Prof. Alexandre Leme Sanches, MSc.

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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).
• Doesnt have total control over the input
  variables of the system, BRYMAN (1989).
• Theres a non-random treatment of the experiment,
  TROCHIN (2001).
• Where the human behavior is present, TROCHIN
  (apud GONÇALVES (2003)).

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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
L O G I C
F U Z Z Y
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 machines 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
A
a
b
c
d
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
a4
a2
a3
0
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4.4. Triangular Fuzzy Number (TFN) If
and are linear functions and a2
a 3
?A (x)
1
a1
a3
a2
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
0
ROR
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18
10
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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
0
Reais

2,6 2,7 3,0
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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

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Subtraction 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
B

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Multiplication Using the line equations A B
Al(y) Bl(y), Ar(y)Br(y) is not a
TFN. Example
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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An (where n is a real number)
AB (where B is a TFN (b1, b2, b3))

undefined
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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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Fuzzyfications example
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Defuzzyfications example
Bad
Medium
Good
1
Very good
Very bad
0 1000 3000 5000
7000 9000 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 (i1...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 8.211.191,38.
Therefore, in a simple Deterministic evaluation,
the investiment could be acepted.
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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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10.6. Analysing the results.
Fuzzyinvest 1.0 Main Screen
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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 doesnt 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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• Questions?