3
Fuzzy Numbers
What we are going to learn today
●
Fuzzy numbers
●
Linguistic Variables
●
Intervals
●
Fuzzy Arithmetic
Fuzzy Numbers
●
e.g.: About 90 degree, 22 years old, ...
●
Fuzzy sets define on the set of real number
A : ℝ  [0,1]
●
@LimCK
Must posses the following properties:
–
A must be a normal set
–
α
–
The support of A must be bounded
A must be a closed interval for every α ∊ (0,1] (convex!)
3 / 25
Fuzzy Numbers
●
@LimCK
The concept can be extended to fuzzy intervals
4 / 25
MF of Fuzzy Numbers
●
●
@LimCK
Not necessarily triangular/trapezoidal and
symmetric.
Generally:
5 / 25
MF of Fuzzy Numbers
@LimCK
6 / 25
MF of Fuzzy Numbers
●
@LimCK
Generally, l(x) and r(x) can be piecewise
function as well.
7 / 25
Linguistic Variables
●
●
●
@LimCK
Fuzzy numbers can be connected to linguistic concepts, e.g. very
small, small, medium, …
Result – linguistic variables
A linguistic variable is fully characterized by a quintuple (v, T, X, g,
m):
–
v : name of the variable
–
T : set of linguistic terms/values of v
–
X : universal set
–
g : syntactic rule (a grammar) for generating linguistic terms
–
m : semantic rule that assigns to each linguistic term
8 / 25
Linguistic Variables
Linguistic variable,
v = “Age”
Linguistic term,
T = {very young,
young, old, … }
Universe,
X = [0, 100]
@LimCK
9 / 25
Linguistic Variables
g – not shown here,
is the rule/grammar
to generate some
other terms such as
“not very young”,
“very very old”, …
m – rules/functions
to define “old”,
“young”, ...
e.g.:
@LimCK
10 / 25
Linguistic hedges / Modifiers
●
Compare:
“cold”
●
●
@LimCK
vs “very cold”
vs
“fairly cold”
Strong modifier reduces the truth value of a
proposition, e.g: “very”
Weak modifier increase the truth value of a
proposition, e.g.: “fairly”
11 / 25
Linguistic hedges / Modifiers
●
Commonly used class of modifiers is:
α
hα(a) = a
●
+
where α ∈ R and a ∈ [0,1]
If α < 1, hα is weak modifier
e.g.: “fairly” : h(a) = √a
●
If α > 1, hα is strong modifier
e.g.: “very” : h(a) = a
@LimCK
2
12 / 25
Arithmetic Operations on Intervals
[a, b] + [c, d] = [a+c, b+d]
[a, b] - [c, d] = [a-d, b-c]
[a,b] • [c,d] = [min(ac,ad,bc,bd) , max(ac,ad,bc,bd)]
[a,b]/[c,d] = [min(a/c,a/d,b/c,b/d) , max(a/c,a/d,b/c,b/d)]
@LimCK
note: / is not defined if 0 ∊ [c,d]
13 / 25
Example
[2, 5] + [1, 3] = [3, 8]
[2,5] - [1,3] = [ -1, 4]
[-1,1] • [-2, -0.5] = [-2, 2]
[-1,1] / [-2, -0.5] = [-2,2]
@LimCK
[0,1] + [-6,5] = [-6, 6]
[0,1] - [-6, 5] = [-5, 7]
[3,4] • [2, 5] = [6, 20]
[4,10] / [1, 2] = [2,10]
14 / 25
Arithmetic Operations on Fuzzy Numbers
●
●
One of the method is based on interval
arithmetic.
If A and B are fuzzy numbers and ◊ is denote
any of the elementary arithmetic operations:
α
α
α
(A ◊ B) = A ◊ B
for α in (0,1]
α
(again, / is not defined if 0 ∊ B)
@LimCK
15 / 25
Arithmetic Operations on Fuzzy Numbers
●
●
@LimCK
With alpha-cut decomposition theorem:
The result of the arithmetic operation is also a
fuzzy number
16 / 25
Example ( before + - * / )
Consider lower bound of A(x)
Consider upper bound of A(x)
Same goes to B(x). So, their alpha cuts are:
@LimCK
17 / 25
Example (addition)
Addition of intervals,
Lower bound,
Upper bound,
α is in (0, 1]
When α=0, x=-10
When α=1, x=5
So x in (-10, 5]
@LimCK
α is in (0, 1]
When α=0, x=25
When α=1, x=10
So x in [10, 25)
18 / 25
Example (subtraction)
Subtraction of intervals,
Lower bound,
Upper bound,
α is in (0, 1]
When α=0, x=15
When α=1, x=30
So x in (15, 30]
@LimCK
α is in (0, 1]
When α=0, x=50
When α=1, x=35
So x in [35, 50)
19 / 25
Example (multiplication)
always negative
for 0 ≤ α ≤ 1
always negative
for 0 ≤ α ≤ 1
always positive
for 0 ≤ α ≤ 1
always positive
for 0 ≤ α ≤ 1
For this specific case, since A is always negative and B is always positive,
BA is entirely in negative region.
Lower bound – the “most negative”
→ negative with largest value
Upper bound – the “least negative”
→ negative with smallest value
(30 – 10α)(5α-20)
= -50α2 + 350α - 600
(10α+10)(-5-5α)
= -50α2 - 100α - 50
@LimCK
20 / 25
Example (multiplication)
Choose – or + ?
Depends on after
the operation, is α
still in [0,1]
@LimCK
21 / 25
Example (division)
always negative
for 0 ≤ α ≤ 1
always negative
for 0 ≤ α ≤ 1
always positive
for 0 ≤ α ≤ 1
always positive
for 0 ≤ α ≤ 1
For this specific case, since A is always negative and B is always positive,
B/A is entirely in negative region.
Lower bound – the “most negative”
→ negative with largest value
(30 – 10α)/(-5-5α)
Upper bound – the “least negative”
→ negative with smallest value
(10α+10)/(5α-20)
[ (30 – 10α)/(-5-5α) , (10α+10)(5α-20) ]
@LimCK
22 / 25
Example (division)
[ (30 – 10α)/(-5-5α) , (10α+10)(5α-20) ]
(30 – 10α)/(-5-5α) = x
@LimCK
(10α+10)/(5α-20) = x
23 / 25
Example
@LimCK
24 / 25
Questions?
@LimCK
25 / 25