FUZZY SETS
Membership Value Assignment
• There are possible more ways to assign
membership values or function to fuzzy
variables than there are to assign probability
density functions to random variables [Dubois
and Prade, 1980]
Fuzzy Logic with Engineering Applications: Timothy J. Ross
Membership Value Assignment
•
•
•
•
•
•
•
•
Intuition
Inference
Rank ordering
Angular fuzzy sets
Neural networks
Genetic algorithms
Inductive reasoning
Soft partitioning
Fuzzy Logic with Engineering Applications: Timothy J. Ross
Intutition
• Derived from the capacity of humans to
develop membership functions through their
own innate intelligence and understanding.
• Involves contextual and semantic knowledge
about an issue; it can also involve linguistic
truth values about this knowledge.
Fuzzy Logic with Engineering Applications: Timothy J. Ross
Types of Membership Functions
• The most commonly used in practice are
–
–
–
–
–
Triangles
Trapezoids
Bell curves
Gaussian, and
Sigmoidal
Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall
Triangular MF
b
a
c
Specified by three parameters {a,b,c} as follows:
0
x<a 

(x − a) (b − a) a ≤ x ≤ b
triangle(x : a,b,c) = 

(c − x) (c − b) b ≤ x ≤ c


0
x >c 

Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall
Trapezoidal MF
c
b
a
d
Specified by four parameters {a,b,c,d} as follows:
0
x<a 

(x − a) (b − a) a ≤ x < b 


1
b ≤ x < c
trapezoidal(x : a,b,c,d) = 
(d − x) (d − c) c ≤ x ≤ d 

0
x ≥ d 

Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall
Gaussian MF
σ
m
Specified by two parameters {m,σ} as follows:
 (x − m)2 
gaussian(x : m,σ ) = exp −

2
σ


Where m and σ denote the center and width of the function, respectively
A small σ will generate a “thin”MF, while a big σ will lead to a “flat”MF.
Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall
Bell-shaped MF
c
Specified by three parameters {a,b,c} as follows:
1
bell(x : a,b,c) =
2b
x−c
1+
a
Where the parameter b is usually positive and we can adjust c and a to vary the
center and width of the function and then use b to control the slopes.
Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall
Bell-shaped MF
1
bell(x : a,b,c) =
2b
x−c
1+
a
c
Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall
Sigmoidal MF
Specified by two parameters {a, c} as follows:
Sigmoidal(x : a,c) =
1
1+ e
− a(x − c)
Where c is the center of the function and a control the slope.
Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall
Sigmoidal MF
Sigmoidal(x : a,c) =
Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall
1
1+ e
− a(x − c)
Hedges: a modifier to a fuzzy set
• Hedge modifies the meaning of the original
set to create a compound fuzzy set
– Example:
• Very (Concentration)
• More or Less(Dilation)
Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall
Hedges: Very & MoreOrLess
Very :
µveryA ( x ) = [ µ A ( x )]
2
MoreorLess :
µMoreOrLessA ( x) = µ A ( x )
Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall
Hedges: Very
Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall
Hedges: VeryVeryVery (Extreme)
Fuzzy Logic:Intelligence, Control, and Information, J. Yen and R. Langari, PrenticeHall
Inference
• Use knowledge to perform deductive
reasoning, i.e . we wish to deduce or infer a
conclusion, given a body of facts and
knowledge.
Fuzzy Logic with Engineering Applications: Timothy J. Ross
Inference : Example
• In the identification of a triangle
– Let A, B, C be the inner angles of a triangle
• Where A ≥ B≥C
– Let U be the universe of triangles, i.e.,
• U = {(A,B,C) | A≥B≥C≥0; A+B+C = 180˚}
– Let ‘s define a number of geometric shapes
•
•
•
•
•
I
R
IR
E
T
Approximate isosceles triangle
Approximate right triangle
Approximate isosceles and right triangle
Approximate equilateral triangle
Other triangles
Fuzzy Logic with Engineering Applications: Timothy J. Ross
Inference : Example
• We can infer membership values for all of
these triangle types through the method of
inference, because we possess knowledge
about geometry that helps us to make the
membership assignments.
• For Isosceles,
µi (A,B,C) = 1- 1/60* min(A-B,B-C)
– If A=B OR B=C THEN µi (A,B,C) = 1;
– If A=120˚,B=60˚, and C =0˚ THEN µi (A,B,C) = 0.
Fuzzy Logic with Engineering Applications: Timothy J. Ross
Inference : Example
• For right triangle,
µR (A,B,C) = 1- 1/90* |A-90˚|
– If A=90˚ THEN µi (A,B,C) = 1;
– If A=180˚ THEN µi (A,B,C) = 0.
• For isosceles and right triangle
– IR = min (I, R)
µIR (A,B,C) = min[µI (A,B,C), µR (A,B,C)]
= 1 - max[1/60min(A-B, B-C), 1/90|A-90|]
Fuzzy Logic with Engineering Applications: Timothy J. Ross
Inference : Example
• For equilateral triangle
µE (A,B,C) = 1 - 1/180* (A-C)
– When A = B = C then µE (A,B,C) = 1,
A = 180 then µE (A,B,C) = 0
• For all other triangles
– T = (I.R.E)’ = I’.R’.E’
= min {1 - µI (A,B,C) , 1 - µR (A,B,C) , 1 - µE (A,B,C)
Fuzzy Logic with Engineering Applications: Timothy J. Ross
Inference : Example
– Define a specific triangle:
• A = 85˚ ≥ B = 50˚ ≥ C = 45˚
µR = 0.94
µI = 0.916
µIR = 0.916
µE = 0. 7
µT = 0.05
Fuzzy Logic with Engineering Applications: Timothy J. Ross
Rank ordering
• Assessing preferences by a single individual, a
committee, a poll, and other opinion methods
can be used to assign membership values to a
fuzzy variable.
• Preference is determined by pairwise
comparisons, and these determine the
ordering of the membership.
Fuzzy Logic with Engineering Applications: Timothy J. Ross
Rank ordering: Example
Fuzzy Logic with Engineering Applications: Timothy J. Ross