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ACC JOURNAL 2022, Volume 28, Issue 2 DOI: 10.15240/ ul/004/2022-2-002
FUZZY PAYBACK PERIOD OF INVESTMENT INTO MODERNIZATION
OF PRODUCTION NETWORK
Simona Haško á1; Jiří Kuče a2; Róbe Kuchá 3
Ins i u e o Technology and Business in České Budějo ice,
School o Expe ness and Valua ion, Resea ch G oup o Business Finance,
Ok užní 517/10, 370 01 České Budějo ice, Czech Republic
e-mail: 1[email p o ec ed]b.cz; 2[email p o ec ed]b.cz; 3[email p o ec ed]z
Abs ac
The di e ence be ween he esul o manage ial calcula ions and eali y can be la gely
a ibu ed o unce ain y. In he case o discoun ed payback pe iod (DPP), i conce ns
unce ain capi al expendi u es, posi i e cash lows, and discoun a es. To esol e his
p oblem he in e als o possible alues ins ead o unce ain poin alues should be ega ded.
This idea is p ojec ed in de ining he signi ican poin s o he inpu pa ame e s o he DPP
calcula ion om which he signi ican poin s o he uzzy payback pe iod (FPP) in he sense
o iangula uzzy numbe s (TFN) a e de i ed. Fo he TFN anking, he weigh ed me hod is
used. The FPP nume ical o mula hus becomes lexible in e ms o he possibili y o
exp essing ai h in he incidence a e o he inpu da a. The exis ing li e a u e omi s o ega d
nega i e weigh s o posi i e and/o nega i e cash lows in he FPP calcula ion. In he
applica ion, he ela ions a e applied o he quan i ica ion o he FPP in e al o possible
alues, by means o which he in es men plan o he mode niza ion o he ligni e powe
plan is e alua ed.
Keywo ds
Fuzzy payback pe iod; Discoun ed payback pe iod; In es men decision; P oduc ion ne wo k;
Unce ain y.
In oduc ion
Managemen p ac ice uses se e al me hods o decide on in es ing in a long- e m business
plan. Some o hem a e based on a one-c i e ion e alua ion o economic e iciency. The mos
signi ican ep esen a i es include ne p esen alue, in e nal a e o e u n, and discoun ed
payback pe iod. When i comes o he inpu componen s o hese c i e ia, hey a e o en
associa ed wi h unce ain y, which usually has wo main sou ces, he agueness o he ules
and ex e nal ci cums ances beyond he decision-make . [1]
1 Resea ch Subjec
The aim o his a icle is o show how unce ain y s emming om uncon ollable
ci cums ances can be deal wi h, a leas pa ially. Fo example, he discoun ed payback
pe iod (DPP) de e mined in a con en ional manne e.g. in [2] will almos ce ainly di e om
he ac ual payback pe iod. This is because he calcula ion is based on a ma hema ical model
o ague no ions ha he a he o iskie a p ojec 's cash low (CFi) is, he less signi ican
alue is oday. [3] This is because oday's money has a bigge alue han he same amoun
expec ed in he u u e. Howe e , mos o he di e ence be ween he DPP calcula ion esul
and eali y is due o unce ain y associa ed wi h igno ance o he exac u u e capi al
expendi u e poin s (CF0), CFi and discoun a e ( ). The e is a g ea e chance o es ima ing he
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in e als in which he espec i e poin alues o he inde e mina e a iables will be loca ed
han co ec ly es ima ing he poin alues hemsel es. I ollows ha we will be mo e
success ul in es ima ing he esul o he c i e ion calcula ion i we use he in e als o
possible alues ins ead o inde ini e poin alues. [4]
The a icle p esen ed builds on his idea and u he de elops i , inspi ed by he wo ks o
Kah aman [5] and Bane jee and Roy [6]. Fo he pu poses o calcula ing he discoun ed
payback pe iod (DPP), he “con en ional” ela ionship alid o he poin alues CF0, CFi,
and is i s e o mula ed o he ela ionship alid o he in e als o possible alues CF0,
CFi and .
Wi hin he calculus o in e als ep esen ed by hei signi ican poin s CF0, CFi, and , he
signi ican poin s o he uzzy discoun ed e u n (FPP) in e al a e de i ed. This p ocedu e is
demons a ed in he e alua ion o he pe spec i e o he in es men plan o he mode niza ion
o a con en ional ligni e powe plan by he FPP c i e ion. The uzzy e u n c i e ion is
calcula ed o a speci ic implemen a ion in he gi en egional condi ions.
The con ibu ion o he a icle is a new pe spec i e o using he anking unc ion o anking
he iangula uzzy numbe s acco ding o Chiu and Pa k [7], which is based on he weigh
pa ame e w, he alue o which is de e mined by he e alua o . This allows him/he , based on
cu en ly a ailable in o ma ion, knowledge, and expe ience, o e alua e he inpu da a lexibly
and hus en e in o an o he wise “mechanical” compu a ional p ocess. The me hod p esen ed
in his way is meaning ul o p ac ice and use - iendly.
2 Li e a u e Re iew
The in es men in he p ojec is cha ac e ized by an ini ial capi al expendi u e wi h he
subsequen assump ion o i s g adual e u n. Ma ić e al. [8] p o ide an o e iew o se e al
s a ic and dynamic me hods designed o he economic e alua ion o p ojec s. Bhanda i [9]
and o he s in hei wo k p o e ha s a ic me hods do no achie e as good esul s as dynamic
me hods. The basic dynamic me hods o in es men e alua ion include payback pe iod (DPP),
ne p esen alue (NPV), and in e nal a e o e u n (IRR). The e is a ela ionship be ween
hem [8]. Shinoda [10] encou ages companies o selec he e alua ion me hods ha a e mos
app op ia e and accu a e o de e mining he e u n on in es men gi en he size o he
p ojec .
F om a heo e ical poin o iew, Bhanda i [9] deal wi h he discoun ed payback me hod.
This is he i s pe iod in which he accumula ed alue o ne discoun ed cash lows equals o
exceeds he capi al in es men . The DPP esul is compa ed wi h he maximum allowable
payback pe iod o o he c i e ia, such as he economic li e o he p ojec . Bhanda i [9]
compa ed his me hod wi h o he in es men e alua ion me hods. He hen p esen ed
a gumen s abou i s ad an ages, which a e he simplici y o he me hod, easy calcula ion, he
abili y o measu e he p o i abili y o he in es men , liquidi y de e mina ion o he
in es men , and isk e lec ion. The DPP c i e ion is widely used in many a eas. Fo example,
e u n on in es men in pho o ol aic powe plan s [11], e u n on in es men in a ious
ecological in es men p ojec s ocused on building hea ing (insula ion, low-ene gy buildings,
biomass boile s, sola he mal sys ems, and hea pumps) [12], e u n on in es men in
ag icul u e in poul y a ming and dai y p oduc ion [13].
The use o mul ic i e ia analyzes has been shown o be e y e ec i e in de e mining DPP
in es men [14]. Fuzzy a i hme ic has a place in mul ic i e ia e alua ion [15]. Dick [16]
in oduces a comp ehensi e uzzy app oach as a new opic o compu a ional in elligence. All
esea ch in he ield o complex uzzy sys ems has so a ocused on conjunc ion, disjunc ion,
and nega ion ope a o s. In his app oach, Py hago ean uzzy se s a e ex ended by he e ms
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an i-membe ship and an i-non-membe ship o he al eady known e ms membe ship and non-
membe ship.
Fuzzy payback pe iod (FPP) is an ex ension o he concep o he payback me hod o eal
cash lows [17]. The cons uc ion o he eal cash lows gene a ed by he p ojec equi es an
es ima e o u u e e enues and cos s, which depends on many pa ame e s such as he size o
in la ion, he in e es a e, e c. [18]. These a iables a e unce ain in na u e and, unless eliable
in o ma ion on hei p obabili y occu s, s a is ical me hods a e no app op ia e o es ima e
hem. I he sou ce o unce ain y is incomple e in o ma ion, i is possible o ep esen he
alues o a iables using uzzy numbe s, which can be in e p e ed as uzzy subse s o a se o
eal numbe s sa is ying some o he condi ions. Fuzzy numbe s make i possible o model
imp obabili y phenomena in a simple way. [19]
Ra iu e al. [20] show ha he uzzy app oach is able o cap u e unce ain y in he
de elopmen o cash low and in e es a es. I s ad an age is he abili y o conside bo h
inancial and non- inancial indica o s; i allows, o example, o combine isk dimension,
inancial e u n, and non- inancial ac o s [21]. In e ms o de e mining he isk o an
in es men p ojec , i is possible o use a uzzy app oach o e alua e quan i a i e and
quali a i e cha ac e is ics h ough he in e p e a ion o inpu pa ame e s by uzzy se s. In his
case, he uzzy app oach can unc ion as a decision-making sys em ha assesses whe he he
isk conce n o a gi en p ojec is jus i ied o no . [22]
Vijayakuma e al. [23] compiled a anking o e alua ion c i e ia o in es men p ojec s. Fo
he o e all e alua ion o he p ojec , i is good o conside he NPV, DPP, in es men size,
cos and p o i abili y, and ime o make a p o i . Fo he inal e alua ion o p ojec s, hey used
a uzzy app oach, which p ocessed he poin esul s o he abo e c i e ia and quan i ies. Se gi
e al. [24] p opose uzzy ex ensions o he mos used capi al budge ing echniques. I is an
ex ension o NPV me hods, equi alen uni o m annual alue, and bene i -cos a io (B / C) o
in e al e alua ion o in es men s using in e al- alued Fe ma ean uzzy se s and algeb aic
and agg ega ion ope a ions.
B iozzo e al. [25] deal wi h he modeling o missing da a using a uzzy app oach and, o he
pu poses o applying adi ional me hods o p ojec e alua ion, including he DPP me hod,
analyzed i s use. They add ha using he uzzy app oach has many ad an ages. The uzzy
me hod con ibu es addi ional in o ma ion o he esul ob ained by he adi ional me hod.
The uzzy me hod can be applied as a complex me hod o de e mining and es ima ing all
inpu quan i ies and o indi idual e alua ion o indi idual componen s sepa a ely, as shown
by Ak e al. [26] when e alua ing he in es men in he was ewa e ea men sys em.
Banda [27] applied a uzzy payback pe iod o e alua e in es men in mining p ojec s and o
e alua e mining me hods. The inancial and echnological aspec s o indi idual p ojec
a ian s we e e alua ed. Kah aman e al. [28] deal wi h he implemen a ion o uzzy logic
in o o he me hods used o e alua e in es men p ojec s, including FPP. In [29], Kah aman
ocused on so wa e de elopmen , which includes, among o he hings, a unc ion o
e alua ing an in es men p ojec using FPP. Compu a ional so wa e capable o syn hesizing
he esul s o dynamic me hods NPV, IRR, cos -bene i a io and DPP was designed and
de eloped by Sama ki and Pull eap [30]. Fuzzy app oach was subsequen ly used o e alua e
he le el o p obabili y o he a e o e u n and he payback pe iod o he in es men p ojec .
3 Me hodology
The discoun ed payback pe iod me hod (DPP) conside s he ime needed o co e he ini ial
in es men cos s o he p ojec . The calcula ion o he DPP conside s he ime alue o money,
and hus makes i possible o p o ide a mo e objec i e esul wi h ega d o he ime and isk
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ac o han he calcula ion o he undiscoun ed e u n. In he ollowing, le us deno e C0 as he
in es men expense, CFi as he ne e u n gene a ed by he in es men o he pe iod i and as
he discoun a e o he p ojec . I CFi > 0 hen:
∑𝐶𝐹𝑖(1+𝑟)−1≥𝐶𝐹0
𝑚
𝑖=1 , (1)
whe e m ep esen s he e u n ho izon conside ing he ime alue o money. Acco ding o he
DPP c i e ion, he in es men is ealized when discoun ed payback pe iod is sho e han i s
economic li e ime.
In he case o unce ain alues CF0, CFi and , we subs i u e in o (1) o CF0, CFi and he
symbols o he iples o eal numbe s CF0 = (CF0l, CF0, CF0 ), CFi = (CFil, CFi, CFi ) and
= ( l, , ), composed o signi ican poin s o in e als o possible unce ain alues. Le
index l, espec i ely he igh index in he espec i e io indica es he smalles , espec i ely
he la ges elemen o he se o alues. The middle numbe indica es he alue o he mos
common o unexpec ed elemen - i is a numbe whose alue we es ima e unde he s anda d
app oach in isk condi ions.
Fo he T iangula Fuzzy Numbe (TFN) gene ally ep esen ed by he h ee pa ame e s
A = (al, a, a ), o which al ≤ a ≤ a applies, and de ining he payback pe iod i holds (2):
(∑(𝐶𝐹𝑖𝑙(𝑦)
(1+𝑟𝑟(𝑦))𝑖)
𝑚
𝑖=1 ,∑(𝐶𝐹𝑖𝑟(𝑦)
(1+𝑟𝑙(𝑦))𝑖)
𝑚
𝑖=1 )
≥((𝐶𝐹2(0)−𝐶𝐹1(0))𝑦+𝐶𝐹1(0) ,(𝐶𝐹2(0)−𝐶𝐹3(0))𝑦+𝐶𝐹3(0)),
(2)
whe e CFk(0) ep esen s he k- h pa ame e o he iangula uzzy numbe CF0, CFil(y) is he
le ep esen a ion o he iangula uzzy CFi, CFi (y) is he igh ep esen a ion o he
iangula uzzy CFi, (y) is he igh ep esen a ion o he discoun a e, l(y) is he le
ep esen a ion o he discoun a e.
I he discoun a e a ies ac oss pe iods, hen o (1+ (y))i and (1+ l(y))i i holds (3):
∏(1+𝑟𝑖′
𝑟(𝑦))
𝑖𝑖′=1 ,∏(1+𝑟𝑖′
𝑙(𝑦))
𝑖𝑖′=1 (3)
The e a e a ew me hods o anking T iangula Fuzzy Numbe s (TFNs), o example, Jain
[31], Chiu and Pa k [7], Kau mann & Gup a [32], and o he s.
Me hods can ake di e en anking esul s and mos o hem equi e complex ma hema ical
calcula ions. Chiu and Pa k [7] p esen a weigh ed me hod o anking T iangula Fuzzy
Numbe s wi h pa ame e s (al, a, a ) as ollows (4):
(𝑎𝑙+𝑎+𝑎𝑟
3)+𝑤𝑎, (4)
whe e w ∈〈0,1〉is he alue de e mined by he na u e and size o he alue a. Ano he
anking me hod ha does no equi e complex ma hema ical calcula ions is he g aded means
me hod (5):
(𝑎𝑙+4𝑎+𝑎𝑟
6) (5)
which Shanmugasunda i & Ganesan [33] used o sol e he uzzy anspo a ion p oblem.
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4 Da a
The da a o he in es men plan o he mode niza ion o a con en ional ligni e powe plan
consis ing o he implemen a ion o P G echnology we e ob ained om he a icle by S aka
[34]. Based on he knowledge o echnological pa ame e s and p oduc ion possibili ies o he
u u e ins alla ion o P G echnology (Powe o gas), he au ho es ima es he annual cash low
(CF), which is cap u ed in Table 1.
To ca y ou he p ocess o so-called me hana ion (spli ing in o me hane) o emi ed CO2
S aka in [34] es ima es elec ici y consump ion o one yea a 56,134 MWh, while he
cu en p ice o 1MWh on he ma ke is a ound 131 € (a he ime o he in es men
calcula ion in 2021). The au ho es ima es he wa e consump ion o he me hana ion p ocess
due o he supply o hyd ogen a 4,806.5 m3 a an a e age p ice o 3.5 € / m3. The ixed
ope a ing cos s o he en i e acili y a e es ima ed a € 430,500. The o al annual expendi u e
is calcula ed a € 7,800,877.
The au ho calcula es annual e enues om he ope a ion o he acili y as ollows: sales o
p oduced oxygen due o he p oduc ion possibili ies o he echnology a e es ima ed a
€ 12,819,384, sales o me hane a € 1,551,184, sales o was e hea he au ho es ima es a
€ 645,227. Recycling o was e CO2 educes i s emissions in o he a mosphe e, which esul s in
a educ ion in he cos o ob aining emission allowances. The p ice o he emission allowance
pe 1 on o CO2 emi ed was a he le el o € 25 a he ime o he in es men calcula ion.
Due o he amoun o ecycled CO2, his ep esen s a sa ing o € 146,905. The budge ed alue
o he annual cash low is € 7,361,823.
Tab. 1: Es ima ion o cash lows
I em
Amoun
€ / uni
€ / yea
Elec ici y consump ion
56,134 MWh
131 € / MWh
7,353,554.00 €
Wa e consump ion
4,806.5 m3
3.5 € / m3
16,823.00 €
Fixed ope a ing cos s
430,500
€
430,500.00 €
To al expendi u e
7,800,877.00 €
Oxygen sales
8,546,256 kg
1.5 € / kg
12,819,384.00 €
Sales o me hane
1,666,625 kg =
23,152 MWh CH4
67 € / MWh NG
1,551,184.00 €
Sales o was e hea
11,731 MWh
55 € / MWh
645,227.00 €
Sa ing CO2 emission
allowances
5,876,208 kg
25 € / on o CO2
146,905.00 €
To al e enue
15,162,700.00 €
Annual cash low
7,361,823.00 €
Sou ce: Own p ocessing based on S aka [34]
Capi al expendi u es o he implemen a ion o P G echnology a e es ima ed a € 21,525,000.
A de ailed b eakdown o in es men i ems is gi en in Table 2.

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Tab. 2: B eakdown o capi al expendi u es
In es men i em
P ice (€)
No e
3 alkaline elec olyze s o 3.56 MW, a o al o 10.68 MW
12,390,000.00
1,180 € / kW
2 medium p essu e gas anks (O2 and H2)
2,940,000.00
490 € / kg
Me hana ion eac o
2,478,000.00
20 % ALE*
CO2 cap u e uni
2,478,000.00
20 % ALE*
O he and un o eseen expenses
1,239,000.00
10 % ALE*
To al in es men cos s
21,525,000.00 €
*Pu chase p ice o alkaline elec olyze
Sou ce: Own p ocessing based on S aka [34]
S aka [34] used he con en ional app oach o calcula e he discoun ed payback pe iod o 3.8
yea s a a discoun a e o 8% and 5.1 yea s a a discoun a e o 15%. Gi en he es ima ed
li e ime o he p ojec since i s commissioning, he esul s o he e u n show a p omising
in es men .
As pa o he e alua ion o he p ojec , S aka [34] admi ed he occu ence o isk and
p ojec ed i in o wo di e en alues o discoun a es. Fu he on, we will de ia e om he
au ho 's con en ional app oach and conside he occu ence o ac o s ha a e inhe en ly
unce ain, and hei impac on he e u n esul may be signi ican . Unce ain y in he ossil
uel ma ke (N1), unce ain y on he pa o he EU owa ds con en ional powe plan s
(emission allowance p ices, ines, axes) (N2), unce ain y abou ene gy ma ke p ices (N3),
unce ain y in p oduc gas p ices (N4) and/o unce ain y in echnology acquisi ion p ices
(N5). These unce ain ac o s a e e lec ed in he unce ain alues o CF, capi al expendi u es,
and p ojec discoun a e.
The logic o e lec ing he e ec s o unce ain y on he e alua ion c i e ion is di ec ly o e ed
a he ime o he ongoing “ene gy e olu ion”. Le us conside he apid de elopmen o
ene gy ma ke p ices, caused by socie al p essu e o inc ease he ca bon neu ali y o EU
Membe S a es and he ansi ion o ully enewable ene gy sou ces [35], which is cu en ly
s imula ed by he Russian-Uk ainian wa [36], he u bulen de elopmen o p ices in he
ma ke o echnologies, ene gy, building ma e ials o se ices, d i en by pandemic
es ic ions and he slowdown in e o s o cu b mining ea lie . [37]
Unce ain y ac o s N1-N5 ha e an impac on he unce ain y o he p ojec inpu da a, which
can in ac be p ojec ed in he in e als o possible alues o he inpu in es men CF0, annual
cash low CFi gene a ed o he li e ime o he p ojec and discoun a e as CF0 = (CF0l, CF0,
CF0 ), CFi = (CFil, CFi, CFi ) and = ( l, , ), whe e index 0, espec i ely I, indica es he
pe iod o capi al expendi u es in yea 0, espec i ely, he pe iod o posi i e cash lows. Le
index l, espec i ely he igh index , indica es he smalles , espec i ely, he la ges elemen
o he se . The middle numbe indica es he alue o he mos common o mos expec ed
elemen ( he alue we es ima e unde he s anda d isk app oach).
Table 3 shows he in e als o possible CF0 alues, annual CFi and iden ical o he
expec ed li e ime o he in es men .
Tab. 3: In e als o limi alues o unce ain a iables
Unce ain a iable
CF0 in housands €
Annual CFi in housands €
Annual in %
Range o possible
alues
(-25,502; -21,252;
-17,001)
(5,889, 7,361, 8,833)
(8; 11.5; 15)
Sou ce: Own
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The ange o possible alues o he in e al CF0 and CFi was de e mined by a de ia ion o +/-
20% om he budge ed alue, see Tables 1 and 2, he ange o possible alues o he in e al
is de ined by he minimum and maximum a e e lec ed in he s anda d DPP calcula ion.
The minimum li e ime o he in es men , esul ing om he o ecas o coal mining in he
Eu ozone, is es ima ed a 9 yea s i he in es men is pu in o ope a ion in 2022. [38] The
es ima e was made based on da a ep esen ing he de elopmen o uel coal p oduc ion in
housands o ons in he pe iod 2012-2020 in he Eu o a ea egion o 19 coun ies. The end
o his de elopmen and i s u u e app oxima ion is shown in Figu e 1.
Sou ce: Own p ocessing based on [38]
Fig. 1: De elopmen o mining in he Eu o egion o 19 coun ies
5 Resul s
In acco dance wi h he me hodology in pa 2 and acco ding o (4), we use he classi ica ion
me hod o so he T iangula Fuzzy Numbe s (TFN) acco ding o Chiu & Pa k [7].
In case ha o w = 0.2 o posi i e and nega i e CF (Table 3) we ge (in housands €):
X0=(−(25,502+21,252+17,001)/3)+0.2(−21,252)=−25,500
TFN1=1,472𝑦+5,889
1.15−0.035 ,−1,472𝑦+8,833
1.08+0.035 =(5,281; 6,602; 7,923)
X1=7,922
Due o he iden ical in e al o annual CF´s ac oss he p ojec li e ime, he in e al o possible
alues o TFNi, hence he uzzy alue Xi, whe e i = 1… m, and m ep esen s he yea o he
p ojec in which he in es men is epaid, is he same. In his case, m = 3.21
The esul o he me hod is a dependen a iable o he subjec i e choice o weigh w. Gi en
ha in es men lows end o ha e a di e en na u e o “ce ain y” han he income lows
gene a ed by he in es men , his is conside ed in he ollowing such ha w0 > wCF, whe e
index 0, espec i ely, CF, is he weigh o capi al expendi u e, espec i ely o a posi i e cash
low. We base on he indings o p o en p ac ice ha p ojec ed capi al expendi u es a e a
sa e low han p ojec ed e enue lows. In he case whe e w0 = wCF, espec i ely, w0 < wCF,
exp esses he e alua o neu al a i ude o he occu ence o CF0 and CFi, espec i ely, he
e alua o ends o belie e in a highe alue o he occu ence o u u e posi i e lows
compa ed o nega i e lows.
Conside he u bulen pe iod 2021/2022, wi hin which he unce ain ies N1-N5 can be
iden i ied. As a esul , he e alua o chooses w0 = 0.5 and wCF = 0. Then
y = -25539x + 509573
0,
100000,
200000,
300000,
400000,
500000,
600000,
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
Eu o a ea - 19 coun ies
26
X0=(−(25,502+21,252+17,001)/3)+0.5(−21,252)=−31,878,
TFN1=1,472𝑦+5,889
1.15−0.035 ,−1,472𝑦+8,833
1.08+0.035 =(5,281; 6,602; 7,923),
X1=6,602.
In his case m = 4.82.
Table 4 p esen s he esul s o m when w0 = 0.5 o X0 = -31,878 and wCFi = 0, 0.1, 0.2, 0.3,
0.4 and 0.5 o calcula e Xi om he ange o possible TFNi alues.
Tab. 4: Fuzzy discoun ed payback pe iod o he p ojec in he numbe o yea s m depending
on he weigh w o he calcula ion o Xi om he ange o possible alues TFNi
wCFi
Xi
m
0.0
6,602
4.82
0.1
7,262
4.39
0.2
7,922
4.02
0.3
8,582
3.71
0.4
9,244
3.45
0.5
9,903
3.22
Sou ce: Own
6 Discussion
The con ibu ion a he heo e ical and p ac ical le el shows how i is possible o, a leas
pa ially, cope wi h ci cums ances ha canno be desc ibed in a con en ional way. Unde
ce ain ci cums ances, he DPP de e mined by he de e minis ic ela ion (1) may di e
signi ican ly om he ac ual payback pe iod. The e o e, ela ion (1) alid o poin alues
CF0, CFi and was e o mula ed o ela ion alid o in e als o possible alues CF0, CFi and
and de ined by TFN ela ion (2). The weigh ed me hod acco ding o Chiu & Pa k [7] was
used o TFN anking. The eason o choosing he Chiu & Pa k´s me hod is he possibili y o
choosing he weigh w, which allows he e alua o o exp ess his / he subjec i e opinion
abou he occu ence o he middle cash low “a” o he in e al.
The esul o he FPP in es men is a dependen o a weigh change w. Wi h he e alua o 's
equi alen expec a ion o posi i e and nega i e lows ( o w = 0.2 and 0.5), he payback
pe iod is 3.2 yea s ( he same uzzy e u n is calcula ed acco ding o (5)). In compa ison wi h
he esul s acco ding o S aka [34] i is a e u n o 7 mon hs, espec i ely almos 2 yea s
sho e (depending on he choice o he minimum o maximum discoun ed a e conside ed by
he au ho ).
The app oach o equal access o in es men and income lows is con adic ed by Ko ha i e al.
[39]. Based on empi ical da a, abou 50,000 obse a ions o he pe iod 1972-1997 hey
analyzed he ela i e con ibu ions o cu en R&D in es men s and long- e m, angible asse s
in es men s o u u e e enue a iabili y. The conclusions showed ha bo h ypes o
in es men gene a e u u e bene i s ha a e mo e unce ain compa ed o in es men
expendi u es, wi h he bene i s o R&D in es men being less ce ain han he bene i s o
in es ing in long- e m, angible asse s.
Re lec ing he g ea e unce ain y abou u u e e enues compa ed o capi al expendi u es and
conside ing se e al speci ic unce ain ies N ela ed o he ask, he weigh o he a e age
in es men expendi u e “a” was u he se a 0.5. The posi i e mean low “a” in line wi h he
expec a ions associa ed wi h he g ea e unce ain y was e alua ed wi h a weigh less han 0.5
(wCFi = 0, 0.1, 0.2, 0.3, 0.4). Unde hese condi ions, he uzzy payback pe iod FPP = (3.45,
27
4.14, 4.82) yea s, whe e he middle numbe can be in e p e ed as he subjec i ely mos
expec ed alue o he payback pe iod. Gi en he es ima ed li espan o he in es men o
9 yea s, his is a e y p omising p ojec .
A wCFi = 0, uzzy Xi is equal o he a i hme ic mean o he CF limi s. A weigh w > 0
indica es a highe expec a ion o he occu ence o he mean alue “a” wi h an impac on he
uzzy in ake Xi owa ds he igh limi alue o exceeding i . The same applies o he beha io
o uzzy expendi u es X0 - wi h he inc eased w he expendi u es g ow in he di ec ion o he
le o he mean alue “a”.
Using he weigh ed anking me hod assumes a posi i e w. Nega i e alue o w would ha e
he opposi e e ec on X0 and Xi, bu no on he FPP esul . The s a egy o “weighing”
nega i e and posi i e lows in he same di ec ion (wi h a “+” o “-” sign) is unsus ainable
om a p ac ical poin o iew. The na u e o he sign w o a gi en ype o low and i s
magni ude has o be p ima ily gi en by he e alua o 's easonable belie in he occu ence o
he mos p omising alue “a”.
In ou case, when w0 = 0.5 and wCFi = 0, 0.1, 0.2, 0.3, 0.4 and 0.5, he choice o he posi i e
sign ype o bo h lows can be jus i ied as ollows: due o he iden i ied unce ain ies, bo h
in es men cos s, especially echnology acquisi ion p ices, and income lows, mainly due o
ising elec ici y and hea p ices, may inc ease.
The anking me hod wi h weighs de e mining he mos p omising middle alue o he CF o
he p ojec allows he decision make o lexibly e alua e he inpu da a acco ding o cu en ly
a ailable in o ma ion, acco ding o his/he knowledge and expe ience unde iden i ied
unce ain ies ela ed o a pa icula ask.
The cu en li e a u e conside s only posi i e w ∈〈0, 1〉 o posi i e and nega i e lows o
he pu poses o FPP calcula ion, hus no conside ing he possibili y o nega i e weigh s o
lows o which a dec ease compa ed o he mean alue “a” can be expec ed.
Conclusion
In he wo ld o economics and managemen , mos decision-making p oblems a e
cha ac e ized as a complex p ocess o which comple e in o ma ion is o en no a ailable. The
esul is usually he di e ence be ween he esul s o he nume ical c i e ia on which he
decision-make s ely and he eali y. This is la gely due o he unce ain y associa ed wi h no
knowing he exac poin inpu alues o he manage ial o inancial c i e ion. This ac is
ci cum en ed in he a icle by he ac ha ins ead o unce ain poin alues we s a om he
in e als o possible inpu alues. Me hodologically, his idea is sol ed by de ining signi ican
poin s o in e als CF0, CFi and and signi ican poin s o discoun ed payback in e al (DPP)
de i ed om hem. T iangula Fuzzy numbe o calcula ing uzzy discoun ed payback pe iod
(FPP) is de ined, which is a pa ame e o e alua ing an in es men plan o ligni e powe
plan mode niza ion.
Fo he TFN anking, a weigh ed me hod was used, based on he weigh w, which e alua es
he mean alue o he in e al. This allows he e alua o o e lec i s own subjec i e opinion
abou he occu ence o he middle alue o lows. The nume ical o mula o FPP hus
becomes lexible in e ms o he possibili y o exp essing ai h in he incidence a e o inpu
da a, depending on he cu en ly a ailable in o ma ion, knowledge, and expe ience o he
e alua o wi hin iden i ied unce ain ies ela ed o a pa icula ask. The cu en li e a u e does
no conside nega i e weigh s o he subjec i e e alua ion o he mean alues o posi i e
and/o nega i e cash lows, which can cause signi ican inaccu acies in he ou come in
con on a ion wi h eali y.