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This document is prepared by ThiangSistem Cerdas: Fuzzy Set and Fuzzy Logic - 1
Sistem Cerdas (TE 4485)
Instructor: ThiangRoom: I.201
Phone: 031-2983115Email: [email protected]
Fuzzy Set, Fuzzy Logic, and its Applications
Introduction
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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
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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:
5.3/03/2.075.2/3.05.2/4.02/15.1/4.025.1/3.01/2.05.0/0
++++++++=B
2 1 3
( )xBµ
1
Fuzzy Set: Basic Concept
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Support of fuzzy set
( ) ( ){ }0/ >= xxFSupp Fµ
Core of fuzzy set
( ) ( ){ }1/ == xxFCore Fµ
Height of fuzzy set
( ) ( ){ }xFh Fµmax=
A fuzzy set F is called normal when h(F) = 1; it is calledsubnormal when h(F) < 1
Fuzzy Set: Basic Concept
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α-cut of fuzzy set
( ){ }αµα ≥= xxF F/
Strong α-cut of fuzzy set
( ){ }αµα >=+ xxF F/
Complement of fuzzy set ( )
( ) ( ) ( )xFhx FF µµ −=
F
Fuzzy Set: example
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( ) ( )60,202 =ASupp
( ) [ ]45,352 =ACore
( ) 12 =Ah
[ ]5.52,5.2725.0 =A
( )5.52,5.2725.0 =+A
)( colorredareasolidF =
Fuzzy Set: Basic Concept
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Fuzzy Subset
( ) ( ) xallforxx BA µµ ≤A fuzzy set, A, is said to be a subset of fuzzy set, B, if
Fuzzy Union (Logic “OR”)
( ) ( ) ( ) ( )[ ]xxxx BABABA µµµµ ,max== ∪+
commutative, associative
Fuzzy Set: Basic Concept
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Fuzzy Intersection (Logic “AND”)
Associativity (1)
( ) ( ) ( ) ( )[ ]xxxx BABABA µµµµ ,min== ∩•
commutative, associative
Min-Max fuzzy logic has intersection distributive over union
( ) ( )xx CABACBA )()()( ⋅+⋅+⋅ = µµ
[ ] [ ]),min(),,min(max),max(,min CABACBA =
Fuzzy Set: Basic Concept
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Associativity (2)Min-Max fuzzy logic has union distributive over intersection
( ) ( )xx CABACBA )()()( +⋅+⋅+ = µµ
[ ] [ ]),max(),,max(min),min(,max CABACBA =
DeMorgan’s Law (1)Min-Max fuzzy logic obeys DeMorgan’s Law #1
( ) ( )xx CBCB +• = µµ
[ ])1(),1(max),min(1 CBCB −−=−
Fuzzy Set: Basic Concept
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DeMorgan’s Law (2)Min-Max fuzzy logic obeys DeMorgan’s Law #2
( ) ( )xx CBCB •+ = µµ
[ ])1(),1(min),max(1 CBCB −−=−
The Law of Excluded MiddleMin-Max fuzzy logic fails the law of excluded middle
oAA /≠•0)1,min( ≠− AA
The Law of ContradictionMin-Max fuzzy logic fails the law of contradiction
UAA ≠+1)1,max( ≠− AA
Fuzzy Set: Basic Concept
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Fuzzy Set: Basic Concept
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The intersection and union operations can also be used to assign memberships on the Cartesian product of two sets
Consider, as an example, the fuzzy membership of a set, G, of liquids that taste good and the set, LA, of cities far from Los Angeles
µG = 0.0/Swamp Water + 0.5/Radish Juice + 0.9/Grape Juice
µLA = 0.0/LA + 0.5/Chicago + 0.8/New York + 0.9/London
Cartesian Product
Fuzzy Set: Basic Concept
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We form the set, E, of Liquids that taste good AND cities that are far from Los Angeles
Cartesian Product
LAGE •=
The following table results
Fuzzy Set: Example
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Determine: 31 AA ∩
)()( 3221 AAAA ∩∪∩
Fuzzy Set: Answers
31 AA ∩
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)()( 3221 AAAA ∩∪∩
Fuzzy Set: Answers
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Fuzzy Arithmetic: Fuzzy number
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A fuzzy set is a fuzzy number if the fuzzy set meets the following properties:
• The fuzzy set must be a normal fuzzy set
• α-cut of the fuzzy set must be a closed interval
•Support of the fuzzy set must be an open interval
Example of fuzzy number
and fuzzy interval
Arithmetic Operation on Interval
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Four arithmetic operations on closed intervals:
[a, b] + [d, e] = [a + d, b + e]
[a, b] – [d, e] = [a – e, b – d]
[a, b] · [d, e] = [min(ad, ae, bd, be), max(ad, ae, bd, be)]
[a, b] / [d, e] = [min(a/d, a/e, b/d, b/e), max(a/d, a/e, b/d, b/e)]
Example:[-3, 4] + [-1, 2] = [-4, 6][-3, 3] – [-4, 3] = [-6, 7][-4, 2] · [-2, 4] = [-16, 8]
[-1, 3] / [2, 4] = [-0.5, 1.5]
[-3, 4] + [-1, 2] = [?, ?][-3, 3] – [-4, 3] = [?, ?][-4, 2] · [-2, 4] = [?, ?][-1, 3] / [2, 4] = [?, ?]
Arithmetic Operation on Fuzzy Number
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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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Matrices
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
=
15.04.02.01.05.017.05.03.04.07.018.06.02.05.08.019.01.03.06.09.01
R
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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⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
5.010007.09.02.00017.0
P
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
4.01008.01003.0005.0
Q
5.0)]0,0min(),1,0min(),3.0,1min(),5.0,7.0max[min()],min(),,min(),,min(),,max[min(
11
411431132112111111
===
rqpqpqpqpr
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡==
4.08.0107.07.0005.0
QPR o
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡==
4.08.0107.07.0005.0
QPR o
Fuzzy Relation: Basic Operation
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a
b
c
A
B
C
X Z
P ◦ Q
Result of composition of fuzzy relation P and Q:
Fuzzy Inference
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Crisp Input
Fuzzification
Rules
Defuzzification
Crisp Output
Antecedent
Consequent
InputMembership
Function
OutputMembership
Function
Fuzzy Inference
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• Assume that we need to evaluate student applicants based on their
GPA and GRE score.
• For simplicity, there are three categories for each score [High (H),
Medium (M), and Low (L)].
• Assume that the decision should be Excellent (E), Very Good (VG),
GOOD (G), Fair (F), and Poor (P).
• An expert will associate the decisions to the GPA and GRE score.
They are then tabulated in Fuzzy If-then Rules form.
Example: Student Applicants Evaluation
Fuzzy Inference
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Example of Fuzzy If-Then Rules
If the GRE is HIGH and the GPA is HIGH then the STUDENT will be EXCELLENT
If the GRE is LOW and the GPA is HIGH then the STUDENT will be FAIR
Antecedent
Consequent
Fuzzy Linguistic Variables
Fuzzy Inference
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Fuzzy If-Then Rules Table
Fuzzy Inference
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Membership Function for GRE
1
1200 1600
LOW MEDIUM HIGH
800
µGRE
Typical shapes of membership function are triangular, trapezoidal, and Gaussian
Fuzzy Inference
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Membership Function for GPA
1
3.0 3.8
LOW MEDIUM HIGH
2.2
µGPA
Fuzzy Inference
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Membership Function for Consequent (Student)
1
70 80
P
60
µC
90 100
F G VG E
Example:Evaluate a student who has GRE of 900 and GPA of 3.6!
Fuzzification
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Convert the crisp inputs (antecedents) into vector of fuzzy membership values
1
1200 1600
LOW MEDIUM HIGH
800
µGRE
0.25
900
0.75
{ }0,25.0,75.0 ==== HMLGRE µµµµResult:
Fuzzification
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0.25
3.6
0.75
{ }75.0,25.0,0 ==== HMLGPA µµµµ
1
3.0 3.8
LOW MEDIUM HIGH
2.2
µGPA
Result:
Rule Evaluation: Min-Max Strategy
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0.75 0.25 0
0
0.25
0.75
0 0 0
0
0
0.25
0.25
0.25
0.75
{ }0,0,25.0,75.0,25.0 ====== EVGGFPC µµµµµµResult:
Defuzzification
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1
70 80
P
60
µC
90 100
F G VG E
0.25
0.75
Result: Student is Fair
{ }0,0,25.0,75.0,25.0 ====== EVGGFPC µµµµµµ
Center of Area