Additions of Completely
Correlated Fuzzy Numbers
Christer Carlsson
Robert Fullér
Péter Majlender
IAMSR
Åbo Akademi
Department of OR
Eötvös University
TUCS / IAMSR
Åbo Akademi
Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
Abstract
In this paper we shall consider additions of interactive fuzzy
numbers, where the interactivity relation between fuzzy
numbers is defined by their joint possibility distribution.
We will prove that Nguyen’s theorem remains valid in this
environment and present the explicit formulas for the γ-level
sets of the extended sum of two completely correlated fuzzy
numbers.
Furthermore, we will show that the interactive and the noninteractive sums have the same membership function for any
pair of completely positively correlated fuzzy numbers.
FUZZ – IEEE 2004
Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
Introduction
A fuzzy number A is a fuzzy set of the real line ú with a normal,
fuzzy convex and continuous membership function of
bounded support. The family of all fuzzy numbers will be
denoted by ö.
An n-dimensional possibility distribution C is a fuzzy set in ún
with a normal membership function of bounded support. The
γ-level set of C is defined by
if γ > 0 and
[C ]γ = {x ∈ ú n | C ( x) ≥ γ }
[C ]γ = cl{x ∈ ú n | C ( x) > γ }
(the closure of the support of C) if γ = 0.
FUZZ – IEEE 2004
Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
If A1, …, An 0 ö are fuzzy numbers then C is said to be their
joint possibility distribution if the following relationship holds
Ai ( xi ) = max C ( x1 , K , xn ), ∀xi ∈ ú, i = 1, K , n,
x j ∈ú , j ≠ i
and in this case Ai is called the i-th marginal possibility
distribution of C.
Let C be a joint possibility distribution of A1, …, An 0 ö. Then
and
C ( x1 , K , xn ) ≤ min{ A1 ( x1 ),K, An ( xn )}
[C ]γ ⊆ [ A1 ]γ × L × [ An ]γ
hold for all x = (x1, …, xn) 0 ún and γ 0 [0,1].
FUZZ – IEEE 2004
Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
Fuzzy numbers A1, …, An are said to be non-interactive if their
joint possibility distribution C satisfies the relationship
C ( x1 , K , xn ) = min{ A1 ( x1 ),K , An ( xn )}, ∀x = ( x1 , K , xn ) ∈ ú n ,
or equivalently,
[C ]γ = [ A1 ]γ × L × [ An ]γ , ∀γ ∈ [0,1].
FUZZ – IEEE 2004
Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
The extension principle for interactive
fuzzy numbers
Definition. Let C be a joint possibility distribution with marginal
possibility distributions A1, …, An 0 ö, and let f : ún 6 ú be a
continuous function. Then
f C ( A1 , K , An ) ∈ ö
is defined by
f C ( A1 , K, An )( y ) = sup C ( x1 ,K , xn ), y ∈ ú.
y = f ( x1 ,K, xn )
FUZZ – IEEE 2004
Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
Remark. If A1, …, An 0 ö are non-interactive, i.e. their joint
possibility distribution is defined by
C ( x1 ,K , xn ) = min{ A1 ( x1 ),K , An ( xn )},
then we obtain the extension principle introduced by Zadeh in
1965
f C ( A1 , K , An )( y ) = sup min{ A1 ( x1 ),K, An ( xn )}.
y = f ( x1 ,K, xn )
Furthermore, if
C ( x1 ,K , xn ) = T ( A1 ( x1 ),K, An ( xn )),
where T is a triangular norm then we get the t-norm-based
extension principle
f C ( A1 , K , An )( y ) =
sup
y = f ( x1 ,K, xn )
T ( A1 ( x1 ),K , An ( xn )).
FUZZ – IEEE 2004
Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
The following lemma can be viewed as a generalization of
Nguyen’s theorem.
Lemma. Let A1, …, An 0 ö be fuzzy numbers, let C be their joint
possibility distribution, and let f : ún 6 ú be a continuous
function. Then
[ f C ( A1 , K, An )]γ = f ([C ]γ )
= { f ( x) ∈ ú | x ∈ [C ]γ }
for all γ 0 [0,1].
FUZZ – IEEE 2004
Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
Additions of interactive fuzzy numbers
In the following we shall analyze some properties of the addition
operator on completely correlated fuzzy numbers, where the
interactivity relation between the fuzzy numbers is represented
by their joint possibility distribution.
Let C be a joint possibility distribution in ú2 with marginal
possibility distributions A, B 0 ö, and let f (x1, x2) = x1 + x2 be
the addition operator in ú2. Let us introduce the notation
A + C B = f C ( A, B).
Furthermore, if A and B are non-interactive, i.e. for their joint
possibility distribution C = A H B, then let us use the notation
A + B = f A× B ( A, B) = f ( A, B).
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Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
Definition. Fuzzy numbers A, B 0 ö are said to be completely
correlated if there exist q, r 0 ú, q … 0 such that their joint
possibility distribution satisfies the following relationship
C ( x1 , x2 ) = A( x1 ) χ {qu1 + r =u2 } ( x1 , x2 ) = B( x2 ) χ {qu1 + r =u2 } ( x1 , x2 ),
where
1 if qx1 + r = x2 ,
χ{qu1 + r =u2 } ( x1 , x2 ) = 
otherwise
0
stands for the characteristic function of the line
{( x1 , x2 ) ∈ ú 2 | qx1 + r = x2 }.
Furthermore, if q is positive then we say that A and B are
completely positively correlated, and if q is negative then we
say that A and B are completely negatively correlated.
FUZZ – IEEE 2004
Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
In these cases we have
[C ]γ = {( x, qx + r ) ∈ ú 2 | a1 (γ ) ≤ x ≤ a2 (γ )},
where [A]γ = [a1(γ), a2(γ)] and [B]γ = q[A]γ + r, œγ 0 [0,1], and
B( x) = A(( x − r ) q ), ∀x ∈ ú.
FUZZ – IEEE 2004
Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
Now let us consider the extended addition of two completely
correlated fuzzy numbers
( A + C B )( y ) = sup C ( x1 , x2 ) = sup A( x1 ) χ {qu1 + r =u2 } ( x1 , x2 ).
y = x1 + x2
y = x1 + x2
Then we find that
[ A + C B ]γ = (q + 1)[ A]γ + r , ∀γ ∈ [0,1].
In particular, if A and B are completely negatively correlated
fuzzy numbers with q = !1, i.e.
B( x) = A(r − x), ∀x ∈ ú,
then
[ A + C B ]γ = 0 [ A]γ + r = r , ∀γ ∈ [0,1],
that is, A +C B is a crisp number, and
A + C B = {r}.
FUZZ – IEEE 2004
Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
Remark. Let A, B 0 ö be two completely negatively correlated
fuzzy numbers with q = !1 and r = 0, i.e.
B( x) = A(− x), ∀x ∈ ú.
Then, it is clear that the interactive sum A +C B is crisp zero
A + C B = {0}.
On the other hand, we have for a γ-level set of the noninteractive sum of A and B
[ A + B ]γ = [ A + A×B B]γ = [a1 (γ ) − a2 (γ ), a2 (γ ) − a1 (γ )],
where [A]γ = [a1(γ), a2(γ)] and [B]γ = ![A]γ = [!a2(γ), !a1(γ)],
γ 0 [0,1]. Hence, we find that in this case
{0} = A + C B ≠ A + B = A − A.
FUZZ – IEEE 2004
Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
Note. If A, B 0 ö are two completely negatively correlated fuzzy
numbers with q = !1 then any γ-level set of the joint
possibility distribution C is included by a certain level set of
the addition operator. Namely,
[C ]γ ⊆ {( x1 , x2 ) ∈ ú 2 | x1 + x2 = r}, ∀γ ∈ [0,1].
FUZZ – IEEE 2004
Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
On the other hand, if q … !1 then
[ A + C B ]γ = (q + 1)[ A]γ + r ≠ const., ∀γ ∈ [0,1],
and in this case the set
[C ]γ ∩ {( x1 , x2 ) ∈ ú 2 | x1 + x2 = y}
at most consists of a single point for any y 0 ú and γ 0 [0,1].
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Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
Let A, B 0 ö be two completely positively correlated fuzzy
numbers, i.e.
B( x) = A(( x − r ) q ), ∀x ∈ ú
holds for some q, r 0 ú, q > 0. Then we find
[ A + C B ]γ = [ A + B]γ , ∀γ ∈ [0,1],
that is,
A + C B = A + B.
Thus, the membership function of the interactive sum of two
completely positively correlated fuzzy numbers is always
equal to the membership function of their non-interactive sum.
FUZZ – IEEE 2004
Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
FUZZ – IEEE 2004
Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
However, if A, B 0 ö are two completely negatively correlated
fuzzy numbers, i.e.
B( x) = A(( x − r ) q ), ∀x ∈ ú
for some q, r 0 ú, q < 0, then we find
[ A + C B]γ ⊂ [ A + B]γ , ∀γ ∈ [0,1],
that is,
A + C B ⊂ A + B.
Thus, (in the sense of subsethood of fuzzy sets) the interactive
sum of two completely negatively correlated fuzzy numbers is
always a proper subset of their non-interactive sum.
FUZZ – IEEE 2004
Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
FUZZ – IEEE 2004
Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
Subtractions of interactive fuzzy numbers
In the following we shall review some properties of the
subtraction operator on completely correlated fuzzy numbers.
Let C be a joint possibility distribution in ú2 with marginal
possibility distributions A, B 0 ö, and let
g ( x1 , x2 ) = f ( x1 ,− x2 ) = x1 − x2
be the subtraction operator in ú2. Let us introduce the
notations
A − C B = g C ( A, B).
and
A − B = g A×B ( A, B) = g ( A, B ).
FUZZ – IEEE 2004
Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
Let A, B 0 ö be two completely correlated fuzzy numbers, and
let us consider their extended subtraction
( A − C B)( y ) = sup C ( x1 , x2 ) = sup A( x1 ) χ {qu1 + r =u2 } ( x1 , x2 ).
y = x1 − x2
y = x1 − x2
Then we find for a γ-level set of A !C B
[ A − C B]γ = (−q + 1)[ A]γ − r , ∀γ ∈ [0,1].
In particular, if A and B are completely positively correlated
fuzzy numbers with q = 1, i.e.
B( x) = A( x − r ), ∀x ∈ ú,
then
[ A − C B ]γ = 0 [ A]γ − r = − r , ∀γ ∈ [0,1],
that is, A !C B is a crisp number, and
A − C B = {− r}.
FUZZ – IEEE 2004
Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
Remark. Let A, B 0 ö be two completely positively correlated
fuzzy numbers with q = 1 and r = 0, i.e.
B( x) = A( x), ∀x ∈ ú.
Then, the interactive difference A !C B is crisp zero
A − C B = {0}.
On the other hand, we have for a γ-level set of the noninteractive difference of A and B
[ A − B]γ = [ A − A× B B]γ = [a1 (γ ) − a2 (γ ), a2 (γ ) − a1 (γ )],
where [A]γ = [B]γ = [a1(γ), a2(γ)], γ 0 [0,1]. Hence, we find that
in this case
{0} = A − C B ≠ A − B = A − A.
FUZZ – IEEE 2004
Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
Let A, B 0 ö be two completely negatively correlated fuzzy
numbers, i.e.
B( x) = A(( x − r ) q ), ∀x ∈ ú
holds for some q, r 0 ú, q < 0. Then we find
[ A − C B]γ = [ A − B ]γ , ∀γ ∈ [0,1],
that is,
A − C B = A − B.
Thus, the membership function of the interactive difference of
two completely negatively correlated fuzzy numbers is always
equal to the membership function of their non-interactive
difference.
FUZZ – IEEE 2004
Carlsson, Fullér and Majlender
Additions of Completely Correlated Fuzzy Numbers
However, if A, B 0 ö are two completely positively correlated
fuzzy numbers, i.e.
B( x) = A(( x − r ) q ), ∀x ∈ ú
for some q, r 0 ú, q > 0, then we find
[ A − C B ]γ ⊂ [ A − B]γ , ∀γ ∈ [0,1],
that is,
A − C B ⊂ A − B.
Thus, the interactive difference of two completely positively
correlated fuzzy numbers is always a proper subset of their
non-interactive difference.
FUZZ – IEEE 2004