Fuzzy Payback Period of Investment into Modernization of Production Network

19 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 20 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 21 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 22 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. 23 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. 24 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 25 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.