Chapter 8 OFN Capital Budgeting Under Uncertainty and Risk

Transcription

1 Chapter 8 OFN Captal Budgetng Under Uncertanty and Rsk Anna Chwastyk and Iwona Psz Abstract The am of ths chapter s to propose a new approach to ncorporatng uncertanty nto captal budgetng. The chapter presents methods that can be used by an nvestor when the decson maker wants to be able to make an nvestment decson where there are alternatve nvestment projects. Ths knd of problem s undertaken under the condtons of uncertanty and rsk usng Ordered Fuzzy Numbers (OFN). The startng pont s the concept of Ordered Fuzzy Numbers. The chapter llustrates the mplementaton of the proposed approach wth an example where two alternatve nvestment projects are analyzed. The authors present the captal budgetng problem usng a numercal example. The descrbed methods dedcated to nvestment project selecton lay the foundatons for a fuzzy decson-makng system. We utlze computer software such as MATLAB to demonstrate how the proposed methods can be appled to assessng the proftablty of alternatve nvestment projects. 8.1 Introducton The captal budgetng problem s concerned wth allocaton of an organzaton s captal to a sutable combnaton of projects (alternatve projects) that can brng maxmal proft to the organzaton [12]. In the lterature we can fnd a varety of methods used n captal budgetng (see, e.g., [1, 2, 6, 22]). The man methods are: the net present value method (NPV), proftablty ndex (PI), and nternal rate of return (IRR). Based on the lterature revew we can state that the classcal forms of these methods do not take nto account the uncertanty and rsk whch may be nherent n the nformaton used n them. Ths nformaton ncludes future cash nflows, cash outflows and avalable nvestment captal, the requred rate of return of the nvestment or cost of captal, and the duraton of the project [21]. A. Chwastyk (B) Opole Unversty of Technology, Prószkowska 76, Opole, Poland e-mal: a.chwastyk@po.opole.pl I. Psz Opole Unversty, Ozmska 46a, Opole, Poland e-mal: psz@un.opole.pl The Author(s) 2017 P. Prokopowcz et al. (eds.), Theory and Applcatons of Ordered Fuzzy Numbers, Studes n Fuzzness and Soft Computng 356, DOI / _8 157

2 158 A. Chwastyk and I. Psz Tradtonally, these nvestment parameters are assumed as a crsp value. As we know, the captal budgetng problem s accompaned by uncertanty and rsk, whch, n general, stem from the lack of access to certan data (mprecse data) [11, 21]. In practce, ths nvolves, above all, the nablty to predct the behavor of the market durng the tmeframe of the project s executon, ncludng weather condtons, the level of prces and costs, avalablty of resources, exchange rates, nterest rates, behavor of competton, changes n the demand/supply level for a gven product or servce, and so on. Therefore several authors began to use fuzzy set theory to help solve the captal budgetng problem n a fuzzy envronment. In the lterature we can fnd another approach to captal budgetng, that s, fuzzy captal budgetng. Several authors studed fuzzy set theory and ts applcaton n captal budgetng [3, 5, 7, 11, 13, 14, 21]. Some authors ndcated certan problems to solve the captal budgetng problem wth fuzzy numbers [3, 5, 6, 21]. The noton of Ordered Fuzzy Numbers (OFN) was proposed by Kosńsk, Prokopowcz and Ślȩżak, [20] to elmnate several drawbacks of classcal convex fuzzy numbers (CFN) such as the loss of precson ncreasng wth the number of performed operatons and the fact that even lnear equatons cannot be solved n the set of fuzzy numbers. A new fuzzy number does not requre any exstence of a membershp functon and can be regarded as an extenson of a parametrc representaton of a fuzzy number. Ordered Fuzzy Numbers were frst used as a tool for a decson- support system concernng fnancal project evaluaton n the paper [18] and the research was contnued n [8]. Ther dea was based on the determnaton of the nternal rate of return of an nvestment project n whch all expendtures and ncome were mprecse and vague. In ths chapter we present a captal budgetng problem usng OFNs. We contnue the research started n the artcle by [9], whch concerned the use of the net present value method to estmate the attractveness of an nvestment opportunty. We now modfy the method presented n the prevous paper by transferrng the defuzzfcaton process to another stage of calculatons and present the next dscount methodsproftablty ndex and nternal rate of return-to make an evaluaton of alternatve nvestment projects more precse. We can see that the descrbed methods dedcated to the nvestment project selecton problem lay the foundatons for a fuzzy decsonmakng system. The chapter s organzed as follows. In Sect. 8.2 we dscuss the concept of fuzzy numbers and Ordered Fuzzy Numbers, whch allow modelng usng uncertan nformaton. Secton 8.3 s dedcated to the nvestment project s estmaton problem. It contans the man defntons of dscounted values of cash flows, net present value method, proftablty ndex, and nternal rate of return. Secton 8.4 presents the authors approach based on OFNs. In Sect. 8.5 we llustrate the ssue on a computatonal example, demonstratng how the methods can be used for the captal budgetng problem. We utlze a MATLAB envronment to demonstrate how the proposed methods can be appled to assess the proftablty of an alternatve nvestment project. Fnal remarks and conclusons are contaned n Sect. 8.6.

3 8 OFN Captal Budgetng Under Uncertanty and Rsk Ordered Fuzzy Numbers The ntroducton of the concepts of fuzzy sets and fuzzy numbers was propelled by the need to descrbe mathematcally mprecse and ambguous phenomena. The above concepts were descrbed n the paper of Lotf A. Zadeh [26] as a generalzaton of classcal set theory. A fuzzy set A n a nonempty space X s a set of pars A = {(x,μ A (x)); x X}, where μ A (x) : X [0, 1] s the membershp functon of a fuzzy set. Ths functon assgns to each element x X ts membershp degree to a fuzzy set. A fuzzy set, and hence ts membershp functon, has two basc nterpretatons. It can be understood as a degree to whch x possesses a certan feature, or as a probablty wth whch a certan, and at ths pont not entrely known, value wll assume a value x. A trangular fuzzy number s denoted wth three real numbers [a, b, c], where a < b < c. Its membershp functon assumes the form: 0 f x a; x a f a < x b; μ A (x) = b a c x f b < x c; c b 0 f x > c. (8.1) If an expert generates a trangular fuzzy number as a result of assessng the dstrbuton of possble values of a certan unknown quantty, t means that the expert deems the values below a, and above c, not possble; whereas the value b s possble wth a degree of 1, and the remanng values are possble to a varyng degree that decreases wth ther dstance from b. The noton of OFN, defned by Kosńsk, Prokopowcz, and Ślȩzak, was ntroduced n order to elmnate postulated defcences of fuzzy numbers: the loss of precson ncreasng wth the number of performed operatons and the fact that even lnear equatons cannot be solved n the set of fuzzy numbers. The theorem formulated by Kosńsk [17] concernng the unversal approxmaton of any nonlnear and contnuous defuzzfcaton operator offers tools for the applcaton of OFNs to fuzzy nference and modelng, ncludng assessng the proftablty of nvestment projects. Ordered Fuzzy Numbers gve a precse and elegant framework for dealng wth fuzzy objects (numbers) and many dfferent methods of defuzzfcaton. Defnton 1 An Ordered Fuzzy Number A s an ordered par ( f, g) of contnuous functons f, g :[0, 1] R. Graphcally the curves of ( f, g) and (g, f ) do not dffer. However, ths par of functons determnes dfferent OFNs; they vary n so-called orentaton, whch s denoted on dagrams by an arrow. Let A = ( f A, g A ), B = ( f B, g B ), and C = ( f C, g C ) be OFNs. Sum C = A + B, product C = A B, and dvson C = A B are defned n the set of OFNs as follows.

4 160 A. Chwastyk and I. Psz f C (x) = f A (x) f B (x) and g C (x) = g A (x) g B (x), (8.2) where denotes +,, and, respectvely. Moreover, A B s only defned when f B (x), g B (x) = 0 for each x [0, 1]. In the set of OFN, subtracton, exponentaton, and takng a root can also be defned n the usual fashon, for example: ( f, g) n = ( f n, g n ). (8.3) When consderng the set of OFNs and the assocated operatons of addton and multplcaton, we obtan a commutatve rng wth unty. By augmentng ths wth scalar multplcaton, we obtan a lnear space, that s, an algebra over real numbers. Moreover, ths set consttutes a commutatve Banach algebra wth unty n the supremum norm n each of the factors C[0, 1] C[0, 1] that are the Banach space. By ntroducng an approprate relaton of partal order, we also obtan a lattce [8]. We say that an OFN A = ( f, g) s nonnegatve f f (x) 0 and g(x) 0 for all x [0, 1]; (8.4) postve f f (x) >0 and g(x) >0 for all x [0, 1]. (8.5) Negatve OFNs are defned n a smlar way. It s worthwhle to pont out that the set of pars of contnuous functons, where one functon s ncreasng and the other s decreasng, and, smultaneously, the ncreasng functon always assumes values lower than the second functon, s a subset of the set of OFNs, whch represents the class of all convex fuzzy numbers wth contnuous membershp functons [4, 10, 16, 23, 25]. Defuzzfcaton s a process that converts a fuzzy set or a fuzzy number nto a crsp value. Functonals, whch map a fuzzy number to a real number, play a vtal role n OFN applcatons. Defnton 2 Let A be an OFN and c R. A mappng φ from the space of all OFNs to the set of real numbers s called a defuzzfcaton functonal f t satsfes the followng propertes, 1. φ(c, c) = c, 2. φ(a + (c, c)) = φ(a) + c, 3. φ(ca) = cφ(a), 4. φ(a) 0, f A s nonnegatve (8.4) where (c, c) s a par of constant functons on the nterval [0, 1] representng the constant c. Therefore, a defuzzfcaton functonal must be homogeneous of order 1, as well as beng restrctve, addtve, and normalzed. The model of constructng defuzzfcaton functonals presented n [19] allows us to obtan a number of defuzzfcaton functonals, whether lnear or nonlnear. In ths chapter we appled the nonlnear center of gravty defuzzfcaton functonal, defned by the followng equatons.

5 8 OFN Captal Budgetng Under Uncertanty and Rsk 161 φ COG ( f, g) = 1 ( f (s)+g(s))( f (s) g(s))ds ( f (s) g(s))ds 0 1 f (s)ds 0 1 ds 0, when, when ( f (s) g(s))ds = 0 ( f (s) g(s))ds = 0. (8.6) 8.3 Classc Captal Budgetng Methods In economc practce, net present value s the most commonly used dscount method. In essence, ths method conssts n assessng the present value of an nvestment project based on the forecasted streams of net cash flows, whch are the measure of an nvestor s future benefts. NPV s defned as a sum of net cash flows (NCFs) dscounted separately for each year and executed over the entre calculaton perod, wth a constant level of nterest (dscount) rate. Ths value expresses the updated (on the day of the assessment) value of benefts, whch the undertakng n queston can yeld n the future. The general form of NPV can be expressed as: NPV = n =0 CF (1 + r), (8.7) where n s the number of years, r s the market captalzaton rate, and CF s the cash flow n the th year of nvestment. NPV allows makng an nvestment decson havng analyzed cash flows, reduced by a specfc outlay, and dscounted by a weghted average cost of captal. Therefore, NPV allows the assessment of the economc value of an undertakng. The employment of a gven method requres forecastng future cash flows, whch nvolves forecastng several uncertan varables such as nterest rate, prces of resources and servces, and exchange rate. It affects the relablty and qualty of forecastng future effects and outlay. NPV allows takng the tme factor nto account. If the net present value of an nvestment project s postve, the project wll contrbute to an ncrease n the value of the company and as a result the wealth of ts owners. It s assumed that a gven nvestment s proftable f the value of dscounted cash flows durng the completon of the nvestment s postve. The proftablty ndex (PI), also known as the proft nvestment rato (PIR) or value nvestment rato (VIR), s the rato of payoff to nvestment of a proposed project. It s a useful tool for rankng projects because t allows us to quantfy the amount of value created per unt of nvestment. The proftablty ndex s a rato of dscounted cash nflows to the dscounted cash outflows:

6 162 A. Chwastyk and I. Psz PI = n =0 n =0 CF + (1+r), (8.8) CF (1+r) where n s the number of years, r s the market captalzaton rate, CF + s the cash nflow n the th year of nvestment, and CF s the cash outflow n the th year of nvestment. The PI helps n rankng nvestments and decdng the best nvestment that should be made. A PI greater than one ndcates that the present value of future cash nflows from the nvestment s hgher than the ntal nvestment, thereby ndcatng that t wll earn profts. A PI of less than one ndcates loss from the nvestment, and a PI equal to one means that there are no profts. NPV and PI technques n captal nvestment decsons are closely related to each other. The PI wll be greater than 1 only when the NPV s postve. However, n the case of mutually exclusve proposals havng dfferent scales of nvestment, that s, where the ntal nvestment n the alternatve projects s not the same, a conflct n NPV and PI may occur. Another captal budgetng method s the IRR. Ths method s descrbed as the dscount rate r that equates the present value of the expected future net cash nflows wth ts ntal outlay so we have NPV = 0. The IRR shows drectly the rate of return on the examned projects. The project s cost-effectve f ts IRR s hgher than the lmt rate, whch s the lowest rate of return acceptable to the nvestor. Generally, the hgher the nternal rate of return the nvestment project has the more proftable the project s [15]. Because of the general problem of fndng the roots of the equaton NPV = 0, there are many numercal methods that can be used to estmate the IRR. We use the method [24] consstng of several stages. Frst, we determne the value of the cash flows n subsequent years of an nvestment. Then, by successve approxmatons, we select two rates of return r 1 and r 2 satsfyng the condtons: 1. NPV 1 calculated for the rate r 1 s close to zero and postve. 2. NPV 2 calculated on the bass of the rate r 2 s close to zero but negatve. On the bass of these values we calculate the IRR of the consdered project. We apply the followng formula for ths purpose. IRR= r 1 + NPV 1(r 2 r 1 ) NPV 1 NPV 2. (8.9) In the presented method of calculatng the IRR, the dfference between r 1 and r 2 s partcularly mportant. Wth the ncrease n dfference between r 1 and r 2, calculaton results become less and less accurate as compared to the actual IRR. In practce ths dfference should be less than one percentage pont. In ths case, the mstake n calculatons of the IRR can be consdered to be rrelevant.

7 8 OFN Captal Budgetng Under Uncertanty and Rsk Fuzzy Approach to the Dscount Methods The classc forms of NPV, PI, and IRR do not take uncertan data nto account. When consderng the fuzzy envronment of an nvestment project, modfyng dscount methods to take nto account uncertan data becomes necessary. Ths allows us to take nto consderaton nformaton uncertanty and decreases the rsk of makng a mstake n assessng the proftablty of an nvestment project. For the problem of defnng a generalzaton of the above-mentoned dscount methods to OFNs, we assume that the captalzaton rate R, cash flows CF, cash nflows CF +, and cash outflows CF are Ordered Fuzzy Numbers. The dscounted cash flows n the th year of nvestment are calculated as follows, CF ((1, 1) + R), (8.10) where (1, 1) stands for a par of constant functons that assume a value of one, and + and sgnfy addton and dvson n a set of OFNs defned through the Eq Exponentaton s performed accordng to the formula 8.3. Therefore, we have the formula for ordered fuzzy net present value: OFNPV = n =0 CF ((1, 1) + R). (8.11) And for modfed proftablty ndex: OFPI = n =0 n =0 CF + ((1,1)+R). (8.12) CF ((1,1)+R) Our modfed method of calculatng the nternal rate of return requres selecton of two ordered fuzzy rates of return R 1 and R 2 satsfyng the followng condtons. 1. OFNPV 1 calculated for the rate R 1 s close to ordered fuzzy zero (whch means the par of constant functons (0, 0)) and postve (see (8.5)). 2. OFNPV 2 calculated for the rate R 2 s close to ordered fuzzy zero and negatve. On the bass of these values we calculate the ordered fuzzy nternal rate of return of the consdered project. We use the followng formula for ths am, OFIRR = R 1 + OFNPV 1(R 2 R 1 ) OFNPV 1 OFNPV 2. (8.13) Before presentng the results of the project evaluaton to the nvestor, the selected defuzzfcaton method should be appled n order to obtan real values:

8 164 A. Chwastyk and I. Psz NPV = φ COG (OFNPV), PI = φ COG (OFPI) and IRR= φ COG (OFIRR). 8.5 Computatonal Example of the Investment Project In ths secton we present an example of a captal budgetng problem usng three methods based on OFNs. These methods are: ordered fuzzy net present value method (OFNPV), ordered fuzzy proftablty ndex (OFPI), and ordered fuzzy nternal rate of return (OFIRR). We consder an example of potental alternatves of nvestment project executon: project one P 1 and project two P 2. Investment decsons are made under the condtons of uncertanty and rsk, nasmuch as t s mpossble to prepare an accurate descrpton of economc and fnancal condtons for the functonng of the consdered projects n the future. The use of OFNs allows us to lmt the effects of uncertanty and rsk. In order to defne the fuzzy condtons of the executon of the nvestment project, the decson-makng process nvolves an expert who has approprate knowledge and experence n plannng and executng smlar projects. A major problem related to the use of OFNs was the requrement for the experts to gve an opnon on ndvdual elements of these alternatves of nvestment projects n the form of OFNs, that s, pars of functons. We propose that the expert descrbe project parameters by means of trangular fuzzy numbers, whch wll be subsequently converted nto OFNs. We assume that the consdered projects are planned for the perods of 7 and 5 years, respectvely. The remanng project parameters reman uncertan, therefore they are determned by the expert n the form of trangular fuzzy numbers. The trangular fuzzy captalzaton rate assumes the form of R =[0.04; 0.06; 0.07].Ths means that accordng to the expert the captalzaton rate of below 4% and above 7% s not possble, whereas the value of 6% s the most probable one, and other values are probable to a dfferent degree: the hgher they are, the closer they are to 6%. In a smlar way, the expert determnes the fuzzy values of cash nflows and outflows n subsequent years for project P 1 and project P 2, respectvely, n Tables 8.1 and 8.4. In order to smplfy the analyss, the data are expressed n thousands of arbtrary monetary unts (a.m.u.). To a trangular fuzzy number A =[a, b, c] has a correspondng OFN: A OFN = ((b a)x + a,(b c)x + c), (8.14) whch s the ordered par of lnear functons (Fg. 8.1). By applyng the formula 8.14 we defne OFNs correspondng to the values determned by the expert. For nstance, the captalzaton rate expressed by OFNs assumes the form: R OFN = (0.02x ; 0.01x ). Then we dscount cash nflows and outflows usng the formula Obvously, dscounted cash flows n the th

9 8 OFN Captal Budgetng Under Uncertanty and Rsk 165 Table 8.1 Input data for the nvestment P 1 usng trangular fuzzy numbers Year Cash outflows Cash nflows 0 [450, 450, 450] [0, 0, 0] 1 [19, 20, 22] [68, 70, 72] 2 [5, 5, 6] [70, 75, 80] 3 [4.5, 5, 6] [70, 75, 85] 4 [4, 5, 6] [90, 100, 125] 5 [4, 5, 6] [110, 120, 135] 6 [4, 5, 6] [110, 125, 140] 7 [4, 5, 6] [100, 110, 130] Table 8.2 Cash nflows and dscounted cash nflows for P 1 wth the use of OFNs Year Cash nflows Dscounted cash nflows 0 (0, 0) (0, 0) 1 (2x + 68, 2x + 72) 2 (5x + 70, 5x + 80) 3 (5x + 70, 10x + 85) 4 (10x + 90, 25x + 125) 5 (10x + 110, 15x + 135) 6 (15x + 110, 15x + 140) 7 (10x + 100, 20x + 130) ( 2x x+1.04, ) 2x+72 ( 0.01x+1.07 ) 5x+70 5x+80, (0.02x+1.04) 2 ( 0.01x+1.07) ( 2 ) 5x+70 10x+85, (0.02x+1.04) 3 ( 0.01x+1.07) ( 3 ) 10x+90 25x+125, (0.02x+1.04) 4 ( 0.01x+1.07) ( 4 ) 10x x+135, (0.02x+1.04) 5 ( 0.01x+1.07) ( 5 ) 15x x+140, (0.02x+1.04) 6 ( 0.01x+1.07) ( 6 ) 10x x+130, (0.02x+1.04) 7 ( 0.01x+1.07) 7 Table 8.3 Selected rates of return for the project P 1 Rates of return Trangular fuzzy numbers Ordered Fuzzy Numbers R 1 (OFNPV 1 postve) [0.04; 0.07; 0.1] (0.03x , 0.03x + 0.1) R 2 (OFNPV 2 negatve) [0.06; 0.9; 0.12] (0.03x , 0.03x ) year of nvestment are obtaned by addng dscounted cash nflows and outflows n the th year of an nvestment. Subsequently, based on the formulas presented n the prevous pont we calculate the ndexes OFNPV, OFPI, and OFIRR. Fnally, these values undergo defuzzfcaton usng the functonal 8.6. Thus we obtan crsp values, whch can be presented to the nvestor. Calculatons for the consdered nvestment projects were performed usng the MATLAB computer program. Let us consder the alternatve nvestment project P 1. Table 8.1 presents the cash nflows and outflows of the project usng trangular fuzzy numbers defned by the expert engaged n the decson process.

10 166 A. Chwastyk and I. Psz Table 8.4 Input data for the nvestment project 2 usng trangular fuzzy numbers Year Cash outflows Cash nflows 0 [450, 450, 450] [0, 0, 0] 1 [0, 0, 0] [145, 150, 155] 2 [0, 0, 0] [140, 150, 155] 3 [0, 0, 0] [110, 125, 140] 4 [0, 0, 0] [60, 75, 80] 5 [0, 0, 0] [60, 75, 90] Fg. 8.1 The par of lnear functons correspondng to the trangular fuzzy number A =[a, b, c]. Thearrow denotes the order of functons, the so-called OFN orentaton y (b c)x + c (b a)x + a 1 x Table 8.2 presents the cash nflows for the frst project expressed n OFNs. The cash outflows for ths project and the cash nflows and outflows for the second project were calculated n an analogous way. Frst we calculated the value of OFNPV and OFPI of the project. After defuzzfcaton, the net present value for project P 1 s equal to a.m.u., whch means that the projected earnngs generated by the proposed nvestment exceed the antcpated costs. The proftablty ndex s equal to , whch further confrms the postve evaluaton of the project. In order to calculate OFIRR for the frst project we selected by successve approxmatons of two rates of return R 1 and R 2 for whch ordered fuzzy net present values are close to ordered fuzzy zero and postve (R 1 )ornegatve(r 2 ), respectvely (Table 8.3). The nternal rate of return for ths project s equal to 8.21%. Let us consder the alternatve nvestment project P 2. Table 8.4 presents the cash nflows and outflows of the project usng trangular fuzzy numbers. The NPV for the project P 2 s equal to a.m.u. so the project would be estmated to be a valuable venture. The PI s equal to , whch further valdates the postve evaluaton of the project. The nternal rate of return for the second project calculated on the data presented n Table 8.5 s equal to 9.78%. In Table 8.6 we present the results of calculaton of the modfed dscount methods for consdered projects P 1 and P 2. We compared the obtaned values of the proposed new methods to ad the decson maker n choosng the best nvestment project. Frst we determned the NPV for each project; then we establshed the proftablty ndex

11 8 OFN Captal Budgetng Under Uncertanty and Rsk 167 Table 8.5 Selected rates of return for the project P 2 Rates of return Trangular fuzzy numbers Ordered Fuzzy Numbers R 1 (OFNPV 1 postve) [0.03; 0.08; 0.13] (0.05x , 0.05x ) R 2 (OFNPV 2 negatve) [0.06; 0.11; 0.16] (0.05x , 0.05x ) Table 8.6 Summarzed results of proposed dscount methods for the projects Methods Project P 1 Project P 2 NPV PI IRR 8.21% 9.78% for each nvestment project, and fnally the nternal rates of return. Accordng to the NPV analyss alone, project P 1 s the most approprate choce for the decson maker. The proftablty ndexes for project P 1 and project P 2 vary slghtly and they are greater than 1, whch confrms the proftablty of both projects. Accordng to the IRR analyss alone, project P 2 s the most approprate choce for the decson maker. The NPV and IRR analyss for these two projects gve us conflctng results. Ths s due to the tmng of the cash flows for each project as well as the sze dfference between the two projects. By the NPV rule the decson maker should choose project P1, so t can be executed. The conventon s to use the NPV rule when the two methods are nconsstent. It better reflects the prmary goal: to mprove the fnancal wealth of the company. 8.6 Summary The captal budgetng problem wth Ordered Fuzzy Numbers corresponds to the project selecton problem. We modfed the method presented n our prevous paper by transferrng the defuzzfcaton process to another stage of calculatons and we presented the next dscount methods, the proftablty ndex and nternal rate of return, to make an evaluaton of alternatve nvestment projects more precse. Tools of that knd can be perceved as a decson-support system based on OFNs. We presented the example of alternatve nvestment project selecton usng new dscount methods. The presented methods may be used to represent mprecse nformaton, among others about cash flows and captalzaton rate. They offer a clear smultaneous representaton of several peces of nformaton. In addton, well-defned arthmetc operatons on OFNs make t easy to perform even complex calculatons connected, for example, wth a long perod of nvestment. Moreover, owng to the elmnaton of ssues related to usng classcal fuzzy numbers such as ncreasng fuzzness over subsequent

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