Representation of the revised monotone
functional by the Choquet integral with
respect to signed fuzzy measure
Biljana Mihailovi¶c
a
,Endre Pap
b
a
Faculty of Engineering,University of Novi Sad
Trg Dositeja Obradovi¶ca 6,21000 Novi Sad,Serbia
email:lica@uns.ns.ac.yu
b
Department of Mathematics,University of Novi Sad
Trg Dositeja Obradovi¶ca 4,21000 Novi Sad,Serbia
email:pape@eunet.yu
Abstract:The signed fuzzy measures are considered and some of their properties are
shown.There is introduced the revised monotone functional and there are given con
ditions for its asymmetric Choquet integralbased representation.
Key words and phrases:revised monotonicity,signed fuzzy measure,Choquet integral.
1 Introduction
Due to its special non linear character,the Choquet integral with respect to
a fuzzy measure,is one of the most popular and °exible aggregation operator
[3,5,17].The basic features of Choquet integral,de¯ned for nonnegative
measurable functions,are monotonicity and comonotonic additivity,see [1,3,
10].For an exhaustive overview of applications of Choquet integral in the
decision under uncertainty we recommend [2,8,9,14,15,18].Recall that
nonnegative set function m such that m(;) = 0 and A ½ B implies m(A) ·
m(B) (monotonicity),is called by various names,such as capacity,nonadditive
measure,fuzzy measure.
A generalized fuzzy measure,a signed fuzzy measure,introduced by Liu in
[6],is a revised monotone,realvalued set function,vanishing at the empty set,
see [10].Murofushi et al.in [7] used term nonmonotonic fuzzy measure to
denote a realvalued set function satisfying m(;) = 0.In this paper we deal
with a signed fuzzy measure in the sense of de¯nition given in [6].
The properties of two usual extensions of Choquet integral to the class of
all measurable functions have been studied by various authors [1,3,7,10].
The ¯rst one extension,the symmetric Choquet integral,introduced by
·
Sipo·s,
is homogeneous with respect to multiplication by a real constant and the second
one,the asymmetric Choquet integral is comonotone additive and homogeneous
with respect to multiplication by a nonnegative constant.In both cases the
monotonicity is violated.The asymmetric Choquet integral is de¯ned with
respect to a realvalued set function m,not necessary monotone.
The fuzzy integral de¯ned with the use of maximum and minimum opera
tors was introduced by Sugeno in [16].The Sugeno integral is de¯ned on the
class of functions whose range is contained in [0;1] and with respect to a nor
malized fuzzy measure.It is comonotone_additive (comonotone maxitive),
^homogeneous and monotone functional.An extension of the Sugeno integral
in the spirit of the symmetric extension of Choquet integral is proposed by
M.Grabisch in [4].The symmetric Sugeno integral is neither monotone nor
commonotone _additive in general.In the paper [13] authors considered a
representation by two Sugeno integrals of the functional L de¯ned on the class
of functions f:X![¡1;1] on a ¯nite set X:In the case of in¯nitely countable
set X there was obtained that the symmetric Sugeno integral is comonotone
6additive functional on the class of functions with ¯nite support.
In this contribution we will deal with a revised monotonicity of a realvalued
set function m,m(;) = 0 and asymmetric Choquet integral with respect to m.
In the next section the short overview of basic notions and de¯nitions is given.
In Section 3 we consider a revised monotonicity of realvalued set functions van
ishing at the empty set.Finally,in Section 4 we introduce a revised monotone
functional and discuss the conditions for its asymmetric Choquet integralbased
representation.
2 Preliminaries
Let X = fx
1
;:::;x
n
g be a ¯nite set.Let P(X) be class of subsets of universal
set X:We have by [6,10] the following de¯nition.
De¯nition 1
A realvalued set function m:P(X)!R,is a signed fuzzy
measure if it satis¯es
(i) m(;) = 0
(ii) (RM) If E;F 2 P(X),E\F =;,then
a) m(E) ¸ 0;m(F) ¸ 0;m(E) _m(F) > 0 ) m(E [F) ¸ m(E) _m(F);
b) m(E) · 0;m(F) · 0;m(E) ^m(F) < 0 ) m(E [F) · m(E) ^m(F);
c) m(E) > 0;m(F) < 0 ) m(F) · m(E [F) · m(E):
The conjugate set function of realvalued set function m,m:P(X)!R
is de¯ned by ¹m(E) = m(X) ¡m(
¹
E),where
¹
E denotes the complement set of
E,
¹
E = X n E.Obviously,if m is fuzzy measure,¹m is fuzzy measure,too.
However,if m is a signed fuzzy measure,its conjugate set function ¹m need not
to be a signed fuzzy measure and this fact will be discussed in the next section.
In the next example,introduced in [12],there is introduced a signed fuzzy
measure m and we give an interpretation of the condition (RM) of the revised
monotonicity of m in an application.
Example 1
Let X be a set of 2n elements.Let A;B ½ X such that X =
A [ B,A\B =;and card(A) = card(B) = n.We de¯ne the set function
m:P(X)!R by:
m(E) =
8
<
:
card (X);E = A
¡card (X);E = B
card (A\E) ¡card (B\E);else:
m is a signed fuzzy measure.
We discuss the revised monotonicity of m.Same as in the modi¯ed version
of the example a workshop,given by Murofushi et al.in [7],let us consider the
set X as the set of all workers in a workshop,and sets A and B are the sets of
good and bad workers in sense of their e±ciency,i.e.,ine±ciency.If we sup
pose that workers from group A work two times better if they work all together
(with nobody else),and workers from B two times worse,and in the other
cases"anybody is e®ective in the proportion to its quantitative membership to
the'good'group A or'bad'group B".The set function m is used to denote
the e±ciency of the worker.The interpretation of revised monotonicity is in
the assumptions that for disjoint groups E of'good'and F of'bad'workers,if
they work together,then their productivity is not greater to productivity of E
and not less to productivity of F,for groups E and F of'good'('bad') workers
the simultaneous productivity is not less (not greater) to theirs individual pro
ductivity.Also,we have m(X) = 0,i.e.,the productivity of all workers in the
workshop equals to zero.
Let f be a realvalued function on X.We denote f(x
i
) = f
i
for i = 1;2;:::n
and F denotes class of all realvalued functions on X.The asymmetric Choquet
integral with respect to a set function m:P(X)!R of function f:X!R is
given by
C
m
(f) =
n
X
i=1
(f
®(i)
¡f
®(i¡1)
)m(E
®(i)
);
where f admits a comonotoneadditive representation f =
P
n
i=1
f
®(i)
1
E
®(i)
and ® = (®(1);®(2);:::;®(n)) is a permutation of index set f1;2;:::;ng such
that
f
®(1)
· f
®(2)
· ¢ ¢ ¢ · f
®(n)
;
f
®(0)
= 0;sets E
®(i)
are given by E
®(i)
= fx
®(i)
;:::;x
®(n)
g and 1
E
is char
acteristic function of set E,E ½ X.The asymmetric Choquet integral can be
expressed in the terms of the Choquet integrals of nonnegative functions f
+
and f
¡
,the positive and negative part of function f,i.e.
C
m
(f) = C
m
(f
+
) ¡C
¹m
(f
¡
);(1)
where f
+
= f _ 0 and f
¡
= (¡f) _ 0,and ¹m is the conjugate set function of
m.
Recall that two functions f and g on X are called comonotone [3] if for all
x;x
1
2 X we have f(x) < f(x
1
) ) g(x) · g(x
1
).The asymmetric Choquet
integral is a comononotone additive functional on F,i.e.for all comonotone
functions f;g 2 F we have
C
m
(f +g) = C
m
(f) +C
m
(g):
3 Signed fuzzy measure
In this section we will consider a signed fuzzy measure m with m(X) = 0.We
will examine when its conjugate set function ¹m is a signed fuzzy measure,too.
Note that for a nonnegative (nonpositive) signed fuzzy measure m,condition
m(X) = 0 implies m(E) = 0 for all E 2 P(X).In the sequel we suppose
that m:P(X)!R is a signed fuzzy measure of nonconstant sign.We easily
obtain the next lemma by de¯nition of signed fuzzy measure and the condition
m(X) = 0.
Lemma 1
Let m be a signed fuzzy measure,m(X) = 0.m(E) and m(
¹
E) are
the opposite sign values,i.e.,
(8E 2 P(X)) (m(E) > 0,m(
¹
E) < 0):
De¯nition 2
We say that a realvalued set function m,m(;) = 0 satis¯es
an intersection property if for all E;F 2 P(X),E\F 6=;and E[F = X
we have
a) m(E) ¸ 0;m(F) ¸ 0;m(E) _m(F) > 0 ) m(E\F) ¸ m(E) _m(F);
b) m(E) · 0;m(F) · 0;m(E) ^m(F) < 0 ) m(E\F) · m(E) ^m(F);
c) m(E) > 0;m(F) < 0 ) m(F) · m(E\F) · m(E):
We have the next theorem.
Theorem 1
Let m be signed fuzzy measure such that m(X) = 0.m has an
intersection property if and only if the conjugate set function ¹mof mis a signed
fuzzy measure.
Proof.Let m be a signed fuzzy measure with m(X) = 0.
(=)) First,we suppose that m has an intersection property.We will prove
that ¹m is a signed fuzzy measure.
(i) Directly by de¯nition of ¹m we have ¹m(;) = 0.
(ii) In order to prove condition (RM) a) let E;F 2 P(X) such that E\F =;
and ¹m(E) ¸ 0;¹m(F) ¸ 0;¹m(E) _ ¹m(F) > 0.We have
¹
E [
¹
F = X and
m(
¹
E) · 0;m(
¹
F) · 0 and m(
¹
E) ^ m(
¹
F) < 0.(a)
If we suppose that
¹
E\
¹
F =;then we have F =
¹
E.By Lemma 1.we
obtain that the values m(F) and m(
¹
F) are the opposite sign values and it is
in contradiction with (a).Therefore,
¹
E\
¹
F 6=;.By the intersection property
of m we have:
m(
¹
E\
¹
F) · m(
¹
E) ^ m(
¹
F) () m(
E [ F ) · m(
¹
E) ^m(
¹
F)
() ¡¹m(E [F) · (¡¹m(E)) ^(¡¹m(F))
() ¹m(E [ F) ¸ ¹m(E) _ ¹m(F):
Hence,we have that ¹m satis¯es condition (RM) a).Similarly we obtain that
¹m satis¯es conditions (RM) b) and c),hence,¹m is a signed fuzzy measure.
((=) Let ¹m be a signed fuzzy measure,i.e.¹m is a revised monotone set
function and ¹m(;) = 0.We obtain the claim directly by de¯nition of the
intersection property and the above consideration.¤
Example 2
Let m be a set function de¯ned at the Example 1.m is a signed
fuzzy measure with m(X) = 0.Obviously,m has an intersection property.Its
conjugate set function ¹m:P(X)!R is de¯ned by:
¹m(E) =
8
<
:
card (X);E = A
¡card (X);E = B
card (B n E) ¡card (An E);else:
¹m is a signed fuzzy measure.Moreover,we have m= ¹m.
4 Revised monotone functional
In this section we focus on the asymmetric Choquet integral with respect to
a signed fuzzy measure.As it is mentioned before,the monotonicity is vio
lated.We will discuss the modi¯cation of monotonicity property,the revised
monotonicity of asymmetric Choquet integral.
A real valued functional L,L:F!R,de¯ned on the class of functions
f:X!R,can be viewed as an extension of a signed fuzzy measure m,so
it is reasonable to require that L(1
E
) = m(E),for all E 2 A (1
E
denotes
characteristic function of set E ½ X).In order to examine the properties of a
real valued functional L,under which it can be represented by the asymmetric
Choquet integral w.r.t.a signed fuzzy measure,it is useful to consider the
concept of comonotone functions.
The functional L is comonotone additive i®
L(f +g) = L(f) +L(g)
for all comonotone functions f;g 2 F.We say that functional L is positive
homogeneous i®
L(af) = aL(f)
for all f 2 F and a ¸ 0:
We introduce a revised monotone functional L de¯ned on F,see [12].
De¯nition 3
Let L:F!R be a functional on F.
(i) L is revised monotone if and only if
a) L(f) ¸ 0;L(g) ¸ 0;L(f) _L(g) > 0 ) L(f +g) ¸ L(f) _L(g)
b) L(f) · 0;L(g) · 0;L(f) ^L(g) < 0 ) L(f +g) · L(f) ^L(g)
c) L(f) > 0;L(g) < 0 ) L(g) · L(f +g) · L(f)
for all functions f;g 2 F.
(ii) L is comonotone revised monotone if and only if conditions a),b) and c)
are satis¯ed for all comonotone functions f;g 2 F.
Note that for a nonnegative functional L acting on nonnegative functions
on X,the revised monotonicity ensures the monotonicity.
Directly by de¯nitions of the comonotone additive and the revised monotone
functional L we have the next proposition.
Proposition 1
The asymmetric Choquet integral w.r.t.a signed fuzzy measure
m,C
m
:F!R is a comonotone revised monotone functional.
Remark 1
Note that any additive functional L:F!Ris a revised monotone
functional.The Lebesgue integral with respect to a signed measure ¹ is a re
vised monotone functional.
We have the next theorem.
Theorem 2
Let L be a real valued,revised monotone,positive homogeneoues
and comonotone additive functional on F.Then there exists a signed fuzzy
measure m
L
,such that L can be represented by the asymmetric Choquet integral
w.r.t.m
L
,i.e.,
L(f) = C
m
L
(f):
Proof.Let m be a set function m de¯ned by
m
L
(E) = L(1
E
);for E µ X:
Observe that for comonotone functions 1
X
and ¡1
E
,we have
m
L
(
¹
E) = L(1
¹
E
) = L(1
X
+(¡1
E
)) = L(1
X
) +L(¡1
E
) = m
L
(X) +L(¡1
E
);
hence
L(¡1
E
) = ¡¹m
L
(E);E µ X:
By de¯nition of m
L
and revised monotonicity of functional L we have:
1) m
L
(;) = L(1
;
) = L(0) = 0
2) a) for E;F 2 A,E\F =;,and
m
L
(E) ¸ 0;m
L
(F) ¸ 0;m
L
(E) _m
L
(F) > 0 we have
m
L
(E [F) = L(1
E[F
) = L(1
E
+1
F
)
¸ L(1
E
) _L(1
E
) = m
L
(E) _ m
L
(F):
Analogously,we obtain that m
L
satis¯es conditions (RM) b) and c),hence m
L
is the revised monotone set function,so it is a signed fuzzy measure.Now,we
consider f 2 F and its comonotone additive representation f = f
+
+(¡f
¡
);
where
f
+
=
n
X
i=1
(a
i
¡a
i¡1
)1
E
i
;
¡f
¡
=
n
X
i=1
(b
i
¡b
i+1
)(¡1
F
i
);
a
i
= f
+
®(i)
;a
0
= 0;b
i
= f
¡
®(n+1¡i)
;b
n+1
= 0;
a
i
's are in nondecreasing,b
i
's are in nonincreasing order,® is a permutation,
such that ¡1< f
®(1)
· ¢ ¢ ¢ · f
®(n)
< 1,E
i
= E
®(i)
;
F
i
= E
1
n E
®(n+2¡i)
;E
®(i)
= fx
®(i)
;:::;x
®(n)
g and E
®(n+1)
=;.
For every i and j the functions 1
E
i
and 1
E
j
are comonotone,and by comono
tone additivity and positive homogeneity of the functional L,we have
L(f
+
) =
n
X
i=1
(a
i
¡a
i¡1
)L(1
E
i
)
=
n
X
i=1
(a
i
¡a
i¡1
)m
L
(E
i
)
= C
m
L
(f
+
)
and
L(¡f
¡
) =
n
X
i=1
(b
i
¡b
i+1
)L(¡1
F
i
)
= ¡
n
X
i=1
(b
i
¡b
i+1
)(¡L(¡1
F
i
))
= ¡
n
X
i=1
(b
i
¡b
i+1
) ¹m
L
(F
i
)
= ¡C
¹m
L
(f
¡
):
Therefore by the comonotonicity of functions f
+
and ¡f
¡
we obtain that
L(f) = L(f
+
+(¡f
¡
))
= L(f
+
) +L(¡f
¡
)
= C
m
L
(f
+
) ¡C
¹m
L
(f
¡
)
= C
m
L
(f):
¤
Acknowledgement The work has been supported by the project MNZ
·
ZSS
144012 and the project"Mathematical Models for Decision Making under Un
certain Conditions and Their Applications"supported by Vojvodina Provincial
Secretariat for Science and Technological Development.
References
[1]
P.Benvenuti,R.Mesiar i D.Vivona:Monotone Set FunctionsBased In
tegrals.In:E.Pap ed.Handbook of Measure Theory,Ch 33.,Elsevier,
2002,13291379.
[2]
A.Chateauneuf,P.Wakker:An Aximatization of Cumulative Prospect
Theory for Decision under Risk.J.of Risk and Uncertainty 18 (1999),
137145.
[3]
D.Denneberg:Nonadditive Measure and Integral.Kluwer Academic Pub
lishers,Dordrecht,1994.
[4]
M.Grabisch:The Symmetric Sugeno Integral.Fuzzy Sets and Systems
139 (2003) 473490.
[5]
M.Grabisch,H.T.Nguyen,E.A.Walker:Fundamentals of Uncertainty
Calculi with Applications to Fuzzy Inference.DordrechtBoston London,
Kluwer Academic Publishers (1995)
[6]
X.Liu:Hahn decomposition theorem for in¯nite signed fuzzy measure.
Fuzzy Sets and Systems 57,1993,189212.
[7]
T.Murofushi,M.Sugeno and M.Machida:Nonmonotonic fuzzy measures
and the Choquet integral,Fuzzy Sets and Systems 64,1994,7386.
[8]
Y.Narukawa,T.Murofushi:Choquet integral representation and prefer
ence.Proc.IPMU'02,Annecy,France,2002,747753.
[9]
Y.Narukawa,T.Murofushi,M.Sugeno:Regular fuzzy measure and repre
sentation of comonotonically additive functional.Fuzzy Sets and Systems
112,2000,177186.
[10]
E.Pap:NullAdditive Set Functions.Kluwer Academic Publishers,Dor
drecht,1995.
[11]
E.Pap,ed.:Handbook of Measure Theory.Elsevier,2002.
[12]
E.Pap,B.Mihailovi¶c:Representation of the utility functional by two fuzzy
integrals.Proc.IPMU'06,Paris,France,2006,17101717.
[13]
E.Pap,B.Mihailovi¶c:A representation of a comonotone6additive and
monotone functional by two Sugeno integrals.Fuzzy Sets and Systems 155
(2005) 7788.
[14]
D.Schmeidler:Subjective probability and expected utility without additiv
ity.Econometrica 57,1989,517587.
[15]
D.Schmeidler:Integral representation without additivity.Proc.Amer.
Math.Soc.97,1986,255261.
[16]
M.Sugeno,Theory of fuzzy integrals and its applications.PhD thesis,
Tokyo Institute of Technology,1974.
[17]
K.Tanaka,M.Sugeno:A study on subjective evaluation of color printing
image.Int.J.Approximate Reasoning 5,1991,213222.
[18]
A.Tverski,D.Kahneman:Advances in prospect theory.Cumulative rep
resentation of uncertainty.J.of Risk and Uncertainty 5,1992,297323.
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