# Uncertainty Theory & Uncertain Programming UTLAB Fuzzy

```Credibility Theory
It is a new branch of mathematics that
studies the behavior of fuzzy phenomena.
Baoding Liu
Uncertainty Theory Laboratory
Department of Mathematical Sciences
Tsinghua University
Uncertainty Theory &amp; Uncertain Programming
UTLAB
Fashion of Mathematics
2300 Years Ago: Euclid: “Elements”, First Axiomatic System
1899: Hilbert: Independence, Consistency, Completeness
1931: K. Godel: Incompleteness Theorem
1933: Kolmogoroff: Probability Theory
2004: B. Liu: Credibility Theory
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory &amp; Uncertain Programming
UTLAB
Why I do not possibility measure?
(a) Possibility is not self-dual, i.e., Pos{ A}+Pos{ Ac }  1.
(b) I will spend &quot;about \$300&quot;:   (200,300, 400).
In order to cover my expenses with maximum chance,
how much needed?
Pos{  x}  1 
x  300
A self-dual measure is absolutely needed!
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory &amp; Uncertain Programming
UTLAB
Five Axioms
Axiom 1. Cr{}=1.
Axiom 2. Cr{ A}  Cr{B} whenever A  B.
Axiom 3. Cr is self-dual, i.e., Cr{ A}  Cr{ Ac }  1.
Axiom 4. Cr i Ai   0.5=supi Cr{ Ai } if Cr{ Ai }  0.5 for each i.
Axiom 5. For each A  P (1   2 
  n ), we have
sup min Crk { k },
if sup min Crk { k }  0.5

1

k

n
(1 , , n )A
(1 , , n )A 1 k  n

Cr{ A}  
min Crk { k }  0.5, if sup min Crk { k }  0.5.
1  ( , sup
(1 , , n )A 1 k  n
, n )Ac 1 k  n
1

Independence (Yes) Consistency (?) Completeness (Absolutely No)
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
UTLAB
Uncertainty Theory &amp; Uncertain Programming
Liu (UT, 2004)
Cr{A  B}  Cr{A}+Cr{B}.
A credibility measure is additive if and only if
there are at most two elements in universal set.
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
UTLAB
Uncertainty Theory &amp; Uncertain Programming
Credibility Semicontinuity Laws
Liu (UT, 2004)
Theorem: Let (, P(), Cr) be a credibility space,
and A1 , A2 ,
 P(). Then
lim Cr{ Ai }  Cr{ A}
i 
if one of the following conditions is satisfied:
(a) Cr{A}  0.5 and Ai  A; (b) lim Cr{ Ai }  0.5 and Ai  A;
i 
(c) Cr{A}  0.5 and Ai  A; (d) lim Cr{ Ai }  0.5 and Ai  A.
i 
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
UTLAB
Uncertainty Theory &amp; Uncertain Programming
Credibility Extension Theorem
Li and Liu (2005)
If Cr{ } satisfies the credibility extension condition,
sup Cr{ }  0.5,
 
Cr{ *}+ sup Cr{ }  1 if Cr{ *}  0.5,
  *
then Cr{ } has a unique extension to a credibility
measure on P(),
sup Cr{ },
if sup Cr{ }  0.5


 A
 A
Cr{A}= 
Cr{ }  0.5, if sup Cr{ }  0.5.
1  sup
c
 A
A
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory &amp; Uncertain Programming
UTLAB
Fuzzy Variable
Definition : A fuzzy variable is a function from a credibility
space (,P(),Cr) to the set of real numbers.
Membership function:  ( x)   2Cr   x   1.
Sufficient and Necessary Condition : A function  :   [0,1]
is a membership function iff sup ( x)  1.
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory &amp; Uncertain Programming
UTLAB
Credibility Measure by Membership Function
Liu and Liu (IEEE TFS, 2002)
Let  be a fuzzy variable with membership
function . Then
1

Cr{  A}   sup  ( x)  1  sup  ( x)  .
2  xA
xAc

Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory &amp; Uncertain Programming
UTLAB
Independent Fuzzy Variables
Zadeh (1978), Nahmias (1978), Yager (1992), Liu (2004)
Liu and Gao (2005)
The fuzzy variables 1 , 2 ,
, m are independent if
m

Cr  {i  Bi }  min Cr{i  Bi }
 i 1
 1i  m
for any sets B1 , B2 , , Bm of .
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
UTLAB
Uncertainty Theory &amp; Uncertain Programming
Let 1 ,  2 ,
,  n be independent fuzzy variables with
membership functions 1 ,  2 ,
,  n , respectively.
Then the membership function of f (1 ,  2 ,
 ( x) 
sup
,  n ) is
min i ( xi ).
x  f ( x1 , x2 , , xn ) 1i  n
It is only applicable to independent fuzzy variables.
It is treated as a theorem, not a postulate.
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory &amp; Uncertain Programming
UTLAB
Expected Value
Liu and Liu (IEEE TFS, 2002)
Let  be a fuzzy variable. Then the expected value of 
is defined by
E[ ]  

0
0
Cr{  r}dr   Cr{  r}dr

provided that at least one of the two integrals is finite.
Yager (1981, 2002): discrete fuzzy variable
Dubois and Prade (1987): continuous fuzzy variable
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
UTLAB
Uncertainty Theory &amp; Uncertain Programming
Why the Definition Reasonable?
(i) Since credibility is self-dual, the expected value

0
0

E[ ]   Cr{  r}dr   Cr{  r}dr
is a type of Choquet integral.
(ii) It has an identical form with random case,

0
0

E[ ]   Pr{  r}dr   Pr{  r}dr.
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
UTLAB
Uncertainty Theory &amp; Uncertain Programming
Credibility Distribution
Liu (TPUP, 2002)
The credibility distribution  : ( , )  [0,1]
of a fuzzy variable  is defined by
 ( x)  Cr{   |  ( )  x}.
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
UTLAB
Uncertainty Theory &amp; Uncertain Programming
A Sufficient and Necessary Condition
Liu (UT, 2004)
A function  :   [0,1] is a credibility distribution
if and only if it is an increasing function with
lim  ( x)  0.5  lim ( x)

 x 
x 

lim  ( y )   ( x) if lim ( y )  0.5 or ( x)  0.5.

y x
 y x
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory &amp; Uncertain Programming
Entropy
UTLAB
(Li and Liu, 2005)
What is the degree of difficulty of predicting the specified value
that a fuzzy variable will take?
n
H [ ]   S (Cr{  xi })
S (t )  t ln t  (1  t ) ln(1  t )
i 1
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
UTLAB
Uncertainty Theory &amp; Uncertain Programming
Random Phenomena
Fuzzy Phenomena
(1654)
(1965)


Probability Theory
(1933)
Credibility Theory
(2004)


Probability
Credibility
Three Axioms
Five Axioms
Sum &quot;+&quot;
Maximization &quot;  &quot;
Product &quot;  &quot;
Minimization &quot;  &quot;
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
UTLAB
Uncertainty Theory &amp; Uncertain Programming
Essential of Uncertainty Theory
Probability Theory 
  Measure Theory + Function Theory
Credibility Theory 

Two basic problems?

[1] Measure of Union:  { A  B}   { A}   {B}
&quot; &quot;
 {A  B}   {A}   {B}
&quot; &quot;
[2] Measure of Product:  { A  B}   { A}   {B} &quot; &quot;
 {A  B}   { A}   {B} &quot; &quot;
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
UTLAB
Uncertainty Theory &amp; Uncertain Programming
(+,)-Axiomatic System: Probability Theory
(,)-Axiomatic System: Credibility Theory
(,)-Axiomatic System: Nonclassical Credibility Theory
(+,)-Axiomatic System: Inconsistent
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
UTLAB
Uncertainty Theory &amp; Uncertain Programming
Fuzzy Programming
max f ( x,  )

subject to:
 g ( x,  )  0,
j

j  1, 2,
,m
x - Man proposes
 - God disposes
It is not a mathematical model!
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
UTLAB
Uncertainty Theory &amp; Uncertain Programming
The Simplest
The Most Fundamental
Problem
Given two fuzzy variables  and  ,
which one is greater?
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory &amp; Uncertain Programming
UTLAB
Expected Value Criterion
    E[ ]  E[ ].
Objective: max f ( x,  )  max E[ f ( x,  )]
Constraint: g j ( x,  )  0  E[ g j ( x,  )]  0
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory &amp; Uncertain Programming
UTLAB
Fuzzy Expected Value Model
Liu and Liu (IEEE TFS, 2002)
Find the decision with maximum expected return
subject to some expected constraints.
max E[ f ( x,  )]

subject to :
 E[ g ( x,  )]  0, j  1,2,, p
j

Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory &amp; Uncertain Programming
UTLAB
Optimistic Value Criterion
    sup (  )  sup (  )
Objective:
max f ( x,  )
x
(Undefined)

max max f : Cr{ f ( x,  )  f }  
x
Constraint:
Baoding Liu
f
g j ( x,  )  0  Cr  g j ( x,  )  0  
Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory &amp; Uncertain Programming
UTLAB
(Maximax) Chance-Constrained Programming
Liu and Iwamura (FSS, 1998)
Maximize the optimistic value subject to chance constraints.
max max f
max f
x
f

subject to :

 Cr{ f ( x,  )  f }  
 Crg j ( x,  )  0, j  1,2,, p  

Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory &amp; Uncertain Programming
UTLAB
Pessimistic Value Criterion
    inf (  )  inf (  )
Objective:
max f ( x,  )
x

max min f : Cr{ f ( x,  )  f }  
x
f
Constraint: g j ( x,  )  0  Cr  g j ( x,  )  0  
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory &amp; Uncertain Programming
UTLAB
(Minimax) Chance-Constrained Programming
Liu (IS, 1998)
Maximize the pessimistic value subject to chance constraints.
max min f
f
 x
subject to :

 Cr{ f ( x,  )  f }  
 Crg ( x,  )  0, j  1,2,, p  
j

Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory &amp; Uncertain Programming
UTLAB
Credibility Criterion
    Cr   r  Cr   r
Remark: Different choice of r produces different ordership.
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
UTLAB
Uncertainty Theory &amp; Uncertain Programming
Fuzzy Dependent-Chance Programming
Liu (IEEE TFS, 1999)
Find the decision with maximum chance
to meet the event in an uncertain environment.
max Cr{hk ( x,  )  0, k  1, 2, , q}

subject to:

g j ( x,  )  0, j  1, 2, , p

Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
UTLAB
Uncertainty Theory &amp; Uncertain Programming
Classify Uncertain Programming via  Graph
Information
Random Fuzzy
Fuzzy random
Fuzzy
Stochastic
Single-Objective P
MOP
GP
DP
MLP
Philosophy
EVM
CCP DCP

Maximax Minimax
Structure
Baoding Liu
Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory &amp; Uncertain Programming
UTLAB
Last Words
[1] Liu B., Foundation of Uncertainty Theory.
[2] Liu B., Introduction to Uncertain Programming.
If you want an electronic copy of my book,
or source files of hybrid intelligent algorithms,