On possibilistic mean value and
variance of fuzzy numbers
Supported by the E-MS Bullwhip project TEKES 40965/98
Christer Carlsson
Institute for Advanced Management Systems Research,
e-mail:christer.carlsson@abo.fi
Robert Fullér
Institute for Advanced Management Systems Research,
e-mail: rfuller@ra.abo.fi
TUC S
Turku Centre for Computer Science
TUCS Technical Report No 299
August 1999
ISBN 952-12-0510-5
ISSN 1239-1891
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 number, possibility distribution, mean value, variance
TUCS Research Group
Institute for Advanced Management Systems Research
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]
! 1
! 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.
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
! 1
a1 (γ) + a2 (γ)
! 1
γ·
dγ
2
,
(2)
γ(a1 (γ) + a2 (γ))dγ = 0
M̄ (A) :=
! 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.
1
Second, we can rewrite M̄ (A) as
M̄ (A) =
!
1
γ(a1 (γ) + a2 (γ))dγ
! 1
! 1
γa1 (γ)dγ + 2 ·
γa2 (γ)dγ
2·
0
=
0
2
!
0
!

γa
(γ)dγ
2

1 0
0

= 
+


2
1
1
2
2
! 1
! 1

γa
(γ)dγ
γa
(γ)dγ
1
2

1 0
0
.
= 
+
!
!
1
1

2
γdγ
γdγ
1
γa1 (γ)dγ
1
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
M∗ (A) = 2
!
1
γa1 (γ)dγ =
0
=
!
=
!
0
0
!
0
1
γa1 (γ)dγ
! 1
γdγ
0
1
Pos[A ≤ a1 (γ)]a1 (γ)dγ
! 1
Pos[A ≤ a1 (γ)]dγ
0
1
Pos[A ≤ a1 (γ)] × min[A]γ dγ
,
! 1
Pos[A ≤ a1 (γ)]dγ
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 possibilityweighted 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,
2
as
M ∗ (A) = 2
!
1
γa2 (γ)dγ =
0
=
!
=
!
0
!
0
1
γa2 (γ)dγ
! 1
γdγ
0
1
Pos[A ≥ a2 (γ)]a2 (γ)dγ
! 1
Pos[A ≥ a2 (γ)]dγ
0
1
0
Pos[A ≥ a2 (γ)] × max[A]γ dγ
,
! 1
Pos[A ≤ a2 (γ)]dγ
0
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 γ ≥ 0
[λM ∗ (A), λM∗ (A)] if γ < 0
where the addition and multiplication by a scalar of fuzzy numbers is defined by
the sup-min extension principle [4].
3
Proof. Really, from the equation
[A + B]γ = [a1 (γ) + b1 (γ), a2 (γ) + b2 (γ)],
we have
M∗ (A + B) = 2
!
1
γ(a1 (γ) + b1 (γ))dγ
! 1
! 1
γa1 (γ)dγ + 2
γb1 (γ)dγ
=2
0
0
0
= M∗ (A) + M∗ (B),
and
∗
M (A + B) = 2
!
1
γ(a2 (γ) + b2 (γ))dγ
! 1
! 1
γa2 (γ)dγ + 2
γb2 (γ)dγ
=2
0
0
0
= M ∗ (A) + M ∗ (B),
furthermore, from
[λA] = λ[A] = λ[a1 (γ), a2 (γ)] =
γ
γ
for λ ≥ 0 we get
M∗ (λA) = 2
M ∗ (λA) = 2
!
1
(
γ(λa1 (γ))dγ = 2λ
0
!
M∗ (λA) = 2
M ∗ (λA) = 2
!
1
γ(λa2 (γ))dγ = 2λ
γ(λa2 (γ))dγ = 2λ
0
1
!
1
γa1 (γ)dγ = λM∗ (A).
γa2 (γ)dγ = λM ∗ (A).
!
1
γa2 (γ)dγ = λM ∗ (A).
0
1
γ(λa1 (γ))dγ = 2λ
0
Which ends the proof.
!
0
1
!
[λa2 (γ), λa1 (γ)] if γ < 0
0
0
and for γ < 0
[λa1 (γ), λa2 (γ)] if γ ≥ 0
!
0
1
γa1 (γ)dγ = λM∗ (A).
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.
4
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. First we find
M̄ (A + B) =
!
!
1
γ(a1 (γ) + b1 (γ) + a2 (γ) + b2 (γ))dγ =
0
1
!
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
! 1
! 1
γ(λa1 (γ) + λa2 (γ))dγ = λ
γ(a1 (γ) + a2 (γ))dγ = λM̄ (A).
M̄ (λA) =
0
0
and, finally, for γ < 0 we have
! 1
! 1
γ(λa2 (γ) + λa1 (γ))dγ = λ
γ(a1 (γ) + a2 (γ))dγ = λM̄ (A).
M̄ (λA) =
0
0
Which ends the proof.
Example 2.1. Let A = (a, α, β) be a triangular fuzzy number with center a, leftwidth 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],
that is,
M∗ (A) = 2
M ∗ (A) = 2
!
!
0
1
0
and therefore,
α
γ[a − (1 − γ)αdγ = a − ,
3
β
γ[a + (1 − γ)β]dγ = a + ,
3
*
)
β
α
M (A) = a − , a + ,
3
3
and, finally,
M̄ (A) =
1
!
0
1
γ[a − (1 − γ)α + a + (1 − γ)β]dγ = a +
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.
5
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]
 /
0
q− − u



if q− − α ≤ u ≤ q−
L


α



 1
if u ∈ [q− , q+ ]
0
/
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 nonincreasing, 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
! 1
! 1
∗
E∗ (A) = q− − α
L(u)du, E (A) = q+ + β
R(u)du,
0
0
(note that the support of A is bounded) and the lower and upper possibilistic mean
values are obtained as
! 1
! 1
M∗ (A) = 2
γ(q− − αL−1 (γ))dγ = q− − α
γL−1 (γ)dγ
0
M ∗ (A) = 2
!
0
1
γ(q+ + βR−1 (γ))dγ = q+ + β
0
!
1
γR−1 (γ)dγ
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,
E(A) ⊂ M (A).
6
Proof. From the relationships
!
1
L(u)du =
0
!
1
−1
L
(γ)dγ
and
0
!
1
R(u)du =
0
!
1
R−1 (γ)dγ.
0
we get
!
1
−1
γL
(γ)dγ <
0
!
1
L(u)du
and
0
!
1
γR
−1
(γ)dγ <
0
!
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, leftwidth α > 0 and right-width β > 0 then
*
)
*
)
α
α
β
β
⊂ E(A) = a − , a +
M (A) = a − , a +
3
3
2
2
and
M̄ (A) = a +
β−α
E∗ (A) + E ∗ (A)
β−α
,= Ē(A) =
=a+
.
6
2
4
However, when A is a symmetric fuzzy number then the equation
M̄ (A) = Ē(A).
always holds.
7
4
Variance of fuzzy numbers
We introduce the (possibilistic) variance of A ∈ F as
1)
*2 2
! 1
a1 (γ) + a2 (γ)
dγ
Pos[A ≤ a1 (γ)]
− a1 (γ)
Var(A) =
2
0
1)
*2 2
! 1
a1 (γ) + a2 (γ)
+
Pos[A ≥ a2 (γ)]
− a2 (γ)
dγ
2
0
*2
! 1 /)
a1 (γ) + a2 (γ)
γ
=
− a1 (γ)
2
0
)
*2 0
a1 (γ) + a2 (γ)
dγ
+
− a2 (γ)
2
!
42
1 1 3
=
γ a2 (γ) − a1 (γ) dγ.
2 0
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 possibilityweighted average of the squared distance between the right-hand endpoint and the
arithmetic mean of the endpoints their of its level sets.
The standard deviation of A is defined by
5
σA = Var(A).
For example, if A = (a, α, β) is a triangular fuzzy number then
!
42
(α + β)2
1 1 3
γ a + β(1 − γ) − (a − α(1 − γ)) dγ =
.
Var(A) =
2 0
24
especially, if A = (a, α) is a symmetric triangular fuzzy number then
Var(A) =
α2
.
6
If A is the characteristic function of the crisp interval [a, b] then
0
/
!
1 1
b−a 2
2
γ(b − a) dγ =
Var(A) =
2 0
2
that is,
b−a
a+b
, M̄ (A) =
.
2
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.
σA =
8
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
!
42
1 1 3
γ (a2 (γ) + θ) − (a1 (γ) + θ) dγ
Var(B) =
2 0
!
42
1 1 3
=
γ a2 (γ) − a1 (γ) dγ = Var(A).
2 0
The covariance between fuzzy numbers A and B is defined as
!
1 1
Cov(A, B) =
γ(a2 (γ) − a1 (γ))(b2 (γ) − b1 (γ))dγ
2 0
The covariance measures how much the endpoints of the γ-level sets of two fuzzy
numbers move in tandem.
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. Suppose λ < 0 and µ < 0. Then we find
!
42
1 1 3
γ λa1 (γ) + µb1 (γ) − λa2 (γ) − µb2 (γ) dγ =
Var(λA + µB) =
2 0
!
42
1 1 3
γ λ(a1 (γ) − a2 (γ)) + µ(b1 (γ) − b2 (γ)) dγ =
2 0
! 1
!
1
1 1
2
2
2
λ ×
γ(a1 (γ) − a2 (γ)) dγ + µ ×
γ(b1 (γ) − b2 (γ))2 dγ+
2 0
2 0
!
1 1
2λµ ×
γ(a1 (γ) − a2 (γ))(b1 (γ) − b2 (γ))dγ =
2 0
9
1
λ Var(A) + µ Var(B) + 2λµ ×
2
2
2
!
1
0
γ(a2 (γ) − a1 (γ))(b2 (γ) − b1 (γ))dγ =
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 γ > 0. Then we get
1
Var(λA + µB) =
2
!
0
1
3
42
γ λa1 (γ) + µb2 (γ) − λa2 (γ) − µb1 (γ) dγ =
!
42
1 1 3
γ λ(a1 (γ) − a2 (γ)) + µ(b2 (γ) − b1 (γ) dγ =
2 0
! 1
!
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
!
1 1
γ(a1 (γ) − a2 (γ))(b2 (γ) − b1 (γ))dγ =
2λµ ×
2 0
!
1 1
2
2
γ(a2 (γ) − a1 (γ))(b2 (γ) − b1 (γ))dγ =
λ Var(A) + µ Var(B) − 2λµ ×
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
αβ (|λ|α + |µ|β)2
α2
β2
+ 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).
10
Proof. 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,
!
42
1 1 3
γ a2 (γ) − a1 (γ) dγ
2 0
!
42
1 1 3
≤
γ b2 (γ) − b1 (γ) dγ = Var(B).
2 0
Var(A) =
Which ends the proof.
Remark 4.1. Alternatively, we could also introduce the variance of A ∈ F as
%
Var (A) =
!
1
0
=
!
0
1
3
4
γ [M̄ (A) − a1 (γ)]2 + [M̄ (A) − a2 (γ)]2 dγ
γ(a21 (γ) + a22 (γ))dγ − E 2 (A),
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
Cov% (A, B) =
! 1
3
4
γ [M̄ (A) − a1 (γ)][M̄ (B) − b1 (γ)] + [M̄ (A) − a2 (γ)][M̄ (B) − b2 (γ)] dγ
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
Var% (λA + µB) = λ2 Var% (A) + µ2 Var% (B) + 2λµCov% (A, B)
where the addition and multiplication by a scalar of fuzzy numbers is defined by
the sup-min extension principle.
Proof. Suppose λ < 0 and µ < 0. Using the linearity of the expected value we
11
find
Var% (λA + µB) =
1 3
4
γ (E(λA+µB)−λa2 (γ)−µb2 (γ))2 +(E(λA+µB)−λa1 (γ)−µb1 (γ))2 dγ =
0
! 1
3
42
γ λM̄ (A) + µM̄ (B) − λa2 (γ) − µb2 (γ) dγ +
0
! 1
3
42
γ λM̄ (A) + µM̄ (B) − λa1 (γ) − µb1 (γ) dγ =
0
! 1
3
42
γ λ(M̄ (A) − a2 (γ)) + µ(M̄ (B) − b2 (γ)) dγ +
0
! 1
3
42
γ λ(M̄ (A) − a1 (γ)) + µ(M̄ (B) − b1 (γ)) dγ =
0
! 1
3
4
γ [M̄ (A) − a1 (γ)]2 + [M̄ (A) − a2 (γ)]2 dγ +
λ2
0
! 1
4
3
γ [M̄ (B) − b1 (γ)]2 + [M̄ (B) − b2 (γ)]2 dγ +
µ2
0
! 1
3
4
2λµ
γ [M̄ (A)−a1 (γ)][M̄ (B)−b1 (γ)]+[M̄ (A)−a2 (γ)][M̄ (B)−b2 (γ)] dγ =
!
0
λ2 Var% (A) + µ2 Var% (B) + 2λµCov% (A, B).
Similar reasoning holds for the case λ > 0 and µ > 0. Which ends the proof.
However, nothing can be said about Var% (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
α2 + β 2 + αβ
Var% (A) =
,
18
and, therefore, any triangular subset of A will have smaller variance.
5
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).
6
Acknowledgment
The authors are thankful to Didier Dubois for his comments and suggestions on the
earlier versions of this paper.
12
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.
13
Turku Centre for Computer Science
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FIN-20520 Turku
Finland
http://www.tucs.abo.fi
University of Turku
• Department of Mathematical Sciences
Åbo Akademi University
• Department of Computer Science
• Institute for Advanced Management Systems Research
Turku School of Economics and Business Administration
• Institute of Information Systems Science