CAPITAL BUDGETING USING TRIANGULAR FUZZY ... - IEPG - Unifei

Sanches, Alexandre Leme; Pamplona, Edson de O. e Montevechi, José Arnaldo B. Capital Budgeting Using Triangular Fuzzy Numbers. V Encuentro Internacional de Finanzas. Santiago, Chile, 19 a 21 de janeiro de 2005

CAPITAL BUDGETING USING TRIANGULAR FUZZY NUMBERS Alexandre Leme Sanches Prof. M.Sc. IEPG – Universidade Federal de Itajubá (UNIFEI) E-mail: [email protected]

Edson de Oliveira Pamplona Prof. Dr. IEPG – Universidade Federal de Itajubá (UNIFEI) E-mail: [email protected]

José Arnaldo Barra Montevechi Prof. Dr. IEPG – Universidade Federal de Itajubá (UNIFEI) E-mail: [email protected]

Summary: The economic engineering the analysis involves uncertainty about future cash flows. To deal quantitatively with imprecision or uncertainty, fuzzy set theory is primarily concerned with vagueness in human thoughts and perceptions. As an alternative to conventional cash flow models, where cash flows are defined as either crisp numbers or vagueness, is proposed an engineering economic decision model in which the uncertain cash flow and discount rates, specified as triangular fuzzy numbers. The present value formulation of this fuzzy cash flow model is derived. The result of the present value is also a fuzzy number with nonlinear membership function. The present value can be approximated by a triangular fuzzy number. Risk analysis involves the development of the probability distribution for the measure of effectiveness. The uncertainty associated with an investment alternative is generally either given as the “possibility” of an unfavorable value of the measure of effectiveness or measured by the variance of the measure of effectiveness. Keywords: Investment evaluation; Fuzzy; Capital budgeting.

Code JEL: D81 - Criteria for Decision-Making under Risk and Uncertainty

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Capital Budgeting Using Triangular Fuzzy Numbers

INTRODUCTION Nowadays, the uncertainties associated with “investment analyses”, in all areas, create a demand for alternatives methods to make possible the translation, to a mathematical language, of the intangible values and human experience, improving the available resources in the decision making process. Therefore, it becomes necessary the application of an investment evaluation method which represents, in a realistic fashion, the economic viability of a given investment subject to uncertainties. One of the main problems found in the investment analyses is the measurement of the analyses key variables. Most of the time, the evaluation of the numerical values of these variables is directly associated with the analyst’s own abstraction. Another problem common problem, and also of great relevance, is how people operate with uncertain values and abstract ones, for, if the intuition of a uncertain value is complex, working with them, intuitively, becomes an impractical task. Fuzzy logic has the capability to represent, numerically, linguistic va lues, uncertain and abstract, aiding significantly to the decision making process in investment analyses. Also, it has the capability to work with those values, being, therefore, the resource herein explored. This paper is about, specifically, the use of Fuzzy logic in the financial field, in what concerns the uncertainties associated with predictions of future situations. Such predictions are input data, in the calculation of fuzzy NPV (Net Present Value). When it comes to the financial field, the user’s skill in the utilization of fuzzy logic might enable such tool to surpass the deterministic methods or even the stochastic ones, for its independence from the historical data to be successfully used. OBJECTIVES ? ?

This paper has as its main objective the demonstration of the proper use of fuzzy logic in the evaluation of investment projects, and its capacity to provide relevant information towards the decision making process under uncertain conditions. The secondary objective is the representation of a software prototype to calculate the fuzzy NPV and analyses relative to the enterprise.

METHODOLOGICAL ASPECTS The research method to be used is known as “quasi-experiment”. According to Bryman (1989), the quasi-experiment is a research experience where the researcher doesn’t have total control over the input variables of the system and there’s a non-random treatment of the experiment. According to Trochin (2001), the quasi-experiment is similar to the experiment, nonetheless there is no random designation. In what concerns internal validation, the quasiexperiment is inferior regarding the experimental method, considered to be the highest internally validated research method. Sill according to Trochin (apud Gonçalves (2003)), the quasi-experiment is a method which incorporates a great part of the quantitative research works where the human behavior is present, presenting various scopes. In the current paper it is adopted the Proxy Pretest Design, which is characterized by the realization of a pre-test and one post-test after the program has been implemented. This is possible through the estimation of the studied variables for the group, before the beginning of the program. Here it will be made an application of the deterministic method and latter the application of the possibilistic method, finally both results will be compared.

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Investment Data (selected group) Deterministic NPV Calculation – viability (pre -test) Sensibility Analyses (uncontrolled)

M e t h o d

Definition of the variables to be Fuzzyfied Fuzzyfication of the selected variables (specialist) Fuzzy NPV Calculation

F u z z y

L o g i c

Viability and possibilities analyses associated with the Fuzzyfied NPVs (post-test). Defuzzyfication of the NPV (if necessary)

Comparison with the Deterministic NPV - The Proxy Pretest Design Figure 1 – Job/Work Sequence

Notwithstanding, it is possible to generalize certain research results, is terms of the investment evaluation method used, for there is no great differences in how the models is applied in other cases. This paper has as its focus the “evaluation method” and not the “subject to be evaluated”. Thus, the limitations mentioned above do not reduce, in an significant way, the capacity of exploration and further attest the subject. LITERATURE REVISION The Fuzzy Logic Fuzzy logic is a bridge which connects the human thinking to the machine’s logic. In a fuzzy set, the transitions between a member or a non-member occur in a continuous ray, being a membership degree associated between “0” (totally non- member) and “1” (totally member). The degree of membership “is not probability”, but a measure of compatibility between object and the concept represented by the fuzzy set. The concept of membership function is further better explained. The uncertainty and subjectivity manipulation capacity, without the rigid barriers from the classic logic, characterize the fuzzy numbers, in Martinez Jr. (2002) viewpoint.

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The Fuzzy Numbers Fuzzy numbers are a subset from the real numbers set, representing the uncertain values. All fuzzy numbers are related to degrees of membership which state how true it is to say if something belongs or not to a determined set. Membership Function Source: Montevechi, J. A; Revisão Sobre Lógica Fuzzy A diffuse set, cloudy or fuzzy “A” of an “E” universe of characterized by a membership function ? A:E ? [0,1], which associates to each “x” from “E” a number ? A(x) in the [0,1] interval, representing the degree of membership of “x” in “A”. To ? A(x) it is given the name of membership function or simply membership, as it is called in most publications. To ? A(x) near “1”, it is said that there is a high possibility that “x” ? “A”. On the other hand when ? A(x) is close to “0”, there is a low possibility that “x” ? “A”. As a fuzzy set has a certain number of affinities relative to the “x” elements which constitute it, it is feasible here to show how they must be represented, once the way it is shown in the literature is completely particular. A = [(x, ? A(x))] x ? E

(1)

According to Von Altrock (1995), to a “A” set, the function mA(x) “membership” is defined as: mA(x) = 1 if and only if x ? A

(2)

0 if and only if x ? A mA(x) ? [0,1] ? x’ ? [x1 , xn ]

Analyzing the following diagram, where the left graph shows the Boolean logic, and the one in the right shows, yet in a discrete fashion, the fuzzy logic, it can be stated that in the Boolean logic the degree of pertinence/belonging of an element regarding a set is 0 or 1, which means, the elements “d” and “b” belong to a set “A” (degree of pertinence / belonging = 1), while the elements “c” and “d” do not belong to set “A” (degree of pertinence/belonging = 0). Now, in the fuzzy logic there may be an element “c” which partially belong to set “A” (0 ? degree of pertinence/belonging ? 1), that is, mA(x)= 0,5.

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A

A

a a

c

b

b c

d

d ?A

?A

1

1 0.5

a

b

c

d

x

a

Boolean Logic (binary)

b

c

d

x

Fuzzy Logic

Figure 2 –“Membership” function Source: http://if.kaist.ac.kr/lecture/cs670/lecture- note/Chapter1.ppt

The Many Forms of Fuzzy Numbers Source: http://if.kaist.ac.kr/lecture/cs670/lecture- note/Chapter5.ppt There are various types of fuzzy numbers and its nomenclature is, in general, associated with its format, such as: sine numbers, bell shape, polygonal, trapezoids, triangular, and son on. In this paper it is emphasized the triangular fuzzy numbers, once the are the most interesting for the financial field. General Shape of the Fuzzy Number In Chiu and Park (1994), a fuzzy number is a fuzzy subset characterized by a Membership function, which satisfies the following conditions: ? Normality: ? A(x) = 1, for, at least, each/one point/dot of x ? R ? Convexity: ? A(x’) = ? A(x 1 ) ? ? A(x2 ), where: ? A(x) ? [0 , 1] and ? x’? [x1 , x 2] Adding to Kuchta (1996), where a fuzzy number is an element of hexadimentional quantification in the following manner:

nf = (a1 ,a2 ,a3 ,a4 , f a (?), f a (?)) 1

(3)

2

5


Where: a1, a2, a3, a4, are real numbers and a1 ? a2? a3 ? a4 f 1a (?) is a continuous real function non decreasing defined in the interval [0, 1] such that: f 1a ( 0 ) = a1 and f 1a ( 1 ) = a2 f 2a (?) is a continuous real function non increasing defined in the interval [0, 1] such that: f 2a ( 1 ) = a3 , and f 2a ( 0 ) = a4

? A(x)

nf = (a1,a2,a3,a4, f a , f a ) 1 2

1

?

?

f1a

f 2a

0 a1

x a2

a4

Figure 3 – General Forma t of the Fuzzy Number Triangular Fuzzy Numbers (TFN) a a If f 1 (x) e f2 (x) are linear functions and also “a2 = a 3 ”, One can omit “a2 “ or “a3 ”, then the membership function takes the following shape:

?0, ? x - a 1 , ? ?? a - a 1 µ( A ) ( x ) = ? 2 a3 - x ? , ? a3 - a2 ? ?0,

x < a1 a1 ? x ? a2

(4)

a2 ? x ? a3 x > a3

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? (x) 1

0 a1

a2

x

a3

Figure 4 –Triangular Fuzzy Number Structure

The Fuzzyfication Fuzzyfication is the mapping of the real numbers domain (generally discrete) to the fuzzy domain. Also it represents linguistic values assignments, vague descriptions or qualitative ones, defined by a membership function to the various input variables. Figure 5 displays the fuzzyfication of the ROR “universe” for a determined company.

ROR Very Low

Low

Medium

High

Very High

1

0

5

10

15

20

25

30

35

40

%

Figure 5 - Fuzzyfication Source: Montevechi (1998)

The Defuzzyfication

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Defuzzyfication is the operation/proceeding in which the value of the output linguistic value, inferred by the fuzzy rules/regulations, will be translated to a discrete value (Shaw, 1999). Figure 6 show the “universe” of the NPV’s classification for the investments of a given company, where the real discrete value is obtained from the fuzzy result.

NPV

Bad

0 1000 (R$)

Regular

Good

3000

Great

5000

Excellent

7000

9000

Figure 6 – Defuzzyfication Source: Montevechi (1998)

Traditional Investment Evaluation Methods The investment eva luation methods, most commonly used and that have wider publicity, involve the basic model of Discounted Cash Flow (DCF), with its main variants: NPV (Net Present Value) and IRR (Internal Rate of Return) (Santos 2001). The traditional methods, initially, may be seen as limited, when it comes to uncertainty dimension, nevertheless they are the basis for the development of sophisticated techniques which, at the present time, have been used with great success. The Net Present Value (NPV) According to Santos (2001), using financial mathematics, it is possible to carry a viability analyses of the project. Thus it is necessary that a forecast of all future cash flows is made for all the “n” future periods. This forecast must be made with as much accuracy as possible, taking into which will it be the outflows and inflows in the next “n” periods, then, with a predetermined rate, the ROR (rate of return), in order to obtain the value of that project at period zero, where these values are added to the initial investme nt. The NPV method has its basic format synthesized by the following equation:

(5)

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n

NPV

=?

i? 0

CFi (1 ? r ) i

Where: ?

CF i = Cash Flow, expected for the given i period;

?

r = discount rate

?

i = 0, 1, 2, 3, ... , n (periods).

The existence of a positive NPV is defined as the basic criteria in the acceptance or rejection of the determined project (Ross, Westerfield, 1995). Evaluation of the Investments under Uncertainty Conditions Using Fuzzy Triangular Numbers In this topic it is presented the “possibilistic” method, based in Buckley (1987) and Souza (1996), which use the Fuzzy set theory. The membership function for NPV (Fuzzy), presented by Buckley (1987), is given:

n -j f n,i ((y)P)= ? f j,i ((y)F j )(1+ f k(j) ((y)r f )) j=0

(6)

For: i = 1, 2, where k = i for negative F and k = 3 - i for positive F (1) left side (2) right side

REAL CASE APPLICATION In the current case, a classic problem of investment viability under uncertainty conditions evaluation occurs, in a single investment (no alternatives) scenario. The decision, here, is to accept or reject the project. Many uncertainties are inherent to this endeavor, some easily qua ntified, other not, nonetheless all must be quantified somehow. The specialists forecasts must be added to the historical data as well as a great dosage of “gut feeling”, taking us to adopt a possibility array/ray for the input data of the endeavor, in order to latter manipulate them, finding the possibilities array/ray for the NPV values. The matter itself is to accept or reject a project, in an “undeterministic” fashion, which means, quantify the chances of failure.

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To perform the involved calculation a software is presented (prototype) named “Fuzzyinvest 1.0”, which, at first, meets all the need of the studied case. The main activity of the “Mining” company is the extraction of raw feldspar, the companies which buy feldspar, here called clients of the mining company, purify and refine the feldspar, which is destined to the construction market. Observing the great expansion of its clients business, and having abundant available raw material, the Mining company has shown interest in the feldspar processing, and in entering in the market as a competitor of its clients. Project Data Fixed Investiment

R$12.874.035,00

Working Capital

R$2.376.000,00

Monthly Fized Cost

R$2.304.125,00

Variable Cost / unit

R$ 16/Ton

Forecasted Sales

100.000 Ton/ano

Price

R$ 98,00/Ton

Planning Horizon

10 years

Residual Value

"R$8.582.690,00"

ROR

15% year

Income Tax

35% year

Depreciation

10% year

Table 1 – “Project Data”

Deterministic NPV Calculation and Sensitivity Analyses To calculate the deterministic NPV the Excel™ software is used. In the same sheet, exploring the available resources, it is also carried a sensitivity analyses for the input variables involves in the calculation.

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Figure 7 – “Deterministic NPV Calculation and Sensitivity Analyses Sheet”

On the aforementioned sheet, it is noticeable the presence of scroll bar in the input variables, it can be noticed yet, the presence of macro which calculate the minimum price, the lowest demand and the highest possible investment, where, based on this data, the investment becomes viable. Considering the didactical character/scope of this paper, it becomes relevant the fuzzyfication of all the input data. The NPV found is R$ 8,211,191.38, which justifies the acceptance of the investment, in the case of a simple deterministic analysis. It is advisable to say, as low as they can be, that all uncertainties affect the final result, therefore a group of small uncertainties in all input variables may result in a great uncertainty in the output variable, when it comes to fuzzy NPV. Involved Uncertainties The uncertainties forecasted by the company’s administration are: ?

Fixed Investment (Initial Outlay) : +/- 10%;

?

Working Capital: +/- 10%;

?

Annual Fixed Cost: +/- 10%;

?

Variable Cost / unit: +/- 13%;

?

Sales Forecast: -30% to +20%;

?

Price: -20% to +15%;

?

Life Time: -20% to +50%;

?

ROR: +/- 10%;

All the variables and its uncertainties, mentioned above, are fuzzyfied and used in the calculation of the Fuzzy NPV.

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Presentation of the “Fuzzyinvest 1.0” Software (prototype) and Calculation of the Possibilistic NPV To carry out the calculation it was created a software “Fuzzyinvest 1.0”, in VBA (Excel, MS Office XP) language, based on the aforementioned concepts, where, after the insertion of the fuzzyfied data it calculates the fuzzy NPV as well as its graphic representation. The software sho w, also, as a result, the “failure possibility” of the investment, that is, the percentage of the triangle area (fuzzy NPV) which is in the negative region of the graph, on the left of the vertical axis. As an analyses option, it is possible to alter the endeavor/project’s data, through the graphics themselves, and watch the consequences on the Fuzzy NPV graph, Still, as an option, the graphical response can be altered, where the software questions which variable it is desired to alter in order to obtain that response. The main screen of “Fuzzyinvest 1.0” is the following:

Figure 8 - “Fuzzyinvest 1.0” Main Screen This screen shows the fields where the input data can be inserted, and the fields that show the output data, those output data are: Fuzzy NPV (graph) and the failure possibility of the endeavor/project. Besides the visualization of the fuzzyfied input data, some analyses may also be obtained through the “Gráfico” (Graph) sheet such as the graphical sensibility analysis. Tools such as Achieve Goal where the input data can be calculated by forcing the output, can also be done with the “Gráfico” sheet. The main screen of “Fuzzyinvest 1.0” has a field assigned for the “Gráfico” sheet. It is as follows:

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Figure 9 – “Gráfico” Sheet” To visualize the involved data and further details about the Fuzzy NPV, it is also shown the Sheet “Cálculos”, which can be taken as the most important part of this paper, for in it one can find all the theory that’s been describe throughout the previous topics.

Figure 10 – “Cálculos” Sheet

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In the “Cálculos” sheet are all the “Scenarios” tools and Achieve Goal, which can also be activated from the “Gráfico” sheet as previously mentioned. Defuzzyfication of the Result and Comparison to a Fuzzy Pattern. The fuzzy NPV value found is (-29.722.301, 8.211.191, 54.929.934) and the failure possibility of the project is 27.51% . With these information at hand, the board of directors of the company may take a more solid decision than it would with but a simple deterministic NPV. According to Shaw’s (1999) definition, a calculation of the failure possibility of 27.51% may be considered as a defuzzyfication operation, for a fuzzy output variable is being translated into a real number. From this point on, there is a discrete numerical value which can be compared to a fuzzy pattern. Such a pattern, defined by the high administration of the company, involves the Universe of the Failure Possibilities classified as follows:

Very Low

Low

Medium

High

Very High

1

0

5

10

15

20

(27,51)

35

40

%

Figure 11 – Universe of the Failure Possibilities of a Project

The company also adopts its own acceptance or rejection criteria towards investments according to the fuzzy pattern, such criteria is abstract and depend on the profile of the investor, the amount of investment in question and even the tangible values defined by the company administration itself. The acceptance criteria of the company are: Fuzzy classification array of the failure

Decision of the company

possibility of the investment Very Low

Unconditionally Accept

Low

Accept with caution

Average

Accept under restrictions

High (27.51%)

Reject and review project

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Very High

Unconditionally Reject

Table 2 – “Investment Projects Acceptance Criteria”

Obs.: The endeavors/projects with failure possibility, with intermediary classifications, must be proportionally considered, which means, for instance: a failure possibility of 17.5% is classified as “0.5 Low / 0.5 Medium” in this pattern. Consequently the decision to accept or reject must be wisely taken, which is “To accept with caution and restriction”. RESULT ANALYSIS In the present case, the failure possibility of the endeavor/project is classified as “High”, therefore, the investment must be rejected and the whole project reviewed. CONCLUS IONS The most relevant conclusion, concerns the comparison of the deterministic NPV with the Fuzzy NPV, being the “uncertainty” dimension made a go investment, in the deterministic method, turn into a rejected one. In spite of the low capacity of generalizatio n of the “quasi-experiment”, due to its nonrandomness inherent to the method, it is concluded that a comparison between the deterministic NPV and the Fuzzy one takes us to relatively general conclusions. The conclusions which may be considered general, are due to the exploration of the investment evaluation method and not the object of analyses. The way to evaluate an investment doesn’t change much, when applied to another object of analyses. The real purpose of the “quasi-experiment” is the adaptation of the “experiment” method to the administrative reality of the organizations, where certain characteristics of the “experiment” do not apply. An important conclusion about fuzzy logic, confirmed by Yager (1980), is regarding the noninversion of the operations, what many times can lead to mistakes. For instance: A+B=C does not imply that C-B=A, which happens to real numbers. Another conclusion is concerning the fact that this paper is restricted to triangular fuzzy numbers, which are a restrict field of the fuzzy logic and that simplify a lot the operations, Fuzzy logic, in general, goes way beyond the applications used in this paper. One of the most relevant information, obtained from the fuzzy NPV, is the failure possibility of the project, it is obtained from a proportion of the area seen under the membership curve, which takes us to an analogy with the PDF (Probability Density Function) using statistical methods. There must be extreme caution when using this analogy, for, the origins of the membership function and the PDF are totally distinct. It is also important to emphasize that the total area under the PDF curve, in any given distribution, is always equal to 1, which doesn’t always happen to the area below the membership curve. The uncertainty associated with the fuzzy NPV, is characterized by the amplitude of the fuzzy number that represents the fuzzy NPV, that is, “a3 – a1 ”, therefore, the “uncertainty associated to the investment” and the “investment viability” are totally independent. The concept of uncertainty must also be disassociated from a “bad” condition, once the same way that uncertain conditions can lead to unfavorable conditions, they may also converge to favorable ones.

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It is also important to point out the great visual analyses power of the fuzzy number, the visualization of the membership graph takes us to another analyses dimension, improving even more the decision making resources. The computerized resources allow us to deal with possible difficulties found in the calculation, with speed and accuracy, what happens with “Fuzzyinvest 1.0”. The software, however, is not totally consistent, thus if absurd input data is inserted it won’t tell the user, and it initiates the calculation. The solution to this problem may also be presented as a suggestion to future works. The user friendly feature of the VBA (Visual Basic Applications) makes the whole programming task easier, which helps the calculation and allows them to be manipulated with Excel sheets. The software values the visual aspect and the relevant information, emphasizing the membership graph and the failure possibility. Many other observations may be noted when manipulating the software, according to the application and the involved problem. The fuzzy logic capacity to provide resources in the decision make process is unchallenged. Nevertheless the need for special caution in the fuzzyfication of the input data and in the general application of the method is also imperative. Aware of the limitations and restriction of the method, it is possible to state that, from now on its applications tend to increase significantly. The application of fuzzy logic in other fields, besides the economic, has been increasing year after year, what indicates a strong tendency of growth in the economic field as well. REFERENCES AND COMPLEMENTARY BIBLIOGRAPHY Bryman, Alan.; Research Methods and Organization Studies. Ed. Routledge, 3ª Edição, 1995. Buckley, J.J.; The Fuzzy Mathematics of Finance, Fuzzy Sets and Systems 21 (1987) Chiu, C.Y.;Park, C. S.; Fuzzy cash flow analysis using present worth criterion; The Engeneering Economist 1994 – volume 39, no 2 (113-137). Cox Earl; Fuzzy Logic for Business and Industry, Ed. Charles River Media (1995). Dixit, A. K., Pindyck, R. S. (1994); Investment Under Uncertainty, Princeton University Press, Princeton, N. J. Giachetti, Ronald E.; Young, Robert E.; Analysis of the Error in the Standard Approximation Used for Multiplication of Triangular and Trapezoidal Fuzzy Numbers and Development of a New Approximation, Department of Industrial Engineering, North Carolina State Universitry. Gonçalves, Cleber; Adjusted Present Value (APV): Avaliação de Negócios com Taxas de Desconto Diferenciadas. Dissertação de Mestrado – Dep. Produção, UNIFEI, (2003). Kaufmann, Arnold; Gupta Madan M.; Fuzzy Mathematical Models in Engineering and Management Science. Elsevier Science Publishers B. V. (1988). Kuchta, Dorota; Fuzzy Capital Budgeting, Fuzzy Sets and Systems 111 (2000) 367–385.

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Martines Jr, Luiz Carlos; Modelo de Análise de Investimentos Baseado em Sistemas Especialistas e Lógica Fuzzy. Dissertação de Mestrado – Escola Politécnica da Universidade de São Paulo 2002. Montevechi, J. A; Revisão Sobre Lógica Fuzzy – Apostila disponível no site do autor – www.uinfei.edu.br/arnaldo. Montevechi, J. A; Pesquisa Operacional – Petrobrás, 1998 – PPT. Pamplona, E. O.; Montevechi, J.A.B., Apostila do Curso de Engenharia Econômica Avançada. UNIFEI - Itajubá - MG, 1995. Ross, S. A.; Westerfield, R. W.; Jordan, B. D.; Princípios de Administração Financeira. Segunda Edição – Ed. Atlas, 2002. Santos, Elieber Mateus; Um Estudo Sobre a Teoria das Opções Reais Aplicada à Análise de Investimentos em Projetos de Pesquisa e Desenvolvimento (p&d), Dissertação de Mestrado – Departamento de Produção, UNIFEI, Novembro 2001. Shaw, I. S.; Simões M. G.; Controle e Modelagem Fuzzy, Ed. Edgar Blücher (1995). Souza, Rubem César R.; Planejamento de Suprimento de Energia Elétrica dos Sistemas Descentralizados na Amazônia Incorporando Incertezas, Ed.da Univ. do Amazonas (1996). Trochin, W.; The Research Methods Knowledge Base. Ed. Atomic Dog Publishing, 2001. Von Altrock, C.; Fuzzy Logic & Neurofuzzy Applications in Business & Finance, Prentice Hall PTR-Upper Saddle River, New Jersey 07458 – 1995. Yager, R. R.; On the Lack of Inverses in Fuzzy Arithmetic, Fuzzy Sets and Systems 4, 1980. http://bigbob.sites.uol.com.br/fuzzy_1.htm http://www.dc.ufscar.br/~fernandes/trabalhos/NebXPara.ppt http://www.din.uem.br/ia/controle/fuz_cara.htm http://members.tripod.com/geloneze/Fuzzy.htm FUZZYTECH 5.5; User Manual, Information Software Corporation.

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