Matakuliah
Tahun
Versi
: H0434/Jaringan Syaraf Tiruan
: 2005
:1
Pertemuan 21
MEMBERSHIP FUNCTION
1
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Menjelaskan konsep fungsi keanggotaan
pada logika fuzzy.
2
Outline Materi
• Pengertian Fungsi keanggotaan.
• Derajat keanggotaan.
3
FUZZY LOGIC
• Lotfi A. Zadeh
“Fuzzy Sets”,
Information and Control,
Vol 8, pp.338-353,1965.
Clearly, the “class of all real
numbers which are much greater
than 1,” or “the class of beautiful
women,” or “the class of tall men,”
do not constitute classes or sets in
the usual mathematical sense of
these terms (Zadeh, 1965).
4
PROF. ZADEH
Fuzzy theory should not be regarded as a
single theory, but rather a methodology to
generalize a specific theory from being
discrete, to being more continuous
5
WHAT IS FUZZY LOGIC

Fuzzy logic is a superset of conventional
(boolean) logic

An approach to uncertainty that combines real
values [0,1] and logic operations

In fuzzy logic, it is possible to have partial truth
values

Fuzzy logic is based on the ideas of fuzzy set
theory and fuzzy set membership often found
in language
6
WHY USE FUZZY LOGIC ?
 An Alternative Design Methodology Which Is
Simpler, And Faster
• Fuzzy Logic reduces the design
development cycle
• Fuzzy Logic simplifies design complexity
• Fuzzy Logic improves time to market
 A Better Alternative Solution To Non-Linear
Control
• Fuzzy Logic improves control performance
• Fuzzy Logic simplifies implementation
• Fuzzy Logic reduces hardware costs
7
Precision in the model
WHEN USE FUZZY LOGIC
Mathematical
Equation
Model free methods
Fuzzy System
Complexity of System
Where few numerical data exist and where only
ambiguous or imprecise information maybe
available.
8
FUZZY SET

In natural language, we commonly
employ:





classes of old people
Expensive cars
numbers much greater than 1
Unlike sharp boundary in crisp set, here
boundaries seem vague
Transition from member to nonmember
appears gradual rather than abrupt
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CRISP AND FUZZY
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FUZZY SET AND MEMBERSHIP
FUNCTION





Universal Set X – always a crisp set.
Crisp set assigns value {0,1} to
members in X
Fuzzy set assigns value [0,1] to members in X
These values are called the membership
functions m.
Membership function of a fuzzy set A is
denoted by :
A: X  [0,1]
A: [x1/m1, x2/m2, …, xn/mn}
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HIMPUNAN
CRISP DAN FUZZY
Himpunan kota yang dekat dengan Bogor
• A = { Jakarta, Sukabumi, Cibinong, Depok }
 CRISP
• B = { (0.7 /Jakarta) , (0.6 /Sukabumi) , (0.9
/Cibinong) , (0.8/Depok) }  FUZZY
Angka
0.6
–
0.9
menunjukkan
tingkat
keanggotaan ( degree of membership )
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CONTOH
Grade ( m )
1
1
Dingin
0
Panas
30 oC
CRISP
0
30 oC
FUZZY
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TINGGI BADAN
m
CRISP
sedang
1
0
150
155
160
tinggi
m
sedang
FUZZY
1
0,5
0
150
155
160
tinggi
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SEASONS
Spring
Summer
Autumn
Winter
Membership
1
0.5
0
Time of the year
15
AROUND 4
Membership
1
0.5
0
0
2
4
Measurements
6
8
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AGE
Membership
1
young
very young
old
0.5
more or less old
not very young
0
0
20
40
60
Age
80
100
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MEMBERSHIP FUNCTION
1
0.5
0
(a)
(d)
(g)
(j)
(b)
(e)
(h)
(k)
1
0.5
0
1
0.5
0
-100
0
(c)
100 -100
0
(f)
100 -100
0
(i)
100 -100
0
(l)
100
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SET OPERATION
1
B
Membership
A
0.5
0
0
20
AUB
40
60
AIB
80
AUB
100
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SET OPERATION
A
AB
B
AB
A
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LINGUISTIC VARIABLES
Linguistic variable is ”a variable whose
values are words or sentences in a natural
or artificial language”. Each linguistic
variable may be assigned one or more
linguistic values, which are in turn connected
to a numeric value through the mechanism
of membership functions.
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LINGUISTIC VARIABLES
Fuzzy linguistic terms often consist of two parts:
1) Fuzzy predicate : expensive, old, rare, dangerous,
good, etc.
2) Fuzzy modifier: very, likely, almost impossible,
extremely unlikely, etc.
The modifier is used to change the meaning of predicate
and it can be grouped into the following two classes:
a) Fuzzy truth qualifier or fuzzy truth value: quite true,
very true, more or less true, mostly false, etc.
b) Fuzzy quantifier: many, few, almost, all, usually, etc.
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FUZZY PREDICATE
Fuzzy predicate
– If the set defining the predicates of individual is a fuzzy
set, the predicate is called a fuzzy predicate
Example
– “z is expensive.”
– “w is young.”
– The terms “expensive” and “young” are fuzzy terms.
Therefore the sets “expensive(z)” and “young(w)” are
fuzzy sets
23
FUZZY PREDICATE
When a fuzzy predicate “x is P” is given, we can
interpret it in two ways :
• P(x) is a fuzzy set. The membership degree of x in
the
set P is defined by the membership function
mP(x)
• mP(x) is the satisfactory degree of x for the
property P.
Therefore, the truth value of the fuzzy predicate is
defined by the membership function :
Truth value = mP(x)
24
FUZZY VARIABLES


Variables whose states are defined by
linguistic concepts like low, medium, high.
These linguistic concepts are fuzzy sets
themselves.
Very
Low
High
high
Temperature
Trapezoidal membership functions
25
FUZZY VARIABLES


Usefulness of fuzzy sets depends on our
capability
to
construct
appropriate
membership functions for various given
concepts in various contexts.
Constructing
meaningful
membership
functions is a difficult problem –GAs have
been employed for this purpose.
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EXAMPLE
if speed is interpreted as a linguistic variable, then its
term set T (speed) could be T = { slow, moderate, fast,
very slow, more or less fast, sligthly slow, ……..}.
where each term in T (speed) is characterized by a
fuzzy set in a universe of discourse U = [0; 100]. We
might interpret
• slow as “ a speed below about 40 km/h"
• moderate as “ a speed close to 55 km/h"
• fast as “ a speed above about 70 km/h"
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SPEED
Values of linguistic variable speed.
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NORMALIZED
DOMAIN INPUT
• NB (Negative Big), NM (Negative
Medium)
• NS (Negative Small), ZE (Zero)
• PS (Positive Small), PM (Positive
Medium)
• PB (Positive Big)
A possible fuzzy partition of [-1; 1].
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MEMBERSHIP FUNCTION
m
Positive
Large
1
0
Negative
Medium
m
0
m
NL
-
NM
NS ZERO PS
0
PM
PL
+
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