The Real Transform: Computing Positive Solutions of Fuzzy Polynomial
Systems
Philippe Aubry
1
, J
´
er
´
emy Marrez
1
and Annick Valibouze
1,2
1
Sorbonne Universit
´
e, CNRS, Laboratoire d’Informatique de Paris 6, LIP6, F-75005 Paris, France
2
Sorbonne Universit
´
e, CNRS, Laboratoire de Probabilit
´
es, Statistique et Mod
´
elisation, LPSM, F-75005 Paris, France
Keywords:
Fuzzy Numbers, Fuzzy Systems, Polynomial Systems, Fuzzy Methods.
Abstract:
This paper presents an efficient method for finding the positive solutions of polynomial systems whose coeffi-
cients are symmetrical L-R fuzzy numbers with bounded support and the same bijective spread functions. The
positive solutions of a given fuzzy system are deduced from the ones of another polynomial system with real
coefficients, called the real transform. This method is based on new results that are universal because they are
independent from the spread functions. We propose the real transform T (E) of a fuzzy equation (E), which
positive solutions are the same as those of (E). Then we compare our approach with the existing method of
the crisp form system.
1 INTRODUCTION
Modeling problems with uncertain data has important
applications in engineering, economics and social sci-
ences (Aluja et al., 1994; Lodwick, 2007). As fuzzy
functions involved in equations can be approximated
by fuzzy polynomials (Liu, 2002; Abbasbandy and
Amirfakhrian, 2006), the problem of solving fuzzy
polynomial systems is of main importance and has
motivated many studies, among them (Buckley and
Qu, 1990; Buckley and Eslami, 1997; Buckley et al.,
2002; Kajani et al., 2005).
Some problems are modeled by a system of con-
tinuous fuzzy valued functions defined over R
n
(see
(Lodwick and Santos, 2003)); Liu (Liu, 2002) pro-
poses a polynomial interpolation that provides an in-
terface between solving these problems and solving
fuzzy polynomial systems. Therefore several authors
have been interested in searching for real solutions of
polynomial equations with fuzzy coefficients. Note
that their work, and ours too, is based only on Zadeh’s
extension principle. The methods of resolution were
initially based on local techniques, firstly for one
univariate polynomial (Abbasbandy and Otadi, 2006;
Amirfakhrian, 2008; Rouhparvar, 2007) and later for
multivariate systems (Abbasbandy et al., 2008; Ah-
mad et al., 2011).
Abbasbandy et al. (Abbasbandy et al., 2008) study
in particular systems of the form

e
a
1,1
xy +
e
a
1,2
x
2
y
2
+ ··· +
e
a
1,d
x
d
y
d
=
e
a
1,0
e
a
2,1
xy +
e
a
2,2
x
2
y
2
+ ··· +
e
a
2,d
x
d
y
d
=
e
a
2,0
and they present several systems coming from appli-
cations like cross location of quadratic surfaces or in
economics.
Recently, a global approach using classical alge-
braic techniques has been developed (Molai et al.,
2013; Farahani et al., 2015; Boroujeni et al., 2016;
Farahani et al., 2019). Indeed, despite a name that
may be confusing, fuzzy numbers benefit from a per-
fectly formal definition. We revisit this approach and
we significantly strengthen it. In the past, both local
and global approaches focused on so-called triangu-
lar fuzzy numbers, that is, with linear spread func-
tions. The results presented here consider more gen-
erally fuzzy coefficients with bounded support and the
same bijective spread functions such as, for example,
quadratic fuzzy numbers, that have quadratic spread
functions. As previous works on this topic, our no-
tion of solution lies on the equality of membership
functions.
In a fuzzy algebraic system, the fuzzy coefficients
(coming from the experiments) are generally given
under a representation called ”tuple”. Although the
tuple representation is formal, it cannot be handle
by usual algebraic methods (Gr
¨
obner bases (Becker
and Weispfenning, 1993), triangular decomposition
(Aubry and Maza, 1999), rational univariate represen-
tation (Rouillier, 1999), . . . ) to solve the system.
Aubry, P., Marrez, J. and Valibouze, A.
The Real Transform: Computing Positive Solutions of Fuzzy Polynomial Systems.
DOI: 10.5220/0008362403510359
In Proceedings of the 11th International Joint Conference on Computational Intelligence (IJCCI 2019), pages 351-359
ISBN: 978-989-758-384-1
Copyright
c
 2019 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
351

Nevertheless, for any fuzzy number with bounded
support and of bijective spread functions, this tuple
representation is transformable into another represen-
tation called ”parametric”, where the coefficients are
no longer fuzzy but real. We give the expression of
the parametric representation as a function of the in-
verse of its spread functions (see Proposition 1).
We show how finding the positive solutions of a
system (S) of s equations with k variables and with
symmetrical fuzzy coefficients reduces to computing
positive solutions of a system formed by 3s equations
of k variables with real coefficients (Theorem 2). This
new algebraic real system denoted by T (S) is called
the real transform of (S). We extend this result to so-
called trapezoidal fuzzy numbers to get 4 s equations
instead of 3s (Section 3.5).
In Section 2, we introduce fuzzy numbers, their
different representations, and the transition from one
to the other in the case of a fuzzy number with
bounded support. In Section 3, we define the real
transform of a given fuzzy polynomial equation. We
show that, in the triangular fuzzy case, the real trans-
form is a system equivalent to the collected crisp form
calculated by previous methods. Finally, the study is
extended to the trapezoidal case.
2 FUZZY NUMBERS
After some generalities, this section recalls two clas-
sical representations of a fuzzy number that we will
use to solve the algebraic fuzzy systems. To go fur-
ther the reader may be interested in (Dubois et al.,
2000). We give some formulas which express the
parametric representation of a fuzzy number in func-
tion of its tuple representation when the number has
a bounded support and bijective spread functions
(Proposition 1). These formulas are the key of our
algebraic method to solve in R
+
the algebraic fuzzy
systems.
2.1 Generalities
Let
e
n be a fuzzy number and µ
e
n
its membership func-
tion from R to the real interval [0, 1], continuous and
satisfying µ
−1
e
n
({1}) = {n}, where the value n is called
the core of
e
n. In literature there are more general defi-
nitions than the one given above. They include the so-
called trapezoidal fuzzy numbers, for which the grade
of membership equals 1 over an interval of R contain-
ing the core. The chosen definition excludes them for
the sake of clarity in our study. Note that most appli-
cations make use of non-trapezoidal numbers. How-
ever, we will show in Section 3.5 that our analysis,
once established, simply extends to the trapezoidal
case.
We define the support of
e
n as the support of its
membership function, i.e. the set of a ∈ R such that
µ
e
n
(a) 6= 0. We denote it by Supp(
e
n).
For a real number r in [0,1], the r-cut of
e
n is the
convex set
e
n
r
= {x ∈ R | µ
e
n
(x) ≥ r} when r 6= 0 and
the 0-cut
e
n
0
is the closure of Supp(
e
n).
Throughout the paper, we will consider fuzzy
numbers with bounded support.
2.2 Tuple Representation
The tuple representation of a fuzzy number with
bounded support was proposed in 1978 by D. Dubois
and H. Prade in (Dubois and Prade, 1978). In this rep-
resentation the arithmetic operations have very sim-
ple expressions as soon as they are performed within
a family described in Definition 1, based on spread
functions.
A function H defined on the real interval [0, +∞[
with values in the real interval ] − ∞,1] is called a
spread function if H(0) = 1, H(1) = 0, H is continue
and decreasing on its domain.
Definition 1. Let L and R be two spread functions. A
fuzzy number
e
n with a bounded support is said of type
L-R if there exist two positive real numbers α and β
such that the membership function µ
e
n
of
e
n is given as
follows:
µ
e
n
(x) =









L

n−x
α

for n − α ≤ x < n when α 6= 0
1 for x = n
R

x−n
β

for n < x ≤ n + β when β 6= 0
0 for x ∈] − ∞, n − α[∪]n + β,+∞[.
The triplet (n, α, β) is called the tuple representa-
tion of fuzzy number
e
n. Real numbers α and β are
respectively called the left spread and the right spread
of
e
n.
Note that a real number n is identified to the fuzzy
number
e
n with α = β = 0 and Supp(
e
n) = {n}.
We denote by F(L,R) the family of fuzzy num-
bers of type L-R. Functions L and R are respectively
called the left spread function and the right spread
function of the family F(L, R), and by extension the
spread functions of
e
n itself. When L(x) = R(
x
k
), k > 0,
the fuzzy number is said symmetrical. Inside a given
family F(L,R), a tuple (n, α, β) represents an unique
element
e
n. The addition is an internal law of F(L,R)
defined by (n,α, β)+ (n
0
,α
0
,β
0
) = (n+ n
0
,α +α
0
,β +
β
0
) .
The approximate product “·“ is distributive with
respect to addition. As we study fuzzy equations with
real indeterminates, we consider only products of the
FCTA 2019 - 11th International Conference on Fuzzy Computation Theory and Applications
352

form q ·
e
n, with q ∈ R. In this case, the product de-
scribed by Dubois and Prade becomes exact:
q · (n,α,β) =

(qn,q α,q β) if q ≥ 0
(qn,−q β,−q α) if q ≤ 0 .
(1)
Note that the inversion of the spreads that keeps them
positive when q < 0 : −qβ and −q α are respectively
the left spread and the right spread of q · (n, α,β). In
particular, we have
−
e
n = −1 · (n,α,β) = (−n,β,α) . (2)
2.3 Parametric Representation
The parametric representation introduced in 1986 by
R. Goetschel et W. Voxman (Goetschel and Voxman,
1986) allows them to embed all the trapezoidal fuzzy
numbers into a topological vector space. The fol-
lowing definition is an adaptation for non-trapezoidal
fuzzy numbers:
Definition 2. The parametric form of a fuzzy number
e
n is an ordered pair [n, n] of functions from the real
interval [0, 1] to R which satisfy the following condi-
tions:
(1) n is a bounded left continuous non-increasing
function on [0,1],
(2) n is a bounded left continuous non-decreasing
function on [0,1],
(3) n(1) = n(1) = n.
Fuzzy number
e
n defined by functions n and n
has membership function µ
e
n
: R → [0,1] such that
µ
e
n
(x) = sup{r | n(r) ≤ x ≤ n(r)}.
A fuzzy arithmetic, described in the following
lemma, operates on parametric representation. It is
coherent with those of the tuple representation given
in Section 2.2.
Lemma 1. (Stefanini and Sorini, 2009) Let q ∈ R and
e
m = [m,m] and
e
n = [n,n] two fuzzy numbers. Then
1.
e
m =
e
n if and only if m(r) = n(r) and m(r) = n(r)
for each real r ∈ [0,1],
2.
e
m +
e
n = [m + n, m + n],
3. q ·
e
n =
(
[q · n, q · n] if q ≥ 0,
[q · n, q · n] if q ≤ 0
where, for any function f from R to R, the product g =
q · f represents the function defined as g(r) = q f (r)
for each r ∈ R.
2.4 From Tuple to Parametric
Representation
In this paper, we consider polynomials equations with
coefficients that are fuzzy numbers of a same fam-
ily F(L,R) satisfying the sufficient requirement that
Figure 1: Graph of functions 3 and 3 from the graph of a
linear membership function. Here, the left restriction (resp.
right restriction) µ
e
n
−
(resp. µ
e
n
+
) is the restriction of µ
e
n
to
the left (resp. right) of the core n.
the spread functions L and R are bijective. Our
solving method implies to rewrite algebraically each
fuzzy coefficient
e
n ∈ F(L,R) from tuple represen-
tation (n,α,β) into parametric representation. The
change of representation is given by formulas of
Proposition 1 below.
The parametric representation of
e
n is strongly re-
lated to its r-cuts
e
n
r
since functions n and n defined
by
n(r) = inf
r∈[0,1]
e
n
r
and n(r) = sup
r∈[0,1]
e
n
r
satisfy the requirements of Definition 2. This relation
appears graphically in Figure 1 where x
1
= n(1/2)
and x
2
= n(1/2) for a triangular fuzzy number
e
3 =
(3,2,3). The graph of functions n and n is obtained
by a plane rotation of the graph of the membership
function followed by a vertical symmetry.
Formally, the transformation is described by the
formulas below.
Proposition 1. Let
e
n = (n, α, β) ∈ F(L,R) where L
and R are bijective spread functions. Then the para-
metric representation [n,n] of
e
n satisfies the following
formulas:

n(r) = n − αL
−1
(r)
n(r) = n + βR
−1
(r) .
(3)
In particular, when the fuzzy number
e
n is triangular,
we have:
n(r) = α r + n − α and n(r) = −β r + n + β .
(4)
Proof 1. For the real number r ∈ [0, 1], Definition 1
implies r = L

n−n(r)
α

= R

n(r)−n
β

. As L and R
are bijective, n(r) and n(r) satisfy Identity (3) of the
proposition.
In the triangular case, we obtain formula (4) be-
cause L = R = F where F(x) = 1 − x is bijective with
F
−1
= F.
The Real Transform: Computing Positive Solutions of Fuzzy Polynomial Systems
353

3 THE REAL TRANSFORM OF A
FUZZY EQUATION
The goal of this paper is to find positive solutions of
a system of polynomial equations whose coefficients
are symmetrical fuzzy numbers belonging to a same
family F(L,R) where L and R are bijective. It is per-
formed by transforming independently each equation
in order to obtain a polynomial system with real coef-
ficients so that it can be solved by algebraic methods.
This section is devoted to the transform of only one
equation. Note that, in practice, we do not encounter
one isolated multivariate equation. For a system re-
duced to a unique equation, the number of variable
is generally reduced to only one too. This particu-
lar case can be treated with our method or by others
such as (Abbasbandy and Otadi, 2006), and recently
(Farahani et al., 2019), but it is not the purpose of this
paper.
We will use the following terminology: a vari-
able is said real if its represents any real number; a
real variable is said positive if it represents any pos-
itive real number, i.e. belonging to R
+
; a k-uplet
(b
1
,..., b
k
) of real variables or real numbers is said
positive if each component b
i
is positive. In this sec-
tion we consider an algebraic equation (E) with fuzzy
coefficients and k real variables x
1
,..., x
k
also called
the indeterminates.
Considering only positive real variables, in Sec-
tion 3.2 a crisp form of (E) is constructed in order to
deduce a collected crisp form of (E) ; in other words,
an algebraic system of equations with real coefficients
whose positive solutions are those of (E). In the liter-
ature this collected crisp form is formed by four equa-
tions obtained from (E) by an algorithm that applies
only when the fuzzy coefficients are triangular.
Moreover Section 3.3 establishes a formula that
provides a particular collected crisp form of (E)
formed by only three equations. We call it the real
transform of (E) and denote it by T (E). Section 3.4
compares the real transform T (E) to the usual col-
lected crisp form given in literature for the triangu-
lar case. Section 3.5 finally generalizes the results to
trapezoidal fuzzy numbers.
3.1 Preliminaries
Let d
d
d = (d
1
,..., d
k
) ∈ N
k
, x
x
x = (x
1
,..., x
k
) and
x
x
x
d
d
d
= x
d
1
1
···x
d
k
k
the monomial of multidegree d
d
d in
the variables x
1
,..., x
k
. In the same way, for
a
a
a = (a
1
,..., a
k
) ∈ R
k
, we denote by a
a
a
d
d
d
the prod-
uct a
d
1
1
···a
d
k
k
. For y
y
y = (y
1
,..., y
k
), we denote by
x
x
x × y
y
y the classical product (x
1
y
1
,..., x
k
y
k
). Note that
(x
x
x × y
y
y)
d
d
d
= x
x
x
d
d
d
y
y
y
d
d
d
.
In this section, we consider the polynomial equa-
tion
(E) :
∑
d
d
d∈Expon(E)
e
n
d
d
d
x
x
x
d
d
d
=
e
m , (5)
where Expon(E) is a finite subset of N
k
, with
e
n
d
d
d
∈ F(L,R) and
e
m ∈ F(L, R)\{(0, 0,0)} for all d ∈
Expon(E). The fuzzy numbers in (E) are symmetri-
cal and given under their respective tuple representa-
tion
e
m = (m,α,β) and
e
n
d
d
d
= (n
d
d
d
,α
d
d
d
,β
d
d
d
). For exam-
ple, when k = 3 and (E) is equation
e
3x
2
1
x
2
+
e
1x
4
3
=
e
6,
Expon(E) is the subset {(2,1, 0), (0,0,4)} of N
3
.
We denote by Sol
+
(E) the set of solutions of (E)
in R
+
k
:
Sol
+
(E) = {a
a
a ∈ R
+
k
|
∑
d
d
d∈Expon(E)
e
n
d
d
d
a
a
a
d
d
d
=
e
m } .
We search for Sol
+
(E) by using the r-cuts in order to
obtain an algebraic system with real coefficients that
can be solved with classical computer algebra meth-
ods.
We consider the case where the indeterminates
x
1
,..., x
k
are real and positive. In this context, we
seek the formula of the real transform T (E) of (E).
The positive solutions of the real algebraic system
T (E) are exactly the positive solutions of the fuzzy
algebraic equation (E).
3.2 Crisp Form of (E) to Find Sol
+
(E)
Algebraic solving of fuzzy equation (E) is usually
based on the passage of the L-R representation of
fuzzy numbers to their parametric representation. In
the presentation below we significally strengthen this
classical method for triangular fuzzy coefficients by
applying it to a generic system and by extending it to
more general fuzzy coefficients.
Following Lemma 1, equation (E) rewrites into
two equalities on the r-cuts of the left and right mem-
bers of (E) if all the x
x
x
d
d
d
, d
d
d ∈ Expon(E), are supposed
to represent reals of the same sign. Indeed, accord-
ing to Rule (3) of this lemma, the multiplication of
a fuzzy number by a scalar q splits into two cases:
q ≤ 0 and q ≥ 0. Thus we search only for the solu-
tions a
a
a ∈ R
+
k
since the real q := a
a
a
d
d
d
is then positive
for each d
d
d ∈ N
k
.
For a
a
a ∈ R
+
k
, according to Lemma 1, the following
equivalence applies:
a
a
a ∈ Sol
+
(E) ⇐⇒
"
∑
d
d
d∈Expon(E)
n
d
d
d
(r) a
a
a
d
d
d
,
∑
d
d
d∈Expon(E)
n
d
d
d
(r) a
a
a
d
d
d
#
= [m(r),m(r)] .
(6)
This equivalence leads us to consider C (E), the fol-
lowing system of two equations with real coefficients
FCTA 2019 - 11th International Conference on Fuzzy Computation Theory and Applications
354

and k + 1 variables x
1
,..., x
k
,r, called the crisp form
of (E) :
C (E) :





∑
d
d
d∈Expon(E)
n
d
d
d
(r) x
x
x
d
d
d
= m(r)
∑
d
d
d∈Expon(E)
n
d
d
d
(r) x
x
x
d
d
d
= m(r) .
Let F be a set of equations in R[x
1
,..., x
k
,r]. We put
Sol
+
k
(F) = {a
a
a ∈ R
+
k
| ∀r ∈ [0,1] (a
1
,...,a
k
,r) ∈ Sol(F)}
where Sol(F) is the set of the solutions of F in R
k+1
.
Take a
a
a = (a
1
,..., a
k
) ∈ R
+
k
. According to Equiva-
lence (6), the k-uplet a
a
a is a solution of (E) if and only
if for all real r ∈ [0,1] the (k + 1)-uplet (a
1
,..., a
k
,r)
is a solution of the crisp form C (E). In other words,
Sol
+
k
(C (E)) is the set of the positive solutions of the
fuzzy equation (E):
Sol
+
(E) = Sol
+
k
(C (E)) . (7)
In the particular triangular case, the crisp form has
two equations with linear expressions w.r.t. the vari-
able r in each side. It is a consequence of formulas
(4). The triangular case is easy because the spread
functions are equal to F : x 7→ 1 − x, with F
−1
= F.
The general case, when the spread functions are sim-
ply bijective, requires using inversion formulas (3)
with two indeterminates instead of only one. This
leads to the crisp form with two parameters in the fol-
lowing theorem:
Theorem 1. Let L and R be two spread functions and
(E) :
∑
d
d
d∈Expon(E)
e
n
d
d
d
x
x
x
d
d
d
=
e
m ,
be a fuzzy equation with coefficients in the family
F(L,R) given by their tuple representations as fol-
lows:
e
m = (m, α, β) and
e
n
d
d
d
= (n
d
d
d
,α
d
d
d
,β
d
d
d
) for d
d
d ∈
Expon(E). If the spread functions L and R are bi-
jective then the crisp form of (E) is given by:
C (E) :



∑
d
d
d
n
d
d
d
x
x
x
d
d
d
− m + (α −
∑
d
d
d
α
d
d
d
x
x
x
d
d
d
)u = 0
∑
d
d
d
n
d
d
d
x
x
x
d
d
d
− m + (−β +
∑
d
d
d
β
d
d
d
x
x
x
d
d
d
)v = 0 ,
(8)
where u = L
−1
(r) and v = R
−1
(r) for all r ∈ [0, 1].
For a
a
a ∈ R
+
k
, we have a
a
a ∈ Sol
+
(E) if and only if, for
all r ∈ [0,1], system (8) is satisfied by the (k+2)-uplet
(a
1
,..., a
k
,L
−1
(r), R
−1
(r)).
Proof 2. By definition, a spread function H sends
[0,1] to itself and if moreover H is bijective then its in-
verse H
−1
is continue and decreasing with H
−1
(1) =
0 and H
−1
(0) = 1. Suppose that the spread functions
L and R are bijective. As each r belongs to [0,1], we
can put u = L
−1
(r) and v = R
−1
(r). When r runs
throughout [0,1] in the increasing sens, the parame-
ters u and v run throughout the same interval [0,1]
in the decreasing sens. According to formulas (3), the
parametric form of the coefficients of the equation are
given by
n
d
d
d
(r) = n
d
d
d
− α
d
d
d
u , n
d
d
d
(r) = n
d
d
d
+ β
d
d
d
v
m(r) = m − α u , m(r) = m + βv .
(9)
for d
d
d ∈ Expon(E). Then the crisp form C (E) of (E)
given in (7) is written as a system of two equations
with k + 2 variables x
1
,..., x
k
,u,v, where u and v are
dependent on each other:
C (E) :



∑
d
d
d
n
d
d
d
x
x
x
d
d
d
− α
d
d
d
u x
x
x
d
d
d
= m − α u
∑
d
d
d
n
d
d
d
x
x
x
d
d
d
+ β
d
d
d
v x
x
x
d
d
d
= m + β v
By collecting all the terms on the left-hand-side of the
two equations, we find the crisp form expressed as in
the form (8) of the theorem. Last assertion in the theo-
rem about Sol
+
(E) follows directly from equality (7).
Note that Theorem 1 only requires that the restric-
tions on [0,1] of the two spread functions L and R are
bijective.
Our approach allows at the same time to improve
and to generalize the methods known so far. For in-
stance, results in (Molai et al., 2013) and (Boroujeni
et al., 2016) are restricted to triangular fuzzy num-
bers. Indeed, the crisp form of (E) with two parame-
ters u = L
−1
(r) and v = R
−1
(r) given in Identity (8) is
a generalization of the crisp form known in triangular
case with only one parameter r where r ∈ [0,1].
In the aforementioned articles, for each problem
to be solved, the algorithm computes the system C (E)
in variables x
1
,..., x
k
,r, which is linear w.r.t. r. Then
it is rewritten into an equivalent system of four al-
gebraic equations in x
1
,..., x
k
with real coefficients
called collected crisp form of (E). In next section, we
will show how to get a particular collected crisp form
reduced to three equations, for symmetrical fuzzy co-
efficients of any family F(L,R) such that the spread
functions L and R are bijective. This is the real trans-
form of (E). In addition, we explicitly give its formu-
lation from (E).
3.3 The Real Transform and the
Positive Real Solutions of (E)
We define here the real transform of a fuzzy equation
(E) and show that its positive real solutions are also
those of (E).
Definition 3. Let L and R be two spread functions and
(E) :
∑
d
d
d∈Expon(E)
e
n
d
d
d
x
x
x
d
d
d
=
e
m ,
The Real Transform: Computing Positive Solutions of Fuzzy Polynomial Systems
355

a fuzzy equation with symmetrical coefficients in the
family F(L, R) given by their representations in tu-
ple as follows:
e
n
d
d
d
= (n
d
d
d
,α
d
d
d
,β
d
d
d
) (d
d
d ∈ Expon(E)) and
e
m = (m,α,β). The real transform T (E) of (E) is the
following polynomial system over R:
T (E) :











∑
d
d
d∈Expon(E)
n
d
d
d
x
x
x
d
d
d
= m
∑
d
d
d∈Expon(E)
α
d
d
d
x
x
x
d
d
d
= α
∑
d
d
d∈Expon(E)
β
d
d
d
x
x
x
d
d
d
= β .
(10)
This definition naturally extends to a system (S) of
fuzzy equations such as (E). We denote by T (S) its
real transform, i.e. the system formed by the real
transforms of the equations in (S).
Theorem 2. According to Definition 3, if the two
spread functions L and R are bijective then the set of
positive real solutions of (E) equals the one of its real
transform; in other words:
Sol
+
(E) = Sol
+
(T (E)) .
Proof 3. Let be a
a
a ∈ R
+
k
. As the spread functions
L and R are bijective, we can apply Theorem 1. Ac-
cording to this theorem, we know that a
a
a ∈ Sol
+
(E)
if and only if, for all r ∈ [0,1], the crisp form of
(E) expressed in (8) is satisfied by the (k + 2)-uplet
(a
1
,..., a
k
,L
−1
(r), R
−1
(r)).
With r = 1, we have u = L
−1
(1) = 0. Then a
a
a ∈
Sol
+
(E) satisfies the equation
∑
d
n
d
d
d
x
x
x
d
d
d
= m. Note
that when r = 1 we have v = 0 too because R(0) = 1,
and we find the same equation and not a second one.
This is why we obtain three equations instead of four.
Then, by taking r = 0 we have u = L
−1
(0) = 1 and
v = R
−1
(0) = 1. By replacing in (8) the expression
∑
d
n
d
d
d
x
x
x
d
d
d
− m by 0 and each variable u and v by 1, we
deduce that a positive solution of (E) is also a positive
solution of the real transform T (E) of (E).
For the inverse inclusion, consider the crisp form
C (E) as a polynomial system in the variables of x
x
x
and with coefficients in the ring R[u,v]. Any solution
(a
1
,..., a
k
) ∈ R
k
of T (E) is also a solution of C (E)
in R
k
whatever the parameters u and v may be in the
interval [0,1]. Obviously it remains true when they
are furthermore connected by the constraint L
−1
(u) =
R
−1
(v) ∈ [0, 1]. Hence any positive real solution of
the real transform T (E) is also a positive real solu-
tion of the fuzzy equation (E).
Theorem 2 ensures that finding the positive real
roots of (E) amounts to finding the positive real roots
of its real transform T (E). Therefore it is no use
to develop intermediate computations on parametric
representation like the previous methods did in the
specific triangular case.
3.4 Comparison with Previous Methods
in the Triangular Case
Consider a system (S) formed by s polynomial equa-
tions with symmetrical fuzzy coefficients. In the spe-
cific case of triangular fuzzy numbers as coefficients,
the authors of (Molai et al., 2013) and (Boroujeni
et al., 2016) compute a collected crisp form of (S)
formed by 4s real algebraic equations. In this part we
are interested in the relationship between their col-
lected crisp form with 4 s equations and our collected
crisp system with 3 s equations, that is the real trans-
form of (S). For both systems, the positive real solu-
tions of each of these systems are also those of (S). It
is the principle of any collected crisp form of (S).
Consider below the system F
1
of Section 6 in
(Boroujeni et al., 2016):
F
1
:

(2,1,1)xy + (3,1,1)x
2
y
2
+ (2, 1, 1)x
3
y
3
= (7,3,3)
(5,1,1)xy + (2,3,1)x
2
y
2
+ (2, 2, 1)x
3
y
3
= (9,6,3).
Applied to first equation, the algorithm proposed in
(Boroujeni et al., 2016) produces the following col-
lected crisp form:







xy + x
3
y
3
− 3 + x
2
y
2
= 0,
xy + 2x
2
y
2
− 4 + x
3
y
3
= 0
−xy − x
3
y
3
+ 3 − x
2
y
2
= 0
3xy + 4x
2
y
2
− 10 + 3 x
3
y
3
= 0 ;
and it produces the following collected crisp form of
the second equation of F
1
:







xy + 3x
2
y
2
+ 2x
3
y
3
− 6 = 0,
4xy − x
2
y
2
− 3 = 0,
−xy − x
3
y
3
+ 3 − x
2
y
2
= 0,
6xy + 3x
2
y
2
− 12 + 3 x
3
y
3
= 0 .
Call T
1
the system formed of the eight preceding
equations.
Furthermore, by applying to F
1
our formula (10)
defining the real transform, we get T (F
1
), the follow-
ing system of six equations:
T (F
1
) :















2xy + 3x
2
y
2
+ 2x
3
y
3
= 7
xy + x
2
y
2
+ x
3
y
3
= 3
xy + x
2
y
2
+ x
3
y
3
= 3
5xy + 2x
2
y
2
+ 2x
3
y
3
= 9
xy + 3x
2
y
2
+ 2x
3
y
3
= 6
xy + x
2
y
2
+ x
3
y
3
= 3 .
An easy computation shows the equivalence of the
systems T
1
and T (F
1
), whose set of solutions is
{(x,y) ∈ R | xy = 1}. This phenomenon of equiva-
lence between both approaches may be explained in
a very general way as we show below by considering
the classical computation of the collected crisp form
FCTA 2019 - 11th International Conference on Fuzzy Computation Theory and Applications
356

obtained by an application of the algorithm of (Borou-
jeni et al., 2016) on the generic equation (5) of (E).
Let
e
m = (n,α,β) and
e
n
d
d
d
= (n
d
d
d
,α
d
d
d
,β
d
d
d
) (d
d
d ∈
Expon(E)) be the respective tuple representations of
the fuzzy coefficients of (E) that are assumed to be
triangular. According to formulas (4), in the triangu-
lar case the r-cuts are given by
e
n
d
d
d
(r) = [α
d
d
d
r + n
d
d
d
− α
d
d
d
,−β
d
d
d
r + n
d
d
d
+ β
d
d
d
]
e
m(r) = [αr + m − α ,−βr + m + β] .
for d
d
d ∈ Expon(E). For a triangular fuzzy number,
L = R = F, where F(x) = F
−1
(x) = 1 − x, being
known, the previous methods replace directly L
−1
(r)
and R
−1
(r) by their expression in the variable r in the
equations. That’s how they end up in the crisp form of
(E) below expressed as two polynomials in the vari-
able r:
C (E) :



(
∑
d
d
d
α
d
d
d
x
x
x
d
d
d
− α)r +
∑
d
d
d
(n
d
d
d
− α
d
d
d
) x
x
x
d
d
d
− m + α = 0
(β −
∑
d
d
d
β
d
d
d
x
x
x
d
d
d
)r +
∑
d
d
d
(n
d
d
d
+ β
d
d
d
) x
x
x
d
d
d
− m − β = 0.
A k-uplet (x
1
,..., x
k
) ∈ R
k
is a solution of C (E) for
all r ∈ [0,1] if and only if each coefficient w.r.t. the
variable r of these independent equations is zero. The
collected crisp form of (E) is therefore written















∑
d
d
d
α
d
d
d
x
x
x
d
d
d
= α
∑
d
d
d
(n
d
d
d
− α
d
d
d
) x
x
x
d
d
d
= m − α
∑
d
d
d
β
d
d
d
x
x
x
d
d
d
= β
∑
d
d
d
(n
d
d
d
+ β
d
d
d
) x
x
x
d
d
d
= m + β .
By applying this transformation to each equation in
the system F
1
, we find the collected crisp form T
1
of
our example. For the generic equation (E), by inject-
ing the first equation into the second one and by not-
ing that the last equation is the sum of the three other
ones, we obtain the real transform T (E) with three
equations defined in (10).
3.5 Case of Trapezoidal Fuzzy Numbers
In this section, we extend the definition of fuzzy
numbers to trapezoidal fuzzy numbers by allowing
µ
−1
e
n
({1}) to an interval [a,b]. As mentioned in Sec-
tion 2.1 our results adapt to symmetrical trapezoidal
fuzzy numbers with bounded support.
In this context, a fuzzy number
e
n with bounded
support is of type L-R if its membership function µ
e
n
has the following form:
µ
e
n
(x) =









L

a−x
α

for a − α ≤ x < a when α 6= 0
1 for x ∈ [a,b]
R

x−n
β

for b < x ≤ b + β when β 6= 0
0 for x ∈] − ∞, a − α[∪]b + β,+∞[.
Then the tuple representation of the fuzzy number
e
n
is the quadruplet (a,b,α,β). The expression of the
parametric representation given in Proposition 1 takes
the following form for a trapezoidal number of type
L − R whose spread functions L and R are bijective:

n(r) = a − αL
−1
(r)
n(r) = b + βR
−1
(r) .
(11)
When the equation (E) :
∑
d
d
d∈Expon(E)
e
n
d
d
d
x
x
x
d
d
d
=
e
m has
symmetrical trapezoidal fuzzy coefficients of type L-
R, where
e
n
d
d
d
= (a
d
,b
d
,α
d
,β
d
) and
e
m = (a,b,α,β) are
the tuple representations of the coefficients, the para-
metric forms (9) given in the proof of Theorem 1 be-
come
n
d
d
d
(r) = a
d
d
d
− α
d
d
d
u , n
d
d
d
(r) = b
d
d
d
+ β
d
d
d
v
m(r) = a − α u , m(r) = b + β v .
for d
d
d ∈ Expon(E). Applying the argument of Sec-
tion 3.3 (here L(1) = L(1) = 0 and R(0) = R(0) = 1),
we obtain in the same way a real transform of (E), but
this time with four equations:
T (E) :















∑
d
d
d
a
d
d
d
x
x
x
d
d
d
= a
∑
d
d
d
b
d
d
d
x
x
x
d
d
d
= b
∑
d
d
d
α
d
d
d
x
x
x
d
d
d
= α
∑
d
d
d
β
d
d
d
x
x
x
d
d
d
= β .
(12)
Consequently the use of the real transform for solv-
ing polynomial fuzzy systems will directly transpose
to systems with symmetrical trapezoidal fuzzy coeffi-
cients.
The real transform of a fuzzy system (S) of s equa-
tions, denoted T (S), with 4 s instead of 3 s equations,
is obtained by slightly applying formula (12) to each
equation of (S).
4 CONCLUSION
Up to now, given a fuzzy system (S) of s equations
and k indeterminates, the existing algebraic meth-
ods have performed computations with the paramet-
ric representation of the coefficients to obtain the col-
lected crisp form of (S) formed by 4s real equations.
We show that these computations are superfluous and
exhibit a formula that defines an equivalent system
with 3s real equations. We call it the real transform
T (S) of the system (S). As a main property, it has the
same positive solutions as (S) (Theorem 2).
Unlike the previous methods that were restricted
to triangular fuzzy numbers, our results apply to sym-
metrical fuzzy numbers of any family F(L,R) where
The Real Transform: Computing Positive Solutions of Fuzzy Polynomial Systems
357

the spread functions L and R are bijective. Moreover
there is no use to compute the inverse of the spread
functions since the real transform is a universal for-
mula independent from L and R.
Further work will explore how to obtain the whole
set of real zeros of (E), not only positive zeros.
The problem when computing with real variables and
fuzzy numbers is intrinsic to fuzzy numbers since the
product by a real scalar is expressed differently de-
pending on the sign of this scalar.
One idea to work around the problem of unknown
sign of x
x
x
d
d
d
is to only focus on positive solutions by
putting back the issue on the fuzzy coefficients. We
solve the system by introducing an artificial k-uplet
of signs I ∈ {−1, 1}
k
, and we replace
e
n
d
d
d
x
x
x
d
d
d
where x
x
x
represents any real by I
d
d
d
e
n
d
d
d
|x
x
x|
d
d
d
which equals
e
n
d
d
d
|x
x
x|
d
d
d
or −
e
n
d
d
d
|x
x
x|
d
d
d
depending on the sign of x
x
x
d
d
d
, where |x
x
x|
is positive. The 2
k
possible k-uplets for I induce the
same number of induced equations E(I). Hence, we
recover the solutions of (E) from the positive solu-
tions of its 2
k
induced equations E(I).
To obtain the real solutions of (E), it will be nec-
essary and sufficient to collect the positive real solu-
tions of the 2
k
real transforms T (E(I)). In practice,
since the equations E(I) are not pairwise distinct, a
strategy will be needed to reduce the number of in-
duced systems T (E(I)) to solve, in order to imple-
ment an optimized algorithm that automatizes the re-
search of solutions by avoiding the studies of signs
needed in previous methods.
ACKNOWLEDGMENT
This work was supported by ANR ARRAND 15-
CE39-0002-01
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