Download chapter 5 fuzzy number 5.1.2 fuzzy number 5.4.3 bell shape fuzzy

CHAPTER 5 FUZZY NUMBER
5.1 CONCEPT OF FUZZY NUMBER
5.1.1 INTERVAL OF CONFIDENCE
5.1.2 FUZZY NUMBER
5.1.3 OPERATION OF INTERVAL
5.2 OPERATION OF FUZZY NUMBER
5.2.1 OPERATION OF α–CUT INTERVAL
5.2.2 OPERATION OF FUZZY NUMBER
5.2.3 EXAMPLES OF FUZZY NUMBER OPERATION
5.3 TRIANGULAR FUZZY NUMBER
5.3.1 DEFINITION OF TRIANGULAR FUZZY NUMBER
5.3.2 OPERATION OF TRIANGULAR FUZZY NUMBER
5.3.3 OPERATION BASED ON MEMBERSHIP FUNCTION
5.3.4 APPROXIMATION OF TRIANGULAR FUZZY NUMBER
5.4 OTHER TYPES OF FUZZY NUMBER
5.4.1 TRAPEZOIDAL FUZZY NUMBER
5.4.2 OPERATIONS OF TRAPEZOIDAL FUZZY NUMBER
5.4.3 BELL SHAPE FUZZY NUMBER
Chapter 5 Fuzzy Number
This chapter describes fuzzy numbers. First of all, we’ll look into interval, the fundamental concept of
fuzzy number, and then operation of fuzzy numbers. In addition, we’ll introduce special kind of fuzzy
number such as triangular fuzzy number and trapezoidal fuzzy number.
5.1 Concept of Fuzzy Number
5.1.1 Interval
When interval is defined on real number ℜ, this interval is said to be a subset of ℜ. For instance, if
interval is denoted as A = [a1, a3] a1, a3 ∈ ℜ , a1 < a3, we may regard this as one kind of sets.
Expressing the interval as membership function is shown in the following (Fig 5.1) :
0,

µ A ( x) = 1,
0,

If a1 = a3, this interval indicates a point. That is,
x < a1
a1 ≤ x ≤ a3
x > a3
[a1, a1] = a1
µA(x)
1
a1
Fig 5.1
a3
Interval A = [a1, a3]
x
5.1.2 Fuzzy Number
Fuzzy number is expressed as a fuzzy set defining a fuzzy interval in the real number ℜ . Since the
boundary of this interval is ambiguous, the interval is also a fuzzy set. Generally a fuzzy interval is
represented by two end points a1 and a3 and a peak point a2 as [a1, a2, a3 ] (Fig 5.2). The a-cut
operation can be also applied to the fuzzy number. If we denote a-cut interval for fuzzy number A as
Aα, the obtained interval Aα is defined as
Aα = [a1(α), a3(α)]
We can also know that it is an ordinary crisp interval (Fig 5.3). We review here the definition of fuzzy
number given in section 1.5.4.
Definition(Fuzzy number)
It is a fuzzy set the following conditions :
-
convex fuzzy set
-
normalized fuzzy set
-
it’s membership function is piecewise continuous.
-
It is defined in the real number.
□
Fuzzy number should be normalized and convex. Here the condition of normalization implies that
maximum membership value is 1.
∃x ∈ ℜ,
µA(x) = 1
The convex condition is that the line by α-cut is continuous and α-cut interval satisfies the following
relation.
Aα = [a1(α), a3(α)]
(α′ < α) ⇒ (a1(α′) ≤ a1(α), a3(α′) ≥ a3(α))
µA(x)
1
a1
Fig 5.2
a2
a3
x
Fuzzy Number A = [a1, a2, a3]
The convex condition may also be written as,
(α′ < α) ⇒ (Aα ⊂ Aα′)
µA(x)
1
α
A
α′
a1(0)
a1(α′) a1(α)
a3(α)
a3(α′)
a3(0)
x
Aα = [a1(α), a3(α)]
Aα′ = [a1(α′), a3(α′)]
Fig 5.3
α-cut of fuzzy number (α’ < α) ⇒ (Aα ⊂ Aα′)
5.1.3 Operation of Interval
Operation of fuzzy number can be generalized from that of crisp interval. Let’s have a look at the
operations of interval.
∀a1, a3, b1, b3 ∈ ℜ
A = [a1, a3], B = [b1, b3]
Assuming A and B as numbers expressed as interval, main operations of interval are
i) Addition
[a1, a3] (+) [b1, b3] = [a1 + b1, a3 + b3]
ii) Subtraction
[a1, a3] (–) [b1, b3] = [a1 – b3, a3 – b1]
iii) Multiplication
[a1, a3] (•) [b1, b3] = [a1 • b1 ∧ a1 • b3 ∧ a3 • b1 ∧ a3 • b3, a1 • b1 ∨ a1 • b3 ∨ a3 • b1 ∨ a3 • b3]
iv) Division
[a1, a3] (/) [b1, b3] = [a1 / b1 ∧ a1 / b3 ∧ a3 / b1 ∧ a3 / b3, a1 / b1 ∨ a1 / b3 ∨ a3 / b1 ∨ a3 / b3]
excluding the case b1 = 0 or b3 = 0
v) Inverse interval
[a1, a3]–1 = [1 / a1 ∧ 1 / a3, 1 / a1 ∨ 1 / a3]
excluding the case a1 = 0 or a3 = 0
When previous sets A and B is defined in the positive real number ℜ +, the operations of
multiplication, division, and inverse interval are written as,
iii′) Multiplication
[a1, a3] (•) [b1, b3] = [a1 • b1, a3 • b3]
iv′) Division
[a1, a3] (/) [b1, b3] = [a1 / b3, a3 / b1]
v) Inverse Interval
[a1, a3]–1 = [1 / a3, 1 / a1]
vi) Minimum
[a1, a3] (∧) [b1, b3] = [a1 ∧ b1, a3 ∧ b3]
vii) Maximum
[a1, a3] (∨) [b1, b3] = [a1 ∨ b1, a3 ∨ b3]
Example 5.1
There are two intervals A and B,
A = [3, 5], B = [–2, 7]
Then following operation might be set.
A(+ ) B = [3 − 2, 5 + 7] = [1, 12]
A(−) B = [3 − 7, 5 − (−2)] = [− 4, 7]
A(•) B = [3 • (−2) ∧ 3 • 7 ∧ 5 • (−2) ∧ 5 • 7, 3 • (−2) ∨ L]
= [− 10, 35]
A(/) B = [3 /( −2) ∧ 3 / 7 ∧ 5 /( −2) ∧ 5 / 7, 3 /( −2) ∨ L]
= [− 2.5, 5 / 7]
 1
1
1  1
1
∧ ,
∨  = − ,
B −1 = [−2,7] −1 = 
 (−2) 7 (−2) 7   2
1
7 
□
5.2 Operation of Fuzzy Number
5.2.1 Operation of α-cut Interval
We referred to α-cut interval of fuzzy number A = [a1, a3] as crisp set.
Aα = [a1(α), a3(α)], ∀α ∈ [0, 1], a1, a3, a1(α), a3(α) ∈ ℜ
So Aα is a crisp interval. As a result, the operations of interval reviewed in the previous section can be
applied to the α-cut interval Aα.
If α-cut interval Bα of fuzzy number B is given
B = [b1, b3],
b1, b3, ∈ ℜ
Bα = [b1(α), b3(α)], ∀α ∈ [0, 1], b1(α), b3(α) ∈ ℜ ,
operations between Aα and Bα can be described as follows :
[a1(α), a3(α)] (+) [b1(α), b3(α)] = [a1(α) + b1(α), a3(α) + b3(α)]
[a1(α), a3(α)] (–) [b1(α), b3(α)] = [a1(α) – b3(α), a3(α) – b1(α)]
These operations can be also applicable to multiplication and division in the same manner.
5.2.2 Operation of Fuzzy Number
Previous operations of interval are also applicable to fuzzy number. Since outcome of fuzzy number
(fuzzy set) is in the shape of fuzzy set, the result is expressed in membership function.
∀x, y, z ∈ ℜ
i) Addition: A (+) B
µ A( + ) B ( z ) = ∨ ( µ A ( x) ∧ µ B ( y ))
z = x+ y
ii) Subtraction: A (–) B
µ A( − ) B ( z ) = ∨ ( µ A ( x) ∧ µ B ( y ))
z= x− y
iii) Multiplication: A (•) B
µ A(•) B ( z ) = ∨ ( µ A ( x) ∧ µ B ( y ))
z = x• y
iv) Division: A (/) B
µ A(/) B ( z ) = ∨ ( µ A ( x) ∧ µ B ( y ))
z=x / y
v) Minimum: A (∧) B
µ A( ∧ ) B ( z ) = ∨ ( µ A ( x) ∧ µ B ( y ))
z = x∧ y
vi) Maximum: A (∨) B
µ A( ∨ ) B ( z ) = ∨ ( µ A ( x) ∧ µ B ( y ))
z = x∨ y
We can multiply a scalar value to the interval. For instance, multiplying a ∈ ℜ ,
a[b1, b3] = [a • b1 ∧ a • b3, a • b1 ∨ a • b3]
Example 5.2
There is a scalar multiplication to interval. Note the scalar value is negative.
–4.15 [–3.55, 0.21] = [(–4.15) • (–3.55) ∧ (–4.15) • 0.21, (–4.15) • (–3.55) ∨ (–4.15) • 0.21]
= [14.73 ∧ –0.87, 14.73 ∨ –0.87]
= [–0.87, 14.73]
□
We can also multiply scalar value to α-cut interval of fuzzy number.
∀α ∈ [0, 1], b1(α), b3(α) ∈ ℜ
a[b1(α), b3(α)] = [a • b1(α) ∧ a • b3(α), a • b1(α) ∨ a • b3(α)]
5.2.3 Examples of Fuzzy Number Operation
Example 5.3 : Addition A(+)B
For further understanding of fuzzy number operation, let us consider two fuzzy sets A and B. Note that
these fuzzy sets are defined on discrete numbers for simplicity.
A = {(2, 1), (3, 0.5)}, B = {(3, 1), (4, 0.5)}
First of all, our concern is addition between A and B. To induce A(+)B, for all x ∈ A, y ∈ B, z ∈ A(+)B,
we check each case as follows(Fig 5.4) :
i) for z < 5,
µA(+)B(z) = 0
ii) z = 5
results from x + y = 2 + 3
µA(2) ∧ µB(3) = 1 ∧ 1 = 1
µ A( + ) B (5) = ∨ (1) = 1
5= 2 + 3
iii) z = 6
results from x + y = 3 + 3 or x + y = 2 + 4
µA(3) ∧ µB(3) = 0.5 ∧ 1 = 0.5
µA(2) ∧ µB(4) = 1 ∧ 0.5 = 0.5
µ A( + ) B (6) = 6=∨3+3 (0.5,0.5) = 0.5
6= 2+ 4
iv) z = 7
results from x + y = 3 + 4
µA(3) ∧ µB(4) = 0.5 ∧ 0.5 = 0.5
µ A( + ) B (7) = ∨ (0.5) = 0.5
7 =3+ 4
v) for z > 7
µA(+)B(z) = 0
So A(+)B can be written as
A(+)B = {(5, 1), (6, 0.5), (7, 0.5)}
□
µA(x)
1
0.5
2
3
(a) Fuzzy set A
µB(x)
1
0.5
3
4
(b) Fuzzy number B
µA (+) B(x)
1
0.5
5 6 7
(c) Fuzzy set A (+) B
Fig 5.4
Add operation of fuzzy set
Example 5.5 : Subtraction A(−)B
Let’s manipulate A(−)B between our previously defined fuzzy set A and B. For x ∈ A, y ∈ B, z ∈
A(−)B, fuzzy set A(−)B is defined as follows (Fig5.5).
i) for z < −2,
µA(−)B(z) = 0
ii) z = −2
results from x − y = 2 − 4
µA(2) ∧ µB(4) = 1 ∧ 0.5 = 0.5
µ A( − ) B (−2) = 0.5
iii) z = −1
results from x − y = 2 − 3 or x − y = 3 − 4
µA(2) ∧ µB(3) = 1 ∧ 1 = 1
µA(3) ∧ µB(4) = 0.5 ∧ 0.5 = 0.5
µ A( − ) B (−1) = −1=∨2 −3(1,0.5) = 1
−1= 3 − 4
iv) z = 0
results from x − y = 3 − 3
µA(3) ∧ µB(3) = 0.5 ∧ 1 = 0.5
µA(−)B(0) = 0.5
v) for z ≥ 1
µA(−)B(z) = 0
So A(−)B is expressed as
A(−)B = {(-2, 0.5), (-1, 1), (0, 0.5)}
□
µA (−) B(x)
1
0.5
−2 −1 0
Fig 5.5
Fuzzy number A (−) B
Example 5.6 : Max operation A(∨)B
Let’s deal with the operation Max A(∨)B between A and B.
For x ∈ A, y ∈ B, z ∈ A(∨)B, fuzzy set A(∨)B is defined by µA(∨)B (z).
i) z ≤ 2
µA(∨)B(z) = 0
ii) z = 3
From x ∨ y = 2 ∨ 3 and x ∨ y = 3 ∨ 3
µA(2) ∧ µB(3) = 1 ∧ 1 = 1
µA(3) ∧ µB(3) = 0.5 ∧ 1 = 0.5
µ A( ∨ ) B (3) = 3=∨2∨ 3(1,0.5) = 1
3 = 3∨ 3
iii) z = 4
From x ∨ y = 2 ∨ 4 and x ∨ y = 3 ∨ 4
µA(2) ∧ µB(4) = 1 ∧ 0.5 = 0.5
µA(3) ∧ µB(4) = 0.5 ∧ 0.5 = 0.5
µ A( ∨ ) B (4) = 4 =∨2∨ 4(0.5,0.5) = 0.5
4 = 3∨ 4
v) z > 5
Impossible
µA(∨)B(z) = 0
So A(∨)B is defined to be
A(∨)B = {(3, 1), (4, 0.5)}
□
So far we have seen the results of operations are fuzzy sets, and thus we come to realize that the
extension principle is applied to the operation of fuzzy number.
5.3 Triangular fuzzy number
5.3.1 Definition of triangular fuzzy number
Among the various shapes of fuzzy number, triangular fuzzy number(TFN) is the most popular one.
Definition(Triangular fuzzy number) It is a fuzzy number represented with three points
as follows :
A = (a1, a2, a3)
This representation is interpreted as membership functions(Fig5.6).
0,


 x − a1 ,
 a2 − a1
µ ( A) ( x) = 
a −x
 3
,
 a3 − a2

0,
x < a1
a1 ≤ x ≤ a2
a2 ≤ x ≤ a3
x > a3
□
µA(x)
1
a1
Fig 5.6
a2
a3
x
Triangular fuzzy number A = (a1, a2, a3)
Now if you get crisp interval by α-cut operation, interval Aa shall be obtained as follows ∀α ∈ [0, 1].
From
a − a 3(α )
a1(α ) − a1
=α , 3
=α
a2 − a1
a3 − a 2
we get
a1(α) = (a2 – a1)α + a1
a3(α) = −(a3 − a2)α + a3
Thus
Aα = [a1(α), a3(α)]
= [(a2 − a1)α + a1, −(a3 − a2)α + a3]
Example 5.7
In the case of the triangular fuzzy number A = (−5, −1, 1) (Fig 5.7), the
membership function value will be,
0,


x + 5,
 4
µ ( A) ( x ) = 
1 − x

,
 2

0,
x < −5
− 5 ≤ x ≤ −1
−1 ≤ x ≤ 1
x >1
1
0.5
−6 −5 −4 −3 −2 −1
Fig 5.7
⇒ x = 4α − 5
⇒ x = −2α + 1
Aα = [a1(α), a3(α)] = [4α − 5, −2α + 1]
If α = 0.5, substituting 0.5 for α, we get A0.5
A0.5 = [a1(0.5), a3(0.5)] = [−3, 0]
1
2
A0.5
α = 0.5 cut of triangular fuzzy number A = (−5, −1, 1)
α-cut interval from this fuzzy number is
x+5
=α
4
1− x
=α
2
0
□
5.3.2 Operation of Triangular Fuzzy Number
Same important properties of operations on triangular fuzzy number are summarized.
(1) The results from addition or subtraction between triangular fuzzy numbers result also triangular
fuzzy numbers.
(2) The results from multiplication or division are not triangular fuzzy numbers.
(3) Max or min operation does not give triangular fuzzy number.
But we often assume that the operational results of multiplication or division to be TFNs as
approximation values.
1) Operation of triangular fuzzy number
First, consider addition and subtraction. Here we need not use membership function. Suppose
triangular fuzzy numbers A and B are defined as,
A = (a1, a2, a3), B = (b1, b2, b3)
i) Addition
A(+) B = (a1 , a2 , a3 )(+)(b1 , b2 , b3 )
= (a1 + b1 , a2 + b2 , a3 + b3 )
: triangular fuzzy number
ii) Subtraction
A(−) B = (a1 , a2 , a3 )(−)(b1 , b2 , b3 )
= (a1 − b3 , a2 − b2 , a3 − b1 )
: triangular fuzzy number
iii) Symmetric image
−(A) = (−a3, −a2, −a1)
Example 5.8
: triangular fuzzy number
Let’s consider operation of fuzzy number A, B(Fig 5.8).
A = (−3, 2, 4), B = (−1, 0, 6)
A (+) B = (−4, 2, 10)
A (−) B = (−9, 2, 5)
□
1
A
B
0.5
−3
−1
0
2
6
4
(a) Triangular fuzzy number A, B
1
A (+) B
0.5
−4
10
0
2
(b) A (+) B of triangular fuzzy numbers
1
0.5
−9
A (−) B
5
0
2
(c) A (−) B triangular fuzzy numbers
Fig 5.8 A (+) B and A (−) B of triangular fuzzy numbers
2) Operations with α-cut
Example 5.9
α-level intervals from α-cut operation in the above two triangular fuzzy numbers A
and B are
Aα
Bα
= [a1(α ) , a 3(α ) ] =
=
(α )
(α )
= [b1 , b3 ] =
=
[(a 2 − a1 )α + a1 ,−(a 3 − a 2 )α + a 3 ]
[5α − 3,−2α + 4]
[(b2 − b1 )α + b1 ,−(b3 −b 2 )α + b3 ]
[α − 1,−6α + 6]
Performing the addition of two α-cut intervals Aα and Bα,
Aα (+) Bα = [6α − 4, −8α + 10]
Especially for α = 0 and α = 1,
A0 (+) B0 = [−4, 10]
A1 (+) B1 = [2, 2] = 2
Three points from this procedure coincide with the three points of triangular fuzzy number (-4, 2, 10)
from the result A(+)B given in the previous example.
Likewise, after obtaining Aα(−)Bα, let’s think of the case when α = 0 and α = 1.
Aα (−) Bα = [11α − 9, −3α + 5]
Substituting α = 0 and α = 1 for this equation,
A0 (−) B0 = [−9, 5]
A1 (−) B1 = [2, 2] = 2
These also coincide with the three points of A(−)B = (−9, 2, 5).
□
Consequently, we know that we can perform operations between fuzzy number using α-cut interval.
5.3.3 Operation of general fuzzy numbers
Up to now, we have considered the simplified procedure of addition and subtraction using three points
of triangular fuzzy number. However, fuzzy numbers may have general form, and thus we have to deal
the operations with their membership functions.
Example 5.10
Addition A (+) B
Here we have two triangular fuzzy numbers and will calculate the addition operation using their
membership functions.
A = (−3, 2, 4), B = (−1, 0, 6)
0,


x + 3,
2 + 3
µ ( A) ( x) = 
4 − x

,
4 − 2

0,
x < −3
−3≤ x ≤ 2
2≤ x≤4
x>4
y < −1
0,


 y +1, −1 ≤ y ≤ 0
0 +1
µ ( B ) ( y) = 
6 − y 0 ≤ y ≤ 6

,
6 − 0

y>6
0,
For the two fuzzy number x ∈ A and y ∈ B, z ∈ A (+) B shall be obtained by their membership
functions.
Let’s think when z = 8. Addition to make z = 8 is possible for following cases :
2 + 6, 3 + 5, 3.5 + 4.5, …
So
µ A( + ) B
=
∨
8= x + y
= ∨
= ∨
[ µ A (2) ∧ µ B (6), µ A (3) ∧ µ B (5), µ A (3.5) ∧ µ B (4.5), L]
[1 ∧ 0, 0.5 ∧ 1 / 6, 0.25 ∧ 0.25, L]
[0,1 / 6, 0.25, L]
If we go on these kinds of operations for all z ∈ A (+) B, we come to the following membership
functions, and these are identical to the three point expression for triangular fuzzy number A = (−4, 2,
10).
0,

z + 4

,
 6
µ A( + ) B ( z ) = 
10 − z

,
 8

0,
z < −4
−4≤ z ≤2
2 ≤ z ≤ 10
z > 10
□
There in no simple method using there point expression for multiplication or division operation. So it
is necessary to use membership functions.
Example 5.11
Multiplication A (•) B
Let triangular fuzzy numbers A and B be
A = (1, 2, 4), B = (2, 4, 6)
x <1
0,

 x − 1,

µ ( A) ( x) = 
− 1 x + 2,
 2

0,
1≤ x < 2
2≤ x<4
x≥4
y<2
0,


2≤ y<4
 1 y − 1,
2
µ ( B) ( y) = 
 1
− y + 3, 4 ≤ y < 6
 2

y≥6
0,
Calculating multiplication A (•) B of A and B, z = x • y = 8 is possible when z = 2 • 4 or z = 4 • 2
µ A( • ) B
=
∨ [ µ A (2) ∧ µ B (4), µ A (4) ∧ µ B (2), L]
x• y =8
= ∨ [1 ∧ 1, 0 ∧ 0, L]
= 1
Also when z = x • y = 12, 3 • 4, 4 • 3, 2.5 • 4.8, … are possible.
µ A( • ) B
=
∨ [ µ A (3) ∧ µ B (4),
x• y =12
µ A (4) ∧ µ B (3), µ A (2.5) ∧ µ B (4.8), L]
= ∨ [0.5 ∧ 1, 0 ∧ 0.5, 0.75 ∧ 0.6, L]
= ∨ [0.5, 0, 0.6, L]
= 0.6
From this kind of method, if we come by membership function for all z ∈ A (•) B, we see fuzzy
number as in Fig 5.9. However, since this shape is in curve, it is not a triangular fuzzy number. For
convenience, we can express it as a triangular fuzzy number by approximating A (•) B.
A(•) B ≅ (2, 8, 24)
We can wee that two end points and one peak point are used in this approximation.
A
B
1
A (•) B
0.5
0
1
2
Fig 5.9
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Multiplication A (•) B of triangular fuzzy number
□
5.3.4 Approximation of Triangular Fuzzy Number
Since it is possible to express approximated values of multiplication and division as triangular fuzzy
numbers, we are now up to the fact that how to get this approximated value easily.
Example 5.12
Approximation of multiplication
First, α-cuts of two fuzzy numbers are our main concern.
A = (1, 2, 4), B=(2, 4, 6)
Aα
Bα
=
=
=
=
[(2 − 1)α + 1, − (4 − 2)α + 4]
[α + 1, − 2α + 4]
[(4 − 2)α + 2, − (6 − 4)α + 6]
[2α + 2, − 2α + 6]
For all α ∈ [0, 1], multiply Aα with Bα which are two crisp intervals. Now in α ∈ [0, 1], we see that
elements of each interval are positive numbers. So multiplication operation of the two intervals is
simple.
Aα (•) Bα
= [α + 1, − 2α + 4](•)[2α + 2, − 2α + 6]
= [(α + 1)(2α + 2), (−2α + 4)(−2α + 6)]
= [2α 2 + 4α + 2, 4α 2 − 20α + 24]
When α = 0,
A0 (•) B0
= [2, 24]
When α = 1,
A0(•)B1 = [2+4+2, 4-20+24] = [8, 8] = 8
We obtain a triangular fuzzy number which is an approximation of A (•) B (Fig 5.9).
A(•) B ≅ (2, 8, 24)
Example 5.13
□
Approximation of division
In the similar way, let’s express approximated value of A (/) B in a triangular fuzzy number. First,
divide interval Aα by Bα. We reconsider the sets A and B in the previous example. For α ∈ [0, 1], since
element in each interval has positive number, we get Aα (/) Bα as follows.
Aα (/) Bα
= [(α + 1) /(−2α + 6), ( −2α + 6) /(2α + 2)]
When α = 0,
A0 (/) B0
= [1 / 6, 4 / 2]
= [0.17, 2]
When α = 1,
A1 (/) B1
= [(1 + 1) /( −2 + 6), (−2 + 4) /( 2 + 2)]
= [2 / 4, 2 / 4]
= 0.5
So the approximated value of A (/) B will be
A(/) B = (0.17, 0.5, 2)
□
5.4 Other Types of Fuzzy Number
5.4.1 Trapezoidal Fuzzy Number
Another shape of fuzzy number is trapezoidal fuzzy number. This shape is originated from the fact
that there are several points whose membership degree is maximum (α = 1).
Definition(Trapezoidal fuzzy number)
We can define trapezoidal fuzzy number A as
A = (a1, a2, a3, a4)
The membership function of this fuzzy number will be interpreted as follows(Fig 5.10).
0,
 x − a1
,

 a2 − a1
µ A ( x) = 1,
 a4 − x
,

 a4 − a3
0,
x < a1
a1 ≤ x ≤ a2
a2 ≤ x ≤ a3
a3 ≤ x ≤ a4
□
x > a4
µA(x)
1
a1
Fig 5.10
a2
a3
a4
x
Trapezoidal fuzzy number A = (a1, a2, a3, a4)
α-cut interval for this shape is written below.
∀α ∈ [0, 1]
Aα = [(a2 – a1)α + a1,
–(a4 – a3)α + a4]
When a2 = a3, the trapezoidal fuzzy number coincides with triangular one.
5.4.2 Operations of Trapezoidal Fuzzy Number
Let’s talk about the operations of trapezoidal fuzzy number as in the triangular fuzzy number,
1) Addition and subtraction between fuzzy numbers become trapezoidal fuzzy number.
2) Multiplication, division, and inverse need not be trapezoidal fuzzy number.
3) Max and Min of fuzzy number is not always in the form of trapezoidal fuzzy number.
But in many cases, the operation results from multiplication or division are approximated trapezoidal
shape. As in triangular fuzzy number, addition and subtraction are simply defined, and multiplication
and division operations should be done by using membership functions.
i) Addition
A(+ ) B = (a1 , a2 , a3 , a4 )(+ )(b1 , b2 , b3 , b4 )
= (a1 + b1 , a2 + b2 , a3 + b3 , a4 + b4 )
ii) Subtraction
A(−) B = (a1 − b4 , a2 − b3 , a3 − b2 , a4 − b1 )
Example 5.14
Multiplication
Multiply two trapezoidal fuzzy numbers as following:
A = (1, 5, 6, 9)
B = (2, 3, 5, 8)
For exact value of the calculation, the membership functions shall be used and the result is described
in Fig 5.11. For the approximation of operation results, we use α-cut interval.
Aα = [4α + 1, –3α + 9]
Bα = [α + 2, –3α + 8]
Since, for all α ∈ [0, 1], each element for each interval is positive, multiplication between α-cut
intervals will be
Aα (•) Bα
= [(4α + 1)(α + 2), (−3α + 9)(−3α + 8)]
= [4α 2 + 9α + 2, 9α 2 − 51α + 72]
If α = 0,
A0 (•) B0
= [2, 72]
If α = 1,
A1 (•) B1
= [4 + 9 + 2, 9 − 51 + 72]
= [15, 30]
So using four points in α = 0 and α = 1, we can visualize the approximated value as trapezoidal fuzzy
number as Fig 5.11.
A(•) B ≅ [2, 15, 30, 72]
□
µ
1.0
0.9
0.8
(1,5, 6, 9) (•) (2,3,5,8) = A (•) B
0.7
0.6
0.5
≅ (2,15, 30, 72)
(1,5, 6, 9) = A
0.4
0.3
(2,3, 5, 8) = B
0.2
x
0.1
10
Fig 5.11
20
30
40
50
60
70
Multiplication of trapezoidal fuzzy number A (•) B
Generalizing trapezoidal fuzzy number, we can get flat fuzzy number. In other words, flat fuzzy
number is for fuzzy number A satisfying following.
∃m1, m2 ∈ ℜ,
m1 < m2
µA(x) = 1,
m1 ≤ x ≤ m2
In this case, not like trapezoidal form, membership function in x < m1 and x < m2 need not be a line as
shown in Fig 5.12.
1
m1
Fig 5.12
m2
Flat fuzzy number
5.4.3 Bell Shape Fuzzy Number
Bell shape fuzzy number is often used in practical applications and its function is defined as
follows(Fig 5.13)
 − ( x − m f ) 2 
µ f ( x) = exp

2δ f2


where
µf
is the mean of the function,
δf
Fig 5.13.
is the standard deviation.
Bell shape fuzzy number