Electrical Engineering Department Petra Christian University This document is prepared by Thiang Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 1 Sistem Cerdas (TE 4485) Instructor: Thiang Room: I.201 Phone: 031-2983115 Email: [email protected]Fuzzy Set, Fuzzy Logic, and its Applications
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Electrical Engineering DepartmentPetra Christian University
This document is prepared by ThiangSistem Cerdas: Fuzzy Set and Fuzzy Logic - 1
Electrical Engineering DepartmentPetra Christian University
This document is prepared by ThiangSistem Cerdas: Fuzzy Set and Fuzzy Logic - 2
AA
AA AA
A
A
Group of Apples
Group of Oranges
OO
OO OO
O
O
OA
AA AA
A
A
Group of Apples?
Group of Oranges?
AO
OO OO
O
O
OA
OA AO
O
A
Group of Apples??
Group of Oranges??
AA
OA OA
O
O
Introduction
Electrical Engineering DepartmentPetra Christian University
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Definition: If temperature is higher than 50°C then it is hot
Temperature is 70°C, is it hot?
Temperature is 30°C, is it hot?
Temperature is 51°C, is it hot?
Temperature is 40°C, is it hot??
Temperature is 45°C, is it hot??
Temperature is 49°C, is it hot????
Temperature is 50°C, is it hot??????
Introduction
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Fuzzy Sets theory was introduced by Lotfi A. Zadeh(1965)
Fuzzy Sets are sets with boundaries that are not precise. The membership in a fuzzy set is not a matter of affirmation or denial, but rather a matter of a degree.
Introduction: Crisp set versus Fuzzy set
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The characteristic of Crisp set assigns a value of either 1 or 0 to each individual in the universal set
Fuzzy set assigns a value within a specified range to each individual in the universal set and the value indicates the membership grade of that individual in the set. Larger value denotes higher degree of set membership.
Crisp Fuzzy
0 0 1 1
Introduction: Crisp set versus Fuzzy set
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Fuzzy Set notation
Continuous ( )∫= xxF F /µ
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Example: The set, B, of numbers near to two. Membership function of the set is defined as:
( ) ( )25 −−= xB exµ
( )∫ −−= xeB x /25
2 1 3
( )xBµ
1
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Fuzzy Set notation
Discrete ( )∑= xxF F /µ
Example: The set, B, of numbers near to two. Membership function of the set is defined as:
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⎪⎩
⎪⎨
⎧
≤<−≤<−+
>−≤=
312/)3(112/)1(
310)(
xforxxforx
xandxforxAµ
Fuzzy Number A
⎪⎩
⎪⎨
⎧
≤<−≤<−
>≤=
532/)5(312/)1(
510)(
xforxxforx
xandxforxAµ
Fuzzy Number B
Calculate: A + B, A – B, A · B, A / B
Method for developing fuzzy arithmetic is based on interval arithmetic. Let A and B denote fuzzy numbers and * denotes any of four basicarithmetic. Then,
BABA ααα ∗=∗ )(
Example:
Arithmetic Operation on Fuzzy Number
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[ ]ααα 23,12 −−=A [ ]ααα 25,12 −+=B
[ ] ( ]1,048,4)( ∈−=+ αααα forBAAddition:
⎪⎩
⎪⎨
⎧
≤<−≤<
>≤=+
844/)8(404/
800)(
xforxxforx
xandxforxBAµ
Membership function of fuzzy number of A + B is:
Arithmetic Operation on Fuzzy Number
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[ ] ( ]1,042,64)( ∈−−=− αααα forBASubtraction:
⎪⎩
⎪⎨
⎧
≤<−−−≤<−+
>−≤=−
224/)2(264/)6(
260)(
xforxxforx
xandxforxBAµ
Membership function of fuzzy number of A – B is:
[ ] ( ][ ] ( ]⎪⎩
⎪⎨⎧
∈+−−
∈+−−+−=⋅
1,5.015164,145.0,015164,5124
)(22
22
αααα
αααααα
forfor
BA
Multiplication:
[ ]
[ ]⎪⎪
⎩
⎪⎪
⎨
⎧
<≤+−
<≤+
<≤−−−
≥−<
=⋅
1532/)1(4302/)1(
052/)4(31550
)(
2/1
2/1
2/1
xforxxforx
xforxxandxfor
xBAµ
Membership function of fuzzy number of A · B is:
Arithmetic Operation on Fuzzy Number
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[ ] ( ][ ] ( ]⎩⎨⎧
∈+−−−∈+−+−
=1,5.0)12/()23(),25/()12(5.0,0)12/()23(),12/()12(
)/(ααααααααααα
forfor
BA
Division:
⎪⎪⎩
⎪⎪⎨
⎧
<≤+−<≤++<≤−−+
≥−<
=
33/1)22/()3(3/10)22/()15(
01)22/()1(310
)(/
xforxxxforxx
xforxxxandxfor
xBAµ
Membership function of fuzzy number of A / B is:
Arithmetic Operation on Fuzzy Number
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Jakarta
Singapore
Kuala Lumpur
Bangkok
Manila
Fuzzy Relation
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Example of crisp relation:
Let X denotes a set of cities in Southeast Asia.
X = {Jakarta, Singapore, Kuala Lumpur, Bangkok, Manila}
Crisp relation that attempts to capture the relational concept near, is represented by the following relation
Jakarta
Singapore
Kuala Lumpur
Bangkok
Manila
Fuzzy Relation
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Using the same example as example of crisp relation, Fuzzy relation that attempts to capture the relational concept near, is represented by the following relation
Jakarta
Singapore
Kuala Lumpur
Bangkok
Manila
Jakarta
Singapore
Kuala Lumpur
Bangkok
Manila
1
0.9
0.6
0.3
0.10.1 0.20.4
0.5
1
Fuzzy Relation: Representations
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Consider the previous example, fuzzy relation is concisely represented by the matrix:
J S K B MJ
S
K
B
M
Fuzzy Relation: Representations
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Mapping Diagram
Consider as an example, a set of documents D = {d1, d2, d3, d4, d5} and a set of key terms T = {t1, t2, t3, t4}.
A Fuzzy relation expressing the degree of relevance of each document to each key term can be represented in the following mapping diagram
Fuzzy Relation: Representations
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Directed Graph
Fuzzy relation can be represented by a directed graph.
Fuzzy Relation: Basic Operation
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Inverse of a fuzzy relation (R-1)
Inverse (R-1) of a fuzzy relation (R) represented by a matrix, can be obtained by exchanging the rows of given matrix with the columns. The resulting matrix is called transpose of given matrix.
Example:
Fuzzy Relation: Basic Operation
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Composition of two fuzzy relations
a
b
c
XY
Z
1
2
3
4
A
B
C
P Q
a
b
c
A
B
C
X Z
P ◦ Q
Fuzzy Relation: Basic Operation
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Standard composition of fuzzy relations
Let P = [pij], Q = [qjk], and R = [rik] are matrix representations of fuzzy relations for which R = P ◦ Q. Matrices relation of composition of fuzzy relations is represented by expression:
[rik] = [pij] ◦ [qjk] where rik = max min(pij, qjk)j
Previous example:
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
5.010007.09.02.00017.0
P
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
4.01008.01003.0005.0
Q
Fuzzy Relation: Basic Operation
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