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**“Capital Budgeting Using**

Triangular Fuzzy Numbers” Prof. Alexandre Leme Sanches, MSc. Prof. Edson de Oliveira Pamplona, Dr. Prof. José Arnaldo Barra Montevechi, Dr.

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Itajubá

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Itajubá

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**Universidade Federal de Itajubá**

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1. Introduction 2. Objectives 3. Methodological aspects 4. Literature revision 5. Operations with Triangular Fuzzy Numbers (TFN) 6. Fuzzyfication and Defuzzyfication 7. The Net Present Value 8. Application of Fuzzy Numbers in Investiments Analysis 9. Analyzing the Fuzzy NPV 10. Real Case Aplication 11. Conclusions

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**1.Introduction: Uncertainties associated with “Investment Analyses”**

Alternatives methods Decision making process Optimization of financial resources

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**Objectives: Main Objective:**

Demonstrate the use of fuzzy logic in the evaluation of investment projects under uncertaint conditions; Secondary Objective: Presentation of a software prototype to calculate the fuzzy NPV and relative analyses.

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**Methodological aspects:**

The research method to be used is known as “quasi-experiment”: Pre and Post Test, TROCHIN (2001). Doesn’t have total control over the input variables of the system, BRYMAN (1989). There’s a non-random treatment of the experiment, TROCHIN (2001). Where the human behavior is present, TROCHIN (apud GONÇALVES (2003)).

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**M E T H O D 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 F U Z Y L O G I C Fuzzyfication of the selected variables (specialist) Fuzzy NPV Calculation 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

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**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 continuously; The degree of “membership is not probability”, but a measure of compatibility between object and the concept represented by the fuzzy set.

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**4.1. Membership Fuction - Example:**

d a b c A a b c d A a b c d x 1 1 A a b c d x 0.5 Boolean Logic (binary) Fuzzy Logic (continuous) A(x): Membership

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**4.2. Fuzzy Number – General Definition, KUCHTA (1996)**

Where: are real numbers and : is a continuous real function non decreasing defined in the interval [0,1], such that: and : is a continuous real function non increasing defined in the interval [0,1], such that: and

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4.3. Fuzzy Number: A(x) 1 x a1 a2 a3 a4

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**A = (a1, a2, a3) 4.4. Triangular Fuzzy Number (TFN):**

If and are linear functions and a2 = a 3: A (x) 1 a1 a2 a3 x A = (a1, a2, a3)

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**4.5. Fuzzy Number – Example I:**

A “Fuzzy Set” representing the NPV: “Rates: Low/Medium/High” 1 High Low 0.6 Medium 0.4 10% 18% 26% ROR

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**4.6. Fuzzy Number – Example II:**

A Fuzzy Set representing: (The value of one Dolar on 16/10/03): “Subjectivity” 1 0.5 2,6 2, ,0 Reais

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**Operations with Triangular Fuzzy Numbers (TFN):**

Addition: If A = (a1, a2, a3) and B = (b1, b2, b3), so: A (+) B = (a1, a2, a3) + (b1, b2, b3) = (a1 + b1, a2 + b2, a3 + b3), is a TFN. Example: 0,5 1 11 A B A + B x µ(x)

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**Subtraction: B If A = (a1, a2, a3) and B = (b1, b2, b3), so:**

A (-) B = (a1, a2, a3) - (b1, b2, b3) = (a1 - b3, a2 - b2, a3 - b1), is a TFN. Example: 0,5 1 A - B A B x µ(x) B

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**Aproach by Chiu e Park (1994)**

Multiplication: Using the line equations: A * B = [[Al(y)* Bl(y), Ar(y)*Br(y)] is not a TFN. Example: 0,5 1 A B A x B x µ(x) Aproach by Chiu e Park (1994)

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**Division: (two diferents cases)**

1) If A and B are both positives: A / B = [Al(y)/ Br(y), Ar(y)/Bl(y)] 2) If A is positive and B is negative: A / B = [Al(y)/ Bl(y), Ar(y)/Br(y)] The result in the first case is a positive fuzzy number and in the second case is a negative fuzzy number.

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**undefined An: (where n is a real number)**

AB: (where B is a TFN (b1, b2, b3)) undefined

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**x undefined AB: (where B is a TFN (b1, b2, b3))**

An: (where n is a real number) x AB: (where B is a TFN (b1, b2, b3)) undefined

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**Fuzzyfication and Defuzzyfication:**

Fuzzyfication: Is the maping of real numbers domain (generally discrete) to the fuzzy domain. Defuzzyfication: Is the proceeding in which the value of the output linguistic, inferred by the fuzzy rules, will be transletad to a discrete value. SHAW I. S. (1999)

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**1 0 5 10 15 20 25 30 35 40 ROR (%) Fuzzyfication’s example: Very Low**

Medium High Very High ROR (%)

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**Defuzzyfication’s example:**

1 Very bad Bad Medium Good Very good NPV 8000

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**The Net Present Value: Where: NPV: net present value**

CF0: first cash flow CFi: cash flow on period i (i=1...n) n: number of periods r: discount rate

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**Application of Fuzzy Numbers in Investiments Analysis:**

The Fuzzy Net Present Value: According to BUCKLEY (1987) the Membership Function to NPV is givem: To i = 1, 2, ... where k = i if F is negative and k = 3 - i if F is positive. Comparing:

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**Analyzing the Fuzzy NPV**

“Investiment Sure and Viable”

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**“Investiment Sure and Unviable”**

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**“Investiment Unsure and Viable”**

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**“Investiment Unsure and Unviable”**

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**Negative area Positive area**

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**Real Case Aplication: 10.1. The Problem:**

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.

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**10.3. NPV Calculations Using Software Excel**

The value of NPV found is R$ ,38. Therefore, in a simple Deterministic evaluation, the investiment could be acepted.

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**“Fuzzyinvest 1.0” Main Screen**

10.5. “Fuzzynvest 1.0” presentation: “Fuzzyinvest 1.0” Main Screen

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“Gráfico” Sheet

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“Cálculos” Sheet

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**“Fuzzyinvest 1.0” Main Screen**

10.6. Analysing the results. “Fuzzyinvest 1.0” Main Screen

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**The failure possibility of the project.**

1 Very Low Low Medium High Very High (27,51) % The failure possibility of the project.

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**Decision of the company**

Fuzzy classification array of the failure possibility of the investment Decision of the company Very Low Unconditionally Accept Low Accept with caution Average Accept under restrictions High (27.51%) Reject and review project Very High Unconditionally Reject “Investment Projects Acceptance Criteria”

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Conclusions: 1) 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.

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Conclusions: 2) The way to evaluate an investment doesn’t change much, when applied to another object of analyses. 3) 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.

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Conclusions: 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. 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.

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Conclusions: 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 values the visual aspect and the relevant information, emphasizing the membership graph and the failure possibility.

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