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