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Fuzzy Sets and Systems ••• (••••) •••–•••
www.elsevier.com/locate/fss
Existence, uniqueness and continuity of trapezoidal approximations
of fuzzy numbers under a general condition
Adrian I. Ban ∗ , Lucian Coroianu
Department of Mathematics and Informatics, University of Oradea, Universităţii 1, 410087 Oradea, Romania
Abstract
The main aim of this paper is to characterize the set of real parameters associated to a fuzzy number, represented in a general
form which include the most important characteristics, with the following property: for any given fuzzy number there exists at
least a trapezoidal fuzzy number which preserves a fixed parameter. The uniqueness of the nearest trapezoidal fuzzy number with
this property is proved, the average Euclidean distance being considered. As an important property, each resulting trapezoidal
approximation operator is continuous. The main results are illustrated by examples.
© 2013 Elsevier B.V. All rights reserved.
Keywords: Fuzzy number; Trapezoidal fuzzy number; Trapezoidal approximation; Continuity
1. Introduction
In many papers (see e.g. [1–4,8–10,12,15,31–33,39,41,47–52]) a problem of the following kind was tackled: given
a distance D between fuzzy numbers and a fuzzy number A, find T , the nearest trapezoidal (or interval, or triangular,
or semi-trapezoidal) fuzzy number of A, with respect to D, such that the real parameter p (or real interval, or pair
of real parameters) is preserved, that is p(A) = p(T ). The topic has become very dynamic after the consideration of
trapezoidal approximation as a reasonable compromise between two opposite tendencies: to lose too much information
and to introduce too sophisticated form of approximation from the point of view of computation (see [32]). On the
other hand, it is worth noting that in the least years many papers ([5,26,27,36–38,40,43–46], etc.) were elaborated to
describe methods and algorithms with trapezoidal or triangular fuzzy numbers as input data such that the usefulness
of trapezoidal (or triangular) approximations in the solving of problems is obvious. In the present paper, the existence
and uniqueness of a trapezoidal approximation under preservation of a real parameter associated with a fuzzy number
A in the general form
1
p(A) = a
1
AL (α) dα + b
0
1
AU (α) dα + c
0
1
αAL (α) dα + d
0
αAU (α) dα,
0
* Corresponding author.
E-mail addresses: [email protected] (A.I. Ban), [email protected] (L. Coroianu).
0165-0114/$ – see front matter © 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.fss.2013.07.004
(1)
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which includes the most important characteristics of a fuzzy number (expected value, value, ambiguity, width, left- and
right-hand ambiguity) and linear combinations of them is studied. In other words, we establish for which parameters
the calculus of the trapezoidal approximation operator could be performed.
The continuity is between the criteria which a trapezoidal, triangular, semi-trapezoidal or interval approximation
operator should possess (see [32]). In the papers [2,12–15,31,48,49,51] the continuity was studied for each particular
trapezoidal approximation operator, that is for every real parameter which is preserved. The sophisticated proofs
start from the expression of trapezoidal approximation. The continuity of a trapezoidal approximation operator under
preservation of a parameter in the general form (1) is proved in the present article. The results of continuity in [13,15,
31,48] are immediate consequences.
The paper is organized as follows. In Section 2 several preliminaries about fuzzy numbers and extended trapezoidal
fuzzy numbers – representations, real characteristics, distances – are presented. Some useful distance properties of the
extended trapezoidal approximation of a fuzzy number and properties of convergence are added in Appendices A
and B. In Section 3 we determine a, b, c, d ∈ R in (1) for which there exists a trapezoidal fuzzy number Tp (A)
such that p(A) = p(Tp (A)), for any given fuzzy number A. The uniqueness of Tp (A) in the additional condition
of nearness to A with respect to the average Euclidean distance between fuzzy numbers is proved in Section 4. The
continuity of the trapezoidal approximation operator Tp is studied in Section 5, based on two lemmas with the enough
sophisticated proofs included in Appendices C and D.
2. Preliminaries
We begin by recalling some basic notions and notations used in this paper.
Definition 1. (See [22].) A fuzzy number A is a fuzzy subset of the real line A : R → [0, 1] satisfying the following
properties:
(i)
(ii)
(iii)
(iv)
A is normal (i.e. there exists x0 ∈ R such that A(x0 ) = 1);
A is fuzzy convex (i.e. A(λx1 + (1 − λ)x2 ) min(A(x1 ), A(x2 )), for every x1 , x2 ∈ R and λ ∈ [0, 1]);
A is upper semicontinuous on R (i.e. ∀ε > 0, ∃δ > 0 such that A(x) − A(x0 ) < ε, whenever |x − x0 | < δ);
cl{x ∈ R: μA (x) > 0} is compact, where cl(M) denotes the closure of the set M.
The α-cut, α ∈ (0, 1], of a fuzzy number A is a crisp set defined as
Aα = x ∈ R: A(x) α .
The support supp A and the 0-cut A0 of a fuzzy number A are defined as
supp A = x ∈ R: A(x) > 0
and
A0 = cl x ∈ R: A(x) > 0 .
Every Aα , α ∈ [0, 1], is a closed interval
Aα = AL (α), AU (α)
where
AL (α) = inf x ∈ R: A(x) α ,
AU (α) = sup x ∈ R: A(x) α
(2)
(3)
for any α ∈ (0, 1] and [AL (0), AU (0)] = A0 . We denote by F (R) the set of all fuzzy numbers.
In what follows, we recall the most important characteristics of a fuzzy number. The expected interval EI(A) and
the expected value EV(A) of a fuzzy number A are given by (see [23,35])
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EI(A) = E∗ (A), E ∗ (A) =
1
1
AL (α) dα,
0
3
AU (α) dα ,
(4)
0
1
EV(A) = E∗ (A) + E ∗ (A) .
(5)
2
The ambiguity Amb(A), value Val(A), width w(A), left-hand ambiguity AmbL (A), right-hand ambiguity AmbU (A)
of a fuzzy number A are introduced by (see [16,21,30,34])
1
Amb(A) =
α AU (α) − AL (α) dα,
(6)
0
1
α AU (α) + AL (α) dα,
(7)
AU (α) − AL (α) dα,
(8)
Val(A) =
0
1
w(A) =
0
1
AmbL (A) =
α EV(A) − AL (α) dα,
(9)
α AU (α) − EV(A) dα.
(10)
0
1
AmbU (A) =
0
A well-known metric on the set of fuzzy numbers, which is an extension of the Euclidean distance, is defined by
(see [30])
1
d (A, B) =
2
2
AL (α) − BL (α) dα +
0
1
2
AU (α) − BU (α) dα.
(11)
0
Fuzzy numbers with simple membership functions are preferred in practice. The most often used fuzzy numbers
are so-called trapezoidal fuzzy numbers. A trapezoidal fuzzy number T , Tα = [TL (α), TU (α)], α ∈ [0, 1], is given by
TL (α) = t1 + (t2 − t1 )α
(12)
TU (α) = t4 − (t4 − t3 )α,
(13)
and
with t1 t2 t3 t4 . We denote by
T = (t1 , t2 , t3 , t4 )
a trapezoidal fuzzy number and by F T (R) the set of all trapezoidal fuzzy numbers.
Sometimes (see [50]) another representation of a trapezoidal fuzzy number T is useful, namely
1
TL (α) = l + x α −
,
2
1
TU (α) = u − y α −
2
for every α ∈ [0, 1], with l, u, x, y ∈ R such that
(14)
(15)
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x 0,
(16)
y0
(17)
2u − 2l x + y.
(18)
and
The relationships between t1 , t2 , t3 , t4 and l, u, x, y in the representations of trapezoidal fuzzy numbers are immediate.
We denote by T = [l, u, x, y] a trapezoidal fuzzy number in representation (14)–(15).
The distance introduced in (11) between T = [l, u, x, y] and T = [l , u , x , y ] becomes (see [48])
2 2
2
2
1
1
d 2 T , T = l − l + u − u +
x − x +
y − y .
12
12
(19)
Definition 2. (See [48].) An extended trapezoidal fuzzy number T = [l, u, x, y] is an ordered pair (TL , TU ) of polynomial functions of degree less than or equal to 1, that is TL (α) = l + x(α − 12 ), TU (α) = u − y(α − 12 ), α ∈ [0, 1],
where l, u, x, y ∈ R.
Remark 3. The profiles TL and TU of an extended trapezoidal fuzzy number T have the same form as in (14), (15)
but l, u, x, y ∈ R may fail to satisfy (16)–(18). Even if the same notations for trapezoidal fuzzy numbers and extended
trapezoidal fuzzy numbers are used, no confusion is possible in the present paper.
The distance between two extended trapezoidal fuzzy numbers or between a fuzzy number and an extended trapezoidal fuzzy number is similarly defined as in (11) or (19). We denote by FeT (R) the set of all extended trapezoidal
fuzzy numbers.
The extended trapezoidal approximation Te (A) = [le (A), ue (A), xe (A), ye (A)] of a given fuzzy number A is
the extended trapezoidal fuzzy number which minimizes the distance d(A, X), where X ∈ FeT (R). In the paper [7] the authors proved that Te (A) is not always a fuzzy number. The extended trapezoidal approximation
Te (A) = [le (A), ue (A), xe (A), ye (A)] of A ∈ F (R), Aα = [AL (α), AU (α)], α ∈ [0, 1], is determined by (see [48])
1
le (A) =
AL (α) dα,
(20)
0
1
ue (A) =
AU (α) dα,
(21)
1 1
AL (α) dα,
α−
xe (A) = 12
2
(22)
0
0
1 1
ye (A) = −12
AU (α) dα.
α−
2
(23)
0
It is well known (see [48, Lemma 2.1] or [8, Lemma 1]) that
xe (A) 0
(24)
ye (A) 0
(25)
and
and from the definition of a fuzzy number the inequality le (A) ue (A) is valid for any fuzzy number A.
Some distance properties of the extended trapezoidal approximation operator Te : F (R) → FeT (R) are given in
Appendix A. We also need a few properties of convergence to prove the main results of the paper. They are included
in Appendix B.
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3. Existence of trapezoidal representation of a fuzzy number under a general condition
It is often useful to replace a fuzzy number by a real number, that is to introduce an operator p : F (R) → Rn
(n = 1 or n = 2). For example, ranking indices are introduced to order the fuzzy numbers (see [5,18–20,28]), as a
necessary step towards the development of methods in decision making, expert systems or data mining with fuzzy
data. On the other hand, from a computational point of view, it would be more convenient if for a given A ∈ F (R)
we could find B ∈ Ω ⊂ F (R), where Ω is a set of fuzzy numbers with a simpler form, such that p(A) = p(B).
In the recent literature we can find many examples of this kind (see [4,8–12,15,31,32,48,50–52]), Ω being the set
of real intervals, triangular fuzzy numbers, trapezoidal fuzzy numbers, semi-trapezoidal fuzzy numbers or other and
p(A) = EI(A), p(A) = Amb(A), p(A) = (Val(A), Amb(A)) or other. As a conclusion, it would be important to know
if for a given p : F (R) → Rn the equation p(A) = p(X) has at least a solution in the set Ω ⊂ F (R). Taking into
account the strong motivation in [32] and since in most applications the trapezoidal fuzzy numbers are preferred, in
this paper we consider Ω = F T (R).
On the other hand, the most important characteristics of a fuzzy number A are linear combinations of
le (A), ue (A), xe (A) and ye (A). Indeed, from (4)–(10) and (20)–(23) we obtain
EI(A) = le (A), ue (A) ,
(26)
1
1
EV(A) = le (A) + ue (A),
2
2
1
1
1
1
Amb(A) = − le (A) + ue (A) − xe (A) − ye (A),
2
2
12
12
1
1
1
1
Val(A) = le (A) + ue (A) + xe (A) − ye (A),
2
2
12
12
w(A) = ue (A) − le (A),
1
1
1
AmbL (A) = − le (A) + ue (A) − xe (A),
4
4
12
1
1
1
AmbU (A) = − le (A) + ue (A) − ye (A),
4
4
12
therefore it is justified to consider the operator p in the class
P = p : F (R) → R|p(A) = ale (A) + bue (A) + cxe (A) + dye (A), a, b, c, d ∈ R .
(27)
(28)
(29)
(30)
(31)
(32)
To sum up, our first aim is to characterize p ∈ P with the following property
∀A ∈ F (R), ∃B ∈ F T (R) such that p(A) = p(B)
or, in other words, to find the set
P0 = p ∈ P: ∀A ∈ F (R), ∃B ∈ F T (R) such that p(A) = p(B) .
Before proving, by examples, that P \ P0 = ∅, let us give the following lemma, useful in the proof of the main result
too.
Lemma 4. Let us consider p ∈ P, p(A) = ale (A) + bue (A) + cxe (A) + dye (A), A ∈ F (R). We have p(T ) = al +
bu + cx + dy, for any trapezoidal fuzzy number T = [l, u, x, y].
Proof. The proof is immediate since one can easily see (by direct calculations or from the fact that the operator Te
satisfies the identity criteria, see [32]) that le (T ) = l, ue (T ) = u, xe (T ) = x and ye (T ) = y. 2
Example 5. We consider the operator p ∈ P given by
p(A) = 2ue (A) − 2le (A) − xe (A) − ye (A)
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for every A ∈ F (R). If A is a fuzzy number such that Te (A) = [le (A), ue (A), xe (A), ye (A)] is not a fuzzy number
then it follows (see (16)–(18), (24) and (25)),
2ue (A) − 2le (A) − xe (A) − ye (A) < 0.
This means that any solution X = [l, u, x, y] ∈ F T (R) of the equation p(X) = p(A), that is
2u − 2l − x − y = 2ue (A) − 2le (A) − xe (A) − ye (A)
does not satisfy (18), a contradiction. We obtain p ∈
/ P0 , that is P \ P0 = ∅.
/ P0 , as the following example proves, therefore our study is
It is well known that Amb ∈ P0 (see [15]), but AmbL ∈
again justified.
Example 6. Let us consider the fuzzy number A given by
1
1
1 2
12 α − 2 α , if α ∈ 0, 12 ,
AL (α) =
1 1
if α ∈ 12
,1 ,
288 ,
1
,
288
We obtain (see (9))
AU (α) =
α ∈ [0, 1].
11
.
497 664
Because the left-hand ambiguity of a trapezoidal fuzzy number T = (t1 , t2 , t3 , t4 ) is always a non-negative value given
by
AmbL (A) = −
−t1 − 5t2 + 3t3 + 3t4
24
we obtain AmbL ∈
/ P0 .
AmbL (T ) =
The following general result gives a characterization of the operators p ∈ P0 .
Theorem 7. Let us consider the operator p ∈ P,
p(A) = ale (A) + bue (A) + cxe (A) + dye (A).
Then p ∈ P0 if and only if one of the following requirements holds:
(i)
(ii)
(iii)
(iv)
a + b = 0;
a = b = 0;
a + b = 0, a = 0 and ac > 1/2 or da > 1/2;
a + b = 0, a =
0, ac 1/6 and da 1/6.
Proof. For a fixed A ∈ F (R) the form of the equation p(A) = p(X), X ∈ F T (R) is (see Lemma 4)
al + bu + cx + dy = ale (A) + bue (A) + cxe (A) + dye (A),
(33)
where a solution (l(A), u(A), x(A), y(A)) of the above equation has to satisfy (16)–(18). The proof consists in two
steps. Firstly, we prove that in each one of the cases (i)–(iv) Eq. (33) has at least one solution in F T (R), for any
arbitrary fuzzy number A. Then, for the remaining cases we find a fuzzy number (for each case in part) such that for
this fuzzy number, (33) has no solution in F T (R).
So, we start by considering a, b, c, d in one of the cases (i)–(iv) and we prove that (33) has at least one solution in
F T (R) for a given A ∈ F (R).
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(i) It is immediate that (l(A), u(A), x(A), y(A)) with
a + 3b
2b
le (A) −
ue (A),
a+b
a+b
2a
3a + b
u(A) = −
le (A) +
ue (A),
a+b
a+b
x(A) = xe (A),
l(A) =
y(A) = ye (A),
is a solution of (33) and by (24), (25) and Lemma 23 in Appendix A we can easily check that (16)–(18) are
satisfied, it is immediate that [l(A), u(A), x(A), y(A)] is a trapezoidal fuzzy number.
(ii) In this case (33) becomes
cx + dy = cxe (A) + dye (A).
A solution of this equation is (3le (A), 3ue (A), xe (A), ye (A)) and it is very easy to verify that [3le (A), 3ue (A),
xe (A), ye (A)] is a trapezoidal fuzzy number.
(iii) Eq. (33) becomes
c
c
d
d
u − l − x − y = ue (A) − le (A) − xe (A) − ye (A).
a
a
a
a
(34)
We distinguish two cases: (iii)a ue (A) − le (A) − ac xe (A) − da ye (A) 0 and (iii)b ue (A) − le (A) − ac xe (A) −
d
a ye (A) < 0.
Case (iii)a In this case a solution of (34) is (l(A), u(A), x(A), y(A)) with
l(A) = 0,
c
d
u(A) = ue (A) − le (A) − xe (A) − ye (A),
a
a
x(A) = 0,
y(A) = 0.
Again it is easy to check that [l(A), u(A), x(A), y(A)] is a trapezoidal fuzzy number.
Case (iii)b If ac > 12 then a solution of (34) is (l(A), u(A), x(A), y(A)) with
l(A) = 0,
ue (A) − le (A) − ac xe (A) − da ye (A)
,
2(1/2 − c/a)
x(A) = 2u(A),
u(A) =
y(A) = 0.
Since the hypothesis imply u(A) > 0 it is easy to check that [l(A), u(A), x(A), y(A)] is a trapezoidal fuzzy
number. If da > 12 then a solution of (34) is (l(A), u(A), x(A), y(A)) with
l(A) = 0,
ue (A) − le (A) − ac xe (A) − da ye (A)
,
2(1/2 − d/a)
x(A) = 0,
u(A) =
y(A) = 2u(A)
and [l(A), u(A), x(A), y(A)] is a trapezoidal fuzzy number too.
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(iv) The hypothesis, (24), (25) and Lemma 23 in Appendix A imply
c
d
1
1
ue (A) − le (A) − xe (A) − ye (A) ue (A) − le (A) − xe (A) − ye (A)
a
a
6
6
0.
This means that we can choose the same solution of (33) as in the case (iii)a .
Now, we prove that in the remaining cases we can always find a fuzzy number A such that the equation p(A) =
p(X), X ∈ F T (R) has no solution. Analyzing (i)–(iv) we conclude that the remaining cases are
(a) a + b = 0, a = 0, c/a ∈ (1/6, 1/2], d/a 1/2, and
(b) a + b = 0, a =
0, c/a 1/2, d/a ∈ (1/6, 1/2].
We discuss only case (a) because, due to the symmetry, the reasoning in case (b) is similar. Firstly, we notice again
that the equation under study has the explicit form given in (34). Let β > 0 be such that c/a = 1/6 + β. For any
ε ∈ (0, 1) we consider the fuzzy number Aε , (Aε )α = [(Aε )L (α), (Aε )U (α)], α ∈ [0, 1], where
εα α 2
ε
6 − 2 , if α ∈ 0, 6 ,
(Aε )L (α) =
ε2
if α ∈ 6ε , 1
72 ,
2
ε
and (Aε )U (α) = (Aε )L (1) = 72
, for every α ∈ [0, 1]. Let
Te (Aε ) = le (Aε ), ue (Aε ), xe (Aε ), ye (Aε )
denotes the extended trapezoidal approximation of Aε . Noting that (Aε )L is differentiable and using integration by
parts and relations (20)–(23), we obtain
1
xe (Aε ) + (6 − ε)le (Aε ) =
1
(Aε )L (α)(12α − ε) dα = (6 − ε)(Aε )L (1) −
0
(Aε )L (α) 6α 2 − εα dα
0
1
= (6 − ε)ue (Aε ) −
(Aε )L (α) 6α 2 − εα dα.
0
Since
(Aε )L (α) =
ε
6
− α,
if α ∈ 0, 6ε ,
if α ∈ 6ε , 1
0,
1
it is immediate that 0 (Aε )L (α)(6α 2 − εα) dα < 0, which implies
xe (Aε ) > (6 − ε)ue (Aε ) − (6 − ε)le (Aε ).
In addition, since ye (Aε ) = 0, from (35) we obtain
c
d
ue (Aε ) − le (Aε ) − xe (Aε ) − ye (Aε )
a
a
c
1
ε
< xe (Aε ) + ue (Aε ) − le (Aε ) − xe (Aε )
6
6
a
1
1
ε
= xe (Aε ) + ue (Aε ) − le (Aε ) − xe (Aε ) − βxe (Aε )
6
6
6
ε
= ue (Aε ) − le (Aε ) − βxe (Aε )
6
ε
ε
xe (Aε ) − βxe (Aε ) =
− β xe (Aε ).
(35) is used here 6(6 − ε)
6(6 − ε)
(35)
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ε0
Clearly, we can chose ε0 ∈ (0, 1) such that 6(6−ε
< β (we recall, β > 0) and noting that in (35) we get xe (Aε0 ) > 0
0)
(this is immediate since from the definition of a fuzzy number we have ue (Aε0 ) le (Aε0 )), we obtain that
ε
− β)xe (Aε ) < 0 and this implies
( 6(6−ε)
c
d
ue (Aε0 ) − le (Aε0 ) − xe (Aε0 ) − ye (Aε0 ) < 0.
a
a
(36)
Therefore, if (l(Aε0 ), u(Aε0 ), x(Aε0 ), y(Aε0 )) is a solution of Eq. (34) for the case when A := Aε0 , then since
c/a 1/2 and d/a 1/2 and taking into account (36), in conditions (16) and (17), we get
1
1
c
d
u(A) − l(A) − x(A) − y(A) u(A) − l(A) − x(A) − y(A)
2
2
a
a
c
d
= ue (Aε0 ) − le (Aε0 ) − xe (Aε0 ) − ye (Aε0 ) < 0.
a
a
This means that the quadruple (l(Aε0 ), u(Aε0 ), x(Aε0 ), y(Aε0 )) does not satisfy condition (18) and therefore
[l(Aε0 ), u(Aε0 ), x(Aε0 ), y(Aε0 )] cannot be a fuzzy number. Summarizing, we can conclude that in the case when
a + b = 0, a = 0, c/a ∈ (1/6, 1/2], d/a 1/2 we can find a fuzzy number A such that the equation p(X) = p(A),
X ∈ F T (R) has no solution. As we have said before, by similar reasonings we get to the same conclusion in the case
when a + b = 0, a = 0, c/a 1/2, d/a ∈ (1/6, 1/2]. Now the proof is complete. 2
Remark 8. Taking into account Theorem 7 and (27)–(30) we immediately get EV, Amb, Val, w ∈ P0 .
Example 9. Because (see (31))
1
1
1
AmbL (A) = − le (A) + ue (A) − xe (A),
4
4
12
1
that is a = − 14 , b = 14 , c = − 12
, d = 0, Theorem 7 establishes that AmbL ∈
/ P0 , a result in concordance with
Example 6. The same conclusion is valid in the case of right-hand ambiguity too because
1
1
1
AmbU (A) = − le (A) + ue (A) − ye (A),
4
4
12
1
that is a = − 14 , b = 14 , c = 0, d = − 12
. We conclude that for a given A ∈ F (R) it is not sure that there exists
T ∈ F T (R) with AmbU (A) = AmbU (T ). It is interesting that AmbL + AmbU = Amb ∈ P0 even if AmbL ∈
/ P0 and
/ P0 .
AmbU ∈
If we change the representation of the elements of P then we get the following result.
Corollary 10. If p : F (R) → R is given by
1
p(A) = a
1
AL (α) dα + b
0
1
AU (α) dα + c
0
1
αAL (α) dα + d
0
αAU (α) dα,
0
a, b, c, d ∈ R then p ∈ P. Moreover, p ∈ P0 if and only if one of the following requirements holds:
(i) 2a + 2b + c + d = 0,
(ii) 2a + c = 2b + d = 0,
(iii) 2a + 2b + c + d = 0, 2a + c = 0, and
(iv) 2a + 2b + c + d = 0, 2a + c = 0,
c
2a+c
c
3(2a+c)
> 1 or
1 and
Proof. By (20)–(23), for a fuzzy number A we have
d
2b+d
d
3(2b+d)
1.
> 1,
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10
1
AL (α) dα = le (A),
0
1
AU (α) dα = ue (A),
0
1
αAL (α) dα =
1
1
xe (A) + le (A),
12
2
0
1
αAU (α) dα = −
1
1
ye (A) + ue (A)
12
2
0
therefore we get
c
d
xe (A) − ye (A)
12
12
which proves that p ∈ P. The proof follows by applying Theorem 7. 2
p(A) = (a + c/2)le (A) + (b + d/2)ue (A) +
4. Uniqueness of trapezoidal approximations under a general condition
Let p ∈ P0 , the set characterized in Theorem 7. For any fuzzy number A we denote
Up (A) = T ∈ F T (R): p(A) = p(T ) .
The following natural question arises: which trapezoidal fuzzy number in Up (A) would be the best choice for a
given fuzzy number A? Inspired by the numerous papers treating the trapezoidal approximation operators written in
the last few years [3,8,12,15,31,32,50] we would consider T (A), where T (A) is the nearest trapezoidal fuzzy number
to A with respect to the average Euclidean metric d (see (11)), that is
d A, T (A) = min d(A, T ),
T ∈Up (A)
is the best choice. Taking into account the result of existence in Theorem 7 we present the following result.
Theorem 11. If p ∈ P0 , p(A) = ale (A) + bue (A) + cxe (A) + dye (A) then for each fuzzy number A ∈ F (R) there
exists a unique trapezoidal fuzzy number T (A) ∈ Up (A) such that
d A, T (A) = min d(A, T ).
(37)
T ∈Up (A)
Proof. Let A ∈ F (R). By Proposition 19 in Appendix A it follows that d 2 (A, B) = d 2 (A, Te (A)) + d 2 (Te (A), B),
for any B ∈ F T (R). Since d 2 (A, Te (A)) is constant and taking into account (19) it results that
T (A) = l(A), u(A), x(A), y(A) ∈ Up (A)
and it satisfies (37) if and only if (l(A), u(A), x(A), y(A)) ∈ R4 is a solution of the problem
2 2
2
2
1
1
min l − le (A) + u − ue (A) +
x − xe (A) +
y − ye (A)
12
12
under the conditions
x 0,
(38)
y 0,
(39)
x + y 2(u − l),
(40)
al + bu + cx + dy = ale (A) + bue (A) + cxe (A) + dye (A).
(41)
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In the paper [48] the author proved that D : R4 × R4 → [0, ∞)
2 2
2
2
1
1
D 2 (l, u, x, y), l , u , x , y = l − l + u − u +
x − x +
y − y
12
12
is a distance over the space R4 , which is generated by an inner product. Then since from the hypothesis (in fact from
Theorem 7) it results that there exists a quadruple (l0 , u0 , x0 , y0 ) ∈ R4 which satisfies conditions (38)–(41), it follows
that the set
Φ(A) = (l, u, x, y) ∈ R4 : (l, u, x, y) satisfies (38)–(41)
is nonempty. Moreover, one can easily prove that the set Φ(A) is a closed convex subset of the topological vector
space (R4 , D). Summarizing, we obtain that (l(A), u(A), x(A), y(A)) is the orthogonal projection with respect to
the distance D of (le (A), ue (A), xe (A), ye (A)) onto the set Φ(A). Since (R4 , D) is a Hilbert space and Φ(A) is a
nonempty closed convex set, it is known (see, e.g., [42, Theorem 4.10, p. 79]) that this projection exists and it is
unique. This proves that T (A) = [l(A), u(A), x(A), y(A)] exists and it is unique. 2
Example 12. Taking into account Example 8 and Theorem 11 we obtain that for each given A ∈ F (R) there exists
a unique nearest trapezoidal fuzzy number T (A) (with respect to (11)) of A such that EV, or Amb, or Val, or w are
preserved. The result was already proved in the case of Amb (see [15, Theorems 9 and 10]).
Example 13. Delgado, Vila and Voxman [21] considered that value and ambiguity (see (6) and (7)) appear to be suitable to be used together for comparison of fuzzy numbers. They introduced the ranking index of a fuzzy number A by
riβ,γ (A) = βVal(A) + γ Amb(A),
where β ∈ [0, 1] and γ ∈ [−1, 1] are given such that |γ | β. Instead of comparing riβ,γ (A) and riβ,γ (B) in order to rank the fuzzy numbers A and B we can compare riβ,γ (T (A)) and riβ,γ (T (B)), where T is an operator
which preserves riβ,γ . The idea was already launched in [12]. In this context it is important to mention that, for
every given A ∈ F (R), there exists a unique nearest trapezoidal fuzzy number (t1 (A), t2 (A), t3 (A), t4 (A)) such that
riβ,γ (t1 , t2 , t3 , t4 ) = riα,β (A). Indeed,
1
riβ,γ (A) = (β + γ )
1
αAU (α) dα + (β − γ )
0
αAL (α) dα,
0
and, with the notations in Corollary 10,
a = 0,
b = 0,
c = β − γ,
d = β + γ,
therefore (i) in Corollary 10 is valid if β = 0 and (iv) in Corollary 10 is valid if β = 0. As a conclusion, riβ,γ ∈ P0 and
taking into account Theorem 11 we obtain the uniqueness of the trapezoidal approximation which preserves riβ,γ .
Remark 14. The merit of the results in Theorems 7 and 11 is the finding of the cases when the calculus of a trapezoidal
approximation under preserving of a specific parameter p ∈ P0 can be performed. For fixed a, b, c, d ∈ R such that
p ∈ P0 we obtain the trapezoidal approximation by following the ideas in [33], [8], [9], [10] or [12].
5. Continuity of trapezoidal approximations under a general condition
Throughout this section, for p ∈ P0 we denote by Tp the well-defined (according with Theorem 11) trapezoidal
approximation operator Tp : F (R) →F T (R), where Tp (A) ∈ Up (A) and it satisfies
d A, Tp (A) = min d(A, T ).
T ∈Up (A)
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Our aim is to discuss the continuity of Tp as an important property of this kind of operators (see [32]). In applications, where it is sometimes indicated as robustness, the criterion of continuity is of extreme importance too. Before
giving a general result of continuity for the operators Tp , p ∈ P0 , we propose the following helpful lemmas. The
proofs of them are included in Appendices C and D, respectively.
Lemma 15. Let us consider p ∈ P,
p(A) = ale (A) + bue (A) + cxe (A) + dye (A),
A ∈ F (R).
Suppose that there exists a convergent sequence of fuzzy numbers (An )n1 and a convergent sequence of trapezoidal
fuzzy numbers (Tn )n1 such that limn→∞ An = A and limn→∞ Tn = T . If p(An ) = p(Tn ) for every n 1, then
p(A) = p(T ).
Proof. See Appendix C.
2
Lemma 16. Let us consider p ∈ P0 ,
p(A) = ale (A) + bue (A) + cxe (A) + dye (A),
A ∈ F (R) and T ∈ F T (R), T = [l, u, x, y] be such that p(A) = p(T ). If (An )n1 is a sequence of fuzzy numbers such
that limn→∞ An = A then there exists a sequence of trapezoidal fuzzy numbers, (Tn )n1 such that limn→∞ Tn = T
and p(An ) = p(Tn ), for sufficiently large n ∈ N.
Proof. See Appendix D.
2
A general result of continuity of the trapezoidal approximation operators (see the beginning of this section) is the
following.
Theorem 17. The trapezoidal approximation operator Tp : F (R) →F T (R), with p ∈ P0 , is continuous with respect
to the average Euclidean metric d (see (11)).
Proof. Let us choose an arbitrary A ∈ F (R) and let us consider the sequence of fuzzy numbers (An )n1 such that
limn→∞ An = A with respect to d. We prove that limn→∞ Tp (An ) = Tp (A) with respect to d which by Heine’s criteria
implies the continuity of the operator Tp . But, first of all, we prove that the sequence (Tp (An ))n1 is bounded with
respect to d. Let T be an arbitrary trapezoidal fuzzy number such that p(A) = p(T ) (the existence of T is assured
by Theorem 7). By Lemma 16 it results the existence of a sequence of trapezoidal fuzzy numbers (Tn )n1 such that
limn→∞ Tn = T and, in addition, p(An ) = p(Tn ) for sufficiently large n 1 (see Theorem 11). By the definition of
Tp it follows that d(An , Tp (An )) d(An , Tn ) for sufficiently large n 1. This implies
d Tp (An ), O d Tp (An ), An + d(An , O)
d(An , Tn ) + d(An , O)
d(O, Tn ) + 2d(An , O),
where O is the real number o represented as a fuzzy number, for sufficiently large n 1. Since the sequences (An )n1
and (Tn )n1 are convergent with respect to d it results that they are also bounded with respect to d. This means that
there exists an absolute constant M > 0 such that max{d(O, Tn ), d(An , O)} M, for every n 1. It results that
d(Tp (An ), O) 3M, for every n 1 and hence the sequence (Tp (An ))n1 is bounded with respect to d. To prove
that limn→∞ Tp (An ) = Tp (A) it suffices to prove that any convergent subsequence (Tp (Akn ))n1 of the bounded
sequence (Tp (An ))n1 converges to the same limit Tp (A). We can use this type of reasoning since one can easily
prove that the convergence of (Tp (Akn ))n1 in (F (R), d) is equivalent with the convergence of a sequence from
(R4 , D), where D is generated by a norm, and, in addition, D and d coincide on R4 . So, let (Tp (Akn ))n1 be a
convergent subsequence of the sequence (Tp (An ))n1 and let us denote with T0 its limit. According to Lemma 26(ii)
in Appendix B, T0 is a trapezoidal fuzzy number. Again, let us choose arbitrarily a trapezoidal fuzzy number T so that
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p(A) = p(T ). From Lemma 16 it results the existence of a sequence of trapezoidal fuzzy numbers (Tn )n1 such that
limn→∞ Tn = T and, in addition, p(An ) = p(Tn ) for sufficiently large n 1. By the definition of the operator Tp it
follows that
d Tp (Akn ), Akn d(Tkn , Akn ),
(42)
for every n 1. Since limn→∞ Tp (Akn ) = T0 , limn→∞ Akn = A, limn→∞ Tkn = T and using the continuity of d,
by passing to limit with n → ∞ in (42) it follows that d(A, T0 ) d(A, T ). On the other hand, by Lemma 15 it
follows that p(A) = p(T0 ). Summarizing, we conclude that p(A) = p(T0 ) and for any trapezoidal fuzzy number T
satisfying p(A) = p(T ) we have d(A, T0 ) d(A, T ). By the definition of the operator Tp it results that Tp (A) = T0
(it is also important here the uniqueness of Tp (A), see Theorem 11). Consequently, any convergent subsequence
of the sequence (Tp (An ))n1 converges to Tp (A) which since (Tp (An ))n1 is bounded implies that actually we
have limn→∞ Tp (An ) = Tp (A). By Heine’s criteria we get that the operator Tp is continuous and the proof is complete. 2
Example 18. If p = 0 and p = Amb then the result in Theorem 17 was obtained in [48], Proposition 6.6 and [15],
Theorem 25, respectively, by proving that the trapezoidal approximation operators are Lipschitz continuous.
6. Future research and conclusions
Often the input data in applications are non-trapezoidal fuzzy numbers or calculus with trapezoidal or nontrapezoidal fuzzy numbers lead to non-trapezoidal fuzzy numbers. Taking into account that many methods with
trapezoidal fuzzy numbers were already elaborated, suitable trapezoidal approximations of data or of intermediate results of calculus are useful. In addition, sometimes it is better if at least a characteristic of interest is
preserved. Our results in Theorems 7 and 11 prove when such a unique approximation is possible and Theorem 17 assures the continuity of a trapezoidal approximation under preservation of a parameter in the general form
1
1
1
1
a 0 AL (α) dα + b 0 AU (α) dα + c 0 αAL (α) dα + d 0 αAU (α) dα, where a, b, c, d ∈ R. In this way, some recent
results in this topic are recovered. The computation of a trapezoidal approximation under preservation of a parameter in the above form must be performed for each particular case, but our general results establish when we can
obtain a solution. In addition, the sophisticated proof of continuity for each case is avoided by the general result in
Theorem 17.
At least the following directions will be approached in our future research on the subject matter of the present
paper:
(1) The obtaining of similar results of existence, uniqueness and continuity in the case of symmetric trapezoidal,
triangular and symmetric triangular approximations.
(2) Sometimes two or more real parameters should be preserved by the trapezoidal or triangular approximation of a
fuzzy number to obtain valid results from a theoretical or practical point of view. The existence, uniqueness and
properties of approximations will be studied in these cases too.
(3) Gradual numbers are a new element of fuzzy set theory, introduced and developed from a theoretical point of
r is defined (see [29]) by an assignment function Ar̃ from (0, 1]
view in [6,24,25,29], etc. A gradual number to R. If A ∈ F (R) then the mappings AL and AU are special cases of gradual numbers and A could be seen
as a special gradual interval, that is a pair of gradual numbers. One can think that the results of existence and
uniqueness given in the present paper could be extended to gradual intervals. Nevertheless, the mathematical
framework – metric structure, associated parameters and their interpretations, particular classes corresponding to
the sets of trapezoidal or triangular fuzzy numbers, etc. – with respect to gradual numbers and gradual intervals
has not yet been fully developed such that it is too premature to obtain, at this moment, results as in the case of
fuzzy numbers. The computation with fuzzy numbers (particularly, with trapezoidal fuzzy numbers) can lead to
objects which are not fuzzy numbers but they are gradual intervals, therefore a possible benefit of approximation
of gradual intervals is immediate.
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14
Acknowledgements
The authors are very grateful to the anonymous referees for their detailed comments and valuable suggestions. For
example, the present structure of the paper and the idea of a possible generalization of the results to gradual numbers
and intervals were suggested by referees.
This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCSUEFISCDI, project number PN-II-ID-PCE-2011-3-0861.
Appendix A. Properties of the extended trapezoidal approximation operator
Proposition 19. (See [47, Proposition 4.2].) Let A be a fuzzy number. Then
d 2 (A, B) = d 2 A, Te (A) + d 2 Te (A), B ,
for any trapezoidal fuzzy number B.
Proposition 20. (See [47, Proposition 4.4].) For all fuzzy numbers A and B
d Te (A), Te (B) d(A, B).
Corollary 21. If A ∈ F (R) then d(A, Te (A)) d(A, B), for any trapezoidal fuzzy number B.
Proof. It is immediate from Proposition 19.
2
Lemma 22. (See [47, Lemma 4.1].) We have
1
1
AL (α) dα =
0
Te (A) L (α) dα,
(43)
0
1
1
α Te (A) L (α) dα,
(44)
Te (A) U (α) dα,
(45)
αAL (α) dα =
0
0
1
1
AU (α) dα =
0
0
1
1
αAU (α) dα =
0
α Te (A) U (α) dα,
0
for every A ∈ F (R).
By direct calculation or from the proof of Theorem 5 in [12] we have
Lemma 23.
6ue (A) − 6le (A) xe (A) + ye (A),
for every A ∈ F (R).
(46)
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Appendix B. Properties of convergence
Lemma 24. (See [17, Lemma 12].) Let (An )n∈N , (An )α = [(An )L (α), (An )U (α)], α ∈ [0, 1], be a sequence such that
An ∈ F (R) ∪ FeT (R), for every n ∈ N and limn→∞ An = A (with respect to d), where
Aα = AL (α), AU (α) , α ∈ [0, 1].
Then
1
1
α (An )L (α) dα =
k
lim
n→∞
0
α k AL (α) dα
0
and
1
1
α (An )U (α) dα =
k
lim
n→∞
0
α k AU (α) dα
0
for all k ∈ N.
The following results of convergence are immediate consequences of Lemmas 2 and 3 in [14].
Lemma 25. (i) If (Tn )n∈N , Tn = (t1 (n), t2 (n), t3 (n), t4 (n)) is a sequence of extended trapezoidal fuzzy numbers such
that limn→∞ (ti )(n) = ti < ∞, i ∈ {1, 2, 3, 4} then limn→∞ Tn = T (with respect to d), where T = (t1 , t2 , t3 , t4 ).
(ii) If (Tn )n∈N , Tn = (t1 (n), t2 (n), t3 (n), t4 (n)) is a convergent sequence (with respect to d) of trapezoidal fuzzy
numbers then its limit is a trapezoidal fuzzy number T = (t1 , t2 , t3 , t4 ), where ti = limn→∞ ti (n), i ∈ {1, 2, 3, 4}.
Taking into account the representations (12)–(13) and (14)–(15) we can rewrite the above results as follows
Lemma 26. (i) If (Tn )n∈N , Tn = [ln , un , xn , yn ] is a sequence of extended trapezoidal fuzzy numbers such that
limn→∞ ln = l, limn→∞ un = u, limn→∞ xn = x, limn→∞ yn = y, then limn→∞ Tn = T (with respect to d), where
T = [l, u, x, y].
(ii) If (Tn )n∈N , Tn = [ln , un , xn , yn ] is a convergent sequence (with respect to d) of trapezoidal fuzzy numbers then
its limit is a trapezoidal fuzzy number T = [l, u, x, y], where limn→∞ ln = l, limn→∞ un = u, limn→∞ xn = x and
limn→∞ yn = y.
Appendix C. Proof of Lemma 15
Let us denote
Tn = [ln , un , xn , yn ],
n 1,
T = [l, u, x, y].
It results that (see Lemma 4)
p(Tn ) = aln + bun + cxn + dyn ,
n∈N
(47)
and
p(T ) = al + bu + cx + dy.
By Lemma 26(ii), it results that
lim ln = l,
n→∞
lim un = u,
n→∞
lim xn = x
n→∞
(48)
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and
lim yn = y,
n→∞
which together with (47)–(48) easily imply that limn→∞ p(Tn ) = p(T ). On the other hand, since p(An ) = p(Tn ), for
every n ∈ N, it follows that
ale (An ) + bue (An ) + cxe (An ) + dye (An ) = aln + bun + cxn + dyn ,
for every n ∈ N. By passing to limit with n → ∞ in the above relation we get (Lemma 24 and (20)–(23) are used
here)
ale (A) + bue (A) + cxe (A) + dye (A) = al + bu + cx + dy,
that is p(A) = p(T ) and the proof is complete.
Appendix D. Proof of Lemma 16
We begin the proof by remembering that, in the hypothesis of the lemma, we have p(T ) = al + bu + cx + dy, for
every T ∈ F T (R) (see Lemma 4). According to Theorem 7 we distinguish the following four cases:
(i)
(ii)
(iii)
(iv)
a + b = 0;
a = b = 0;
a + b = 0, a = 0 and c/a > 1/2 or d/a > 1/2;
a + b = 0, a =
0, c/a 1/6 and d/a 1/6.
Before the discussion on cases we have to notice that, in the hypothesis of the lemma, there exists a sequence of
real numbers (αn )n1 with the property that limn→∞ αn = 0 and such that p(An ) = p(T ) + αn for all n ∈ N, n 1.
Indeed, we observe that for every n 1 we can write
p(An ) = p(A) + a le (An ) − le (A) + b ue (An ) − ue (A) + c xe (An ) − xe (A) + d ye (An ) − ye (A)
= p(T ) + a le (An ) − le (A) + b ue (An ) − ue (A) + c xe (An ) − xe (A) + d ye (An ) − ye (A) .
By denoting
αn = a le (An ) − le (A) + b ue (An ) − ue (A) + c xe (An ) − xe (A) + d ye (An ) − ye (A) ,
for every n 1, we obtain the desired relation and limn→∞ αn = 0 because limn→∞ le (An ) = le (A), limn→∞ ue (An )
= ue (A), limn→∞ xe (An ) = xe (A) and limn→∞ ye (An ) = ye (A) in the hypothesis limn→∞ An = A (see Lemma 24
and (20)–(23)).
Case (i). Take
αn
αn
Tn = l +
,u +
, x, y .
a+b
a+b
It is easy to check that each Tn , n 1 is a proper trapezoidal fuzzy number. We observe that p(Tn ) = p(T ) + αn =
p(An ) for each n 1. In addition, from (19) we obtain
d 2 (Tn , T ) =
2αn2
(a + b)2
therefore limn→∞ Tn = T .
Case (ii). We exclude the case c = d = 0 since in this case Tp in the beginning of this section becomes the
trapezoidal approximation operator without any other additional condition and it is well known that this operator
is nonexpansive and therefore it is continuous (see [48, Propositions 6.5 and 6.6]). So, let us suppose that c = 0 or
d = 0. Without any loss of generality let us suppose that c = 0.
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Firstly, we assume that x > 0. We take
|αn |
αn
Tn = l, u +
,x + ,y .
2|c|
c
Since x > 0 and limn→∞ αn = 0, it results that for sufficiently large n we have x + αcn > 0. This easily implies that
for sufficiently large n we have that Tn is a proper trapezoidal fuzzy number. In addition it is easy to check that the
requirements of the lemma hold for the sequence (Tn )n1 .
Now, we assume that x = 0. Firstly, let us consider c > 0. If for some n 1 we have αn 0 then we can take
αn αn
Tn = l, u + , , y .
2c c
If αn < 0 then we cannot have d = 0. Contrariwise, we would have p(T ) = 0 and p(An ) = cxe (An ) 0 which would
imply that αn 0, a contradiction. Now, if y > 0 then since d = 0 we can easily use the same type of reasoning as in
the case when x > 0, that is we can take
αn
αn
Tn = l, u −
, 0, y +
2d
d
and the requirements are satisfied. If y = 0 then we cannot have d > 0 because
0 cxe (An ) + dye (An ) = p(An ) = p(T ) + αn = αn ,
a contradiction. Therefore, we necessarily have that d < 0. In this case we take
αn
αn
Tn = l, u +
, 0,
.
2d
d
It is easy to check that Tn is indeed a trapezoidal fuzzy number and that the requirements hold. Now, let us consider
c < 0. If for some n 1 we have αn 0 then we can take
αn αn
Tn = l, u + , , y
2c c
which is a trapezoidal fuzzy number and satisfies the requirements of the lemma. In the sequel we may assume αn > 0.
If d = 0 then
0 cxe (An ) = p(An ) = p(T ) + αn = αn ,
a contradiction with our assumption αn > 0. This means that if y > 0 then we can take
αn
αn
Tn = l, u +
, 0, y +
.
2d
d
If y = 0 then we cannot have d < 0. Indeed, if d < 0 then
0 cxe (An ) + dye (An ) = p(An ) = p(T ) + αn = αn
a contradiction with our assumption αn > 0. We necessarily have d > 0 and we take
αn
αn
Tn = l, u +
, 0,
2d
d
a trapezoidal fuzzy number and the requirements are satisfied.
Case (iii). We observe that in this case we have
p(T ) = a(l − u) + cx + dy.
Firstly, we assume that a > 0. If for some n we have αn 0 then we take Tn = [l + αan , u, x, y] which obviously is
a trapezoidal fuzzy number satisfying p(Tn ) = p(An ). Therefore, we can suppose that αn > 0. We take
αn
2αn
Tn = l −
, u, x +
,y
(49)
2c − a
2c − a
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if c/a >
1
2
and
Tn = l −
αn
2αn
, u, x, y +
2d − a
2d − a
(50)
if d/a > 12 , which again are trapezoidal fuzzy numbers satisfying p(Tn ) = p(An ). The relation limn→∞ Tn = T
follows after some simple calculations.
Now, we assume that a < 0. If for some n we have αn 0 then we take
αn
Tn = l + , u, x, y
(51)
a
a trapezoidal fuzzy number with p(Tn ) = p(An ). Contrariwise, if αn < 0 then we take Tn as in (49) or (50) if c/a >
or d/a > 12 , respectively (because 2c − a < 0 or 2d − a < 0).
Case (iv). Without any loss of generality we assume that a > 0. If αn 0 then we take
αn
Tn = l + , u, x, y .
a
1
2
In what follows we suppose that αn > 0. If u = l then
0 = 2u − 2l x + y 0
therefore x = y = 0, which implies p(T ) = 0. On the other hand, we have
c
d
p(An ) = a le (An ) − ue (An ) + xe (An ) + ye (An )
a
a
1
1
a le (An ) − ue (An ) + xe (An ) + ye (An )
6
6
which by Lemma 23 implies
p(An ) = p(T ) + αn = αn 0
which leads to a contradiction. Hence it results that necessarily u − l > 0. Now, if 2u − 2l > x + y then we take
αn
Tn = l + , u, x, y
a
and for sufficiently large n we get 2u − 2(l + αan ) > x + y which implies that Tn is a trapezoidal fuzzy number for
sufficiently large n. The last case is when 2u − 2l = x + y. Since 2u − 2l = x + y > 0, it results that either x > 0 or
y > 0. Without any loss of generality let us suppose that x > 0. We take
αn
2αn
Tn = l −
, u, x +
,y .
2c − a
2c − a
Since 2c − a = a(2c/a − 1) −2a/3 < 0 it follows that the coordinates are correctly defined. Then since x > 0 it
2αn
results that x + 2c−a
> 0 for sufficiently large n. Now it is easy to verify that Tn is a proper trapezoidal fuzzy number
for sufficiently large n. Finally, the sequence (Tn )n1 satisfies the requirements of the lemma and this finishes the
proof.
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