FUZZY CAPITAL BUDGETING: INVESTMENT PROJECT VALUATION AND OPTIMISATION

L. Dimova, P. Sevastjanov,* D. Sevastjanov**

* Institute of Comp.& Information Sci., Technical University of Czestochowa, Dabrowskiego 73, 42-200 Czestochowa, Poland 

** Maltex.com, inc., 100 William st., 6 floor, New York, NY, USA,

dmitry.sevastianov@multex.com

Abstract

The other problem is that usually we must consider a set of different local criteria  based on the financial parameters of investments. As one of its possible decision, the numerical method for optimisation of future cash-flows in fuzzy setting when considering the generalised project's quality criterion as the compromise between local criteria of a profit maximisation and financial risk minimisation is proposed.

Keywords: Capital Budgeting;Risk; Fuzzy Sets; Multiobjective Optimisation

Introduction

At first, let us consider the usual non-fuzzy approaches to the capital budgeting problem. There are a lot of financial parameters proposed in literature[1-4] for budgeting .

The main of them are: the Net Present Value (NPV), Internal Rate of Return (IRR), Payback period(PB) , Profitability Index (PI). These parameters are usually used for project quality estimation but in practice they have deferent importance.

It is earnestly shown in [5] that the most important parameters are NPV and IRR (see Table.1).

Therefore, our further consideration will be based on the analysing only NPV and IRR. Good review of other useful financial parameters is in [6].

Table 1. Frequency of the financial parameters using (from 103 largest petroleum and gas companies of  USA in 1983)

Net Present Value is usually calculated as follows:

(1)

where d is a discount rate; t_n is the production starting year; t_c is the year of the ending of investments; KV_t  is a capital investment in a year t, P_t- is an income in a year t, T is a duration of investment project in years.

Usually discount rate is equal to the average bank interest rate in investor's country  or equal to other appropriate value for estimation of return in the case of alternative capital investments to the other projects or securities.

The economic nature of internal rate of return (IRR) can be explained as follows. As an alternative to investments in the analysing project the depositing under some bank interest (distributed in time as in the case of the analysing project) is considered. It is suggested that all the incomes that will be received during realisation of project will also be  deposited with the same interest rate. If discount rate is equal IRR, then the investment in the project will give the same total income, as in the case of depositing. Thus, both alternatives  are economically equivalent. If actual bank discount rate is less then IRR of the considered project, the investment in the project is more preferable. Therefore IRR is the threshold discount rate selecting effective and the ineffective investment projects. The value of IRR is  the   solution of the non-linear equation with respect to d:

(2)

Estimation of  IRR is frequently used as a first step of financial analysis. Only projects  with  IRR which are not below than some accepted threshold value (usually 15-20 %) are choosing for further consideration.

Nowadays, traditional approach to evaluation of NPV, IRR and other financial parameters is subjected to the quite deserved criticism, since the future incomes P_t, capital investments KV_t and rates d are rather uncertain parameters. Uncertainties which we meet in capital budgeting, differ from ones in the case of share prices forecasting and can not be adequately described in probability terms. In real active investment we usually deal with the business-plan which takes a long time-as a rule some years- for its realisation. In such cases, description of  uncertainty within the framework of traditional probability methods usually is impossible because of absence of an objective information about future events probabilities. Thus, what we really have in such cases are some expert's estimations. In real world situations, investors or experts involved are able to predict  confidently only intervals of possible values Pt, KVt and d and the most expected values inside these intervals. That is why, during last two decades the growing interest to the applications of interval arithmetic [7] and fuzzy sets theory methods[8] in  budgeting has being observed.

      Fuzzy NPV and risk assessment connected with it

The technique offered is based on fuzzy extencion principle [8]. Thus, the values of uncertain parameters P_t, KV_t and d are substituted  by corresponding fuzzy  intervals. In practice it means that the expert sets lower - P_t1 (pessimistic value) and upper - P_t4 (optimistic value) boundaries of intervals and internal intervals of the most expected values [P_t2, P_t3] for the parameters analysing (see Fig. 1).. The function m (P_t) is usually interpreted as a membership function,  i.e. the degree to which  the parameter’s values belong to considered interval (in our case [P_t1, P_t4]). The membership function changes continuously from 0 (area out of interval) up to maximum value equal to 1, in area of the most possible values. It is obvious that the membership function is the generalisation of  usual set’s characteristic function which is equal to 1 for all values of parameters inside a set, and is equal to 0 in  all other cases.

Figure 1.  Fuzzy interval of uncertain parameter P_t  and its membership function m(P_t ).

 The linear character of function is not obligatory, but  such a mode is the most used and allows to represent fuzzy intervals is convenient form by quadruple P_t = {P_t1, P_t2, P_t3, P_t4}. Then all the necessary calculations are carried out using the special fuzzy- interval arithmetic rules.

Let us recall some basic principles of fuzzy arithmetic [25].

In general, for arbitrary forms of membership functions the technique of fuzzy- interval calculations is based on the representation of initial fuzzy intervals by so-called a -cuts (Fig.1) that, in fact, are crisp intervals associated with the corresponding degrees of a membership. All further calculations are made with those font-family: Symbol'>a-cuts according with the well known crisp interval-arithmetic rules and the resulting fuzzy- intervals are obtained as a disjunction of corresponding final a -cuts being calculated.

Thus, if f A is a fuzzy number, then

where  A _a    is the  crisp interval {x: m _A (x) ³ a }, a A _a  is fuzzy interval {( x, a ): x Î A _a   }. Thus, if  A, B, Z  are fuzzy numbers (intervals) and @ is an operation from {+, -, *, /} then

 Z = À@Â=. (5)

Since in the case of a -cut presentation, fuzzy arithmetic is based on crisp interval arithmetic rules, the basic definitions of applied interval analysis must be presented too. There are some definitions of interval arithmetic ( see [26,27]), but in practical applications the  so-called  «naive» form proved the best. According to it, if А = [a₁, a₂] and В = [b₁, b₂] are crisp intervals, then

Z = А@В={ z=x@y,  }. (6)

As the direct consequence of the basic definition (6) the next expressions were obtained:  

А+В=[a₁+b₁, b₂+b₂],       А-В=[a₁-b₂, a₂-b₁],

А ·В=[min(a₁·b₁, a₂· b₂, a₁·b₂, a₂·b₁), max(a₁·b₁, a₂· b₂, a₁·b₂, a₂·b₁)],

А/В=[a₁, a₂] · [1/b₂, 1/b₁]

Of course, there are many internal problems within applied interval analysis, like the division by zero-containing interval, but in general it can be considered as the good mathematical tool for modelling under the conditions of uncertainty. 

To illustrate, let us consider an example. Let we have the investment project, in which  building phase proceeds two years with the investments KV₀ and KV₁  accordingly. The profits are expected only after finishing building phase and will be obtained during two years (P₂ and P₃). It is suggested that fuzzy  interval for discount d remains stable during the time of project realisation. The appropriate trapezoidal initial fuzzy intervals were as follows:      KV₀ = {2,2.8,3.5,4}; KV₁ = {0,0.88,1.50,2};      KV₂ = {0,0,0,0}; KV₃ = {0,0,0,0};       P₀ = {0,0,0,0}; P₁ = {0,0,0,0};P₂ = {6.5,7.5,8.0,8.5}; P₃ = {5.5,6.5,7.0,7.5}; d = {0.08,0.13,0.22,0.35}.

The resulting fuzzy interval NPV, calculated  using fuzzy extension of Eq.(1) is presented in Fig. 2.

Figure 2. Resulting fuzzy interval  NPV.

The obtained fuzzy interval allows to estimate the boundaries of possible  values of predicted NPV, the interval of  most expected values, and also- that is very important- to evaluate a degree of financial risk of the investments.

To estimate the financial risk, we have taken  into account the  following inherent property of fuzzy sets. Let A be some fuzzy subset of X,  being described by the membership function m (A) . Then complementary fuzzy subset ` A   has the membership function m ( ` A)=1- m (A). The principal  difference fuzzy subset from usual precise ones is that intersection of fuzzy A and `A is not empty,  that is A Ç ` A= B , where B is a not empty fuzzy subset, too. It is clear that the closer A to `A , the more power of a set B and  more A differ from ordinary  sets.

 Using this circumstance R Yager  [28] proposed a set of grades of nonfuzziness of fuzzy subsets

(7)

Hence, the grade of fuzziness may be defined as

(8)

The definition (8) is in compliance with the obvious requests to the grade of fuzziness. If  A is a fuzzy subset on X , m (A) is its membership function and dd is the corresponding grade of fuzziness, then following properties  should be observed:

1) dd(A) = 0, if A is crisp subset;

2) dd(A) has a maximum value if  m (A) = 1/2 for x Î X;

3) dd(А1)>dd(A), if m (х)< m (y)  (x Î A1; y Î A).

It is proved that the introduced measure is similar to the Shannon entropy measure [28].

 In the most useful case (p = 1) expression (8)  is transformed to

   (9)

It is clear (see Eq.(9))  that grade of fuzziness is rising from 0 if m (A) = 1 (crisp subset) up to 1 if m (A) = 1/2 (maximum degree of fuzziness).

 With respect to our problem the grade of nonfuzziness of fuzzy interval NPV can linguistically be interpreted as a risk or uncertainty of obtaining the Net Present Value in the interval [NPV₁, NPV₄]. Really, the more precise, (more «rectangular») interval we receive, the more degree of uncertainty  and  risk we obtain. Of course, at first  this assertion seems paradoxical. However, any precise (crisp) interval is not containing any additional information about relative preference of values placed inside it. Therefore, it contains less useful information, than any fuzzy interval being constructed on its basis. In the latter case the additional information that reducing uncertainty is derived from the membership function of considered fuzzy interval.

Therefore, problem investments efficiency evaluation on the base of NPV becomes two-criteria and requires the special approach and appropriate technique. Recently, we proposed such technique [29] based on the fuzzy set theory was developed , however its detailed consideration is out of scope of this paper.

3.       The set of real value IRR estimations based on the fuzzy cash flows

In essence, problem of Internal Rate of Return (IRR) evaluation looks as fuzzy interval solution of the Eq.(2) with respect to d.

It is proved that the solution of the equations with fuzzy parameters is possible by expression of these parameters (P_t , KV_t and d in our case) as a sets of corresponding a - cuts. As the result for the problem of evaluating IRR we obtain a system of the non-linear crisp - interval equations:

, (10)

where [P_t] _a , [KV_t] _a and [d]_a   are the crisp intervals on corresponding a - cuts.

In general, we can state that the naiv assumption that in a right hand side of Eq.(10) there should be a degenerated zero interval [0,0], does not ensure deriving of an adequate outcomes since there is non-degenerated interval expression on the left hand side of Eq. (10), but let us consider this situation thoroughly.    

As the simplest example, consider two-years project, when all investments are finished in the first year, and the production and deriving of the incomes begins and ends in the second year. Then each of the equations for a-cuts (10) should  be divided on two :

  - left boundary of an interval NPV

(11)

   - right boundary of an interval NPV

 The formal solution (10) with respect to d₁ and d₂ is trivial: , however it is senseless, as the right boundary of an interval [d1, d2] always appears less than  left one.

This, on the first glance, absurd result should be easily explained from common methodological positions. Really, the rules of interval mathematics are constructed in such away that any arithmetical operations with intervals give us as an interval, too. These rules are in the full correspondence with the well known common methodical position, according to which  any arithmetical operation with uncertainties must to increase the total uncertainty and the  system’s entropy . Therefore,  if  in our case we place to the right hand side of  (10) and (11) a degenerated zero intervals, it will be equivalent to the request of reducing  uncertainty of the left parts up to zero that should be possible only in the case of inverse character of an interval [d₁, d₂] that in turn can be interpreted as a request of entering negative entropy in a system.

 Thus, the  presence a degenerated zero interval in the right hand sides  of the interval equations is incorrect. The more acceptable approach to solving  of this problem has been constructed with the help of following reasons. It is easy to see, when analysing expressions (11) that for any value d₁ the minimal width of an interval NPV is reached if d₂ = d₁. This is in accordance  with common methodical positions: the minimum uncertainty of an outcome (NPV) is reached in the case of minimum uncertainty of all the system’s parameters used. It is clear (see Fig. 3 ) that the most reasonable decision of «zero» problem is the request for the middle of an interval NPV to be placed on a  zero point (request of symmetry of an interval concerning to zero).

The obvious, on the first sight, intention to minimise the sizes of received interval NPV results in deriving positive or negative intervals of minimum width, but not intersecting an zero point, that does not correspond to the natural definition of a zero containing interval. Besides it can be  easily proved that only request of symmetry of a zero containing interval ensures an asymptotically valid outcome in the case of contraction of the boundaries of all considered intervals to their centres.

 Thus, generally problem is reduced to searching such exact (non-interval) values d, which can provide a symmetry with respect to zero resulting intervals NPV on each a -cut in the equations (10), i.e. would guarantee fulfilment of a request (NPV₁ + NPV₂) = 0, for each a = 0,0.1,0.2..., 1.

Of course, the problem is decided by the numerical methods.

Figure3. The discount dependence of an interval NPV  when the investments in the first year is KV₀ = [1,2], income in the second year is P₁ = [2,3]:  and D (NPV) is width of an interval NPV.

To illustrate the  above theoretical considerations, let's compare two investment projects that must be realised during  4 years . Fuzzy  cash flows  K_t = P_t- KV_t  are defined with the help of four-reference points form described above(see Table 2). It worth noting that the data of first project are more certain.

Table 2.

Fuzzy parameters of compered projects.

The  results of estimations  for two comparing investment projects with different  fuzzy cash flows are  presented  on Fig. 4., too.  It is seen that values of IRR _a   obtained for each a -cut can  increase or decrease with growing of a and as the result for each project  own set of possible real number  values of IRR has been obtained. Thus, the problem of  interpretation of the results rises.   

To do this, we propose to reduce the sets of IRR _a  obtained on each a- cut to  the little set of parameters which can be easily  interpreted. The first elementary parameter-  average value IRR_m - is certainly convenient, however it does not take into account that with growing of a   the reliability of an outcome increases too, i.e. IRR _a , obtained on higher a -cuts are more expected, than obtained on lower ones, because of a-cut’s definition. On the other hand, a precise intervals [NPV1, NPV2] _a  corresponding to each of IRR _a has a widths which being in some sense a measure of uncertainty for received non-interval value IRR _a , since the widths of intervals [NPV1, NPV2] _a characterise quantitatively the difference of the left hand side of Eq.(10) from degenerated zero interval [0,0]. This allows us to introduce  two weighted estimations of IRR on a set IRR _a : least expected (least reliable)  IRR_min and most expected  (most reliable)  IRR_max   :

, (12)

, (13)

where n is the number of a -cuts.

In decision making practice, when choosing the best project, it is worthy  to use all three proposed parameters IRR_m, IRR_min, IRR_max.

Interpretation of [NPV1, NPV2] _a   as performance for uncertainty of IRR _a allows to propose the quantitative and expressed in monetary units evaluations of  financial risk of project considered (uncertainty degree  of received values IRR_ср, IRR_min, IRR_max as a consequence  of the  initial data uncertainty ):

(14)

  Parameter R_m can play a key role in project's efficiency estimation.

For our example we get

Project 1:  IRR_min = 0,34;  IRR_max = 0,327; IRR_ср = 0,335; R_ср = 1,56.

Project 2: IRR_min = 0,322; IRR_max = 0,329; IRR_ср = 0,325; R_ср = 3,52.

Thus, the considered projects have rather close values of IRR_ср, IRR_min, IRR_max.  At the same time ,the  risk value R_cp  for the second project is considerably higher than risk of the first one. So, the first project is the bast one.

 Except the parameters described above, there are some other useful estimations: IRR_nr - most reliable value of IRR _a - connected with the minimum interval [NPV1, NPV2] _nr among of all [NPV1, NPV2] _a and IRR _nr - the least reliable value of  IRR _a - connected with the maximum interval [NPV1, NPV2] _r  among all of [NPV1, NPV2] _a . It is clear, that [NPV1, NPV2] _nr and [NPV1, NPV2] _r are the risk estimations  for appropriate IRR_nr and IRR_r.

 It should be noted (see Fig. 4)  that difference between values of IRR_nr for the projects compered is rather negligible, but the difference in risk estimations is considerable.

4. Method for numerical solution of project optimisation problem

Proposed here approach to optimisation problem is based on the consideration of all the initial fuzzy intervals P_t and KV_t as the restrictions on controlled input data and on the assumption that d_t is the a random parameter describing an external, in relation to the considered project, uncertainty. We also take into account that there may be some preferences on the interval of possible values of d which may be expressed by certain  membership function , say m _d (d). Thus, we deal with the random as well as with a fuzzy uncertainties when describing discount factor.

The problem is decided in two steps.

At  first, according to the fuzzy extension principle we substitute in equation (1) all the parameters  P_t , KV_t  and d_t  by  corresponding  fuzzy-intervals. As the result we obtain fuzzy-interval estimation of NPV.

On the next step, we  consider the obtained fuzzy-interval  NPV as the restriction on the profit that can be  derived we built the local criterion of NPV maximisation.

 For mathematical description of  local criteria we use so-called desirability functions which are, in essence, the special interpretation of usual  membership functions. Briefly, the desirability function is rising from 0 (in the field of inadmissible values of its argument) up to 1 (in area of the most preferable argument's values). Thus, the construction of desirability function for NPV is rather obvious: the desirability function m_NPV (NPV) can be considered only on an interval of possible values restricted by the interval [NPV1,NPV4] and, naturally, the more values of NPV, the more degree of desirability  (see Fig. 4).

Figure. 4. Connection between restriction and local criterion:1-intitial fuzzy interval of NPV (fuzzy restriction); 2- the desirability function m_NPV (NPV) .

The initial fuzzy intervals P_t and KV_t, are also considered as the desirability functions ..., which describing the restrictions on the  controlled input variables. It is clear that  initial intervals were already constructed in such a way that in the case of their interpretation as the desirability functions, the more preferable values from intervals  of P_t and KV_t appear those which are more realisable (possible). Since  these desirability functions are connected with the possibility of realisation  of corresponding values of variables P_t and KV_t, they describing implicitly the financial risk of the project.

On  the set of all the desirability functions the general criterion maximising is created:

, (15)

where   a ₁ and a₂   are ranks, characterising the relative importance of  profit maximisation and risk minimisation local criteria; Ù is the minimising operation,  - desirability function of NPV.

There are many different form of general criterion were using in applications. As emphasised in [30], now choosing of concrete realisation of aggregating operator, which is usually called t-norm, is rather the application dependent problem. However, the choosing min-operator in Eq.(15)  is the most straight-out approach when we not permit the compensating small values of some criteria by the great values of other ones. The problem is reduced to searching of non-interval (precise) values of PP1, PP2..., KKV1, KKV2... (changing in  the appropriate fuzzy intervals P1, P2..., KV1, KV2), which have to maximise the general criterion (15).

 The problem is complicated by fact  that the discount d is a random parameter,  distributed in a specific interval.

Procedure of a solution was carried out as follows.

At first, from  interval of discount’s possible  values by random way a fixed number is selected. Further with the help of Nollaw-Furst random  method the  optimum solution is obtained, as the best compromise between uncertainty of basic data and intention to derive the maximum profit,  i.e. the optimisation problem comes to the maximisation of the general criterion (15).  The optimal values PP^d_t and KKV^d_t, , are a local  optimum decision for given discount value. Therefore,  we are repeating the described procedure with the new random discount values until the statistically representative sample of optimum solutions at the various d will had been obtained. The final optimum PP⁰_t , KKV⁰_t values are calculated  as the  weighted evaluations taking into account the degrees of  possibility of various  d_i, which are defined by an initial fuzzy interval d. with a membership function m _d (d_i)

, (16)

where m is the number of discount values used for the solution of a problem. Similarly , all  KKV⁰_t  can be expressed, too.

It is possible also to take into account the values of a general criterion in the optimum points:

, (17)

where b ₁, b ₂  are corresponding weights. The similar expression we have for KKV⁰_t .

It worth noting that last expression enables us  to take into account, apart from of reliability of values d_i, the degree of compatibility (in other words, the degree  of consensus)  in compromise at each selected values of discount.

Obtained  optimal  PP_t⁰ and KK_t⁰  may be used for the final project quality estimation.

For the example considered in previous  Section (Table 2, project 1), using expressions (16), (17) , we have obtained the results presented in Table 3.

Table.3. The results of optimisation.

Further, by substituting these (PP_t⁰ and KK_t⁰ ) and fuzzy interval d in the expression (1), we get the optimal fuzzy value of NPV.

For our example we  obtain

NPV₁₆ = {4.057293, 6.110165, 8.073906, 9.454419},

when using the expression (16) and

NPV₁₇ = {4.065489, 6.109793, 8.064094, 9.436519}

on the base of expression (17).

It is clear that there are no great deference between the results obtained using expressions (16) and (17) in our case.

In Fig. 5, the fuzzy NPV₁₆ obtained with using the  optimal PP_t⁰ and KK_t⁰  is compared with the  initial one obtained using direct account on the initial fuzzy  values P_t and KV_t, without use of optimisation. Of course, in optimal case we have a greater mean value of fuzzy interval NPV.

Figure 5. The comparison of the initial and optimal fuzzy intervals  NPV

Using optimal PP_t⁰ , KK_t⁰  and  method described in Section 2 , the degree of project risk may be estimated , too. This risk can be assumed as the  financial risk of the project as a whole..

For the aims of usual accounting practice it is possible to calculate the average weighted value of NPV by use the  expression:

. (18)

For our example the NPV₁₆= 6.8931 and NPV₁₇ = 6.8942  have been obtained.

4.       Conclusion

The natural way for project risk assessment is to treat it as the degree of  fuzziness of fuzzy valued Net Present Value, NPV. It is clearly shown, why it is impossible to get a fuzzy Internal Rate of Return, IRR. The only real valued IRR may be obtained as a decision of fuzzy equation, but a set of new useful parameters connected with IRR and characterising uncertainty of the problem may be obtained as the additional result.

Multiobjective project optimisation problem in the mixed fuzzy and random setting is formulated as the compromise between the local criteria of profit maximisation and risk minimisation. Numerical method for deciding of this problem is described and tested.

References

[1] B. Belletante, H. Arnaud, Choisir ses investissements, Chotar et Assosies Editeurs, Paris, 1989.

[2] E.F. Brigham, Fundamentals of Financial Management, The Dryden Press, New York, 1992.

[3] Ch. Chansa-ngavej, C.A. Mount-Campbell, Decision criteria in capital budgeting under uncertainties: implications for future research, Int. J. Prod. Economics 23 (1991) 25-35.

4] P. Liang, F. Song, Computer-aided risk evaluation system for capital investment, Omega 22 (4) (1994) 391- 400.

[5]. Bogle H.F., Jehenck G.K. (1985). " Investment Analysis: US Oil and Gas Producers Score High in University Survey ". Proceeding of Hydrocarbon Economics and Evaluation Symposium, Dallas14-15 march 1985, 234-241.

[6].D. Babusiaux, A. Pierru. Capital budgeting, project valuation and financing mix: Methodological proposals. Europian Journal of Operational Research. 135 (2001) 326-337.

[7].  Moore R.E. (1966). Interval analysis. - Englewood Cliffs. N.J.: Prentice-Hall.

[8 ]L.A.Zadeh,Fuzzy sets,Inf.Control , 8 (1965)  338 –353.

[9 ]. T.L.Ward,Discounted fuzzy cash .ow analysis, in: :in 1985 Fall Industrial Engineering

Conference Proceedings,1985, pp.476 –481.

[10 ]. J.U.Buckley,The fuzzy mathematics of .nance,Fuzzy Sets and Systems 21 (1987) 257 –273.

[11] S. Chen, An empirical examination of capital budgeting techniques: impact of investment types and firm characteristics, Eng. Economist 40 (2) (1995), 145-170.

[12]. Ch.-Yu Chiu, Ch.S. Park, Fuzzy cash ow analysis using present worth criterion, Eng. Economist. 39 (2) (1994) 113-138.

[13]. F. Choobineh, A. Behrens, Use of intervals and possibility distributions in economic analysis, J. Oper. Res. Soc. 43 (9) (1992) 907-918.

[14]. M. Li Calzi, Towards a general setting for the fuzzy mathematics of finance, Fuzzy Sets and Systems 35 (1990) 265-280.

[15]. G. Perrone, Fuzzy multiple criteria decision model for the evaluation of AMS, Comput. Integrated Manufacturing Systems 7 (4) (1994) 228-239.

[16 ].C.Y.Chiu, C.S.Park,Fuzzy cash flow analysis using present worth criterion, Eng.Econom. 39 (2) (1994) 113 –138.

[17 ]. C.Kahraman, E.Tolga,  Z.Ulukan, Justification of manufacturing technologies using fuzzy

benefit/cost ratio analysis, Int.J.Product.Econom. 66 (1) (2000) 45 –52

[18 ]. C.Kahraman, Z.Ulukan ,Continuous compounding in capital budgeting using fuzzy concept, in: Proceedings of the 6th IEEE International Conference on Fuzzy Systems (FUZZ-IEEE ’97), Bellaterra, Spain,1997, pp.1451 –1455.

[19 ].C.Kahraman, Z.Ulukan, Fuzzy cash flows under inflation, i n:Proceedings of the Seventh

International Fuzzy Systems Association World Congress (IFSA ’97),University of Econom-ics,Prague,Czech Republic, vol .IV, 1997, pp.104 –108.

[20]. P. Sevastianov, D. Sevastianov. Risk and capital budgeting parameters evaluation from the fuzzy sets theory position. – Reliable software, 1997, 1, pp. 10-19. (in Russian)

[21]. L. Dimova, D. Sevastianov, P. Sevastianov. Application of fuzzy sets theory, methods for the evaluation of investment efficiency parameters. Fuzzy economic review. 2000, Vol. V, N 1, P. 34-48.

[22]. D. Kuchta .Fuzzy capital  budgeting. Fuzzy Sets and Systems 111 (2000) 367-385.

[23]. C. Kahraman. Fuzzy versus probabilistic benefit/cost ratio analisis for public work projects. Int. J. Appl. Math. Comp. Sci.,(2001) V. 11, N 3, 705-718.

[24]. C. Kahraman, D. Ruan, E.Tolga. Capital budgeting techniques using discountedfuzzy versus probabilistic cash fows. Information Sciences. 142 (2002) 57 –76

[25]. Kaufmann A., Gupta M. Introduction to fuzzy arithmetic-theory and applications. - New York: Van Nostrand Reinhold, 1985.

[26]. Moore R.E. (1966). Interval analysis. - Englewood Cliffs. N.J.: Prentice-Hall,

27]. Jaulin L., Kieffir M., Didrit O., Walter E. (2001) Applied Interval Analysis. Springer-Verlag, London.

[28]. Yager  R.A.(1979). " On the measure of fuzziness and negation. Part 1. Membership in the  Unit Interval " . Int. J. Gen. Syst,. 5, 221-229.

[29]. Sevastianov P., Dimova L.,Zhestkova E. Methodology of the multicriteria quality estimation  and its software realizing. Proceedings of the Fourth International Conference on New Information Technologies NITe', Minsk 2000. V. 1, p.50-54.

[30]. Zimmermann H.-J., Zysno P. (1980) . Latest connectives in human decision making. Fuzzy Sets and Systems. 4, 37-51.