Fuzzy Integrals in MultiCriteria Decision Making
Dec. 2011 Jiliang
University China
Contents
Multi-Criteria Decision Making Problem
 Aggregation
• Requirements of aggregation operators
• Common aggregation operators
 Fuzzy Measure and Integrals
 Properties of Fuzzy Integral
 Importance and Interaction of Criteria
 Decision Making in Pattern Recognition
 Summary

Multi-Criteria Decision Making Problem
Set of alternativ es or acts Ω  1 , 2 ,...,  p 
Consquence or result w.r .t. attributes or criteria x  x1 , x2 ,..., xn   X
t
Find the best one in Ω in terms of utility function u : X  R
and build the order relation  that x  y  u ( x)  u ( y ).
Aggregatio n of component utility functions :
u ( x1 , x2 ,..., xn )  H (u1 ( x1 ), u2 ( x2 ),..., un ( xn ))
where H is called an aggregatio n operator.
n
Example : u ( x1 , x2 ,..., xn )   ui ( xi )
i 1
Aggregation in MCDM
Utility
Criteria
C1
Alternativ e i
C2
x1
x2
u1 ( x1 )
Aggregatio n Operator
Final Result
u2 ( x2 )
i  1,... p
H ()
C3
C4
x3
x4
u3 ( x3 )
u4 ( x4 )
of Evaluation
Requirements of Aggregation Operator

Mathematical Properties
• Properties of extreme values
H (0,0,...,0)  0, H (1,1,...,1)  1
• Idempotency
H (a, a,..., a)  a
• Continuity
• Non-decreasing w.r.t. each argument
• Stability under the same positive linear
transform
H (ra1  t ,..., ran  t )  rH (a1 ,..., an )  t , r  0, t  R
Requirements of Aggregation Operator

Behavioral Properties
• Expressing the weights of unequal importance on criteria
• Expressing the behavior of decision maker from perfect
tolerance (disjunctive behavior) to total intolerance
(conjunctive behavior)
• Accept when some criteria are met
• Demand all criteria have to be equally met
• Expressing compensatory effect:
• Redundancy when two criteria express the same things
• Synergy of two criteria: little importance separately but
important jointly
• Easy semantic interpretation of aggregation operator
Common Aggregation Operator
 Quasi-arithmetic
Mean
n
1 n


f
1 
M f (a1 ,..., an )  f   f (ai ) M w1 ,..., wn (a1 ,..., an )  f  wi f (ai )
 n i 1

 i 1

1
Example: Mean and Generalized Mean
n
1 n
f ( x)  x  M f (a1 ,...., an )   ai or M f (a1 ,...., an )   wi ai
n i 1
i 1
1/ p


f ( x)  x p  M f (a1 ,...., an )    wi aip 
 i 1

n
Common Aggregation Operator
Median: mid-ordered data after sorting
 Weighted minimum and maximum
wmin (a1 ,..., an )  in1 (1  wi )  ai 
wmax (a1 ,..., an )   in1 wi  ai 

• When all weights are 1, then weighted minimum becomes
•
•
the min-operator
The larger weight value represents the more degree of
importance in the aggregation process
When all weights are 0, then weighted maximum becomes
the max-operator
Common Aggregation Operator

Ordered weighted averaging (OWA)
• Weighted average of ordered input
n
OWA w1 ,..., wn (a1 ,..., an )   wi a(i )
i 1
n
w
i 1
i
1
a(1)  a( 2 )  ....  a( n )
Note:
1
1
( w1 ,....wn )  ( ,...., )  Average
n
n
1
1
( w1 ,....wn )  (0,
,....,
,0)  Trimmed mean
n2
n2
( w1 ,....wn )  (1,0,....,0,0)  Minimum
( w1 ,....wn )  (0,0,..0,1,0..,0,0)  Median
Fuzzy Measures

Fuzzy measure
 : P( X )  [0,1] satisfying following axioms.
(1)  ()  0,  ( X )  1
(2) A  B  X implies  ( A)   ( B )

Additivity, Super-additivity, Sub-additivity
A B 
 ( A  B)   ( A)   ( B) : additivity in probabilit y measure
 ( A  B)   ( A)   ( B) : super additivity
 ( A  B)   ( A)   ( B) : sub additivity
Fuzzy Measure

Sugeno’s g-lamda measure
g is a fuzzy measure satisfying the following condition.
For all A, B  P ( X ) with A  B  ,
g ( A  B )  g(A)  g(B)  g(A)  g(B) for some   1.
Then g is called Sugeno measure or g   measure.
Note :
1.   0  g 0 satisfies additivity
  0  g  satisfies super - additivity
  0  g  satisfies sub - additivity
2. g ( A  B) 
g ( A)  g ( B) - g ( A  B)  g ( A) g ( B)
1  g ( A  B )
Fuzzy Measure
For X  {x1 , x2 ,..., xn }
g i : g ({ xi }) is called fuzzy density function.
Note : A  B  , B  C  , A  C   
g ( A  B  C )  g ( A)  g ( B)  g (C ) 
 ( g ( A) g ( B)  g ( B) g (C )  g ( A) g (C ))  2 g ( A) g ( B) g (C )
or
g ({ x1 , x2 , x3 })  g 1  g 2  g 3   ( g 1 g 2  g 2 g 3  g 1 g 3 )  2 g 1 g 2 g 3
In general, for X  {x1 , x2 ,..., xn }
n 1
n
g( X )   g  
j
j 1
n
g
j 1 k  j 1


i
   (1  g )  1 /   1
 xi  X

j
g k 2   ...n 1 g 1 g 2 ...g n
Fuzzy Measure
Theorem : The following equation has a unique solution in (-1, )


i
g ( X )    (1  g )  1 /   1
 xi X

Colloary : If {g 1 , g 2 ,...g n } is given, then one can construct a
correspond ing g   measure.
{g 1 , g 2 ,...g n }   calculatio n from equation
 construct a g   measure.
Note: We need only n numbers of fuzzy density instead of 2n.
Fuzzy Measures and Integrals
Sugeno integral of a function f : X  [0,1] w.r.t. 
S   f ( x1 ),..., f ( xn )    in1  f ( x(i ) )   A(i ) 
where 0  f ( x(1) )  f ( x( 2) )  ...  f ( x( n ) ) and A(i )  x(i ) , x(i 1) ,..., x( n ) 
Choquet integral of a function f : X  [0,1] w.r.t. 
C  f ( x1 ),..., f ( xn )    f ( x(i ) )  f ( x(i 1) ))  A(i ) 
n
i 1
where 0  f ( x( 0 ) )  f ( x(1) )  f ( x( 2) )  ...  f ( x( n ) ) and
A(i )  x(i ) , x(i 1) ,..., x( n ) 
Fuzzy Measures and Integrals
Sugeno integral of a function f w.r.t.  :
 a  0.3,  b  0.4,  c  0.1
 a, b  0.82,  a, c  0.43,  b, c  0.54
f a   0.9, f b   0.6, f c   0.3
S   f ( x1 ),..., f ( xn )   0.3  1  0.6  0.82   0.9  0.3  0.82
Chiquet integral of a function f w.r.t.  :
C  f ( x1 ),..., f ( xn )   0.3 1  0.3  0.82  0.3  0.3  0.636
Properties of Fuzzy Integrals
Sugeno and Choquet integral are idempotent,
continuous, and monotonically non-decreasing
operators.
 Choquet integral with additive measure coincide
with a weighted arithmetic mean.
 Choquet integral is stable under positive linear
transforms.
 Choquet
integral is suitable for cardinal
aggregation where numbers have a real meaning.
 Sugeno integral is suitable for ordinal aggregation
where only order makes sense.

Properties of Fuzzy Integrals



Any OWA operator is a Choquet integral.
Sugeno and Choquet integral contains all order
statistics, thus in particular, min, max, and the median.
Weighted minimum and weighted maximum are special
case of Sugeno integral
Importance of Criteria and Interaction
Example:
wmath  w physics  3 wliter  2
Student _ A 
3 18  3 16  2 10
 15.25
8
Rank Order: A > C > B
Importance of Criteria and Interaction
 math   physics   0.45  literartur e  0.3
 math, physics   0.5  0.45  0.45 because of redundancy
 math, literature  0.9  0.45  0.3 because of synergy
 physics , literature  0.9  0.45  0.3 because of synergy
    0  math, physics , literature  1
Rank Order: C > A > B
Index for Importance
Global Importance of x j : Shapley index
Λx j  
   A A  x     A with
A X  x j
X
j
 Λx   1
n
j 1
j
Multiplied by n
X

X
 A 
 A  1! A !
X!
Index of Interaction
Average Interactio n Index between xi and x j :
I  xi , x j  
with  X
   A I  x , x
X
A X  xi , x j

X
 A 
 A  2! A !
X!
i
j
| A
 [-1,1]
and
I  xi , x j | A   A  xi , x j     A  xi    A  x j     A
Note: Redundancy and synergy
Identification of Fuzzy Measure

Identification Based on Semantics
• Importance of criteria
• Interaction between criteria
• Symmetric criteria {math, physics}
• Veto effect
H (a1 ,..., a j ,..., an )  a j  G(a1 ,..., a j 1 , a j 1 ,..., an )
 ( A)  0 for all A  X  {x j }
• Pass effect
H (a1 ,..., a j ,..., an )  a j  G(a1 ,..., a j 1 , a j 1 ,..., an )
 ( A)  1 for all A  x j
Identification of Fuzzy Measure

Identification Based on Learning Data
Identify t he fuzzy measure  that minimize the error
E   C ( z k1 ,..., z kn )  yk  for learning data ( z k , yk ), k  1,..., l
l
2
2
k 1
M. Grabish, H. T. Nguyen, and E. A. Walker, Fundamentals of
Uncertainty Calculi, with Applications to Fuzzy Inference, Kluwer
Academic, 1995
Decision Making in Pattern recognition
Feature level
Aggregation of class
simple classifier memberships
C1
H1 ()
x1
Input
pattern

x
x2
Class
C2
Decision
x3
C3
x4
C4
H 2 ()
Class
label
Decision Making in Pattern recognition
Complex
classifiers
Aggregation of class
memberships
C1
Input
pattern

x
C2
H ()
C3
C4
Class
Decision
Class
label
Summary


Multi-Criteria Decision Making Problem and Aggregation
Operators
Fuzzy Integrals have useful properties required for aggregation
operator in multi-criteria decision making
• Not only degree of importance foe a separate criterion but also
redundancy and synergy effects between criteria


Identification of Fuzzy measure based on
• Semantic involved in the decision making problem
• Learning data
• Semantics and learning data
Application are diverse
• Pattern Recognition
• Multi-sensor Fusion