WP-EMS # 07/01
“On Fuzzy Arithmetic Operations: Some Properties
and Distributive Approximations"
• Luciano Stefanini (U. Urbino)
• Maria Letizia Guerra (U. Bologna & U. Urbino)
WP-EMS # 2007/03
On Fuzzy Arithmetic Operations: Some Properties and Distributive
Approximations
Luciano Stefanini
Faculty of Economics, University of Urbino "Carlo Bo", Italy
e-mail: lucste@uniurb.it
Maria Letizia Guerra
Department of Mathematics for Social and
Economic Sciences, University of Bologna
and Faculty of Economics, University of Urbino "Carlo Bo", Italy
e-mail: letizia@uniurb.it
Abstract
We analyze a decomposition of the fuzzy numbers (or intervals) which seems to be
of interest in the study of some properties of fuzzy arithmetic operations and, in particular, in the analysis of fuzziness, of shape-preservation (symmetry) and distributivity of
multiplication and division.
By the use of the same decomposition, we suggest an approximation of multiplication
and division to reduce the overestimation effect and/or to obtain total-distributivity of
multiplication and left-distributivity of division.
Finally, we compare the proposed approximation with the results of standard (α−cuts
based) fuzzy mathematics and with other new definitions of fuzzy arithmetic operations
that recently appeared in the literature.
1. Introduction and Basic Results
The scientific literature on fuzzy arithmetic operations is rich of several approaches to
define fuzzy operations having many desired properties that are not always present in the
classical extension principle approach (see [14]) or its approximations (see [13]): shape
preservation (e.g. [3], [4], [13]), reduction of the overestimation effect (e.g. [6], [7]),
requisite constraints (e.g. [9], [8]), distributivity of multiplication and division (e.g. [1],
[10], [11], [12]). These problems are essentially approached by joining representations of
fuzzy quantities and fuzzy operations (e.g. [2], [5], [6], [9]).
In this paper, we introduce a decomposition of the fuzzy numbers (or intervals) into
three additive components that isolate a "crisp part", a "symmetric fuzzy part" and a
"profile of symmetry" each capturing precise properties of fuzzy quantities. The decomposition is used to analyze some properties of fuzzy operations and, particularly, distributivity of multiplication (and division) and symmetry of the results of fuzzy operations.
The decomposition is also used to suggest some approximations of fuzzy operations
that reduce the range of fuzziness (with respect to the classical exact operations) and
assure distributivity.
1.1. Fuzzy Intervals and Numbers. As a definition of a fuzzy number or interval,
we adopt the so called a − cut setting:
Definition 1. A fuzzy number (or interval) u is any pair (u− , u+ ) of functions u± :
[0, 1] −→ R satisfying the following conditions:
(i) u− : α −→ u−
α ∈ R is a bounded monotonic increasing (non decreasing) function
1
On Fuzzy Arithmetic Operations: Some Properties and Distributive Approximations2
∀α ∈ [0, 1] ;
(ii) u+ : α −→ u+
α ∈ R is a bounded monotonic decreasing (non increasing) function
∀α ∈ [0, 1] ;
+
(iii) u−
α ≤ uα ∀α ∈ [0, 1] .
+
We denote the (α = 1) − cut [u−
u− , u
b+ ]; if u
b− < u
b+ we have a fuzzy
1 < u1 ] by [b
−
+
interval and if u
b =u
b we have a fuzzy number.
+
We use the notation [u]α = [u−
α , uα ] to denote explicitly the α − cuts of u.
−
+
We will also refer to u and u as the lower and upper branches on u (corresponding
to left and right branches in the membership function), respectively.
We denote by FI the set of fuzzy intervals and by F ⊂ FI the set of fuzzy numbers.
If u−
b− and u+
b+ , ∀α we have a crisp interval or a crisp number; we denote
α = u
α = u
b
b
by FI and by F the corresponding sets.
If u
b− = u
b+ = 0 we obtain a 0−fuzzy number and denote the corresponding set by
F0 .
+
If u−
b+ + u
b− , ∀α then the fuzzy interval is called symmetric; we denote by
α + uα = u
SI and by S the sets of symmetric fuzzy intervals and numbers; S0 = S ∩ F0 will be the
set of symmetric 0−fuzzy numbers.
+
We say that u is positive if u−
α > 0 , ∀α ∈ [0, 1] and that u is negative if uα < 0 ,
∀α ∈ [0, 1]; the sets of positive and negative fuzzy numbers are denoted by F+ and F−
respectively and their symmetric subsets are denoted by S+ and S− .
For the standard triangular fuzzy numbers and trapezoidal fuzzy intervals we will use
the notations ha, b, ci with a ≤ b ≤ c and ha, b, c, di with a ≤ b ≤ c ≤ d; their α-cuts are
[ha, b, ci]α = [a + α(b − a), c − α(c − b)] and [ha, b, c, di]α = [a + α(b − a), d − α(d − c)]
respectively.
1.2. Standard fuzzy arithmetic operations. If u = (u− , u+ ) and v = (v − , v + ) are
two given fuzzy intervals, the standard arithmetic operations are defined as follows:
Definition 2. (Standard addition, scalar multiplication, subtraction)
For α ∈ [0, 1] :
£
¤
−
+
+
[u + v]α = u−
α + vα , uα + vα ,
£
©
ª
© −
ª¤
+
+
[ku]α = min ku−
, k ∈ R;
α , kuα , max kuα , kuα
in particular, if k = −1, [−u]α =
(1)
(2)
−
[−u+
α , −uα ] ,
£
¤
+
+
−
, α ∈ [0, 1] .
[u − v]α = u−
α − vα , uα − vα
Definition 3. (Standard multiplication and division)
+
For α ∈ [0, 1] , [uv]α = [(uv)−
α , (uv)α ] with
©
ª
−
− +
+ −
+ +
= min u−
(uv)−
α vα , uα vα , uα vα , uα vα
α
©
ª
+
−
− +
+ −
+ +
(uv)α = max u−
α vα , uα vα , uα vα , uα vα .
¡ ¢− ¡ ¢+
¤
£
if 0 ∈
/ v0− , v0+ , [ uv ]α = [ uv α , uv α ] with
½ − − + +¾
³ u ´−
uα uα uα uα
= min
,
,
,
v α
vα− vα+ vα− vα+
½
¾
³ u ´+
−
+
+
u−
α uα uα uα
= max
,
,
,
.
v α
vα− vα+ vα− vα+
(3)
(4)
(5)
On Fuzzy Arithmetic Operations: Some Properties and Distributive Approximations3
It is also well known that, in general, distributivity of multiplication and left-distributivity
of division are not valid, except for special cases (see [11] or [3]).
2. A decomposition of fuzzy intervals
+
Consider a generic fuzzy interval u in terms of its α − cuts [u]α = [u−
α , uα ] , α ∈ [0, 1] and
denote the (α = 1) − cut of u by
£ − +¤
u
b= u
b ,u
b
(6)
so that u
b ∈ FbI ; u
b is a crisp number (and u is a fuzzy number) if and only if
b+ = u
b.
u
b− = u
The basic property of the α − cuts is that they are compact intervals and that
α0 ≤ α00 =⇒ [u]α00 ⊆ [u]α0 .
(7)
In particular, it holds:
u
b ⊆ [u]α , ∀α ∈ [0, 1] .
The Hukuhara difference of two compact intervals A = [a1 , a2 ] and B = [b1 , b2 ] is defined
by
A −h B = C ⇐⇒ A = B + C
where B + C = {b + c | b ∈ B, c ∈ C} is the usual Minkowski addition. If B ⊆ A
then the Hukuhara difference is well defined and given by the compact interval C =
[a1 − b1 , a2 − b2 ] , (note that 0 ∈ C).
Consider, for α ∈ [0, 1] , the compact intervals:
£
¤
o
[u]α = uα −h u
b = u−
b− , u+
b+ ;
α −u
α −u
it is immediate to verify that the family of sets
o
no
[u]α | α ∈ [0, 1]
o
o
o
+
defines the α − cuts [u−
α , uα ] of a fuzzy number u ∈ F0 .
16,0
Hukuhara-lower
Hukuhara-upper
12,0
u-lower
u-upper
14,0
10,0
12,0
8,0
10,0
6,0
8,0
4,0
6,0
2,0
4,0
-
2,0
0,0
0,0
-2,0
0,2
0,4
0,6
0,8
1,0
-2,0
0,2
0,4
0,6
0,8
1,0
-4,0
General fuzzy interval u = (u− , u+ ) Hukuhara difference uα −h u
b
On Fuzzy Arithmetic Operations: Some Properties and Distributive Approximations4
In order to capture the asymmetry of u we define the following profile function of
u:
u
eα =
o
u−
α
u
e : [0, 1] −→ R
+
2
o
u+
α
α −→ u
eα
=
It is immediate to verify that:
+
u−
u
b− + u
b+
α + uα
−
2
2
(8)
u ∈ S ⇐⇒e
uα = 0 ∀α ∈ [0, 1]
Observe that for u ∈ FI ,
u+
b+ − u
eα
α −u
eα − u−
u
b− + u
α
−
b−
u+
u
b+ − u
α − uα
−
2
2
−
+
−
u
−
u
b−
u+
u
b
α
α
−
2
2
=
=
As the last step of our decomposition, we define the S0 −component of u as the fuzzy
number u ∈ S0 having α − cuts
[−uα , uα ] ,
with uα ≥ 0 ∀α and uα decreasing, obtained by
uα =
−
b−
u+
u
b+ − u
α − uα
−
.
2
2
u_bar-upper
8,0
(9)
u_hat-low + u-tilde
9,0
u_bar-lower
u_hat-up + u-tilde
u-tilde
8,0
6,0
7,0
4,0
6,0
2,0
5,0
0,0
0,2
0,4
0,6
0,8
1,0
4,0
-2,0
3,0
-4,0
2,0
1,0
-6,0
0,0
-8,0
Fuzzy and profile components
So, ∀α ∈ [0, 1] :
and we write (10) as
½
0,2
0,4
0,6
0,8
1,0
Mixed component (b
u, u
e, 0)
b+ + u
eα + uα
u+
α =u
−
uα = u
b− + u
eα − uα
u = (b
u, u
e, u) or
£ − +¤
[u]α = u
eα + [−uα , uα ]
b ,u
b +u
(10)
(11)
On Fuzzy Arithmetic Operations: Some Properties and Distributive Approximations5
obtaining a decomposition of the form
(f uzzy) = (crisp) + (prof ile) + (0 − symmetric f uzzy).
By use of decomposition (11) we can characterize the fuzzy intervals or numbers as follows:
1. u ∈ F0 ⇐⇒b
u− = u
b+ = 0
2. u ∈ F− ⇐⇒b
u+ + u
eα ≤ −uα , ∀α
3. u ∈ F+ ⇐⇒b
u− + u
eα ≥ uα , ∀α
4. u ∈ S ⇐⇒e
uα = 0 , ∀α
5. u ∈ S0 ⇐⇒b
u− = u
b+ = 0, u
eα = 0 , ∀α
6. u ∈ S− ⇐⇒b
u− ≤ −uα , u
eα = 0 , ∀α
7. u ∈ S+ ⇐⇒b
u+ ≥ uα , u
eα = 0 , ∀α
Note also that u
e1 = 0 and −uα ≤ u
eα ≤ uα ∀α ∈ [0, 1] and that the profile function
induces a linear operator; in fact, if u = (b
u, u
e, u) and v = (b
v , ve, v) then u + v =
f = k u.
]
u, ke
u, |k| u) and u + v = u
e + ve, ku
e
(b
u + vb, u
e + ve, u + v) and, ∀k ∈ R, ku = (kb
Let’s denote by P the set of all possible profile functions:
Theorem 4. Any
u
e : [0, 1] −→ R
with u
e1 = 0.
u = (b
u, u
e, u) ∈ FbI × P × S0
represents a fuzzy interval if and only if the following condition is satisfied by the pair
(e
u, u) ∈ P × S0 :
α0 < α00 =⇒ |e
uα00 − u
eα0 | ≤ uα0 − uα00
(12)
Proof. If u is a fuzzy interval with u
b, u
e and u defined by (6), (8) and (9) respectively,
then (12) is immediate. Suppose now that (12) is valid, then for α0 = α < 1 and α00 = 1
we have
|e
uα | ≤ uα
and necessary condition (7) is satisfied. By definition (10) we then have:
+
u−
α ≤ uα
∀α( as uα ≥ 0).
For α0 < α00 the following is true
and
u+
b+ + u
eα0 + uα0 ≥ u
b+ + u
eα00 + uα00 = u+
α0 = u
α00
u−
b− + u
eα0 − uα0 ≤ u
b− + u
eα00 − uα00 = u−
α0 = u
α00
−
so that u+
α is decreasing (not increasing) and uα is increasing (not decreasing). It follows
that u is a proper fuzzy interval.
On Fuzzy Arithmetic Operations: Some Properties and Distributive Approximations6
Definition 5. (valid pair (e
u, u))
We say that a pair (e
u, u) ∈ P × S0 is a valid pair if it satisfies condition (12), i.e. if
u
e + u ∈ F0 is a fuzzy number.
bI × P × S0
In terms of the above definition, we can say that any u = (b
u, u
e, u) ∈ F
represents a fuzzy interval if and only if (e
u, u) is a valid pair. On the other hand, valid
pairs of P × S0 are the elements of F0 , the 0-fuzzy numbers.
Note that if u
e and u are differentiable functions with respect to α ∈ ]0, 1[ , then
condition (12) can be stated in terms of first derivatives as |e
u0α | ≤ −u0α
∀α ∈ ]0, 1[ .
Furthermore, as u
e1 = u1 = 0 and the absolute local variations of u
e are not greater then
local variations of u, it follows that if (e
u, u) ∈ P × S0 is a valid pair then also |e
uα | ≤ uα ,
∀α ∈ [0, 1].
bI , u
Using decomposition u = (b
u, u
e, u) with u
b∈F
e ∈ P and u ∈ S0 and with (e
u, u) a
valid pair, the following possibilities are of interest:
(b
u, 0, 0) is crisp;
(0, 0, u) is in S0 ;
(b
u, 0, u) is in SI ;
(0, u
e, u) is in F0 ;
(b
u, u
e, u) is in FI ;
(0, u
e, 0) is in P.
Also a configuration (b
u, u
e, 0) with u
b = [b
u− , u
b+ ] may be of interest as it represents
an interval of fixed length but of varying position, depending on the profile function u
e:
at different degree of possibility α, the position of the fixed length interval changes by
following profile u
e.
If ϕu : R −→ [0, 1] is the membership function of fuzzy interval u, of the form

0
for x£≤ u−

0

¤

l

b−
 u (x) for x ∈ u−
0 ,u
1
for x ∈ £[b
u− , u
b+ ¤]
ϕu (x) =

r
+

b , u+
u (x) for x ∈ u

0


0
for x ≥ u+
0
with ul (x) increasing and ur (x) decreasing, we have that
½
l
−
ϕu (u−
α ) = α = u (uα )
∀α ∈ [0, 1] :
+
r
ϕu (uα ) = α = u (u+
α)
In terms of decomposition (10) we can write the following relations, ∀α ∈ ]0, 1[ :
½ l
u (x) = α ⇐⇒ x = u
b− + u
eα − uα
ur (x) = α ⇐⇒ x = u
b+ + u
eα + uα
The fuzziness of u is essentially contained in component u ∈ S0 , while the profile function
u
e is related to the asymmetry of u with respect to crisp component u
b.
Starting with one or more valid pairs, it is not difficult to construct other valid pairs.
For example, if (e
u, u) is given we can consider any function F : P × S0 → P × S0 such
e
u, u), F (e
u, u)) is a valid pair with Fe : P × S0 → P and F : P × S0 → S0 .
that F (e
u, u) = (F (e
A particular case is related to the extension of a given function f : R → R to a 0-fuzzy
number u = (0, u
e, u) ∈ F0 for which
fg
(u) =
f (u) =
min{f (x)|x ∈ [u]α } + max{f (x)|x ∈ [u]α }
− f (0)
2
max{f (x)|x ∈ [u]α } − min{f (x)|x ∈ [u]α }
− f (0).
2
On Fuzzy Arithmetic Operations: Some Properties and Distributive Approximations7
Similarly, starting with n valid pairs (e
ui , ui ), i = 1, 2, ..., n, and setting u
e = (e
u1 , ..., u
en ) ∈
1
n
n
n
n
n
P , u = (u , ..., u ) ∈ (S0 ) we can construct two functions Fe : P × (S0 ) → P and
F : Pn ×(S0 )n → S0 such that (Fe(e
u, u), F (e
u, u)) is a valid pair.
A useful result is given by the following
v , v) ∈ P × S0 be two valid pairs.
Proposition 6. Let (e
u, u) , (e
Then, ∀a, b, c, d, e ∈ R also
¢
¡
v , |a| u + |b| v + (|c| + |d| + |e|)uv
ae
u + be
v + ce
uve + de
uv + eue
is a valid pair.
Proof.
Consider α0 < α00 ∈ [0, 1]; we have the following inequalities (remember that
uα0 ≥ uα00 ≥ 0 and v α0 ≥ v α00 ≥ 0):
eα0 ) + b(e
vα00 − veα0 )+
|a(e
uα00 − u
+c(e
uα00 veα00 − u
eα0 veα0 )+
+d(e
uα00 v α00 − u
eα0 vα0 )+
+e(uα00 veα00 − uα0 veα0 )| ≤
≤ |a||e
uα00 − u
eα0 | + |b||e
vα00 − veα0 |+
+|c| [|e
uα00 ||e
vα00 − veα0 | + |e
vα0 ||uα00 − uα0 |] +
+|d| [|e
uα00 ||v α00 − v α0 | + v α0 |uα00 − uα0 |] +
+|e| [uα00 |e
vα00 − veα0 | + |e
vα0 ||uα00 − uα0 |] ≤
0 − uα00 ) + |b|(v α0 − v α00 )+
≤ |a|(u
α
¸
·
+|c| uα00 ( v α0 −v α00 ) + v α0 (uα0 − uα00 ) +
|{z}
|{z} ¸
·
+|d| uα00 ( v α0 −v α00 ) + v α0 (uα0 − uα00 ) +
|{z}
|{z} ¸
·
+|e| uα00 ( v α0 −v α00 ) + v α0 (uα0 − uα00 ) ≤
|{z}
|{z}
≤ |a|(uα0 − uα00 ) + |b|(v α0 − v α00 ) + (|c| + |d| + |e|)(uα0 v α0 − uα00 v α00 ).
Also,
Proposition 7. (adapted from [[10]]
v , v) are two valid pairs, then also (w,
e w) with
If (e
u, u) and (e
w
eα
wα
is a valid pair.
=
=
min{e
uα − uα , veα − v α } + max{e
uα + uα , veα + v α }
2
max{e
uα + uα , veα + v α } − min{e
uα − uα , veα − v α }
2
Finally, the Hausdorff distance on FI is defined by:
¯ ¯
¯ªª
©
©¯
−¯ ¯ +
+¯
D (u, v) = sup max ¯u−
α − vα , uα − vα
α∈[0,1]
If we use the decomposition
(13)
On Fuzzy Arithmetic Operations: Some Properties and Distributive Approximations8
½
b− + u
eα − uα
u−
α =u
+
uα = u
b+ + u
eα + uα
(and similarly for v), then
½ −
uα − vα− = (b
u− − vb− ) + (e
uα − veα ) + (v α − uα )
;
+
+
uα − vα = (b
u+ − vb+ ) + (e
uα − veα ) + (uα − v α )
we can obtain a modified distance on FI by considering the three components

b (u, v) = max {|b

D
u− − vb− | , |b
u+ − vb+ |}



e (u, v) = sup |e
uα − veα |
D
α∈[0,1]



D (u, v) = sup |uα − v α |

α∈[0,1]
and by defining
b (u, v) + D
e (u, v) + D (u, v) .
D∗ (u, v) = D
(14)
b ≤ D, D
e ≤ 2D, D ≤ 2D
It is possible to see that D∗ (u, v) is equivalent to D (u, v) as D
∗
and then D ≤ D ≤ 5D (the proof in appendix).
3. General Properties of Arithmetic Operations
In this section we analyze standard arithmetic operations between fuzzy numbers in terms
of the decomposition
b × P × S0
u = (b
u, u
e, u) ∈ F
(15)
and we study, in particular, distributivity and symmetry associated to multiplication and
division.
Remember that (15) is equivalent to
½ +
uα = u
b+u
eα + uα
∀α ∈ [0, 1] .
(16)
u−
b+u
eα − uα
α =u
3.1. Addition, scalar multiplication and subtraction. Addition, scalar multiplication and subtraction are immediate. Let
u = (b
u, u
e, u) and v = (b
v , ve, v)
be two fuzzy numbers, then we can write:
u + v = (b
u + vb, u
e + ve, u + v) ;
ku = (kb
u, ke
u, |k| u) , for k ∈ R
u − v = u + (−v) = (b
u − vb, u
e − ve, u + v) .
(17)
(18)
(19)
It may be of interest to note that addition and difference have the same 0-symmetric fuzzy
components u + v and they differ (only) in the crisp and the profile parts. On the other
hand, it is well known that:
u = (0, 0, u) ∈ S0 ⇐⇒ u = −u
and that
u, v ∈ S0 ⇐⇒ u + v = u − v = v − u = −u − v.
On Fuzzy Arithmetic Operations: Some Properties and Distributive Approximations9
These facts can be a little formalized; consider the following relation on F (u ∼ v if they
differ only for the S0 −component):
½
u
b = vb
u ∼ v ⇐⇒
u
eα = veα ∀α ∈ [0, 1]
which is an equivalence relation on F. The equivalence class corresponding to 0 =
(0, 0, 0) ∈ S0 (the null element of addition in F) is the whole S0 , i.e. [0]∼ = S0 ; in fact,
with respect to ∼ all the elements of S0 are equivalent to 0 and in particular (u − u) ∼ 0
∀u ∈ F, as (u − u) = (b
u, u
e, u) − (b
u, u
e, u) = (0, 0, 2u) ∈ S0 .
3.2. Multiplication. Consider now the multiplication of u, v ∈ F where u = (b
u, u
e, u)
and v = (b
v , ve, v) are fuzzy numbers with
½ +
½ +
uα = u
vα = vb + veα + v α
b+u
eα + uα
and
u−
=
u
b
+
u
e
−
u
vα− = vb + veα − v α
α
α
α
The α − cuts of the fuzzy product uv are given (see the Appendix) by:
£
¤ £ − +¤
+
[uv]α = u−
vα , vα
α , uα
¯ ¯
¯ ¯ª
£
©
= min (b
u+u
eα ) vα− − uα ¯vα− ¯ , (b
u+u
eα ) vα+ − uα ¯vα+ ¯ ,
¯ ¯
¯ ¯ª¤
©
max (b
u+u
eα ) vα− + uα ¯vα− ¯ , (b
u+u
eα ) vα+ + uα ¯vα+ ¯ .
(20)
Distributivity of multiplication. We use (20) to analyze the distributivity property of the multiplication.
Given three fuzzy numbers in terms of their components
u = (b
u, u
e, u) ; v = (b
v , ve, v) ; z = (b
z , ze, z)
we obtain from (20) the α − cuts
·
½
¾
(b
u + vb + u
eα + veα ) zα− − (uα + v α ) |zα− | ,
[(u + v) z]α =
min
,
(b
u + vb + u
eα + veα ) zα+ − (uα + v α ) |zα+ |
½
¾¸
(b
u + vb + u
eα + veα ) zα− − (uα + v α ) |zα− | ,
max
(b
u + vb + u
eα + veα ) zα+ + (uα + v α ) |zα+ |
and
[uz + vz]α
We then have:
=
·
min {(b
u+u
eα ) zα− − uα |zα− | , (b
u+u
eα ) zα+ − uα |zα+ |}
−
−
+ min {(b
v + veα ) zα − v α |zα | , (b
v + veα ) zα+ − v α |zα+ |} ,
¸
max {(b
u+u
eα ) zα− + uα |zα− | , (b
u+u
eα ) zα+ + uα |zα+ |}
.
+ max {(b
v + veα ) zα− + v α |zα− | , (b
v + veα ) zα+ + v α |zα+ |}
Theorem 8. The distributivity (u + v) z = uz + vz is valid if and only if, for each
α ∈ [0, 1] , at least one of the following four systems of inequalities is satisfied:

(b
u+u
eα )zα− − uα |zα− | ≤ (b
u+u
eα )zα+ − uα |zα+ |



−
−
(b
v + veα )zα − v α |zα | ≤ (b
v + veα )zα+ − v α |zα+ |
(21)
+
+
(b
u+u
eα )zα + uα |zα | ≤ (b
u+u
eα )zα− + uα |zα− |



+
+
−
−
(b
v + veα )zα + v α |zα | ≤ (b
v + veα )zα + v α |zα |
On Fuzzy Arithmetic Operations: Some Properties and Distributive Approximations10

(b
u+u
eα )zα− − uα |zα− | ≤ (b
u+u
eα )zα+ − uα |zα+ |



−
−
(b
v + veα )zα − v α |zα | ≤ (b
v + veα )zα+ − v α |zα+ |
−
−
u+u
eα )zα + uα |zα | ≤ (b
u+u
eα )zα+ + uα |zα+ |
 (b


−
−
(b
v + veα )zα + v α |zα | ≤ (b
v + veα )zα+ + v α |zα+ |

(b
u+u
eα )zα+ − uα |zα+ | ≤ (b
u+u
eα )zα− − uα |zα− |



+
+
(b
v + veα )zα − v α |zα | ≤ (b
v + veα )zα− − v α |zα− |
+
+
(b
u+u
eα )zα + uα |zα | ≤ (b
u+u
eα )zα− + uα |zα− |



+
+
(b
v + veα )zα + v α |zα | ≤ (b
v + veα )zα− + v α |zα− |

(b
u+u
eα )zα+ − uα |zα+ | ≤ (b
u+u
eα )zα− − uα |zα− |



+
+
(b
v + veα )zα − v α |zα | ≤ (b
v + veα )zα− − v α |zα− |
−
−
(b
u+u
eα )zα + uα |zα | ≤ (b
u+u
eα )zα+ + uα |zα+ |



−
−
(b
v + veα )zα + v α |zα | ≤ (b
v + veα )zα+ + v α |zα+ |
(22)
(23)
(24)
Proof. (see Appendix)
We have the following dependencies:
(21)=⇒ |zα− | ≥ |zα+ | ;
(22)=⇒ u
b+u
eα ≥ 0, vb + veα ≥ 0;
(23)=⇒ u
b+u
eα ≤ 0, vb + veα ≤ 0;
(24)=⇒ |zα− | ≤ |zα+ | .
The following propositions give relevant cases of validity of the distributivity and are
immediately verified (at our knowledge, some of them are new in the fuzzy literature).
The first two properties say that the distributivity is valid if z ∈ S0 and, independently
of their sign, the fuzzy numbers u and v have "homogeneous" asymmetries, i.e. (b
u+e
uα )(b
v+
veα ) ≥ 0 .
b+u
eα ≥ 0, vb + veα ≥ 0 then system (22) is satisfied.
Proposition 9. Given z ∈ S0 and u
Proof. In fact, |zα− | = −zα− = zα+ > 0 and (22) reduces simply to (b
u+u
eα )zα+ ≥ 0 and
+
(b
v + veα )zα ≥ 0.
Proposition 10. Given z ∈ S0 and u
b+u
eα ≤ 0, vb + veα ≤ 0 then system (23) is satisfied.
Proof. In fact, |zα− | = −zα− = zα+ > 0 and (23) reduces simply to (b
u+u
eα )zα+ ≤ 0 and
+
(b
v + veα )zα ≤ 0.
Proposition 11. Given z ∈ F+ and u, v ∈ F0 then system (24) is satisfied.
Proof.
In this case, z has zα− = |zα− |, zα+ = |zα+ |, and zα− ≤ zα+ ; u and v have −uα ≤
u
b+u
eα ≤ uα and −v α ≤ vb + veα ≤ v α ; then system (24) is immediate.
Proposition 12. Given z ∈ F− and u, v ∈ F0 then system (21) is satisfied.
Proof.
In this case, z has zα− = −|zα− |, zα+ = −|zα+ |, and zα− ≤ zα+ ; u and v have
−uα ≤ u
b+u
eα ≤ uα and −v α ≤ vb + veα ≤ v α ; then system (21) is immediate.
Proposition 13. ([11]) Given z ∈ F and u, v ∈ S0 then either system (21) or (24) is
satisfied.
Proposition 14. ([11]) Given z ∈ F and u, v ∈ F+ then system (22) is satisfied.
Proposition 15. ([11]) Given z ∈ F and u, v ∈ F− then system (23) is satisfied.
So, the cases where the distributivity of multiplication is valid require that at least
one of the two factors are of special nature; in particular, distributivity is valid within S0 ,
F+ or F− .
On Fuzzy Arithmetic Operations: Some Properties and Distributive Approximations11
Symmetry in Multiplication. By the use of the decomposition (11) we can analyze
the effect of the multiplication on the symmetry of the product fuzzy number.
It is immediate to see that if u, v ∈ S0 i.e. u = (0, 0, u) and v = (0, 0, v), then
uv = (0, 0, uv) ∈ S0 is the exact product.
If we take u, v ∈ F0 i.e. u = (0, u
e, u) and v = (0, ve, v) then the α − cuts of the product
uv ∈ F0 are given by
[uv]α
= [e
uα veα − uα v α − |uα veα − u
eα v α |,
u
eα veα + uα v α + |uα veα + u
eα v α |];
(25)
it follows that the elements of the decomposition of uv are:
g
(uv)
α
and
(uv)α
where wα
|uα veα + u
eα v α | − |uα veα − u
eα v α |
2
= u
eα veα + max{uα veα , u
eα v α }
= u
eα veα +
|uα veα − u
eα v α | + |uα veα + u
eα v α |
2
= uα v α + wα .
∈ {uα veα , u
eα v α , −uα veα , −e
uα v α }
= uα v α +
and we see how the profile functions of u and v contribute to the definition of the product.
We also see from (25) that S0 absorbs F0 in the sense that if u ∈ S0 and v ∈ F0 then
uv = (0, 0, uv + |ue
v |) ∈ S0 (in a similar way, F0 absorbs F as if u ∈ F0 , v ∈ F then
uv ∈ F0 ).
Consider now the case u, v ∈ S i.e. u = (b
u, 0, u) and v = (b
v , 0, v). Then the product
uv is given (after some algebra) by
[uv]α
= u
bvb + [min{uα v α − |b
uv α + uα vb|, −uα v α − |b
uv α − uα vb|},
max{uα v α + |b
uv α + uα vb|, −uα v α + |b
uv α − uα vb|}].
(26)
Note that, in this case, min{., .} ≤ 0 and max{., .} ≥ 0 and if min{., .} = uα v α −
|b
uv α + uα vb| then max{., .} = uα v α + |b
uv α + uα vb|.
minα
minα
If we compute (f
uv)α = maxα +
and (uv)α = maxα −
we obtain the following
2
2
three cases:
Case 1. min{., .} = uα v α − |b
uv α + uα vb| and max{., .} = uα v α + |b
uv α + uα vb| : then
(f
uv)α = uα v α and (uv)α = |b
uv α + uα vb| and a positive asymmetry is introduced by the
operation;
Case 2. min{., .} = −uα v α − |b
uv α − uα vb| and max{., .} = −uα v α + |b
uv α − uα vb| :
then (f
uv)α = −uα v α and (uv)α = |b
uv α + uα vb| and a negative asymmetry is introduced;
Case 3. min{., .} = −uα v α − |b
uv α − uα vb| and max{., .} = uα v α + |b
uv α + uα vb| : then
(f
uv)α = |b
uv α + uα vb| − |b
uv α − uα vb| and (uv)α = uα v α + 12 (|b
uv α + uα vb| + |b
uv α + uα vb|) and
an unsigned asymmetry is introduced.
On Fuzzy Arithmetic Operations: Some Properties and Distributive Approximations12
3.3. Division. We complete this section by analyzing the division between fuzzy numbers u ∈ F and z ∈ F− ∪ F+ .
The reciprocal of z is the fuzzy number given by z −1 with α − cuts
·
¸
1 1
−1
[z ]α = + , −
zα zα
and the division is defined as
u
= uz −1 .
z
Note that if u = (0, 0, uα ) ∈ S0 then uz ∈ S0 as the α − cuts are
·
½
½
¾
¾¸
u
1
1
1
1
[ ]α = −uα max
,
−
u
max
,
−
,
.
α
z
zα+ zα−
zα+ zα−
−1 , z −1 ) is
z −1 , zg
If z = (b
z , 0, z) ∈ S− ∪ S+ is symmetric, then it is easy to see that z −1 = (b
not symmetric as
−1
zg
α
zα−1
=
=
z 2α
zb(b
z 2 − z 2α )
zα
.
2
zb − z 2α
The distributivity of the division, i.e. the equality
u+v
u v
= +
z
z
z
can be written in terms of the distributivity of the multiplication:
(u + v) z −1 = uz −1 + vz −1 .
Proposition 16. If z ∈ F+ ∪ F− and u, v ∈ F0 (or in particular u, v ∈ S0 ) then the
division is distributive.
Proof.
In fact, if z ∈ F+ then z −1 ∈ F+ or if z ∈ F− then z −1 ∈ F− ; the application
of propositions 13 and 14 gives the distributivity.
Proposition 17. ([11]) If z ∈ F+ ∪ F− and u, v ∈ F+ (or u, v ∈ F− ) then the division is
distributive.
4. Approximate Distributive Operations
In this section we will use the decomposition of the fuzzy numbers introduced in section
2 and the concept of valid pair, to define approximate multiplication and division; in
particular, we will focus on approximate operations with the distributivity property.
4.1. Approximate Multiplication. By the help of the propositions in section 2. it
is immediate to verify that, given two fuzzy numbers u = (b
u, u
e, u) and v = (b
v , ve, v) , then
also (b
uvb, u
eve, uv) is a valid fuzzy number (use a = b = d = e = 0 and c = 1) ..
Definition 18. Define an approximate multiplication as
[u ∗ v]α = [b
uvb + u
eα veα − uα v α , u
bvb + u
eα veα + uα v α ]
or, in terms of the decomposition,
u ∗ v = (b
uvb, u
eve, uv) .
On Fuzzy Arithmetic Operations: Some Properties and Distributive Approximations13
We will now compare the standard multiplication with the new multiplication above;
considering the standard product uv, after some algebra, we can write its α − cuts as
[uv]α = u
bvb + u
bveα + vbu
eα + u
eα veα + Aα
where
Aα
=
It also holds:
·
½
¾
uα v α − |(b
u+u
eα ) v α + (b
v + veα ) uα | ,
min
,
−uα v α − |(b
u+u
eα ) v α − (b
v + veα ) uα |
½
¾¸
uα v α + |(b
u+u
eα ) v α + (b
v + veα ) uα | ,
max
.
−uα v α + |(b
u+u
eα ) v α − (b
v + veα ) uα |
[u ∗ v]α
where Bα
Note that, in any case, Bα ⊆ Aα .
= u
bvb + u
eα veα + Bα
= [−uα v α , uα v α ] .
Case 19. (S0 − case)
If u, v ∈ S0 then ∀α ∈ [0, 1] the following is immediate:
(u ∗ v)α = (uv)α .
Case 20. (S − case)
If u, v ∈ S then ∀α ∈ [0, 1] the following is true:
(u ∗ v)α ⊆ (uv)α .
Proof.
In this case we have that u
eα = veα = 0 ∀α then (uv)α = u
bvb + Aα and
(u ∗ v)α = u
bvb + Bα . Then (u ∗ v)α ⊆ (uv)α as Bα ⊆ Aα .
Case 21. (F0 − case)
If u, v ∈ F0 then ∀α ∈ [0, 1] we have
(u ∗ v)α ⊆ (uv)α
Proof.
As u
b = vb = 0 it follows that u−
eα − uα ≤ 0, vα− = veα − v α ≤ 0,
α = u
+
u+
=
u
e
+
u
≥
0
and
v
=
v
e
+
v
≥
0.
In
this
case
α
α
α
α
α
α
£
© − + + −ª
©
ª¤
−
+ +
(uv)α = min uα vα , uα vα , max u−
α vα , uα vα
Consider the following averages of the terms in the min and the max above
wα− =
+
+ −
+
u−
u− v − + u+
α vα + uα vα
α vα
, wα+ = α α
;
2
2
it is obvious that [wα− , wα+ ] ⊆ (uv)α as min ≤ average ≤ max . We now see that (u ∗ v)α =
[wα− , wα+ ] ; in fact, by computing the decomposition of the fuzzy number w defined by the
w− +w+
u− +u+ v − +v+
w+ −w−
α − cuts [wα− , wα+ ] we find w
eα = α 2 α = α 2 α α 2 α = u
eα veα and wα = α 2 α =
− +
−
u+
α −uα vα −vα
2
2
= uα v α .
On Fuzzy Arithmetic Operations: Some Properties and Distributive Approximations14
Case 22. If u ∈ S0 , v ∈ F then
(u ∗ v)α ⊆ (uv)α .
Proof.
In fact, as u
b = 0 and u
eα = 0 ∀α, then
(uv)α
= [min {uα v α − |b
v + veα | uα , −uα v α − |b
v + veα | uα } ,
max {uα v α + |b
v + veα | uα , −uα v α + |b
v + veα | uα }]
= [−uα v α − |b
v + veα | uα , uα v α + |b
v + veα | uα ] ∈ S0
and (u ∗ v)α = [−uα v α , uα v α ] so that (u ∗ v)α ⊆ (uv)α .
The properties above suggest that the fuzzy number
u ∗ v = (b
uvb, u
eve, uv)
can be used as an approximation of the standard multiplication uv.
In particular, the approximation is exact if u, v ∈ S0 and it reduces the range of the
fuzziness (i.e. the range of the α − cuts) when u, v ∈ S , when u, v ∈ F0 and when
u ∈ S0 , v ∈ F.
Proposition 23. (Distributivity of ∗)
If u, v, z ∈ F then (u + v) ∗ z = u ∗ z + v ∗ z.
Proof.
In fact (u + v) ∗ z = ((b
u + vb) ∗ zb, (e
u + ve) ∗ ze, (u + v) ∗ z) = u ∗ z + v ∗ z as
distributivity is valid for each of the three components.
4.2. Approximate division. In a similar way as we have seen for the multiplication,
we can apply the decomposition u = (b
u, u
e, u) to the division u/v where v ∈ F− or v ∈ F+ .
Note that, for such v we always have vb 6= 0.
In the special case where u = (1, 0, 0) we obtain the fuzzy reciprocal of v, having the
following α − cuts :
·
¸
1 1
−1
(v )α = + , − , ∀α ∈ [0, 1].
vα vα
The decomposition of v −1 is then given by
vα
1 g
vb + veα
1
−1 =
, vα−1 = − + − and vα−1 = − + .
vd
vb
vb
vα vα
vα vα
−1 , uv −1 ) is a valid pair and
Using the results that we have seen for the multiplication, (e
uvg
we can approximate the division by using it:
Definition 24. For u = (b
u, u
e, u) ∈ F and v = (b
v , ve, v) F− ∪ F+ define an approximate
division as
u
b g
u : v = u ∗ v −1 = ( , u
ev −1 , uv −1 ).
vb
The following properties are then immediate:
¡ ¢
1. u ∈ S0 =⇒ (u : v)α ⊆ uv α ∀α ;
2. (u + v) : z = u : z + v : z .
v2α
−1
Note also that, if v ∈ S ∪S , then vg
=
6= 0 so that v −1 ∈
/ S (the reciprocal
−
+
α
v
e2 −v 2α
operation introduces a form of asymmetry as a nonzero profile).
On Fuzzy Arithmetic Operations: Some Properties and Distributive Approximations15
5. Examples
In Multiplication 1, the two fuzzy numbers u and v are both trapezoidal with linear left
and right branches u = h−2, 0, 1, 2i and v = h−2, −1, 0, 2i .
In Multiplication 2. we use u and v in the decomposed form with u
b = 1, u
eα =
−(1 − α)2 /2, uα = (1 − α)(2 − α)/2 and vb = 2, veα = −2e
uα , v α = 1.5uα .
(uv)-left
(uv)-right
(u*v)-left
(u*v)-right
5,0
4,0
(uv)-left
(uv)-right
(u*v)-left
(u*v)-right
8,0
7,0
6,0
3,0
5,0
2,0
4,0
1,0
3,0
0,0
0,0
0,2
0,4
0,6
0,8
1,0
2,0
-1,0
1,0
-2,0
0,0
0,0
-3,0
0,2
0,4
0,6
0,8
1,0
-1,0
-4,0
-2,0
-5,0
-3,0
Multiplication 1
Multiplication 2
In Division 1, the two fuzzy numbers u and v are both trapezoidal with linear left and
right branches u = h1, 2, 3, 4i and v = h2, 3, 5, 7i .
In Division 2, the two fuzzy numbers u and v are both trapezoidal with linear left and
right branches u = h−1, 0, 0, 5i and v = h1, 2, 2, 3i . Note that u ∈ F0 .
(u/v)-left
(u/v)-right
(u:v)-left
(u:v)-right
6,0
(u/v)-left
(u/v)-right
(u:v)-left
(u:v)-right
6,0
5,0
5,0
4,0
4,0
3,0
3,0
2,0
1,0
2,0
0,0
1,0
0,0
0,2
0,4
0,6
0,8
1,0
-1,0
0,0
0,0
0,2
0,4
0,6
Division 1
0,8
1,0
-2,0
Division 2
In the calculus of fuzzy expressions we take an example with triangular u = h0, 1, 2i and
uv
v = h1, 2, 3i and compute the Klir ([8]) operation ( u+v
)E compared to the approximation
App = (u ∗ v) : (u + v).
Finally, we compare the multiplication ∗ with the arithmetic introduced by Ma et
al. in [10]. In our notation, the input fuzzy numbers of their last example are u =
On Fuzzy Arithmetic Operations: Some Properties and Distributive Approximations16
(1, α(1−α)
, 2−α(α+1)
) and v = (2, α(α−1)
, 2−α(α+1)
); their multiplication produces the lin2
2
2
2
ear
symmetric
uv
=
(2,
0,
1
−
α)
having
α
−
cuts
[uv]α = [1 + α, 3 − α] while u ∗ v =
µ
³
´2 ³
´2 ¶
α(α−1)
2−α(α+1)
2, −
,
.
2
2
App-left
1,4
Ma-left
Ma-right
(u*v)-left
(u*v)-right
3,5
App-right
Klir-left
1,2
3,0
Klir-right
1,0
2,5
0,8
2,0
0,6
1,5
0,4
1,0
0,2
0,5
-
0,0
0,0
0,2
0,4
0,6
Klir example
0,8
1,0
0,0
0,2
0,4
0,6
0,8
1,0
Ma et al. example
6. Conclusions
A decomposition of fuzzy numbers is introduced that isolates three basic components with
specific interpretation in the fuzzy context. The decomposition is used to study properties
of fuzzy arithmetic, in particular distributivity and symmetry in multiplication and division. We show that distributivity may depend on the form of symmetry of the involved
fuzzy quantities and we characterize necessary and sufficient condition for distributivity of
multiplication. Further, we use the decomposition to construct possible approximations
of fuzzy multiplication and division with the distributivity property. The approximations
are compared with other proposed operations in the fuzzy literature and some preliminary
results in this direction appear quite interesting for further investigation in modeling fuzzy
quantities and understanding properties of fuzzy arithmetic.
7. Appendix
We give now some of the (simple but tedious) proofs of the results illustrated in the paper.
7.1.
D∗ is equivalent to D. We have defined in (14)
b (u, v) + D
e (u, v) + D (u, v) .
D∗ (u, v) = D
We see that D ≤ D∗ ≤ 5D; first, we have D ≤ D∗ as
¯ ¯ +
¯ª
©¯ −
D (u, v) ≤ max ¯u
b − vb− ¯ , ¯u
b − vb+ ¯ + sup |e
uα − veα | + sup |uα − v α |
α∈[0,1]
α∈[0,1]
b (u, v) + D
e (u, v) + D (u, v) = D∗ (u, v) .
≤ D
On the other hand,
¯ ¯ +
¯ª
©¯ −
b (u, v) = max ¯u
D
b − vb− ¯ , ¯u
b − vb+ ¯ ≤ D (u, v) ;
On Fuzzy Arithmetic Operations: Some Properties and Distributive Approximations17
e (u, v) = sup |e
D
uα − veα |
α
¯·
¸ ·
¸¯
¯ 1¡ −
¢ 1¡ −
¢
¢ 1¡ −
¢ ¯
1¡ −
+ ¯
+
+
+
¯
= sup ¯
b
−
u + uα +
u
b +u
v + vα +
v + v1 ¯
2 α
2
2 α
2 1
α
¾
½
¯ 1¯ +
¯ 1¯ −
¯ 1¯ +
¯
1 ¯¯ −
−¯
+¯
−¯
+¯
¯
¯
¯
≤ sup
u − vα + uα − vα + u
b − vb + u
b − vb
2 α
2
2
2
α
¯
¯
¯
¯
¯ +
¯ªª
ª
©¯
©
©¯ −
+¯
≤ sup max ¯uα − vα− ¯ , ¯u+
+ max ¯u
b− − vb− ¯ , ¯u
b − vb+ ¯
α − vα
α
b (u, v) ≤ 2D (u, v) ;
≤ D (u, v) + D
D (u, v) = sup |uα − v α |
α
¯
¯
¯1 ¡ +
¢ 1¡ +
¢ 1¡ +
¢ 1¡ +
¢¯
−
−
−
− ¯
¯
= sup ¯ uα − uα −
b −
u
b −u
v − vα +
vb − vb ¯
2
2
2 α
2
α
¯
¯
¯
¯
¯
¯ªª
ª
©¯
©
©¯ +
−¯
≤ sup max ¯uα − vα+ ¯ , ¯u−
+ max ¯u
b+ − vb+ ¯ , ¯u
b− − vb− ¯
α − vα
α
b (u, v) + D (u, v) ≤ 2D (u, v) ;
≤ D
b +D
e + D ≤ 5D.
so, D∗ = D
7.2. Computation of α − cuts of uv. In terms of the decomposition, the α − cuts
of the fuzzy product uv are given by (20):
£
¤ £ − +¤
+
[uv]α = u−
vα , vα
α , uα
¯ ¯
¯ ¯ª
£
©
u+u
eα ) vα+ − uα ¯vα+ ¯ ,
= min (b
u+u
eα ) vα− − uα ¯vα− ¯ , (b
¯ ¯
¯ ¯ª¤
©
max (b
u+u
eα ) vα− + uα ¯vα− ¯ , (b
u+u
eα ) vα+ + uα ¯vα+ ¯ .
In fact,
£ − +¤ £ − +¤
−
− +
+ −
+ +
uα , uα vα , vα
= [min{u−
α vα , uα vα , uα vα , uα vα },
max{u−
v − , u− v + , u+ v − , u+ v + }]
·
½α α α α − α α −α α
¾
(b
u+u
eα )vα − uα vα , (b
u+u
eα )vα+ − uα vα+ ,
=
min
,
(b
u+u
eα )vα− + uα vα− , (b
u+u
eα )vα+ + uα vα+
½
¾¸
(b
u+u
eα )vα− − uα vα− , (b
u+u
eα )vα+ − uα vα+ ,
max
(b
u+u
eα )vα− + uα vα− , (b
u+u
eα )vα+ + uα vα+
·
½
¾
(b
u+u
eα )vα− + min{−uα vα− , +uα vα− },
=
min
,
(b
u+u
eα )vα+ + min{−uα vα+ , +uα vα+ }
½
¾¸
(b
u+u
eα )vα− + max{−uα vα− , +uα vα− },
max
;
(b
u+u
eα )vα+ + max{−uα vα+ , +uα vα+ }
But uα ≥ 0 and then min{−uα vα− , +uα vα− } = −uα |vα− | , min{−uα vα+ , +uα vα+ } = −uα |vα+ |
, max{−uα vα− , +uα vα− } = uα |vα− | and max{−uα vα+ , +uα vα+ } = uα |vα+ |.
7.3. Proof of distributivity conditions. We have distributivity if and only if the
minimum (and resp. the maximum) in [(u + v) z]α is equal to the sums of the two minima
(and resp. the two maxima) in [uz + vz]α .
On Fuzzy Arithmetic Operations: Some Properties and Distributive Approximations18
To simplify the notation we pose
aα
bα
cα
= u
b+u
eα , a0α = vb + veα ,
= uα ≥ 0, b0α = v α ≥ 0,
= zα− , dα = zα+
and define
mα
nα
n0α
Mα
Nα
Nα0
=
=
=
=
=
=
min{(aα + a0α )cα − (bα + b0α )|cα |, (aα + a0α )dα − (bα + b0α )|dα |}
min{aα cα − bα |cα |, aα dα − bα |dα |}
min{a0α cα − b0α |cα |, a0α dα − b0α |dα |}
max{(aα + a0α )cα + (bα + b0α )|cα |, (aα + a0α )dα + (bα + b0α )|dα |}
max{aα cα + bα |cα |, aα dα + bα |dα |}
max{a0α cα + b0α |cα |, a0α dα + b0α |dα |}
so that
[(u + v)z]α = [mα , Mα ] and [uz + vz]α = [nα + n0α , Nα + Nα0 ].
We then have to verify in which cases mα = nα + n0α and Mα = Nα + Nα0 . For the
minimum, we have the two cases (we omit the subscript α)

n = ac − b|c| , n0 = a0 c − b0 |c|

⇓
(i)

m = (a + a0 )c − (b + b0 )|c| = n + n0

n = ad − b|d| , n0 = a0 d − b0 |d|

⇓
(i)’

m = (a + a0 )d − (b + b0 )|d| = n + n0
and, for the maximum, the two cases

N = ac + b|c| , N 0 = a0 c + b0 |c|

⇓
(ii)

M = (a + a0 )c + (b + b0 )|c| = N + N 0

N = ad + b|d| , N 0 = a0 d + b0 |d|

⇓
(ii)’

M = (a + a0 )d + (b + b0 )|d| = N + N 0 .
Combining (i) or (i)’ with (ii) or (ii)’, gives the four systems of inequalities:

ac − b |c| ≤ ad − b |d|


 0
a c − b0 |c| ≤ a0 d − b0 |d|
(i) and (ii)
ad + b |d| ≤ ac + b |c|


 0
a d + b0 |d| ≤ a0 c + b0 |c|

ac − b |c| ≤ ad − b |d|


 0
a c − b0 |c| ≤ a0 d − b0 |d|
(i) and (ii)’
ac + b |c| ≤ ad + b |d|


 0
0
0
0
a c + b |c| ≤ a d + b |d|
On Fuzzy Arithmetic Operations: Some Properties and Distributive Approximations19




ad − b |d| ≤ ac − b |c|
a0 d − b0 |d| ≤ a0 c − b0 |c|
 ad + b |d| ≤ ac + b |c|

 0
a d + b0 |d| ≤ a0 c + b0 |c|

ad − b |d| ≤ ac − b |c|


 0
a d − b0 |d| ≤ a0 c − b0 |c|
ac + b |c| ≤ ad + b |d|


 0
a c + b0 |c| ≤ a0 d + b0 |d| .
(i)’ and (ii)
(i)’ and (ii)’
References
[1] P. Benvenuti, R. Mesiar, Pseudo-arithmetical operations as a basis for the general
measure and integration theory, Information Sciences, 160 (2004), 1-11.
[2] M. Delgado, M. A. Vila, W. Voxman, On a canonical representation of fuzzy nymbers,
Fuzzy Sets ans Systems, 93 (1998), 125-135.
[3] D. Dubois, H. Prade, Fuzzy numbers, an overview. In J. C. Bezdek (ed.), Analysis
of Fuzzy Information (Mathematics), CRC Press, 1988, 3-39.
[4] M.L. Guerra, L. Stefanini, Approximate Fuzzy Arithmetic Operations Using
Monotonic Interpolations, Fuzzy Sets and Systems,150/1 (2005), 5-33.
[5] A. Kandel, Fuzzy mathematical techniques with applications, Addison Wesley, Reading, Mass, 2000.
[6] A. Kaufman M. M. Gupta, Introduction to fuzzy arithmetic, theory and applications,
Van Nostrand Reinhold Company, New York, 1985.
[7] G. J. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice
Hall, 1995.
[8] G. J. Klir, Fuzzy arithmetic with requisite constraints, Fuzzy Sets and Systems, 91
(1997), 165-175.
[9] G. J. Klir, Uncertainty Analysis in Engineering and Sciences, Kluwer, 1997.
[10] M. Ma, M. Friedman, A. Kandel, A new fuzzy arithmetic, Fuzzy Sets and Systems,
108 (1999), 83-90.
[11] M. Mares, Weak arithmetics of fuzzy numbers, Fuzzy Sets and Systems, 91 (1997),
143-153.
[12] R. Mesiar, J. Rybarik, Pan-operations structure, Fuzzy Sets and Systems, 74 (1995),
365-369.
[13] L. Stefanini, L.Sorini, M.L.Guerra, Parametric Representations of Fuzzy Numbers
and Application to Fuzzy Calculus, Fuzzy Sets and Systems, 157 (2006), 2423-2455.
[14] L. A. Zadeh, Fuzzy Sets, Information and Control, 8 (1965), 338-353.