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Computers and Mathematics with Applications 61 (2011) 1379–1401
Contents lists available at ScienceDirect
Computers and Mathematics with Applications
journal homepage: www.elsevier.com/locate/camwa
Approximations of fuzzy numbers by trapezoidal fuzzy numbers
preserving the ambiguity and value
A. Bana,∗, A. Brândaşb, L. Coroianua,e,1, C. Negruţiuc, O. Nicad,e,1
aDepartment of Mathematics and Informatics, University of Oradea, 410087 Oradea, Romania
bMihai Veliciu High School, 315100 Chişineu-Criş, Romania
cTraian Vuia High School, 410191 Oradea, Romania
dOşorhei School, 417360 Oşorhei, Romania
eDepartment of Mathematics, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania
a r t i c l ei n f o
Article history:
Received 9 May 2010
Received in revised form 2 January 2011
Accepted 3 January 2011
Keywords:
Fuzzy number
Trapezoidal fuzzy number
Approximation
Ambiguity
Value
a b s t r a c t
Value and ambiguity are two parameters which were introduced to represent fuzzy
numbers. In this paper, we find the nearest trapezoidal approximation and the nearest
symmetrictrapezoidalapproximationtoagivenfuzzynumber,withrespecttotheaverage
Euclidean distance, preserving the value and ambiguity. To avoid the laborious calculus
associatedwiththeKarush–Kuhn–Tuckertheorem,theworkingtoolinsomerecentpapers,
a less sophisticated method is proposed. Algorithms for computing the approximations,
many examples, proofs of continuity and two applications to ranking of fuzzy numbers
and estimations of the defect of additivity for approximations are given.
© 2011 Elsevier Ltd. All rights reserved.
1. Introduction
Uncertainty and incomplete information in decision making, linguistic controllers, expert systems, data mining, pattern
recognition,etc.,areoftenrepresentedbyfuzzynumbers.Inrecentyears,severalresearchershavefocusedonthecomputing
of different approximations of fuzzy numbers and new approaches for ranking of fuzzy numbers (see, e.g., [1–17]).
To capture the relevant information, to simplify the task of representing and handling fuzzy numbers, the value and
the ambiguity of a fuzzy number were introduced in [18]. In the same paper, the authors discussed how to approximate
a given fuzzy number by a suitable trapezoidal one preserving the value and ambiguity. Because it is not possible to
uniquely determine a trapezoidal fuzzy number, which is characterized by four numbers, from two conditions, some
additional conditions must be introduced. In the present paper, we completely solve the problems of finding the nearest
trapezoidal approximation and the nearest symmetric trapezoidal approximation of a fuzzy number with respect to the
average Euclidean distance, such that the value and ambiguity are preserved.
The paper is organized as follows. In Section 2, we recall basic definitions and results. The nearest trapezoidal
approximation of a fuzzy number preserving the value and ambiguity is determined in Section 3. The method is less
sophisticated than previous methods; it avoids the laborious calculus associated with the Karush–Kuhn–Tucker theorem
and, in addition, it allows us to prove the continuity of the trapezoidal approximation. The symmetric case is tackled in
∗Corresponding author.
E-mail addresses: aiban@uoradea.ro (A. Ban), adapelea@yahoo.com (A. Brândaş), lcoroianu@uoradea.ro (L. Coroianu), corina-negrutiu@yahoo.com
(C. Negruţiu), octavia.nica@math.ubbcluj.ro (O. Nica).
1Partially supported by the project POSDRU/CPP107/DMI1.5/S/76841.
0898-1221/$ – see front matter © 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.camwa.2011.01.005
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Section 4. Taking into account the solving in the general case, the results are easily obtained. The expected value, width,
left-hand ambiguity and right-hand ambiguity are parameters used in Section 5 to express the main results of the paper
in a more compact form and to give some algorithms for calculating the trapezoidal approximations. Sections 6 and 7 are
dedicatedtoexamplesandproperties.Weinsistonthecontinuityofthetrapezoidalapproximationoperatorsandweignore
the elementary proofs of other properties. Value and ambiguity are used together to rank fuzzy numbers [18]; therefore
we can compare the trapezoidal approximations preserving the value and ambiguity instead, to compare fuzzy numbers
(Section 8). The defects of additivity of trapezoidal approximation operators determined in Sections 3 and 4 are estimated
in Section 8. The paper is completed by some conclusions and open problems.
2. Preliminaries
We consider the following well-known description of a fuzzy number A:
called the left side of the fuzzy number and rA : [a3,a4] −→ [0,1] is a non-increasing upper semicontinuous function,
rA(a3) = 1,rA(a4) = 0, called the right side of the fuzzy number. The α-cut, α ∈ (0,1], of a fuzzy number A is a crisp set
defined as
A(x) =
0,
lA(x),
1
rA(x),
0,
if x ≤ a1,
if a1≤ x ≤ a2,
if a2≤ x ≤ a3,
if a3≤ x ≤ a4,
if a4≤ x,
(1)
where a1,a2,a3,a4∈ R,lA: [a1,a2] −→ [0,1] is a non-decreasing upper semicontinuous function, lA(a1) = 0,lA(a2) = 1,
Aα= {x ∈ R : A(x) ≥ α}.
The support or 0-cut A0of a fuzzy number is defined as
A0= {x ∈ R : A(x) > 0}.
Every α-cut, α ∈ [0,1], of a fuzzy number A 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},
for anyα ∈ (0,1]. If the sides of the fuzzy number A are strictly monotone, then one can see easily that ALand AUare inverse
functions of lAand rA, respectively. We denote by F(R) the set of all fuzzy numbers.
Some important parameters of a fuzzy number A, Aα= [AL(α),AU(α)], α ∈ [0,1], are the ambiguity Amb(A) and the
value Val(A). They are given by (see [18])
∫1
∫1
A metric on the set of fuzzy numbers, which is an extension of the Euclidean distance, is defined by [19]
∫1
Fuzzy numbers with simple membership functions are preferred in practice. The most used such 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
and
Amb(A) =
0
α(AU(α) − AL(α))dα,
(2)
Val(A) =
0
α(AU(α) + AL(α))dα.
(3)
d2(A,B) =
0
(AL(α) − BL(α))2dα +
∫1
0
(AU(α) − BU(α))2dα.
(4)
TU(α) = t4− (t4− t3)α,
where t1,t2,t3,t4∈ R,t1≤ t2≤ t3≤ t4. When t2= t3, we obtain a triangular fuzzy number. When t2− t1= t4− t3, we
obtain a symmetric trapezoidal fuzzy number. We denote
T = (t1,t2,t3,t4)
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a trapezoidal fuzzy number, and we denote by FT(R) the set of all trapezoidal fuzzy numbers, and by FS(R) the set of all
symmetric trapezoidal fuzzy numbers.
Sometimes (see [16]), it is useful to denote a trapezoidal fuzzy number by
T = [l,u,x,y],
with l,u,x,y ∈ R such that x,y ≥ 0,x + y ≤ 2(u − l),
TU(α) = u − y
TL(α) = l + x
α −
1
2
1
2
,
α −
,
for every α ∈ [0,1].
It is immediate that
l =
t1+ t2
2
t3+ t4
2
,
(5)
u =
x = t2− t1,
y = t4− t3,
,
(6)
(7)
(8)
and T ∈ FS(R) if and only if x = y. Also, by direct calculation, we get
Amb(T) =−6l + 6u − x − y
12
6l + 6u + x − y
12
,
(9)
Val(T) =
.
(10)
The distance between T,T′∈ FT(R),T = [l,u,x,y] and T′= [l′,u′,x′,y′] becomes [15]
1
12(x − x′)2+
d2(T,T′) = (l − l′)2+ (u − u′)2+
1
12(y − y′)2.
(11)
Let A,B ∈ F (R),Aα= [AL(α),AU(α)],Bα= [BL(α),BU(α)], α ∈ [0,1] and λ ∈ R. We consider the sum A + B and
the scalar multiplication λ · A by (see e. g. [20, p. 40])
(A + B)α= Aα+ Bα= [AL(α) + BL(α),AU(α) + BU(α)]
and
respectively, for every α ∈ [0,1]. In the case of the trapezoidal fuzzy numbers T = (t1,t2,t3,t4) and S = (s1,s2,s3,s4), we
obtain
(λ · A)α= λAα=
[λAL(α),λAU(α)],
[λAU(α),λAL(α)],
if λ ≥ 0,
if λ < 0,
T + S = (t1+ s1,t2+ s2,t3+ s3,t4+ s4).
An extended trapezoidal fuzzy number [15] is an order pair of polynomial functions of degree less than or equal to 1. An
extended trapezoidal fuzzy number may not be a fuzzy number, but the distance between two extended trapezoidal fuzzy
numbers is similarly defined as in (4) or (11). In addition, we define the value and the ambiguity of an extended trapezoidal
fuzzy number in the same way as in the case of a trapezoidal fuzzy number. We denote by FT
trapezoidal fuzzy numbers; that is,
where TLand TUhave the same meaning as above.
The extended trapezoidal approximation Te(A) = [le,ue,xe,ye] of a fuzzy number A is the extended trapezoidal fuzzy
number which minimizes the distance d(A,X), where X ∈ FT
e(R) the set of all extended
FT
e(R) =
T = [l,u,x,y] : TL(α) = l + x
α −
1
2
,TU(α) = u − y
α −
1
2
,α ∈ [0,1],l,u,x,y ∈ R
,
e(R). In [3], the authors proved that Te(A) is not always a fuzzy
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number. The extended trapezoidal approximation Te(A) = [le,ue,xe,ye] of a fuzzy number A is determined [15] by the
following equalities:
∫1
∫1
∫1
∫1
The real numbers xeand yeare non-negative (see [15]), and from the definition of a fuzzy number we have le≤ ue.
In [14], the author proved two very important distance properties for the extended trapezoidal approximation operator,
as follows.
le=
0
AL(α)dα,
(12)
ue=
0
AU(α)dα,
(13)
xe= 12
0
α −
1
2
1
2
AL(α)dα,
(14)
ye= −12
0
α −
AU(α)dα.
(15)
Proposition 1 ([14, Proposition 4.2]). Let A be a fuzzy number. Then
d2(A,B) = d2(A,Te(A)) + d2(Te(A),B)
for any trapezoidal fuzzy number B.
(16)
Proposition 2 ([14, Proposition 4.4]). d(Te(A),Te(B)) ≤ d(A,B) for all fuzzy numbers A,B.
Remark 3. Let A,B ∈ F (R), and let Te(A) = [le,ue,xe,ye],Te(B) = [l′
tions of A and B. Proposition 2 and (11) imply that
(le− l′
and
(xe− x′
e,u′
e,x′
e,y′
e] be the extended trapezoidal approxima-
e)2+ (ue− u′
e)2≤ d2(A,B)
e)2+ (ye− y′
e)2≤ 12d2(A,B).
The following result is immediate.
Proposition 4. If A is a fuzzy number and Te(A) = [le,ue,xe,ye] is the extended trapezoidal approximation of A, then
Amb(A) = Amb(Te(A))
and
(17)
Val(A) = Val(Te(A)).
Proof. By direct calculation, we get
(18)
Amb(Te(A)) =
1
12(6ue− 6le− xe− ye) =
∫1
∫1
1
2
∫1
∫1
0
AU(α)dα −
1
2
∫1
0
AL(α)dα
−
0
α −
1
2
AL(α)dα +
0
α −
1
2
AU(α)dα
=
0
α(AU(α) − AL(α))dα = Amb(A).
The proof of the second equality is similar.
?
At the end of this section, let us remark that a more general framework could be considered in this paper, taking
into account the value and ambiguity of A ∈ F (R) with respect to a reducing function S (that is, an increasing function
S : [0,1] → [0,1] such that S(0) = 0,S(1) = 1) introduced by (see [18])
∫1
AmbS(A) =
0
ValS(A) =
0
∫1
S (α)(AL(α) + AU(α))dα,
S (α)(AU(α) − AL(α))dα
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and the distance between two fuzzy numbers A and B by (see [19])
∫1
wherep ≥ 1.Nevertheless,theparticularcasesS (α) = α andp = 2hereallowthevalidityof(16)–(18),thebasicproperties
in the proofs of the main results in the paper. In addition, the case S (α) = α and p = 2 is the most important one, and it is
the one that is almost exclusively considered in the scientific literature. In the general case, or in any other particular case,
the solving of the proposed problem and/or the proof of properties is not possible, at least with the present tools.
dp(A,B) =
p
0
|AL(α) − BL(α)|pdα +
∫1
0
|AU(α) − BU(α)|pdα,
3. Approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the ambiguity and value
Inthissectionwecompute,foranygivenfuzzynumberA,thenearest(withrespecttometricd)trapezoidalfuzzynumber
T(A)such thatAmb(A) = Amb(T(A))and Val(A) = Val(T(A)). ByPropositions 1and4, itfollows thatthe problem offinding
the nearest trapezoidal fuzzy number preserving the ambiguity and value of a fuzzy number A is equivalent to the problem
of finding a trapezoidal fuzzy number T(A) such that
Amb(T(A)) = Amb(Te(A)),
Val(T(A)) = Val(Te(A))
and
d(T(A),Te(A)) ≤ d(B,Te(A)),
for all B ∈ FT(R) satisfying
Amb(B) = Amb(Te(A))
and
Val(B) = Val(Te(A)),
where Te(A) denotes the nearest extended trapezoidal fuzzy number of A.
Therefore, T(A) = [lT,uT,xT,yT] if and only if (lT,uT,xT,yT) ∈ R4is a solution of the problem
min
(l − le)2+ (u − ue)2+
under the conditions
x ⩾ 0,
y ⩾ 0,
x + y ⩽ 2u − 2l,
−6l + 6u − x − y = −6le+ 6ue− xe− ye,
6l + 6u + x − y = 6le+ 6ue+ xe− ye,
where le,ue,xe,yeare given by (12)–(15). We immediately obtain that (19)–(24) is equivalent to
min(x − xe)2+ (y − ye)2
x ⩾ 0,
y ⩾ 0,
1
2xe−
In addition,
l = −1
and
1
6(y − ye) + ue.
Let us consider the set
and let us denote by PM(Z) the orthogonal projection of Z ∈ R2on non-empty set M ⊂ R2, with respect to dE, the Euclidean
metric on R2.
1
12(x − xe)2+
1
12(y − ye)2
(19)
(20)
(21)
(22)
(23)
(24)
(25)
under the conditions
(26)
(27)
x + y ⩽ 3ue− 3le−
1
2ye.
(28)
6(x − xe) + le
(29)
u =
(30)
MA=
(x,y) ∈ R2: x ⩾ 0,y ⩾ 0,x + y ⩽ 3ue− 3le−
1
2xe−
1
2ye
,
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Fig. 1. Cases for (xe,ye).
Theorem 5. Problem (25)–(28) has an unique solution.
Proof. Taking into account (12)–(15), we have
3ue− 3le−
1
2xe−
1
2ye= 6
∫1
0
α(AU(α) − AL(α))dα ≥ 0;
therefore, MA ̸= ∅. Because MAis a closed convex subset of the Hilbert space R2, there exists a unique element (see [21,
Theorem 4.10, p. 79]), PMA(Ce), where Ce= (xe,ye), such that
dE
C∈MA
Ce,PMA(Ce)= inf
As a conclusion, T(A) = [lT,uT,xT,yT] is the nearest (with respect to metric d) trapezoidal fuzzy number to a given fuzzy
number A such that Amb(A) = Amb(T(A)) and Val(A) = Val(T(A)), if and only if (xT,yT) is the orthogonal projection of
(xe,ye) on MAand
lT= −1
1
6(yT− ye) + ue.
Becausexe≥ 0andye≥ 0,thefollowingcases(correspondingto(i),(ii),(iii),(iv)inFig.1)tofindPMA(Ce),theorthogonal
projection of (xe,ye) on MA, are possible.
(i) (xe,ye) ∈ MA; that is, xe+ ye⩽ 3ue− 3le−1
The inequality is equivalent to xe+ ye⩽ 2(ue− le), and we get PMA(xe,ye) = (xe,ye); that is,
xT= xe,
yT= ye.
(ii)
Then PMA(xe,ye) =
xT= 3ue− 3le−
yT= 0.
(iii)
Then PMA(xe,ye) =
1
2xe−
dE(Ce,C).
?
6(xT− xe) + le,
(31)
uT=
(32)
2xe−1
2ye.
3
2xe−1
2ye− 3ue+ 3le> 0.
3ue− 3le−1
1
2xe−
2xe−1
2ye,0
; that is,
1
2ye,
1
2xe−3
2ye+ 3ue− 3le< 0.
0,3ue− 3le−1
2xe−1
2ye
; that is,
xT= 0,
yT= 3ue− 3le−
1
2ye.
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(iv) (xe,ye) is not in the cases (i)–(iii); that is,
xe+ ye> 2(ue− le),
3
2xe−
1
2xe−
1
2ye− 3ue+ 3le≤ 0,
3
2ye+ 3ue− 3le≥ 0.
Then (xT,yT) is the orthogonal projection of (xe,ye) on the line x + y = 3ue− 3le−1
3
2ue−
3
2ue−
2xe−1
2ye; that is,
xT=
3
2le+
3
2le−
1
4xe−
3
4xe+
3
4ye,
1
4ye.
yT=
Example 6. Let us consider the fuzzy number A given in parametric form by
AL(α) = 2α − 2,
AU(α) = 1 −√α,
After elementary calculus from (12)–(15), we obtain
4
5,
Then
and the orthogonal projection of(xe,ye) =
BL(α) = 2α − 20,
BU(α) = 1 −√α,
then, from (12)–(15), we get
4
5,
and
Because (xe,ye) =
Taking into account (12)–(15), (31) and (32), we obtain the following result.
α ∈ [0,1].
xe= 2,
ye=
le= −1,
ue=
1
3.
MA=
(x,y) ∈ R2: x ≥ 0,y ≥ 0,x + y ≤
13
5
on the set MAis the orthogonal projection of
,
2,4
5
2,4
5
on the line x+y =
13
5;
that is, the point (xT,yT) =19
10,
7
10
(the above case (iv) is applicable). If the fuzzy number B is given by
α ∈ [0,1],
xe= 2,
ye=
le= −19,
ue=
1
3
MB=
(x,y) ∈ R2: x ≥ 0,y ≥ 0,x + y ≤
∈ MB, the orthogonal projection of (xe,ye) on the set MBis even (xe,ye); that is, (xT,yT) =
283
5
.
2,4
5
2,4
5
(the above case (i) is applicable).
Theorem 7. Let A ∈ F (R),Aα= [AL(α),AU(α)], α ∈ [0,1], and let T(A) = [lT,uT,xT,yT] be the nearest trapezoidal fuzzy
number to A which preserves the ambiguity and value.
(i) If
∫1
0
(3α − 1)AL(α)dα −
∫1
0
(3α − 1)AU(α)dα ≤ 0(33)
then
xT= 6
∫1
0
(2α − 1)AL(α)dα,
∫1
yT= −6
0
(2α − 1)AU(α)dα,
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lT=
∫1
∫1
0
AL(α)dα,
uT=
0
AU(α)dα.
(ii) If
∫1
0
(3α − 1)AL(α)dα +
∫1
0
(α − 1)AU(α)dα > 0(34)
then
xT= −6
yT= 0
lT= 3
∫1
0
αAL(α)dα + 6
∫1
0
αAU(α)dα
∫1
∫1
0
αAL(α)dα −
∫1
0
αAU(α)dα
uT= 2
0
αAU(α)dα.
(iii) If
∫1
0
(α − 1)AL(α)dα +
∫1
0
(3α − 1)AU(α)dα < 0(35)
then
xT= 0
yT= −6
∫1
αAL(α)dα
∫1
0
αAL(α)dα + 6
∫1
0
αAU(α)dα
lT= 2
∫1
0
uT= −
0
αAL(α)dα + 3
∫1
0
αAU(α)dα.
(iv) If
∫1
∫1
0
(3α − 1)AL(α)dα −
∫1
∫1
0
(3α − 1)AU(α)dα > 0 (36)
0
(3α − 1)AL(α)dα +
0
(α − 1)AU(α)dα ≤ 0(37)
and
∫1
0
(α − 1)AL(α)dα +
∫1
0
(3α − 1)AU(α)dα ≥ 0(38)
then
xT= 3
∫1
0
(α − 1)AL(α)dα + 3
∫1
∫1
uT= −1
2
0
∫1
0
(3α − 1)AU(α)dα
∫1
∫1
1
2
0
yT= −3
0
(3α − 1)AL(α)dα − 3
0
(α − 1)AU(α)dα
lT=
1
2
0
(3α + 1)AL(α)dα −
∫1
1
2
0
(3α − 1)AU(α)dα
∫1
(3α − 1)AL(α)dα +
(3α + 1)AU(α)dα.
Taking into account (5)–(8), we immediately get t1 = l −
consequence of Theorem 7.
x
2,t2 = l +
x
2,t3 = u −
y
2,t4 = u +
y
2, and the following
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Corollary 8. Let A ∈ F (R),Aα= [AL(α),AU(α)], α ∈ [0,1], and let T(A) = (t1,t2,t3,t4) be the nearest trapezoidal fuzzy
number to A which preserves the ambiguity and value.
(i) If (33) is satisfied then
∫1
∫1
∫1
∫1
(ii) If (34) is satisfied then
∫1
∫1
(iii) If (35) is satisfied then
∫1
∫1
(iv) If (36)–(38) are satisfied then
∫1
∫1
∫1
t1=
0
(4 − 6α)AL(α)dα,
(39)
t2=
0
(6α − 2)AL(α)dα,
(40)
t3=
0
(6α − 2)AU(α)dα,
(41)
t4=
0
(4 − 6α)AU(α)dα.
(42)
t1= 6
0
αAL(α)dα − 4
∫1
0
αAU(α)dα
(43)
t2= t3= t4= 2
0
αAU(α)dα.
(44)
t1= t2= t3= 2
0
αAL(α)dα
(45)
t4= 6
0
αAU(α)dα − 4
∫1
0
αAL(α)dα.
(46)
t1= 2
0
AL(α)dα −
∫1
0
(6α − 2)AU(α)dα
∫1
∫1
(47)
t2= t3=
0
(3α − 1)AL(α)dα +
0
(3α − 1)AU(α)dα
(48)
t4=
0
(2 − 6α)AL(α)dα + 2
0
AU(α)dα.
(49)
Example 9. Case (iv) in Corollary 8 is applicable to the fuzzy number A in Example 6 and case (i) is applicable to the fuzzy
number B in Example 6. The nearest trapezoidal fuzzy numbers to A and B preserving the value and ambiguity of A and B
are, respectively,
T(A) =−29
15,−
1
30,−
1
30,2
3
,
T(B) =−20,−18,−
1
15,11
15
.
4. Approximation of fuzzy numbers by symmetric trapezoidal fuzzy numbers preserving the ambiguity and
value
Symmetric fuzzy numbers are often used in practice (see, e.g., [22,23]) because they have a more perceptive natural
interpretation and are easy to handle. According to Theorem 7, the nearest trapezoidal fuzzy number which preserves the
ambiguity and value of a symmetric fuzzy number is symmetric too. Indeed, if A ∈ F (R) is symmetric, that is
AL(1) − AL(α) = AU(α) − AU(1),
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for every α ∈ [0,1], then
∫1
=
0
0
(3α − 1)AL(α)dα +
∫1
∫1
AL(1) + AU(1)
2
0
(α − 1)AU(α)dα
2αAL(α)dα − ≤ −AL(1) +
∫1
0
2αAL(α)dα ≤ 0
and
∫1
0
(α − 1)AL(α)dα +
AL(1) + AU(1)
2
∫1
∫1
0
(3α − 1)AU(α)dα
=−
0
2αAL(α)dα ≥ AL(1) −
∫1
0
2αAL(α)dα ≥ 0;
thereforecases (ii)and (iii)inTheorem7arenot applicable. Onthe otherhand, ifcases(i) or(iv) inTheorem7areapplicable,
then xT= yT; therefore T(A) is a symmetric trapezoidal fuzzy number.
In this section, we compute the nearest symmetric trapezoidal fuzzy number to a given fuzzy number, preserving
the ambiguity and value. Taking into account the previous section, S(A) = [lS,uS,xS,yS] is the nearest (with respect to
metric d) symmetric trapezoidal fuzzy number of a given fuzzy number A with the extended trapezoidal approximation
Te(A) = [le,ue,xe,ye] such that Amb(A) = Amb(S(A)) and Val(A) = Val(S(A)) if and only if (xS,yS) is the solution of the
problem (see (25)–(28))
min(x − xe)2+ (y − ye)2,
y ≥ 0,
x + y ≤ 3ue− 3le−
x = y,
and (see (29)–(30))
(50)
x ≥ 0,
(51)
(52)
1
2xe−
1
2ye,
(53)
(54)
lS= −1
6(xS− xe) + le
(55)
uS=
1
6(yS− ye) + ue.
(56)
The problem (50)–(54) is equivalent to
min
3
2ue−
2x2− 2(xe+ ye)x + x2
x ≥ 0,
x ≤
e+ y2
e
(57)
3
2le−
1
4xe−
1
4ye;
therefore, the following cases are possible.
(i)
xe+ ye
2
≤
3
2ue−
3
2le−
1
4xe−
1
4ye;
that is,
xe+ ye≤ 2(ue− le).
Then the solution of problem (57) is the minimum point of the function g,
g(x) = 2x2− 2(xe+ ye)x + x2
that is,
xe+ ye
2
e+ y2
e;
xS=
.
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From (54)–(56), we obtain
xe+ ye
yS=
2
,
uS= ue+
1
12xe−
1
12xe−
1
12ye
1
12ye.
lS= le+
(ii)
xe+ ye
2
>
3
2ue−
3
2le−
1
4xe−
1
4ye;
that is,
xe+ ye> 2(ue− le).
Because the function g, g(x) = 2x2− 2(xe+ ye)x + x2
of problem (57) is
3
2ue−
From (54)–(56), we obtain
3
2ue−
5
4ue−
lS= −1
e+ y2
e, is decreasing on
0,3
2ue−3
2le−1
4xe−1
4ye
, the solution
xS=
3
2le−
1
4xe−
1
4ye.
yS=
3
2le−
1
4le−
5
4le+
1
4xe−
1
24xe−
5
24xe+
1
4ye,
5
24ye,
1
24ye.
uS=
4ue+
Example 10. If A is the fuzzy number in Example 6 then
g(x) = 2x2−
28
5
x +
116
25
(58)
and
3
2ue−
3
2le−
1
4xe−
1
4ye=
13
10.
283
10.
Because g is decreasing on−∞,7
3
2ue−
Because the minimum of g is attained in x =
The following results are immediate from (12)–(15) and (5)–(8), respectively.
5
, we obtain that the minimum of g on
0,13
10
is attained in x =
13
10; that is, xT=
13
10(the
above case (ii) is applicable). If B is the fuzzy number in Example 6 then g is given in (58) too, and
3
2le−
7
1
4xe−
1
4ye=
5∈
0,283
10
, we get xT=
7
5(the above case (i) is applicable).
Theorem 11. Let A ∈ F (R),Aα = [AL(α),AU(α)], α ∈ [0,1], and let S(A) = [lS,uS,xS,yS] be the nearest symmetric
trapezoidal fuzzy number to A which preserves the ambiguity and value.
(i) If
∫1
then
∫1
lS=
2
0
2
0
1
2
0
2
0
0
(3α − 1)AL(α)dα −
∫1
0
(3α − 1)AU(α)dα ≤ 0(59)
xS= yS= 3
1
0
(2α − 1)AL(α)dα − 3
∫1
(2α − 1)AU(α)dα,
∫1
0
(2α − 1)AU(α)dα,
∫1
∫1
(2α + 1)AL(α)dα +
1
∫1
uS=
(2α − 1)AL(α)dα +
1
(2α + 1)AU(α)dα.
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(ii) If
∫1
0
(3α − 1)AL(α)dα −
∫1
0
(3α − 1)AU(α)dα > 0(60)
then
xS= yS= 3
lS= −1
5
2
∫1
αAU(α)dα +
0
αAU(α)dα − 3
∫1
αAL(α)dα,
0
αAL(α)dα,
2
∫1
∫1
0
5
2
∫1
∫1
0
uS=
0
αAU(α)dα −
1
2
0
αAL(α)dα.
Corollary 12. Let A ∈ F (R),Aα = [AL(α),AU(α)], α ∈ [0,1], and let S(A) = (s1,s2,s3,s4) be the nearest symmetric
trapezoidal fuzzy number to A which preserves the ambiguity and value.
(i) If (59) is satisfied then
∫1
∫1
∫1
∫1
(ii) If (60) is satisfied then
∫1
∫1
∫1
s1= 2
0
(2α − 1)AU(α)dα − 2
∫1
∫1
0
(α − 1)AL(α)dα
(61)
s2= −
0
(2α − 1)AU(α)dα +
0
(4α − 1)AL(α)dα
(62)
s3=
0
(4α − 1)AU(α)dα −
∫1
0
(2α − 1)AL(α)dα
∫1
(63)
s4= −2
0
(α − 1)AU(α)dα + 2
0
(2α − 1)AL(α)dα.
(64)
s1= −2
0
αAU(α)dα + 4
∫1
∫1
0
αAL(α)dα
(65)
s2= s3=
0
αAU(α)dα +
0
αAL(α)dα
(66)
s4= 4
0
αAU(α)dα − 2
∫1
0
αAL(α)dα.
(67)
Example 13. Case (ii) in Corollary 12 is applicable for computing the nearest symmetric trapezoidal fuzzy number to fuzzy
number A in Example 6, preserving the value and ambiguity of A, and
Case (i) in Corollary 12 is applicable for computing the nearest symmetric trapezoidal fuzzy number to fuzzy number B in
Example 9, preserving the value and ambiguity of B, and
S(A) =−23
15,−
7
30,−
7
30,16
15
.
S(B) =−98
5,−91
5,−
4
15,17
15
.
5. Algorithms
Conditions (33)–(38) seem to be very technical and difficult to interpret. To make these conditions more clear and the
trapezoidalapproximationoperatorT morecompact,weexpresstheformulaeusingsomeparametersassociatedwithfuzzy
numbers instead of the integrals given in Theorem 7 and Corollary 8. The idea is not new: it was proposed in [9,10] for the
trapezoidal approximation which preserves the expected interval and continued in [24].
The typical value of a fuzzy number A, Aα = [AL(α),AU(α)], α ∈ [0,1], called the expected value of A, is given by
(see [25,26])
∫1
EV(A) =
1
2
0
AL(α)dα +
∫1
0
AU(α)dα
.
(68)
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The non-specificity of fuzzy number A, called the width of A, is introduced by (see [27])
∫1
To describe the spread of the left-hand part and the right-hand part of a fuzzy number with respect to the expected value,
the concepts of left-hand ambiguity and right-hand ambiguity were introduced in [10] as follows:
∫1
∫1
We need the following result to give suitable interpretations and algorithms for the nearest trapezoidal approximation
preserving the ambiguity and value.
w(A) =
0
AU(α)dα −
∫1
0
AL(α)dα.
(69)
AmbL(A) =
0
α (EV(A) − AL(α))dα,
(70)
AmbU(A) =
0
α (AU(α) − EV(A))dα.
(71)
Proposition 14. Let A be a fuzzy number.
(i) If 4AmbL(A) < Amb(A) then w(A) > 3Amb(A).
(ii) If 4AmbU(A) < Amb(A) then w(A) > 3Amb(A).
Proof. (i) Because (see e.g. [4, Lemma 1, (ii)])
∫1
and the hypothesis is equivalent to (34), we obtain
0
AU(α)dα − 2
∫1
0
αAU(α)dα ≥ 0
w(A) − 3Amb(A) = 3
∫1
∫1
−4
0
αAL(α)dα − 3
∫1
∫1
0
αAU(α)dα −
∫1
∫1
0
AL(α)dα +
∫1
∫1
0
AU(α)dα
= 3
0
αAL(α)dα +
∫1
0
αAU(α)dα −
∫1
0
AL(α)dα −
0
AU(α)dα
0
αAU(α)dα + 2
0
AU(α)dα > 0.
(ii) Because (see e.g. [4, Lemma 1, (i)])
∫1
and the hypothesis is equivalent to (35), we obtain
2
0
αAL(α)dα −
∫1
0
AL(α)dα ≥ 0
w(A) − 3Amb(A) = 3
∫1
0
αAL(α)dα − 3
∫1
∫1
∫1
∫1
0
αAU(α)dα −
∫1
∫1
0
AL(α)dα +
∫1
∫1
0
AU(α)dα
= −3
0
αAU(α)dα −
0
∫1
αAL(α)dα +
0
AL(α)dα +
0
AU(α)dα
+4
0
αAL(α)dα − 2
0
AL(α)dα > 0.
?
We can rewrite condition (33) as
∫1
and, according to (2) and (69),
w(A) ≤ 3Amb(A).
Otherwise, that is, when
w(A) > 3Amb(A),
0
AU(α)dα −
∫1
0
AL(α)dα ≤ 3
∫1
0
αAU(α)dα − 3
∫1
0
αAL(α)dα,
the approximation is always a triangular fuzzy number (see Corollary 8).
Condition (34) is equivalent to
3AmbL(A) < AmbU(A)
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or
4AmbL(A) < Amb(A),
and condition (35) is equivalent to
3AmbU(A) < AmbL(A)
or
4AmbU(A) < Amb(A).
As a conclusion, for fuzzy numbers that are more vague, the approximation is a trapezoidal fuzzy number computed by
(39)–(42). For fuzzy numbers that are less vague, the approximation is as follows:
– a left-side triangular fuzzy number if the ambiguity of the fuzzy number is located in the left part of the support;
– a right-side triangular fuzzy number if the ambiguity of the fuzzy number is located in the right part of the support;
– a proper triangular fuzzy number if the left-hand and right-hand ambiguities rise above a certain level of ambiguity.
Tosumup,wegetthefollowingalgorithmforcomputingthenearesttrapezoidalapproximationpreservingtheambiguity
and value.
Algorithm 1. Step 1: If w(A) ≤ 3Amb(A) then apply (39)–(42) to compute the approximation; else
Step 2: if 4AmbL(A) < Amb(A) then apply (43)–(44) to compute the approximation; else
Step 3: if 4AmbU(A) < Amb(A) then apply (45)–(46) to compute the approximation; else
Step 4: apply (47)–(49).
To avoid checking unnecessary requirements, we use Algorithm 1 if the fuzzy number is almost symmetrical or
moderately asymmetrical. If the fuzzy number A is strongly asymmetric to the right or to the left, then we can use the
following algorithm.
Algorithm 2. Step 1: If 4AmbL(A) < Amb(A) then apply (43)–(44) to compute the approximation; else
Step 2: if 4AmbU(A) < Amb(A) then apply (45)–(46) to compute the approximation; else
Step 3: if w(A) > 3Amb(A) then apply (47)–(49) to compute the approximation; else
Step 4: apply (39)–(42).
According to Theorem 11, Corollary 12 and the above discussions, we get the following algorithm for computing the
nearest symmetric trapezoidal approximation of a fuzzy number preserving the ambiguity and value.
Algorithm 3. Step 1: If w(A) ≤ 3Amb(A) then apply (61)–(64) to compute the approximation; else
Step 2: apply (65)–(67).
6. Examples
An important kind of fuzzy number was introduced in [28] as follows. Let a,b,c,d ∈ R such that a < b ≤ c < d. A fuzzy
number A such that
Aα= [AL(α),AU(α)] =
a + (b − a)α1/r,d − (d − c)α1/r,α ∈ [0,1],
where r > 0, is denoted by A = (a,b,c,d)r.
Corollary 15.
(i) If
(1 − r)(a − d) + 2r (r + 2)(b − c) ⩽ 0
then
(72)
T ((a,b,c,d)r)
(5r + 1)a + 2r (r − 1)b
=
(1 + r)(1 + 2r)
,(1 − r)a + 2r (r + 2)b
(1 + r)(1 + 2r)
,2r (r + 2)c + (1 − r)d
(1 + r)(1 + 2r)
,2r (r − 1)c + (5r + 1)d
(1 + r)(1 + 2r)
.
(ii) If
(1 − r)a +
2r2+ 4r
b − 2r2c − (3r + 1)d > 0 (73)
then
T ((a,b,c,d)r) =
3a − 2d + 6rb − 4rc
1 + 2r
,d + 2rc
1 + 2r,d + 2rc
1 + 2r,d + 2rc
1 + 2r
.
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(iii) If
(3r + 1)a + 2r2b −
2r2+ 4r
a + 2rb
c + (r − 1)d > 0 (74)
then
T ((a,b,c,d)r) =
1 + 2r,a + 2rb
1 + 2r,a + 2rb
1 + 2r,−2a + 3d − 4rb + 6rc
1 + 2r
.
(iv) If
(1 − r)(a − d) + 2r (r + 2)(b − c) > 0
(1 − r)a +
(75)
2r2+ 4r
b − 2r2c − (3r + 1)d ≤ 0
2r2+ 4r
(76)
(3r + 1)a + 2r2b −
T ((a,b,c,d)r) = (t1,t2,t3,t4),
c + (r − 1)d ≤ 0(77)
then
where
t1=(2 + 4r)a +
t2= t3=(1 − r)a +
t4=(r − 1)a −
Proof. If A = (a,b,c,d)rthen
∫1
∫1
∫1
and
∫1
such that the proof is immediate from Corollary 8.
2r + 4r2
4r + 2r2
4r + 2r2
b −
2r2+ 4r
b +
b +
c + (r − 1)d
4r + 2r2
4r2+ 2r
(1 + 2r)(1 + r)
,
c + (1 − r)d
2(1 + 2r)(1 + r)
,
c + (2 + 4r)d
(1 + 2r)(1 + r)
.
0
AL(α)dα =
a + rb
r + 1,
rc + d
r + 1,
a + 2rb
2(2r + 1)
0
AU(α)dα =
0
αAL(α)dα =
0
αAU(α)dα =
2rc + d
2(2r + 1)
?
Example 16. Case (i) in Corollary 15 is applicable to fuzzy number (1,2,3,4)2, and
19
The fuzzy numbers (1,200,201,220)2and (1,20,30,320)2satisfy conditions (73) and (74), respectively, and
355
T ((1,20,30,320)2) =
5
Case (iv) in Corollary 15 is applicable to fuzzy number (1,2,4,35)2, and
7
In the case of symmetric approximation, we obtain the following result.
T ((1,2,3,4)2) =
15,31
15,44
15,56
15
.
T ((1,200,201,220)2) =
5
,1024
5
,1024
5
,1024
5
,
81
5,81
5,81
5,1158
.
T ((1,2,4,35)2) =
5,2,2,133
5
.
Corollary 17. (i) If
(1 − r)(a − d) + 2r (r + 2)(b − c) ⩽ 0(78)
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then
S ((a,b,c,d)r) = (s1,s2,s3,s4),
where
s1=(3r + 1)a + 2r2b + 2rc − 2rd
(1 + r)(1 + 2r)
s2=
(1 + r)(1 + 2r)
s3=
(1 + r)(1 + 2r)
s4=−2ra + 2rb + 2r2c + (3r + 1)d
(1 + r)(1 + 2r)
,
a +
ra − rb +
2r2+ 3r
b − rc + rd
2r2+ 3r
,
c + d
,
.
(ii) If
(1 − r)(a − d) + 2r (r + 2)(b − c) > 0(79)
then
S ((a,b,c,d)r) = (s1,s2,s3,s4),
where
s1=
2a + 4rb − 2rc − d
1 + 2r
s2= s3=
s4=−a − 2rb + 4rc + 2d
1 + 2r
,
a + 2rb + 2rc + d
2(1 + 2r)
,
.
Example 18. The fuzzy number (1,2,3,4)2satisfies (78), and
19
The fuzzy numbers (1,200,201,220)2and (1,2,4,35)2satisfy (79), and
578
S ((1,2,4,35)2) =
S ((1,2,3,4)2) =
15,31
15,44
15,56
15
.
S ((1,200,201,220)2) =
5
,365
2
,365
2
,1247
5
,
−33
5,6,6,93
5
.
7. Properties
The properties of translation invariance (i.e., U (A + z) = U(A) + z, for every z ∈ R,A ∈ F (R)), scale invariance (i.e.,
U (λ · A) = λ·U(A), foreveryλ ∈ R,A ∈ F (R)) andidentity (i.e.,U(A) = A, foreveryA ∈ FT(R)orA ∈ FS(R), respectively)
of the approximation operators T and S are immediate (see, e.g., [4, Theorem 12]). Carlsson and Fullér [29] defined the lower
and upper possibilistic mean values of a fuzzy number A,Aα= [AL(α),AU(α)], α ∈ [0,1], as
∫1
and
∫1
Because
M∗(A) = Val(A) − Amb(A)
and
M∗(A) = Val(A) + Amb(A),
M∗(A) = 2
0
αAL(α)dα
M∗(A) = 2
0
αAU(α)dα.
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we obtain that T and S preserve the lower and upper possibilistic mean values, that is M∗(T(A)) = M∗(S(A)) = M∗(A) and
M∗(T(A)) = M∗(S(A)) = M∗(A), for every A ∈ F (R). The continuity of the operator T is an immediate consequence of the
following result.
Theorem 19. The nearest trapezoidal approximation operator preserving the ambiguity and value T : F(R) → FT(R), given
in Theorem 7, satisfies
d(T(A),T(B)) ⩽ (2√
2 + 1)d(A,B)
(80)
for all A,B ∈ F(R).
Proof. Let us consider A,B ∈ F(R),
T(A) = [lT,uT,xT,yT],
T(B) = [l′
the trapezoidal approximations preserving the ambiguity and value of A and B,
T,u′
T,x′
T,y′
T],
Te(A) = [le,ue,xe,ye],
Te(B) = [l′
the extended trapezoidal approximations of A and B, and
e,u′
e,x′
e,y′
e],
Ae(xe,ye),
Be(x′
A0(xT,yT),
B0(x′
e,y′
e),
T,y′
T).
Then
d2(T(A),T(B)) =
where dEdenotes the Euclidean metric on R2,
lT= −1
T= −1
6
1
6(yT− ye) + ue,
and
1
6
The Cauchy–Buniakowski–Schwarz inequality implies that
=
1
=
12
lT− l′
T
2+
uT− u′
T
2+
d2
E(A0,B0)
12
,
(81)
6(xT− xe) + le,
l′
x′
T− x′
e
+ l′
e,
uT=
u′
T=
y′
T− y′
e
+ u′
e.
lT− l′
T
−1
2+
6(xT− xe) + le+
x′
36
2
E(A0,B0)
+ d2(Te(A),Te(B)) + 2
uT− u′
T
2
1
6
x′
T− x′
+
2
e
− l′
le− l′
e
2
2
2+
le− l′
2
+
1
1
6(yT− ye) + ue−
yT− y′
12
2+ 2
1
6
y′
T− y′
+
2
e
− u′
ue− u′
e
2
=
6
x′
T− xT
+
2
1
6
xe− x′
36
xe− x′
ee
+
6
yT− y′
T
+
2
2.
1
6
y′
y′
e− ye
e
2
2
⩽ 3
T− xT
+
e
+
le− l′
e
+ 3
yT− y′
36
2
ue− u′
T
+
e− ye
36
2
+
ue− u′
ue− u′
e
=
x′
T− xT
12
d2
+
xe− x′
12
e
2
+ 3
le− l′
e
T
+
y′
e− ye
12
+ 3
e
2
ee
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Fig. 2. Two cases in the finding of the Lipschitz constant.
We easily get (see Proposition 2 and Remark 3)
lT− l′
T
2+
uT− u′
T
2≤ 3d2(A,B) +
d2
E(A0,B0)
12
.
Substituting in (81), we obtain
d2(T(A),T(B)) ≤ 3d2(A,B) +
Let us assume (contrariwise the proof is similar) that
6u′
d2
E(A0,B0)
6
.
(82)
e− 6l′
e− x′
e− y′
e≥ 6ue− 6le− xe− ye.
We consider
MA=
(x,y) ∈ R2: x ≥ 0,y ≥ 0,x + y ≤ 3ue− 3le−
1
2xe−
1
2x′
1
2ye
1
2y′
,
MB=
(x,y) ∈ R2: x ≥ 0,y ≥ 0,x + y ≤ 3u′
e− 3l′
e−
e−
e
and
C
3ue− 3le−
1
2xe−
1
2ye,0
,
C′
0,3ue− 3le−
1
2xe−
1
2ye
,
G
3u′
e− 3l′
e−
1
2x′
e−
1
2x′
1
2y′
e,0
,
G′
0,3u′
e− 3l′
e−
e−
1
2y′
e
,
the points which determine the closed convex sets MAand MB, MA⊆ MB, in the Euclidean space R2(see Fig. 2).
We have (see the proof of Theorem 5)
A0= PMA(Ae)
and
B0= PMB(Be).
We denote by B1 the projection of B0 on the convex set MA, that is, the unique element in MA which minimizes
dE(B0,Q), where Q ∈ MA. It is easy to check that B1is the projection of Beon the set MA, that is, B1 ∈ MAand
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minR∈MAdE(Be,R) = dE(Be,B1). Also, it is immediate that (in Fig. 2 two particular cases are represented: Aein case (iv),
Bein case (ii) and Ae,Bein case (iv), respectively; see Fig. 1 too)
1
2(x′
∫1
∫1
= 24
0
0
= 24d2(A,B);
therefore
dE(B1,B0) ≤ 2√
Because MAis a closed convex subset of the Hilbert space R2we obtain (see [15, Appendix C])
dE(B1,B0) ≤ dE(C,G) = dE
We have (see (12)–(15) and Fig. 2)
C′,G′.
(83)
d2
E(C,G) = [3(u′
e− ue) − 3(l′
∫1
e− le) −
e− xe) −
∫1
2
1
2(y′
e− ye)]2
=
6
0
α (BU(α) − AU(α))dα − 6
0
α (BL(α) − AL(α))dα
∫1
∫1
(BU(α) − AU(α))2dα
2
≤ 72
0
α (BU(α) − AU(α))dα
∫1
(BL(α) − AL(α))2dα +
+
0
α (BL(α) − AL(α))dα
∫1
2
≤ 72
0
α2dα
0
(BU(α) − AU(α))2dα + 72
0
α2dα
0
(BL(α) − AL(α))2dα
∫1
∫1
6d(A,B).
dE(PMA(Ae),PMA(Be)) ≤ dE(Ae,Be);
that is,
dE(A0,B1) ≤ dE(Ae,Be).
Since by Remark 3 we get dE(Ae,Be) ≤ 2√
dE(A0,B0) ≤ dE(A0,B1) + dE(B1,B0)
≤ dE(Ae,Be) + 2√
≤
Substituting in (82), we obtain
(84)
3d(A,B), it follows that (see Fig. 2)
6d(A,B)
2√
3 + 2√
6
d(A,B).
d(T(A),T(B)) ⩽
2√
2 + 1
?
d(A,B),
and the proof is complete.
Remark 20. The problem of finding the best Lipschitz constant of the trapezoidal approximation operator given in
Theorem 7 is not easy to study. It is more sophisticated than the case of the trapezoidal approximation operator preserving
the expected interval, completely solved in [30]. Following the same ideas as above, estimations can be found in other
interesting case, when we compare d(T(A),T(B)) and d(Te(A),Te(B)), with the observation that the calculus of the best
constant is sophisticated too.
From the proof of the above theorem, with the same notation,
lT− l′
T
2+
le− l′
2+
uT− u′
2+
uT− u′
T
2≤
e
d2
E(A0,B0)
12
2≤ d2(Te(A),Te(B)), we get
2≤ 3d2(Te(A),Te(B)) +
+ d2(Te(A),Te(B)) + 2
le− l′
e
2+ 2
ue− u′
e
2,
and since
e
ue− u′
lT− l′
TT
d2
E(A0,B0)
12
.
Then (81) implies that
d2(T(A),T(B)) ≤ 3d2(Te(A),Te(B)) +
d2
E(A0,B0)
6
.
(85)
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Taking into account (83) and (84), we have
dE(A0,B0) ≤ dE(A0,B1) + dE(B1,B0)
≤ dE(Ae,Be) + dE(C,G)
=
On the other hand,
[
≤ 36(u′
= 12
= 12d2(Te(A),Te(B)) + 24(u′
Using (86) and the immediate inequality
dE(Ae,Be) ≤ 2√
we get
Substituting in (85), we obtain
The continuity of the trapezoidal approximation operator S given in Theorem 11 is an immediate consequence of the
following result.
(x′
e− xe)2+ (y′
e− ye)2+ dE(C,G).
(86)
d2
E(C,G) =
3(u′
e− ue) − 3(l′
e− ue)2+ (l′
(u′
e− le) −
e− le)2+ (x′
1
2(x′
e− xe) −
e− xe)2+ (y′
1
12(x′
e− ue)2+ (l′
1
2(y′
e− ye)
e− ye)2
1
12(y′
]2
e− ue)2+ (l′
e− le)2+
e− xe)2+
e− le)2
e− ye)2
+ 24(u′
e− ue)2+ (l′
e− le)2
≤ 36d2(Te(A),Te(B)).
3d(Te(A),Te(B)),
dE(A0,B0) ≤
2√
3 + 6
d(Te(A),Te(B)).
d(T(A),T(B)) ≤
11 + 4√
3d(Te(A),Te(B)).
Theorem 21. The nearest symmetric trapezoidal approximation operator preserving the ambiguity and value S : F(R) → FS(R)
satisfies
√
11d(A,B)
d(S(A),S(B)) ⩽
for all A,B ∈ F(R).
Proof. Let us consider A,B ∈ F(R),
S(A) = [lS,uS,xS,xS],
S(B) = [l′
the nearest symmetric trapezoidal approximations preserving the ambiguity and value of A and B,
S,u′
S,x′
S,x′
S],
Te(A) = [le,ue,xe,ye],
Te(B) = [l′
the extended trapezoidal approximations of A and B, and
e,u′
e,x′
e,y′
e],
Ae(xe,ye),
Be(x′
A0(xS,xS),
B0(x′
e,y′
e),
S,x′
S).
Following the same reasoning as in the proof of Theorem 19, we get
d2(S(A),S(B)) ≤ 3d2(A,B) +
Without any loss of generality, let us assume that
6u′
d2
E(A0,B0)
6
.
(87)
e− 6l′
e− x′
e− y′
e≥ 6ue− 6le− xe− ye.
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We consider the closed convex sets
MB=
MA=
(x,y) ∈ R2: x ≥ 0,x = y,x + y ≤ 3ue− 3le−
1
2xe−
1
2x′
1
2ye
1
2y′
,
(x,y) ∈ R2: x ≥ 0,x = y,x + y ≤ 3u′
e− 3l′
e−
e−
e
and the points
C
3ue− 3le−
1
2xe−
1
2ye,0
,
C′
0,3ue− 3le−
1
2xe−
1
2ye
,
G
3u′
e− 3l′
e−
1
2x′
e−
1
2x′
1
2y′
e,0
,
G′
0,3u′
e− 3l′
e−
e−
1
2y′
e
.
According to (50)–(54), it follows that
A0= PMA(Ae)
and
B0= PMB(Be).
As in the proof of Theorem 19, let us consider B1= PMA(B0). Again, it is immediate that B1is the projection of Beon the set
MA. In addition, we have
dE(B1,B0) ≤
Since by the proof of Theorem 19 we have dE(C,G) ≤ 2√
(88) are important here)
1
√
2
dE(C,G) =
1
√
2
dE
C′,G′.
(88)
6d(A,B) and dE(Ae,Be) ≤ 2√
3d(A,B), it follows that ((84) and
dE(A0,B0) ≤ dE(A0,B1) + dE(B1,B0)
≤ dE(Ae,Be) + 2√
≤ 4√
Substituting in (87), we obtain
3d(A,B)
3d(A,B).
d(S(A),S(B)) ⩽
√
11d(A,B),
which proves the theorem.
?
8. Applications
8.1. Ranking of fuzzy numbers by the value–ambiguity indices
A method of ranking fuzzy numbers is to introduce a real number R(A), called the ranking index, for every fuzzy number
A, and then to define
A < B ⇔ R(A) < R(B),
A ∼ B ⇔ R(A) = R(B)
and
A > B ⇔ R(A) > R(B).
Delgado et al. [18] considered that any comparison procedure ought to take into account the magnitude assessment as well
as the imprecision involved in any fuzzy number; therefore, value and ambiguity appear to be good parameters to be used
together for this purpose. They introduced the following definition.
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Definition 22. For every fuzzy number A,
ri(A) = αVal(A) + βAmb(A),
where α ∈ [0,1] and β ∈ [−1,1] are given, will be called rank index for A.
When Val(A) = Val(B) and Amb(A) = Amb(B), two fuzzy numbers A and B should be considered to be equal. The
comparison procedure should be mainly determined by the value; therefore |β| ≪ α must be imposed and β represents
the decision-maker’s attitude towards the uncertainty.
Because the operator T preserves the ambiguity and value, we can compare ri(T(A)) and ri(T(B)) instead to compare
ri(A) and ri(B) to rank fuzzy numbers A and B; in this way the calculus is simplified.
Example 23. Let us consider the fuzzy numbers A = (1,200,201,220)2and B = (1,20,30,320)2. Taking into account the
results in Example 16 and (5)–(10), for α =
ri(A) = 91.473 > 26.409 = ri(B);
therefore A > B.
1
2and β =
1
100, we obtain
8.2. Estimation of the defect of additivity
The trapezoidal approximation operator T is not additive, as the following example proves.
Example 24. Let us consider the fuzzy numbers A = (1,2,3,4)2and B = (1,2,4,35)2given in Example 16. Because
A + B = (2,4,7,39)2,
condition (72) is satisfied, and
38
We obtain
40
T (A + B) =
15,62
15,73
15,457
15
.
T (A + B) ̸= T(A) + T(B) =
15,61
15,74
15,455
15
.
Following the ideas in [31], the notion of defect of additivity of a trapezoidal approximation operator with respect to a
given fuzzy number is introduced.
Definition 25. Let A ∈ F (R), and let U : F (R) → FT(R) be a trapezoidal approximation operator. The defect of additivity
of the operator U with respect to fuzzy number A is given by
δU(A) = sup
B∈F(R)
d(U(A) + U(B),U(A + B)).
Theorem 19 helps us to find an estimation of the defect of additivity of the approximation operator T in Theorem 7.
Indeed, if O is the trapezoidal fuzzy number (0,0,0,0) then T(O) = O and
d(T(A) + T(B),T(A + B)) ≤ d(T(A) + T(B),T(B)) + d(T(B),T(A + B))
= d(O,T(A)) + d(T(B),T(A + B))
= d(T(O),T(A)) + d(T(B),T(A + B))
≤ (2√
= (2√
= 2(2√
0
2 + 1)d(O,A) + (2√
2 + 1)d(O,A) + (2√
∫1
2 + 1)d(B,A + B)
2 + 1)d(O,A)
U(α)
2 + 1)
A2
L(α) + A2
dα
1/2
,
for every A,B ∈ F (R).
The reasoning above can be repeated for the non-additive approximation operator S in Theorem 11. The following result
is immediate.
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Proposition 26. Let A be a fuzzy number. Then
δT(A) ≤ 2(2√
2 + 1)
∫1
0
A2
L(α) + A2
U(α)
dα
1/2
and
δS(A) ≤ 2√
11
∫1
0
A2
L(α) + A2
U(α)
dα
1/2
.
9. Conclusion
The main results ofthis paper areTheorems 7and11, where operators ofapproximation of fuzzy numbers by trapezoidal
fuzzy numbers are given. The finding of the best Lipschitz constant in Theorems 19 and 21 and the computation of the defect
of additivity for the trapezoidal approximation operators T and S are open problems.
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