菅野積分の変遷
福島大学 藤本勝成
2015年12月5日(土)
Type-1・Type-2ファ ジィシステム研究部会 研究会＠関西学院大学大阪梅田キャンパス
timeline
1975
Sugeno intergaral (Sugeno 1974)
1980
1985
1990
1995
Opposite-Sugeno intergaral (Imaoka 1995)
2000
Symmetric Sugeno integral (Grabisch 2003)
2005
2010
2015
Lattice valued Sugeno intergaral (Marichal 2009)
Cumulative Prospect Theory type (CPT) Sugeno-integral
BC Sugeno-integral (Sugeno integral
w.r.t.bi-capacity)
(Grabich
and Labreuche 2010)
(Grabich and Labreuche 2010)
timeline
1975
1995
Sugeno integral (Sugeno 1974)
Unipolar scale
Opposite-Sugeno integral (Imaoka 1995)
2000
Symmetric Sugeno integral (Grabisch 2003)
2005
Ordinal
model!
Lattice valued Sugeno intergaral (Marichal 2009)
2010
2015
Cumulative Prospect Theory type (CPT) Sugeno-integral
(Grabich and Labreuche 2010)
BC Sugeno-integral (Sugeno integral w.r.t.bi-capacity)
(Grabich and Labreuche 2010)
Bipolar scale
timeline
1975
1995
Sugeno intergaral (Sugeno 1974)
a1
Opposite-Sugeno integral (Imaoka 1995)
a2
a3
2000
Symmetric Sugeno integral (Grabisch 2003)
2005
a2 a3
a1
a1
Lattice valued Sugeno intergaral (Marichal 2009)
2010
CPT Sugeno-integral (Grabich and Labreuche 2010)
BC Sugeno-integral (Grabich and Labreuche 2010)
2015
a2 a3
a2 a3
a1
timeline
1975
1995
Sugeno intergaral (Sugeno 1974)
a1
Opposite-Sugeno integral (Imaoka 1995)
a2
a3
2000
Symmetric Sugeno integral (Grabisch 2003)
2005
a1
2010
a1
CPT Sugeno-integral (Grabich and Labreuche 2010)
BC Sugeno-integral (Grabich and Labreuche 2010)
2015
a2 a3
a2 a3
a2 a3
a1
timeline
1975
1995
Sugeno intergaral (Sugeno 1974)
a1
Opposite-Sugeno intergaral (Imaoka 1995)
a2
a3
2000
Symmetric Sugeno integral (Grabisch 2003)
2005
a2 a3
a1 a2
a3
a1
2010
2015
negative
CPT Sugeno-integral (Grabich and Labreuche 2010)
∅
BC Sugeno-integral (Grabich and Labreuche 2010) 12
3
12
positive 13
23
123
∅ 1 2 3 12 13 23 123
Setting up a model
Which choice is preferred?
Attributes/Criteria
Choices
: Preference score of choice C3 with regard to attribute a2.
Setting up a model
Attributes/Criteria
Choices
Attributes/Criteria
Choices
weighted average
Score(choice c2)
Score(choice c5)
Attributes/Criteria
Choices
C2
1.00
0.75
C3
0.50
C5
0.25
a1
a2
a3
0.00
a1
a2
a3
a1
a2
a3
Attributes/Criteria
Choices
C2
1.00
0.75
Score(choice c2)
0.50
0.25
a1
a2
a3
0.00
weighted average
Integral w.r.t. weight
Attributes/Criteria
Choices
C2
1.00
0.75
C3
0.50
C5
0.25
a1
a2
a3
0.00
a1
a2
a3
a1
a2
a3
Fuzzy measure and Sugeno integral
Def. (Fuzzy measure by Sugeno(1974))
boundary condition
monotonicity
continuity
Fuzzy measure and Sugeno integral
Def. (Fuzzy measure (later))
monotonicity
Continuity from below
Setting up a model
Attributes/Criteria
Choices
1.0
e.g.,
0.0
0.8
0.0
0.5
0.5
1.0
Def. (Sugeno integral (original ver.))
Def. (Sugeno integral (later))
: measurable
1.00
1.00
The Sugeno integral
Tradeoff between preference score and importance
Def. (Choquet integral )
: measurable
Def. (opposite-Sugeno integral )
: measurable
Def. (co-monotonic)
: measurable
: co-monotonic
or
Def. (co-monotonic)
: measurable
: co-monotonic
or
Def. (co-monotonic)
: measurable
: co-monotonic
or
Def. (co-monotonic)
: measurable
: co-monotonic
or
Th. (characterization of the Sugeno integral)
: sugeno integral
(i.e.,
)
1)
2)
whenever
co-monotonically maxitive/minitive
: co-monotonic
Th. (characterization of the Choquet integral)
: Choquet integral (i.e.,
)
1)
2)
whenever
co-monotonically additive
: co-monotonic
Fuzzy integrals as extension (interpolation) of fuzzy measures.
Fuzzy integrals as extension (interpolation) of fuzzy measures.
Fuzzy integrals as extension (interpolation) of fuzzy measures.
Fuzzy integrals as extension (interpolation) of fuzzy measures.
extension
Fuzzy integrals as extension (interpolation) of fuzzy measures.
extend to
1
0
1
Fuzzy integrals as extension (interpolation) of fuzzy measures.
1
0
1
Fuzzy integrals as extension (interpolation) of fuzzy measures.
1
0
1
Fuzzy integrals as extension (interpolation) of fuzzy measures.
Fuzzy integrals as extension (interpolation) of fuzzy measures.
Fuzzy integrals as extension (interpolation) of fuzzy measures.
Break:
Lattice valued Sugeno integral
M.Couceiro, J.-L. Marichal, Characterizations of discrete Sugeno integrals as polynomial
functions over distributive lattice, Fuzzy sets and systems, 161, 694-707(2010)
J.-L. Marichal, Weighted lattice polynomials, Discrete Mathj. 309, 814-820 (2009)
A.Rico, Sugeno integral in a finite Boolean algebra, Fuzzy sets and systems, 159, 17091719 (2008)
lattice
12/34
1/2/34
134/2
12/3/4
123/4
13/2/4
1234
13/24
1/2/3/4
1/24/3
124/3
14/2/3
1/234
1/23/4
14/23
Lattice of partitions of {1,2,3,4}
lattice
lattice
timeline
1975
1995
Sugeno intergaral (Sugeno 1974)
Opposite-Sugeno intergaral (Imaoka 1995) Unipolar scale
2000
Symmetric Sugeno integral (Grabisch 2003)
2005
Ordinal
model!
Lattice valued Sugeno intergaral (Marichal 2009)
2010
2015
Cumulative Prospect Theory type (CPT) Sugeno-integral
(Grabich and Labreuche 2010)
BC Sugeno-integral (Sugeno integral w.r.t.bi-capacity)
(Grabich and Labreuche 2010)
Bipolar scale
Extension of Sugeno integral onto bipolar scale
timeline
1975
1995
Sugeno intergaral (Sugeno 1974)
Opposite-Sugeno intergaral (Imaoka 1995) Unipolar scale
2000
Symmetric Sugeno integral (Grabisch 2003)
2005
symmetric
a2 a3
a2 a3
a1
a1
a2 a3
a1
2010
a2 a3
2015
a1
=
+
a1 a2 a3
a2 a3
a1
asymmetric
timeline
1975
1995
Sugeno intergaral (Sugeno 1974)
Opposite-Sugeno intergaral (Imaoka 1995) Unipolar scale
2000
Symmetric Sugeno integral (Grabisch 2003)
2005
symmetric
a2 a3
a2 a3
a1
a1
a2 a3
a1
2010
a2 a3
2015
a1
a2 a3
a1
a2 a3
a1
timeline
1975
Sugeno intergaral (Sugeno 1974)
Unipolar scale
2000
Symmetric Sugeno integral (Grabisch 2003)
2005
a2 a3
a2 a3
a1 a2 a3
a1
a1
2010
a2 a3
2015
a1
=
+
a1 a2 a3
a2 a3
a1
asymmetric
?
Which have been introduced by Grabisch(2003)
as an extension of minmam and miximam operators
onto bipolar scale.
Which are called
symmetric minimum, symmetric maximums.
[0,1] X [0,1]
[-1,1] X [-1,1]
M. Grabisch:
The symmetric Sugeno integral, Fuzzy Sets and Systems,139, pp. 473-490, 2003.
extend
under considering the ring structure :
・,
+ on [-1, 1] x [-1,1]
extend
extend
under considering the ring structure :
・,
+
on [-1, 1] x [-1,1]
extend
the binary operator, the symmetric maximum , are extended to n-ary operator :
[0,1] X [0,1]
[-1,1] X [-1,1]
M. Grabisch:
The symmetric Sugeno integral, Fuzzy Sets and Systems,139, pp. 473-490, 2003.
the binary operator, the symmetric maximum , are extended to n-ary operator :
timeline
1975
1995
Sugeno intergaral (Sugeno 1974)
Opposite-Sugeno intergaral (Imaoka 1995) Unipolar scale
2000
Symmetric Sugeno integral (Grabisch 2003)
2005
Ordinal
model!
Lattice valued Sugeno intergaral (Marichal 2009)
2010
2015
Cumulative Prospect Theory type (CPT) Sugeno-integral
(Grabich and Labreuche 2010)
BC Sugeno-integral (Sugeno integral w.r.t.bi-capacity)
(Grabich and Labreuche 2010)
Bipolar scale
timeline
1975
1995
Sugeno intergaral (Sugeno 1974)
a1
Opposite-Sugeno intergaral (Imaoka 1995)
a2
a3
2000
Symmetric Sugeno integral (Grabisch 2003)
2005
a1
2010
a1
CPT Sugeno-integral (Grabich and Labreuche 2010)
BC Sugeno-integral (Grabich and Labreuche 2010)
2015
a2 a3
a2 a3
a2 a3
a1
timeline
1975
1995
Sugeno intergaral (Sugeno 1974)
a1
Opposite-Sugeno intergaral (Imaoka 1995)
a2
a3
2000
Symmetric Sugeno integral (Grabisch 2003)
2005
a1
2010
a1
CPT Sugeno-integral (Grabich and Labreuche 2010)
BC Sugeno-integral (Grabich and Labreuche 2010)
2015
a2 a3
a2 a3
a2 a3
a1
timeline
1975
1995
Sugeno intergaral (Sugeno 1974)
a1
Opposite-Sugeno intergaral (Imaoka 1995)
a2
a3
2000
Symmetric Sugeno integral (Grabisch 2003)
2005
a2 a3
a1 a2
a3
a1
2010
2015
negative
CPT Sugeno-integral (Grabich and Labreuche 2010)
∅
BC Sugeno-integral (Grabich and Labreuche 2010) 12
3
12
positive 13
23
123
∅ 1 2 3 12 13 23 123
Prof. Sugeno’s ambition!
Any admissible preference
orders in MCDM(MADM)
can be represented by
my hierarchical bipolar
Sugeno integral!
That is, controversies
over modeling-capability
for MCDM
will be terminated!
Setting up a model
Attributes/Criteria
A
Choices
: Preference score of choice C3 with regard to attribute a2.
Attributes/Criteria
Choices
C2
(1pt.)
a1
C3
(2pt.)
(1pt.)
a2
C5
(1pt.)
a3
a1
a2
a3
a1
a2
a3
(-1pt.)
(-1pt.)
(-2pt.)
(-1pt.) (-2pt.)
Attributes/Criteria
Choices
DM has an aggregated preference order
on the set of choices {c1, c2, c3, c4,c5}.
How preference orders are inadmissible ?
Comparing choices c2 with c5,
c2 dominates c5 with regard to all attributes.
Attributes/Criteria
Choices
C2
(1pt.)
a1
C5
(2pt.)
(1pt.)
a2
a3
a1
a2
a3
(-1pt.)
(-1pt.) (-2pt.)
Comparingc5
with
choices c2 and
c5,
Nevertheless,
is preferred
to c2!
c2 dominates c5 with regard to all attributes.
Attributes/Criteria
Choices
C2
(1pt.)
a1
C5
(2pt.)
(1pt.)
a2
a3
a1
a2
a3
(-1pt.)
Inadmissible!!
(-1pt.) (-2pt.)
Comparing with choices c2 and c5,
c2 dominates c5 with regard to all attributes.
Attributes/Criteria
Choices
C2
C5
(2pt.)
Nevertheless,
c2
and
c5
are
equally
preferred
!
(1pt.)
(1pt.)
a1
a2
a3
a1
a2
a3
(-1pt.)
Inadmissible!!
(-1pt.) (-2pt.)
Inadmissible preference orders
Attributes/Criteria
A
Choices
C
Admissible preference orders
Attributes/Criteria
A
Choices
C
def
Prof. Sugeno’s ambition!
Any admissible preference orders
can be represented
by some types of his (i.e., the Sugeno) integrals!
Hierarchical Bipolar Sugeno Integral
Prof. Sugeno’s ambition!
属性
選
択
肢
m p1
Bipolar
Sugeno
Integral
Bipolar
Sugeno
Integral
m p2
Bipolar
Sugeno
Integral
Bipolar
Sugeno
Integral
Integral
Sugeno
Integral
a1
a2
Bipolar
Sugeno
Integral Score
So far,
a3Sugeno and his colleagues have proved the case where
Bipolar
Bipolar
# ofmchoices are
at
most
3
with
any
#
of attributes.
Sugeno
p3
What is Sugeno’s
“hierarchical bipolar Sugeno integral”?
bi-polar Sugeno integral
( integer valued function on N )
: Set of criteria
: Set of integer valued
bipolar preference scores
bi-polar Sugeno integral
( integer valued function on N )
Rearrange criteria/attributes in increasing order
w.r.t. |f|, not to the original f.
: Set of criteria
Extension toward bi-polar cases
( integer valued function on N )
Rearrange criteria/attributes in increasing order
w.r.t. |f|, not to the original f.
: Set of criteria
: Set of criteria
Negative part
Positive part
Negative part
Positive part
( integer valued function on N )
: Set of criteria
Positive part
Negative part
N
( integer valued function on 3N )
: Set of criteria
Positive part
Negative part
( integer valued function on 3N )
The Sugeno integral
Tradeoff between preference score and importance
Bipolar Sugeno integral
Tradeoff between preference score and importance
Tradeoff between preference score and importance
What is
the “hierarchical bipolar
Sugeno integral”?
Hierarchical Bipolar Sugeno Integral
Hierarchical Bipolar Sugeno Integral
Hierarchical Bipolar Sugeno Integral
Aggregated
preference score
timeline
1975
1995
Sugeno intergaral (Sugeno 1974)
Unipolar scale
Opposite-Sugeno intergaral (Imaoka 1995)
2000
Symmetric Sugeno integral (Grabisch 2003)
2005
Ordinal
model!
Lattice valued Sugeno intergaral (Marichal 2009)
2010
2015
Unipolar measure
(fuzzy measure)
Bipolar scale
Cumulative Prospect Theory type (CPT) Sugeno-integral
(Grabich and Labreuche 2010)
BC Sugeno-integral (Sugeno integral w.r.t.bi-capacity)
(Grabich and Labreuche 2010)
Bipolar measure
(bi-capacity)
[Opposite S-integral]
H. Imaoka: A proposal of opposite-Sugeno integral and a uniform expression
of fuzzy integrals, Proc 1995 IEEE Int Conf Fuzzy Syst., Vol 2, pp. 583-590,
March 1995
[Lattice valued S-integral]
J.-L. Marichal: Weighted lattice polynomials, Discrete mathematics, Vol. 309,
814-820, March 2009.
[SS-integral]
M. Grabisch: The symmetric Sugeno integral, Fuzzy Sets and Systems,
139, pp. 473-490, 2003.
[CPTS-integral and BCS-integral]
M. Grabisch and C. Labreuche: A decade of application of the Choquet and
Sugeno integrals in multi-criteria decision aid,
Annals of Operations Research, Vol. 175, 247-286, March 2010.
[Hierarchical S-integral model]
M. Sugeno: Ordinal preference models based on S-integrals, Proc. of
Nonlinear Mathematics for Uncertainty and its Applications, Beijing, 2011.