# 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?

 Measure of Union:  { A  B}   { A}   {B}
&quot; &quot;
 {A  B}   {A}   {B}
&quot; &quot;
 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
 Liu B., Foundation of Uncertainty Theory.
 Liu B., Introduction to Uncertain Programming.
If you want an electronic copy of my book,
or source files of hybrid intelligent algorithms,