On weighted possibilistic mean and variance of
fuzzy numbers ∗
Robert Fullér
rfuller@abo.fi
Péter Majlender
peter.majlender@abo.fi
Abstract
Dubois and Prade defined an interval-valued expectation of fuzzy numbers, viewing them as consonant random sets. Carlsson and Fullér defined
an interval-valued mean value of fuzzy numbers, viewing them as possibility distributions. In this paper we shall introduce the notation of weighted
interval-valued possibilistic mean value of fuzzy numbers and investigate its
relationship to the interval-valued probabilistic mean. We shall also introduce the notations of crisp weighted possibilistic mean value, variance and
covariance of fuzzy numbers, which are consistent with the extension principle. Furthermore, we show that the weighted variance of linear combination
of fuzzy numbers can be computed in a similar manner as in probability theory.
1
Weighted possibilistic mean values
In 1987 Dubois and Prade [2] defined an interval-valued expectation of fuzzy numbers, viewing them as consonant random sets. They also showed that this expectation remains additive in the sense of addition of fuzzy numbers.
In this paper introducing a weighting function measuring the importance of
γ-level sets of fuzzy numbers we shall define the weighted lower possibilistic
and upper possibilistic mean values, crisp possibilistic mean value and variance
of fuzzy numbers, which are consistent with the extension principle and with the
well-known defintions of expectation and variance in probability theory. We shall
also show that the weighted interval-valued possibilistic mean is always a subset
(moreover a proper subset excluding some special cases) of the interval-valued
probabilistic mean for any weighting function.
∗
The final version of this paper appeared in: Fuzzy Sets and Systems, 139(2003) 297-312.
1
The theory developed in this paper is fully motivated by the principles introduced in [2, 4, 6] and by the possibilistic interpretation of the ordering introduced
in [5].
A fuzzy number A is a fuzzy set of the real line R with a normal, fuzzy convex
and continuous membership function of bounded support. The family of fuzzy
numbers will be denoted by F. A γ-level set of a fuzzy number A is defined by
[A]γ = {t ∈ R|A(t) ≥ γ} if γ > 0 and [A]γ = cl{t ∈ R|A(t) > 0} (the closure
of the support of A) if γ = 0. It is well-known that if A is a fuzzy number then
[A]γ is a compact subset of R for all γ ∈ [0, 1].
Definition 1.1 Let A ∈ F be fuzzy number with [A]γ = [a1 (γ), a2 (γ)], γ ∈ [0, 1].
A function f : [0, 1] → R is said to be a weighting function if f is non-negative,
monotone increasing and satisfies the following normalization condition
Z 1
f (γ)dγ = 1.
(1)
0
We define the f -weighted possibilistic mean (or expected) value of fuzzy number A
as
Z 1
a1 (γ) + a2 (γ)
M̄f (A) =
f (γ)dγ.
(2)
2
0
It should be noted that if f (γ) = 2γ, γ ∈ [0, 1] then
Z 1
Z 1
a1 (γ) + a2 (γ)
[a1 (γ) + a2 (γ)] γdγ = M̄ (A).
2γdγ =
M̄f (A) =
2
0
0
That is the f -weighted possibilistic mean value defined by (2) can be considered as
a generalization of possibilistic mean value introduced in [1]. From the definition
of a weighting function it can be seen that f (γ) might be zero for certain (unimportant) γ-level sets of A. So by introducing different weighting functions we can
give different (case-dependent) importances to γ-levels sets of fuzzy numbers.
Remark 1.1 The f -weighted possibilistic mean value of fuzzy number A coincides
with the value of A (with respect to reducing function f ) introduced by Delgado,
Vila and Woxman in ([4], page 127). In fact, their paper has inspired us to introduce the notation of f -weighted possibilistic mean.
In the following we will consider two weighting functions f1 and f2 to be equal
if the integral of their absolute difference |f1 − f2 | is zero, that is
Z 1
|f1 (γ) − f2 (γ)|dγ = 0.
0
2
Let us introduce a family of weighting function 1 : [0, 1] → R defined as
1 if γ ∈ (0, 1]
1(γ) =
a if γ = 0
where a ∈ [0, 1] is an arbitrary real number. It is clear that 1 is a weighting function
and
Z 1
Z 1
a1 (γ) + a2 (γ)
a1 (γ) + a2 (γ)
M̄1 (A) =
1(γ)dγ =
dγ.
(3)
2
2
0
0
Remark 1.2 We note here that the 1-weighted possibilistic mean value defined
by (3) coincides with the generative expectation of fuzzy numbers introduced by
Chanas and M. Nowakowski in ([3], page 47).
Definition 1.2 Let f be a weighting function and let A be a fuzzy number. Then
we define the f -weighted interval-valued possibilistic mean of A as
Mf (A) = [Mf− (A), Mf+ (A)],
where
Mf− (A) =
R1
1
Z
a1 (γ)f (γ)dγ =
0
0
R1
0
=
a1 (γ)f (Pos[A ≤ a1 (γ)])dγ
R1
0 f (Pos[A ≤ a1 (γ)])dγ
R1
min[A]γ f (Pos[A ≤ a1 (γ)])dγ
,
R1
0 f (Pos[A ≤ a1 (γ)])dγ
0
=
a1 (γ)f (γ)dγ
R1
0 f (γ)dγ
and
Mf+ (A)
R1
1
Z
=
a2 (γ)f (γ)dγ =
0
R1
=
0
R1
=
0
0
a2 (γ)f (γ)dγ
R1
0 f (γ)dγ
a2 (γ)f (Pos[A ≥ a2 (γ)])dγ
R1
0 f (Pos[A ≥ a2 (γ)])dγ
min[A]γ f (Pos[A ≥ a2 (γ)])dγ
.
R1
f
(Pos[A
≥
a
(γ)])dγ
2
0
Here we used the relationships
Pos[A ≤ a1 (γ)] = sup A(u) = γ,
u≤a1 (γ)
3
and
Pos[A ≥ a2 (γ)] = sup A(u) = γ.
u≥a2 (γ)
So Mf− (A) is the f -weighted average of the minimum of the γ-cuts and it is
why we call it the f -weighted lower possibilistic mean value of fuzzy number A.
Similarly, Mf+ (A) is the f -weighted average of the maximum of the γ-cuts and it
is why we call it the f -weighted upper possibilistic mean value of fuzzy number
A.
The following two theorems can directly be proven using the definition of f weighted interval-valued possibilistic mean.
Theorem 1.1 Let A, B ∈ F and let f be a weighting function, and let λ be a real
number. Then
Mf (A + B) = Mf (A) + Mf (B),
Mf (λA) = λMf (A).
Remark 1.3 The f -weighted possibilistic mean of A, defined by (2), is the arithmetic mean of its f -weighted lower and upper possibilistic mean values, i.e.
M̄f (A) =
Mf − (A) + Mf+ (A)
2
.
(4)
Theorem 1.2 Let A and B be two fuzzy numbers, and let λ ∈ R. Then we have
M̄f (A + B) = M̄f (A) + M̄f (B),
2
M̄f (λA) = λM̄f (A).
Relation to upper and lower probability mean values
In this Section we will show an important relationship between the interval-valued
expectation E(A) = [E∗ (A), E ∗ (A)] introduced by Dubois and Prade in [2] and
the f -weighted interval-valued possibilistic mean Mf (A) = [Mf− (A), Mf+ (A)]
for any fuzzy number with strictly decreasing shape functions.
An LR-type fuzzy number A can be described with the following membership
function:
 
 L q− − u

if q− − α ≤ u ≤ q−


α



 1
if u ∈ [q− , q+ ]
A(u) =
u − q+



R
if q+ ≤ u ≤ q+ + β


β




0
otherwise
4
where [q− , q+ ] is the peak of fuzzy number A; q− and q+ are the lower and upper modal values; L, R : [0, 1] → [0, 1] with L(0) = R(0) = 1 and L(1) =
R(1) = 0 are non-increasing, continuous functions. We will use the notation A =
(q− , q+ , α, β)LR . Hence, the closure of the support of A is exactly [q− −α, q+ +β].
If L and R are strictly decreasing functions then the γ-level sets of A can easily be
computed as
[A]γ = [q− − αL−1 (γ), q+ + βR−1 (γ)], γ ∈ [0, 1].
The lower and upper probability mean values of the fuzzy number A are computed
by [2]
Z 1
Z 1
∗
R(u)du.
(5)
L(u)du, E (A) = q+ + β
E∗ (A) = q− − α
0
0
The f -weighted lower and upper possibilistic mean values are computed by
Z 1
−
Mf (A) =
q− − αL−1 (γ) f (γ)dγ
0
Z 1
Z 1
=
q− f (γ)dγ −
αL−1 (γ)f (γ)dγ
0
0
Z 1
L−1 (γ)f (γ)dγ,
= q− − α
0
(6)
Z 1
+
−1
Mf (A) =
q+ + βR (γ) f (γ)dγ
0
Z 1
Z 1
=
q+ f (γ)dγ +
βR−1 (γ)f (γ)dγ
0
0
Z 1
= q+ + β
R−1 (γ)f (γ)dγ.
0
We can state the following theorem.
Theorem 2.1 Let f be a weighting function and let A be a fuzzy number of type
LR with strictly decreasing and continuous shape functions. Then, the f -weighted
interval-valued possibilistic mean value of A is a subset of the interval-valued
probabilistic mean value, i.e.
Mf (A) ⊆ E(A).
Furthermore, Mf (A) is a proper subset of E(A) whenever f 6= 1.
5
Proof 1 see Fuzzy Sets and Systems, 139(2003) 297-312.
Example 2.1 Let f (γ) = (n + 1)γ n and let A = (a, α, β) be a triangular fuzzy
number with center a, left-width α > 0 and right-width β > 0 then a γ-level of A
is computed by
[A]γ = [a − (1 − γ)α, a + (1 − γ)β], ∀γ ∈ [0, 1].
Then the power-weighted lower and upper possibilistic mean values of A are computed by
Z 1
Mf− (A) =
[a − (1 − γ)α](n + 1)γ n dγ
0
Z 1
Z 1
n
(1 − γ)γ n dγ
γ dγ − α(n + 1)
= a(n + 1)
0
0
=a−
α
,
n+2
and,
Mf+ (A) =
Z
1
[a + (1 − γ)β](n + 1)γ n dγ
0
Z 1
Z 1
n
= a(n + 1)
γ dγ + β(n + 1)
(1 − γ)γ n dγ
0
=a+
0
β
,
n+2
and therefore,
"
#
α
β
Mf (A) = a −
,a +
.
n+2
n+2
That is,
1
α
β
a−
+a+
M̄f (A) =
2
n+2
n+2
!
=a+
β−α
.
2(n + 2)
So,
lim M̄f (A) = lim
n→∞
n→∞
6
β−α
a+
2(n + 2)
= a.
Example 2.2 Let A = (a, b, α, β) be a fuzzy number of trapezoidal form with peak
[a, b], left-width α > 0 and right-width β > 0, and let f (γ) = (n + 1)γ n , n ≥ 0.
A γ-level of A is computed by
[A]γ = [a − (1 − γ)α, b + (1 − γ)β], ∀γ ∈ [0, 1],
then the power-weighted lower and upper possibilistic mean values of A are computed by
Z 1
Mf− (A) =
[a − (1 − γ)α](n + 1)γ n dγ
0
Z 1
Z 1
n
(1 − γ)γ n dγ
γ dγ − α(n + 1)
= a(n + 1)
0
0
=a−
α
,
n+2
and,
Mf+ (A) =
Z
1
[b + (1 − γ)β](n + 1)γ n dγ
0
Z 1
Z 1
n
= b(n + 1)
γ dγ + β(n + 1)
(1 − γ)γ n dγ
0
0
β
,
=b+
n+2
and therefore,
"
α
β
Mf (A) = a −
,b +
n+2
n+2
#
That is,
α
β
1
a−
+b+
M̄f (A) =
2
n+2
n+2
!
=
a+b
β−α
+
.
2
2(n + 2)
So,
lim M̄f (A) = lim
n→∞
n→∞
a+b
β−α
+
2
2(n + 2)
=
a+b
.
2
Example 2.3 Let f (γ) = (n + 1)γ n , n ≥ 0 and let A = (a, α, β) be a triangular
fuzzy number with center a, left-width α > 0 and right-width β > 0 then
α
β
α
β
Mf (A) = a −
,a +
⊂ E(A) = a − , a +
n+2
n+2
2
2
7
and for n > 0 we have
M̄f (A) = a +
β−α
β−α
6= Ē(A) = a +
.
2(n + 2)
4
Example 2.4 Let A = (a, b, α, β) be a fuzzy number of trapezoidal form and let
!
1
f (γ) = (n − 1) √
−1 ,
n
1−γ
where n ≥ 2. It is clear that f is a weighting function with f (0) = 0 and
lim f (γ) = ∞.
γ→1−0
Then the f -weighted lower and upper possibilistic mean values of A are computed
by
"
#
Z 1
1
[a − (1 − γ)α](n − 1) √
Mf− (A) =
− 1 dγ
n
1−γ
0
=a−
(n − 1)α
,
2(2n − 1)
and
Mf+ (A) =
Z
1
0
=b+
"
#
1
[b + (1 − γ)β](n − 1) √
− 1 dγ
n
1−γ
(n − 1)β
,
2(2n − 1)
and therefore
"
#
(n − 1)α
(n − 1)β
Mf (A) = a −
,b +
.
2(2n − 1)
2(2n − 1)
That is,
!
1
(n − 1)α
(n − 1)β
a + b (n − 1)(β − α)
+b+
+
.
M̄f (A) =
a−
=
2
2(2n − 1)
2(2n − 1)
2
4(2n − 1)
So,
lim M̄f (A) = lim
n→∞
n→∞
a + b (n − 1)(β − α)
+
2
4(2n − 1)
8
=
a+b β−α
+
.
2
8
Remark 2.1 When A is a symmetric fuzzy number then the equation M̄f (A) =
Ē(A) holds for any weighting function f . In the limit case, when A = (a, b, 0, 0) is
the characteristic function of interval [a, b], the f -weighted possibilistic and probabilistic interval-valued means are equal, E(A) = Mf (A) = [a, b].
3
Weighted possibilistic variance
Definition 3.1 Let A and B be fuzzy numbers and let f be a weighting function.
We define the f -weighted possibilistic variance of A by
Z 1
Varf (A) =
0
a2 (γ) − a1 (γ)
2
2
f (γ)dγ,
(7)
and the f -weighted covariance of A and B is defined as
1
Z
a2 (γ) − a1 (γ) b2 (γ) − b1 (γ)
·
f (γ)dγ.
2
2
Covf (A, B) =
0
(8)
It should be noted that if f (γ) = 2γ, γ ∈ [0, 1] then
Z 1
Varf (A) =
0
1
=
2
Z
1
a2 (γ) − a1 (γ) 2
2γdγ
2
[a2 (γ) − a1 (γ)]2 γdγ = Var(A),
0
and
1
Z
Covf (A, B) =
0
1
=
2
a2 (γ) − a1 (γ) b2 (γ) − b1 (γ)
·
f (γ)dγ
2
2
Z
1
(a2 (γ) − a1 (γ)) · (b2 (γ) − b1 (γ)) 2γdγ = Cov(A, B).
0
That is the f -weighted possibilistic variance and covariance defined by (7) and (8)
can be considered as a generalization of possibilistic variance and covariance introduced in [1]. It can easily be verified that the weighted covariance is a symmetrical
bilinear operator.
Example 3.1 Let A = (a, b, α, β) be a trapezoidal fuzzy number and let f (γ) =
9
(n + 1)γ n be a weighting function. Then,
#2
Z 1"
a2 (γ) − a1 (γ)
γ n dγ
Varf (A) = (n + 1)
2
0
Z 1
n+1
=
[(b − a) + (α + β)(1 − γ)]2 γ n dγ
4
0
b−a
(n + 1)(α + β)2
α+β 2
=
+
+
.
2
2(n + 2)
4(n + 2)2 (n + 3)
So,
α+β 2
b−a
+
lim Varf (A) = lim
n→∞
n→∞
2
2(n + 2)
2
(n + 1)(α + β)
b−a
+
.
=
2
4(n + 2) (n + 3)
2
The following theorem can be proven in a similar way as Theorem 4.1 from
[1].
Theorem 3.1 Let f be a weighting function, let A, B and C be fuzzy numbers and
let x and y be real numbers. Then the following properties hold,
Covf (A, A) = Varf (A), Covf (A, B) = Covf (B, A),
Varf (x̄) = 0, Covf (x̄, A) = 0,
Covf (xA + yB, C) = |x|Covf (A, C) + |y|Covf (B, C),
Varf (xA + yB) = x2 Varf (A) + y 2 Varf (B) + 2|x||y|Covf (A, B),
where x̄ and ȳ denote the characteristic functions of the sets {x} and {y}, respectively.
Example 3.2 Let A = (a, b, α, β) and B = (a0 , b0 , α0 , β 0 ) be fuzzy numbers of
trapezoidal form. Let f (γ) = (n + 1)γ n , n ≥ 0, be a weighting function then the
power-weighted covariance between A and B is computed by
Covf (A, B)
Z 1
(b − a) + (1 − γ)(α + β) (b0 − a0 ) + (1 − γ)(α0 + β 0 )
=
·
(n + 1)γ n dγ
2
2
"0
#"
#
b−a
α+β
b0 − a0
α0 + β 0
(n + 1)(α + β)(α0 + β 0 )
=
+
+
+
.
2
2(n + 2)
2
2(n + 2)
4(n + 2)2 (n + 3)
10
So,
"
#
b0 − a0
α0 + β 0
lim Covf (A, B) = lim
+
n→∞
n→∞
2
2(n + 2)
(n + 1)(α + β)(α0 + β 0 )
b − a b0 − a0
+
=
·
.
4(n + 2)2 (n + 3)
2
2
b−a
α+β
+
2
2(n + 2)
#"
If a = b and a0 = b0 , i.e. we have two triangular fuzzy numbers, then their
covariance becomes
Covf (A, B) =
4
(α + β)(α0 + β 0 )
.
2(n + 2)(n + 3)
Summary
In this paper we have introduced the notations of the weighted lower possibilistic
and upper possibilistic mean values, crisp possibilistic mean value and variance
of fuzzy numbers, which are consistent with the extension principle and with the
well-known defintions of expectation and variance in probability theory. We have
also showed that the weighted interval-valued possibilistic mean is always a subset
(moreover a proper subset excluding some special cases) of the interval-valued
probabilistic mean for any weighting function.
References
[1] C. Carlsson, R. Fullér, On possibilistic mean value and variance of fuzzy
numbers, Fuzzy Sets and Systems, 122(2001) 315-326.
[2] D. Dubois, H. Prade, The mean value of a fuzzy number, Fuzzy Sets and
Systems, 24(1987) 279-300.
[3] S. Chanas and M. Nowakowski, Single value simulationof fuzzy variable,
Fuzzy Sets and Systems, 25(1988) 43-57.
[4] M. Delgado, M. A. Vila and W. Woxman, On a canonical reprsentation of
fuzzy numbers, Fuzzy Sets and Systems. 93(1998) 125-135.
[5] R. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and
Systems, 18(1986) 31-43.
[6] S. Heilpern, The expected value of a fuzzy number, Fuzzy Sets and Systems,
47(1992) 81-86.
11
5
Citations
[A9] Robert Fullér and Péter Majlender, On weighted possibilistic mean and
variance of fuzzy numbers, FUZZY SETS AND SYSTEMS, 136(2003) 363-374.
[MR1984582]
in journals
A9-c22 Martin O, Klir G J, Defuzzification as a special way of dealing with retranslation, International Journal of General Systems, 36: (6), pp. 683-701.
2007
http://dx.doi.org/10.1080/03081070701456088
A9-c21 Zhang W -G, Wang Y -L, A comparative analysis of possibilistic variances and covariances of fuzzy numbers Fundamenta Informaticae, 79: (12), pp. 257-263. 2007
A9-c20 Xinwang Liu and Hsinyi Lin, Parameterized approximation of fuzzy number with minimum variance weighting functions, MATHEMATICAL AND
COMPUTER MODELLING, 46 (2007) 1398-1409. 2007
http://dx.doi.org/10.1016/j.mcm.2007.01.011
Apart from the weighting function that put different emphasis on
the possible values of a fuzzy number, we can also put different
emphasis on the cut-level set of a fuzzy number. This method
was proposed by Fullér and Majlender [A9]. The method was
extended to a more general form without the monotonic condition in the weighting function definition by Liu recently [19], and
a parameterized weighting function with maximum entropy was
proposed. (page 1406)
A9-c19 Thavaneswaran A, Thiagarajah K, Appadoo SS Fuzzy coefficient volatility (FCV) models with applications MATHEMATICAL AND COMPUTER
MODELLING, 45 (7-8): 777-786 APR 2007
http://dx.doi.org/10.1016/j.mcm.2006.07.019
A9-c18 Vercher E, Bermudez JD, Segura JV Fuzzy portfolio optimization under
downside risk measures FUZZY SETS AND SYSTEMS 158 (7): 769-782
APR 1 2007
http://dx.doi.org/10.1016/j.fss.2006.10.026
12
Weighted mean values introduced in [A9] are a generalization of
those possibilistic ones and allow us to incorporate the importance of α-level sets. It shows that for LR-fuzzy numbers, any f
-weighted interval-valued possibilistic mean value is a subset of
the interval-valued mean of a fuzzy number in the sense of Dubois
and Prade. (page 770)
A9-c17 Supian Sudradjat, The Weighted Possibilistic Mean Variance and Covariance of Fuzzy Numbers, JOURNAL OF APPLIED QUANTITATIVE
METHODS, Volume 2, Issue 3, pp. 349-356, September 30, 2007
A9-c16 Thiagarajah, K., Appadoo, S.S., Thavaneswaran, A. Option valuation
model with adaptive fuzzy numbers, COMPUTERS AND MATHEMATICS
WITH APPLICATIONS, 53 (5), pp. 831-841. 2007
http://dx.doi.org/10.1016/j.camwa.2007.01.011
In the next section, in the line of Fuller and Majlender [A9] we
discuss weighted possibilistic moment of the adaptive fuzzy number. (page 834)
Viewing the fuzzy numbers as random sets, Dubois and Prade [5]
defined their interval-valued expectation. However, Carlsson and
Fuller [A14] defined a possibilistic interval-valued mean value
of fuzzy numbers by viewing them as possibility distributions.
Furthermore, weighted possibilistic mean, variance and weighted
interval-valued possibilistic mean value of fuzzy numbers are all
introduced in Fuller and Majlender [A9].
..
.
Furthermore, weighted possibilistic mean, variance and weighted
interval-valued possibilistic mean value of fuzzy numbers are all
introduced in Fuller and Majlender [A9]. (page 835)
A9-c15 Dubois D Possibility theory and statistical reasoning COMPUTATIONAL
STATISTICS & DATA ANALYSIS, 51 (1): 47-69 NOV 1 2006
http://dx.doi.org/10.1016/j.csda.2006.04.015
Fullér and colleagues (Carlsson and Fuller, 2001; Fullér and Majlender, 2003) consider introducing a weighting function on [0, 1]
in order to account for unequal importance of cuts when computing upper and lower expectations.
13
..
.
The notion of variance has been extended to fuzzy random variables (Koerner, 1997), but little work exists on the variance of a
fuzzy interval. Fuller and colleagues (Carlsson and Fuller, 2001;
Fullér and Majlender, 2003) propose a definition as follows
Z 1
sup Mγ − inf Mγ 2
f (γ) dγ
V̄ (M ) =
2
0
(page 63)
A9-c14 Stefanini L, Sorini L, Guerra ML Parametric representation of fuzzy numbers and application to fuzzy calculus FUZZY SETS AND SYSTEMS 157
(18): 2423-2455 SEP 16 2006
http://dx.doi.org/10.1016/j.fss.2006.02.002
Other functions can be calculated numerically; examples are the
covariance and variance of fuzzy numbers u, v (see [3, A9]);
(page 2437)
A9-c13 Matia F, Jimenez A, Al-Hadithi BM, et al. The fuzzy Kalman filter: State
estimation using possibilistic techniques, FUZZY SETS AND SYSTEMS,
157 (16): 2145-2170 AUG 16 2006
http://dx.doi.org/10.1016/j.fss.2006.05.003
A9-c12 Xinwang Liu, On the maximum entropy parameterized interval approximation of fuzzy numbers, FUZZY SETS AND SYSTEMS, 157(2006) 869878. 2006
http://dx.doi.org/10.1016/j.fss.2005.09.010
Fullér and Majlender [A9] further extended these results to the
notion of weighted possibilistic mean with the weighting function
method.
..
.
Based on this interval aggregation idea, the definition of the weighting function in [A9] is extended without the monotonic increasing assumption. In fact, the increasing weighting function in [A9]
can be seen as the weighted average of the level sets which places
more emphasis on the level sets with higher membership grades.
(page 870)
14
The property of the f-weighted interval-valued possibilistic mean
is also extended directly:
Theorem 1 (Fullér and Majlender [A9]). Let A, B ∈ F and let f
be a weighting function, and let µ be a real number, then,
Mf (A + B) = Mf (A) + Mf (B),
Mf (µA) = µMf (A).
(page 871)
Regarding the weighting function as an aggregation operator for
the fuzzy number level sets, the paper extends the concept of a
weighting function proposed by Fullér and Majlender to a general
form without the monotonic increasing assumption. (page 878)
A9-c11 Ayala G, Leon T, Zapater V, Different averages of a fuzzy set with an
application to vessel segmentation, IEEE TRANSACTIONS ON FUZZY
SYSTEMS, 13(3): 384-393 JUN 2005
http://dx.doi.org/10.1109/TFUZZ.2004.839667
A9-c10 Cheng CB, Fuzzy process control: construction of control charts with
fuzzy numbers, FUZZY SETS AND SYSTEMS, 154 (2): 287-303 SEP 1
2005
http://dx.doi.org/10.1016/j.fss.2005.03.002
Fullér and Majlender [A9] further generalized the above definition by introducing a weighting function, ω(γ), to measure the
importance of the γ-level set of F , and defined the ω-weighted
possibilistic variance of F as
..
.
However, to allow the use of the given scores in this estimation,
the present study re-defines the γ-weighted possibilistic variance
of a triangular fuzzy number based on the definitions of Carlsson
and Fullér [A14] and Fullér and Majlender [A9],
..
.
In Eq. (4), the mode of the fuzzy number, m, replaces the arithmetic mean of the γ-level set used in the definition of Carlsson
and Fullér [A14]. When F is symmetric, the γ-weighted possibilistic variance defined in Eq. (4) equals to the variance defined
by Fullér and Majlender [A9]. (pages 291-292)
15
A9-c9 Bodjanova S, Median value and median interval of a fuzzy number, INFORMATION SCIENCES, 172 (1-2): 73-89 JUN 9 2005
http://dx.doi.org/10.1016/j.ins.2004.07.018
Weighted possibilistic mean, variance, and covariance of fuzzy
numbers can be found in the work of Fuller and Majlender [A9].
(page 74)
A9-c8 Hong DH, Kim KT, A note on weighted possibilistic mean, FUZZY SETS
AND SYSTEMS, 148 (2): 333-335 DEC 1 2004
http://dx.doi.org/10.1016/j.fss.2004.04.011
in proceedings
A9-c7 Cheng, Jao-Hong Lee, Chen-Yu , Product Outsourcing under Uncertainty:
an Application of Fuzzy Real Option Approach, IEEE International Fuzzy
Systems Conference, 2007 (FUZZ-IEEE 2007), 23-26 July 2007, London,
[ISBN: 1-4244-1210-2], pp. 1-6. 2007
http://dx.doi.org/10.1109/FUZZY.2007.4295675
A9-c6 Wang X, Xu WJ, Zhang WG, et al., Weighted possibilistic variance of
fuzzy number and its application in portfolio theory, in: Fuzzy Systems
and Knowledge Discovery, LECTURE NOTES IN ARTIFICIAL INTELLIGENCE, vol. 3613, pp. 148-155. 2005
http://dx.doi.org/10.1007/11539506 18
Abstract. Dubois and Prade defined an interval-valued expectation of fuzzy numbers, viewing them as consonant random sets.
Fullér and Majlender then proposed an weighted possibility mean
value, variance and covariance of fuzzy numbers, viewing them
as weighted possibility distributions. In this paper, we define a
new weighted possibilistic variance and covariance of fuzzy numbers based on Fullér and Majlenders’ notations.
..
.
In 2003, Fullér and Majlender proposed an weighted possibility
mean value of fuzzy numbers, viewing them as weighted possibility distributions [A9]. (page 148)
16
A9-c5 Zhang WG, Wang YL, Portfolio selection: Possibilistic mean-variance
model and possibilistic efficient frontier, in: Megiddo, N.; Xu, Y.; Alonstioti,
N.; Zhu, B. (Eds.) Algorithmic Applications in Management First International Conference, AAIM 2005, Xian, China, June 22-25, 2005, Proceedings
Series: Lecture Notes in Computer Science , Vol. 3521 Sublibrary: Information Systems and Applications, incl. Internet/Web, and HCI, Springer,
[ISBN: 978-3-540-26224-4], 2005, pp. 203-213. 2005
http://dx.doi.org/10.1007/11496199 23
Fullér and Majlender [A9] defined the weighted possibilistic mean
value of A as
Z 1
MfL (A) + MfU (A)
a(γ) + b(γ)
M̄f (A) =
f (γ)
dγ =
.
2
2
0
(page 204)
A9-c4 Zhang, J.-P., Li, S.-M. Portfolio selection with quadratic utility function
under fuzzy enviornment 2005 International Conference on Machine Learning and Cybernetics, ICMLC 2005, pp. 2529-2533. 2005
http://dx.doi.org/10.1109/ICMLC.2005.1527369
A9-c3 Blankenburg, B., Klusch, M. BSCA-F: Efficient fuzzy valued stable coalition forming among agents Proceedings - 2005 IEEE/WIC/ACM International Conference on Intelligent Agent Technology, IAT’05, 2005, art. no.
1565632, pp. 732-738. 2005
http://dx.doi.org/10.1109/IAT.2005.48
A9-c2 Garcia, F.A.A. Fuzzy real option valuation in a power station reengineering
project, Soft Computing with Industrial Applications - Proceedings of the
Sixth Biannual World Automation Congress, pp. 281-287. 2004
http://ieeexplore.ieee.org/iel5/9827/30981/01439379.pdf?
A9-c1 Wang, X., Xu, W.-J., Zhang, W.-G. A class of weighted possibilistic meanvariance portfolio selection problems Proceedings of 2004 International Conference on Machine Learning and Cybernetics, 4, pp. 2036-2040. 2004
http://ieeexplore.ieee.org/iel5/9459/30104/01382130.pdf?arnumber=1382130
17