International Journal of Analysis and Applications

Volume 16, Number 2 (2018), 290-305

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-16-2018-290

AN APPROXIMATION OF FUZZY NUMBERS BASED ON POLYNOMIAL FORM

FUZZY NUMBERS

SH. YEGANEHMANESH AND M. AMIRFAKHRIAN∗

Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran

∗Corresponding author: amirfakhrian@iauctb.ac.ir

Abstract. In this paper, we approximate an arbitrary fuzzy number by a polynomial fuzzy number through

minimizing the distance between them. Throughout this work, we used a distance that is a meter on the

set of all fuzzy numbers with continuous left and right spread functions. To support our claims analytically,

we have proven some theorems and given supplementary corollaries.

1. Introduction

Comparison of fuzzy numbers is an indispensable part of most systems using such numbers. To this end,

many researchers active in the Fuzzy Theory domain have tried to make fuzzy numbers comparable. Some

authors have approximated a fuzzy number by a single crisp number. This method which is called ranking

suffers from loss of some useful information.

Some authors such as [8] convert a given fuzzy number into an interval and solve an interval arithmetic

problem instead of a more complicated fuzzy computation. However, the fuzzy central concept fades here.

Finding the nearest triangular or trapezoidal fuzzy number associated to an arbitrary given fuzzy number is

another method on which some authors such as [2], [4], [6], [10] and [11] have concentrated. However, this

method fails to guarantee the same modal value (or interval). Also, some authors such as [12] and [13] have

made a considerable contribution to the coefficients of polynomial, the concept that we have used in this

research.

Received 2017-10-28; accepted 2018-01-08; published 2018-03-07.

2010 Mathematics Subject Classification. 00A00.

Key words and phrases. fuzzy number; parametric form; distance; polynomial form.

c©2018 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

290

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-16-2018-290


Int. J. Anal. Appl. 16 (2) (2018) 291

In this paper, we propose two methods for approximating a given arbitrary fuzzy number with a polynomial

fuzzy number to a great degree of accuracy. The first method splits the approximation problem into two

sub-problems and solves them separately whereas the second one solves the problem in a general form.

2. Basic Concepts

In this section, the basic concepts used throughout the paper are given. Let F(R) be the set of all fuzzy

numbers (the set of all normal and convex fuzzy sets) on the real line.

Definition 2.1. A generalized LR fuzzy number ũ with the membership function µũ(x), x ∈ R can be defined

as [1]:

µũ(x) =




Lũ(x), a ≤ x ≤ b,

1, b ≤ x ≤ c,

Rũ(x), c ≤ x ≤ d,

0, otherwise,

(2.1)

where Lũ is the left membership function and Rũ is the right membership function. It is assumed that Lũ is in-

creasing in [a,b] and Rũ is decreasing in [a,b], and that

Lũ(a) = Rũ(d) = 0 and Lũ(b) = Rũ(c) = 1. In addition, if Lũ and Rũ are linear, then ũ is a trape-

zoidal fuzzy number, which is denoted by ũ = (a,b,c,d). If b = c, we denoted it by ũ = (a,c,d), which is a

triangular fuzzy number.

The parametric form of a fuzzy number is given by

ũ = (u,u), where u and u are functions defined over [0, 1] and satisfy the following requirements:

(1) u is a monotonically increasing left continuous function.

(2) u is a monotonically decreasing left continuous function.

(3) u ≤ u, in [0, 1].

We name u and u, left and right spread functions, respectively. If a is a crisp number, then u(r) = u(r) = a,

for ∀r ∈ [0, 1].

Definition 2.2. We say that a fuzzy number ṽ has an m−degree polynomial form, if there exist two poly-

nomials p and q of degree at most m such that ṽ = (p,q) [3].

Let ṽ ∈ Fm(R) be the set of all m−degree polynomial form fuzzy numbers. For 0 < α ≤ 1, α-cut of a

fuzzy number ũ is defined by [5] as follows:

[ũ]α = {t ∈ R | µũ(t) ≥ α}. (2.2)



Int. J. Anal. Appl. 16 (2) (2018) 292

The core of a fuzzy number is defined by [5] as follows:

core(ũ) = {t ∈ R | µũ(t) = 1}. (2.3)

Let Fc(R) be the set of all fuzzy numbers with continuous left and right spread functions and let Fm(R)

be the set of all m−degree polynomial form fuzzy numbers [3]. We also consider Πm as the set of all poly-

nomials of degree at most m.

We can write a fuzzy number ũ ∈Fm(R) as follows:

ũ = (u,u), (2.4)

where u, u ∈ Πm.

3. A Parametric Distance

In order to measure the distance between two fuzzy numbers, here, we propose a new definition.

Definition 3.1. For ũ, ṽ ∈F(R), the distance of ũ and ṽ is defined by

Dp,q(ũ, ṽ) =

(∫ 1
0

q|u(r) −v(r)|p dr +
∫ 1

0

(1 −q)|u(r) −v(r)|p dr
)1

p

, (3.1)

where q ∈ [0, 1] and p > 0.

Theorem 3.1. Dp,q is a metric on Fc(R).

Proof. It can be found in [7].

�

As the q changes in (3.1), the distance Dp,q gets biased towards either the left spread function or the right

one.

4. The Best Polynomial Fuzzy Number to an Arbitrary Fuzzy Number

Knowing the fact that a fuzzy number can be approximated in terms of an m-degree polynomial, it is

now aimed at finding the nearest m-degree polynomial to a given fuzzy number. To this end, the proposed

parametric distance defined in Section 3 is used.

Assume that ũ is an arbitrary fuzzy number and ṽ is an approximated m-degree polynomial form fuzzy

number. For p = 2 in (3.1), the distance becomes as:

D2,q(ũ, ṽ) =

(∫ 1
0

q|u(r) −v(r)|2 dr +
∫ 1

0

(1 −q)|u(r) −v(r)|2 dr
)1

2

, (4.1)



Int. J. Anal. Appl. 16 (2) (2018) 293

where q ∈ [0, 1].

Now, the approximation problem becomes as:


min
ṽ∈Fm

D2,q(ũ, ṽ),

s.t.

ṽ(1) = ũ(1),

(4.2)

which can be expanded as follows:




min
v,v∈Πm

∫ 1
0
q|u(r) −v(r)|2 dr +

∫ 1
0

(1 −q)|u(r) −v(r)|2 dr,

s.t.

v(1) = u(1),

v(1) = u(1).

(4.3)

Before solving this problem, let’s present the Lemma 4.1 which will come in handy in our approximation

method.

Lemma 4.1. Let f and g be two arbitrary functions defined on a domain D ⊆ R. Then over this domain

we have:

min(f(x) + g(x)) ≥ min f(x) + min g(x). (4.4)

Proof. Straightforward. �

In the following, we propose our two new approximation methods which minimize the distance first based

on splitting the problem and second based on general form.

4.1. Minimization by splitting the problem. From Lemma 4.1, it is clear that splitting the problem

(4.3) into two sub-problems will lead us to have a less objective value. Since q is constant and both terms of

the objective functions in (4.3) are non-negative, by Lemma 4.1 the problem is divided into two independent

sub-problems:




min
v∈Πm

∫ 1
0
|u(r) −v(r)|2 dr,

s.t.

v(1) = u(1).

(4.5)

and 


min
v∈Πm

∫ 1
0
|u(r) −v(r)|2 dr,

s.t.

v(1) = u(1).

(4.6)



Int. J. Anal. Appl. 16 (2) (2018) 294

Assume that v(r) =
∑m
j=0 ajr

j, for solving the problem (4.5) with Lagrangian method we define the

following function:

F(a,λ) =

∫ 1
0

|u(r) −v(r)|2 dr −λ(u(1) −v(1)), (4.7)

where a = (a0,a1, ...,am)
t.

The necessary condition to minimize the function F is that the gradient of function should be zero. The

gradient of function F can be shown as follows:

5F(a,λ) =




2Ha + λ1 − 2m

1ta −u(1)


 , (4.8)

where 1 = (1, ..., 1)t, H is the m + 1 Hermitian matrix which its elements defined as Hij = (i + j + 1)
−1

and m is the momentum u :

m =

[∫ 1
0

riu(r) dr

]t
i=0,...,m

. (4.9)

5F = 0 gives following: 


Ha +
1

2
λ1 − m = 0,

v(1) = u(1).

(4.10)

Thus, we define QF : Rm+2 −→ Rm+2 as follows:

QF (a,λ) =



Ha +

1

2
λ1 − m

1ta −u(1)


 . (4.11)

Hence, we try to solve QF (x) = 0 which is a linear system such that:

x = (a,λ)t. (4.12)

To solve this system, we have:

QF (a,λ) = A x − R, (4.13)

where

R =




m

u(1)


 , (4.14)

And A is as follows:

A =



H

1

2
1

1t 0


 . (4.15)



Int. J. Anal. Appl. 16 (2) (2018) 295

Hence, we have:

x = A−1 R, (4.16)

Theorem 4.1. The inverse matrix of A has the following form:

A−1 =



H−1 − vvt

1

m + 1
v

2

m + 1
vt −

2

(m + 1)2


 , (4.17)

where v = 1
m+1

H−11.

Proof. It is straightforward. �

We consider the solution of minimization problem (4.7) as x∗ = (a∗,λ∗)t such that:


a∗ = (H−1 − vvt)m +
u(1)

m + 1
v,

λ∗ =
2

m + 1
vtm −

2u(1)

(m + 1)2
.

(4.18)

Now, in the same way, we solve the Problem (10).

Let v(r) =
m∑
j=0

bjr
j, b = (b0,b1, ...,bm)

t. For solving with Lagrangian method , we continue by defining G as

follows:

G(b,µ) =

∫ 1
0

|u(r) −v(r)|2 dr −µ(u(1) −v(1)). (4.19)

Let define the momentum vector of u as:

m =

[∫ 1
0

riu(r) dr

]t
i=0,...,m

, (4.20)

Thus, we define QG : Rm+2 −→ Rm+2 as follows:

QG(b,µ) =



Hb +

1

2
µ1 − m

1tb −u(1)


 . (4.21)

as we did it before we have a linear system az follows:

QG(b,µ) = A x − R, (4.22)

where

x = (b,µ)t, (4.23)

thus:

x = A−1 R, (4.24)



Int. J. Anal. Appl. 16 (2) (2018) 296

Considering the solution of minimization problem (4.7) as x∗ = (b∗,µ∗)t such that:


b∗ = (H−1 − vvt)m +
u(1)

m + 1
v,

µ∗ =
2

m + 1
vtm −

2u(1)

(m + 1)2
.

(4.25)

In summary, assuming ũ ∈F(R) be an arbitrary fuzzy number, we find the best approximation of ũ out

of Fm for a fixed integer m. In this case, ũ∗m is the best approximation of ũ, such that:

u∗(r) =
m∑
j=0

a∗jr
j and u∗(r) =

m∑
j=0

b∗jr
j,

where a∗ = (a∗0,a
∗
1, ...,a

∗
m)

t and b∗ = (b∗0,b
∗
1, ...,b

∗
m)

t.

We denote the best approximation of ũ ∈ F out of Fm by ũ∗m. In following theorem we show that the

best approximation of an arbitrary polynomial fuzzy number is itself.

Theorem 4.2. If ũ ∈Fm then its best approximation ũ∗m, out of Fm(R) with respect to distance (3.1) exists

and ũ∗m = ũ.

Proof. Straightforward. �

Corollary 4.2. Best approximation of an arbitrary trapezoidal fuzzy number is itself.

Proof. It can obtained by Theorem 4.2. �

4.2. Minimization of the problem in general form. In this section, we try to solve problem (4.3) in

general form. Assume that v(r) =
m∑
j=0

ajr
j and v(r) =

m∑
j=0

bjr
j. To this end, for solving the problem (4.3)

with Lagrangian method we define the following function:

E(a, b,λ,µ) =

∫ 1
0

q(u(r) −v(r))2 dr +
∫ 1

0

(1 −q)(u(r) −v(r))2 dr

−λ(u(1) −v(1)) −µ(u(1) −v(1)),
(4.26)

where a = (a0,a1, ...,am)
t, b = (b0,b1, ...,bm)

t.

The necessary condition to minimize the function E is that the gradient of function should be zero.

The gradient of function E can be shown as follows:

5E(a, b,λ,µ) =




2qHa + λ1 − 2qm

2(1 −q)Hb + µ1 − 2(1 −q)m

1ta −u(1)

1tb −u(1)



, (4.27)



Int. J. Anal. Appl. 16 (2) (2018) 297

where H is the m + 1 Hermitian matrix, 1 = (1, ..., 1)t, m and m respectively are the momentum vectors of

u and u defined in (4.9) and (4.20).

5E = 0 gives following:




qHa +
1

2
λ11 −qm = 0,

(1 −q)Hb +
1

2
µ11 − (1 −q)m = 0,

v(1) = u(1),

v(1) = u(1).

(4.28)

Now, we define QE as follows:

QE(a,λ, b,µ) =




qHa +
1

2
λ1 −qm

1ta −u(1)

(1 −q)Hb +
1

2
µ1 − (1 −q)m

1tb −u(1)



. (4.29)

Hence, we try to solve QE(t) = 0 which is a linear system such that:

t = (a,λ, b,µ)t. (4.30)

To solve this system, we let QE(t) = AE t−Z = 0 where:

AE =




qH
1

2
1 0 0

1t 0 0 0

0 0 (1 −q)H
1

2
1

0 0 1t 0




4×4

, (4.31)



Int. J. Anal. Appl. 16 (2) (2018) 298

and

Z =




qm

u(1)

(1 −q)m

u(1)



. (4.32)

Now we countinue with finding the invers of coefficient matrix, AE. By considering AE,γ as

AE,γ =



γH

1

2
1

1t 0


 , (4.33)

where γ ∈ [0, 1] and we have

AE =



AE,q 0

0 AE,(1−q)


 . (4.34)

Lemma 4.3. A−1E,γ has the following form:

A−1E,γ =




1

γ
(H−1 − vvt)

1

m + 1
v

2

m + 1
vt −γ

2

(m + 1)2


 , (4.35)

such that v = 1
m+1

H−11.

Proof. Straightforward. �

Theorem 4.3. The inverse matrix of AE has the following form:

A−1E =



A−1E,q 0

0 A−1
E,(1−q)


 , (4.36)

Proof. It is straightforward. �

To do this end, with Theorem 4.3 we have:

t = A−1E Z, (4.37)



Int. J. Anal. Appl. 16 (2) (2018) 299

We consider the solution of minimization problem (4.26) as xE
∗ = (a∗,λ∗, b∗,µ∗)t such that:



a∗ = (H−1 − vvt)m +
u(1)

m + 1
v,

b∗ = (H−1 − vvt)m +
u(1)

m + 1
v,

λ∗ =
2q

m + 1
vtm −

2qu(1)

(m + 1)2
,

µ∗ =
2(1 −q)
m + 1

vtm −
2(1 −q)u(1)

(m + 1)2
.

(4.38)

Analogous to the Theorem 4.2, in following theorem we again show that the best approximation of an

arbitrary polynomial fuzzy number is itself.

Theorem 4.4. If ũ ∈Fm then its best approximation ũ∗m, out of Fm(R) with respect to distance (3.1) exists

and ũ∗m = ũ.

Proof. It can be proved by (4.38). �

Corollary 4.4. If ũ ∈Fl where (l ≤ m), then ũ∗m = ũ .

Proof. straightforward. �

Note that if the obtained approximated coefficients yield a polynomial form fuzzy number, this polynomial

is the best approximation of the given fuzzy number.

5. Convergence of Approximation

In this section, the convergence of the proposed approximation methods are shown.

Lemma 5.1. Let m ∈ N

lim
m→∞

1

m

m∑
j=1

1

j
= 0.

Proof. It is trivial. �

Theorem 5.1. If u and u are integrable functions in [0, 1] and ũ∗m is the best approximation of ũ by splitting

the problem in Subsection 4.1 out of Fm, then

lim
m→∞

ũ∗m = ũ.



Int. J. Anal. Appl. 16 (2) (2018) 300

Proof. From (4.18), we have

lim
m→∞

λ∗ = lim
m→∞

(
2

m + 1
vtm −

2u(1)

(m + 1)2

)

= 2 lim
m→∞

1

m + 1

m∑
j=0

1∫
0

rju(r) dr

(5.1)

Since rj is nonnegative in [0, 1], according to Midpoint Theorem for integrals there exists θj ∈ (0, 1), such

that
1∫

0

rju(r) dr =
u(θj)

j + 1
, j = 0, ...,m (5.2)

Therefore, from (5.1) and Lemma 5.1 we have

lim
m→∞

λ∗ = 2 lim
m→∞

1

m + 1

m∑
j=0

u(θj)

j + 1

≤‖u‖∞2 lim
m→∞

1

m + 1

m+1∑
j=1

1

j
= 0.

(5.3)

From (4.15), (4.18) and (5.3), when m →∞, a∗ is the solution of Ha = m, where H is a Hermitian matrix.

In this case, a∗ is the solution of a common crisp problem and for this solution we have the convergence.

Similarly from (4.25) , we have

lim
m→∞

µ∗ = 0 (5.4)

and these claims hold for b∗ in Hb = m.

Since a∗ and b∗ are both convergent, therefore, u∗ and u∗ are also convergent and this completes the

proof. �

Theorem 5.2. If u and u are integrable functions in [0, 1] and ũ∗m is the best approximation of ũ of general

form in Subsection 4.2 out of Fm, then

lim
m→∞

ũ∗m = ũ.

Proof. It was obtained by (4.38) and Lemma 5.1. �

Corollary 5.2. If ũ ∈Fm, the approximation sequence converges to the exact solution in the first iteration.

Proof. straightforward. �

Considering (4.18), (4.25) and (4.38) for an arbitrary fuzzy number, the best approximation regarding

both methods are identical. According to following lemma we present an explicit formula to approximate an

arbitrary fuzzy number with a trapezoidal fuzzy number.



Int. J. Anal. Appl. 16 (2) (2018) 301

Theorem 5.3. If ũ = (u,u) is an arbitrary fuzzy number, then its best linear approximation ũ∗1 regarding

the distance (3.1) is ũ∗1 = (a0 + a1r,b0 + b1r) where:

a0 =
1

2
(6

∫ 1
0

u(r) dr − 6
∫ 1

0

ru(r) dr −u(1)), (5.5)

a1 = −
3

2
(2

∫ 1
0

u(r) dr − 2
∫ 1

0

ru(r) dr −u(1)), (5.6)

b0 =
1

2
(6

∫ 1
0

u(r) dr − 6
∫ 1

0

ru(r) dr −u(1)), (5.7)

b1 = −
3

2
(2

∫ 1
0

u(r) dr − 2
∫ 1

0

ru(r) dr −u(1)). (5.8)

Proof. straightforward. �

In the following, let’s present the Lemma 5.3 which will come in handy in showing our best linear approx-

imation of an arbitrary fuzzy number is a trapezoidal fuzzy number.

Lemma 5.3. For any arbitrary function g, if g is a monotonically increasing left continuous function then:∫ 1
0

xg(x) dx−
∫ 1

0

g(x) dx +
1

2
g(1) ≥ 0,

and if g is a monotonically decreasing left continuous function then:∫ 1
0

xg(x) dx−
∫ 1

0

g(x) dx +
1

2
g(1) ≤ 0,

Proof. Straightforward. �

Lemma 5.4. The best linear approximation of an arbitrary fuzzy number ũ = (u,u) regarding the distance

(3.1) is a trapezoidal fuzzy number.

Proof. As regards to distance (3.1) and by Lemma 5.3 and 5.3 it was obtained. �

Due to the Theorem 5.3 and Lemma 5.4 for any arbitrary fuzzy number, the nearest trapezoidal fuzzy

number regarding the distance (3.1) can be obtained from equations (5.5) - (5.8).

6. Numerical Examples

In this section we present some examples which have been solved by Mathematica software using 10

decimal digits.

Example 6.1. Let ũ = (2r2 + 1, 5 − r2). By assuming m = 2 and each q ∈ [0, 1] the best approximation of

ũ is itself. According to Theorem 4.2 it could be foretold.

Example 6.2. Let ũ = (r2 + 1, 3 −r2). By assuming m = 1 and each q ∈ [0, 1] the best approximation of ũ

can be found by Lemma 5.3. The best approximation is ( 3
4

+ 5
4
r, 13

4
− 5

4
r).



Int. J. Anal. Appl. 16 (2) (2018) 302

Example 6.3. Let ũ = (er,e2−r). For m = 1 and m = 3, the best approximations of ũ are ũ∗1 and ũ
∗
3,

where:



u∗1(r) =

1

2
(−12 + 5e) +

3

2
(4 −e)r

u∗1(r) =
5

2
e−

3

2
er



u∗3(r) =
1

4
(−4560 + 1679e) −

15

4
(−3216 + 1183e)r

+
15

4
(−7168 + 2637e)r2 −

35

4
(−1824 + 671e)r3

u∗3(r) =
1

4
(3599e− 1320e2) +

15

4
(−2719e + 1000e2)r

−
15

4
(−6285e + 2312e2)r2 +

35

4
(−1631e + 600e2)r3

and for q = 0.5 the distance (4.1) between ũ and ũ∗1 is D(ũ, ũ
∗
1) = 0.185451 and the distance between ũ and

ũ∗3 is D(ũ, ũ
∗
3) = 0.000835893.

Example 6.4. Let ũ = (ln [(e− 1)r + 1] , 2−ln [(e− 1)r + 1]), pi(r) = ri and an arbitrary q . The distances

(4.1) between ũ and ũ∗m, for m = 1, · · · , 7, are shown in Table 1.

m D(ũ, ũ∗m)

1 4.85342 × 10−2

2 7.2679 × 10−3

3 1.27813 × 10−3

4 2.44005 × 10−4

5 4.89405 × 10−5

6 1.04687 × 10−5

7 5.88816 × 10−6

Table 1: Distances for different values of m

Regarding Theorem 5.1, it was predictable that increasing the variable m would reduce the associated

error. Since ũ is a symmetric fuzzy number and by (4.38) the best approximation of it is independent of q,

the distance (4.1) between ũ and ũ∗m is independent of q.

Example 6.5. In this example, we approximate a fuzzy number with m = 1 by a trapezoidal one and compare

the results from our method with the results obtained from other four methods proposed in [2, 9, 11, 14] in a

tabular format in Table 2.



Int. J. Anal. Appl. 16 (2) (2018) 303

u(r) / u(r) (1) (2) (3) (4) (5)

1 − 0.3
√
− ln r

2 + 0.7
√
− ln r

0.484391

1

2

3.20309

0.50052

0.96775

2.07526

3.16546

0.06790

1.67725

1.67725

3.2866

0.52195

0.89256

2.10743

2.9722

0.50052

0.96775

2.07526

3.16546

3r

7 − 3r

0.740495

3

4

6.2595

0.84003

2.80092

4.19908

6.15997

0.4905

3.5

3.5

6.5095

0.48633

2.97779

4.02221

6.51367

0.84003

2.80092

4.19908

6.15997

1

2

1

1

2

2

1

1

2

2

0.75

1.5

1.5

2.25

1

1

1

2

1

1

2

2

r + 1

5 − 3r

1

2

2

5

1

2

2

5

−0.5

2

2

4.5

1

2

2

2.5

1

2

2

5

1

3 −r

1

1

2

3

1

1

2

3

0.5

1.75

1.75

3

−0.33333

0.66667

2.33333

2.33333

1

1

2

3

Table 2. Numerical results of examples

As it is obvious, our method yields trapezoidal fuzzy numbers with closer cores in comparison with the

ones obtained from the other four methods. While the other four methods fail in approximating the m-degree

polynomial form fuzzy numbers, our method can approximate all the trapezoidal, triangular and m-degree

polynomial form fuzzy numbers.

Example 6.6. Let ũ = (2 + er−1, 4 − ln [(e− 1)r + 1]), pi(r) = ri. The distances between ũ and ũ∗m, for

m = 0, · · · , 4 and q = {0, 0.25, 0.5, 0.75, 1}, are shown in Table 3.

D(ũ, ũ∗m) q = 0 q = 0.25 q = 0.5 q = 0.75 q = 1

m = 0 0.504053 0.482261 0.459435 0.435415 0.409989

m = 1 0.0485342 0.0458563 0.0430119 0.0399657 0.0366673

m = 2 0.0072679 0.00643446 0.0054756 0.00430839 0.00267248

m = 3 0.00127813 0.00110962 0.000910436 0.000653097 0.000155493

m = 4 0.000244005 0.000211315 0.000172538 0.000122003 0.000000000

Table 3: Distances for different values of m and q



Int. J. Anal. Appl. 16 (2) (2018) 304

This table shows that as the variable m increases, the distance (4.1) between exact given fuzzy number

and our approximated polynomial form fuzzy number reduces(it can be predicted by Theorem 5.1). In this

example ũ is not a symmetric fuzzy number. Hence, the distance between ũ and ũ∗m depends on q. As we can

see in the table, whenever q increases from 0 to 1, the distance decreases. Considering the distance equation

(4.1), it can be deduced that the approximation of right spread is more precise than the approximation of

the left one.

7. Conclusion

In this paper, a new distance metric was proposed on the set of all fuzzy numbers with continuous left

and right spread functions. Using this metric, a given fuzzy number can be approximated through finding

the nearest polynomial form fuzzy number out of the set of all m-degree polynomial form fuzzy numbers.

Hence, two methods were proposed to solve the approximation problem. We showed that both of the

methods not only yield the same results, but also are convergent. Finally, we investigated our theorems in

some numerical examples.

References

[1] S. Abbasbandy, M. Amirfakhrian, The nearest trapezoidal form of a generalized left right fuzzy number, Int. J. Approx.

Reason. 43 (2006), 166-178.

[2] S. Abbasbandy, B. Asady, The nearest trapezoidal fuzzy number to a fuzzy quantity, Appl. Math. Comput. 156 (2004),

381-386.

[3] M. Amirfakhrian, Numerical Solution of a System of Polynomial Parametric form Fuzzy Linear Equations, Book Chapter

of Ferroelectrics, INTECH Publisher, Austria, (2010).

[4] A. I. Ban, L. Coroianu, Existence, uniqueness and continuity of trapezoidal approximations of fuzzy numbers under a

general condition, Fuzzy Sets Syst. 3 (2014) 3-22.

[5] D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Application, Academic Press, New York, (1980).

[6] L. Coroianu, M. Gagolewski, P. Grzegorzewski, Nearest piecewise linear approximation of fuzzy numbers, Fuzzy Sets Syst.

233 (2013), 26-51.

[7] P. Grzegorzewski, Metrics and orders in space of fuzzy numbers, Fuzzy Sets Syst. 97 (1998), 83-94.

[8] P. Grzegorzewski, Nearest interval approximation of a fuzzy number, Fuzzy Sets Syst. 130 (2002), 321-330.

[9] P. Grzegorzewski, P. Mró wka, Trapezoidal approximations of fuzzy numbers, Fuzzy Sets Syst. 153 (2005), 115-135.

[10] P. Grzegorzewski, K. Pasternak-Winiarska,Natural trapezoidal approximations of fuzzy numbers, Fuzzy Sets Syst. 250

(2014), 90-109.

[11] M. Ma and A. Kandel and M. Friedman, Correction to ”A new approach for defuzzification”, Fuzzy Sets Syst. 128 (2002),

133-134.

[12] V. Powers, B. Reznick, Polynomials that are positive on an interval, Trans. Amer. Math. Soc. 352 (10) (2000), 4677C4692.

[13] J. Stolfi, M.V.A. Andrade, J.L.D. Comba, and R.Van Iwaarden. Affine arithmetic: a correlation-sensitive variant of interval

arithmetic, accessed January 17, (2008).

[14] W. Voxman, Some remarks on distance between fuzzy numbers, Fuzzy Sets Syst. 100 (1998), 353-365.



Int. J. Anal. Appl. 16 (2) (2018) 305

[15] H. J. Zimmermann, Fuzzy Set Theory and Its Applications, 2nd Edition, Kluwer Academic, Boston, (1991).


	1. Introduction
	2. Basic Concepts
	3. A Parametric Distance
	4. The Best Polynomial Fuzzy Number to an Arbitrary Fuzzy Number
	4.1. Minimization by splitting the problem
	4.2. Minimization of the problem in general form

	5. Convergence of Approximation
	6. Numerical Examples
	7. Conclusion
	References