A Normative View on Possibility Distributions
Christer Carlsson1 , Robert Fullér1,2 , and Péter Majlender3,1,2
1
2
3
Institute for Advanced Management Systems Research, Åbo Akademi
University, Lemminkäinengatan 14C, FIN-20520 Åbo, Finland,
christer.carlsson@abo.fi,
Department of Operations Research, Eötvös Loránd University, Pázmány
Péter sétány 1C, P. O. Box 120, H-1518 Budapest, Hungary,
robert.fuller@abo.fi,
Turku Centre for Computer Science, Åbo Akademi University,
Lemminkäinengatan 14B, FIN-20520 Åbo, Finland, peter.majlender@abo.fi
Abstract
In this paper we will consider fuzzy numbers from a normative point of
view and will illustrate the concepts of possibilistic mean value, covariance, variance and correlation by several examples. We will show that
zero correlation does not always imply non-interactivity. We will also
discuss the limitations of direct definitions of joint possibility distributions (for example, when one simple aggregates the membership values
of two fuzzy numbers by a triangular norm).
1 Probability and Possibility
The concept of independence (dependence) has been studied in depth
in possibility theory, for good surveys see, e.g. (Campos and Huete
1999). Dubois and Prade (1987) 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. Carlsson and Fullér (2001) introduced the possibilistic mean value, variance and covariance of fuzzy numbers. Fullér
2
C. Carlsson, R. Fullér and P. Majlender
and Majlender (2003) introduced the notations of crisp weighted possibilistic mean value, variance and covariance of fuzzy numbers, that are
consistent with the extension principle. Fullér and Majlender (2003a)
introduced a measure of interactivity between marginal distributions of
a joint possibility distribution C as the expected value of their interactivity relation on C. Carlsson et al (2003) showed a possibilistic analog
of the probabilistic Cauchy-Schwarz inequality.
In this paper we will illustrate some important feautures of possibilistic mean value, covariance, variance and correlation by several examples.
In probability theory, the dependency between two random variables
can be characterized through their joint probability density function.
Namely, if X and Y are two random variables with probability density functions fX (x) and fY (y), respectively, then the density function,
fX,Y (x, y), of their joint random variable (X, Y ), should satisfy the
following properties
Z
Z
fX,Y (t, y)dt = fY (y),
(1)
fX,Y (x, t)dt = fX (x),
R
R
for all x, y ∈ R. Furthermore, fX (x) and fY (y) are called the the
marginal probability density functions of random variable (X, Y ). The
covariance between X and Y is defined as Cov(X, Y ) = E(XY ) −
E(X)E(Y ), where E is the expected value operator, and if X and Y
are independent then Cov(X, Y ) = 0. For any random variables X and
Y and real numbers λ and µ the following relationship holds
Var(λX + µY ) = λ2 Var(X) + µ2 Var(Y ) + 2λµCov(X, Y ),
where Var(X) denotes the variance of X. The correlation coefficient
between X and Y is defined by
Cov(X, Y )
,
ρ(X, Y ) = p
Var(X)Var(Y )
and it is clear that −1 ≤ ρ(X, Y ) ≤ 1.
A Normative View on Possibility Distributions
3
A fuzzy set A in R is said to be a fuzzy number if it is normal,
fuzzy convex and has an upper semi-continuous membership function
of bounded support. The family of all fuzzy numbers will be denoted
by F . A γ-level set of a fuzzy set A in Rm is defined by [A]γ = {x ∈
Rm : A(x) ≥ γ} if γ > 0 and [A]γ = cl{x ∈ Rm : A(x) > γ} (the
closure of the support of A) if γ = 0. If A ∈ F is a fuzzy number
then [A]γ is a convex and compact subset of R for all γ ∈ [0, 1]. Fuzzy
numbers can be considered as possibility distributions (Zadeh 1978).
A fuzzy set B in Rm is said to be a joint possibility distribution of
fuzzy numbers Ai ∈ F, i = 1, . . . , m, if it satisfies the relationship
max
xj ∈R, j6=i
B(x1 , . . . , xm ) = Ai (xi ), ∀xi ∈ R, i = 1, . . . , m.
Furthermore, Ai is called the i-th marginal possibility distribution of
B, and the projection of B on the i-th axis is Ai for i = 1, . . . , m.
If Ai ∈ F, i = 1, . . . , m, and B is their joint possibility distribution
then the relationships B(x1 , . . . , xm ) ≤ min{A1 (x1 ), . . . , Am (xm )}
and [B]γ ⊆ [A1 ]γ × · · · × [Am ]γ hold for all x1 , . . . , xm ∈ R and
γ ∈ [0, 1].
Fuzzy numbers Ai ∈ F, i = 1, . . . , m, are said to be non-interactive
if their joint possibility distribution, B, is given by
B(x1 , . . . , xm ) = min{A1 (x1 ), . . . , Am (xm )},
or, equivalently, [B]γ = [A1 ]γ × · · · × [Am ]γ , for all x1 , . . . , xm ∈ R
and γ ∈ [0, 1].
Note 1. If A, B ∈ F are non-interactive then their joint membership
function is defined by A × B. It is clear that in this case any change in
the membership function of A does not effect the second marginal possibility distribution and vice versa. On the orther hand, if A and B are
said to be interactive then they can not take their values independently
of each other (Dubois, Prade 1988).
Note 2. Marginal probability distributions are determined from the joint
one by the principle of ’falling integrals’ and marginal possibility distributions are determined from the joint possibility distribution by the
principle of ’falling shadows’.
4
C. Carlsson, R. Fullér and P. Majlender
Fig. 1. Non-interactive possibility distributions.
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 nonnegative, monotone increasing and satisfies the following normalization
condition
Z 1
f (γ)dγ = 1.
(2)
0
Different weighting functions can give different (case-dependent) importances to γ-levels sets of fuzzy numbers.
2 Weighted Possibilistic Mean Value and Variance
The notations of weighted lower possibilistic and upper possibilistic
mean values, crisp possibilistic mean value and variance of fuzzy numbers were introduced in (Fullér, Majlender 2003). Namely, they intro-
A Normative View on Possibility Distributions
5
duced the f -weighted possibilistic mean value of fuzzy number A as
Ef (A) =
Z
1
a1 (γ) + a2 (γ)
f (γ)dγ.
2
0
(3)
We note that if f (γ) = 2γ, γ ∈ [0, 1] then
Ef (A) =
Z
0
1
a1 (γ) + a2 (γ)
2γdγ =
2
Z
1
[a1 (γ) + a2 (γ)] γdγ.
0
That is the f -weighted possibilistic mean value defined by (3) can be
considered as a generalization of possibilistic mean value introduced in
(Carlsson, Fullér 2001). 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.
Example 1 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. The γ-level of A is computed by [A]γ = [a − (1 −
γ)α, b + (1 − γ)β], ∀γ ∈ [0, 1]. Then the weighted possibilistic mean
values of A are computed by
Ef (A) =
β−α
a+b
+
.
2
2(n + 2)
So,
lim Ef (A) = lim
n→∞
n→∞
a+b
β−α
a+b
=
+
.
2
2(n + 2)
2
Let A, B ∈ F and let f be a weighting function. In (Fullér, Majlender 2003) the f -weighted possibilistic variance of A was introduced as
Varf (A) =
Z 1
0
a2 (γ) − a1 (γ) 2
f (γ)dγ,
2
(4)
6
C. Carlsson, R. Fullér and P. Majlender
and the f -weighted covariance of A and B was defined by
Z 1
a2 (γ) − a1 (γ) b2 (γ) − b1 (γ)
·
f (γ)dγ.
Covf (A, B) =
2
2
0
(5)
It should be noted that if f (γ) = 2γ, γ ∈ [0, 1] then
Varf (A) =
Z 1
0
=
1
2
Z
1
a2 (γ) − a1 (γ) 2
2γdγ
2
2
[a2 (γ) − a1 (γ)] γdγ = Var(A),
0
and
Covf (A, B) =
Z
0
1
a2 (γ) − a1 (γ) b2 (γ) − b1 (γ)
·
2γdγ
2
2
Z
1 1
=
(a2 (γ) − a1 (γ)) · (b2 (γ) − b1 (γ)) γdγ
2 0
= Cov(A, B).
Where Var(A) and Cov(A, B) denote the possibilistic variance and covariance introduced by (Carlsson, Fullér 2001). That is the f -weighted
possibilistic variance and covariance defined by (4) and (5) can be considered as a generalization of the concepts introduced by (Carlsson,
Fullér 2001).
It can easily be verified that the weighted covariance is a symmetrical bilinear operator.
Example 2 Let A = (a, b, α, β) be a trapezoidal fuzzy number and let
f (γ) = (n + 1)γ n be a weighting function. Then,
#2
a2 (γ) − a1 (γ)
γ n dγ
Varf (A) = (n + 1)
2
0
b−a
(n + 1)(α + β)2
α+β 2
=
+
+
.
2
2(n + 2)
4(n + 2)2 (n + 3)
Z
1
"
A Normative View on Possibility Distributions
7
Example 3 Let A = (a, b, α, β) and B = (a′ , b′ , α′ , β ′ ) be fuzzy numbers of trapezoidal form. Let f (γ) = (n + 1)γ n , n ≥ 0, be a weighting
function, then the power-weighted covariance of A and B is computed
by
"
#"
#
b−a
b ′ − a′
α+β
α′ + β ′
Covf (A, B) =
+
+
2
2(n + 2)
2
2(n + 2)
+
(n + 1)(α + β)(α′ + β ′ )
.
4(n + 2)2 (n + 3)
So,
lim Covf (A, B) =
n→∞
b − a b ′ − a′
·
.
2
2
If a = b and a′ = b′ , i.e. we have two triangular fuzzy numbers,
then their covariance becomes
Covf (A, B) =
(α + β)(α′ + β ′ )
.
2(n + 2)(n + 3)
The following theorem (Fullér, Majlender 2003) shows that the variance of linear combinations of fuzzy numbers can easily be computed
(in a similar manner as in probability theory).
Theorem 1 Let f be a weighting function, let A and B be fuzzy numbers and let λ and µ be real numbers. Then the following properties
hold,
Covf (λA + µB, C) = |λ|Covf (A, C) + |µ|Covf (B, C),
and
Varf (λA + µB) = λ2 Varf (A) + µ2 Varf (B) + 2|λ||µ|Covf (A, B),
where the operations additions and multiplication by scalar of fuzzy
numbers are defined by the sup-min extension principle (Zadeh 1965).
8
C. Carlsson, R. Fullér and P. Majlender
3 Expected Value of Functions on Fuzzy Sets
The main drawback of definition (5) is that Covf (A, B) is always nonnegative for any pair of fuzzy numbers, A and B. So even though definition (5) is consistent with the extension principle it may not always be
consistent with the principle of ’falling shadows’. In this section we will
use the concept of average values of well-chosen real-valued function
on γ-level sets of a joint possibility distribution to measure covariance
between γ-level sets of its marginal distributions.
Let B be a joint possibility distribution in Rn , let γ ∈ [0, 1] and
let g : Rn → R be a function. It is well-known from analysis that the
average value of function g on [B]γ can be computed by
Z
1
Z
C[B]γ (g) =
g(x)dx
γ
dx [B]
[B]γ
=Z
1
dx1 . . . dxn
Z
g(x1 , . . . , xn )dx1 . . . dxn .
[B]γ
[B]γ
Following (Fullér, Majlender 2003a) will call C as the central value operator.
If g : R → R is a single-variable function and A ∈ F is a fuzzy
number then the average value of function g on [A]γ is defined by
Z
1
Z
C[A]γ (g) =
g(x)dx.
γ
dx [A]
[A]γ
Especially, if g(x) = x, for all x ∈ R is the identity function and
A ∈ F is a fuzzy number with [A]γ = [a1 (γ), a2 (γ)], then the average
value of the identity function on [A]γ is computed by
Z
a1 (γ) + a2 (γ)
1
.
xdx =
C[A]γ (id) = Z
2
γ
dx [A]
[A]γ
A Normative View on Possibility Distributions
9
Because C[A]γ (id) is nothing else, but the center of [A]γ we will use the
shorter notation C([A]γ ) for C[A]γ (id).
It is clear that C[B]γ is linear for any arbitrarily fixed joint possibility
distribution B and for any γ ∈ [0, 1]. Let us denote the projection
functions on R2 by πx and πy , that is, πx (u, v) = u and πy (u, v) = v
for u, v ∈ R. The following two theorems (Fullér, Majlender 2003a)
show two important properties of central value operator.
Theorem 2 If A, B ∈ F are non-interactive and g = πx + πy is the
addition operator on R2 then
C[A×B]γ (πx + πy ) = C[A]γ (id) + C[B]γ (id) = C([A]γ ) + C([B]γ ),
for all γ ∈ [0, 1].
Theorem 3 If A, B ∈ F are non-interactive and p = πx πy is the multiplication operator on R2 then
C[A×B]γ (πx πy ) = C[A]γ (id) · C[B]γ (id) = C([A]γ ) · C([B]γ ),
for all γ ∈ [0, 1].
The following defintion (Fullér, Majlender 2003a) is crucial for the
theory of possibilistic dependencies.
Definition 1 Let C be the joint possibility distribution with marginal
possibility distributions A, B ∈ F, and let γ ∈ [0, 1]. The measure of
interactivity between the γ-level sets of A and B is defined by
R[C]γ (πx , πy ) = C[C]γ (πx − C[C]γ (πx ))(πy − C[C]γ (πy )) .
Using the definition of central value we have
Z
1
(x − C[C]γ (πx ))(y − C[C]γ (πy ))dxdy
R[C]γ (πx , πy ) = Z
γ
[C]
dxdy
[C]γ
= C[C]γ (πx πy ) − C[C]γ (πx ) · C[C]γ (πy ),
for all γ ∈ [0, 1].
10
C. Carlsson, R. Fullér and P. Majlender
Note 3. The interactivity relation computes the average value of the interactivity function
g(x, y) = (x − C[C]γ (πx ))(y − C[C]γ (πy )),
on [C]γ .
We can also use the principle of central values to introduce the notation of expected value of functions on fuzzy sets. Let g : R → R
be a function and A ∈ F. Let us consider again the average value of
function g on [A]γ
Z
1
Z
C[A]γ (g) =
g(x)dx.
γ
dx [A]
[A]γ
In (Fullér, Majlender 2003a) the expected value of function g on A with
respect to a weighting function f was defined by
Z
Z 1
Z 1
1
Z
g(x)dxf (γ)dγ.
Ef (g; A) =
C[A]γ (g)f (γ)dγ =
γ
0
0
dx [A]
[A]γ
Especially, if g(x) = x, for all x ∈ R is the identity function then we
get
Z 1
a1 (γ) + a2 (γ)
Ef (id; A) = Ef (A) =
f (γ)dγ,
2
0
which is thef -weighted possibilistic mean value of A introduced in
(Fullér, Majlender 2003).
Note 4. The expected value of a function on a fuzzy number A is nothing else but the expected value of its average values on all gamma level
sets of A.
The variance of A was defined in (Fullér, Majlender 2003a) as the
expected value of function g(x) = (x − C([A]γ ))2 on A. That is,
Z 1
(a2 (γ) − a1 (γ))2
f (γ)dγ.
Varf (A) = Ef (g; A) =
12
0
A Normative View on Possibility Distributions
11
The covariance between marginal possibility distributions was defined in (Fullér, Majlender 2003a) as the expected value of their interactivity function on their joint possibility distribution. Namely, if C is
a joint possibility distribution in R2 and A, B ∈ F denote its marginal
possibility distributions then the covariance of A and B with respect to
a weighting function f (and with repect to their joint possibility distributioin C) was defined by
Covf (A, B) =
=
Z
1
0
Z
1
R[C]γ (πx , πy )f (γ)dγ
0
(6)
C[C]γ (πx πy ) − C[C]γ (πx ) · C[C]γ (πy ) f (γ)dγ.
The next theorem (Fullér, Majlender 2003a) states the bilinearity of
the interactivity relation operator.
Theorem 4 Let C be a joint possibility distribution in R2 , and let λ, µ ∈
R. Then
R[C]γ (λπx + µπy , λπx + µπy ) =
λ2 R[C]γ (πx , πx ) + µ2 R[C]γ (πy , πy ) + 2λµR[C]γ (πx , πy ).
The following theorem (Carlsson et al 2003) shows a very important
property of the correlation operator.
Theorem 5 Let A, B ∈ F be fuzzy numbers with joint possibility distribution C. Then, their correlation coefficient defined by
Covf (A, B)
,
ρf (A, B) = p
Varf (A)Varf (B)
satisfies the relationship
−1 ≤ ρf (A, B) ≤ 1
for any weighting function f .
12
C. Carlsson, R. Fullér and P. Majlender
4 Illustrations of Possibilistic Correlation
Let us consider several interesting cases. In (Carlsson et al 2003) we
proved that if A and B are non-interactive, that is, their joint possibility
distribution is A × B then ρf (A, B) = 0.
Example 1 Consider now the case depicted in Fig. 2. It can be shown
(Carlsson et al 2003) that in this case ρf (A, B) = 1.
Fig. 2. The case of ρf (A, B) = 1.
Note 5. In this case ρf (A, B) = 1 for any weighting function f ; and if
A(u) = γ for some u ∈ R then there exists a unique v ∈ R that B can
take (see Fig. 2), furthermore, if u is moved to the left (right) then the
corresponding value (that B can take) will also move to the left (right).
Loosely speaking, in this case the shadows move in tandem that is, if
one moves A to the left (right) then B will also move to the left (right).
A Normative View on Possibility Distributions
13
Example 2 Consider now the case depicted in Fig. 3. It can be shown
(Carlsson et al 2003) that in this case ρf (A, B) = −1.
Note 6. In this case ρf (A, B) = −1 for any weighting function f ; and
if A(u) = γ for some u ∈ R then there exists a unique v ∈ R that B
can take (see Fig. 3), furthermore, if u is moved to the left (right) then
the corresponding value (that B can take) will move to the right (left).
Loosely speaking, in this case the shadows move oppositively that is, if
one moves A to the left (right) then B will move to the right (left).
Fig. 3. The case of ρf (A, B) = −1.
Example 3 Let A(x) = B(x) = x · χ[0,1] (x) for x ∈ R. Thus, the
γ-level sets of A and B are [A]γ = [B]γ = [γ, 1] for γ ∈ [0, 1]. Let
their joint possibility distribution I be defined by
I(x, y) = (x + y − 1) · χT (x, y),
14
C. Carlsson, R. Fullér and P. Majlender
where
T = {(x, y) ∈ R2 |x ≤ 1, y ≤ 1, x + y ≥ 1}.
It is easy to see that,
[I]γ = cl{(x, y) ∈ R2 |I(x, y) > γ}
= {(x, y) ∈ R2 |x ≤ 1, y ≤ 1, x + y ≥ 1 + γ}.
This situation is depicted on Fig. 4, where we have shifted the fuzzy sets
Fig. 4. The case of ρf (A, B) = −1/3.
to get a better view of the situation.
We shall compute the correlation coefficient between A and B. From
the equations,
Z
Z
(1 − γ)2
(1 − γ)2 (1 + γ)(5 + γ)
dxdy =
xydxdy =
,
,
2
24
[I]γ
[I]γ
Z
Z
Z
1 1
(1 − γ)2 (2 + γ)
xdxdy =
ydxdy =
,
(1 − x2 )dx =
2 γ
6
[I]γ
[I]γ
A Normative View on Possibility Distributions
15
and
Z
dx = 1 − γ,
[A]γ
Z
xdx =
[A]γ
Z
x2 dx =
[A]γ
(1 − γ)(1 + γ + γ 2 )
,
3
(1 − γ)(1 + γ)
2
we get
1
Covf (A, B) = −
36
Z
1
(1 − γ)2 f (γ)dγ,
0
and
Varf (A) = Varf (B) =
1
12
Z
1
(1 − γ)2 f (γ)dγ,
0
Therefore,
ρf (A, B) = −1/3,
for any weighting function f .
Example 4 Let A(x) = B(1 − x) = x · χ[0,1] (x) for x ∈ R. So, the γlevel set of A and B are computed by [A]γ = [γ, 1] and [B]γ = [0, 1−γ]
for γ ∈ [0, 1]. Let
H(x, y) = (x − y) · χV (x, y),
where
V = {(x, y) ∈ R2 |x ≤ 1, y ≥ 0, x − y ≥ 0}.
This situation is depicted on Fig. 5, where we have shifted the fuzzy sets
to get a better view of the situation. After some calculations we get
ρf (A, B) = 1/3,
for any weighting function f .
16
C. Carlsson, R. Fullér and P. Majlender
Fig. 5. The case of ρf (A, B) = 1/3.
5 Interactive Fuzzy Numbers can be Uncorrelated
Zero correlation does not always imply non-interactivity. Let A, B ∈ F
be fuzzy numbers, let C be their joint possibility distribution, and let
γ ∈ [0, 1]. Suppose that [C]γ is symmetrical, i.e. there exists a ∈ R
such that
C(x, y) = C(2a − x, y),
for all x, y ∈ [C]γ (hence, line defined by {(a, t)|t ∈ R} is the axis
of symmetry of [C]γ ). We shall show that in this case the interactivity
relation of [A]γ and [B]γ vanishes, i.e. R[C]γ (πx , πy ) = 0. Indeed, let
H = {(x, y) ∈ [C]γ |x ≤ a},
then
Z
[C]γ
xydxdy =
Z
H
xy + (2a − x)y dxdy = 2a
Z
H
ydxdy,
A Normative View on Possibility Distributions
Z
Z
xdxdy =
Z
ydxdy = 2
[C]γ
[C]γ
H
x + (2a − x) dxdy = 2a
Z
ydxdy,
Z
dxdy = 2
[C]γ
H
Z
17
dxdy,
H
Z
dxdy,
H
therefore, we obtain
R[C]γ (πx , πy ) = Z
−Z
1
[C]γ
dxdy
Z
[C]γ
1
dxdy
Z
xydxdy
[C]γ
[C]γ
xdxdy Z
1
dxdy
Z
ydxdy = 0.
[C]γ
[C]γ
So, if all γ-level sets of joint possibility distribution C are symmetrical
Fig. 6. The case of ρf (A, B) = 0 for interactive fuzzy numbers.
18
C. Carlsson, R. Fullér and P. Majlender
then
Covf (A, B) = 0,
for any weighting function f .
6 Explicitly Given Joint Possibility Distributions
In many papers authors consider joint possibility distributions that are
derived from given marginal distributions by aggregating their membership values. Namely, let A, B ∈ F. We will say that their joint possibility distribution C is directly defined from its marginal distributions
if
C(x, y) = T (A(x), B(y)), x, y ∈ R,
where T : [0, 1] × [0, 1] → [0, 1] is a function satisfying the properties
max T (A(x), B(y)) = A(x), ∀x ∈ R,
(7)
max T (A(x), B(y)) = B(y), ∀y ∈ R,
(8)
y
and
x
for example a triangular norm.
Note 7. In this case the joint distribution depends barely on the membership values of its marginal distributions.
We will show that in this case the covariance (and, consequently, the
correlation) between its marginal distributions will be zero whenever at
least one of its marginal distributions is symmetrical.
Theorem 6 Let A, B ∈ F and let their joint possibility distribution C
be defined by
C(x, y) = T (A(x), B(y)),
for x, y ∈ R, where T is a function satisfying conditions (7) and (8). If
A is a symmetrical fuzzy number then
Covf (A, B) = 0,
for any fuzzy number B, aggregator T , and weighting function f .
A Normative View on Possibility Distributions
19
Proof. If A is a symmetrical fuzzy number with center a such that
A(x) = A(2a − x) for all x ∈ R then,
C(x, y) = T (A(x), B(y)) = T (A(2a − x), B(y)) = C(2a − x, y),
that is, C is symmetrical. Hence, considering the results obtained above
we have
R[C]γ (πx , πy ) = 0,
and, therefore,
Covf (A, B) = 0,
for any weighting function f . Which ends the proof.
Summary
We have illustrated some important feautures of possibilistic mean value,
covariance, variance and correlation by several examples. We have
shown that zero correlation does not always imply non-interactivity. We
have also shown the limitations of direct definitions of joint possibility
distributions from individual fuzzy numbers, for example, when one
simply aggregates the membership values of two fuzzy numbers by a
triangular norm
References
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C. Carlsson, R. Fullér and P. Majlender
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