Extension Principle
Adriano Cruz, adriano@nce.ufrj.br
PPGI-UFRJ
September 2011
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle
September 2011
1 / 62
September 2011
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Summary
1
Introduction
2
Crisp Functions, Mappings and Relations
3
Functions of Fuzzy Sets
4
Fuzzy Arithmetic
5
Interval Analysis in Arithmetic
6
Approximate Methods of Extension
Vertex Method
DSW Algorithm
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle
Section Summary
1
Introduction
2
Crisp Functions, Mappings and Relations
3
Functions of Fuzzy Sets
4
Fuzzy Arithmetic
5
Interval Analysis in Arithmetic
6
Approximate Methods of Extension
Vertex Method
DSW Algorithm
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle
September 2011
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September 2011
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Would a precise model be a contradiction?
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle
Bibliography
Kevin M. Passino, Stephen Yurkovich, Fuzzy Control in Chapter 5, Addison
Wesley Longman, Inc, USA, 1998.
Timothy J. Ross , Fuzzy Logic with Engineering Applications, John Wiley
and Sons, Inc, USA, 2010.
R. R. Yager, A characterization of the extension principle, Fuzzy Sets Syst.,
18, 205-217, 1986
John Yen, Reza Langari, Fuzzy Logic: Intelligence, Control and Information,
Prentice Hall, USA, 1999
L. Zadeh, The concept of a linguistic variable and its application to
approximate reasoning, Part I. Inf Sci., 8, 199-249, 1975
W. Dong and H. Shah, Vertex Method for computing functions of fuzzy
variables. Fuzzy Sets Syst., 24, 65-78, 1987.
W. Dong and H. Shah and F. Wong, Fuzzy computations in risk and decision
analysis, Civ. Eng. Syst., 2, 201-208, 1985.
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle
September 2011
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Background
Consider a function y = f (x).
If we known x it is possible to determine y .
Is it possible to extend this mapping when the input, x, is a fuzzy
value.
The extension principle developed by Zadeh (1975) and later by Yager
(1986) establishes how to extend the domain of a function on a fuzzy
sets.
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle
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Section Summary
1
Introduction
2
Crisp Functions, Mappings and Relations
3
Functions of Fuzzy Sets
4
Fuzzy Arithmetic
5
Interval Analysis in Arithmetic
6
Approximate Methods of Extension
Vertex Method
DSW Algorithm
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle
September 2011
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September 2011
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Crisp Mappings
X
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
f(X)
Extension Principle
Y
Functions Applied to Intervals
An interval I is a crisp set, I ∈ X .
Compute the image of the interval, which is a crisp set in Y .
Presumably, sets in the power set of X can be mapped to the power
set of Y , that is f : P(X ) → P(Y ).
The image B ∈ Y of a set A ∈ X can be calculated as B = f (A) or
for all x ∈ A, y = f (x)
B is defined by its characteristic value
χB (y ) = χf (A) (y ) =
_
χA (x)
y =f (x)
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Extension Principle
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Functions Applied to Intervals
y
f(I)
I
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Extension Principle
x
September 2011
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Functions Applied to Intervals - Example I
Consider the Universe X = {−2, −1, 0, 1, 2}
Consider the set A = {0, 1}
0
Using the Zadeh notation A = { −2
+
0
−1
+
1
0
+
1
1
+ 20 }
Consider the mapping y = |4x| + 2
What is the resulting set B on the Universe Y = {2, 6, 10}
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Extension Principle
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September 2011
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Functions Applied to Intervals - Example II
Using χB (y ) = χf (A) (y ) =
and y = |4x| + 2.
W
y =f (x) χA (x)
χB (2) = ∨{χA (0)} = 1.
χB (6) = ∨{χA (−1), χA (1)} = ∨{0, 1} = 1.
χB (10) = ∨{χA (−2), χA (2)} = ∨{0, 0} = 0.
B = { 12 +
1
6
+
0
10 }
or B = {2, 10}.
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle
Using Relations
It is possible to achieve the results using a relation that express the
mapping y = |4x| + 2.
Lets X = {−2, −1, 0, 1, 2}.
Lets Y = {0, 1, 2, . . . , 9, 10}
The relation

−2
−1
0
1
2
R=





0
0
0
0
0
0
1
0
0
0
0
0
2
0
0
1
0
0
3
0
0
0
0
0
4
0
0
0
0
0
5
0
0
0
0
0
6
0
1
0
1
0
7
0
0
0
0
0
8
0
0
0
0
0
9
0
0
0
0
0
10

1
0 

0 

0 
1
B =A◦R
0
A = { −2
+
0
−1
1
0
+
+
1
1
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
+ 02 } or more conveniently A = {0, 0, 1, 1, 0}
Extension Principle
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Applying the Relation
Using χB (y ) =
we find
W
x∈X (χA (x)
∧ χR (x, y ))
1, for y = 2, 6
χB (y ) =
0, otherwise
.
Or
B=
0 0 1 0 0 0 1 0 0 0
0
+ + + + + + + + + +
0 1 2 3 4 5 6 7 8 9 10
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Extension Principle
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Section Summary
1
Introduction
2
Crisp Functions, Mappings and Relations
3
Functions of Fuzzy Sets
4
Fuzzy Arithmetic
5
Interval Analysis in Arithmetic
6
Approximate Methods of Extension
Vertex Method
DSW Algorithm
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle
September 2011
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Extension Principle
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Fuzzy Mappings
ABC
D
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Starting Point
Consider two universes of discourse X and Y and a function y = f (x).
Suppose that elements in universe X form a fuzzy set A.
What is the image (defined as B) of A on Y under the mapping f ?
Similarly to the crisp definition, B is obtained as
_
µB (y ) = µf (A) (y ) =
µA (x)
y =f (x)
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Extension Principle
September 2011
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Simplifying the Notation
Fuzzy vector is a convenient shorthand for calculations that use
matrix relations.
Fuzzy vector is a vector containing only the fuzzy membership values.
Consider the fuzzy set:
0
0 0.2 0.3 0.5 0.7 0.9 1 0 0 0
+
+
+
+
+
+ + + + +
B=
0
1
2
3
4
5
6 7 8 9 10
The fuzzy set B may be represented by the fuzzy vector b:
b = 0, 0.2, 0.3, 0.5, 0.7, 0.9, 1, 0, 0, 0, 0
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Extension Principle
September 2011
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Extension Principle
Suppose that f is a function from X to Y and A is a fuzzy set on X
defined as
A = µA (x1 )/x1 + µA (x2 )/x2 + . . . + µA (xn )/xn
.
The extension principle states that the image of fuzzy set A under the
mapping f (.) can be expressed as a fuzzy set B defined as
B = f (A) = µA (x1 )/y1 + µA (x2 )/y2 + . . . + µA (xn )/yn
where yi = f (xi )
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Extension Principle
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Many-to-one mappings
If f (.) is a many-to-one mapping, then, for instance, there may exist
x1 , x2 ∈ X , x1 6= x2 , such that f (x1 ) = f (x2 ) = y ∗ , y ∗ ∈ Y .
The membership degree at y = y ∗ is the maximum of the
membership degrees at x1 and x2 more generally, we have
µB (y ∗ ) = max µA (x)
y =f (xi )
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Extension Principle
September 2011
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Monotonic Continuous Functions
For each point in the interval:
Compute the image of the interval.
The membership degrees are carried through.
y
B
µB(x)
µA(x)
A
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Extension Principle
x
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Monotonic Continuous Functions Ex.
Function: y = f (x) = 0.6 ∗ x + 4.
Input: Fuzzy number - around-5.
around − 5 = { 0.3
3 +
f (around − 5) =
f (around − 5) =
f (around − 5) =
1.0
0.3
5 + 7 }.
1
0.3
{ f0.3
(3) + f (5) + f (7) }.
0.3
1
0.3
{ 0.6∗3+4
+ 0.6∗5+4
+ 0.6∗7+4
}.
1
0.3
{ 0.3
5.8 + 7 + 8.2 }.
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle
x
Monotonic Continuous Functions Ex.
8.2
y
x
8
7
7
5.8
6
5
4
µA(x)
3
2
0.3
1.0
µB(x)
1
1
2
3
4
5
6
7
1
2
3
4
5
6
7
x
µ A(x)
1.0
0.3
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x
Extension Principle
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September 2011
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Non-Monotonic Continuous Functions Ex.
Function: y = f (x) = x 2 − 6 ∗ x + 11.
Input: Fuzzy number - around-4.
around − 4 = { 0.3
2 +
f (around − 4) =
f (around − 4) =
f (around − 4) =
f (around − 4) =
0.6
1
0.6
0.3
3 + 4 + 5 + 6 }.
0.6
1
0.6
0.3
{ f0.3
(2) + f (3) + f (4) + f (5) + f (6) }.
0.6
1
0.6
0.3
{ 0.3
3 + 2 + 3 + 6 + 11 }.
0.6
0.6
0.3
{ 0.3∨1
3 + 2 + 6 + 11 }.
1
0.6
0.3
{ 0.6
2 + 3 + 6 + 11 }.
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle
Generalizing
Suppose the input universe is composed of the Cartesian product of
many universes.
The mapping f is defined on the power set of this universe as
f : P(X1 × X2 × · · · × Xn ) → P(Y ).
Let the fuzzy sets A1 , A2 , . . . , An be defined on X1 , X2 , . . . , Xn
then B = f (A1 , A2 , . . . , An ).
The membership function of B is defined as
µB (y ) =
max
y =f (x1 ,x2 ,...,xn )
{min [µA1 (x1 ), µA2 (x2 ), . . . , µAn (xn )]}
This equation is usually called the Zadeh’s extension principle.
If the function f is a continous-valued expression, the max operator is
replaced by the sup (supremum) which is the least upper bound.
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Extension Principle
September 2011
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Example
Inputs: A = { 0.2
1 +
1
2
+
0.7
4 }
1
and B = { 0.5
1 + 2}
Output: f (A, B) = A × B (arithmetic product).
min(0.2, 0.5)
max[min(0.2, 1), min(0.5, 1)]
+
+
1
2
min(0.7, 1)
max[min(0.7, 0.5), min(1, 1)]
+
4
8
0.5
1
0.7
0.2
+
+ +
=
1
2
4
8
A×B =
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Extension Principle
September 2011
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Fuzzy Transform
Fuzzy transform happens when the input of a single element
(nonfuzzy) maps to a fuzzy set in the output universe.
An element x in universe X is mapped to a fuzzy set B in universe Y .
B = f (x), where f is a fuzzy mapping.
If X and Y are finite f can be expressed as a fuzzy relation R or
R=
x1
x2
..
.
xi
..
.
xn
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)










y1 y2 . . . yj . . .
r11 r12 . . . r1j . . .
r21 r22 . . . r2j . . .
..
..
..
..
..
.
.
.
.
.
ri 1 ri 2 . . . rij . . .
..
..
..
..
..
.
.
.
.
.
rn1 rn2 . . . rnj . . .
Extension Principle
ym
r1m
r2m
..
.
rim
..
.
rnm










September 2011
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Fuzzy Transform Singleton
For a particular singleton xi its fuzzy image is the fuzzy set Bi = f (xi )
µBi (yj ) = rij or in fuzzy vector notation
bi = {ri 1 , ri 2 , . . . , rim }.
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Extension Principle
September 2011
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Fuzzy Transform Generalized
For a particular fuzzy input set A its fuzzy image is B = f (A)
W
µB (y ) = x∈X (µA (x) ∧ µR (x, y ))
b = a ◦ R.
bj = maxi (min(ai , rij ))
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Extension Principle
September 2011
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September 2011
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Section Summary
1
Introduction
2
Crisp Functions, Mappings and Relations
3
Functions of Fuzzy Sets
4
Fuzzy Arithmetic
5
Interval Analysis in Arithmetic
6
Approximate Methods of Extension
Vertex Method
DSW Algorithm
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle
Fuzzy Numbers
A fuzzy number is fuzzy subset of the universe of a numerical number.
A fuzzy real number is a fuzzy subset of the domain of real numbers.
A fuzzy integer number is a fuzzy subset of the domain of integers.
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September 2011
Extension Principle
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Examples of Fuzzy Numbers
µ( x)
1.0
Fuzzy Integer Number 5
1
2
3
4
5
6
7
8
9
10
x
µ( x)
1.0
Fuzzy Real Number 5
1
2
3
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4
5
6
Extension Principle
7
8
9
10
x
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Fuzzy Arithmetic
Applying the extension principle to arithmetic operations it is possible
to define fuzzy arithmetic operations
Let x and y be the operands, z the result.
Let A, B and C denote the fuzzy sets that represent the operands x,
y and z respectively.
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Extension Principle
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Fuzzy Arithmetic
Using the extension principle a fuzzy arithmetic operation denoted by
∗ ∈ {+, −, ×, ÷} is defined as
µC (z) = max {min [µA (x), µB (y )]}
z=x∗y
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Extension Principle
September 2011
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Example of Problem
We will calculate the product of two fuzzy sets defined as:
0.66
0.66
0.33
0
1
X = 10 + 0.33
+
+
+
+
+
2
3
4
5
6
7
0 0.33
0.66
0.33
1
0.66
Y = 2 + 3 + 4 + 5 + 6 + 7 + 08
The result would be:
X ×Y =
0 0 0 0 0.33 0.33 0.33 0.33 0.66
+ + + +
+
+
+
+
2 3 4 5
6
8
9
10
12
1
0.33 0.66 0.33 0.66
0.33 0.66
+
+
+
+
+
+
+
14
15
16
18
20
21
24
0.66 0.33 0.66
0
0.33 0.33
0
+
+
+
+
+
+
+
25
28
30 32
35
36
40
0
0
0
0.33
+
+
+
+
42
48
49 56
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September 2011
Extension Principle
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Example of Problem
µ(X)
1
0.5
0
0
1
2
3
4
5
6
7
8
9
5
6
7
8
9
X
µ(Y)
1
0.5
0
0
1
2
3
4
Y
µ(X × Y)
1
0.5
0
0
10
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
20
30
X×Y
Extension Principle
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50
60
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Section Summary
1
Introduction
2
Crisp Functions, Mappings and Relations
3
Functions of Fuzzy Sets
4
Fuzzy Arithmetic
5
Interval Analysis in Arithmetic
6
Approximate Methods of Extension
Vertex Method
DSW Algorithm
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle
September 2011
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Definitions I
Let I1 and I2 two interval numbers defined ordered pair of real
numbers with lower and upper bounds.
I1 = [a, b] where a ≤ b and I2 = [c, d] where c ≤ d.
I1 ∗ I2 = [a, b] ∗ [c, d] where ∗ ∈ {+, −, ×, ÷} is another interval.
When adding or multiplying two intervals we are performing these
operations on the infinite number of combinations of pairs from each
of the two intervals.
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Extension Principle
September 2011
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Definitions II
[a, b] + [c, d] =[a + c, b + d]
[a, b] − [c, d] =[a − d, b − c]
[a, b] × [c, d] =[min(ac, ad , bc, bd), max(ac, ad , bc, bd)]
1 1
,
since 0 ∈
/ [c, d]
[a, b] ÷ [c, d] =[a, b] ×
d c
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Extension Principle
September 2011
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Conclusions
When adding or multiplying two intervals we are performing these
operations on the infinite number of combinations of pairs from each
of the two intervals.
We only need to find the endpoints of the intervals to find the
endpoints of the solutions.
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle
September 2011
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Section Summary
1
Introduction
2
Crisp Functions, Mappings and Relations
3
Functions of Fuzzy Sets
4
Fuzzy Arithmetic
5
Interval Analysis in Arithmetic
6
Approximate Methods of Extension
Vertex Method
DSW Algorithm
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
September 2011
Extension Principle
41 / 62
Vertex Method
The method is based on a combination of the λ-cut and standard
interval analysis [Dong and Shah, 1987].
The algorithm is easy to implement and can be computationally
efficient.
µ
1
A
λ
I
0
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
a
λ
b
Extension Principle
x
September 2011
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Vertex Algorithm
Any continuous membership function can be represented by a
continuous sweep of λ-cut intervals from λ = 0+ to λ = 1.
Let y = f (x) be extended for fuzzy sets, or B = f (A).
A will be decomposed into a series for λ-cut intervals.
When f (x) is continuous and monotonic on Iλ = [a, b] the interval
representing B at a particular value of λ(Bλ ) is
Bλ = f (Iλ ) = [min(f (a), f (b)), max(f (a), f (b))]
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Extension Principle
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Vertex Algorithm for n-inputs
Let y = f (x1 , x2 , . . . , xn ).
The input space is represented by n-dimensional Cartesian region.
N = 2n is the number of vertices of the region.
Each of the input variables is described by an interval Ii λ at a specific
λ-cut where Ii λ = [ai , bi ], i = 1, 2, . . . , n.
Bλ =f (I1λ , I2λ , . . . , Inλ )
Bλ = min(f (cj )), max(f (cj )) , j = 1, 2, . . . , N
j
j
(1)
(2)
where cj is the coordinate of the jth vertex representing the
n-dimensional Cartesian region.
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Extension Principle
September 2011
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Vertex Algorithm for n-inputs
The method is accurate only when the conditions of continuity and
no extreme points are satisfied.
An extreme point is a point of maximum or minimum.
Extreme points should be treated as additional vertices Ek .
Bλ = min(f (cj ), f (Ek )), max(f (cj ), f (Ek )) ,
j,k
j,k
j = 1, 2, . . . , N and k = 1, 2, . . . , m
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September 2011
Extension Principle
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Example of Problem
We will use Vertex Method to determine the output of the function
y = x(2 − x) to an input fuzzy set A = µ(x).
We will use the three λ-cuts: λ = 0+ , 0.5, 1.0
The corresponding intervals are: I0+ = [0.5, 2], I0.5 = [0.75, 1.5],
I1 = [1, 1].
The extreme point x = 1, y = 1 can be calculated by derivatives.
y=f(x)
y
1
0.5
0
0
0.5
1
1.5
2
2.5
1.5
2
2.5
x
A
µ(x)
1
0.5
0
0
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
0.5
0.75
1
x
Extension Principle
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Calculating
I0+ = [0.5, 2] ⇒ c1 = 0.5, c2 = 2, E1 = 1
f (c1 ) = 0.75, f (c2 ) = 0, f (E1 ) = 1
B0+ = [min(0.75, 0, 1), max(0.75, 0, 1)] = [0, 1]
I0.5 = [0.75, 1.5] ⇒ c1 = 0.75, c2 = 1.5, E1 = 1
f (c1 ) = 0.9375, f (c2 ) = 0.75, f (E1 ) = 1
B0.5 = [min(0.9375, 0.75, 1), max(0.9375, 0.75, 1)] = [0.75, 1]
I1 = [1, 1] ⇒ c1 = 1, c2 = 1, E1 = 1
f (c1 ) = f (c2 ) = f (E1 ) = 1
B1 = [min(1, 1, 1), max(1, 1, 1)] = [1, 1]
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September 2011
Extension Principle
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Results
y=f(x)
y
1
0.5
0
0
0.5
1
1.5
2
2.5
1.5
2
2.5
1.5
2
2.5
y=µ(x)
x
A
1
0.5
0
0
0.5
0.75
1
x
B
µ(y)
1
0.5
0
0
0.5
1
y
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle
September 2011
48 / 62
Another Example of Problem
We will use Vertex Method to calculate the product of two fuzzy sets
defined as:
1
0.66
0.66
0.33
0
X = 10 + 0.33
+
+
+
+
+
2
3
4
5
6
7
0 0.33
0.66
0.33
1
0.66
Y = 2 + 3 + 4 + 5 + 6 + 7 + 08
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle
September 2011
49 / 62
September 2011
50 / 62
Interval I0+
Support for X is the interval [1, 7].
Support for Y is the interval [2, 8].
x
1
1
7
7
y
2
8
2
8
f()
f(a)=2
f(b)=8
f(c)=14
f(d)=56
min=2, max=56 and B0+ = [2, 56]
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle
Interval I0.33
Support for X is the interval [2, 6].
Support for Y is the interval [3, 7].
x
2
2
6
6
y
3
7
3
7
f()
f(a)=6
f(b)=14
f(c)=18
f(d)=42
min=6, max=42 and B0.33 = [6, 42]
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle
September 2011
51 / 62
September 2011
52 / 62
Interval I0.66
Support for X is the interval [3, 5].
Support for Y is the interval [4, 6].
x
3
3
5
5
y
4
6
4
6
f()
f(a)=12
f(b)=18
f(c)=20
f(d)=30
min=12, max=30 and B0.66 = [12, 30]
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle
Interval I1.0
Support for X is the interval [4, 4].
Support for Y is the interval [5, 5].
x
4
y
5
f()
f(a)=20
min=20, max=20 and B1.0 = [20, 20]
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
September 2011
Extension Principle
53 / 62
Vertex Method
µ
1
A
0.66
0.33
0
10
20
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
30
40
Extension Principle
50
60
XxY
September 2011
54 / 62
DSW Algorithm
It uses λ-cut and standard interval analysis [Dong, Shah and Wong,
1985].
The algorithm:
Repeat for different values of λ where 0 ≤ λ ≤ 1:
Find the interval(s) in the input membership function(s) that
correspond to this λ;
Using standard binary interval operations, compute the interval for the
output membership function for the selected λ;
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
September 2011
Extension Principle
55 / 62
Example of Problem
We will use DSW Method to determine the output of the function
y = x(2 + x) to an input fuzzy set A = µ(x).
We will use the three λ-cuts: λ = 0+ , 0.5, 1.0
The corresponding intervals are: I0+ = [0.5, 2], I0.5 = [0.75, 1.5],
I1 = [1, 1].
y=f(x)
8
y
6
4
2
0
0
0.5
1
1.5
2
2.5
1.5
2
2.5
x
A
µ(x)
1
0.5
0
0
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
0.5
0.75
1
x
Extension Principle
September 2011
56 / 62
Calculating
I0+ = [0.5, 2]
B0+ = 2 · [0.2, 2] + [0.52 , 22 ] = [1, 4] + [0.25, 4] = [1.25, 8]
I0.5 = [0.75, 1.5]
B0.5 = 2 · [0.75, 1.5] + [0.752 , 1.52 ] = [1.5, 3] + [0.5625, 2.25] =
[2.0625, 5.25]
I1 = [1, 1]
B1 = 2 · [1, 1] + [12 , 12 ] = [2, 2] + [1, 1] = [3, 3] = 3
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
September 2011
Extension Principle
57 / 62
Results
y
y=f(x)
5
0
0
0.5
1
1.5
2
2.5
1.5
2
2.5
x
A
µ(x)
1
0.5
0
0
0.5
0.75
1
x
B
µ(y)
1
0.5
0
0
1
2
3
4
5
6
7
8
9
y
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle
September 2011
58 / 62
Another Example of Problem
We will use DSW Method to calculate the product of two fuzzy sets
defined as:
1
0.66
0.66
0.33
0
X = 10 + 0.33
+
+
+
+
+
2
3
4
5
6
7
0 0.33
0.66
0.33
1
0.66
Y = 2 + 3 + 4 + 5 + 6 + 7 + 08
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle
September 2011
59 / 62
Interval Arithmetic
I0+ : [1, 7] · [2, 8] = [min(2, 14, 8, 56), max(2, 14, 8, 56)] = [2, 56].
I0.33 : [2, 6] · [3, 7] = [min(6, 18, 14, 42), max(6, 18, 14, 42)] = [6, 42].
I0.66 : [3, 5] · [4, 6] = [min(12, 20, 18, 30), max(12, 20, 18, 30)] =
[12, 30].
I1 : [4, 4] · [5, 5] = [min(20, 20, 20, 20), max(20, 20, 20, 20)] = [20, 20].
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle
September 2011
60 / 62
DSW Algorithm
µ
1
A
0.66
0.33
0
10
20
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
30
40
Extension Principle
50
60
XxY
September 2011
61 / 62
September 2011
62 / 62
The End
Adriano Cruz, adriano@nce.ufrj.br (PPGI-UFRJ)
Extension Principle