Il working paper è inserito nell'archivio RePEc.
http://ideas.repec.org/p/vnm/wpaper/170.htm

CORAZZA M.

GIOVE S

Archivio Ricerca Ca'Foscari

Fuzzy Interval NEt present value

The evaluation of investment projects is an ever-present problem in everyday firm life and in several economic research fields. In this paper we propose a model in which we conjugate the operative usability of the net present value with the capability of the fuzzy and the interval approaches to manage uncertainty. In particular, our fuzzy interval net present value can be interpreted, besides the usual present value of an investment project, also as the present value of a contract in which the buyer lets the counterpart the possibility to release goods/services for money amounts that can vary in prefixed intervals, at time instants that can also vary in prefixed intervals. Given this interpretation, it is
"natural" to represent the good/service money amounts and the time instants by means of triangular fuzzy numbers, and the cost of the buyer as a strictly-increasing function of the level alpha belonging to [0; 1] associated to the generic cut of the fuzzy interval net present value. In her/his turn, the buyer can reduce the widths of both these kinds of intervals by paying a proper cost. Of course, the buyer is characterized by an utility function
that, in our case, depends on the features of the net present value of the contract and negatively depends on the cost. So, the buyer has to determine the optimal value of ® which maximizes her/his utility. As far the interest rates regard, noticing that usually the economic operators are ignorant of the future, we assume that they are only able to specify a variability range for each of the considered time period interest rate.
This is why we represent the interest rates by means of interval numbers. Besides proposing our model, we formulate and solve the nonlinear optimization programming problems which have to be coped with in order to determine the extremals of the generic cut of the fuzzy interval net present value, and deal with some questions related to the utility function of the buyer.The evaluation of investment projects is an ever-present problem in firm life and in several economic research fields. In this paper we propose a model in which we conjugate the operative usability of the net present value with the capability of the fuzzy and the interval approaches to manage uncertainty. In particular, our fuzzy interval net present value can be interpreted, besides the usual present value of an investment project, as the present value of a contract in which the buyer lets the counterpart the possibility to release goods/services for money amounts that can vary in prefixed intervals, at time instants that can also vary in prefixed intervals. In her/his turn, the buyer can reduce the widths of both these kinds of intervals by paying a proper cost. Given this interpretation, it is “natural” to represent the good/service money amounts and the time instants by means of triangular fuzzy numbers, and the cost of the buyer as a strictly-increasing function of the level α ∈ [0, 1] associated to the generic cut of the fuzzy interval net present value. Of course, the buyer is characterized by an utility function that, in our case, depends on the features of the net present value of the contract and on the cost. So, the buyer has to determine the optimal value of α which maximizes her/his utility. As far the interest rates regard, noticing that usually the economic operators are ignorant of the future, we assume that they are only able to specify a variability range for each of the considered period interest rate. This is why we represent the interest rates by means of interval numbers. Besides proposing our model, we formulate and solve the nonlinear optimization programming problems which have to be coped with in order to determine the extremals of the generic cut of the fuzzy interval net present value, and we deal with some questions related to the utility function of the buyer

CORAZZA, Marco

GIOVE, Silvio

Archivio istituzionale della ricerca - Università degli Studi di Venezia Ca' Foscari

Fuzzy interval net present value

Il working paper è inserito nell'archivio RePEc.
http://ideas.repec.org/p/vnm/wpaper/170.htmlIn this paper we conjugate the operative usability of the net present value with the capability of the fuzzy and the interval approaches to manage uncertainty. Our fuzzy interval net present value can be interpreted, besides the usual present value of an investment project, as the present value of a contract in which the buyer lets the counterpart the possibility to release goods/services for money amounts that can vary, at time instants that can also vary. The buyer can reduce the widths of these variations by paying a cost. So, it is "natural" to represent the good/service money amounts  and the time instants by means of triangular fuzzy numbers, and the cost of the buyer as a strictly-increasing function of the level α ε [0; 1] associated to the generic cut of the fuzzy interval net present value. As usual, the buyer is characterized by an utility function, depending on α and
on the cost, that he/she has to maximize. As far the interest rates regard, we assume that the economic operators are only able to specify a variability range for each of the considered period interest rate. So, we represent the interest rates by means of interval numbers. Besides proposing our model, we formulate and solve the programming problems which have to be coped with to determine the extremals of the cut of the fuzzy interval net present value, and we deal with some questions related to the utility function of the buyer

Department of Applied Mathematics,  University of Venice     WORKING PAPER SERIES          Marco Corazza and Silvio Giove   Fuzzy interval net present value          Working Paper n. 170/2008 November 2008  ISSN: 1828-6887                                                This Working Paper is published under the auspices of the Department of Applied Mathematics of the Ca’ Foscari University of Venice. Opinions expressed herein are those of the authors and not those of the Department. The Working Paper series is designed to divulge preliminary or incomplete work, circulated to favour discussion and comments. Citation of this paper should consider its provisional nature. Fuzzy Interval Net Present ValueMarco Corazza Silvio Giove<corazza@unive.it> <sgiove@unive.it>Dept. of Applied Mathematics Dept. of Applied MathematicsCa’ Foscari University of Venice Ca’ Foscari University of VeniceandAdvanced School of Economics(November 2008)Abstract. In this paper we conjugate the operative usability of the net present valuewith the capability of the fuzzy and the interval approaches to manage uncertainty. Ourfuzzy interval net present value can be interpreted, besides the usual present value of aninvestment project, as the present value of a contract in which the buyer lets the counterpartthe possibility to release goods/services for money amounts that can vary, at time instantsthat can also vary. The buyer can reduce the widths of these variations by paying a cost.So, it is “natural” to represent the good/service money amounts and the time instantsby means of triangular fuzzy numbers, and the cost of the buyer as a strictly-increasingfunction of the level α ∈ [0, 1] associated to the generic cut of the fuzzy interval net presentvalue. As usual, the buyer is characterized by an utility function, depending on α andon the cost, that he/she has to maximize. As far the interest rates regard, we assumethat the economic operators are only able to specify a variability range for each of theconsidered period interest rate. So, we represent the interest rates by means of intervalnumbers. Besides proposing our model, we formulate and solve the programming problemswhich have to be coped with to determine the extremals of the cut of the fuzzy interval netpresent value, and we deal with some questions related to the utility function of the buyer.Keywords: Net present value, fuzzy set theory, interval number theory, α–cut, utilityfunction.JEL Classification Numbers: C61, G19.MathSci Classification Numbers: 90C99.Correspondence to:Marco Corazza Dept. of Applied Mathematics, Ca’ Foscari University of VeniceDorsoduro 3825/E30123 Venezia, ItalyPhone: [++39] (041)-234-6921Fax: [++39] (041)-522-1756E-mail: corazza@unive.it1 IntroductionThe correct evaluation of investment projects is an ever-present problem in everyday firm lifeand in several economic research fields. In order to deal with this problem, many approachesand tools have been developed, from the classical deterministic net present value (NPV)1and internal interest rate (IIR) to the refined stochastic real options.Of course, a central question concerns the capability of the considered approaches/toolsto manage the ignorance of the future. Whereas the classical models simply do not considerthis aspect (with conceivable consequences), on the other hand the majority of the stochasticapproaches face it in a too much technical way for real-life applications. Because of that,several academic and professional researches presented and present intermediate tools.In this paper we propose a model of this latter kind, in which we conjugate the operativeusability of the NPV with the capability of the fuzzy and the interval approaches to manageuncertainty. It is important to notice that we have chosen to not deal with uncertainty bymeans of probability and stochastic tools because, coherently with the real life, these toolsare usually beyond the knowledge and the computational ability of standard economicoperators.The literature concerning this topic is rich enough. Some of the first models werepresented in [2], in which fuzzy cash amounts, a fuzzy interest rate, and a fuzzily definednumber of time periods are used. These models are mainly investigated from a theoreticalpoint of view. More operative approaches are proposed in [3] and [4]. In the former, atriangular fuzzy NPV is used for fuzzifying the Myers-Cohn model of property-liabilityinsurance pricing. In the latter, an NPV based on triangular fuzzy cash amounts andtriangular fuzzy interest rates is used for specifying dynamic replacement model. In [5], afuzzy NPV-based procedure is considered as an effective tool for the selection of aviationtechnologies. In [1], a triangular fuzzy NPV is introduced as intermediate step for thedetermination of the related fuzzy IRR. In [7], trapezoidal fuzzy cash amounts, crisp interest1A commonly used formulation of the NPV isPnj=0 Sj.Qjk=0 (1 + ik) , where n is the number of (equallength) time periods, Sj is the cash amount at time instant j, and ik is the interest rate during the timeperiod (k − 1, k].1rates, and probabilistic methods are jointly used for developing a fuzzy-probabilistic NPValgorithm for the evaluation of investment projects.Of course, this review does not intend to be exhaustive but only exemplificative.With respect to the cited contributions, and the referred ones therein, our model differsfrom a technical standpoint, from the economic interpretability, and from its usability.In short, the NPV we detail in section 2 can be interpreted, besides the usual presentvalue of a generic investment project, as the present value of a contract in which the buyerlets the counterpart the possibility to release goods/services for money amounts that canvary in prefixed intervals, at time instants that can also vary in prefixed intervals. In her/histurn, the buyer can reduce the widths of both these kinds of intervals by paying a propercost. Given this interpretation of the NPV, also coherently with the cited contributions andwith several other fuzzy modelings, it is “natural” enough to represent the good/servicemoney amounts and the time instants by means of triangular fuzzy numbers, and the costof the buyer as a strictly-increasing function of the level α ∈ [0, 1] associated to the genericcut of the fuzzy interval NPV.Of course, the buyer – as any economic operator – is characterized by an utility functionthat, in our case, depends on the features of the NPV of the contract and negatively dependson the cost. So, recalling that all these quantities depend on α, the buyer has to determinethe optimal value of such an α which maximizes her/his utility.As far the interest rates regard, recalling that the economic operators involved in theinvestment project/contract are ignorant of the future, we reasonably assume that they areonly able to specify a suitable variability range for each of the considered period interestrate. This is why we represent the interest rates by means of interval numbers. Moreover,it is to notice that, to the best of our knowledge, ours is the first model in which thetime instants in which one releases the cash amounts/goods/services are fuzzified, and theinterest rates are represented in terms of interval numbers.The remainder of the paper is organized as follows. In the next session we propose ourFuzzy Interval Net Present Value (FINe) model and some of its main features. In section 32we formulate and solve the nonlinear optimization programming problems which have to becoped with in order to determine the extremals of the generic cut of the FINe. In section4 we briefly deal with some questions related to the utility function of the buyer. Finally,in section 5 we list some open items.2 The FINe modelLet we start by specifying the discrete-time frame associated to the considered investmentproject/contract:t0, t˜1, . . . , t˜j , . . . , t˜n,wheret0 is the current time instant, which we represent by a crisp number;t˜1, . . ., t˜j , . . ., t˜n are the future time instants in which the cash amounts/goods/servicesare released, that we represent by triangular fuzzy numbers;2 as known, the membershipfunctions of this typology of numbers areµetj (t) =0 if t < tj,1(t− tj,1) /(tj,2 − tj,1) if tj,1 ≤ t < tj,2(tj,3 − t) /(tj,3 − tj,2) if tj,2 ≤ t < tj,30 if t ≥ tj,3, with j = 1, . . . , n,in which tj,1, tj,2 and tj,3 are real numbers such that tj,1 ≤ tj,2 ≤ tj,3 (a commonly usednotation for this kind of numbers is t˜j = (tj,1, tj,2 , tj,3));n is the number of time periods, which we represent by a crisp number.In particular, we assume that, as in standard NPV models, the time instants in whichthe cash amounts/goods/services are released constitute a non decreasing sequence withrespect to the deponent, that is we assume thatt0 ≤ t1,1 and tj,3 ≤ tj+1,1, with j = 1, . . . , n− 1. (1)2Throughout the paper, we use the overwriting e· for indicating fuzzy numbers. Moreover, for notationalconvenience, thereafter we represent the crisp number t0 in terms of the (degenerate) triangular fuzzy numberet0 = (t0, t0, t0).3Let we continue by specifying the fuzzy investment project/contract:A˜ ={(−S0, t˜0) ,(S˜1, t˜1) , . . . ,(S˜j , t˜j) , . . . ,(S˜n, t˜n)} ,whereS0 > 0 is the current cash amount/good/service value, which we represent by a crispnumber;S˜1, . . . S˜j , . . . , S˜n are the future positive cash amount/good/service values, which we repre-sent by positive triangular fuzzy number.3Finally, let we assume the following time period frame associated to the interest ratestructure:(τ1, τ2] , (τ2, τ3] , . . . , (τj−1, τj ] , . . . , (τn, τn+1] , (2)whereτ1 = t0, τ2 ∈ (t1,3, t2,1], . . ., τj ∈ (tj−1,3, tj,1],. . ., τn ∈ (tn−1,3, tn,1], τn+1 > tn,3 are the crisptime instants in which the interest rates can change.4It is important to notice that, by construction, such time instants constitute a strictly-increasing sequence with respect to the deponent, that is thatτ1 < τ2 < . . . < τj < . . . < τn. (3)Of course, it is possible to choose more articulated time period frames than the onespecified by (2) and (3). In case of such a choice, one should obtain a model which shouldbe more difficult to algebraically treat than ours, but that at the while should not givesignificant improvements from the economic meaning standpoint. This is why we opt forthe proposed time period frame.Given this notation, we formulate as follows our FINe model:3A triangular fuzzy number ex = (x1, x2, x3) is defined positive if x1 > 0.4It is to notice that, in general, this (crisp) discrete-time frame differs from the (fuzzy) one associated tothe investment project/contract.4FINe(A˜)= −S0 +n∑j=1S˜j(1 + îj)−(etj−τj) j−1∏k=1(1 + îk)−(τk+1−τk), (4)whereî1, . . ., îj , . . ., în are the current and the future positive interest rates, which we represent bypositive interval numbers;5 as known, the membership functions of this typology of numbersareµbij (i) =0 if i < ij,11 if ij,1 ≤ i ≤ ij,20 if i > ij,2, with j = 1, . . . , n,in which ij,1 and ij,2 are real numbers such that ij,1 ≤ ij,2 (a commonly used notation forthis kind of numbers is îj = (ij,1, ij,2)).63 Determination of the α–cuts for the FINe modelOnce the NPV of A˜ is formulated in terms of fuzzy and interval numbers, we go on to thedetermination of the shape of this number. In order to carry out that, we have to determinethe extremals of the α–cuts, with α ∈ [0, 1], of (4), that is the minimum and the maximumof the sets:XFINe(eA),α ={x : µFINe(eA)(x) ≥ α},that we respectively indicate by FINe(A˜)(α)L and FINe(A˜)(α)R.In their turn, FINe(A˜)(α)L and FINe(A˜)(α)R depend on the values the argumentsof (4) take in their corresponding α–cuts. Therefore, in order to determine the extremalsof XFINe(eA),α we have to cope with the following nonlinear optimization programmingproblems:5Throughout the paper, we use the overwriting b· for indicating interval numbers.6An interval number bx = (x1, x2) is defined positive if x1 > 0.5min /max −S0 +n∑j=1Sj (1 + ij)−(tj−τj)j−1∏k=1(1 + ik)−(τk+1−τk)S1,...,Sn,i1,...,in,t1,...,tns.t.Sj,L(α) ≤ Sj ≤ Sj,R(α), with j = 1, . . . , n,ij,L(α) ≤ ij ≤ ij,R(α), with j = 1, . . . , n,tj,L(α) ≤ tj ≤ tj,R(α), with j = 1, . . . , n,, (5)whereSj,L = Sj,1+(Sj,2 − Sj,1)α is the minimum of the α–cut of the triangular fuzzy number S˜j ;Sj,R = Sj,3 + (Sj,3 − Sj,2)α is the maximum of the α–cut of the triangular fuzzy numberS˜j ;ij,L = ij,1 is the minimum of the α–cut of the interval number îj ;ij,R = ij,2 is the maximum of the α–cut of the interval number îj ;tj,L = tj,1 + (tj,2 − tj,1)α is the minimum of the α–cut of the triangular fuzzy number t˜j ;tj,R = tj,3 + (tj,3 − tj,2)α is the maximum of the α–cut of the triangular fuzzy number t˜j .It is important to notice that the minimum programming problem is related to thedetermination of FINe(A˜)(α)L, and that the maximum programming problem is relatedto the determination of FINe(A˜)(α)R.After some calculations and some arrangements, the following partial derivatives of thetarget function (TF) of (5) with respect to Sj , ij , and tj are obtained:∂∂SjTF = (1 + ij)−(tj−τj)j−1∏k=1(1 + ik)−(τk+1−τk) , with j = 1, . . . , n;∂∂ijTF = − (tj − τj) (1 + ij)−(tj−τj−1)Sjj−1∏k=1(1 + ik)−(τk+1−τk)−− (τj+1 − τj) (1 + ij)−(τj+1−τj−1)n∑k=j+1Sk (1 + ik)−(tk−τk) ··k−1∏l=1,l 6=j(1 + il)−(τl+1−τl), with j = 1, . . . , n;∂∂tjTF = − (1 + ij)−(tj−τj) ln (1 + ij)Sjj−1∏k=1(1 + ik)−(τk+1−τk), with j = 1, . . . , n.6Given the positions (1), the results (3), and the positivity of the cash amount/good/ser-vice values and of the interest rates, it is easy to verify that ∂∂SjTF > 0 for all j, that∂∂ijTF < 0 for all j, and that ∂∂tjTF < 0 for all j.At this point, given the strict monotony of the TF of (5) with respect to Sj , ij and tj ,and given the latter results, for each α ∈ [0, 1] we can determine as follows the respectivepoints of optimum of the programming problems (5):Minimum programming problem Maximum programming problemS∗j,min = Sj,L(α) == Sj,1 + (Sj,2 − Sj,1)α,with j = 1, . . . , n;S∗j,max = Sj,R(α) == Sj,3 + (Sj,3 − Sj,2)α,with j = 1, . . . , n;i∗j,min = ij,R(α) = ij,2,with j = 1, . . . , n;i∗j,max = ij,L(α) = ij,1,with j = 1, . . . , n;t∗j,min = tj,R(α) == tj,3 + (tj,3 − tj,2)α,with j = 1, . . . , n;t∗j,max = tj,L(α) == tj,1 + (tj,2 − tj,1)α,with j = 1, . . . , n;Of course, exploiting the usual extension principle (for details see [6]):FINe(A˜)(α)L = −S0 +n∑j=1S∗j,min(1 + i∗j,min)−(t∗j,min−τj) ··j−1∏k=1(1 + i∗k,min)−(τk+1−τk)andFINe(A˜)(α)L = −S0 +n∑j=1S∗j,max(1 + i∗j,max)−(t∗j,max−τj) ··j−1∏k=1(1 + i∗k,max)−(τk+1−τk).It is to notice that, if α = 1, then S∗j,min = S∗j,max = Sj,2, i∗j,min = ij,1 < i∗j,max = ij,2and t∗j,min = t∗j,max = tj,2. So, the width of the 1–cut of FINe(A˜)depends only on theextremals of the 1–cut of îj .74 About the buyer’s utility functionAs premised, in case (4) is the FINe of a contract like the one described in the introduction,the utility function of the buyer depends on the features of this FINe and negatively dependson the cost the buyer has to pay for reducing the widths of the α–cuts related to S˜j and t˜j ,that is for reducing the width of the α–cut related to FINe(A˜). Recalling that all thesequantities depend on α, in order to determine the optimal value of α itself which maximizesthe buyer’s utility, at first we have to detail such a function.Coherently with the rationality of the generic economic operator, for each α we assumethat the buyer’s utility function positively depends on some return measure of the consideredα–cut (that we indicated by r(α)), negatively depends on some variability measure of theconsidered α–cut (that we indicated by v(α)), and negatively depends on the cost c(α). Inparticular, for these quantities we propose the following specifications:r(α) =FINe(A˜)(α)L + FINe(A˜)(α)R2,v(α) = FINe(A˜)(α)R − FINe(A˜)(α)Landc(α) = bα ∈ [0, b], with b > 0.So, we can indicate the buyer’s utility function as U (r(α), v(α), c(α)).It is important to notice that v(α) and c(α) are, respectively, a strictly-decreasingfunction and a strictly-increasing function of the level α, whereas the monotony of r(α)depends of the shape of FINe(A˜). In fact, it is easy to verify that r(α) is a strictly-decreasing/constant/strictly-increasing function of α according to the occurrence thatFINe(A˜)is a right-skewed/symmetric/left-skewed fuzzy interval number with respectto r(1) (see figure 1).77A fuzzy interval number ebx is defined right-skewed/symmetric/left-skewed with respect to r(1) if r(α) </ = / > r(1) for all α ∈ [0, 1).8Figure 1: From up to down: a right-skewed/symmetric/left-skewed fuzzy interval numberwith respect to r(1). In all the figures, the continuous line, the dashed line, and the dottedline represent, respectively, the shape of the considered fuzzy interval number, of its r(α),and of its r(1).At this point, in order to determine the optimal value of α which maximizes the buyer’sutility we have to deal with the following nonlinear programming problem:max U (r(α), v(α), c(α))α {c(α) < FINe(A˜)(α)L0 ≤ α ≤ 1. (6)It is to notice that, due to the first constraint, the non emptiness of the feasible regionof (6) is not ensured.5 Some open itemsGiven to its youth, our approach leaves open interesting items that we intend to investigatein future researches. Among the ones we consider the most significant, we list the following:9– specifying some analytical/numerical procedure by which to determine the optimalvalue of the level α which maximizes the buyer’s utility;– extending our approach in order to be able to apply it, besides to investment projects/contracts, to whatever financial operation;– given two, or more, fuzzy financial operations/contracts, specifying some methodologyby which to order them on the basis of suitable comparisons of their respective FINe.References[1] I.E.G. Banholzer, A.L. Sanches, E.O. Pamplona and J.A.B. Montevechi, An interpre-tation for the internal rate of returns through fuzzy calculation, Proceedings of theXII Congress of International Association for Fuzzy–Set Management and Economy(2005), 33–51[2] J.J. Buckley, The fuzzy mathematics of finance, Fuzzy Sets and Systems 21 (1987),257–273.[3] J.D. Cummins and R.A. Derrig, Fuzzy financial pricing of property-liability insurance[and Discussions and Authors’ reply], North American Actuarial Journal 4 (1997) 21–44.[4] A.O. Esogbue and W.E. II Hearnes, On replacement models via a fuzzy set theoreticframework, IEEE Transaction on Systems, Man, and Cybernetics – Part C: Applica-tions and Reviews 28 (1998) 549–560.[5] O. Houseman, A. Tiwari and R. Roy, A methodology for the selection of new tech-nologies in the aviation industry, Decision Engineering Report Series of the CranfieldUniversity (2004) i–iii and 1–36.[6] A. Kaufmann, Introduction to the Theory of Fuzzy Subsets. Volume I, Academic Press,New York, NY, 1975.[7] C.-T. Tsao, Assessing the probabilistic fuzzy net present value for a capital investmentchoice using fuzzy arithmetic, Journal of the Chinese Institute of Industrial Engineers22 (2005) 106–118.10

Archivio Istituzionale della Ricerca

The evaluation of investment projects is an ever-present problem in everyday firm life and in several economic research fields. In this paper we propose a model in which we conjugate the operative usability of the net present value with the capability of the fuzzy and the interval approaches to manage uncertainty. In particular, our fuzzy interval net present value can be interpreted, besides the usual present value of an investment project, also as the present value of a contract in which the buyer lets the counterpart the possibility to release goods/services for money amounts that can vary in prefixed intervals, at time instants that can also vary in prefixed intervals. Given this interpretation, it is\ud
"natural" to represent the good/service money amounts and the time instants by means of triangular fuzzy numbers, and the cost of the buyer as a strictly-increasing function of the level alpha belonging to [0; 1] associated to the generic cut of the fuzzy interval net present value. In her/his turn, the buyer can reduce the widths of both these kinds of intervals by paying a proper cost. Of course, the buyer is characterized by an utility function\ud
that, in our case, depends on the features of the net present value of the contract and negatively depends on the cost. So, the buyer has to determine the optimal value of ® which maximizes her/his utility. As far the interest rates regard, noticing that usually the economic operators are ignorant of the future, we assume that they are only able to specify a variability range for each of the considered time period interest rate.\ud
This is why we represent the interest rates by means of interval numbers. Besides proposing our model, we formulate and solve the nonlinear optimization programming problems which have to be coped with in order to determine the extremals of the generic cut of the fuzzy interval net present value, and deal with some questions related to the utility function of the buyer

GIOVE S.

Name not available

The evaluation of investment projects is an ever-present problem in everyday firm life and in several economic research fields. In this paper we propose a model in which we conjugate the operative usability of the net present value with the capability of the fuzzy and the interval approaches to manage uncertainty. In particular, our fuzzy interval net present value can be interpreted, besides the usual present value of an investment project, also as the present value of a contract in which the buyer lets the counterpart the possibility to release goods/services for money amounts that can vary in prefixed intervals, at time instants that can also vary in prefixed intervals. Given this interpretation, it is

"natural" to represent the good/service money amounts and the time instants by means of triangular fuzzy numbers, and the cost of the buyer as a strictly-increasing function of the level alpha belonging to [0; 1] associated to the generic cut of the fuzzy interval net present value. In her/his turn, the buyer can reduce the widths of both these kinds of intervals by paying a proper cost. Of course, the buyer is characterized by an utility function

that, in our case, depends on the features of the net present value of the contract and negatively depends on the cost. So, the buyer has to determine the optimal value of ® which maximizes her/his utility. As far the interest rates regard, noticing that usually the economic operators are ignorant of the future, we assume that they are only able to specify a variability range for each of the considered time period interest rate.

This is why we represent the interest rates by means of interval numbers. Besides proposing our model, we formulate and solve the nonlinear optimization programming problems which have to be coped with in order to determine the extremals of the generic cut of the fuzzy interval net present value, and deal with some questions related to the utility function of the buyer

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Fuzzy Interval NEt present value

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