Transcript Fuzzy Sets and Fuzzy Reasoning

 Fuzzy sets and Crisp sets
 Fuzzy set operations
 Representation of Fuzzy Sets
 Fuzzy Relations
 Fuzzy Rules
 Fuzzy Inferencing
Types of Uncertainty and the
Modeling of Uncertainty
Stochastic Uncertainty:
 The Probability of Hitting the Target Is 0.8
Lexical Uncertainty:
 "Tall Men", "Hot Days", or "Stable Currencies"
 We Will Probably Have a Successful Business Year.
 The Experience of Expert A Shows That B Is Likely to
Occur. However, Expert C Is Convinced This Is Not True.
Most Words and Evaluations We Use in Our Daily Reasoning Are
Not Clearly Defined in a Mathematical Manner. This Allows
Humans to Reason on an Abstract Level!
Probability and Uncertainty
“... a person suffering from hepatitis shows in
60% of all cases a strong fever, in 45% of all
cases yellowish colored skin, and in 30% of all
cases suffers from nausea ...”
Stochastics and Fuzzy Logic
Complement Each Other !
© INFORM 1990-1998
Slide 4
Fuzzy versus Probability
Example #1
 A bottle of liquid has a probability of
#1
½ of being rat poison and ½ of being
pure water.
 A second bottle’s contents, in the
fuzzy set of liquids containing lots of
rat poison, is ½.
 The meaning of ½ for the two bottles
clearly differs significantly and would
impact your choice should you be
dying of thirst.
#2
(cite: Bezdek)
5
Fuzzy versus Probability
Example #2
 Fuzzy is said to measure “possibility” rather than “probability”.
 Difference


All things possible are not probable.
All things probable are possible.
 Contrapositive


All things impossible are improbable
Not all things improbable are impossible
6
Fuzzy sets


The concept of a set is fundamental to
mathematics.
However, our own language is also the supreme
expression of sets. For example, car indicates
the set of cars. When we say a car, we mean one
out of the set of cars.
Fuzzy Set Theory
Conventional (Boolean) Set Theory:
38.7°C
38°C
40.1°C
41.4°C
Fuzzy Set Theory:
42°C
39.3°C
“Strong Fever”
37.2°C
38.7°C
38°C
40.1°C
39.3°C
“More-or-Less” Rather Than “Either-Or” !
© INFORM 1990-1998
41.4°C
42°C
“Strong Fever”
37.2°C
Slide 8

The classical example in fuzzy sets is tall men.
The elements of the fuzzy set “tall men” are all
men, but their degrees of membership depend
on their height.
Degree of Membership
Crisp
Fuzzy
Name
Height, cm
Chris
Mark
John
208
205
198
1
1
1
1.00
1.00
0.98
Tom
David
181
179
1
0
0.82
0.78
Mike
Bob
Steven
172
167
158
0
0
0
0.24
0.15
0.06
Bill
Peter
155
152
0
0
0.01
0.00
Crisp and fuzzy sets of “tall men”
Degree of
Membership
1.0
Crisp Sets
0.8
Tall Men
0.6
0.4
0.2
0.0
150
160
170
180
190
200
210
Height, cm
Degree of
Membership
1.0
Fuzzy Sets
0.8
0.6
0.4
0.2
0.0
150
160
170
180
190
200
210
Height, cm

The x-axis represents the universe of discourse
 the range of all possible values applicable to a
chosen variable. In our case, the variable is the
man height. According to this representation,
the universe of men’s heights consists of all tall
men.

The y-axis represents the membership value of
the fuzzy set. In our case, the fuzzy set of “tall
men” maps height values into corresponding
membership values.

A fuzzy set is a set with fuzzy boundaries.
Let X be the universe of discourse and its
elements be denoted as x. In the classical set
theory, crisp set A of X is defined as function
fA(x) called the characteristic function of A
fA(x): X  {0, 1}, where
1, if x  A
f A ( x)  
0,if x  A

In the fuzzy theory, fuzzy set A of universe X is
defined by function A(x) called the
membership function of set A
A(x): X  [0, 1], where A(x) = 1 if x is totally in A;
A(x) = 0 if x is not in A;
0 < A(x) < 1 if x is partly in A.
For any element x of universe X, membership
function A(x) equals the degree to which x is an
element of set A.
degree of membership (membership value)
of element x in set A.
Crisp and fuzzy sets of short, average and tall
men
Degree of
Membership
1.0
Crisp Sets
Short
0.8
Average
Short
Tall
Tall Men
0.6
0.4
0.2
0.0
150
160
170
Degree of
Membership
1.0
180
190
200
210
Height, cm
200
210
Fuzzy Sets
0.8
Short
0.6
Tall
Average
0.4
Tall
0.2
0.0
150
160
170
180
190
Fuzzy Sets
Sets with fuzzy boundaries
A = Set of tall people
Crisp set A
1.0
Fuzzy set A
1.0
.9
Membership
.5
5’10’’
Heights
function
5’10’’
6’2’’
Heights
Membership Functions (MFs)
Characteristics of MFs:
 Subjective measures
 Not probability functions
“tall” in Asia
MFs
.8
“tall” in the US
.5
“tall” in Basketball team
.1
5’10’’
Heights
Fuzzy Sets
Formal definition:
A fuzzy set A in X is expressed as a set of ordered pairs:
A  {( x,  A ( x ))| x  X }
Fuzzy set
Membership
function
(MF)
Universe or
universe of discourse
A fuzzy set is totally characterized by a
membership function (MF).
Fuzzy Sets with Discrete Universes
Fuzzy set C = “desirable city to live in”
X = {SF, Boston, LA} (discrete and nonordered)
C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)}
Fuzzy set A = “sensible number of children”
X = {0, 1, 2, 3, 4, 5, 6} (discrete universe)
A = {(0, .1), (1, .3), (2, .7), (3, 1), (4, .6), (5, .2), (6, .1)}
Fuzzy Sets with Cont. Universes
Fuzzy set B = “about 50 years old”
X = Set of positive real numbers (continuous)
B = {(x, B(x)) | x in X}
B(x) 
1
 x  50 
1 

 10 
2
Alternative Notation
A fuzzy set A can be alternatively denoted as follows:
X is discrete
X is continuous
A

A
( xi ) / xi
xi X
A    A( x) / x
X
Note that S and integral signs stand for the union of
membership grades; “/” stands for a marker and does not
imply division.
Fuzzy Partition
Fuzzy partitions formed by the linguistic values “young”,
“middle aged”, and “old”:
More Definitions
Support
Core
Normality
Crossover points
Fuzzy singleton
a-cut, strong a-cut
Convexity
Fuzzy numbers
Bandwidth
Symmetricity
Open left or right, closed
MF Terminology
MF
1
.5
a
0
Core
Crossover points
a - cut
Support
X
Operations of fuzzy sets
(x)
(x)
B
1
1
A
A
0
1
x
0
B
1
Not A
0
Complement
x
Containment
(x)
1
1
A
B
0
1
x
AB
A
0
(x)
x
A
x
B
0
x
1
AB
0
Intersection
x
0
Union
x
Set-Theoretic Operations
Subset:
A  B  A  B
Complement:
A  X  A  A ( x )  1  A ( x )
Union:
C  A  B  c ( x )  max( A ( x ), B ( x ))  A ( x ) B ( x )
Intersection:
C  A  B  c ( x )  min( A ( x ), B ( x ))  A ( x ) B ( x )
Set-Theoretic Operations
subset.m
MF Formulation

x a c  x 
,
 , 0
b a c b 
Triangular MF:
trimf ( x ; a , b , c )  max min 
Trapezoidal MF:

d  x 
x a
trapmf ( x ; a , b , c , d )  max min
, 1,
 , 0
b a
d c  

Gaussian MF:

gaussmf ( x ; a , b , c )  e
Generalized bell MF:
gbellmf ( x ; a , b , c ) 
1  x c 
 

2  
2
1
x c
1
b
2b
MF Formulation
Properties of Set Operations
Most of the properties that hold for classical sets (e.g., commutativity, associativity and idempotence)
hold also for fuzzy sets except for following two properties:
Law of contradiction
A A  
~
~
the intersection of a fuzzy set and its complement results in a fuzzy set with membership values of up to ½
and thus does not equal the empty set (as in the case of classical sets)
Law of excluded middle
A A U
~
~
Representation of Fuzzy Set
 Explicitly -- as Table
 Implicitly – as Function to Compute Membership
How to represent a fuzzy set in a
computer?


First, we determine the membership functions.
In our “tall men” example, we can obtain fuzzy
sets of tall, short and average men.
The universe of discourse  the men’s heights 
consists of three sets: short, average and tall
men. As you will see, a man who is 184 cm tall is
a member of the average men set with a degree
of membership of 0.1, and at the same time, he
is also a member of the tall men set with a
degree of 0.4.
Linguistic Variables
A numerical variables takes numerical values:
Age = 65
A linguistic variables takes linguistic values:
Age is old
A linguistic values is a fuzzy set.
All linguistic values form a term set:
T(age) = {young, not young, very young, ...
middle aged, not middle aged, ...
old, not old, very old, more or less old, ...
not very yound and not very old, ...}
Linguistic Values (Terms)
complv.m
Linguistic Hedges
Very:
Very( A)  A
Somewhat:
Somewhat( A)  A
Extremely
Extremely( A)  A
2
0.5
8
Fuzzy rules
In 1973, Lotfi Zadeh published his second most
influential paper. This paper outlined a new
approach to analysis of complex systems, in which
Zadeh suggested capturing human knowledge in
fuzzy rules.
What is a fuzzy rule?
A fuzzy rule can be defined as a conditional
statement in the form:
IF
x is A
THEN y is B
where x and y are linguistic variables; and A and B
are linguistic values determined by fuzzy sets on
the universe of discourses X and Y, respectively.
What is the difference between classical
and fuzzy rules?
A classical IF-THEN rule uses binary logic, for
example,
Rule: 1
IF
speed is > 100
THEN stopping_distance is long
Rule: 2
IF
speed is < 40
THEN stopping_distance is short
The variable speed can have any numerical value
between 0 and 220 km/h, but the linguistic
variable stopping_distance can take either value
long or short. In other words, classical rules are
expressed in the black-and-white language of
Boolean logic.
We can also represent the stopping distance rules
in a fuzzy form:
Rule: 1
IF
speed is fast
THEN stopping_distance is long
Rule: 2
IF
speed is slow
THEN stopping_distance is short
In fuzzy rules, the linguistic variable speed also
has the range (the universe of discourse) between
0 and 220 km/h, but this range includes fuzzy
sets, such as slow, medium and fast. The universe
of discourse of the linguistic variable
stopping_distance can be between 0 and 300 m
and may include such fuzzy sets as short, medium
and long.
Fuzzy rules relate fuzzy sets.
 In a fuzzy system, all rules fire to some
extent, or in other words they fire partially.
If the antecedent is true to some degree of
membership, then the consequent is also
true to that same degree.

Product set
40
Product set
 A={a1,a2} B={b1,b2} C={c1,c2}
 AxBxC =
{(a1,b1,c1),(a1,b1,c2),(a1,b2,c1),(a1,b2,c2),(a2,b1,c1),(a2,
b1,c2),(a2,b2,c1), (a2,b2,c2)}
41
Crisp relation
 A relation among crisp sets A1 , A2 ,
, An is a subset of
the Cartesian product. It is denoted by R
R  A1  A2   An
 Using the membership function defines the crisp relation
R:
1 iff (x1 , x2 , ..., xn )  R,
 R ( x1 , x2 , ,xn )  
0 otherwise
where x1  A1 , x2  A2 ,..., xn  An
42
Fuzzy relation
 A fuzzy relation is a fuzzy set defined on the Cartesian
product of crisp sets A1, A2, ..., An where tuples (x1, x2, ...,
xn) may have varying degrees of membership within the
relation.
 The membership grade indicates the strength of the
relation present between the elements of the tuple.
R : A1  A2  ...  An  [0,1]
R  (( x1 , x2 ,..., xn ), R ) | R ( x1 , x2 ,..., xn )  0, x1  A1 , x2  A2 ,..., xn  An 
43
Representation methods
 Bipartite graph
(Crisp)
(Fuzzy)
44
Representation methods
 Matrix
B
y1
y2
y3
B
y4
x1
x1
x2
x2
x3
x3
x4
x4
(Crisp)
y1
y2
y3
y4
(Fuzzy)
45
Representation methods
 Digraph
(Crisp)
(Fuzzy)
46
Domain and range of fuzzy
relation  ( x)  max  ( x, y)
 Domain:
 Range :
dom ( R )
yB
R
ran ( R ) ( y )  max  R ( x, y )
xA
domain
range
47
Domain and range of fuzzy
relation
 Fuzzy matrix
dom ( R ) ( x1 )  1.0
dom ( R ) ( x2 )  0.4
dom ( R ) ( x3 )  1.0
dom ( R ) ( x4 )  1.0
dom ( R ) ( x5 )  0.5
dom ( R ) ( x6 )  0.2
48
Operations on fuzzy matrices
 Sum:
A  B  max[aij , bij ]
 Example
49
Operations on fuzzy matrices
 Max product: C = A・B=AB=
 Example
C12  ?
50
Max product
 Example
C12  0.1
51
Max product
 Example
C13  0.5
52
Max product
 Example
C
53
Operations on fuzzy matrices
 Scalar product:
 A where 0    1
 Example
a
0.5 A 
b
c
a
b
0.1 0.25 0.0
0.2 0.5 0.05
c
0.0
0.5
0.0
54
Operations on fuzzy relations
 Union relation
( x, y)  A  B
RS ( x, y)  max( R ( x, y), s ( x, y))
 R ( x, y)  s ( x, y)
 For n relations
( x, y )  A  B
 R  R ... R ( x, y )    R ( x, y )
1
2
n
Ri
i
55
Union relation
 Example
56
Operations on fuzzy relations
 Intersection relation
( x, y)  A  B
 RS ( x, y)  min(  R ( x, y), s ( x, y))
  R ( x, y)  s ( x, y)
 For n relations
( x, y )  A  B
 R  R ... R ( x, y )    R ( x, y )
1
2
n
Ri
i
57
Intersection relation
 Example
58
Operations on fuzzy relations
 Complement relation:
( x, y)  A  B
R ( x, y)  1  R ( x, y)
 Example
59
Composition of fuzzy relations
 Max-min composition ( x, y)  A  B, ( y, z )  B  C
S R ( x, z)  max[min(R ( x, y), S ( y, z))]
y
 [R ( x, y)  S ( y, z)]
y
 Example
60
Composition of fuzzy relations
61
Composition of fuzzy relations
 Example
S R (1, a )  max[min(0.1,0.9), min(0.2,0.2), min(0.0,0.8), min(1.0,0.4)]
 max[0.1,0.2,0.0,0.4]  0.4
62
Composition of fuzzy relations
 Example
S R (1,  )  max[min(0.1,0.0), min(0.2,1.0), min(0.0,0.0), min(1.0,0.2)]
 max[0.0,0.2,0.0,0.2]  0.2
63
Composition of fuzzy relations
64
α-cut of fuzzy relation

 Example
65
α-cut of fuzzy relation
66
Fuzzy rules relate fuzzy sets.
 In a fuzzy system, all rules fire to some
extent, or in other words they fire partially.
If the antecedent is true to some degree of
membership, then the consequent is also
true to that same degree.

Fuzzy sets of tall and heavy men
Degree of
Membership
1.0
Degree of
Membership
1.0
Heavy men
0.8
Tall men
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
160
0.0
180
190
200
Height, cm
70
80
100
120
Weight, kg
These fuzzy sets provide the basis for a weight
estimation model. The model is based on a
relationship between a man’s height and his weight:
IF
height is tall
THEN weight is heavy
The value of the output or a truth membership
grade of the rule consequent can be estimated
directly from a corresponding truth membership
grade in the antecedent. This form of fuzzy
inference uses a method called monotonic
Degree of
Degree of
selection.
Membership
1.0
Membership
1.0
Tall men
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
160
180
200
190
Height, cm
Heavy men
70
80
120
100
Weight, kg
A fuzzy rule can have multiple antecedents, for
example:
IF
AND
AND
THEN
project_duration is long
project_staffing is large
project_funding is inadequate
risk is high
IF
service is excellent
OR
food is delicious
THEN tip is generous
Extension of fuzzy set
A crisp function
f : X Y
f ( A)  { y | y  f ( x), x  A}
f ( B)  {x | f ( x)  B}
Let A  {a1, a 2, a3}
B  {b1, b2}
then
f : a1  b1
f : a 2  b2
f : a 3  b3
71
Extension of fuzzy set
There are two universal sets
X  {x1, x 2,...,xn}
Y  { y1, y 2,..., yn}
And
A X,B Y
A  {( x1,1), ( x2,1), ( x3,1)}
We can obtain B by A and R, use
R  {((x1, y1),1), ((x2, y 2),1), ((x3, y 2),1)}
 ( y)  Max[min( ( x), R( x, y))],x  A, y  B
72
Extension of fuzzy set
By
 ( y)  Max[min( ( x), R( x, y))],x  A, y  B
 ( y1)  Max[min( ( x1), R ( x1, y1))]  max[min(1,1)]  1
 ( y 2)  Max[min( ( x 2), R ( x 2, y 2)),min( ( x 3), R ( x 3, y 2))]
 max[min(1,1), min(1,1)]  1
get B  {( y1,1), ( y 2,1)}
73
Extension of fuzzy set
If A is a fuzzy set
and R is
A  {( x1,0.2), ( x2,0.7), ( x3,0.6)}
R  {((x1, y1),1), ((x2, y 2),1), ((x3, y 2),1)}
We can also get B by A an R, use
 ( y)  Max[min( ( x), R( x, y))],x  X , y Y
74
Extension of fuzzy set
By use
 ( y)  Max[min( ( x), R( x, y))],x  X , y Y
 ( y1)  Max[min( ( x1), R ( x1, y1))]  Max[min(0.2,1)]  0.2
 ( y 2)  Max[min( ( x 2), R ( x 2, y 2)),min( ( x3), R ( x3, y 2))]
 Max[min(0.7,1), min(0.6,1)]  0.7
so we can get B  {( y1,0.2), ( y 2,0.7)}
75
Extension of fuzzy set
A  {( x1,0.2), ( x2,0.7), ( x3,0.6)}
If A is a fuzzy set
and R is a fuzzy relation
R  {((x1, y1),0.5), ((x2, y 2),0.4), ((x3, y 2),0.9)}
We can get B by using
 ( y)  Max[min( ( x), R( x, y))],x  X , y Y
76
Extension of fuzzy set
By
 ( y)  Max[min( ( x), R( x, y))],x  X , y Y
 ( y1)  Max[min( ( x1), R ( x1, y1))]  Max[min(0.2,0.5)]  0.2
 ( y 2)  Max[min( ( x 2), R ( x 2, y 2)),min( ( x3), R ( x 3, y 2))]
 Max[min(0.7,0.4), min(0.6,0.9)]  0.6
so we can get B  {( y1,0.2), ( y 2,0.6)}
77
Extension of fuzzy set
Extension of a crisp relation
78
Extension of fuzzy set
79
Extension by fuzzy relation
80
Extension by fuzzy relation
81
Extension by fuzzy relation
MR 2
82
Extension by fuzzy relation
MR 2
B ' (b1 )  max[min(0.8, 0.3), min(0.3, 0.8)]  0.3
B ' (b2 )  max[min(0.8,1.0), min(0.3, 0.0)]  0.8
B ' (b3 )  max[min(0.8, 0.0), min(0.3, 0.0)]  0.0
83
Extension by fuzzy relation
B '  {(b1,0.3),(b2 ,0.8),(b3 ,0.0)}
MR 2
C ' (c1 )  max[min(0.3,0.7), min(0.8,0.2), min(0.0, 0.0)]  0.3
C ' (c2 )  max[min(0.3,0.4), min(0.8,0.0), min(0.0, 0.3)]  0.3
C ' (c3 )  max[min(0.3,1.0), min(0.8,0.3), min(0.0, 0.9)]  0.8
84
Extension by fuzzy relation
85
Fuzzy Inference
member
ship
values
Fuzzy
Controller
Defuzzifier
Rule
Engine
real
numbers
Fuzzifier
System
real
numbers
fuzzy
sets
Basic Elements of a
Fuzzy Reasoning System
Fuzzy Logic Defines
the Control Strategy
on a Linguistic Level!
Fuzzification, Fuzzy Inference, Defuzzification:
Measured Variables
(Linguistic Values)
2. Fuzzy-Infe re nce
Command Variables
(Linguistic Values)
Linguistic
Lev e l
Nume rical
Lev e l
3. De fuzzification
1. Fuzzification
Measured Variables
(Numerical Values)
Plant
Command Variables
(Numerical Values)
Basic Elements of a
Fuzzy Logic System
Container Crane Case Study:
Two Measured
Variables and One
Command Variable !
© INFORM 1990-1998
Slide 88
Basic Elements of a
Fuzzy Logic System
Control Loop of the Fuzzy Logic Controlled Container Crane:
Angle, Distance
(Numerical Values)
2. Fuzzy-Infe re nce
Closing the Loop
With Words !
Power
(Linguistic Variable)
Linguistic
Lev e l
Nume rical
Lev e l
Angle, Distance
(Numerical Values)
© INFORM 1990-1998
3. De fuzzification
1. Fuzzification
Containe r Crane
Power
(Numerical Values)
Slide 89
1. Fuzzification:
- Linguistic Variables The Linguistic
Term Definitions:
Variables Are the
Distance
:= {far, medium, close, zero, neg_close}
Angle
:= {pos_big, pos_small, zero, neg_small, neg_big} “Vocabulary” of a
Fuzzy Logic System !
Power
:= {pos_high, pos_medium, zero, neg_medium,
neg_high}
Membership Function Definition:
µ neg_close zero close medium
µ
zero
far
neg_big
neg_small
pos_small
pos_big
1
1
0.9
0.8
0.2
0.1
0
0
-90°
-45°
0°
4°
Angle
© INFORM 1990-1998
45°
90°
-10
0
10
12m
Distance [yards]
20
30
Slide 90
2. Fuzzy-Inference:
- “IF-THEN”-Rules Computation of the “IF-THEN”-Rules:
#1: IF Distance = medium AND Angle = pos_small THEN Power = pos_medium
#2: IF Distance = medium AND Angle = zero THEN Power = zero
#3: IF Distance = far AND Angle = zero THEN Power = pos_medium
 Aggregation:
 Composition:
© INFORM 1990-1998
Computing the “IF”-Part
Computing the “THEN”-Part
Slide 91
2. Fuzzy-Inference:
- Aggregation Boolean Logic Only
Defines Operators for 0/1:
A
0
0
1
1
B
0
1
0
1
AvB
0
0
0
1
Fuzzy Logic Delivers
a Continuous Extension:
 AND: µAvB = min{ µA; µB }
 OR:
µA+B = max{ µA; µB }
 NOT: µ-A = 1 - µA
Aggregation of the “IF”-Part:
#1: min{ 0.9, 0.8 } = 0.8
#2: min{ 0.9, 0.2 } = 0.2
#3: min{ 0.1, 0.2 } = 0.1
© INFORM 1990-1998
Aggregation Computes How
“Appropriate” Each Rule Is for
the Current Situation !
Slide 92
2. Fuzzy-Inference:
Composition
Result for the Linguistic Variable "Power":
pos_high
with the degree 0.0
pos_medium
with the degree 0.8
zero
with the degree 0.2
neg_medium
with the degree 0.0
neg_high
with the degree 0.0
( = max{ 0.8, 0.1 } )
Composition Computes
How Each Rule Influences
the Output Variables !
© INFORM 1990-1998
Slide 93
3. Defuzzification
Finding a Compromise Using “Center-of-Maximum”:
µ
neg_high
neg_medium zero pos_medium
pos_high
1
“Balancing” Out
the Result !
0
-30
-15
0
Power [Kilowatts]
© INFORM 1990-1998
15
30
6.4 KW
Slide 94
Defuzzification methods
Center of Gravity Method (COG)
yCOG 
 y
y
( y)dy
yY

yY
y
( y)dy
Defuzzification methods
Center of Sums Method (COS)
r
yCOS 
  y
i 1 yY
r
 
i 1 yY
yi
yi
( y )dy
( y )dy
Defuzzification methods
Mean of Maxima Method (MOM)
y MOM 
 ydy

yMAX (
y)
 dy
yMAX (
y)
Fuzzification- rule evaluationdefuzzification
1
y'(MOM)
0.7
y'(COG)
0
1
1.5
2
3
4
y
1.9
y'(COG) =
(0 × 0) + (1 × 1) + (1 × 2) + (0.7 × 3)
 1.9
1 + 1 + 0.7
y'(MOM) =
1+2
= 1.5
2
Another Example
 Assume that we need to evaluate student applicants based on their GPA and GRE
scores.
 For simplicity, let us have three categories for each score [High (H), Medium (M), and
Low(L)]
 Let us assume that the decision should be Excellent (E), Very Good (VG), Good (G),
Fair (F) or Poor (P)
 An expert will associate the decisions to the GPA and GRE score. They are then
Tabulated.
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99
Fuzzy Rule Table
GRE
H
M
L
H
GPA
M
L
100
Fuzzification
 Fuzzifier converts a crisp input into a vector of fuzzy
membership values.
 The membership functions
 reflects the designer's knowledge
 provides smooth transition between fuzzy sets
 are simple to calculate
 Typical shapes of the membership function are
Gaussian, trapezoidal and triangular.
Robert Jackson Marks II
101

GRE

GRE
= {L , M , H }
102

GPA

GPA
= {L , M , H }
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Membership Function of the
Consequent
F
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 Transform the crisp antecedents into a vector of fuzzy
membership values.
 Assume a student with GRE=900 and GPA=3.6.
Examining the membership function gives


GRE
= {L = 0.8 , M = 0.2 , H = 0}
GPA
= {L = 0 , M = 0.6 , H = 0.4}
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F
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 Converts the output fuzzy numbers into a unique (crisp) number
 Method: Add all weighted curves and find the center of mass
F
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 Fuzzy set with the largest membership value is selected.
 Fuzzy decision:
F = {B, F, G,VG, E}
F = {0.6, 0.4, 0.2, 0.2, 0}
 Final Decision (FD) = Bad Student
 If two decisions have same membership max, use the average of the two.
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F
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