ion, as de ailed in Table 2.
No e ha as we gene a e he yea ly ixed cos o acili y as a pe cen age
o he opening cos , his da a is also di ec ly a ec ed by he economic
index.
•The P ocessing cos s (𝜇𝑗) o small-sized acili ies is de ined wi h
a uni o m dis ibu ion on an in e al depending on he economic
index o he ma ke whe e acili y 𝑗lies (see line 2 o Table 2).
The ic i ious mone a y uni used in he es o his a icle is
called ela i e money uni (𝑟𝑚𝑢). To model economies o scale,
he p ocessing cos s a a e age-sized and la ge-sized acili ies a e
ob ained by mul iplying hese alues by 0.98 and 0.96, espec-
i ely. These pa ame e s ha e been adjus ed in such a way ha
he ela i e pa o p ocessing cos s oughly ep esen s a ound
35% –45% o he o al cos s in each ins ance.
•The alue o he ixed opening cos (𝑂𝑗), ep esen ed in mone-
a y alue, is s ongly ela ed o he alue o he eal es a e. To
model economies o scale as he capaci y g ows, ollowing [47],
we assume ha he ixed opening cos o a acili y 𝑗is oughly
p opo ional o he squa e oo o i s capaci y. We se 𝑂𝑗=
𝜑𝑗√𝐶𝑗, whe e 𝐶𝑗 ep esen s he capaci y o acili y 𝑗and 𝜑𝑗
is he opening cos pa ame e a loca ion 𝑗(mainly de e mined
by he cos o he local eal es a e). The alue o 𝜑𝑗is gene a ed
andomly wi h a uni o m dis ibu ion be ween 625 and 750 (see
line 3 o Table 2). These in e als ha e been se by successi e
adjus men s in such a way ha he ela i e pa o he ixed
opening cos s oughly ep esen 25% –35% o he o al logis ics
cos s (see, e.g. [47,48] o simila app oaches).
•The Fixed yea ly unning cos (𝐹𝑗) is se a 5% o he ixed
opening cos o each acili y, 𝑂𝑗, pe yea . Thus, i ep esen s
a ound 10% –15% o he o al cos s.
•The anspo a ion cos s a e conside ed p opo ional o he
Euclidean dis ance a eled. We assume hey a e simila in all
ma ke s. To ha e he anspo a ion cos ep esen ing 10% o 20%
o o al cos s [49], he uni anspo a ion cos , 𝜔, is se a 0.002
𝑟𝑚𝑢 in all he ins ances.
Ope a ions Resea ch Pe spec i es 14 (2025) 100319
7
H. Rezaei e al.
Table 2
Gene a ion o logis ical da a.
Economic index [50, 70[ [70, 90[ [90, 110[ [110, 130[ [130, 150]
P ocessing cos 𝜇( mu∕uni ) [1, 1.1] [1.1, 1.2] [1.2, 1.3] [1.3, 1.4] [1.4,1.5]
Opening cos pa ame e 𝜑[625, 650] [650, 675] [675, 700] [700, 725] [725, 750]
Selling p ice 𝑃( mu∕uni ) [3, 3.4] [3.4, 3.8] [3.8, 4.2] [4.2, 4.6] [4.6, 5]
Table 3
Calcula ion o he cos o loan 𝐾 𝐷𝑡.
Deb a io in e al Cos o loan
(in %) (in %)
(0,30] 3.2
(30,40] 3.5
(40,50] 4.0
(50,60] 5.0
(60,70] 6.6
(70,80] 9.6
Selling p ice. The p ice 𝑃𝑖p oposed o cus ome 𝑖∈depends on he
ma ke in which 𝑖lies. I s alue lies in he in e al [3,5] (see line 4 o
Table 2).
Accessibili y pa ame e s. The accessibili y pa ame e 𝑉𝑖𝑗 is se o alue 1
i he dis ance be ween acili y 𝑗∈and cus ome 𝑖∈is less han o
equal o 750 ( oughly hal o he longes possible dis ance in he g id),
and 0 o he wise.
5.2. Financial pa ame e s
The a e age cos o equi y 𝐾𝐸is se a 9% [50]. In o de o es ima e
he cos o loan 𝐾 𝐷𝑡in ela ion o he p obabili y o bank up cy,
we employ an a bi age equa ion used by a isk-neu al bank1. This
app oach allows us o quan i y he expec ed a e o e u n on loans by
aking in o accoun hei associa ed isks.
Using he a bi age equa ion, we modeled he cos o loan as a piece-
wise linea unc ion a ying be ween 3.2% and 9.6%. We conside ed six
deb a io in e als as de ined in Table 3.
In all ins ances, we conside a li e ime alue 𝐿and a numbe
o annui ies 𝑁 ha a e bo h equal o 10 yea s. The dep ecia ion
𝐷 𝑒𝑝𝑗calcula ed wi h his li e ime alue, and he sal age alue 𝑆 𝑉𝑗
is assumed o be negligible [27]. The alue o he bank up cy cos
pa ame e 𝛾is se o 0.5 and he uppe bound o he deb a io 𝜁is se
o 0.8 in o de o ensu e an accep able inancial si ua ion. The alue o
he bank up cy p obabili y pa ame e 𝛽in cons ain (25) is se o 3.
The co po a e ax a e 𝜂is se a 30%.
5.3. Se o ins ances
Following he p inciples desc ibed abo e, 32 ins ances we e gene -
a ed: 16 ins ances ha e andom loca ions and 16 ins ances ha e clus-
e ed loca ions, 16 ins ances ha e andom demands and 16 ins ances
ha e g owing demands.
1The a bi age equa ion used by a isk-neu al bank can be exp essed in
he o m o he ollowing equa ion:
𝐹(1 +𝑟𝑓)𝑁= (1 +𝐾 𝐷)𝑁(𝑝𝑅 + (1 −𝑝)𝐹).
Whe e 𝐹,𝑟𝑓,𝑁,𝐾 𝐷,𝑝, and 𝑅a e loan acial alue, isk- ee a e, loan
du a ion, cos o loan, p obabili y o bank up cy, and alue gi en de aul o he
amoun ha lende eco e s i he company de aul s on i s deb , espec i ely.
No e ha , we se he isk- ee a e o 3% in acco dance wi h he cos o deb
epo ed by he KPMG epo [50]. Acco ding o his epo , he cos o deb o
indus ial manu ac u ing companies wi h a deb a io o 25%–30% is a ound
3%. Se ing 𝑟𝑓 o 3% le s us ob ain almos he same cos o deb o he same
amoun o deb a ios.
Ins ance names a e o med by hei size ollowed by one o he
le e s A, B, C, o D. No e ha he e a e 4 di e en ypes o ins ances
due o applying ou pa e ns o gene a e he cus ome s’ and acili ies’
coo dina es as well as he cus ome s’ demands.
Fo each size o ins ance, Table 4enume a es he numbe o bina y
and con inuous a iables associa ed o he logis ical and inancial pa
o he model, as well as he numbe o cons ain s. No e ha he bina y
a iables in he inancial pa o he model come om he linea iza ion
o nonlinea cons ain s.
6. Nume ical expe imen s
6.1. In eg a ed and sequen ial app oaches
As ep esen ed in Fig. 2, APV can be decomposed in o a logis ical
pa (OGV) and a inancial pa (FGV). The whole ma hema ical model
can be decomposed in o wo sub-p oblems p esen ed in Sec ions 4.1
and 4.2, espec i ely. The logis ical sub-p oblem de e mines which acil-
i ies should be opened, which cus ome s should be se ed as well as he
associa ed p oduc lows. Once he logis ical decisions ha e been ixed,
he inancial sub-p oblem op imizes he alue o APV by maximizing FGV
(see Fig. 4).
The Sequen ial App oach mimics he decision p ocess ollowed in
classical acili y loca ion models. The logis ical and inancial sub-
p oblems a e sol ed sequen ially: Fi s , he 𝑂 𝐺 𝑉 o mula de ined
by Eq. (16) is maximized, subjec o cons ain s (1)—(15). Then,
he 𝑂 𝐺 𝑉is conside ed cons an and he 𝐹 𝐺 𝑉is op imized. This
amoun s o maximizing he APV o mula de ined in cons ain (28),
subjec o cons ain s (17)—(26). The In eg a ed App oach conside s
he whole ma hema ical model (1)–(28) a once and maximizes APV
by simul aneously de e mining he alue o all logis ical and inancial
a iables.
By na u e, he objec i e unc ion alues o he op imal solu ions
in he in eg a ed app oach a e highe han hose in he sequen ial
app oach. The ela i e gap be ween he op imal solu ions o he wo
app oaches measu es he bene i o in oducing inancial conside a ions
in o acili y loca ion models. This compa ison unde sco es he angible
ad an ages o in eg a ing inancial decisions in o classical acili y lo-
ca ion models, demons a ing he bene i s o adop ing an in eg a ed
inancial pe spec i e. In e u n, he in eg a ed app oach is likely o
yield compu a ional di icul ies due o he size o he ma hema ical
model. An objec i e o he nume ical expe imen s is o explo e he
p ac ical limi s o such a model.
6.2. Assessmen o he sequen ial and he in eg a ed app oaches
All nume ical expe imen s we e un on an In el(R) Xeon(R) Gold
6230 CPU @ 2.10 GHz using en co es. The ma hema ical model was
sol ed wi h he IBM ILOG CPLEX Op imiza ion S udio 20.1.0 sol e ,
wi h a ime limi o 6 h (21 600 seconds).
Table 5compa es he compu a ional ime needed o sol e he in-
eg a ed and he sequen ial app oaches, espec i ely. Columns 2 and 4
indica e he compu a ional ime (in seconds) o ge an op imal solu ion.
Columns 3 and 5 indica e he op imali y gap when no op imal solu ion
could be ob ained a e 3 h o compu a ion.
This able shows ha bo h app oaches a e ac able o ins ances
wi h up o 150 cus ome s. As expec ed, he sequen ial app oach is
easie o sol e han he in eg a ed app oach: 29 ins ances could be
sol ed o op imali y wi h he sequen ial app oach wi hin 6 h, only
Ope a ions Resea ch Pe spec i es 14 (2025) 100319
8
H. Rezaei e al.
Fo p ac ical implemen a ion o he in eg a ed decision model, com-
pany managemen mus conside he op imal mix o deb and equi y
inancing o i s s a egic p ojec s. The bene i s om deb - ela ed ax
sa ings a e ela i ely s aigh o wa d o calcula e, as hey depend on
he easily obse able co po a e ax a es in he coun ies whe e he
loca ions a e es ablished. In con as , es ima ing bank up cy cos s is
a mo e challenging. The p obabili y o a company’s bank up cy can
be es ima ed using se e al ac o s: (i) he c edi a ing assigned o
he company’s deb by a ing agencies, (ii) he in e es a e se by
lende s (wi h highe a es indica ing a g ea e isk o bank up cy), (iii)
econome ic models based on he company’s inancial a ios (sco ing
unc ions). Bank up cy cos s will be assessed by conside ing he cos s o
bank up cy p ocedu es in each coun y o di ec cos s and by compa -
ing he bank up cy cos s o p e iously de aul ed companies o indi ec
cos s. Ano he manage ial challenge is b inging he ope a ional and
inancial eams oge he o collabo a i ely de elop he implemen a ion
s a egy.
Since he goal o his pape was o highligh he in e ac ions be-
ween logis ical and inancial decisions a he han desc ibing a pa icu-
la supply chain, we conside ed a simple wo-echelon supply chain. This
model can be ex ended in many di ec ions. Conside ing a supply chain
wi h al eady ac i e acili ies, cus ome s and cu en loans only equi es
modi ying he ini ial condi ions. Relaxing he inc emen al cus ome
sa is ac ion, allowing pa ial cus ome sa is ac ion o se ing single-
sou cing cons ain s would sligh ly modi y he logis ics cons ain s, bu
hese ules do no a ec he inancial pa o he model. Ex ending he
model o mo e complex supply chains (e.g., wi h addi ional echelons,
selec ion o aw ma e ial supplie s, esizing o acili ies, selec ion o
subcon ac o s, selec ion o anspo a ion modes [51]) would equi e
a se o mo e elabo a ed logis ical cons ain s bu , once again, hese
logis ical ules do no a ec he inancial pa o he model.
Howe e , in oducing he possibili y o closing acili ies has di ec
inancial consequences. A ealis ic assump ion is ha , once opened,
acili ies should be ope a ing o a minimum numbe o pe iods (which
is gene ally la ge han 𝑇). Assuming ha acili ies al eady ope a ing in
pe iod 0 can be closed be o e he end o hei li e ime has consequences
bo h on he alue o OGV and FGV. Closing a acili y and selling i
modi ies he cash lows. All associa ed loans mus be he objec o ea ly
eimbu semen , wi h possible inancial penal ies. Finally, cons ain s
(15), which assumed 𝑦𝑗 𝑇= 1 o any ope a ing acili y, should be
adap ed o he case o closing acili ies. Ou model can easily apply o a
ealis ic case whe e he axes and cos o deb di e om one loca ion
o ano he . In ha case, he di e ence be ween bo h app oaches would
p obably be mo e signi ican .
A u he possible ex ension is he conside a ion o s ochas ic da a o
pa ame e s. Se e al ac o s, such as demand, logis ics cos s, dis up ion
p obabili ies, and inancial pa ame e s, a e subjec o unce ain y. A
s ochas ic model inco po a ing mul iple unce ain pa ame e s would
p o ide a mo e ealis ic ep esen a ion. Howe e , managing se e al
s ochas ic pa ame e s simul aneously emains a signi ican challenge in
s ochas ic op imiza ion. Robus op imiza ion migh be a mo e ac able
app oach, hough i may yield e y conse a i e esul s. A possible so-
lu ion is o u ilizing me hods like App oxima e Dynamic P og amming
and S ochas ic Dynamic Op imiza ion, which handle bo h unce ain y
and ime-dependen decision-making.
O he possible ex ensions o he model would be o explo e inancial
isk-awa e acili y loca ion models [52], o o ex end he model by
in eg a ing ac ical a iables such as he wo king capi al.
The cos ly linea iza ion mechanism p e en s MILP sol e s o sol e
much la ge ins ances han hose p esen ed in his pape .
Table 5illus a es he model’s limi ed scalabili y. While he solu-
ions ob ained om he in eg a ed app oach a e ela i ely close o hose
om he sequen ial app oach, a signi ican gap be ween he lowe and
uppe bounds eme ges o ins ances wi h mo e han 210 cus ome s. Fo
e en la ge ins ances, such as hose wi h 270 cus ome s, he sequen ial
app oach also s uggles o ind op imal solu ions. To ackle la ge
ins ances o mo e complex logis ics ne wo ks no add essed in his
s udy, al e na i e solu ion me hods a e equi ed. U ilizing heu is ic
me hods o a combina ion o exac and heu is ic app oaches could
p o ide an e ec i e s a egy o sol ing hese la ge p oblems.
CRediT au ho ship con ibu ion s a emen
Hamid eza Rezaei: W i ing – e iew & edi ing, W i ing – o iginal
d a , Valida ion, So wa e, Me hodology, In es iga ion, Fo mal anal-
ysis, Concep ualiza ion. Na halie Bos el: W i ing – e iew & edi ing,
W i ing – o iginal d a , Valida ion, Supe ision, Me hodology, In es i-
ga ion, Fo mal analysis, Concep ualiza ion. Vincen Ho elaque: W i -
ing – e iew & edi ing, W i ing – o iginal d a , Supe ision, P ojec
adminis a ion, Me hodology, In es iga ion, Funding acquisi ion, Con-
cep ualiza ion. Oli ie Pé on: W i ing – e iew & edi ing, W i ing
– o iginal d a , Valida ion, Supe ision, Me hodology, In es iga ion,
Fo mal analysis, Concep ualiza ion. Jean-Lau en Vi iani: W i ing –
e iew & edi ing, W i ing – o iginal d a , Valida ion, Me hodology.
Decla a ion o compe ing in e es
None
Acknowledgmen s
This wo k has been suppo ed by ANR unde he FILEAS FOG (ANR
17-CE10-0001-01) p ojec .
Appendix A. No a ions
Se s
Se o cus ome s (𝑖)
Se o candida e acili ies (𝑗)
Se o ime pe iods (𝑡)
′Ex ended se o ime pe iods (𝑡)
Decision a iables
𝑦𝑗 𝑡= 1i 𝑗∈is ope a ing in pe iod 𝑡∈∪′, 0
o he wise
{0,1}
𝑏𝑜𝑟𝑟𝑜𝑤𝑡Amoun bo owed in pe iod 𝑡∈[0,∞[
𝐸 𝐸𝑡Ex e nal equi y in pe iod 𝑡∈[0,∞[
𝐼 𝐸𝑡In es ed amoun o in e nal equi y in pe iod
𝑡∈
[0,∞[
In e media e logis ical a iables
𝑥𝑖𝑡 = 1i 𝑖∈is se ed in pe iod 𝑡∈, 0 o he wise {0,1}
𝑒𝑗 𝑡Expenses occu ed in pe iod 𝑡∈𝑇 o 𝑗∈[0,∞[
𝑞𝑖𝑗 𝑡Quan i y shipped om 𝑗∈ o 𝑖∈in pe iod 𝑡∈𝑇[0,∞[
𝑟𝑗 𝑡Re enues ob ained in pe iod 𝑡∈𝑇 ela ed o 𝑗∈[0,∞[
Ope a ions Resea ch Pe spec i es 14 (2025) 100319
15

H. Rezaei e al.
In e media e inancial a iables
𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒
𝑡𝑡 Value o loan balance 𝑙
𝑡in pe iod 𝑡∈[0,∞[
𝑐 𝑎𝑠ℎ𝑡Cash le el in pe iod 𝑡∈] − ∞,∞[
𝐶 𝐹𝑗 𝑡Cash low gene a ed by 𝑗∈𝐽in pe iod
𝑡∈𝑇
] − ∞,∞[
𝐸 𝐵 𝐼 𝑇𝑗 𝑡Ea nings be o e in e es and ax by 𝑗∈
in pe iod 𝑡∈
] − ∞,∞[
𝑒𝑞 𝑢𝑖𝑡𝑦𝑡Amoun o equi y in pe iod 𝑡∈[0,∞[
𝐹 𝐶 𝐹𝑗 𝑡F ee cash low associa ed wi h 𝑗∈in
pe iod 𝑡∈
] − ∞,∞[
𝐹 𝐺 𝑉Financially gene a ed alue o he i m ] − ∞,∞[
𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑡𝑡 In e es o loan 𝑙
𝑡in pe iod 𝑡∈[0,∞[
𝐾 𝐷𝑡Cos o loan 𝑙𝑡[0,1]
𝑂 𝐺 𝑉Ope a ionally gene a ed alue o he i m ] − ∞,∞[
𝑁 𝑂 𝑃 𝐴𝑇𝑡Ne ope a ing p o i a e ax in pe iod
𝑡∈
] − ∞,∞[
𝑝P obabili y o bank up cy [0,1]
𝑟𝑒𝑝𝑎𝑦
𝑡𝑡 Repaymen o loan 𝑙
𝑡in pe iod 𝑡∈[0,∞[
Logis ical pa ame e s
𝐶𝑗Yea ly p oduc ion capaci y o acili y 𝑗∈
𝐷𝑖𝑡 Demand o 𝑖∈in pe iod 𝑡∈
𝐷 𝑒𝑝𝑗Dep ecia ion o 𝑗∈
𝐹𝑗Fixed yea ly unning cos o 𝑗∈
𝐽𝑚𝑎𝑥 Max numbe o acili ies o be loca ed
𝐿Facili y li e ime
𝑂𝑗Opening cos o 𝑗∈
𝑃𝑖P oduc selling p ice o 𝑖∈
𝑆 𝑉𝑗Sal age Value o 𝑗∈
𝑉𝑖𝑗 Equals 1 i 𝑗∈can se e 𝑖∈, 0 o he wise
𝜇𝑗P ocessing cos o 𝑗∈
𝜔Uni anspo a ion cos
Financial pa ame e s
𝐾𝐸Cos o equi y a e
𝑁Numbe o annui ies
𝛽Bank up cy p obabili y pa ame e
𝛾Bank up cy cos pa ame e
𝛿Di idend payou a io
𝜂Fi m ax a e
𝜁Uppe bound o deb a io
Appendix B. De ailed esul s: accumula ed capaci y and o al in-
es men
Table B.12 epo s he accumula ed capaci y ins alled a each pe iod
along wi h he o al amoun o in es men in each solu ion (sum o he
ixed cos o he selec ed acili ies). Fo he sake o cla i y, he capaci y
and in es men alues in Table B.12 ep esen housands o uni s.
Appendix C. Linea iza ion p ocedu es
In he p oposed ma hema ical model, se e al cons ain s a e non-
linea due o he p oduc o wo decision a iables. We i s ecall
se e al well-known linea iza ion echniques (Appendix C.1) and hen
explain how hese echniques a e applied o ou model.
C.1. Classical linea iza ion p ocedu es
•Linea iza ion 1: p oduc o a eal and a bina y a iable
Le 𝑢and 𝑣be wo eal posi i e a iables and 𝑏a bina y a iable.
Gi en an uppe bound 𝑈o a iable 𝑢, he exp ession 𝑣=𝑏𝑢 can
be linea ized by:
𝑣≤𝑢
𝑣≥𝑢−𝑈(1 −𝐵)
𝑣≤𝑈 𝑏
𝑣≥0
•Linea iza ion 2: p oduc o wo con inuous a iables
The e is no exac way o linea ize a p oduc o wo con inuous
a iables 𝑢and 𝑣. In ou model, mos con inuous a iables ep e-
sen la ge mone a y alues. The consequence o ounding down
hese a iables o he nea es in ege alue is hen negligible.
Assuming ha a iable 𝑢is ounded down, we use a powe -o - wo
decomposi ion o ep esen 𝑢as a se o bina y a iables 𝑏𝑖:
𝑢=𝑏0+ 2𝑏1+ 4𝑏2+ 8𝑏3+⋯+ 2⌊𝑙 𝑜𝑔2𝑈⌋𝑏𝑙 𝑜𝑔2𝑈,
whe e U is an uppe bound o 𝑣. Fo example, he alue 100 = 4
+ 32 + 64 can be ep esen ed by he ec o (0,0,1,0,0,1,1).
The p oduc 𝑢𝑣 can now be ew i en as ollows:
𝑢𝑣 =𝑣
𝑖=⌊𝑙 𝑜𝑔2𝑈⌋
∑
𝑖=0
2𝑖𝑏𝑖.
I is a weigh ed sum o he e ms 𝑣𝑏𝑖, whe e 𝑣is a con inuous a i-
able and 𝑏𝑖is an in ege a iable. These e ms can be linea ized
wi h Linea iza ion 1.
•Linea iza ion 3: Piece-wise linea iza ion
Conside a gene al non-linea unc ion 𝑓(𝑢)o a single a iable 𝑢,
whe e 𝑢∈ [𝑢0, 𝑢ℎ].
We conside in e media e alues 𝑢1,…, 𝑢ℎ−1 and in e als o he
o m [𝑢𝑖, 𝑢𝑖+1]0≤𝑖≤ℎ−1. Le 𝜆𝑖be a ec o o bina y a iables, whe e
𝜆𝑖= 1i and only i 𝑢∈ [𝑢𝑖, 𝑢𝑖+1].
We de ine a ec o 𝜉o con inuous a iables such ha
𝑢=
ℎ−1
∑
𝑖=0
𝜉𝑖
𝜆𝑖𝑢𝑖≤𝜉𝑖≤𝜆𝑖𝑢𝑖+1
ℎ−1
∑
𝑖=0
𝜆𝑖= 1.
Only one alue in ec o 𝜉is s ic ly posi i e and i co esponds o
he alue 𝑢. Hence, he unc ion 𝑓(𝑢)is app oxima ed by selec ing
he app op ia e in e al and conside ing he piece-wise linea
app oxima ion o 𝑓(𝑢)in his in e al:
𝑓(𝑢) =∑
𝑖
𝜆𝑖𝑓(𝑢𝑖) +∑
𝑖((𝜉𝑖−𝜆𝑖𝑢𝑖)𝑓(𝑢𝑖+1) −𝑓(𝑢𝑖)
𝑢𝑖+1 −𝑢𝑖).
Since only one alue o ec o s 𝜆and 𝜉is s ic ly posi i e, his
exp ession educes o
𝑓(𝑢) =𝜆𝑖𝑓(𝑢𝑖) + (𝜉𝑖−𝑢𝑖)𝑓(𝑢𝑖+1) −𝑓(𝑢𝑖)
𝑢𝑖+1 −𝑢𝑖
o some 0≤𝑖≤ℎ− 1.
•Linea iza ion 4: Logical cons ain s
Logical cons ain s allow he exp ession o logical ope a o s such
as logical-o , logical-and, and condi ional s a emen s (i ... hen
...) in he linea p og amming con ex .
– O -condi ion: Conside a gene al ma hema ical exp ession
𝑔(𝑥)and wo pa ame e s 𝑏1and 𝑏2. The exp ession
(𝑔(𝑥)≤𝑏1) ∨ (𝑔(𝑥)≤𝑏2)
Ope a ions Resea ch Pe spec i es 14 (2025) 100319
16
H. Rezaei e al.
Table B.12
Accumula ed capaci y ins alled a each pe iod and o e all in es men wi h he sequen ial and he in eg a ed app oaches.
1s pa
Ins ance App oach To al capaci y ins alled (×103) In es men
=1 =2 =3 =4 =5 (×103 mu)
60-A In eg a ed 10.5 10.5 10.5 14 14 146.1
Sequen ial 10.5 10.5 10.5 10.5 10.5 104.7
60-B In eg a ed 10.5 10.5 10.5 16.5 16.5 162.4
Sequen ial 4.5 9 9 15 15 154.7
60-C In eg a ed 12 12 12 12 12 111.9
Sequen ial 6 6 6 12 12 111.9
60-D In eg a ed 6 9.5 9.5 15.5 15.5 153.3
Sequen ial 6 6 6 12 12 111.9
90-A In eg a ed 16.5 16.5 16.5 22.5 22.5 211.3
Sequen ial 12 12 12 18 18 165.7
90-B In eg a ed 12 12 16.5 22.5 22.5 211.3
Sequen ial 6 12 12 22.5 22.5 211.3
90-C In eg a ed 10.5 10.5 10.5 16.5 16.5 154.8
Sequen ial 10.5 10.5 10.5 16.5 16.5 154.8
90-D In eg a ed 6 10.5 10.5 16.5 16.5 154.8
Sequen ial 6 10.5 10.5 16.5 16.5 154.8
120-A In eg a ed 16.5 22.5 22.5 28.5 28.5 250.6
Sequen ial 16.5 16.5 22.5 28.5 28.5 250.6
120-B In eg a ed 12 16.5 22.5 28.5 28.5 250.6
Sequen ial 12 16.5 22.5 28.5 28.5 250.6
120-C In eg a ed 18 18 18 28.5 28.5 243.7
Sequen ial 12 12 12 22.5 22.5 191.8
120-D In eg a ed 12 18 22.5 28.5 28.5 243.7
Sequen ial 6 12 18 24 24 201.4
150-A In eg a ed 32 32 36.5 41 41 341.2
Sequen ial 27.5 27.5 32 36.5 36.5 296.6
150-B In eg a ed 21.5 27.5 32 41 41 341.2
Sequen ial 21.5 27.5 32 36.5 36.5 296.6
150-C In eg a ed 33 33 36.5 41 41 370.5
Sequen ial 28.5 28.5 33 37.5 37.5 331.2
150-D In eg a ed 22.5 28.5 33 41 41 370.5
Sequen ial 22.5 28.5 33 37.5 37.5 331.2
2nd pa
Ins ance App oach To al capaci y ins alled (×103) In es men
=1 =2 =3 =4 =5 (×103 mu)
180-A In eg a ed 33 37.5 37.5 43.5 43.5 391.5
Sequen ial 33 33 33 39 39 346.9
180-B In eg a ed 28.5 33 39 45 45 400.7
Sequen ial 22.5 28.5 33 45 45 400.7
180-C In eg a ed 34.5 34.5 34.5 42.5 42.5 385.4
Sequen ial 34.5 34.5 34.5 39 39 346.6
180-D In eg a ed 30 33.5 41.5 46 46 424.2
Sequen ial 24 30 34.5 42.5 42.5 385.4
210-A In eg a ed 28.5 33 43.5 53 53 502.1
Sequen ial 28.5 28.5 37.5 49.5 49.5 463.1
210-B In eg a ed 28.5 33 39 55.5 55.5 514.2
Sequen ial 22.5 28.5 33 49.5 49.5 463.1
210-C In eg a ed 30 30 39.5 53.5 53.5 505
Sequen ial 30 30 39.5 49 49 460.7
210-D In eg a ed 24 30 36 53.5 53.5 505
Sequen ial 24 27.5 33.5 50 50 466
240-A In eg a ed 40.5 52.5 63 63 63 559.1
Sequen ial 57 57 57 63 63 559.1
240-B In eg a ed 39 43.5 49.5 59 59 539.4
Sequen ial 39 45 51 57 57 505.8
240-C In eg a ed 57 57 57 63 63 553.9
Sequen ial 57 57 57 63 63 553.9
(con inued on nex page)
Ope a ions Resea ch Pe spec i es 14 (2025) 100319
17
H. Rezaei e al.
Table B.12 (con inued).
2nd pa
Ins ance App oach To al capaci y ins alled (×103) In es men
=1 =2 =3 =4 =5 (×103 mu)
240-D In eg a ed 36 45 51 54.5 60.5 541.9
Sequen ial 40.5 45 51 57 57 503.2
270-A In eg a ed 15 34.5 40.5 57 57 521.4
Sequen ial 39 39 39 45 45 402.1
270-B In eg a ed 33 43.5 48 60 60 543
Sequen ial 28.5 33 39 51 51 453.8
270-C In eg a ed 45 45 54 58.5 58.5 535.9
Sequen ial 39 39 39 54 54 497.6
270-D In eg a ed 33 43.5 49.5 64.5 64.5 592.2
Sequen ial 34.5 40.5 45 64.5 64.5 592.2
can be linea ized by:
𝑔(𝑥)≤𝑏1+𝑀×𝜗
𝑔(𝑥)≤𝑏2+𝑀× (1 −𝜗)
whe e 𝑀is a la ge enough posi i e alue and 𝜗a bina y
a iable.
– Condi ional exp ession: Conside he ollowing exp ession
{𝑔(𝑥)≥0⇒𝜌= 1
𝑔(𝑥)<0⇒𝜌= 0,
whe e 𝜌is a bina y a iable. This condi ional exp ession is
linea ized by:
{𝑔(𝑥) +𝜖≤𝜌×𝑀
−𝑔(𝑥)≤(1 −𝜌) ×𝑀
whe e 𝜖is a ela i ely small posi i e alue.
C.2. Linea iza ion o he OGV second e m
The second e m o he OGV o mula (16) is non-linea due o he
p oduc o con inuous a iables 𝐶 𝐹𝑗 𝑇and bina y a iables 𝑦𝑗 𝑡. This
exp ession can be linea ized wi h Linea iza ion 1 o Appendix C.1.
Since he cash low a iables 𝐶 𝐹𝑗 𝑇measu e he cash lows o acili y
𝑗∈a e pe iod 𝑇, i s maximal alue co esponds o he case whe e
all cus ome s’ demands a e se ed by acili y 𝑗. Hence, he uppe bound
𝑈can be se o ∑𝑖∈𝑃𝑖𝐷𝑖𝑇 .
C.3. Linea iza ion o cons ain s (19)
Acco ding o he cons ain s (19), he epaymen o loan 𝑙
𝑡can be
a posi i e alue only o pe iods 
𝑡 < 𝑡≤
𝑡+𝑁. This amoun depends on
he amoun bo owed (𝑏𝑜𝑟𝑟𝑜𝑤
𝑡), and i s associa ed in e es a e, 𝐾 𝐷
𝑡.
We linea ize he p oduc o 𝑏𝑜𝑟𝑟𝑜𝑤
𝑡and 𝐾 𝐷
𝑡. Fi s , 𝐾 𝐷
𝑡is w i en as
ollows:
𝐾 𝐷
𝑡=
𝑛
∑
𝑚=1
𝑖𝑟𝑚×𝜃
𝑡𝑚 ∀
𝑡∈,
whe e 𝑛 ep esen s he numbe o deb a io in e als and 𝑖𝑟𝑚is he
in e es a e associa ed wi h he in e al 𝑚∈ [1, 𝑛].𝜃
𝑡𝑚 is a bina y
a iable ha akes he alue o 1 i he deb a io a e ac i a ing loan
𝑙
𝑡belongs o he in e al 𝑚, and 0 o he wise.
Thus, i 𝜃
𝑡𝑚 = 1, he in e es a e o loan 𝑙
𝑡is 𝑖𝑟𝑚. Now he epaymen
o mula can be ew i en as ollows:
𝑟𝑒𝑝𝑎𝑦
𝑡𝑡 =𝑏𝑜𝑟𝑟𝑜𝑤
𝑡×𝐾 𝐷
𝑡(1 +𝐾 𝐷
𝑡)𝑁
(1 +𝐾 𝐷
𝑡)𝑁− 1
=𝑏𝑜𝑟𝑟𝑜𝑤
𝑡×
𝑛
∑
𝑚=1
𝑖𝑟𝑚(1 +𝑖𝑟𝑚)𝑁
(1 +𝑖𝑟𝑚)𝑁− 1×𝜃
𝑡𝑚
=
𝑛
∑
𝑚=1
𝑖𝑟𝑚(1 +𝑖𝑟𝑚)𝑁
(1 +𝑖𝑟𝑚)𝑁− 1×𝜃
𝑡𝑚 ×𝑏𝑜𝑟𝑟𝑜𝑤
𝑡,
whe e he e m 𝜃
𝑡𝑚 ×𝑏𝑜𝑟𝑟𝑜𝑤
𝑡is a p oduc o eal and bina y a iables,
ha can be linea ized acco ding o he Linea iza ion 1 o Appendix C.1,
whe e he uppe bound 𝑈can be se o he maximum in es men size
which equals ∑𝑗∈𝐽𝑂𝑗.
C.4. Linea iza ion o cons ain s (20)
The linea iza ion o p oduc 𝐾 𝐷
𝑡×𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒
𝑡,𝑡−1 ollows he same
p inciple as in Appendix C.3.
We ew i e 𝐾 𝐷
𝑡=∑𝑛
𝑚=1 𝑖𝑟𝑚×𝜃
𝑡𝑚, and ob ain he ollowing equa ion:
𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑡𝑡 =
𝑛
∑
𝑚=1
𝑖𝑟𝑚×𝜃
𝑡𝑚 ×𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒
𝑡,𝑡−1.
Using again he p ocedu e explained in Appendix C.1 concludes he
linea iza ion. Fo he p oduc o bina y (𝜃
𝑡𝑚) and eal (𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒
𝑡,𝑡−1)
a iables, he uppe bound 𝑈is se o he maximum possible amoun
o bo ow i.e., ∑𝑗∈𝐽𝑂𝑗.
C.5. Linea iza ion o cons ain s (23)
In cons ain s (23), he exp ession max(0, 𝑐 𝑎𝑠ℎ𝑡−1)is non-linea due
o he max ope a o . We in oduce he con inuous a iables 𝐼 𝐸 𝑀 𝐴𝑋𝑡
and auxilia y bina y a iables 𝑤𝑡sa is ying:
𝐼 𝐸 𝑀 𝐴𝑋𝑡≥𝑐 𝑎𝑠ℎ𝑡−1 ∀𝑡∈
𝐼 𝐸 𝑀 𝐴𝑋𝑡≤𝑐 𝑎𝑠ℎ𝑡−1 +𝑈(1 −𝑤𝑡) ∀𝑡∈
𝐼 𝐸 𝑀 𝐴𝑋𝑡≤𝑈 𝑤𝑡∀𝑡∈
𝐼 𝐸 𝑀 𝐴𝑋𝑡≥0 ∀𝑡∈,
whe e 𝑈=∑𝑖∈∑𝑗∈∑𝑡∈𝑃𝑖𝐷𝑖𝑡.
C.6. Linea iza ion o he ax shield bene i - objec i e unc ion’s second e m
The ax shield bene i is calcula ed as:
(1 −𝑝)𝜂∑
𝑡∈∪′∑
𝑡∈𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑡𝑡
(1 +𝐾𝐸)𝑡
whe e bo h 𝑝and ∑
𝑡∈𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑡𝑡 a e con inuous a iables. We use
Linea iza ion 2 p esen ed in Appendix C.1.
𝑈can be se o he maximum alue o in e es i.e., he in e es paid
i all candida e acili ies a e opened using deb inancing. Igno ing he
impac o he ime alue o money: 𝑈=𝑁×𝐾 𝐷
𝑡×∑𝑗∈𝐽𝑂𝑗, whe e 𝐾 𝐷
𝑡
is se o i s maximum possible alue.
C.7. Linea iza ion o he expec ed bank up cy cos (EBC) - objec i e unc-
ion’s hi d e m
EBC is es ima ed as 𝑝×𝛾×𝑂 𝐺 𝑉whe e 𝑝and 𝑂 𝐺 𝑉a e con inuous
a iables. We use Linea iza ion 2 o Appendix C.1.
The maximum alue o OGV can be ob ained i all cus ome s a e
se ed a all pe iods wi hou any cos . Thus:
Ope a ions Resea ch Pe spec i es 14 (2025) 100319
18
H. Rezaei e al.
Table D.13
Pa ame e alues o sensi i i y analysis.
Pa ame e Baseline Al e na i e Al e na i e
alue 1 alue 2
Tax a e (𝜂) 30% 20% 40%
Deb a io uppe bound (𝜁) 0.8 0.7 0.9
Bank up cy cos pa ame e (𝛾) 0.5 0.4 0.6
Facili y ixed yea ly unning cos (𝐹) 5% 7.5% 10%
Uni anspo cos (𝜔) 0.002 mu 0.001 mu 0.003 mu
𝑈= (1 −𝜂) ×(∑
𝑖∈∑
𝑗∈∑
𝑡∈
𝑃𝑖𝐷𝑖𝑡 +∑
𝑖∈∑
𝑗∈∑
𝑡∈′
𝑃𝑖𝐷𝑖𝑇 ).
C.8. Linea iza ion o he p obabili y o bank up cy 𝑝- eq. (25)
In Eq. (25), he p obabili y o bank up cy is gi en by
𝑝=(∑
𝑡∈𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒
𝑡𝑇
∑
𝑡∈𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒
𝑡𝑇 +𝑒𝑞 𝑢𝑖𝑡𝑦𝑇)𝛽
.
We i s conside he deb a io:
𝜅=∑
𝑡∈𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒
𝑡𝑇
∑
𝑡∈𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒
𝑡𝑇 +𝑒𝑞 𝑢𝑖𝑡𝑦𝑇
.
Then, we in oduce he no a ion 𝑇 𝐴=∑
𝑡∈𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒
𝑡𝑇 +𝑒𝑞 𝑢𝑖𝑡𝑦𝑇 ep e-
sen ing he o al asse s in pe iod 𝑡.
Using a powe -o - wo decomposi ion, 𝑇 𝐴can be ep esen ed by a
se o bina y a iables 𝑏𝑖.
The deb a io 𝜅can hen be ew i en as:
𝜅=∑
𝑡∈𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒
𝑡𝑇
∑𝑗=𝑈
𝑗=0 2𝑗𝑏𝑗
.
Using a c oss mul iplica ion, we ha e
𝜅×∑𝑗=𝑈
𝑗=0 2𝑗𝑏𝑗=∑
𝑡∈𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒
𝑡𝑇 .
This o mula ion does no con ain any a io, bu s ill mul iplies
he con inuous a iable 𝜅wi h a se o bina y a iables 𝑏𝑗. Thus,
Linea iza ion 1 o Appendix C.1 can be used, wi h 𝑈= 1. Finally,
we use a piece-wise linea iza ion (Linea iza ion 3 o Appendix C.1) o
linea ize he exp ession 𝑝(𝜅) =𝜅𝛽.
C.9. Linea iza ion o deb a io in e als
The deb a io in e als impose di e en in e es a es, i.e., 𝐾 𝐷
𝑡, o
loan 𝑙
𝑡.
𝛼1𝑚≤𝜅
𝑡≤𝛼2𝑚⇒𝐾 𝐷
𝑡=𝑖𝑟𝑚,
whe e 𝑚 ep esen s an in e al numbe , 𝜅
𝑡deno es he deb a io a 
𝑡,
and 𝛼1𝑚and 𝛼2𝑚a e he deb a io’s lowe and uppe bounds associa ed
wi h in e al 𝑚.
In e al 𝑚is a condi ional exp ession ha can be linea ized using
Linea iza ion 4 o Appendix C.1. To his end, he in e es a e o loan
𝑙
𝑡, is eplaced by he e m ∑𝑛
𝑚=1 𝑖𝑟𝑚×𝜃
𝑡𝑚, whe e 𝜃
𝑡𝑚 akes he alue o
1 i 𝜅
𝑡, he deb a io a e ac i a ing loan 𝑙
𝑡, s ands in he in e al 𝑚,
and 0 o he wise.
Appendix D. Sensi i i y analysis
We conduc a sensi i i y analysis o assess he impac o a ious
pa ame e s on he objec i e unc ion (APV) and ill a e wi hin he
in eg a ed app oach. Fi e key pa ame e s a e examined: Tax Ra e (𝜂),
Deb Ra io Uppe Bound (𝜁), Bank up cy Cos Pa ame e (𝛾), Facili y
Fixed Yea ly Running Cos (𝐹), and Uni T anspo Cos (𝜔). Fo
each pa ame e , h ee alues a e es ed: he baseline alue used in
p io nume ical expe imen s and wo al e na i e alues as de ined in
Table D.13. In his able, he Facili y Fixed Yea ly Running Cos (𝐹) is
calcula ed as a pe cen age o he acili y opening cos (𝑂).
The sensi i i y o hese pa ame e s is es ed on a ep esen a i e
subse o six ins ances. These selec ed ins ances — 60-A, 60-C, 120-B,
120-D, 210-A, 210-B — ep esen di e en sizes and pa e ns wi h an
equal dis ibu ion ac oss ins ance ypes. All ins ances we e sol ed unde
he same se ings as hose speci ied in Sec ion 6.2. The key obse a ions
ega ding he in luence o hese pa ame e s on APV and ill a e a e
ou lined in he ollowing subsec ions. Fig. D.7 isualizes he de ailed
esul s o his analysis.
D.1. Tax a e
APV:. The impac o he ax a e on APV is almos linea . A educ ion
in he ax a e will esul in an inc ease in APV, due o he dec ease in
he ax bu den on he i m’s ne income, which will in u n lead o an
inc ease in OGV. Howe e , he ad an age o he ax shield is educed
as he ax a e is lowe ed. While lowe deb le els educe bank up cy
cos s, he o e all impac on APV emains posi i e a lowe ax a es.
Con e sely, highe ax a es inc ease he bene i om he ax shield,
pa ially o se ing he decline in OGV. This is accompanied by highe
deb and a g ea e isk o bank up cy, which ul ima ely educes APV a
highe ax a es.
Fill a e:. Lowe ing he ax a e gene ally educes he ill a e o mos
ins ances, excep o 210-A. Con e sely, aising he ax a e inc eases
he ill a e o hal o he ins ances, while he o he s emain close o
he baseline.
When he ax a e dec eases, he i m expe iences a lowe ax
bu den, which leads o an inc ease in OGV. Consequen ly, he expec ed
bank up cy cos – di ec ly linked o OGV – ends o ise as a esul o
his inc ease, making he i m mo e cau ious abou inc easing i s deb
a io.
This educes he incen i e o aise deb , while he diminished ax
shield u he lowe s FGV. These combined e ec s educe he mo i-
a ion o open new acili ies, aligning wi h ea lie indings (ob ained
h ough compa ison o he in eg a ed and sequen ial app oaches) ha
inancial conside a ions make in es men mo e appealing. Simila ea-
soning applies o he cases whe e he ill a e inc eases when he ax
a e is aised. Thus, he obse ed decline in he ill a e unde lowe
and highe ax a es can be explained by he in e play be ween ising
bank up cy cos s, educed deb , and educed ax shield bene i s.
D.2. Deb a io uppe bound
Conside ing alue 0.9 as deb a io uppe bound amoun o c ea e a
new deb a io in e al (80%,90%] in Table 3. We associa e his in e al
wi h a 15.9% cos o loan.
APV:. The esul s o he sensi i i y analysis e eal a posi i e co ela-
ion be ween APV and he deb a io uppe bound. While inc easing he
i m’s deb a io aises he p obabili y o bank up cy, i also inc eases
he bene i o he ax shield. In all ins ances, he inc ease in he ax
shield bene i ou weighs he inc ease in expec ed bank up cy cos s,
esul ing in a highe APV when he i m is allowed o bo ow mo e.
Con e sely, dec easing he uppe bound o 0.7 educes APV ac oss all
ins ances.
The inc ease in APV when aising he uppe bound o 0.9 is la ge
han he dec ease when lowe ing i o 0.7. This asymme y is due o he
sha p ise in loan cos s as he deb a io inc eases. A highe deb a io,
like 0.9, aises loan in e es a es and boos s he ax shield bene i . In
con as , lowe ing he uppe bound o 0.7 educes loan cos s, bu he
dec ease in he ax shield is less signi ican han he bene i gained a
a 0.9 deb a io.
Fill a e:. Dec easing he deb a io uppe bound gene ally esul s in
a dec eased o unchanged ill a e, while inc easing he uppe bound
ypically leads o a highe ill a e. A highe ax shield makes opening
new acili ies mo e appealing, while a lowe one has he opposi e e ec .
The g ea e ill a e inc ease wi h a highe deb a io uppe bound is
due o he la ge ax shield bene i ou weighing he educ ion om a
lowe bound.
Ope a ions Resea ch Pe spec i es 14 (2025) 100319
19
H. Rezaei e al.
Fig. D.7. Impac o a ia ion o ax a e 𝜂on APV (a) and on he ill a e (b)
Fig. D.8. Impac o a ia ion o he deb a io uppe bound 𝜁on APV (a) and on he ill a e (b)
Table D.14
Calcula ion o he cos o loan 𝐾 𝐷𝑡 o he al e na i e 𝛾le els.
Deb a io in e al Cos o Loan (in %)
𝛾= 0.5𝛾= 0.4𝛾= 0.6
(baseline)
(0,30] 3.2 3.2 3.2
(30,40] 3.5 3.4 3.5
(40,50] 4.0 3.9 4.1
(50,60] 5.0 4.9 5.1
(60,70] 6.6 6.5 6.8
(70,80] 9.6 9.4 9.8
D.3. Bank up cy cos pa ame e
Changing he bank up cy cos pa ame e impac s he loan cos . A
highe (o lowe ) bank up cy cos pa ame e esul s in a lowe (o
highe ) eco e y alue in he e en o de aul , a ec ing how much he
company can eco e . As a esul , he loan cos adjus s acco dingly, and
we upda ed i using he a bi age equa ion, as shown in Table D.14.
APV:. Reducing he bank up cy cos pa ame e inc eases bo h OGV
and he deb a io, po en ially esul ing in he same o highe APV. In
all ins ances, lowe bank up cy cos s lead o highe APV. Con e sely,
inc easing bank up cy cos s consis en ly educes APV, con i ming a
nega i e co ela ion be ween he bank up cy cos pa ame e and APV.
Fill a e:. Dec easing he bank up cy cos pa ame e may ha e a pos-
i i e e ec on he ill a e. I p o ides an oppo uni y o in es mo e,
which in u n can lead o se ing mo e cus ome s. This is e lec ed in
he esul s, whe e lowe ing he bank up cy cos ei he inc eases he ill
a e o keeps i unchanged. Con e sely, inc easing he bank up cy cos
educes he a ac i eness o u he in es men s. As he cos inc eases,
he impac o FGV diminishes, making i less p o i able o in es u he
o se e addi ional cus ome s. This esul s in a dec eased o unchanged
ill a e. The sha p educ ion in ill a e o ins ance 60-C is p ima ily
due o he limi ed numbe o candida e acili ies in his ins ance. When
he bank up cy cos inc eases, he FGV e ec dec eases and he e a e
no iable al e na i e acili ies o eplace he one(s) ha a e closed.
This inding suppo s he ea lie obse a ion ha he in eg a ion
o inancial dimensions can enhance he a ac i eness o in es men ,
ul ima ely leading o imp o ed cus ome sa is ac ion.
D.4. Facili y ixed yea ly unning cos
APV:. The e is a nega i e linea ela ionship be ween APV and he
acili y’s ixed yea ly unning cos . This cos , gene a ed as a pe cen age
o he acili y’s opening cos , di ec ly impac s APV. The highe he
pe cen age, he highe he yea ly unning cos , leading o a educ ion
in OGV and, consequen ly, a lowe APV.
Fill a e:. A simila end can be obse ed be ween he inc ease in ixed
yea ly unning cos s and he ill a e. As his cos ises, he ill a e ends
o dec ease o emain cons an ac oss all ins ances. Fo mos ins ances,
inc easing he ixed yea ly unning cos om 5% o 7.5% and hen o
10% consis en ly educes he ill a e, indica ing ha ewe acili ies
a e being opened as he cos s ise (see Figs. D.8–D.11).
Ope a ions Resea ch Pe spec i es 14 (2025) 100319
20

H. Rezaei e al.
Fig. D.9. Impac o a ia ion o he bank up cy cos pa ame e 𝛾on APV (a) and on he ill a e (b)
Fig. D.10. Impac o a ia ion o he acili y ixed yea ly cos 𝐹on APV (a) and on he ill a e (b)
Fig. D.11. Impac o a ia ion o he uni anspo cos 𝜔on APV (a) and on he ill a e (b)
D.5. Uni anspo cos
APV:. This pa ame e has a nega i e e ec on APV ac oss all ins ances.
Dec easing he uni anspo cos leads o a highe APV, while inc eas-
ing i educes APV. This is because inc easing anspo cos s aises he
i m’s o al expenses, leading o a lowe OGV, which di ec ly lowe s
APV.
Fill a e:. The ela ionship be ween uni anspo cos and ill a e is
mo e complex. Lowe ing he uni anspo cos ei he inc eases he ill
a e o keeps i cons an o all ins ances. De ailed solu ions e eal wo
easons o he highe ill a e: o hal o he ins ances, lowe anspo
cos s make i mo e p o i able o se e addi ional cus ome s, while o
he o he hal , mo e acili ies a e opened, con ibu ing o he inc ease
in ill a e. Con e sely, inc easing he uni anspo cos educes he
ill a e in mos ins ances, excep o 210-A. In he emaining se en
ins ances, he opened capaci y is ei he less han o equal o ha
o he baseline, indica ing insu icien p o i abili y o jus i y opening
addi ional acili ies despi e lowe anspo cos s.
Ope a ions Resea ch Pe spec i es 14 (2025) 100319
21
H. Rezaei e al.
Da a a ailabili y
Da a will be made a ailable on eques .
Re e ences
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