On possibilistic mean value and variance of fuzzy numbers ∗
Christer Carlsson
christer.carlsson@abo.fi
Robert Fullér
rfuller@abo.fi, rfuller@cs.elte.hu
Abstract
Dubois and Prade introduced the mean value of a fuzzy number as a closed interval bounded by
the expectations calculated from its upper and lower distribution functions. In this paper introducing
the notations of lower possibilistic and upper possibilistic mean values we definine the interval-valued
possibilistic mean and investigate its relationship to the interval-valued probabilistic mean. We also
introduce the notation of crisp possibilistic mean value and crisp possibilistic variance of continuous
possibility distributions, which are consistent with the extension principle. We also show that the
variance of linear combination of fuzzy numbers can be computed in a similar manner as in probability
theory.
Keywords: Fuzzy numbers; Possibility theory
1
Introduction
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 the notations of lower possibilistic and upper possibilistic mean values we
definine the interval-valued possibilistic mean, crisp possibilistic mean value and crisp (possibilistic) variance of a continuous possibility distribution, which are consistent with the extension principle and with
the well-known defintions of expectation and variance in probability theory. The theory developed in this
paper is fully motivated by the principles introduced in [2] and by the possibilistic interpretation of the
ordering introduced in [3].
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].
Let A and B ∈ F be fuzzy numbers with [A]γ = [a1 (γ), a2 (γ)] and [B]γ = [b1 (γ), b2 (γ)], γ ∈ [0, 1].
In 1986 Goetschel and Voxman introduced a method for ranking fuzzy numbers as [3]
Z 1
Z 1
A ≤ B ⇐⇒
γ(a1 (γ) + a2 (γ)) dγ ≤
γ(b1 (γ) + b2 (γ)) dγ
(1)
0
0
As was pointed out by Goetschel and Voxman this definition of ordering given in (1) was motivated in part
by the desire to give less importance to the lower levels of fuzzy numbers.
∗
The final version of this paper appeared in: C. Carlsson and R. Fullér, On possibilistic mean value and variance of fuzzy
numbers, Fuzzy Sets and Systems, 122(2001) 315-326. doi: 10.1016/S0165-0114(00)00043-9
1
2
Possibilistic mean value of fuzzy numbers
We explain now the way of thinking that has led us to the introduction of notations of lower and upper
possibilitistic mean values. First, we note that from the equality
Z 1
a1 (γ) + a2 (γ)
Z 1
γ·
dγ
2
,
(2)
γ(a1 (γ) + a2 (γ))dγ = 0
M̄ (A) :=
Z 1
0
γ dγ
0
it follows that M̄ (A) is nothing else but the level-weighted average of the arithmetic means of all γ-level
sets, that is, the weight of the arithmetic mean of a1 (γ) and a2 (γ) is just γ.
Second, we can rewrite M̄ (A) as
Z 1
Z 1
Z 1
γa2 (γ)dγ
γa1 (γ)dγ + 2 ·
2·
0
0
γ(a1 (γ) + a2 (γ))dγ =
M̄ (A) =
2
0
Z 1
Z 1
Z 1
Z 1


γa2 (γ)dγ 
γa1 (γ)dγ
γa2 (γ)dγ 
γa1 (γ)dγ
1 0
1
0
0
.
 =  0Z
+
= 
+
Z 1
1

 2
2
1
1
γdγ
γdγ
2
2
0
0
Third, let us take a closer look at the right-hand side of the equation for M̄ (A). The first quantity,
denoted by M∗ (A) can be reformulated as
Z 1
Z 1
γa1 (γ)dγ
0
M∗ (A) = 2
γa1 (γ)dγ =
Z 1
0
γdγ
0
Z 1
Z 1
Pos[A ≤ a1 (γ)]a1 (γ)dγ
Pos[A ≤ a1 (γ)] × min[A]γ dγ
= 0Z 1
= 0 Z 1
,
Pos[A ≤ a1 (γ)]dγ
Pos[A ≤ a1 (γ)]dγ
0
0
where Pos denotes possibility, i.e.
Pos[A ≤ a1 (γ)] = Π((−∞, a1 (γ]) = sup A(u) = γ.
u≤a1 (γ)
(since A is continuous!) So M∗ (A) is nothing else but the lower possibility-weighted average of the
minima of the γ-sets, and it is why we call it the lower possibilistic mean value of A.
In a similar manner we introduce M ∗ (A) , the upper possibilistic mean value of A, as
Z 1
Z 1
γa2 (γ)dγ
M ∗ (A) = 2
γa2 (γ)dγ = 0 Z 1
0
γdγ
0
Z 1
Z 1
Pos[A ≥ a2 (γ)]a2 (γ)dγ
Pos[A ≥ a2 (γ)] × max[A]γ dγ
0
0
=
=
,
Z 1
Z 1
Pos[A ≥ a2 (γ)]dγ
Pos[A ≤ a2 (γ)]dγ
0
0
2
where we have used the equality
Pos[A ≥ a2 (γ)] = Π([a2 (γ), ∞)) = sup A(u) = γ.
u≥a2 (γ)
Let us introduce the notation
M (A) = [M∗ (A), M ∗ (A)].
that is, M (A) is a closed interval bounded by the lower and upper possibilistic mean values of A.
Definition 2.1 We call M (A) the interval-valued possibilistic mean of A.
If A is the characteristic function of the crisp interval [a, b] then M ((a, b, 0, 0)) = [a, b], that is, an
interval is the possibilistic mean value of itself. We will now show that M is a linear function on F in the
sense of the extension principle.
Theorem 2.1 Let A and B be two non-interactive fuzzy numbers and let λ ∈ R be a real number. Then
M (A + B) = M (A) + M (B),
M (λA) = λM (A),
i.e.
M ∗ (A + B) = M ∗ (A) + M ∗ (B),
M∗ (A + B) = M∗ (A) + M∗ (B),
and
(
[M∗ (λA), M ∗ (λA)] =
[λM∗ (A), λM ∗ (A)] if λ ≥ 0
[λM ∗ (A), λM∗ (A)] if λ < 0
where the addition and multiplication by a scalar of fuzzy numbers is defined by the sup-min extension
principle [4].
Proof 1 Really, from the equation
[A + B]γ = [a1 (γ) + b1 (γ), a2 (γ) + b2 (γ)],
we have
1
Z
M∗ (A + B) = 2
γ(a1 (γ) + b1 (γ))dγ
Z 1
Z 1
γa1 (γ)dγ + 2
γb1 (γ)dγ
=2
0
0
0
= M∗ (A) + M∗ (B),
and
M ∗ (A + B) = 2
Z
1
γ(a2 (γ) + b2 (γ))dγ
Z 1
Z 1
=2
γa2 (γ)dγ + 2
γb2 (γ)dγ
0
0
0
= M ∗ (A) + M ∗ (B),
furthermore, from
(
γ
γ
[λA] = λ[A] = λ[a1 (γ), a2 (γ)] =
3
[λa1 (γ), λa2 (γ)]
if γ ≥ 0
[λa2 (γ), λa1 (γ)] if γ < 0
for λ ≥ 0 we get
1
Z
M∗ (λA) = 2
γ(λa1 (γ))dγ = 2λ
γa1 (γ)dγ = λM∗ (A).
0
0
1
Z
∗
1
Z
1
Z
γ(λa2 (γ))dγ = 2λ
M (λA) = 2
γa2 (γ)dγ = λM ∗ (A).
0
0
and for λ < 0
1
Z
γ(λa2 (γ))dγ = 2λ
M∗ (λA) = 2
Z
γa2 (γ)dγ = λM ∗ (A).
0
0
M ∗ (λA) = 2
1
Z
1
1
Z
γa1 (γ)dγ = λM∗ (A).
γ(λa1 (γ))dγ = 2λ
0
0
Which ends the proof.
We introduce the crisp possibilistic mean value of A by (2) as the arithemtic mean of its lower possibilistic and upper possibilistic mean values, i.e.
M̄ (A) =
M∗ (A) + M ∗ (A)
.
2
The following theorem shows two very important properties of M̄ : F → R.
Theorem 2.2 Let [A]γ = [a1 (γ), a2 (γ)] and [B]γ = [b1 (γ), b2 (γ)] be fuzzy numbers and let λ ∈ R be a
real number. Then
M̄ (A + B) = M̄ (A) + M̄ (B),
and
M̄ (λA) = λM̄ (A).
Proof 2 First we find
Z
1
γ(a1 (γ) + b1 (γ) + a2 (γ) + b2 (γ))dγ =
M̄ (A + B) =
0
Z
1
Z
γ(a1 (γ) + a2 (γ))dγ +
0
1
γ(b1 (γ) + b2 (γ))dγ = M̄ (A) + M̄ (B),
0
and for λ ≥ 0 we get
Z
M̄ (λA) =
1
Z
γ(λa1 (γ) + λa2 (γ))dγ = λ
0
1
γ(a1 (γ) + a2 (γ))dγ = λM̄ (A).
0
and, finally, for λ < 0 we have
Z 1
Z 1
M̄ (λA) =
γ(λa2 (γ) + λa1 (γ))dγ = λ
γ(a1 (γ) + a2 (γ))dγ = λM̄ (A).
0
0
Which ends the proof.
4
Example 2.1 Let A = (a, α, β) be a triangular fuzzy number with center a, left-width α > 0 and rightwidth β > 0 then a γ-level of A is computed by
[A]γ = [a − (1 − γ)α, a + (1 − γ)β], ∀γ ∈ [0, 1],
that is,
1
Z
M∗ (A) = 2
0
M ∗ (A) = 2
Z
0
1
α
γ[a − (1 − γ)αdγ = a − ,
3
β
γ[a + (1 − γ)β]dγ = a + ,
3
and therefore,
β
α
M (A) = a − , a + ,
3
3
and, finally,
Z
1
γ[a − (1 − γ)α + a + (1 − γ)β]dγ = a +
M̄ (A) =
0
β−α
.
6
Specially, when A = (a, α) is a symmetric triangular fuzzy number we get M̄ (A) = a. If A is a symmetric
fuzzy number with peak [q− , q+ ] then the equation
M̄ (A) =
q− + q+
.
2
always holds.
3
Relation to upper and lower probability mean values
We show now an important relationship between the interval-valued expectation E(A) = [E∗ (A), E ∗ (A)]
introduced in [2] and the interval-valued possibilistic mean M (A) = [M∗ (A), M ∗ (A)] for LR-fuzzy numbers with strictly decreasing shape functions.
An LR-type fuzzy number A ∈ F can be described with the following membership function [1]
 q− − u



if q− − α ≤ u ≤ q−
L


α



 1
if u ∈ [q− , q+ ]
A(u) =
u − q+



if q+ ≤ u ≤ q+ + β
R


β




0
otherwise
where [q− , q+ ] is the peak of 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 mappings. We shall use the
notation A = (q− , q+ , α, β)LR . The closure of the support of A is exactly [q− − α, q+ + β].
If L and R are strictly decreasing functions then we can easily compute the γ-level sets of A. That is,
[A]γ = [q− − αL−1 (γ), q+ + βR−1 (γ)], γ ∈ [0, 1].
Following [2] (page 293) the lower and upper probability mean values of A ∈ F are computed by
Z 1
Z 1
∗
E∗ (A) = q− − α
L(u)du, E (A) = q+ + β
R(u)du,
0
0
5
(note that the support of A is bounded) and the lower and upper possibilistic mean values are obtained as
Z 1
Z 1
−1
M∗ (A) = 2
γ(q− − αL (γ))dγ = q− − α
γL−1 (γ)dγ
0
M ∗ (A) = 2
Z
0
1
γ(q+ + βR−1 (γ))dγ = q+ + β
1
Z
γR−1 (γ)dγ
0
0
Therefore, we can state the following lemma.
Lemma 3.1 If A ∈ F is a fuzzy number of LR-type with strictly decreasing (and continuous) shape functions then its interval-valued possibilistic mean is a proper subset of its interval-valued probabilistic mean,
M (A) ⊂ E(A).
Proof 3 From the relationships
Z 1
Z 1
L−1 (γ)dγ
L(u)du =
0
Z
1
Z
R−1 (γ)dγ.
0
0
0
1
R(u)du =
and
we get
1
Z
γL−1 (γ)dγ <
0
Z
1
Z
L(u)du and
1
γR−1 (γ)dγ <
0
0
Z
1
R(u)du.
0
Which ends the proof.
Lemma 3.1 reflects on the fact that points with small membership degrees are considered to be less important in the definition of lower and upper possibilistic mean values than in the definition of probabilistic
ones.
In the limit case, when A = (q− , q+ , 0, 0), the possibilistic and probablistic mean values are equal, and
the equality
E(A) = M (A) = [q− , q+ ]
holds.
Example 3.1 Let A = (a, α, β) be a triangular fuzzy number with center a, left-width α > 0 and rightwidth β > 0 then
α
β
α
β
M (A) = a − , a +
⊂ E(A) = a − , a +
3
3
2
2
and
β−α
E∗ (A) + E ∗ (A)
β−α
6= Ē(A) =
=a+
.
6
2
4
However, when A is a symmetric fuzzy number then the equation
M̄ (A) = a +
M̄ (A) = Ē(A).
always holds.
6
4
Variance of fuzzy numbers
We introduce the (possibilistic) variance of A ∈ F as
2 !
a1 (γ) + a2 (γ)
Var(A) =
Pos[A ≤ a1 (γ)]
− a1 (γ)
dγ
2
0
2 !
Z 1
a1 (γ) + a2 (γ)
Pos[A ≥ a2 (γ)]
− a2 (γ)
+
dγ
2
0
2 2 !
Z 1
a1 (γ) + a2 (γ)
a1 (γ) + a2 (γ)
γ
− a1 (γ) +
− a2 (γ)
dγ
=
2
2
0
Z
2
1 1
=
γ a2 (γ) − a1 (γ) dγ.
2 0
Z
1
The variance of A is defined as the expected value of the squared deviations between the arithmetic
mean and the endpoints of its level sets, i.e. the lower possibility-weighted average of the squared distance
between the left-hand endpoint and the arithmetic mean of the endpoints of its level sets plus the upper
possibility-weighted average of the squared distance between the right-hand endpoint and the arithmetic
mean of the endpoints their of its level sets.
Remark 4.1 From a probabilistic viewpoint, the possibilistic mean and variance of a fuzzy number A can
be viewed as the expected values, when the level gamma is treated as a random variable having a beta
distribution Beta(2,1) (for which the higher the level, the higher its weight or density), of the conditional
mean value and variance given gamma, respectively, of the random variable taking on values a1 (γ) and
a2 (γ) with probabilities 0.5. This could be an additional argument for the ”coherence” between the
definitions of E(A) and V ar(A) introduced in this paper.
The standard deviation of A is defined by
σA =
p
Var(A).
For example, if A = (a, α, β) is a triangular fuzzy number then
Var(A) =
1
2
Z
0
1
2
(α + β)2
γ a + β(1 − γ) − (a − α(1 − γ)) dγ =
.
24
especially, if A = (a, α) is a symmetric triangular fuzzy number then
α2
Var(A) = .
6
In the limit case, when A = (a, 0) is a fuzzy point (i.e. a1 (γ) = a2 (γ) = a = const., ∀γ) we get
Var(A) = 0.
If A is the characteristic function of the crisp interval [a, b] then
Z
1 1
b−a 2
2
Var(A) =
γ(b − a) dγ =
2 0
2
that is,
σA =
b−a
,
2
M̄ (A) =
7
a+b
.
2
In probability theory, the corresponding result is: if the two possible outcomes of a probabilistic variable have equal probabilities then the expected value is their arithmetic mean and the standard deviation is
the half of their distance.
We show now that the variance of a fuzzy number is invariant to shifting. Let A ∈ F and let θ be a
real number. If A is shifted by value θ then we get a fuzzy number, denoted by B, satisfying the property
B(x) = A(x − θ) for all x ∈ R. Then from the relationship
[B]γ = [a1 (γ) + θ, a2 (γ) + θ]
we find
1
Var(B) =
2
Z
0
1
2
1
γ (a2 (γ) + θ) − (a1 (γ) + θ) dγ =
2
Z
1
2
γ a2 (γ) − a1 (γ) dγ = Var(A).
0
The covariance between fuzzy numbers A and B is defined as
Z
1 1
Cov(A, B) =
γ(a2 (γ) − a1 (γ))(b2 (γ) − b1 (γ))dγ.
2 0
Taking into consideration that supp(A) = [a1 (0), a2 (0)] and supp(B) = [b1 (0), b2 (0)] we find that the
covariance measures how much the products of γ-levels sets of two fuzzy numbers A and B are close to
the product of γ-levels sets of the universal fuzzy sets in supp(A) and in supp(B). From
(a2 (0) − a1 (0))(b2 (0) − b1 (0)) > (a2 (γ) − a1 (γ))(b2 (γ) − b1 (γ)) > 0,
for any γ ∈ (0, 1) it follows that
(a2 (0) − a1 (0))(b2 (0) − b1 (0))
> Cov(A, B) > 0
4
holds for any pair of (continuous) fuzzy numbers A and B in supp(A) and in supp(B), respectively. In
the limit case, when A is constant (i.e. a1 (γ) = a2 (γ) = a = const., ∀γ) we find that Cov(A, B) = 0.
Let A = (a, α) and B = (b, β) be symmetric trianglar fuzzy numbers. Then
Cov(A, B) =
αβ
.
6
The following theorem shows that the variance of linear combinations of fuzzy numbers can easily be
computed (in the same manner as in probability theory).
Theorem 4.1 Let λ, µ ∈ R and let A and B be fuzzy numbers. Then
Var(λA + µB) = λ2 Var(A) + µ2 Var(B) + 2|λµ|Cov(A, B)
where the addition and multiplication by a scalar of fuzzy numbers is defined by the sup-min extension
principle.
Proof 4 Suppose λ < 0 and µ < 0. Then we find
Z
2
1 1
Var(λA + µB) =
γ λa1 (γ) + µb1 (γ) − λa2 (γ) − µb2 (γ) dγ =
2 0
Z
2
1 1
γ λ(a1 (γ) − a2 (γ)) + µ(b1 (γ) − b2 (γ)) dγ =
2 0
8
Z
1 1
γ(a1 (γ) − a2 (γ)) dγ + µ ×
γ(b1 (γ) − b2 (γ))2 dγ+
2
0
0
Z 1
1
2λµ ×
γ(a1 (γ) − a2 (γ))(b1 (γ) − b2 (γ))dγ =
2 0
Z
1 1
2
2
λ Var(A) + µ Var(B) + 2λµ ×
γ(a2 (γ) − a1 (γ))(b2 (γ) − b1 (γ))dγ =
2 0
1
λ ×
2
2
Z
1
2
2
λ2 Var(A) + µ2 Var(B) + 2λµCov(A, B) =
λ2 Var(A) + µ2 Var(B) + 2|λµ|Cov(A, B).
Similar reasoning holds for the case λ > 0 and µ > 0. Suppose now that λ > 0 and µ < 0. Then we get
Var(λA + µB) =
1
2
Z
1
2
γ λa1 (γ) + µb2 (γ) − λa2 (γ) − µb1 (γ) dγ =
0
Z
2
1 1
γ λ(a1 (γ) − a2 (γ)) + µ(b2 (γ) − b1 (γ) dγ =
2 0
Z 1
Z
1 1
1
2
2
2
γ(a1 (γ) − a2 (γ)) dγ + µ ×
γ(b2 (γ) − b1 (γ))2 dγ+
λ ×
2 0
2 0
Z
1 1
2λµ ×
γ(a1 (γ) − a2 (γ))(b2 (γ) − b1 (γ))dγ =
2 0
Z
1 1
2
2
λ Var(A) + µ Var(B) − 2λµ ×
γ(a2 (γ) − a1 (γ))(b2 (γ) − b1 (γ))dγ =
2 0
λ2 Var(A) + µ2 Var(B) + 2|λµ|Cov(A, B).
Which ends the proof.
As a special case of Theorem 4.1 we get Var(λA) = λ2 Var(A) for any λ ∈ R and
Var(A + B) = Var(A) + Var(B) + 2Cov(A, B).
Let A = (a, α) and B = (b, β) be symmetric triangular fuzzy numbers and λ µ be real numbers. Then
Var(λA + µB) = λ2
α2
β2
αβ (|λ|α + |µ|β)2
+ µ2 + 2|λµ|
=
,
6
6
6
6
which coincides with the variance of the symmetric triangular fuzzy number
λA + µB = (λa + µb, |λ|α + |µ|β).
Another important question is the relationship between the subsethood and the variance of fuzzy numbers. One might expect that A ⊂ B (that is A(x) ≤ B(x) for all x ∈ R) should imply the relationship
Var(A) ≤ Var(B) because A is considered a ”stronger restriction” than B.
The following theorem shows that subsethood does entail smaller variance.
Theorem 4.2 Let A, B ∈ F with A ⊂ B. Then Var(A) ≤ Var(B).
9
Proof 5 From A ⊂ B it follows that b1 (γ) ≤ a1 (γ) ≤ a2 (γ) ≤ b2 (γ), for all γ ∈ [0, 1]. That is,
a2 (γ) − a1 (γ) ≤ b2 (γ) − b1 (γ)
for all γ ∈ [0, 1], and therefore,
Z
Z
2
2
1 1
1 1
γ a2 (γ) − a1 (γ) dγ ≤
γ b2 (γ) − b1 (γ) dγ = Var(B).
Var(A) =
2 0
2 0
Which ends the proof.
Remark 4.2 Alternatively, we could also introduce the variance of A ∈ F as
Z 1
Var0 (A) =
γ [M̄ (A) − a1 (γ)]2 + [M̄ (A) − a2 (γ)]2 dγ
0
Z 1
=
γ(a21 (γ) + a22 (γ))dγ − E 2 (A),
0
i.e. the possibility-weighted average of the squared distance between the expected value and the left hand
and right hand endpoints of its level sets; and the covariance as
Z 1
γ [M̄ (A) − a1 (γ)][M̄ (B) − b1 (γ)] + [M̄ (A) − a2 (γ)][M̄ (B) − b2 (γ)] dγ
Cov0 (A, B) =
0
Then the following theorem would hold
Theorem 4.3 Let λ, µ ∈ R such that λµ > 0 and let A and B be fuzzy numbers. Then
Var0 (λA + µB) = λ2 Var0 (A) + µ2 Var0 (B) + 2λµCov0 (A, B)
where the addition and multiplication by a scalar of fuzzy numbers is defined by the sup-min extension
principle.
Proof 6 Suppose λ < 0 and µ < 0. Using the linearity of the expected value we find
Var0 (λA + µB) =
Z 1
γ (E(λA + µB) − λa2 (γ) − µb2 (γ))2 + (E(λA + µB) − λa1 (γ) − µb1 (γ))2 dγ =
0
Z 1
2
γ λM̄ (A) + µM̄ (B) − λa2 (γ) − µb2 (γ) dγ+
0
Z 1
2
γ λM̄ (A) + µM̄ (B) − λa1 (γ) − µb1 (γ) dγ =
0
Z 1
2
γ λ(M̄ (A) − a2 (γ)) + µ(M̄ (B) − b2 (γ)) dγ+
0
Z 1
2
γ λ(M̄ (A) − a1 (γ)) + µ(M̄ (B) − b1 (γ)) dγ =
0
Z 1
2
λ
γ [M̄ (A) − a1 (γ)]2 + [M̄ (A) − a2 (γ)]2 dγ+
0
Z 1
µ2
γ [M̄ (B) − b1 (γ)]2 + [M̄ (B) − b2 (γ)]2 dγ+
0
Z 1
2λµ
γ [M̄ (A) − a1 (γ)][M̄ (B) − b1 (γ)] + [M̄ (A) − a2 (γ)][M̄ (B) − b2 (γ)] dγ =
0
λ2 Var0 (A) + µ2 Var0 (B) + 2λµCov0 (A, B).
Similar reasoning holds for the case λ > 0 and µ > 0. Which ends the proof.
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However, nothing can be said about Var0 (λA + µB) if λµ < 0, and it is not clear if subsethood entails
smaller variance. However, if A = (a, α, β) is a triangular fuzzy number then
Var0 (A) =
α2 + β 2 + αβ
,
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and, therefore, any triangular subset of A will have smaller variance.
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Summary
We have introduced the notation of interval-valued possibilistic mean of fuzzy numbers and investigated
its relationship to interval-valued probabilistic mean. We have proved that the proposed concepts ”behave
properly” (in a similar way as their probabilistic counterparts).
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Acknowledgment
The authors are thankful to Didier Dubois and Maria Angeles Gil for their useful comments and suggestions on the earlier versions of this paper.
References
[1] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications (Academic Press, New
York, 1980).
[2] D. Dubois and H. Prade, The mean value of a fuzzy number, Fuzzy Sets and Systems 24(1987)
279-300.
[3] R. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18(1986) 31-43.
[4] L.A. Zadeh, Fuzzy Sets, Information and Control, 8(1965) 338-353.
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Follow ups
The results of this paper have been improved and/or generalized in 411 works. see: http://uni-obuda.hu/users/fuller.robert/c
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