on some fuzzy optimization problems
DESCRIPTION
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 PresentationTRANSCRIPT
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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
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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
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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
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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。
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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。
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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)(
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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。
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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~
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Example
Ex: The weather is “good”
20 25 30 35 x
A(x)A
fuzzy set
25 30 x
A(x)A
crisp set
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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
, ~
~……………...
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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
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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
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Ex:
A: The weather is “hot”
......
23
4.0
22
3.0
21
2.0A~
Example
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模糊集合之運算 聯集( Union )
交集( Intersection )
補集( Complement )
)}(),(max{)( ~~~~ uuuBABA
)}(),(min{)( ~~~~ uuuBABA
)(1)( ~~ uuAAC
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Example
Ex: two fuzzy set and find BA~
and ~
1
15 20 x
BA~~
~A B
~
BA~~
)(~ xA
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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
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- 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 , )( ~
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Fuzzy numbers
Two classes
One class has 30 students
One class has 25 students
~~{
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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)
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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)
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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
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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
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鐘形模糊數
0
1
X
(x)
2
2)(
~ )(
x
Aex
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不規則模糊數
a b c0
1
X
(x)
L(x) R(x)
cxb
bc
bxR
bxaab
axL
xA
)(
)()(~
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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
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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
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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
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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
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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
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Fuzzy Ranking
(>) ??M N
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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
~~
{
{
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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
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Example
7~
3~
4~
0~
243
21
3221
xx
xxx
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How to rank fuzzy numbers?
The study of fuzzy ranking began in 1970's Over 20 ranking methods were proposed No \best" method agreed
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How to Select Fuzzy Ranking
Easy to compute Consistency Ability to discriminate Go with intuition Fits your model Consider combination of different
rankings
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Optimization
Optimization models can be very useful.
..
max
ts
0,
1002
yx
yx
x y
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Optimization models for Decision making
max
xf
xf
p
i
throughput
profit
..ts
skqxh
rjdxg
kk
jj
,,1
,,1
,
,
resource
demand
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Past Industrial Experience
Optimization models can be very useful.
Problems are harden to define than to solve.
Most decision are made under uncertainty.
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Fuzzy Optimization
max
..ts
x y
0,
100~
2
yx
yx
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Fuzzy Optimization and Decision making
fuzzy vector :
~
,,1 ,~~,
,,1 ,~~
, ..
~,
~,
maximize
1
skqxh
rjdxgts
xf
xf
kk
ji
p
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Solution Methods
-level approach Parametric approach Semi-infinite programming approach Set-inclusion approach Possibilistic programming approach
……
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Recent Development
System of Fuzzy Inequalities
Fuzzy Variational Inequalities
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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
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Motivation
NLP
where is a convex set and is a
smooth real-valued function defined on .
, xs.t.
xmin
K
h
nRK fK
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Variational Inequalities
Find such that
for each
where means the inner product operation.
x Kx
,0xx,x h ,x K
,
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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
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Fuzzy Inequalities – System I
“ ” means “approximately less than or equal to”.
~
ljxg
mixf
tsRx
j
i
n
,,2,1 ,0
,,2,1 ,0~
.. Find
*
~
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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
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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
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Equivalently,
When is invertible
nRx
ljxg
mix
j
iF
,10
,,1 ,0
,,1 , s.t.
max
~
iF
~
1~~
iFxfx
iiF
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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
~
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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
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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 ,
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Extension to solving fuzzy inequalities with piecewise linear membership functions
iF
~
it
xfi
:xf ii
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(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.)
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Fuzzy Inequalities – Systems II
Find such thatx X
Iixfi
,0~~
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Fundamental Problem
No universally accepted theory for ranking two fuzzy sets.
?~~?
~~
ab
ba
R
a~b~
R
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Simple Case
Solving
is to find optimal solutions to the semi-infinite programming problem
njx
mibxa
j
jij
n
ji
,,1 ,0
,,1 ,~~
1
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(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
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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
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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
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Fuzzy VI Problem
Vz ,,z0
F ,V
such that , Find
:F,VVI
consider 1,0Given
yx
xyx
RRyx
xnn
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Maximizing Solution to
F~
,V~
VI
10
Vz ,,z0
F ,V s.t.
max
yx
xyx
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Optimization with parameterized equilibrium constraints
Bi-level programming
— Gap function
— Penalty method Maximum feasible problem
— Bisection with auxiliary program
— Analytic center cutting plane
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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.
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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.
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Many
Thanks