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On Some Fuzzy Optimization Problems. 主講人:胡承方博士 義守大學工業工程與管理學系 April 16, 2010. 模糊理論. Zadeh (1965) 首創模糊集合 (Fuzzy Set) 何謂「 Fuzzy 」 今天天氣「有點熱」 顧客的滿意度「頗高」 從清華大學到竹科的距離「很近」 義守大學是一所「不錯」的大學. 模糊. 機率. 模糊且隨機. 模糊與機率不同處之比較. 模糊理論. 將人類認知過程中(主要為思考與推理)之不確定性,以數學模式表之。 - PowerPoint PPT Presentation
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On Some Fuzzy Optimization Problems
主講人:胡承方博士義守大學工業工程與管理學系
April 16, 2010
模糊理論 Zadeh (1965) 首創模糊集合 (Fuzzy Set)
何謂「 Fuzzy 」今天天氣「有點熱」顧客的滿意度「頗高」 從清華大學到竹科的距離「很近」義守大學是一所「不錯」的大學
模 糊 機 率
元素歸屬程度 集合的發生率
不涉及統計 使用統計
訊息愈多 模糊仍存在
訊息愈多 不確定性遞減
處理真的程度 是可能性 或預期的情形
模糊 機率
模糊且隨機
模糊與機率不同處之比較
模糊理論 將人類認知過程中 ( 主要為思考與推理 )之不確定性,以數學模式表之。
把傳統的數學從只有『對』與『錯』的二值邏輯 (Binary logic) 擴展到含有灰色地帶的連續多值 (Continuous multi-value)邏輯。
模糊理論 利用『隸屬函數』 (Membership Function)值來描述一個概念的特質,亦即使用 0與 1 之間的數值來表示一個元素屬於某一概念的程度,這個值稱為該元素對集合的隸屬度 (Membership grade) 。
當隸屬度為 1 或 0 時便如同傳統的數學中的『對』與『錯』,當介於兩者之間便屬於對與錯之間的灰色地帶。
傳統集合 (Crisp Sets)
傳統集合是以二值邏輯 (Binary Logic) 為基礎的方式來描述事物,元素 x 和集合A 的關係只能是 A 或 A ,是一種『非此即彼』的概念。以特徵函數表示為:
Ax
AxxA
,0
,1)(
7
模糊集合 (Fuzzy Sets)
而模糊集合則是指在界限或邊界不分明且具有特定事物的集合,以建立隸屬函數 (Membership Function) 來表示模糊集合,也就是一種『亦此亦彼』的概念。
隸屬函數 (Membership Functions) 假設宇集 (universe)U={x1, x2,…, xn} , 是定義在 U 之下的模糊集合,
為模糊集合之隸屬函數(Membership Function) 。
表示模糊集合 中 xi的隸屬程度(Degree of Membership) 。
A~
1 1 2 2{ ( , ( )) , ( , ( )) ,..., ( , ( )) }.n nA A A
A x x x x x x
]1,0[:~ UA
)(~ iA
x A~
Example
Ex: The weather is “good”
20 25 30 35 x
A(x)A
fuzzy set
25 30 x
A(x)A
crisp set
Example
10 toclose numbers real
~numbers real
A
X
0
0.5
1
1.5
0 5 10 15 20
2101
1~
xx
A
RxxxAA
, ~
~……………...
傳統集合 模糊集合Characteristic function
特徵函數
A(x)
X{0,1}
Membership function隸屬函數
X[0,1]
傳統與模糊集合不同處之比較
)(~ xA
模糊集合表示法 宇集 U 為有限集合
宇集 U 無限集合或有限連續
一般的表示方法
iiA
xxA /)(~
~
i
Ux
i xxA /)( ~
A~
} ))(,{(~
~ UxxxA iiA
i
Ex:
A: The weather is “hot”
......
23
4.0
22
3.0
21
2.0A~
Example
模糊集合之運算 聯集( Union )
交集( Intersection )
補集( Complement )
)}(),(max{)( ~~~~ uuuBABA
)}(),(min{)( ~~~~ uuuBABA
)(1)( ~~ uuAAC
Example
Ex: two fuzzy set and find BA~
and ~
1
15 20 x
BA~~
~A B
~
BA~~
)(~ xA
Example
(15)= (15) (15)
=min( (15), (15))
=min(1,0)=0
(20)= (20) (20)
=min( (20), (20))
=min(0.7,0.2)=0.2
BA~~
BA~~
)(~ xA
)(~ x
A
)(~ xA
)(~ x
A
)(~ xB
)(~ x
B
)(~ xB
)(~ xB
- 截集 ( -cut 或 -level)
模糊集合 的 - 截集定義為 :
而模糊集合 取 - 截集所形成的區間範圍為
]1,0[
, )( ~
UxxxA iiA
i
A~
A~
ULA
AAxxA , )( ~
Fuzzy numbers
Two classes
One class has 30 students
One class has 25 students
~~{
模糊數 (Fuzzy Numbers)
If is a normal fuzzy set on R and is a closed interval for each then is a fuzzy number.
(Note that: is a normal, if
I
( ) 1, . )I
x x R
I0 1, I
I
a b c0
1
X
(x)
L(x) R(x)
模糊數的種類 三角形模糊數 (Triangular Fuzzy Number) 梯形模糊數 (Trapezoidal Fuzzy Number) 鐘形模糊數 (Bell Shaped Fuzzy Number) 不規則模糊數 (Non-Symmetric Fuzzy
Number)
三角形模糊數
a b c0
1
X
(x)
cx ,
cxb ,bc
xc
bxa ,ab
ax x<a ,
xA
0
0
)(~
( , , )A a b c
梯形模糊數
a b c0
1
X
(x)
d
otherwise,
dxc ,c-d
x-dcxb,
x<ba ,b-a
x-a
xA
0
1)(~
( , , , )A a b c d
鐘形模糊數
0
1
X
(x)
2
2)(
~ )(
x
Aex
不規則模糊數
a b c0
1
X
(x)
L(x) R(x)
cxb
bc
bxR
bxaab
axL
xA
)(
)()(~
模糊運算 (Fuzzy Arithmetic)
模糊數加法 模糊數乘法 模糊數除法 模糊數倒數 模糊數開根號運算
模糊數加法 三角形模糊數
:模糊數加法運算子 梯形模糊數
),,,(
),,,(),,,(
21212121
22221111
ddccbbaa
dcbadcba
),,(),,(),,( 212121222111 ccbbaacbacba
模糊數乘法 三角形模糊數 (k>0)
:模糊數乘法運算子 梯形模糊數
),,(),,( ckbkakcbak
),,,(),,,( dkckbkakdcbak
模糊數乘法 三角形模糊數 (a1>0,a2>0)
:模糊數乘法運算子 梯形模糊數
),,(),,(),,( 212121222111 ccbbaacbacba
),,,(
),,,(),,,(
21212121
22221111
ddccbbaa
dcbadcba
模糊數除法 三角形模糊數
:模糊數除法運算子 梯形模糊數
)/,/,/(),,(),,( 212121222111 acbbcacbacba
)/,/,/,/(
),,,(),,,(
21212121
22221111
adbccbda
dcbadcba
Fuzzy Ranking
(>) ??M N
Why ranking fuzzy numbers ?
Two classrooms to be preassigned to two classes
One large room
One small room
One class has 30 students
One class has 25 students
~~
{
{
Fuzzy Ranking
Solving
is to find optimal solutions to the system of
fuzzy linear inequalities problem
njx
mibxa
j
jij
n
ji
,,1 ,0
,,1 ,~~
1
Example
7~
3~
4~
0~
243
21
3221
xx
xxx
How to rank fuzzy numbers?
The study of fuzzy ranking began in 1970's Over 20 ranking methods were proposed No \best" method agreed
How to Select Fuzzy Ranking
Easy to compute Consistency Ability to discriminate Go with intuition Fits your model Consider combination of different
rankings
Optimization
Optimization models can be very useful.
..
max
ts
0,
1002
yx
yx
x y
Optimization models for Decision making
max
xf
xf
p
i
throughput
profit
..ts
skqxh
rjdxg
kk
jj
,,1
,,1
,
,
resource
demand
Past Industrial Experience
Optimization models can be very useful.
Problems are harden to define than to solve.
Most decision are made under uncertainty.
Fuzzy Optimization
max
..ts
x y
0,
100~
2
yx
yx
Fuzzy Optimization and Decision making
fuzzy vector :
~
,,1 ,~~,
,,1 ,~~
, ..
~,
~,
maximize
1
skqxh
rjdxgts
xf
xf
kk
ji
p
Solution Methods
-level approach Parametric approach Semi-infinite programming approach Set-inclusion approach Possibilistic programming approach
……
Recent Development
System of Fuzzy Inequalities
Fuzzy Variational Inequalities
Motivation
LP
K-K-T Optimality Conditions
0wbxc
0 w
cw
0 x
bx
such thatwx, Find
0 w 0x
cw s.t.D bx s.t.P
wbmax x cmin
TT
T
T
TT
A
A
AA
Motivation
NLP
where is a convex set and is a
smooth real-valued function defined on .
, xs.t.
xmin
K
h
nRK fK
Variational Inequalities
Find such that
for each
where means the inner product operation.
x Kx
,0xx,x h ,x K
,
System of Fuzzy Inequalities
“ ” means “approximately less than or equal to”.
Examples :
~
RRgf
Jjxg
Iixf
ni
i
j
j
:,
,0~~
,0~
7~
3~
4~
0~
243
21
3221
xx
xxx
Fuzzy Inequalities – System I
“ ” means “approximately less than or equal to”.
~
ljxg
mixf
tsRx
j
i
n
,,2,1 ,0
,,2,1 ,0~
.. Find
*
~
1, if 0
, if 0
0, if
i
Fi i i i
i i
i
f x
x f x f x t
f x t
decreasingstrictly
and continuous:xfu ii
iF~
xfii
t
Each fuzzy inequality 0 determines a fuzzy set ~
in with
i i
n
f x F
R
Fuzzy Decision Making
(Bellman/Zadeh,1970) Decision Making Model
Solving(*) is to find optimal solutions to
x
FD
iFD
i
mi~~
1min
~~
ljxg
x
j
iFn miRx
,,2,1 ,0 s.t.
min max ~
1
Equivalently,
When is invertible
nRx
ljxg
mix
j
iF
,10
,,1 ,0
,,1 , s.t.
max
~
iF
~
1~~
iFxfx
iiF
If , are convex and are concave, then a solution to (*) can be obtained by solving a convex programming problem
xfi xg
i i
n
iF
Rx
ljxg
mixf
j
i
1,0
,,1 ,0
,,1 ,0 s.t.
max1
~
Huard’s “Method of Centers” + Entropic Regularization Method reduce the problem to solving a sequence of unconstrained smooth convex programs
with a sufficiently large p.
( Hu, C.-F. and Fang, S.-C., “Solving Fuzzy Inequalities with Concave Membership Functions”, Fuzzy Sets and Systems, vol. 99 (2),pp. 233-240,1998 )
l
j
m
iii
k
x
ppxgp
xfppp
i
1
1
1
,
1expexpexp
expexpln1
min
Semi-infinite programming extension for
(Hu, C.-F. and Fang, S.-C., “A Relaxed
Cutting Plane Algorithm for Solving Fuzzy Inequality Systems ”, Optimization, vol. 45, pp. 89-106, 1999)
JjIi ,
Extension to solving fuzzy inequalities with piecewise linear membership functions
iF
~
it
xfi
:xf ii
(Hu, C.-F. and Fang, S.-C., “Solving Fuzzy Inequalities with Piecewise Linear Membership Functions”, IEEE Transactions on Fuzzy Systems, vol. 7 (2),pp. 230-235,April, 1999.
Hu, C.-F. and Fang, S.-C., “Solving a System
of Infinitely Many Fuzzy Inequalities with Piecewise Linear Membership Functions”, Computers and Mathematics with Applications, vol.40,pp. 721-733, 2000.)
Fuzzy Inequalities – Systems II
Find such thatx X
Iixfi
,0~~
Fundamental Problem
No universally accepted theory for ranking two fuzzy sets.
?~~?
~~
ab
ba
R
a~b~
R
Simple Case
Solving
is to find optimal solutions to the semi-infinite programming problem
njx
mibxa
j
jij
n
ji
,,1 ,0
,,1 ,~~
1
(Fang, S.-C., Hu, C.-F., Wang H.-F. and Wu, S.-Y., “Linear Programming with Fuzzy Coefficients in Constraints”, Computers and Mathematics with Applications, vol. 37 (10),pp. 63-76, 1999.)
.0,1 ,,,2,1 ,0
,,2,1 ,1,
,,,2,1 ,1, , s.t.
1 max
,
1
1
~~
~~
njx
mittRxtR
mittLxtL
j
ibija
ijbjija
j
n
j
n
j
Fuzzy Variational Inequalities
An Optimization problem can be cast into a variational inequality problem
Find such that
where V is a nonempty, closed, convex subset of and is a point-to-point mapping.
V,FVI
VX
Vz 0Fz T xx
nR
)(,F~
F
,V~
V
)(~
V~
xF
nn RR:F
Problem such that
As difficult as an optimization problem with parameterized equilibrium constraints.
n nVI V,F : Find (x,y) R R
V
~z, ,0yz
xF~
yy,
,V~
xx,
(x)V~
(x)F~
V~
T
x
Fuzzy VI Problem
Vz ,,z0
F ,V
such that , Find
:F,VVI
consider 1,0Given
yx
xyx
RRyx
xnn
Maximizing Solution to
F~
,V~
VI
10
Vz ,,z0
F ,V s.t.
max
yx
xyx
Optimization with parameterized equilibrium constraints
Bi-level programming
— Gap function
— Penalty method Maximum feasible problem
— Bisection with auxiliary program
— Analytic center cutting plane
Hu, C.-F., 2000, “Solving Variational Inequalities in a Fuzzy Environment”, Journal of Mathematical Analysis and Applications, Vol. 249, No. 2, pp. 527-538.
Hu, C.-F., 2001, “Solving Fuzzy Variational Inequalities over a Compact Set”, Journal of Computational and Applied Mathematics,Vol. 129, pp. 185-193.
Fang, S.-C. and Hu, C.-F.,“Solving Fuzzy Variational Inequalities”, Journal of Fuzzy Optimization and Decision Making, vol. 1, No. 1,pp. 134-143, 2002.
Hu, C.-F., “Generalized Variational Inequalities with Fuzzy Relations”, Journal of Computational and Applied Mathematics, vol. 146, No. 1,pp. 47-56, 2002.
Many
Thanks