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Credibility Theory
Baoding LiuUncertainty Theory Laboratory
Department of Mathematical Sciences
Tsinghua University
It is a new branch of mathematics that studies the behavior of fuzzy phenomena.
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A BFashion 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 & Uncertain Programming U T L A B
Why I do not possibility measure?
(a) Possibility is not self-dual, i.e., Pos{ }+Pos{ } 1.
(b) I will spend "about $300": (200,300,400).
In order to cover my expenses with maximum chance,
how much needed?
Pos{ } 1
cA A
x x
A self-dual measure is absolutely ne
300
eded!
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
Five Axioms
1 2
Axiom 1. Cr{ }=1.
Axiom 2. Cr{ } Cr{ } whenever .
Axiom 3. Cr is self-dual, i.e., Cr{ } Cr{ } 1.
Axiom 4. Cr 0.5=sup Cr{ } if Cr{ } 0.5 for each .
Axiom 5. For each ( ), we have
Cr{
c
i i i i i
n
A B A B
A A
A A A i
A P
A
1 1
11
1 1( , , ) ( , , )
1 1( , , )( , , )
sup min Cr { }, if sup min Cr { } 0.5
}1 sup min Cr { } 0.5, if sup min Cr { } 0.5
Independence (Yes) Consistency (?) Completeness
.
(Ab
n n
cnn
k k k kk n k nA A
k k k kk n k nAA
solutely No)
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
Credibility Subadditivity Theorem
A credibility measure
Liu (UT, 2004)
Credibil
is additive if and onl
ity measure is subadditive, i.e.,
Cr{ } Cr{
y if
there are at most two elements in un
}+
iversal set.
Cr{ }.A B A B
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
Credibility Semicontinuity Laws
1 2
Liu (UT, 2004)
Theorem: Let ( , ( ),Cr) be a credibility space,
and , , ( ). Then
lim Cr{ } Cr{ }
if one of the following conditions is satisfied:
(a) Cr{ } 0.5 and ; (b) lim Cr{ } 0.5 an
ii
i ii
P
A A P
A A
A A A A
d ;
(c) Cr{ } 0.5 and ; (d) lim Cr{ } 0.5 and .
i
i i ii
A A
A A A A A A
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
Credibility Extension Theorem
*
* *
Li and Liu (2005)
If Cr{ } satisfies the credibility extension condition,
supCr{ } 0.5,
Cr{ }+supCr{ } 1 if Cr{ } 0.5,
then Cr{ } has a unique extension to a credib
ility
measure on P( ),
supCr{ }, if supCr{ } 0.5Cr{A}=
1 supCr{ } 0.5, if supCr{ } 0.5.c
A A
AA
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
Fuzzy Variable
A fuzzy variable is a function from a credibility
space ( ,P( ),Cr) to the set of real nu
Membership functi
mbers.
( ) 2Cr 1.
A function : [0,1]
i
on:
:
Definition :
Sufficient and Necessary Condit
x
io
x
n
s a membership function iff sup ( ) 1.x
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
Credibility Measure by Membership Function
Let be a fuzzy variable with membership
function . Then
1Cr{ } sup ( ) 1 sup ( ) .
2 cx A x A
A x x
Liu and Liu (IEEE TFS, 2002)
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
Independent Fuzzy Variables
1 2
11
1 2
Zadeh (1978), Nahmias (1978), Yager (1992), Liu (2004)
Liu and Gao (2005)
The fuzzy variables , , , are independent if
Cr { } min Cr{ }
for any sets , , , of .
m
m
i i i ii m
i
m
B B
B B B
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
Theorem: Extension Principle of Zadeh
1 2
1 2
1 2
1 2
1( , , , )
Let , , , be independent fuzzy variables with
membership functions , , , , respectively.
Then the membership function of ( , , , ) is
( ) sup min
It is only a
( ).
pplic
n
n
n
n
i ii nx f x x x
f
x x
able to independent fuzzy variables.
It is treated as a theorem, not a postulate.
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
Expected Value
0
0
Liu and Liu (IEEE TFS, 2002)
Let be a fuzzy variable. Then the expected value of
is defined by
E[ ] Cr{ }d Cr{ }d
provided that at least one of the two integrals is finite.
Yage
r r r r
r (1981, 2002): discrete fuzzy variable
Dubois and Prade (1987): continuous fuzzy variable
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
Why the Definition Reasonable?
0
0
(i) Since credibility is self-dual, the expected value
E[ ] Cr{ }d Cr{ }d
is a type of Choquet integral.
(ii) It has an identical form with random case,
E[ ] Pr{
r r r r
r
0
0}d Pr{ }d .r r r
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
Credibility Distribution
Liu (TPUP, 2002)
The credibility distribution : ( , ) [0,1]
of a fuzzy variable is defined by
( ) Cr{ | ( ) }.x x
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
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 ( ) 0.5 lim ( )
lim ( ) ( ) if lim ( ) 0.5 or ( ) 0.5.x x
y x y x
x x
y x y x
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
Entropy (Li and Liu, 2005)
1
What is the degree of difficulty of predicting the specified value
that a fuzzy variable will take?
[ ] (Cr{ })n
ii
H S x
( ) ln (1 ) ln(1 )S t t t t t
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
Random Phenomena Fuzzy Phenomena
(1654) (1965
Probability Credibility
Three Axioms Five Axioms
Sum "+" Maximiza
)
Probability Theory Credibility Theory
(1933) (2004)
tion " "
Product " " Minimization " "
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
Essential of Uncertainty Theory
Probability Theory Function Theory
Credibility Theory
Measure Theory
Two basic problems?
[1] Measure o
f Unio
{ } }
+
{n: A B A
[2] Measure of Produ
{ } " "
{ } { } { } " "
{ } { } { } " "
{ } { } { }
t
"
:
"
c
B
A B A B
A B A B
A B A B
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
What Mathematics Made?
(+, )-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
Uncertainty Theory & Uncertain Programming U T L A B
Fuzzy ProgrammingFuzzy Programming
max ( , )
subject to:
( , ) 0, 1,2, ,
- Man proposes
- God disposes
It is not a mathematical model!
j
f x
g x j m
x
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
The Simplest
Given two fuzzy variables and ,
which one is greater?
The Most FundamentalProblem
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
Expected Value Criterion
[ ] [ ].
Objective: max ( , ) max [ ( , )]
Constraint: ( , ) 0 [ ( , )] 0j j
E E
f x E f x
g x E g x
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
pjxgE
xfE
j ,,2,1,0)],([
:subject to
)],([max
Liu and Liu (IEEE TFS, 2002)Find the decision with maximum expected returnsubject to some expected constraints.
Fuzzy Expected Value Model
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
Optimistic Value Criterion
sup sup( ) ( )
Objective: max ( , ) (Undefined)
max max : Cr{ ( , ) }
Constraint: ( , ) 0 Cr ( , ) 0
x
x f
j j
f x
f f x f
g x g x
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
(Maximax) Chance-Constrained Programming
(Maximax) Chance-Constrained Programming
Liu and Iwamura (FSS, 1998)Maximize the optimistic value subject to chance constraints.
ffx
maxmax
pjxg
fxf
f
j ,,2,1,0),(Cr
}),({Cr
:subject to
max
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
Pessimistic Value Criterion
inf inf( ) ( )
Objective: max ( , )
max min : Cr{ ( , ) }
Constraint: ( , ) 0 Cr ( , ) 0
x
x f
j j
f x
f f x f
g x g x
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
(Minimax) Chance-Constrained Programming
(Minimax) Chance-Constrained Programming
Liu (IS, 1998)Maximize the pessimistic value subject to chance constraints.
pjxg
fxf
f
j
fx
,,2,1,0),(Cr
}),({Cr
:subject to
minmax
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
Credibility Criterion
Cr Cr
Remark: Different choice of produces different ordership.
r r
r
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
Fuzzy Dependent-Chance ProgrammingFuzzy Dependent-Chance Programming
j
max Cr{ ( , ) 0, 1,2, , }
subject to:
g ( , ) 0, 1,2, ,
kh x k q
x j p
Liu (IEEE TFS, 1999)Find the decision with maximum chance to meet the event in an uncertain environment.
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
Classify Uncertain Programming via Graph
Information
PhilosophySingle-Objective PMOP
GPDP
MLP
Structure
EVM CCP DCP
Random FuzzyFuzzy random
FuzzyStochastic
Maximax Minimax
Baoding Liu Tsinghua University
http://orsc.edu.cn/~liu
Uncertainty Theory & Uncertain Programming U T L A B
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,please download them from
http://orsc.edu.cn/~liu
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