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
模糊與機率不同處之比較
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：
1 , x  A
 A( x )  
0 , x  A
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。
7
隸屬函數(Membership Functions)
~
 假 設 宇 集 (universe)U={x1, x2,…, xn}，A
是定義在U之下的模糊集合，
A  { ( x1,  A ( x1)) ,( x2,  A ( x2)) ,...,( xn,  A ( xn)) }.

 A~ : U  [0,1] 為模糊集合之隸屬函數
(Membership Function)。
~
  ~ ( xi ) 表示模糊集合 A 中xi的隸屬程度
A
(Degree of Membership)。
Example
Ex: The weather is “good”
A(x)
A(x)
A
A
25
30
crisp set
x
20 25
30 35
fuzzy set
x
Example
X  real numbers
~
A  real numbersclose to10
~
A   x,  A~ x  x  R
1.5
……………...
1
 A~  x  
0.5
0
0
5
10
15
20
1
2
1  x  10
傳統與模糊集合不同處之比較
傳統集合
模糊集合
Characteristic function Membership function
特徵函數
隸屬函數
A(x)
 A~ ( x)
X{0,1}
X[0,1]
模糊集合表示法
 宇集U為有限集合
~
A    A~ ( xi ) / xi
 宇集U無限集合或有限連續
~
A    A~ ( xi ) / xi
xU
 一般的表示方法
~
A  {( xi,  A~ ( xi )) xi U }
Example
Ex:
A: The weather is “hot”
 0.2 0.3 0.4

A


 ...... 
~
23
 21 22

模糊集合之運算
 聯集（Union）
 A~B~ (u)  max{ A~ (u), B~ (u)}
 交集（Intersection）
 A~B~ (u)  min{ A~ (u), B~ (u)}
 補集（Complement）
 A~ (u)  1   A~ (u)
C
Example
~
~
~ ~
Ex: two fuzzy set A and B and find A
B
 A~ ( x)
~
A
~
B
1
15 20
~ ~
AB
x
Example
 A~B~
 A~B~
(15)=  A~ ( x) (15)   ~ ( x) (15)
B
=min(  A~ ( x) (15), B~ ( x) (15))
=min(1,0)=0
(20)=  A~ ( x) (20)  B~ ( x) (20)
=min(  A~ ( x) (20), B~ ( x) (20))
=min(0.7,0.2)=0.2
a-截集(a -cut或a -level)
~
 模糊集合 A 的a-截集定義為:

Aa  xi  A~ ( xi )  a , xi  U

a  [0,1]
~
 而模糊集合 A 取a -截集所形成的區間範
圍為

Aa  x  A~ ( x)  a
 A
L
a
, AU
a

Fuzzy numbers
 Two classes
{
~
One class has 30 students
~
One class has 25 students
模糊數(Fuzzy Numbers)
 If I is a normal fuzzy set on R and
Ia is a
closed interval for each 0  a  1, then I is a
fuzzy number.
(Note that: I is a normal, if I ( x)  1,  x  R. )
(x)
1
L(x)
0
a
R(x)
b
c
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
(x)
1
0
a
b
c
A  (a, b, c)
x<a
 0 ,
x a
 b  a , a  x  b
 A~ ( x)  
cx

, b xc
c b

X
 0 ,
xc
梯形模糊數
(x)
1
0
a
b
c
A  (a, b, c, d )
d
 x-a
 b-a , a  x<b
 1,
bxc
 A~ ( x)  
x-d

, cxd
 c-d
X
 0, otherwise
鐘形模糊數
(x)
1
 A~ ( x)  e
0
s 
X

( x )2

2
不規則模糊數
(x)
1
L(x)
0
a
R(x)
b
c
 xa
 L( b  a ) a  x  b
 A~ ( x)  
x b
 R(
) bxc
 c b
X
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
(a1 , b1 , c1 )  (a2 , b2 , c2 )  (a1  a2 , b1  b2 , c1  c2 )
：模糊數加法運算子
 梯形模糊數
(a1 , b1 , c1 , d1 )  (a2 , b2 , c2 , d 2 ) 
(a1  a2 , b1  b2 , c1  c2 , d1  d 2 )
模糊數乘法
 三角形模糊數(k>0)
k  (a, b, c)  (k  a, k  b, k  c)
 ：模糊數乘法運算子
 梯形模糊數
k  (a, b, c, d )  (k  a, k  b, k  c, k  d )
模糊數乘法
 三角形模糊數(a1>0,a2>0)
(a1 , b1 , c1 )  (a2 , b2 , c2 )  (a1  a2 , b1  b2 , c1  c2 )
 ：模糊數乘法運算子
 梯形模糊數
(a1 , b1 , c1 , d1 )  (a2 , b2 , c2 , d 2 ) 
(a1  a2 , b1  b2 , c1  c2 , d1  d 2 )
模糊數除法
 三角形模糊數
(a1 , b1 , c1 )(a2 , b2 , c2 )  (a1 / c2 , b1 / b2 , c1 / a2 )
 ：模糊數除法運算子
 梯形模糊數
(a1 , b1 , c1 , d1 )(a2 , b2 , c2 , d 2 ) 
(a1 / d 2 , b1 / c2 , c1 / b2 , d1 / a2 )
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
~
 ~
 aij x j  bi , i  1,  , m
 j 1
 x  0,
j  1,  , n
j

n
is to find optimal solutions to the system of
fuzzy linear inequalities problem
Example
3x1  4 x  2 x3  0
~
~
~
~
4 x1  3 x 2
7
2
2
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 x y
s.t. x  2 y  100
x, y  0
Optimization models for Decision making
max
s.t.

f i x 

f p x 
profit
throughput
 resource 
h x   q , k  1,, s  demand 
g j x   d j , j  1,, r
k
k
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
s.t.
xy
x  2 y  100
~
x, y  0
Fuzzy Optimization and Decision
making
 f1  x,~ 

 
maximize 
  ~
 fp  x, 
~ ~


s.t.
g i  x,   d j , j  1,  , r
~ ~

hk  x,   qk , k  1,  , s
~
:
fuzzy vector
Solution Methods
a
 a -level approach
 Parametric approach
 Semi-infinite programming approach
 Set-inclusion approach
 Possibilistic programming approach
……
Recent Development
 System of Fuzzy Inequalities
 Fuzzy Variational Inequalities
Motivation
min cT x
LP
P 
s.t. Ax  b
x0
max bT w
D  s.t. AT w  c
w 0
K-K-T Optimality Conditions
Find x,w  such that
Ax  b
x 0
AT w  c
w 0
cT x  b T w  0
Motivation
NLP
min hx 
s.t. x K ,
where K  R n is a convex set and f  is a
smooth real-valued function defined on K .
Variational Inequalities
Find x such that x  K
hx , x  x  0,
where
, 
for each x   K ,
means the inner product operation.
System of Fuzzy Inequalities
 f i  x  
~ 0, i  I
~
~
 g j  x   0 , j  J
fi , g j : R n  R
“ ” means “approximately less than or equal to”.
~
Examples：
3x1  4 x  2 x3  0
~
~
~
~
4 x1  3 x 2
7
2
2
Fuzzy Inequalities – System I
Find x  R
n
s.t.
0, i  1,2,, m

 fi x  
~
* 

 g j  x   0, j  1,2,, l

~
“~ ” means “approximately less than or
equal to”.

Each fuzzy inequality f i  x   0 determines a fuzzy set Fi
~
in R n with
  u  f i  x  : continuous and
~
Fi
if f i  x  0
1,

 Fi  x   i  f i  x , if 0  f i  x   ti
0,
if f i  x   ti

i
strictly decreasing
ti
f i x 
Fuzzy Decision Making
 (Bellman/Zadeh,1970) Decision Making Model
~
~
D   Fi
  min  x 
~
D
1i  m
~
Fi
 Solving(*) is to find optimal solutions to

~  x 
max
min

Fi
n
xR
1i  m
s.t. g j  x   0, j  1,2,, l
 Equivalently,
max a
s.t .  F~i  x   a , i  1, , m
g j  x   0,
j  1,, l
0  a  1, x  R n
 When
  is invertible
~
Fi
1




F~i x  a  fi x  F~i a 
If f i x  , g i x  are convex and  i  are concave, then a
solution to (*) can be obtained by solving a convex
programming problem
max a
s.t.  a   f i  x   0, i  1,, m
1
~
Fi
 g j  x   0,
j  1,, l
0  a  1,
x  Rn
Huard’s “Method of Centers” + Entropic Regularization
Method reduce the problem to solving a sequence of
unconstrained smooth convex programs


m
1 
k
min ln exp p a  a   exp p f i  x   i1 a  
x ,a p
i 1


exp pg i  x   exp p a   exp pa  1

j 1

l
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 )
 Semi-infinite programming extension for
i  I, j  J
(Hu, C.-F. and Fang, S.-C., “A Relaxed
Cutting Plane Algorithm for Solving Fuzzy
Inequality Systems ”, Optimization, vol. 45,
pp. 89-106, 1999)
 Extension to solving fuzzy inequalities with
piecewise linear membership functions
 
~
Fi
  f i  x  :
i
ti
f i x 
 (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
x  X such that
~
~
fi x  0,
iI
Fundamental Problem
No universally accepted theory for ranking
two fuzzy sets.
 
~
a
~
~
a b?
~ ~
b a?
~
b
R
R
Simple Case
 Solving
~
 n ~
 aij x j  bi , i  1,  , m
 j 1
 x  0,
j  1,  , n
j

is to find optimal solutions to the semi-infinite
programming problem
max1  a
n
s.t.  La~ij t   x j  Lb~ij t , t  a ,1, i  1,2,  , m,
j 1
n
R
j 1
a~ij
t   xj  Rb~i t , t  a ,1, i  1,2,  , m
x j  0, j  1,2,  , n,
a  0,1.
 (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.)
Fuzzy Variational Inequalities
 An Optimization problem can be cast into a
variational inequality problem VI V,F
Find X  V such that
z  xT Fx  0
z  V
where V is a nonempty, closed, convex subset
n
n
n
of R and F : R  R is a point-to-point
mapping.
~
V  V,  V~ 
~
F  F,  F~ ( x ) ()
 Problem


VI V,F : Find (x,y)  R n  R n
such that
~
x, V~ x  V,
~


y,~F(x)  y   Fx 
~
z  x  y  0, z, V~(x)  V
T
As difficult as an optimization problem with
parameterized equilibrium constraints.
Fuzzy VI Problem
Given a  0,1 consider
VI Va , Fx a  :
Find x, y   R  R such t hat
x  Va , y  Fx a
n
n
0  z  x, y , z  Va
 
~ ~
Maximizing Solution to VI V, F

max a
s.t . x  Va , y  Fx a
0  z  x, y , z  Va
0 a 1
 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