# Nonlinear Triangular Intuitionistic Fuzzy Number and Its Application in Linear Integral Equation

Authors: Sankar Prasad Mondal; Adrijit Goswami; Sujit Kumar De
Journal: Advances in Fuzzy Systems
 (2019)

---

## Abstract

In this paper we introduce the different arithmetic operations on nonlinear intuitionistic fuzzy number (NIFN). All the arithmetic operations are done by max-min principle method which is nothing but the application of interval analysis. We also define the nonlinear intuitionistic fuzzy function which is used for finding the values, ambiguities, and ranking of nonlinear intuitionistic fuzzy number. The de-i-fuzzification of the corresponding intuitionistic fuzzy solution is done by average of(
α
,
β
)-cut method. Finally we solve integral equation with NIFN by the help of intuitionistic fuzzy Laplace transform method.


---

## Body

## 1. Introduction

### 1.1. Intuitionistic Fuzzy Sets Theory

Lotfi A. Zadeh [1] published the theory on fuzzy sets and systems. Chang and Zadeh [2] introduced the concept of fuzzy numbers. Different mathematicians have been studying them (dimension of one or dimension of n, see, for example, [3–6]). With the improvements of theories and applications of fuzzy numbers, this concept becomes more and more significant.Generalization of [1] is taken to be one of intuitionistic fuzzy set (IFS) theory. IFS was first introduced by Atanassov [7] and has been found to be suitable for dealing with various important areas. The fuzzy set considers only the degree of belongingness but not the nonbelongingness. Fuzzy set theory does not incorporate the degree of hesitation (i.e., degree of nondeterminacy defined). To handle such facts, Atanassov [7] explored the concept of fuzzy set theory by IFS theory. The degree of acceptance in fuzzy sets is considered only, but on the other hand IFS is characterized by a membership function and a nonmembership function so that the sum of both values is less than one [8]. Various results on intuitionistic fuzzy set theory are discussed in the papers [9, 10]. The uncertainty theory and calculus constitute a very popular topic nowadays [11–16].


### 1.2. Fuzzy Integral Equation

Integral equation is very important in the theory of calculus. Nowadays it is very important for application. Now if it is with uncertainty, then its behavior changes. In this paper the idea of intuitionistic fuzzy integral equation is given when the intuitionistic fuzzy number is taken as nonlinear in the membership concept. Before going to the main topic we need to study previous works related to the topic which are done by different researchers. Intuitionistic fuzzy integral is discussed in [17]. There exist several literature sources where fuzzy integral equation is solved such as fuzzy Fredholm integral equation [18–23] and fuzzy Volterra integral equation [24–29].


### 1.3. Motivation

Many authors consider intuitionistic fuzzy number in different articles and apply it in different areas. But the point is that they considered the intuitionistic fuzzy number with only the linear membership and nonmembership function. But it is not always necessary to consider the membership and nonmembership functions as linear functions. Linear membership and nonmembership function can be a special case. In this paper we consider the intuitionistic fuzzy number with nonlinear membership and nonmembership functions. Previously, many researchers found arithmetic operation on intuitionistic fuzzy number by different methods. Most of them consider the resultant number as an approximated intuitionistic fuzzy number. Now how can we find some operation between two said numbers using interval arithmetic concept? If we consider the number with integral equation, then what is its solution? How can we find approximated crisp value of the intuitionistic fuzzy numbers? Few questions arise on the researcher’s mind. From that motivation we try to find the best possible work on this paper.


### 1.4. Novelties

In spite of the few above-mentioned developments, other few developments can still be done in this paper, which are(i)
formulation of the concept of nonlinear intuitionistic fuzzy number;
(ii)
arithmetic operation of nonlinear intuitionistic fuzzy number by max-min principle;
(iii)
applying this number with integral equation problem;
(iv)
using intuitionistic fuzzy Laplace transform for solving intuitionistic integral equation;
(v)
finding the valuation, ambiguities, and ranking of intuitionistic fuzzy function;
(vi)
de-i-fuzzification of said number, being done here by average of(
α
,
β
)-cut method.


### 1.5. Structure of the Paper

The structure of the paper is as follows: In the first section we imitate the previously published work on fuzzy and intuitionistic fuzzy integral equations. The second section presents the basic preliminary concept. We define intuitionistic fuzzy Laplace transform and its properties. In the third section we introduce nonlinear intuitionistic fuzzy number and find the arithmetic operation on that number using max-min principle method. The concept of ranking of the number is also addressed in this section. The de-i-fuzzification of the number is done by mean of(
α
,
β
)-cut method in the fourth section. The intuitionistic fuzzy distance and integral are defined in the fifth section. The sixth section provides the construction and solution of integral equation in intuitionistic fuzzy environment. The conclusion is given in the seventh section.


## 1.1. Intuitionistic Fuzzy Sets Theory

Lotfi A. Zadeh [1] published the theory on fuzzy sets and systems. Chang and Zadeh [2] introduced the concept of fuzzy numbers. Different mathematicians have been studying them (dimension of one or dimension of n, see, for example, [3–6]). With the improvements of theories and applications of fuzzy numbers, this concept becomes more and more significant.Generalization of [1] is taken to be one of intuitionistic fuzzy set (IFS) theory. IFS was first introduced by Atanassov [7] and has been found to be suitable for dealing with various important areas. The fuzzy set considers only the degree of belongingness but not the nonbelongingness. Fuzzy set theory does not incorporate the degree of hesitation (i.e., degree of nondeterminacy defined). To handle such facts, Atanassov [7] explored the concept of fuzzy set theory by IFS theory. The degree of acceptance in fuzzy sets is considered only, but on the other hand IFS is characterized by a membership function and a nonmembership function so that the sum of both values is less than one [8]. Various results on intuitionistic fuzzy set theory are discussed in the papers [9, 10]. The uncertainty theory and calculus constitute a very popular topic nowadays [11–16].


## 1.2. Fuzzy Integral Equation

Integral equation is very important in the theory of calculus. Nowadays it is very important for application. Now if it is with uncertainty, then its behavior changes. In this paper the idea of intuitionistic fuzzy integral equation is given when the intuitionistic fuzzy number is taken as nonlinear in the membership concept. Before going to the main topic we need to study previous works related to the topic which are done by different researchers. Intuitionistic fuzzy integral is discussed in [17]. There exist several literature sources where fuzzy integral equation is solved such as fuzzy Fredholm integral equation [18–23] and fuzzy Volterra integral equation [24–29].


## 1.3. Motivation

Many authors consider intuitionistic fuzzy number in different articles and apply it in different areas. But the point is that they considered the intuitionistic fuzzy number with only the linear membership and nonmembership function. But it is not always necessary to consider the membership and nonmembership functions as linear functions. Linear membership and nonmembership function can be a special case. In this paper we consider the intuitionistic fuzzy number with nonlinear membership and nonmembership functions. Previously, many researchers found arithmetic operation on intuitionistic fuzzy number by different methods. Most of them consider the resultant number as an approximated intuitionistic fuzzy number. Now how can we find some operation between two said numbers using interval arithmetic concept? If we consider the number with integral equation, then what is its solution? How can we find approximated crisp value of the intuitionistic fuzzy numbers? Few questions arise on the researcher’s mind. From that motivation we try to find the best possible work on this paper.


## 1.4. Novelties

In spite of the few above-mentioned developments, other few developments can still be done in this paper, which are(i)
formulation of the concept of nonlinear intuitionistic fuzzy number;
(ii)
arithmetic operation of nonlinear intuitionistic fuzzy number by max-min principle;
(iii)
applying this number with integral equation problem;
(iv)
using intuitionistic fuzzy Laplace transform for solving intuitionistic integral equation;
(v)
finding the valuation, ambiguities, and ranking of intuitionistic fuzzy function;
(vi)
de-i-fuzzification of said number, being done here by average of(
α
,
β
)-cut method.


## 1.5. Structure of the Paper

The structure of the paper is as follows: In the first section we imitate the previously published work on fuzzy and intuitionistic fuzzy integral equations. The second section presents the basic preliminary concept. We define intuitionistic fuzzy Laplace transform and its properties. In the third section we introduce nonlinear intuitionistic fuzzy number and find the arithmetic operation on that number using max-min principle method. The concept of ranking of the number is also addressed in this section. The de-i-fuzzification of the number is done by mean of(
α
,
β
)-cut method in the fourth section. The intuitionistic fuzzy distance and integral are defined in the fifth section. The sixth section provides the construction and solution of integral equation in intuitionistic fuzzy environment. The conclusion is given in the seventh section.


## 2. Preliminaries

### 2.1. Basic Concept Intuitionistic Fuzzy Set Theory

Definition 1 (intuitionistic fuzzy set: [8]).
An IFSA
~
i in X is an object having the form A
~
i
=
<
x
,
μ
A
~
i
x
,
ϑ
A
~
i
x
>
:
x
∈
X, where the μ
A
~
i
x
:
X
→
[
0,1
] and ϑ
A
~
i
x
:
X
→
[
0,1
] define the degree of membership and degree of nonmembership, respectively, of the element x
∈
X to the set A
~
i, which is a subset of X, for every element of x
∈
X, 0
≤
μ
A
~
i
x
+
ϑ
A
~
i
x
≤
1.Definition 2 (triangular intuitionistic fuzzy number: [30]).
A TIFNA
~
i is a subset of IFN in R with following membership function and nonmembership function as follows:(1)
μ
A
~
i
x
=
x
-
a
1
a
2
-
a
1
for
a
1
≤
x
<
a
2
,
a
3
-
x
a
3
-
a
2
for
a
2
<
x
≤
a
3
0
o
t
h
e
r
w
i
s
e
,
ϑ
A
~
i
x
=
a
2
-
x
a
2
-
a
1
′
for
a
1
′
≤
x
<
a
2
x
-
a
2
a
3
′
-
a
2
for
a
2
<
x
≤
a
3
′
1
o
t
h
e
r
w
i
s
ewhere a
1
′
≤
a
2
≤
a
3
′ and a
1
≤
a
2
≤
a
3.
The TIFN is denoted byA
~
i
T
I
F
N
=
(
a
1
,
a
2
,
a
3
;
a
1
′
,
a
2
,
a
3
′
).Definition 3.
Let us consider intuitionistic fuzzy-valued functionf
i
(
t
) defined in the parametric form(2)
f
i
t
α
,
β
=
f
1
t
,
α
,
f
2
t
,
α
;
g
1
t
,
β
,
g
2
t
,
β
.Therefore if we consider the above said intuitionistic fuzzy number, the parametric form is as follows.(3)
A
~
i
α
,
β
=
a
1
+
α
a
2
-
a
1
,
a
3
-
α
a
3
-
a
2
;
a
2
-
β
a
2
-
a
1
′
,
a
2
+
β
a
3
′
-
a
2


### 2.2. Intuitionistic Fuzzy Laplace Transform

Suppose thatf
i
(
t
) is an intuitionistic fuzzy-valued function and s is a real parameter. We define the intuitionistic fuzzy Laplace transform of f as follows.Definition 4 (see [31]).
The intuitionistic fuzzy Laplace transform of an intuitionistic fuzzy-valued functionf
i
(
t
) is defined as follows.(4)
L
f
i
t
=
∫
0
∞
e
-
s
t
⊙
f
i
t
d
t
=
lim
ϵ
→
∞
⁡
∫
0
ϵ
e
-
s
t
⊙
f
i
t
d
t
,
where
s
>
0Consider the intuitionistic fuzzy-valued function f
i
t; then the lower and upper intuitionistic fuzzy Laplace transform of this function are denoted based on the lower and upper intuitionistic fuzzy-valued function f
i
t as follows.
L
f
i
(
t
)
α
,
β = [
lim
ϵ
→
∞
⁡
∫
0
ϵ
e
-
s
t
f
1
(
t
,
α
)
d
t, lim
ϵ
→
∞
⁡
∫
0
ϵ
e
-
s
t
f
2
(
t
,
α
)
d
t; lim
ϵ
→
∞
⁡
∫
0
ϵ
e
-
s
t
g
1
(
t
,
β
)
d
t, lim
ϵ
→
∞
⁡
∫
0
ϵ
e
-
s
t
g
2
(
t
,
β
)
d
t
]
i.e.,L
f
i
(
t
)
α
,
β
=
[
l
f
1
t
,
α
,
l
f
2
t
,
α; l
g
1
t
,
β
,
l
g
2
t
,
β
]Now we define the absolute value of an intuitionistic fuzzy-valued function as follows.Basic Property
(1)
Linearity property: Let h
1
i
(
t
)
,
h
2
i
(
t
) be two continuous intuitionistic fuzzy-valued functions; then
L
c
1
h
1
i
t
⊕
c
2
h
2
i
t = c
1
L
h
1
i
(
t
) ⊕c
2
L
h
2
i
(
t
).Remark 5.
Letf
i
t be a continuous intuitionistic fuzzy-valued function on [
0
,
∞
) and ≥
0, then
L
ρ
⨀
f
i
t = ρ
⨀
L
f
i
t.(2)
First Translation Theorem: Let f
i
t be a continuous intuitionistic fuzzy-valued function and L
f
i
t
=
F
s, then
L
f
i
t
⊙
e
a
t
=
F
s
-
a, where e
a
t is real-valued function and s
-
a
>
0.Definition 6.
In order to solve intuitionistic fuzzy differential equations, it is necessary to know the intuitionistic fuzzy Laplace transform of the derivative off
i
,
f
i
′. The virtue of L
f
i
′
(
t
) is that it can be written in terms of L
f
i
t.Theorem 7.
Suppose thatf
i
t is continuous intuitionistic fuzzy-valued function on 0
,
∞ and exponential order p and that f
i
(
t
) is piecewise continuous intuitionistic fuzzy-valued function on 0
,
∞; then(a)
L
f
i
′
(
t
)
=
s
L
f
i
t
⊝
f
i
0, if f
i
(
t
) is (i)-differentiable,
(b)
L
f
i
′
(
t
)
=
-
f
i
(
0
)
⊝
(
-
s
L
(
f
i
t
), if f
i
(
t
) is (ii)-differentiable.Proof.
(a)L.H.S:
L
f
i
′
(
t
)
=
f
1
′
(
t
,
α
)
,
f
2
′
(
t
,
α
)
;
g
1
′
(
t
,
β
)
,
g
2
′
(
t
,
β
)
andR.H.S:
s
L
f
i
t
⊝
f
i
0
=
s
l
f
1
(
t
,
α
)
,
l
f
2
(
t
,
α
)
;
l
g
1
(
t
,
β
)
,
l
g
2
(
t
,
β
)⊝f
1
(
0
,
α
)
,
f
2
(
0
,
α
)
;
g
1
(
0
,
β
)
,
g
2
(
0
,
β
)
=
s
l
f
1
(
t
,
α
)
-
f
1
(
0
,
α
)
,
s
l
f
2
(
t
,
α
)
-
f
2
(
0
,
α
)
;
s
l
g
1
(
t
,
β
)
-
g
1
(
0
,
β
)
,
s
l
g
2
(
t
,
β
)
-
g
2
(
0
,
β
)
=
f
1
′
(
t
,
α
)
,
f
2
′
(
t
,
α
)
;
g
1
′
(
t
,
β
)
,
g
2
′
(
t
,
β
).
Hence,L.H.S=R.H.S.Proof.
(b)L.H.S:
L
f
i
′
(
t
)
=
f
2
′
(
t
,
α
)
,
f
1
′
(
t
,
α
)
;
g
2
′
(
t
,
β
)
,
g
1
′
(
t
,
β
) and
R.H.S:
-
f
i
(
0
)
⊝
(
-
s
L
f
i
t
=-
f
1
(
0
,
α
)
,
f
2
(
0
,
α
)
;
g
1
(
0
,
β
)
,
g
2
(
0
,
β
) ⊝(
-
s
)
l
f
1
t
,
α
,
f
2
t
,
α
;
g
1
t
,
β
,
g
2
(
t
,
β
)
=
-
f
2
(
0
,
α
)
+
s
l
f
2
(
t
,
α
)
,
-
f
1
(
0
,
α
)
+
s
l
f
1
(
t
,
α
)
;
-
g
2
(
0
,
β
)
+
s
l
g
2
(
t
,
β
)
,
-
g
1
0
,
β
+
s
l
g
1
t
,
β = f
2
′
(
t
,
α
)
,
f
1
′
(
t
,
α
)
;
g
2
′
(
t
,
β
)
,
g
1
′
(
t
,
β
)
=
L
f
i
′
(
t
).
Hence,L.H.S=R.H.S.


## 2.1. Basic Concept Intuitionistic Fuzzy Set Theory

Definition 1 (intuitionistic fuzzy set: [8]).
An IFSA
~
i in X is an object having the form A
~
i
=
<
x
,
μ
A
~
i
x
,
ϑ
A
~
i
x
>
:
x
∈
X, where the μ
A
~
i
x
:
X
→
[
0,1
] and ϑ
A
~
i
x
:
X
→
[
0,1
] define the degree of membership and degree of nonmembership, respectively, of the element x
∈
X to the set A
~
i, which is a subset of X, for every element of x
∈
X, 0
≤
μ
A
~
i
x
+
ϑ
A
~
i
x
≤
1.Definition 2 (triangular intuitionistic fuzzy number: [30]).
A TIFNA
~
i is a subset of IFN in R with following membership function and nonmembership function as follows:(1)
μ
A
~
i
x
=
x
-
a
1
a
2
-
a
1
for
a
1
≤
x
<
a
2
,
a
3
-
x
a
3
-
a
2
for
a
2
<
x
≤
a
3
0
o
t
h
e
r
w
i
s
e
,
ϑ
A
~
i
x
=
a
2
-
x
a
2
-
a
1
′
for
a
1
′
≤
x
<
a
2
x
-
a
2
a
3
′
-
a
2
for
a
2
<
x
≤
a
3
′
1
o
t
h
e
r
w
i
s
ewhere a
1
′
≤
a
2
≤
a
3
′ and a
1
≤
a
2
≤
a
3.
The TIFN is denoted byA
~
i
T
I
F
N
=
(
a
1
,
a
2
,
a
3
;
a
1
′
,
a
2
,
a
3
′
).Definition 3.
Let us consider intuitionistic fuzzy-valued functionf
i
(
t
) defined in the parametric form(2)
f
i
t
α
,
β
=
f
1
t
,
α
,
f
2
t
,
α
;
g
1
t
,
β
,
g
2
t
,
β
.Therefore if we consider the above said intuitionistic fuzzy number, the parametric form is as follows.(3)
A
~
i
α
,
β
=
a
1
+
α
a
2
-
a
1
,
a
3
-
α
a
3
-
a
2
;
a
2
-
β
a
2
-
a
1
′
,
a
2
+
β
a
3
′
-
a
2


## 2.2. Intuitionistic Fuzzy Laplace Transform

Suppose thatf
i
(
t
) is an intuitionistic fuzzy-valued function and s is a real parameter. We define the intuitionistic fuzzy Laplace transform of f as follows.Definition 4 (see [31]).
The intuitionistic fuzzy Laplace transform of an intuitionistic fuzzy-valued functionf
i
(
t
) is defined as follows.(4)
L
f
i
t
=
∫
0
∞
e
-
s
t
⊙
f
i
t
d
t
=
lim
ϵ
→
∞
⁡
∫
0
ϵ
e
-
s
t
⊙
f
i
t
d
t
,
where
s
>
0Consider the intuitionistic fuzzy-valued function f
i
t; then the lower and upper intuitionistic fuzzy Laplace transform of this function are denoted based on the lower and upper intuitionistic fuzzy-valued function f
i
t as follows.
L
f
i
(
t
)
α
,
β = [
lim
ϵ
→
∞
⁡
∫
0
ϵ
e
-
s
t
f
1
(
t
,
α
)
d
t, lim
ϵ
→
∞
⁡
∫
0
ϵ
e
-
s
t
f
2
(
t
,
α
)
d
t; lim
ϵ
→
∞
⁡
∫
0
ϵ
e
-
s
t
g
1
(
t
,
β
)
d
t, lim
ϵ
→
∞
⁡
∫
0
ϵ
e
-
s
t
g
2
(
t
,
β
)
d
t
]
i.e.,L
f
i
(
t
)
α
,
β
=
[
l
f
1
t
,
α
,
l
f
2
t
,
α; l
g
1
t
,
β
,
l
g
2
t
,
β
]Now we define the absolute value of an intuitionistic fuzzy-valued function as follows.Basic Property
(1)
Linearity property: Let h
1
i
(
t
)
,
h
2
i
(
t
) be two continuous intuitionistic fuzzy-valued functions; then
L
c
1
h
1
i
t
⊕
c
2
h
2
i
t = c
1
L
h
1
i
(
t
) ⊕c
2
L
h
2
i
(
t
).Remark 5.
Letf
i
t be a continuous intuitionistic fuzzy-valued function on [
0
,
∞
) and ≥
0, then
L
ρ
⨀
f
i
t = ρ
⨀
L
f
i
t.(2)
First Translation Theorem: Let f
i
t be a continuous intuitionistic fuzzy-valued function and L
f
i
t
=
F
s, then
L
f
i
t
⊙
e
a
t
=
F
s
-
a, where e
a
t is real-valued function and s
-
a
>
0.Definition 6.
In order to solve intuitionistic fuzzy differential equations, it is necessary to know the intuitionistic fuzzy Laplace transform of the derivative off
i
,
f
i
′. The virtue of L
f
i
′
(
t
) is that it can be written in terms of L
f
i
t.Theorem 7.
Suppose thatf
i
t is continuous intuitionistic fuzzy-valued function on 0
,
∞ and exponential order p and that f
i
(
t
) is piecewise continuous intuitionistic fuzzy-valued function on 0
,
∞; then(a)
L
f
i
′
(
t
)
=
s
L
f
i
t
⊝
f
i
0, if f
i
(
t
) is (i)-differentiable,
(b)
L
f
i
′
(
t
)
=
-
f
i
(
0
)
⊝
(
-
s
L
(
f
i
t
), if f
i
(
t
) is (ii)-differentiable.Proof.
(a)L.H.S:
L
f
i
′
(
t
)
=
f
1
′
(
t
,
α
)
,
f
2
′
(
t
,
α
)
;
g
1
′
(
t
,
β
)
,
g
2
′
(
t
,
β
)
andR.H.S:
s
L
f
i
t
⊝
f
i
0
=
s
l
f
1
(
t
,
α
)
,
l
f
2
(
t
,
α
)
;
l
g
1
(
t
,
β
)
,
l
g
2
(
t
,
β
)⊝f
1
(
0
,
α
)
,
f
2
(
0
,
α
)
;
g
1
(
0
,
β
)
,
g
2
(
0
,
β
)
=
s
l
f
1
(
t
,
α
)
-
f
1
(
0
,
α
)
,
s
l
f
2
(
t
,
α
)
-
f
2
(
0
,
α
)
;
s
l
g
1
(
t
,
β
)
-
g
1
(
0
,
β
)
,
s
l
g
2
(
t
,
β
)
-
g
2
(
0
,
β
)
=
f
1
′
(
t
,
α
)
,
f
2
′
(
t
,
α
)
;
g
1
′
(
t
,
β
)
,
g
2
′
(
t
,
β
).
Hence,L.H.S=R.H.S.Proof.
(b)L.H.S:
L
f
i
′
(
t
)
=
f
2
′
(
t
,
α
)
,
f
1
′
(
t
,
α
)
;
g
2
′
(
t
,
β
)
,
g
1
′
(
t
,
β
) and
R.H.S:
-
f
i
(
0
)
⊝
(
-
s
L
f
i
t
=-
f
1
(
0
,
α
)
,
f
2
(
0
,
α
)
;
g
1
(
0
,
β
)
,
g
2
(
0
,
β
) ⊝(
-
s
)
l
f
1
t
,
α
,
f
2
t
,
α
;
g
1
t
,
β
,
g
2
(
t
,
β
)
=
-
f
2
(
0
,
α
)
+
s
l
f
2
(
t
,
α
)
,
-
f
1
(
0
,
α
)
+
s
l
f
1
(
t
,
α
)
;
-
g
2
(
0
,
β
)
+
s
l
g
2
(
t
,
β
)
,
-
g
1
0
,
β
+
s
l
g
1
t
,
β = f
2
′
(
t
,
α
)
,
f
1
′
(
t
,
α
)
;
g
2
′
(
t
,
β
)
,
g
1
′
(
t
,
β
)
=
L
f
i
′
(
t
).
Hence,L.H.S=R.H.S.


## 3. Nonlinear Triangular Intuitionistic Fuzzy Number and Its Arithmetic Operations

Definition 8 (see [32]).
A NTIFNA
~
n
i is a subset of IFN in R with the following membership function and nonmembership function:(5)
μ
A
~
n
i
x
=
x
-
a
1
a
2
-
a
1
p
for
a
1
≤
x
<
a
2
1
for
x
=
a
2
a
3
-
x
a
3
-
a
2
q
for
a
2
<
x
≤
a
3
0
o
t
h
e
r
w
i
s
e
,
ϑ
A
~
n
i
x
1
=
a
2
-
x
a
2
-
a
1
′
r
for
a
1
′
≤
x
<
a
2
0
for
x
=
a
2
x
-
a
2
a
3
′
-
a
2
s
for
a
2
<
x
≤
a
3
′
1
o
t
h
e
r
w
i
s
ewhere a
1
′
≤
a
2
≤
a
3
′ and a
1
≤
a
2
≤
a
3.
The TIFN is denoted byA
~
n
i
T
I
F
N
=
(
a
1
,
a
2
,
a
3
;
a
1
′
,
a
2
,
a
3
′
).Definition 9 (α-cut set).
A
α-cut set of A
~
n
i
T
I
F
N
=
(
a
1
,
a
2
,
a
3
;
a
1
′
,
a
2
,
a
3
′
) is a crisp subset of R which is defined as follows.(6)
A
α
=
x
:
μ
A
~
i
x
≥
α
,
∀
α
∈
0,1
=
A
1
α
,
A
2
α
=
a
1
+
α
1
/
p
a
2
-
a
1
,
a
3
-
α
1
/
q
a
3
-
a
2Definition 10 (β-cut set).
Aβ-cut set of A
~
n
i
T
I
F
N
=
(
a
1
,
a
2
,
a
3
;
a
1
′
,
a
2
,
a
3
′
) is a crisp subset of R which is defined as follows. (7)
A
β
=
x
:
ϑ
A
~
i
x
≤
β
,
∀
β
∈
0,1
=
A
1
′
β
,
A
2
′
β
=
a
2
-
β
1
/
r
a
2
-
a
1
′
,
a
2
+
β
1
/
s
a
3
′
-
a
2Definition 11 ((
α
,
β
)-cut set).
A
(
α
,
β
)-cut set of A
~
i
G
T
r
I
F
N
=
A
~
n
i
T
I
F
N
=
(
a
1
,
a
2
,
a
3
;
a
1
′
,
a
2
,
a
3
′
) is a crisp subset of R which is defined as follows. (8)
A
α
,
β
=
A
1
α
,
A
2
α
;
A
1
′
β
,
A
2
′
β
,
α
+
β
≤
1
,
α
∈
0,1
,
β
∈
0,1Theorem 12.
The sum of the membership and the nonmembership function at any particular point is between 0 and 1.
That is, if for a nonlinear intuitionistic fuzzy numberA
~
n
i
T
I
F
N
=
(
a
1
,
a
2
,
a
3
;
a
1
′
,
a
2
,
a
3
′
)(see Figure 1), membership and nonmembership function are denoted by μ
A
~
n
i
(
x
) and ϑ
A
~
n
i
(
x
1
); then
0
≤
μ
A
~
n
i
x
+
ϑ
A
~
n
i
(
x
1
)
≤
1
.Figure 1
Nonlinear intuitionistic fuzzy number.Proof.
From Figure1 we prove the theorem by splitting up the region.
Now we split up the region into different intervals and points as(9)
a
1
′
,
a
1
′
,
a
1
,
a
1
,
a
1
,
a
2
,
a
2
,
a
2
,
a
3
,
a
3
,
a
3
,
a
3
′
,
a
3
′and first take n
=
1. Then in each of the above points and intervals, the values of membership function and nonmembership function are assumed as in Table 1.Table 1
Membership and nonmembership value for different region.
Interval
μ
A
~
n
i
x
ϑ
A
~
n
i
x
1
μ
A
~
n
i
x
+
ϑ
A
~
n
i
x
1
Conclusion
x
=
a
1
′
0
≤
β
≤
β
≤
1
a
1
′
<
x
<
a
1
0
<
β
<
β
<
1
x
=
a
1
0
≤
β
≤
β
≤
1
a
1
<
x
<
a
2
<
α
<
β
<
α
+
β
≤
1
x
=
a
2
α
0
α
1
a
2
<
x
<
a
3
<
α
<
β
<
α
+
β
≤
1
x
=
a
3
0
<
β
<
β
≤
1
a
3
<
x
<
a
3
′
0
<
β
<
α
+
β
≤
1
x
=
a
3
′
0
≤
β
≤
β
≤
1Note 13.
The above proof is done for takingn
=
1. It can be proved by all values of n. For other values other than 1, we can take numerical examples for proving the theorem.


### 3.1. Max-Min Principle Method for Arithmetic Operation on Intuitionistic Fuzzy Number

LetA
~
n
i and B
~
n
i be two fuzzy numbers and ∗ be an arbitrary operation such that(10)
C
~
n
i
=
A
~
n
i
∗
B
~
n
i
.We know that, for any arbitrary operation between two fuzzy numbers, the resulting fuzzy numbers are not the same as the typed fuzzy numbers in nature. Many authors consider the approximated resulting fuzzy number.Our aim is to first convert the fuzzy number into parametric fuzzy number, and using interval arithmetic operation we find the resulting fuzzy number in parametric form.Let(
α
,
β
)-cut of C
~
n
i be c
1
α
,
c
2
α
;
c
1
′
β
,
c
1
′
(
β
), A
~
n
i be [
a
1
α
,
a
2
α
;
a
1
′
β
,
a
2
′
(
β
)
], and B
~
n
i be [
b
1
α
,
b
2
α
;
b
1
′
β
,
b
2
′
(
β
)
].Now the component of the resulting fuzzy number in parametric form is written as(11)
c
1
α
=
min
⁡
a
1
α
∗
b
1
α
,
a
1
α
∗
b
2
α
,
a
2
α
∗
b
1
α
,
a
2
α
∗
b
2
α
c
2
α
=
max
⁡
a
1
α
∗
b
1
α
,
a
1
α
∗
b
2
α
,
a
2
α
∗
b
1
α
,
a
2
α
∗
b
2
α
c
1
′
β
=
min
⁡
a
1
′
β
∗
b
1
′
β
,
a
1
′
β
∗
b
2
′
β
,
a
2
′
β
∗
b
1
′
β
,
a
2
′
β
∗
b
2
′
β
c
1
′
β
=
max
⁡
a
1
′
β
∗
b
1
′
β
,
a
1
′
β
∗
b
2
′
β
,
a
2
′
β
∗
b
1
′
β
,
a
2
′
β
∗
b
2
′
βwhere c
1
α, c
1
′
β are increasing functions and c
2
α, c
1
′
β are decreasing functions.


### 3.2. Arithmetic Operation on Nonlinear Intuitionistic Fuzzy Number

IfA
~
n
i
T
I
F
N
=
(
a
1
,
a
2
,
a
3
;
a
1
′
,
a
2
,
a
3
′
) and B
~
n
i
T
I
F
N
=
(
b
1
,
b
2
,
b
3
;
b
1
′
,
b
2
,
b
3
′
) are two nonlinear triangular fuzzy numbers with the membership function and the nonmembership function as(12)
μ
A
~
n
i
x
=
x
-
a
1
a
2
-
a
1
p
1
for
a
1
≤
x
<
a
2
1
for
x
=
a
2
a
3
-
x
a
3
-
a
2
q
1
for
a
2
<
x
≤
a
3
0
o
t
h
e
r
w
i
s
e
,
ϑ
A
~
n
i
x
=
a
2
-
x
a
2
-
a
1
′
r
1
for
a
1
′
≤
x
<
a
2
0
for
x
=
a
2
x
-
a
2
a
3
′
-
a
2
s
1
for
a
2
<
x
≤
a
3
′
1
o
t
h
e
r
w
i
s
eand(13)
μ
B
~
n
i
x
=
x
-
b
1
b
2
-
b
1
p
2
for
b
1
≤
x
<
b
2
1
for
x
=
b
2
b
3
-
x
b
3
-
b
2
q
2
for
b
2
<
x
≤
b
3
0
o
t
h
e
r
w
i
s
e
,
(14)
ϑ
B
~
n
i
x
=
b
2
-
x
b
2
-
b
1
′
r
2
for
b
1
′
≤
x
<
b
2
0
for
x
=
b
2
x
-
b
2
b
3
′
-
b
2
s
2
for
b
2
<
x
≤
b
3
′
1
o
t
h
e
r
w
i
s
ewith (
α
,
β
)-cut (15)
A
α
,
β
=
a
1
+
α
1
/
p
1
a
2
-
a
1
,
a
3
-
α
1
/
q
1
a
3
-
a
2
;
a
2
-
β
1
/
r
1
a
2
-
a
1
′
,
a
2
+
β
1
/
s
1
a
3
′
-
a
3and(16)
B
α
,
β
=
b
1
+
α
1
/
p
2
b
2
-
b
1
,
b
3
-
α
1
/
q
2
b
3
-
b
2
;
b
2
-
β
1
/
r
2
b
2
-
b
1
′
,
b
2
+
β
1
/
s
2
b
3
′
-
b
3for a particular case we consider that p
1
=
p
2
=
q
1
=
q
2
=
n
1 and r
1
=
r
2
=
s
1
=
s
2
=
n
2.


#### 3.2.1. Addition of Two Normal Fuzzy Numbers Using(
α
,
β
)-Cut

(17)
C
α
,
β
=
A
α
,
β
+
B
α
,
β
=
a
1
+
b
1
+
α
1
/
n
1
a
2
+
b
2
-
a
1
-
b
1
,
a
3
+
b
3
-
α
1
/
n
1
a
3
+
b
3
-
a
2
-
b
2
;
a
2
+
b
2
-
β
1
/
n
2
a
2
+
b
2
-
a
1
′
-
b
1
′
,
a
2
+
b
2
+
β
1
/
n
2
a
3
′
+
b
3
′
-
a
2
-
b
2The membership and the nonmembership function are defined as of C
~
n
i
T
I
F
N
(18)
μ
B
~
n
i
x
=
x
-
a
1
-
b
1
a
2
+
b
2
-
a
1
-
b
1
n
1
for
a
1
+
b
1
≤
x
<
a
2
+
b
2
1
for
x
=
a
2
+
b
2
a
3
+
b
3
-
x
a
3
+
b
3
-
a
2
-
b
2
n
1
for
a
2
+
b
2
<
x
≤
a
3
+
b
3
0
o
t
h
e
r
w
i
s
e
,and(19)
ϑ
B
~
n
i
x
=
a
2
+
b
2
-
x
a
2
+
b
2
-
a
1
′
-
b
1
′
n
2
for
a
1
′
+
b
1
′
≤
x
<
a
2
+
b
2
0
for
x
=
a
2
+
b
2
2
x
-
a
2
-
b
2
a
3
′
+
b
3
′
-
a
2
-
b
2
n
2
for
a
2
+
b
2
<
x
≤
a
3
′
+
b
3
′
1
o
t
h
e
r
w
i
s
e
.


#### 3.2.2. Subtraction of Two Normal Fuzzy Numbers Using(
α
,
β
)-Cut

(20)
C
α
,
β
=
A
α
,
β
-
B
α
,
β
=
a
1
-
b
3
+
α
1
/
n
1
a
2
-
b
2
-
a
1
+
b
3
,
a
3
-
b
1
-
α
1
/
n
1
a
3
-
b
1
-
a
2
+
b
2
;
a
2
-
b
2
-
β
1
/
n
2
a
2
-
b
2
-
a
1
′
+
b
3
′
,
a
2
-
b
2
+
β
1
/
n
a
3
′
-
b
1
′
-
a
2
+
b
2The membership and the nonmembership function are defined as for C
~
n
i
T
I
F
N
(21)
μ
B
~
n
i
x
=
x
-
a
1
+
b
3
a
2
-
b
2
-
a
1
+
b
3
n
1
for
a
1
-
b
3
≤
x
<
a
2
-
b
2
1
for
x
=
a
2
-
b
2
a
3
-
b
1
-
x
a
3
-
b
1
-
a
2
-
b
2
n
1
for
a
2
-
b
2
<
x
≤
a
3
-
b
1
0
o
t
h
e
r
w
i
s
e
,and(22)
ϑ
B
~
n
i
x
=
a
2
-
b
2
-
x
a
2
-
b
2
-
a
1
′
+
b
3
′
n
2
for
a
1
′
-
b
3
′
≤
x
<
a
2
-
b
2
0
for
x
=
a
2
-
b
2
x
-
a
2
+
b
2
a
3
′
-
b
1
′
-
a
2
+
b
2
n
2
for
a
2
-
b
2
<
x
≤
a
3
′
-
b
1
′
1
o
t
h
e
r
w
i
s
e
.


#### 3.2.3. Multiplication by a Scalar

Ifk
>
0 is a positive scalar, then(23)
k
A
α
,
β
=
k
a
1
+
α
1
/
n
1
a
2
-
a
1
,
k
a
3
-
α
1
/
n
1
a
3
-
a
2
;
k
a
2
-
β
1
/
n
2
a
2
-
a
1
′
,
k
a
2
+
β
1
/
n
2
a
3
′
-
a
3
.Therefore, k
A
~
n
i
T
I
F
N
=
(
k
a
1
,
k
a
2
,
k
a
3
;
k
a
1
′
,
k
a
2
,
k
a
3
′
).Ifk
<
0 is a negative scalar, then(24)
k
A
α
,
β
=
k
a
3
-
α
1
/
n
1
a
3
-
a
2
,
k
a
1
+
α
1
/
n
1
a
2
-
a
1
;
k
a
2
+
β
1
/
n
2
a
3
′
-
a
3
,
k
a
2
-
β
1
/
n
2
a
2
-
a
1
′
.Therefore, k
A
~
n
i
T
I
F
N
=
(
k
a
3
,
k
a
2
,
k
a
1
;
k
a
3
′
,
k
a
2
,
k
a
1
′
).


#### 3.2.4. Multiplication and Division of Two Nonlinear Fuzzy Numbers Using Interval Rule Base Method

Consider two intervals[
a
,
b
] and [
c
,
d
]wherea
,
b
,
c
,
d may be positive or negative.Therefore we can writea
,
b
∙
c
,
d = [
min
⁡
a
c
,
a
d
,
b
c
,
b
d, max
⁡
a
c
,
a
d
,
b
c
,
b
d
] and a
,
b
/
c
,
d = [
min
⁡
a
/
c
,
a
/
d
,
b
/
c
,
b
/
d, max
⁡
a
/
c
,
a
/
d
,
b
/
c
,
b
/
d
].For intuitionistic fuzzy multiplication or division we can use this concept on the(
α
,
β
)-cut interval.(25)
A
1
α
,
A
2
α
;
A
1
′
β
,
A
2
′
βMultiplication: C
α
,
β
=
A
α
,
β
∙
B
α
,
β
=
[
P
1
α
,
P
2
α
;
P
1
′
β
,
P
2
′
(
β
)
]where(26)
P
1
α
=
min
⁡
A
1
α
B
1
α
,
A
1
α
B
2
α
,
A
2
α
B
1
α
,
A
2
α
B
2
α
P
2
α
=
max
⁡
A
1
α
B
1
α
,
A
1
α
B
2
α
,
A
2
α
B
1
α
,
A
2
α
B
2
α
P
1
′
β
=
min
⁡
A
1
′
β
B
1
′
β
,
A
1
′
β
B
2
′
β
,
A
2
′
β
B
1
′
β
,
A
2
′
β
B
2
′
β
P
2
′
β
=
max
⁡
A
1
′
β
B
1
′
β
,
A
1
′
β
B
2
′
β
,
A
2
′
β
B
1
′
β
,
A
2
′
β
B
2
′
βDivision: C
α
,
β
=
A
α
,
β
/
B
α
,
β
=
[
Q
1
α
,
Q
2
α
;
Q
1
′
β
,
Q
2
′
(
β
)
]where(27)
P
1
α
=
min
⁡
A
1
α
B
1
α
,
A
1
α
B
2
α
,
A
2
α
B
1
α
,
A
2
α
B
2
α
P
2
α
=
max
⁡
A
1
α
B
1
α
,
A
1
α
B
2
α
,
A
2
α
B
1
α
,
A
2
α
B
2
α
P
1
′
β
=
min
⁡
A
1
′
β
B
1
′
β
,
A
1
′
β
B
2
′
β
,
A
2
′
β
B
1
′
β
,
A
2
′
β
B
2
′
β
P
2
′
β
=
max
⁡
A
1
′
β
B
1
′
β
,
A
1
′
β
B
2
′
β
,
A
2
′
β
B
1
′
β
,
A
2
′
β
B
2
′
βFor every α
,
β
∈
[
0,1
]   (28)
C
α
,
β
=
A
α
,
β
∙
B
α
,
β
=
=
min
⁡
R
α
,
max
⁡
R
α
;
min
⁡
S
α
,
max
⁡
S
αwhere R
α = [
A
1
α
B
1
α
,
A
1
α
B
2
α, A
2
α
B
1
α
,
A
2
α
B
2
α
] and S
α = [
A
1
′
β
B
1
′
β
,
A
1
′
β
B
2
′
β, A
2
′
β
B
1
′
β
,
A
2
′
β
B
2
′
β
].Example 14.
Ifa
~
=
(
40,45,50
;
38,45,52
) and b
~
=
(
-
14
,
-
10
,
-
7
;
-
12
,
-
10
,
-
8
), then find a
~
∙
b
~ and a
~
/
b
~.
Solution
(29)
a
~
α
,
β
=
40
+
5
α
1
/
3
,
50
-
5
α
1
/
3
;
45
-
7
β
1
/
3
,
45
+
7
β
1
/
3
b
~
α
,
β
=
-
14
+
4
α
1
/
3
,
-
7
-
3
α
1
/
3
;
-
10
-
2
β
1
/
3
,
-
10
+
2
β
1
/
3Now let e
~
=
a
~
∙
b
~, h
~
=
a
~
/
b
~, [
e
~
]
α
,
β
=
[
e
1
α
,
e
2
α
;
e
1
′
β
,
e
2
′
(
β
)
], and [
h
~
]
α
,
β
=
[
h
1
α
,
h
2
α
;
h
1
′
β
,
h
2
′
(
β
)
].
Now by interval rule base system we find the following.(30)
e
1
α
=
min
⁡
40
+
5
α
1
/
3
-
14
+
4
α
1
/
3
,
40
+
5
α
1
/
3
-
7
-
3
α
1
/
3
,
50
-
5
α
1
/
3
-
14
+
4
α
1
/
3
,
50
-
5
α
1
/
3
-
7
-
3
α
1
/
3
e
2
α
=
max
⁡
40
+
5
α
1
/
3
-
14
+
4
α
1
/
3
,
40
+
5
α
1
/
3
-
7
-
3
α
1
/
3
,
50
-
5
α
1
/
3
-
14
+
4
α
1
/
3
,
50
-
5
α
1
/
3
-
7
-
3
α
1
/
3
e
1
′
β
=
min
⁡
45
-
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
-
7
β
1
/
3
-
10
+
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
+
2
β
1
/
3
e
2
′
β
=
max
⁡
45
-
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
-
7
β
1
/
3
-
10
+
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
+
2
β
1
/
3From Figure 2 we can conclude that the e
1
α and e
2
α should be chosen as follows.  (31)
e
1
α
=
50
-
5
α
1
/
3
-
14
+
4
α
1
/
3
,
e
2
α
=
40
+
5
α
1
/
3
-
7
-
3
α
1
/
3From Figure 3 we can conclude that the e
1
′
β and e
2
′
β should be chosen as the following.   (32)
e
1
′
β
=
45
+
7
β
1
/
3
-
10
-
2
β
1
/
3
,
e
2
′
β
=
45
-
7
β
1
/
3
-
10
+
2
β
1
/
3Hence by interval rule base the (
α
,
β
)-cut of e
~ is given by the following.  (33)
e
~
α
,
β
=
50
-
5
α
1
/
3
-
14
+
4
α
1
/
3
,
40
+
5
α
1
/
3
-
7
-
3
α
1
/
3
;
45
+
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
-
7
β
1
/
3
-
10
+
2
β
1
/
3Similarly,(34)
h
1
α
=
min
⁡
40
+
5
α
1
/
3
-
14
+
4
α
1
/
3
,
40
+
5
α
1
/
3
-
7
-
3
α
1
/
3
,
50
-
5
α
1
/
3
-
14
+
4
α
1
/
3
,
50
-
5
α
1
/
3
-
7
-
3
α
1
/
3
h
2
α
=
max
⁡
40
+
5
α
1
/
3
-
14
+
4
α
1
/
3
,
40
+
5
α
1
/
3
-
7
-
3
α
1
/
3
,
50
-
5
α
1
/
3
-
14
+
4
α
1
/
3
,
50
-
5
α
1
/
3
-
7
-
3
α
1
/
3
h
1
′
β
=
min
⁡
45
-
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
-
7
β
1
/
3
-
10
+
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
+
2
β
1
/
3
h
2
′
β
=
max
⁡
45
-
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
-
7
β
1
/
3
-
10
+
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
+
2
β
1
/
3
.From Figure 5 we can conclude that the h
1
α and h
2
α should be chosen as follows.(35)
h
1
α
=
50
-
5
α
1
/
3
-
7
-
3
α
1
/
3
,
h
2
α
=
40
+
5
α
1
/
3
-
14
+
4
α
1
/
3From Figure 6 we can conclude that the h
1
′
β and h
2
′
β should be chosen as(36)
h
1
′
β
=
45
+
7
β
1
/
3
-
10
+
2
β
1
/
3and(37)
h
2
′
β
=
45
-
7
β
1
/
3
-
10
-
2
β
1
/
3
.Hence by interval rule base the (
α
,
β
)-cut of e
~ is given by the following.(38)
h
~
α
,
β
=
50
-
5
α
1
/
3
-
7
-
3
α
1
/
3
,
40
+
5
α
1
/
3
-
14
+
4
α
1
/
3
;
45
+
7
β
1
/
3
-
10
+
2
β
1
/
3
,
45
-
7
β
1
/
3
-
10
-
2
β
1
/
3
Remark. We recommend seeing graphical representation Figures 2, 3, 4, 5, 6, and 7Figure 2
Possiblee
1
α and e
2
α as α
∈
[
0,1
].Figure 3
Possiblee
1
′
β and e
2
′
β as β
∈
[
0,1
].Figure 4
e
1
α, e
2
α, e
1
′
β, and e
2
′
β for α
,
β
∈
[
0,1
].Figure 5
Possibleh
1
α and h
2
α as α
∈
[
0,1
].Figure 6
Possibleh
1
′
β and h
2
′
β as β
∈
[
0,1
].Figure 7
h
1
α, h
2
α, h
1
′
β, and h
2
′
β for α
,
β
∈
[
0,1
].


### 3.3. Intuitionistic Fuzzy Function

Considering thatG
1
(
α
), G
2
(
α
), G
1
′
(
β
), and G
2
′
(
β
) are the continuous functions on the interval I.The setG
~ can be determined by membership and nonmembership functions as follows:(39)
μ
G
~
t
m
α
=
α
,
m
α
=
G
1
α
,
0
≤
α
≤
1
α
,
m
α
=
G
2
α
,
0
≤
α
≤
1
0
,
o
t
h
e
r
w
i
s
eand(40)
ϑ
G
~
t
n
β
=
β
,
n
β
=
G
1
′
β
,
0
≤
β
≤
1
β
,
n
β
=
G
2
′
β
,
0
≤
β
≤
1
1
,
o
t
h
e
r
w
i
s
ewhere, 0
≤
α
+
β
≤
1.The intuitionistic fuzzy function is denoted asG
~
t, and the (
α
,
β
)-cut of G
~
t is as follows.(41)
G
1
α
,
G
2
α
;
G
1
′
β
,
G
2
′
βExample 15.
Consider the intuitionistic fuzzy-valued functionF
~
(
t
)
=
a
~
t
2
+
t
-
1.
Due to presence of the intuitionistic coefficienta
~, the function becomes intuitionistic fuzzy function (see also the graphical representation Figure 8).
The(
α
,
β
)-cut of F
~
t is(42)
F
~
t
α
,
β
=
3
+
α
1
/
4
t
2
+
t
-
1
,
6
-
2
α
1
/
4
t
2
+
t
-
1
;
4
-
2
β
1
/
4
t
2
+
t
-
1
,
4
+
β
1
/
4
t
2
+
t
-
1i.e., [
F
~
t
]
α
,
β
=
=
[
(
3
+
α
1
/
4
)
,
(
6
-
2
α
1
/
4
)
;
(
4
-
2
β
1
/
4
)
,
(
4
+
β
1
/
4
)
]
t
2
+
t
-
1.Figure 8
F
1
α
,
F
2
α
;
F
1
′
β and F
2
′
(
β
) for α
,
β
∈
[
0,1
] at t
=
5.


### 3.4. Values and Ambiguities of NIFN

LetA
α
n
i and A
β
n
i be any α-cut and β-cut set of a triangular IFN A
~
n
i
=
(
a
1
,
a
2
,
a
3
;
a
1
′
,
a
2
,
a
3
′
), respectively. The values of the membership function μ
A
~
n
i
(
x
) and the nonmembership function ϑ
A
~
n
i
(
x
) for the triangular IFN A
~
n
i are defined as follows:(43)
V
μ
A
~
n
i
=
∫
0
1
L
α
A
n
i
+
R
α
A
n
i
f
α
d
αand(44)
V
ϑ
A
~
n
i
=
∫
0
1
L
β
A
n
i
+
R
β
A
n
i
g
β
d
β
.The f
(
α
) is (i) a nonnegative and nondecreasing function on the interval [
0,1
] and (ii) f
0
=
0 and ∫
0
1
f
(
α
)
d
α
=
1
/
2. The function g
(
β
) is (i) a nonnegative and nonincreasing function on interval [
0,1
] and (ii) g
1
=
0 and ∫
0
1
g
(
β
)
d
β
=
1
/
2.If we choosef
α
=
α and g
β
=
1
-
β, then the above assumption is correct.Hereα
∈
[
0
,
w
] and β
∈
[
v
,
1
]The ambiguities of the membership function and the nonmembership function are defined as(45)
G
μ
A
~
n
i
=
∫
0
1
R
α
A
n
i
-
L
α
A
n
i
f
α
d
αand(46)
G
ϑ
A
~
n
i
=
∫
0
1
R
β
A
n
i
-
L
β
A
n
i
g
β
d
βrespectively.Theorem 16.
The values of the membership function and the nonmembership function of nonlinear IFNA
~
n
i are calculated as follows:(47)
V
μ
A
~
n
i
=
1
-
n
a
1
+
4
n
a
2
+
1
-
n
a
3
n
+
1and(48)
V
ϑ
A
~
n
i
=
n
2
+
n
+
2
a
2
+
n
a
3
′
+
a
1
′
n
+
1
n
+
2
.Proof.
(49)
V
μ
A
~
n
i
=
∫
0
1
L
α
A
n
i
+
R
α
A
n
i
f
α
d
α
=
∫
0
1
a
1
+
α
1
/
n
a
2
-
a
1
+
a
3
-
α
1
/
n
a
3
-
a
2
f
α
d
α
=
∫
0
1
a
1
+
a
3
+
α
1
/
n
2
a
2
-
a
1
-
a
3
α
d
α
=
1
-
n
a
1
+
4
n
a
2
+
1
-
n
a
3
n
+
1and(50)
V
ϑ
A
~
n
i
=
∫
0
1
L
β
A
n
i
+
R
β
A
n
i
g
β
d
β
=
∫
0
1
a
2
-
β
1
/
n
a
2
-
a
1
′
+
a
2
+
β
1
/
n
a
3
′
-
a
2
g
β
d
β
=
∫
0
1
2
a
2
+
β
1
/
n
a
3
′
+
a
1
′
-
2
a
2
1
-
β
d
β
=
n
2
+
n
+
2
a
2
+
n
a
3
′
+
a
1
′
n
+
1
n
+
2
.Theorem 17.
The ambiguities of the membership function and the nonmembership function of nonlinear IFNA
~
n
i are calculated as follows:(51)
G
μ
A
~
n
i
=
a
3
-
a
1
2
2
n
+
1and(52)
G
ϑ
A
~
n
i
=
n
2
a
3
′
-
a
1
′
n
+
1
2
n
+
1Proof.
(53)
G
μ
A
~
n
i
=
∫
0
1
R
α
A
n
i
-
L
α
A
n
i
f
α
d
α
=
∫
0
1
a
3
-
α
1
/
n
a
3
-
a
2
-
a
1
-
α
1
/
n
a
2
-
a
1
f
α
d
α
=
∫
0
1
a
3
-
a
1
-
α
1
/
n
a
3
-
a
1
α
d
α
=
∫
0
1
a
3
-
a
1
1
-
α
1
/
n
α
d
α
=
a
3
-
a
1
2
2
n
+
1
.and(54)
G
ϑ
A
~
n
i
=
∫
0
1
R
β
A
n
i
-
L
β
A
n
i
g
β
d
β
=
∫
0
1
a
2
+
β
1
/
n
a
3
′
-
a
2
-
a
2
+
β
1
/
n
a
2
-
a
1
′
g
β
d
β
=
∫
0
1
a
3
′
-
a
1
′
β
1
/
n
1
-
β
d
β
=
n
2
a
3
′
-
a
1
′
n
+
1
2
n
+
1
.


### 3.5. Ranking of Intuitionistic Fuzzy Number Using Valuation and Ambiguity

If we wish to find the ranking of an intuitionistic fuzzy number, then we need to define valuation and ambiguity index.The valuation index is denoted as follows(55)
V
A
~
n
i
,
λ
=
V
μ
A
~
n
i
+
λ
V
μ
A
~
n
i
-
V
ϑ
A
~
n
iand the ambiguity index is denoted as follows(56)
A
A
~
n
i
,
λ
=
G
ϑ
A
~
n
i
-
λ
G
ϑ
A
~
n
i
-
G
μ
A
~
n
ifor λ
∈
[
0,1
].Nowλ can be chosen by different ways. It depends upon decision making choice.The range ofλ, λ
∈
[
0
,
1
/
2
) indicates the decision maker’s pessimistic attitude and λ
∈
(
1
/
2
,
1
] indicates the decision maker’s optimistic attitude towards uncertainty.Generally it is better to takeλ
=
1
/
2.The ranking of the given number can be written as follows.(57)
R
A
~
n
i
=
V
A
~
n
i
,
1
2
+
A
A
~
n
i
,
1
2Basically we cannot compare between two intuitionistic fuzzy numbers, that is, determining which one is greater than, equal to, or less than the other. But using the concept of ranking we can give some approximate relation between them. There exist different literature sources where researchers define ranking in their way. For different definition the comparison may be different.For our definition we can conclude that(1)
IfR
A
~
n
i
>
~
R
B
~
n
i then A
~
n
i
>
~
B
~
n
i
(2)
IfR
A
~
n
i
=
~
R
B
~
n
i then A
~
n
i
=
~
B
~
n
i
(3)
IfR
A
~
n
i
<
~
R
B
~
n
i then A
~
n
i
<
~
B
~
n
iWhere the symbol “>
~”, “=
~” and “<
~” defines the greater than, equal to, and less than relation in fuzzy sense.Example 18.
LetA
~
n
i
=
(
22,25,27
;
21,25,28
) and B
~
n
i
=
(
21,25,26
;
20,25,28
), n
=
1
/
2; then compare the numbers.Solution. See Table 2.Table 2
Comparison of two intuitionistic fuzzy numbers
Valuation of the membership function and the nonmembership function
V
μ
A
~
n
i
=
49.66
V
μ
B
~
n
i
=
49
V
ϑ
A
~
n
i
=
24.86
V
ϑ
B
~
n
i
=
24.73
Ambiguities of membership function and nonmembership function
G
μ
A
~
n
i
=
1.12
G
μ
B
~
n
i
=
1.12
G
ϑ
A
~
n
i
=
0.58
G
ϑ
B
~
n
i
=
0.66
Valuation index
V
A
~
n
i
,
λ
=
62.06
V
B
~
n
i
,
λ
=
61.13
Ambiguity index
A
A
~
n
i
,
λ
=
0.85
A
B
~
n
i
,
λ
=
0.89
Ranking
R
A
~
n
i
=
62.91
R
B
~
n
i
=
62.02HenceA
~
n
i
>
~
B
~
n
i, i.e., A
~
n
i is greater than B
~
n
i in fuzzy sense.


## 3.1. Max-Min Principle Method for Arithmetic Operation on Intuitionistic Fuzzy Number

LetA
~
n
i and B
~
n
i be two fuzzy numbers and ∗ be an arbitrary operation such that(10)
C
~
n
i
=
A
~
n
i
∗
B
~
n
i
.We know that, for any arbitrary operation between two fuzzy numbers, the resulting fuzzy numbers are not the same as the typed fuzzy numbers in nature. Many authors consider the approximated resulting fuzzy number.Our aim is to first convert the fuzzy number into parametric fuzzy number, and using interval arithmetic operation we find the resulting fuzzy number in parametric form.Let(
α
,
β
)-cut of C
~
n
i be c
1
α
,
c
2
α
;
c
1
′
β
,
c
1
′
(
β
), A
~
n
i be [
a
1
α
,
a
2
α
;
a
1
′
β
,
a
2
′
(
β
)
], and B
~
n
i be [
b
1
α
,
b
2
α
;
b
1
′
β
,
b
2
′
(
β
)
].Now the component of the resulting fuzzy number in parametric form is written as(11)
c
1
α
=
min
⁡
a
1
α
∗
b
1
α
,
a
1
α
∗
b
2
α
,
a
2
α
∗
b
1
α
,
a
2
α
∗
b
2
α
c
2
α
=
max
⁡
a
1
α
∗
b
1
α
,
a
1
α
∗
b
2
α
,
a
2
α
∗
b
1
α
,
a
2
α
∗
b
2
α
c
1
′
β
=
min
⁡
a
1
′
β
∗
b
1
′
β
,
a
1
′
β
∗
b
2
′
β
,
a
2
′
β
∗
b
1
′
β
,
a
2
′
β
∗
b
2
′
β
c
1
′
β
=
max
⁡
a
1
′
β
∗
b
1
′
β
,
a
1
′
β
∗
b
2
′
β
,
a
2
′
β
∗
b
1
′
β
,
a
2
′
β
∗
b
2
′
βwhere c
1
α, c
1
′
β are increasing functions and c
2
α, c
1
′
β are decreasing functions.


## 3.2. Arithmetic Operation on Nonlinear Intuitionistic Fuzzy Number

IfA
~
n
i
T
I
F
N
=
(
a
1
,
a
2
,
a
3
;
a
1
′
,
a
2
,
a
3
′
) and B
~
n
i
T
I
F
N
=
(
b
1
,
b
2
,
b
3
;
b
1
′
,
b
2
,
b
3
′
) are two nonlinear triangular fuzzy numbers with the membership function and the nonmembership function as(12)
μ
A
~
n
i
x
=
x
-
a
1
a
2
-
a
1
p
1
for
a
1
≤
x
<
a
2
1
for
x
=
a
2
a
3
-
x
a
3
-
a
2
q
1
for
a
2
<
x
≤
a
3
0
o
t
h
e
r
w
i
s
e
,
ϑ
A
~
n
i
x
=
a
2
-
x
a
2
-
a
1
′
r
1
for
a
1
′
≤
x
<
a
2
0
for
x
=
a
2
x
-
a
2
a
3
′
-
a
2
s
1
for
a
2
<
x
≤
a
3
′
1
o
t
h
e
r
w
i
s
eand(13)
μ
B
~
n
i
x
=
x
-
b
1
b
2
-
b
1
p
2
for
b
1
≤
x
<
b
2
1
for
x
=
b
2
b
3
-
x
b
3
-
b
2
q
2
for
b
2
<
x
≤
b
3
0
o
t
h
e
r
w
i
s
e
,
(14)
ϑ
B
~
n
i
x
=
b
2
-
x
b
2
-
b
1
′
r
2
for
b
1
′
≤
x
<
b
2
0
for
x
=
b
2
x
-
b
2
b
3
′
-
b
2
s
2
for
b
2
<
x
≤
b
3
′
1
o
t
h
e
r
w
i
s
ewith (
α
,
β
)-cut (15)
A
α
,
β
=
a
1
+
α
1
/
p
1
a
2
-
a
1
,
a
3
-
α
1
/
q
1
a
3
-
a
2
;
a
2
-
β
1
/
r
1
a
2
-
a
1
′
,
a
2
+
β
1
/
s
1
a
3
′
-
a
3and(16)
B
α
,
β
=
b
1
+
α
1
/
p
2
b
2
-
b
1
,
b
3
-
α
1
/
q
2
b
3
-
b
2
;
b
2
-
β
1
/
r
2
b
2
-
b
1
′
,
b
2
+
β
1
/
s
2
b
3
′
-
b
3for a particular case we consider that p
1
=
p
2
=
q
1
=
q
2
=
n
1 and r
1
=
r
2
=
s
1
=
s
2
=
n
2.


### 3.2.1. Addition of Two Normal Fuzzy Numbers Using(
α
,
β
)-Cut

(17)
C
α
,
β
=
A
α
,
β
+
B
α
,
β
=
a
1
+
b
1
+
α
1
/
n
1
a
2
+
b
2
-
a
1
-
b
1
,
a
3
+
b
3
-
α
1
/
n
1
a
3
+
b
3
-
a
2
-
b
2
;
a
2
+
b
2
-
β
1
/
n
2
a
2
+
b
2
-
a
1
′
-
b
1
′
,
a
2
+
b
2
+
β
1
/
n
2
a
3
′
+
b
3
′
-
a
2
-
b
2The membership and the nonmembership function are defined as of C
~
n
i
T
I
F
N
(18)
μ
B
~
n
i
x
=
x
-
a
1
-
b
1
a
2
+
b
2
-
a
1
-
b
1
n
1
for
a
1
+
b
1
≤
x
<
a
2
+
b
2
1
for
x
=
a
2
+
b
2
a
3
+
b
3
-
x
a
3
+
b
3
-
a
2
-
b
2
n
1
for
a
2
+
b
2
<
x
≤
a
3
+
b
3
0
o
t
h
e
r
w
i
s
e
,and(19)
ϑ
B
~
n
i
x
=
a
2
+
b
2
-
x
a
2
+
b
2
-
a
1
′
-
b
1
′
n
2
for
a
1
′
+
b
1
′
≤
x
<
a
2
+
b
2
0
for
x
=
a
2
+
b
2
2
x
-
a
2
-
b
2
a
3
′
+
b
3
′
-
a
2
-
b
2
n
2
for
a
2
+
b
2
<
x
≤
a
3
′
+
b
3
′
1
o
t
h
e
r
w
i
s
e
.


### 3.2.2. Subtraction of Two Normal Fuzzy Numbers Using(
α
,
β
)-Cut

(20)
C
α
,
β
=
A
α
,
β
-
B
α
,
β
=
a
1
-
b
3
+
α
1
/
n
1
a
2
-
b
2
-
a
1
+
b
3
,
a
3
-
b
1
-
α
1
/
n
1
a
3
-
b
1
-
a
2
+
b
2
;
a
2
-
b
2
-
β
1
/
n
2
a
2
-
b
2
-
a
1
′
+
b
3
′
,
a
2
-
b
2
+
β
1
/
n
a
3
′
-
b
1
′
-
a
2
+
b
2The membership and the nonmembership function are defined as for C
~
n
i
T
I
F
N
(21)
μ
B
~
n
i
x
=
x
-
a
1
+
b
3
a
2
-
b
2
-
a
1
+
b
3
n
1
for
a
1
-
b
3
≤
x
<
a
2
-
b
2
1
for
x
=
a
2
-
b
2
a
3
-
b
1
-
x
a
3
-
b
1
-
a
2
-
b
2
n
1
for
a
2
-
b
2
<
x
≤
a
3
-
b
1
0
o
t
h
e
r
w
i
s
e
,and(22)
ϑ
B
~
n
i
x
=
a
2
-
b
2
-
x
a
2
-
b
2
-
a
1
′
+
b
3
′
n
2
for
a
1
′
-
b
3
′
≤
x
<
a
2
-
b
2
0
for
x
=
a
2
-
b
2
x
-
a
2
+
b
2
a
3
′
-
b
1
′
-
a
2
+
b
2
n
2
for
a
2
-
b
2
<
x
≤
a
3
′
-
b
1
′
1
o
t
h
e
r
w
i
s
e
.


### 3.2.3. Multiplication by a Scalar

Ifk
>
0 is a positive scalar, then(23)
k
A
α
,
β
=
k
a
1
+
α
1
/
n
1
a
2
-
a
1
,
k
a
3
-
α
1
/
n
1
a
3
-
a
2
;
k
a
2
-
β
1
/
n
2
a
2
-
a
1
′
,
k
a
2
+
β
1
/
n
2
a
3
′
-
a
3
.Therefore, k
A
~
n
i
T
I
F
N
=
(
k
a
1
,
k
a
2
,
k
a
3
;
k
a
1
′
,
k
a
2
,
k
a
3
′
).Ifk
<
0 is a negative scalar, then(24)
k
A
α
,
β
=
k
a
3
-
α
1
/
n
1
a
3
-
a
2
,
k
a
1
+
α
1
/
n
1
a
2
-
a
1
;
k
a
2
+
β
1
/
n
2
a
3
′
-
a
3
,
k
a
2
-
β
1
/
n
2
a
2
-
a
1
′
.Therefore, k
A
~
n
i
T
I
F
N
=
(
k
a
3
,
k
a
2
,
k
a
1
;
k
a
3
′
,
k
a
2
,
k
a
1
′
).


### 3.2.4. Multiplication and Division of Two Nonlinear Fuzzy Numbers Using Interval Rule Base Method

Consider two intervals[
a
,
b
] and [
c
,
d
]wherea
,
b
,
c
,
d may be positive or negative.Therefore we can writea
,
b
∙
c
,
d = [
min
⁡
a
c
,
a
d
,
b
c
,
b
d, max
⁡
a
c
,
a
d
,
b
c
,
b
d
] and a
,
b
/
c
,
d = [
min
⁡
a
/
c
,
a
/
d
,
b
/
c
,
b
/
d, max
⁡
a
/
c
,
a
/
d
,
b
/
c
,
b
/
d
].For intuitionistic fuzzy multiplication or division we can use this concept on the(
α
,
β
)-cut interval.(25)
A
1
α
,
A
2
α
;
A
1
′
β
,
A
2
′
βMultiplication: C
α
,
β
=
A
α
,
β
∙
B
α
,
β
=
[
P
1
α
,
P
2
α
;
P
1
′
β
,
P
2
′
(
β
)
]where(26)
P
1
α
=
min
⁡
A
1
α
B
1
α
,
A
1
α
B
2
α
,
A
2
α
B
1
α
,
A
2
α
B
2
α
P
2
α
=
max
⁡
A
1
α
B
1
α
,
A
1
α
B
2
α
,
A
2
α
B
1
α
,
A
2
α
B
2
α
P
1
′
β
=
min
⁡
A
1
′
β
B
1
′
β
,
A
1
′
β
B
2
′
β
,
A
2
′
β
B
1
′
β
,
A
2
′
β
B
2
′
β
P
2
′
β
=
max
⁡
A
1
′
β
B
1
′
β
,
A
1
′
β
B
2
′
β
,
A
2
′
β
B
1
′
β
,
A
2
′
β
B
2
′
βDivision: C
α
,
β
=
A
α
,
β
/
B
α
,
β
=
[
Q
1
α
,
Q
2
α
;
Q
1
′
β
,
Q
2
′
(
β
)
]where(27)
P
1
α
=
min
⁡
A
1
α
B
1
α
,
A
1
α
B
2
α
,
A
2
α
B
1
α
,
A
2
α
B
2
α
P
2
α
=
max
⁡
A
1
α
B
1
α
,
A
1
α
B
2
α
,
A
2
α
B
1
α
,
A
2
α
B
2
α
P
1
′
β
=
min
⁡
A
1
′
β
B
1
′
β
,
A
1
′
β
B
2
′
β
,
A
2
′
β
B
1
′
β
,
A
2
′
β
B
2
′
β
P
2
′
β
=
max
⁡
A
1
′
β
B
1
′
β
,
A
1
′
β
B
2
′
β
,
A
2
′
β
B
1
′
β
,
A
2
′
β
B
2
′
βFor every α
,
β
∈
[
0,1
]   (28)
C
α
,
β
=
A
α
,
β
∙
B
α
,
β
=
=
min
⁡
R
α
,
max
⁡
R
α
;
min
⁡
S
α
,
max
⁡
S
αwhere R
α = [
A
1
α
B
1
α
,
A
1
α
B
2
α, A
2
α
B
1
α
,
A
2
α
B
2
α
] and S
α = [
A
1
′
β
B
1
′
β
,
A
1
′
β
B
2
′
β, A
2
′
β
B
1
′
β
,
A
2
′
β
B
2
′
β
].Example 14.
Ifa
~
=
(
40,45,50
;
38,45,52
) and b
~
=
(
-
14
,
-
10
,
-
7
;
-
12
,
-
10
,
-
8
), then find a
~
∙
b
~ and a
~
/
b
~.
Solution
(29)
a
~
α
,
β
=
40
+
5
α
1
/
3
,
50
-
5
α
1
/
3
;
45
-
7
β
1
/
3
,
45
+
7
β
1
/
3
b
~
α
,
β
=
-
14
+
4
α
1
/
3
,
-
7
-
3
α
1
/
3
;
-
10
-
2
β
1
/
3
,
-
10
+
2
β
1
/
3Now let e
~
=
a
~
∙
b
~, h
~
=
a
~
/
b
~, [
e
~
]
α
,
β
=
[
e
1
α
,
e
2
α
;
e
1
′
β
,
e
2
′
(
β
)
], and [
h
~
]
α
,
β
=
[
h
1
α
,
h
2
α
;
h
1
′
β
,
h
2
′
(
β
)
].
Now by interval rule base system we find the following.(30)
e
1
α
=
min
⁡
40
+
5
α
1
/
3
-
14
+
4
α
1
/
3
,
40
+
5
α
1
/
3
-
7
-
3
α
1
/
3
,
50
-
5
α
1
/
3
-
14
+
4
α
1
/
3
,
50
-
5
α
1
/
3
-
7
-
3
α
1
/
3
e
2
α
=
max
⁡
40
+
5
α
1
/
3
-
14
+
4
α
1
/
3
,
40
+
5
α
1
/
3
-
7
-
3
α
1
/
3
,
50
-
5
α
1
/
3
-
14
+
4
α
1
/
3
,
50
-
5
α
1
/
3
-
7
-
3
α
1
/
3
e
1
′
β
=
min
⁡
45
-
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
-
7
β
1
/
3
-
10
+
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
+
2
β
1
/
3
e
2
′
β
=
max
⁡
45
-
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
-
7
β
1
/
3
-
10
+
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
+
2
β
1
/
3From Figure 2 we can conclude that the e
1
α and e
2
α should be chosen as follows.  (31)
e
1
α
=
50
-
5
α
1
/
3
-
14
+
4
α
1
/
3
,
e
2
α
=
40
+
5
α
1
/
3
-
7
-
3
α
1
/
3From Figure 3 we can conclude that the e
1
′
β and e
2
′
β should be chosen as the following.   (32)
e
1
′
β
=
45
+
7
β
1
/
3
-
10
-
2
β
1
/
3
,
e
2
′
β
=
45
-
7
β
1
/
3
-
10
+
2
β
1
/
3Hence by interval rule base the (
α
,
β
)-cut of e
~ is given by the following.  (33)
e
~
α
,
β
=
50
-
5
α
1
/
3
-
14
+
4
α
1
/
3
,
40
+
5
α
1
/
3
-
7
-
3
α
1
/
3
;
45
+
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
-
7
β
1
/
3
-
10
+
2
β
1
/
3Similarly,(34)
h
1
α
=
min
⁡
40
+
5
α
1
/
3
-
14
+
4
α
1
/
3
,
40
+
5
α
1
/
3
-
7
-
3
α
1
/
3
,
50
-
5
α
1
/
3
-
14
+
4
α
1
/
3
,
50
-
5
α
1
/
3
-
7
-
3
α
1
/
3
h
2
α
=
max
⁡
40
+
5
α
1
/
3
-
14
+
4
α
1
/
3
,
40
+
5
α
1
/
3
-
7
-
3
α
1
/
3
,
50
-
5
α
1
/
3
-
14
+
4
α
1
/
3
,
50
-
5
α
1
/
3
-
7
-
3
α
1
/
3
h
1
′
β
=
min
⁡
45
-
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
-
7
β
1
/
3
-
10
+
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
+
2
β
1
/
3
h
2
′
β
=
max
⁡
45
-
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
-
7
β
1
/
3
-
10
+
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
+
2
β
1
/
3
.From Figure 5 we can conclude that the h
1
α and h
2
α should be chosen as follows.(35)
h
1
α
=
50
-
5
α
1
/
3
-
7
-
3
α
1
/
3
,
h
2
α
=
40
+
5
α
1
/
3
-
14
+
4
α
1
/
3From Figure 6 we can conclude that the h
1
′
β and h
2
′
β should be chosen as(36)
h
1
′
β
=
45
+
7
β
1
/
3
-
10
+
2
β
1
/
3and(37)
h
2
′
β
=
45
-
7
β
1
/
3
-
10
-
2
β
1
/
3
.Hence by interval rule base the (
α
,
β
)-cut of e
~ is given by the following.(38)
h
~
α
,
β
=
50
-
5
α
1
/
3
-
7
-
3
α
1
/
3
,
40
+
5
α
1
/
3
-
14
+
4
α
1
/
3
;
45
+
7
β
1
/
3
-
10
+
2
β
1
/
3
,
45
-
7
β
1
/
3
-
10
-
2
β
1
/
3
Remark. We recommend seeing graphical representation Figures 2, 3, 4, 5, 6, and 7Figure 2
Possiblee
1
α and e
2
α as α
∈
[
0,1
].Figure 3
Possiblee
1
′
β and e
2
′
β as β
∈
[
0,1
].Figure 4
e
1
α, e
2
α, e
1
′
β, and e
2
′
β for α
,
β
∈
[
0,1
].Figure 5
Possibleh
1
α and h
2
α as α
∈
[
0,1
].Figure 6
Possibleh
1
′
β and h
2
′
β as β
∈
[
0,1
].Figure 7
h
1
α, h
2
α, h
1
′
β, and h
2
′
β for α
,
β
∈
[
0,1
].


## 3.2.1. Addition of Two Normal Fuzzy Numbers Using(
α
,
β
)-Cut

(17)
C
α
,
β
=
A
α
,
β
+
B
α
,
β
=
a
1
+
b
1
+
α
1
/
n
1
a
2
+
b
2
-
a
1
-
b
1
,
a
3
+
b
3
-
α
1
/
n
1
a
3
+
b
3
-
a
2
-
b
2
;
a
2
+
b
2
-
β
1
/
n
2
a
2
+
b
2
-
a
1
′
-
b
1
′
,
a
2
+
b
2
+
β
1
/
n
2
a
3
′
+
b
3
′
-
a
2
-
b
2The membership and the nonmembership function are defined as of C
~
n
i
T
I
F
N
(18)
μ
B
~
n
i
x
=
x
-
a
1
-
b
1
a
2
+
b
2
-
a
1
-
b
1
n
1
for
a
1
+
b
1
≤
x
<
a
2
+
b
2
1
for
x
=
a
2
+
b
2
a
3
+
b
3
-
x
a
3
+
b
3
-
a
2
-
b
2
n
1
for
a
2
+
b
2
<
x
≤
a
3
+
b
3
0
o
t
h
e
r
w
i
s
e
,and(19)
ϑ
B
~
n
i
x
=
a
2
+
b
2
-
x
a
2
+
b
2
-
a
1
′
-
b
1
′
n
2
for
a
1
′
+
b
1
′
≤
x
<
a
2
+
b
2
0
for
x
=
a
2
+
b
2
2
x
-
a
2
-
b
2
a
3
′
+
b
3
′
-
a
2
-
b
2
n
2
for
a
2
+
b
2
<
x
≤
a
3
′
+
b
3
′
1
o
t
h
e
r
w
i
s
e
.


## 3.2.2. Subtraction of Two Normal Fuzzy Numbers Using(
α
,
β
)-Cut

(20)
C
α
,
β
=
A
α
,
β
-
B
α
,
β
=
a
1
-
b
3
+
α
1
/
n
1
a
2
-
b
2
-
a
1
+
b
3
,
a
3
-
b
1
-
α
1
/
n
1
a
3
-
b
1
-
a
2
+
b
2
;
a
2
-
b
2
-
β
1
/
n
2
a
2
-
b
2
-
a
1
′
+
b
3
′
,
a
2
-
b
2
+
β
1
/
n
a
3
′
-
b
1
′
-
a
2
+
b
2The membership and the nonmembership function are defined as for C
~
n
i
T
I
F
N
(21)
μ
B
~
n
i
x
=
x
-
a
1
+
b
3
a
2
-
b
2
-
a
1
+
b
3
n
1
for
a
1
-
b
3
≤
x
<
a
2
-
b
2
1
for
x
=
a
2
-
b
2
a
3
-
b
1
-
x
a
3
-
b
1
-
a
2
-
b
2
n
1
for
a
2
-
b
2
<
x
≤
a
3
-
b
1
0
o
t
h
e
r
w
i
s
e
,and(22)
ϑ
B
~
n
i
x
=
a
2
-
b
2
-
x
a
2
-
b
2
-
a
1
′
+
b
3
′
n
2
for
a
1
′
-
b
3
′
≤
x
<
a
2
-
b
2
0
for
x
=
a
2
-
b
2
x
-
a
2
+
b
2
a
3
′
-
b
1
′
-
a
2
+
b
2
n
2
for
a
2
-
b
2
<
x
≤
a
3
′
-
b
1
′
1
o
t
h
e
r
w
i
s
e
.


## 3.2.3. Multiplication by a Scalar

Ifk
>
0 is a positive scalar, then(23)
k
A
α
,
β
=
k
a
1
+
α
1
/
n
1
a
2
-
a
1
,
k
a
3
-
α
1
/
n
1
a
3
-
a
2
;
k
a
2
-
β
1
/
n
2
a
2
-
a
1
′
,
k
a
2
+
β
1
/
n
2
a
3
′
-
a
3
.Therefore, k
A
~
n
i
T
I
F
N
=
(
k
a
1
,
k
a
2
,
k
a
3
;
k
a
1
′
,
k
a
2
,
k
a
3
′
).Ifk
<
0 is a negative scalar, then(24)
k
A
α
,
β
=
k
a
3
-
α
1
/
n
1
a
3
-
a
2
,
k
a
1
+
α
1
/
n
1
a
2
-
a
1
;
k
a
2
+
β
1
/
n
2
a
3
′
-
a
3
,
k
a
2
-
β
1
/
n
2
a
2
-
a
1
′
.Therefore, k
A
~
n
i
T
I
F
N
=
(
k
a
3
,
k
a
2
,
k
a
1
;
k
a
3
′
,
k
a
2
,
k
a
1
′
).


## 3.2.4. Multiplication and Division of Two Nonlinear Fuzzy Numbers Using Interval Rule Base Method

Consider two intervals[
a
,
b
] and [
c
,
d
]wherea
,
b
,
c
,
d may be positive or negative.Therefore we can writea
,
b
∙
c
,
d = [
min
⁡
a
c
,
a
d
,
b
c
,
b
d, max
⁡
a
c
,
a
d
,
b
c
,
b
d
] and a
,
b
/
c
,
d = [
min
⁡
a
/
c
,
a
/
d
,
b
/
c
,
b
/
d, max
⁡
a
/
c
,
a
/
d
,
b
/
c
,
b
/
d
].For intuitionistic fuzzy multiplication or division we can use this concept on the(
α
,
β
)-cut interval.(25)
A
1
α
,
A
2
α
;
A
1
′
β
,
A
2
′
βMultiplication: C
α
,
β
=
A
α
,
β
∙
B
α
,
β
=
[
P
1
α
,
P
2
α
;
P
1
′
β
,
P
2
′
(
β
)
]where(26)
P
1
α
=
min
⁡
A
1
α
B
1
α
,
A
1
α
B
2
α
,
A
2
α
B
1
α
,
A
2
α
B
2
α
P
2
α
=
max
⁡
A
1
α
B
1
α
,
A
1
α
B
2
α
,
A
2
α
B
1
α
,
A
2
α
B
2
α
P
1
′
β
=
min
⁡
A
1
′
β
B
1
′
β
,
A
1
′
β
B
2
′
β
,
A
2
′
β
B
1
′
β
,
A
2
′
β
B
2
′
β
P
2
′
β
=
max
⁡
A
1
′
β
B
1
′
β
,
A
1
′
β
B
2
′
β
,
A
2
′
β
B
1
′
β
,
A
2
′
β
B
2
′
βDivision: C
α
,
β
=
A
α
,
β
/
B
α
,
β
=
[
Q
1
α
,
Q
2
α
;
Q
1
′
β
,
Q
2
′
(
β
)
]where(27)
P
1
α
=
min
⁡
A
1
α
B
1
α
,
A
1
α
B
2
α
,
A
2
α
B
1
α
,
A
2
α
B
2
α
P
2
α
=
max
⁡
A
1
α
B
1
α
,
A
1
α
B
2
α
,
A
2
α
B
1
α
,
A
2
α
B
2
α
P
1
′
β
=
min
⁡
A
1
′
β
B
1
′
β
,
A
1
′
β
B
2
′
β
,
A
2
′
β
B
1
′
β
,
A
2
′
β
B
2
′
β
P
2
′
β
=
max
⁡
A
1
′
β
B
1
′
β
,
A
1
′
β
B
2
′
β
,
A
2
′
β
B
1
′
β
,
A
2
′
β
B
2
′
βFor every α
,
β
∈
[
0,1
]   (28)
C
α
,
β
=
A
α
,
β
∙
B
α
,
β
=
=
min
⁡
R
α
,
max
⁡
R
α
;
min
⁡
S
α
,
max
⁡
S
αwhere R
α = [
A
1
α
B
1
α
,
A
1
α
B
2
α, A
2
α
B
1
α
,
A
2
α
B
2
α
] and S
α = [
A
1
′
β
B
1
′
β
,
A
1
′
β
B
2
′
β, A
2
′
β
B
1
′
β
,
A
2
′
β
B
2
′
β
].Example 14.
Ifa
~
=
(
40,45,50
;
38,45,52
) and b
~
=
(
-
14
,
-
10
,
-
7
;
-
12
,
-
10
,
-
8
), then find a
~
∙
b
~ and a
~
/
b
~.
Solution
(29)
a
~
α
,
β
=
40
+
5
α
1
/
3
,
50
-
5
α
1
/
3
;
45
-
7
β
1
/
3
,
45
+
7
β
1
/
3
b
~
α
,
β
=
-
14
+
4
α
1
/
3
,
-
7
-
3
α
1
/
3
;
-
10
-
2
β
1
/
3
,
-
10
+
2
β
1
/
3Now let e
~
=
a
~
∙
b
~, h
~
=
a
~
/
b
~, [
e
~
]
α
,
β
=
[
e
1
α
,
e
2
α
;
e
1
′
β
,
e
2
′
(
β
)
], and [
h
~
]
α
,
β
=
[
h
1
α
,
h
2
α
;
h
1
′
β
,
h
2
′
(
β
)
].
Now by interval rule base system we find the following.(30)
e
1
α
=
min
⁡
40
+
5
α
1
/
3
-
14
+
4
α
1
/
3
,
40
+
5
α
1
/
3
-
7
-
3
α
1
/
3
,
50
-
5
α
1
/
3
-
14
+
4
α
1
/
3
,
50
-
5
α
1
/
3
-
7
-
3
α
1
/
3
e
2
α
=
max
⁡
40
+
5
α
1
/
3
-
14
+
4
α
1
/
3
,
40
+
5
α
1
/
3
-
7
-
3
α
1
/
3
,
50
-
5
α
1
/
3
-
14
+
4
α
1
/
3
,
50
-
5
α
1
/
3
-
7
-
3
α
1
/
3
e
1
′
β
=
min
⁡
45
-
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
-
7
β
1
/
3
-
10
+
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
+
2
β
1
/
3
e
2
′
β
=
max
⁡
45
-
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
-
7
β
1
/
3
-
10
+
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
+
2
β
1
/
3From Figure 2 we can conclude that the e
1
α and e
2
α should be chosen as follows.  (31)
e
1
α
=
50
-
5
α
1
/
3
-
14
+
4
α
1
/
3
,
e
2
α
=
40
+
5
α
1
/
3
-
7
-
3
α
1
/
3From Figure 3 we can conclude that the e
1
′
β and e
2
′
β should be chosen as the following.   (32)
e
1
′
β
=
45
+
7
β
1
/
3
-
10
-
2
β
1
/
3
,
e
2
′
β
=
45
-
7
β
1
/
3
-
10
+
2
β
1
/
3Hence by interval rule base the (
α
,
β
)-cut of e
~ is given by the following.  (33)
e
~
α
,
β
=
50
-
5
α
1
/
3
-
14
+
4
α
1
/
3
,
40
+
5
α
1
/
3
-
7
-
3
α
1
/
3
;
45
+
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
-
7
β
1
/
3
-
10
+
2
β
1
/
3Similarly,(34)
h
1
α
=
min
⁡
40
+
5
α
1
/
3
-
14
+
4
α
1
/
3
,
40
+
5
α
1
/
3
-
7
-
3
α
1
/
3
,
50
-
5
α
1
/
3
-
14
+
4
α
1
/
3
,
50
-
5
α
1
/
3
-
7
-
3
α
1
/
3
h
2
α
=
max
⁡
40
+
5
α
1
/
3
-
14
+
4
α
1
/
3
,
40
+
5
α
1
/
3
-
7
-
3
α
1
/
3
,
50
-
5
α
1
/
3
-
14
+
4
α
1
/
3
,
50
-
5
α
1
/
3
-
7
-
3
α
1
/
3
h
1
′
β
=
min
⁡
45
-
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
-
7
β
1
/
3
-
10
+
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
+
2
β
1
/
3
h
2
′
β
=
max
⁡
45
-
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
-
7
β
1
/
3
-
10
+
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
-
2
β
1
/
3
,
45
+
7
β
1
/
3
-
10
+
2
β
1
/
3
.From Figure 5 we can conclude that the h
1
α and h
2
α should be chosen as follows.(35)
h
1
α
=
50
-
5
α
1
/
3
-
7
-
3
α
1
/
3
,
h
2
α
=
40
+
5
α
1
/
3
-
14
+
4
α
1
/
3From Figure 6 we can conclude that the h
1
′
β and h
2
′
β should be chosen as(36)
h
1
′
β
=
45
+
7
β
1
/
3
-
10
+
2
β
1
/
3and(37)
h
2
′
β
=
45
-
7
β
1
/
3
-
10
-
2
β
1
/
3
.Hence by interval rule base the (
α
,
β
)-cut of e
~ is given by the following.(38)
h
~
α
,
β
=
50
-
5
α
1
/
3
-
7
-
3
α
1
/
3
,
40
+
5
α
1
/
3
-
14
+
4
α
1
/
3
;
45
+
7
β
1
/
3
-
10
+
2
β
1
/
3
,
45
-
7
β
1
/
3
-
10
-
2
β
1
/
3
Remark. We recommend seeing graphical representation Figures 2, 3, 4, 5, 6, and 7Figure 2
Possiblee
1
α and e
2
α as α
∈
[
0,1
].Figure 3
Possiblee
1
′
β and e
2
′
β as β
∈
[
0,1
].Figure 4
e
1
α, e
2
α, e
1
′
β, and e
2
′
β for α
,
β
∈
[
0,1
].Figure 5
Possibleh
1
α and h
2
α as α
∈
[
0,1
].Figure 6
Possibleh
1
′
β and h
2
′
β as β
∈
[
0,1
].Figure 7
h
1
α, h
2
α, h
1
′
β, and h
2
′
β for α
,
β
∈
[
0,1
].


## 3.3. Intuitionistic Fuzzy Function

Considering thatG
1
(
α
), G
2
(
α
), G
1
′
(
β
), and G
2
′
(
β
) are the continuous functions on the interval I.The setG
~ can be determined by membership and nonmembership functions as follows:(39)
μ
G
~
t
m
α
=
α
,
m
α
=
G
1
α
,
0
≤
α
≤
1
α
,
m
α
=
G
2
α
,
0
≤
α
≤
1
0
,
o
t
h
e
r
w
i
s
eand(40)
ϑ
G
~
t
n
β
=
β
,
n
β
=
G
1
′
β
,
0
≤
β
≤
1
β
,
n
β
=
G
2
′
β
,
0
≤
β
≤
1
1
,
o
t
h
e
r
w
i
s
ewhere, 0
≤
α
+
β
≤
1.The intuitionistic fuzzy function is denoted asG
~
t, and the (
α
,
β
)-cut of G
~
t is as follows.(41)
G
1
α
,
G
2
α
;
G
1
′
β
,
G
2
′
βExample 15.
Consider the intuitionistic fuzzy-valued functionF
~
(
t
)
=
a
~
t
2
+
t
-
1.
Due to presence of the intuitionistic coefficienta
~, the function becomes intuitionistic fuzzy function (see also the graphical representation Figure 8).
The(
α
,
β
)-cut of F
~
t is(42)
F
~
t
α
,
β
=
3
+
α
1
/
4
t
2
+
t
-
1
,
6
-
2
α
1
/
4
t
2
+
t
-
1
;
4
-
2
β
1
/
4
t
2
+
t
-
1
,
4
+
β
1
/
4
t
2
+
t
-
1i.e., [
F
~
t
]
α
,
β
=
=
[
(
3
+
α
1
/
4
)
,
(
6
-
2
α
1
/
4
)
;
(
4
-
2
β
1
/
4
)
,
(
4
+
β
1
/
4
)
]
t
2
+
t
-
1.Figure 8
F
1
α
,
F
2
α
;
F
1
′
β and F
2
′
(
β
) for α
,
β
∈
[
0,1
] at t
=
5.


## 3.4. Values and Ambiguities of NIFN

LetA
α
n
i and A
β
n
i be any α-cut and β-cut set of a triangular IFN A
~
n
i
=
(
a
1
,
a
2
,
a
3
;
a
1
′
,
a
2
,
a
3
′
), respectively. The values of the membership function μ
A
~
n
i
(
x
) and the nonmembership function ϑ
A
~
n
i
(
x
) for the triangular IFN A
~
n
i are defined as follows:(43)
V
μ
A
~
n
i
=
∫
0
1
L
α
A
n
i
+
R
α
A
n
i
f
α
d
αand(44)
V
ϑ
A
~
n
i
=
∫
0
1
L
β
A
n
i
+
R
β
A
n
i
g
β
d
β
.The f
(
α
) is (i) a nonnegative and nondecreasing function on the interval [
0,1
] and (ii) f
0
=
0 and ∫
0
1
f
(
α
)
d
α
=
1
/
2. The function g
(
β
) is (i) a nonnegative and nonincreasing function on interval [
0,1
] and (ii) g
1
=
0 and ∫
0
1
g
(
β
)
d
β
=
1
/
2.If we choosef
α
=
α and g
β
=
1
-
β, then the above assumption is correct.Hereα
∈
[
0
,
w
] and β
∈
[
v
,
1
]The ambiguities of the membership function and the nonmembership function are defined as(45)
G
μ
A
~
n
i
=
∫
0
1
R
α
A
n
i
-
L
α
A
n
i
f
α
d
αand(46)
G
ϑ
A
~
n
i
=
∫
0
1
R
β
A
n
i
-
L
β
A
n
i
g
β
d
βrespectively.Theorem 16.
The values of the membership function and the nonmembership function of nonlinear IFNA
~
n
i are calculated as follows:(47)
V
μ
A
~
n
i
=
1
-
n
a
1
+
4
n
a
2
+
1
-
n
a
3
n
+
1and(48)
V
ϑ
A
~
n
i
=
n
2
+
n
+
2
a
2
+
n
a
3
′
+
a
1
′
n
+
1
n
+
2
.Proof.
(49)
V
μ
A
~
n
i
=
∫
0
1
L
α
A
n
i
+
R
α
A
n
i
f
α
d
α
=
∫
0
1
a
1
+
α
1
/
n
a
2
-
a
1
+
a
3
-
α
1
/
n
a
3
-
a
2
f
α
d
α
=
∫
0
1
a
1
+
a
3
+
α
1
/
n
2
a
2
-
a
1
-
a
3
α
d
α
=
1
-
n
a
1
+
4
n
a
2
+
1
-
n
a
3
n
+
1and(50)
V
ϑ
A
~
n
i
=
∫
0
1
L
β
A
n
i
+
R
β
A
n
i
g
β
d
β
=
∫
0
1
a
2
-
β
1
/
n
a
2
-
a
1
′
+
a
2
+
β
1
/
n
a
3
′
-
a
2
g
β
d
β
=
∫
0
1
2
a
2
+
β
1
/
n
a
3
′
+
a
1
′
-
2
a
2
1
-
β
d
β
=
n
2
+
n
+
2
a
2
+
n
a
3
′
+
a
1
′
n
+
1
n
+
2
.Theorem 17.
The ambiguities of the membership function and the nonmembership function of nonlinear IFNA
~
n
i are calculated as follows:(51)
G
μ
A
~
n
i
=
a
3
-
a
1
2
2
n
+
1and(52)
G
ϑ
A
~
n
i
=
n
2
a
3
′
-
a
1
′
n
+
1
2
n
+
1Proof.
(53)
G
μ
A
~
n
i
=
∫
0
1
R
α
A
n
i
-
L
α
A
n
i
f
α
d
α
=
∫
0
1
a
3
-
α
1
/
n
a
3
-
a
2
-
a
1
-
α
1
/
n
a
2
-
a
1
f
α
d
α
=
∫
0
1
a
3
-
a
1
-
α
1
/
n
a
3
-
a
1
α
d
α
=
∫
0
1
a
3
-
a
1
1
-
α
1
/
n
α
d
α
=
a
3
-
a
1
2
2
n
+
1
.and(54)
G
ϑ
A
~
n
i
=
∫
0
1
R
β
A
n
i
-
L
β
A
n
i
g
β
d
β
=
∫
0
1
a
2
+
β
1
/
n
a
3
′
-
a
2
-
a
2
+
β
1
/
n
a
2
-
a
1
′
g
β
d
β
=
∫
0
1
a
3
′
-
a
1
′
β
1
/
n
1
-
β
d
β
=
n
2
a
3
′
-
a
1
′
n
+
1
2
n
+
1
.


## 3.5. Ranking of Intuitionistic Fuzzy Number Using Valuation and Ambiguity

If we wish to find the ranking of an intuitionistic fuzzy number, then we need to define valuation and ambiguity index.The valuation index is denoted as follows(55)
V
A
~
n
i
,
λ
=
V
μ
A
~
n
i
+
λ
V
μ
A
~
n
i
-
V
ϑ
A
~
n
iand the ambiguity index is denoted as follows(56)
A
A
~
n
i
,
λ
=
G
ϑ
A
~
n
i
-
λ
G
ϑ
A
~
n
i
-
G
μ
A
~
n
ifor λ
∈
[
0,1
].Nowλ can be chosen by different ways. It depends upon decision making choice.The range ofλ, λ
∈
[
0
,
1
/
2
) indicates the decision maker’s pessimistic attitude and λ
∈
(
1
/
2
,
1
] indicates the decision maker’s optimistic attitude towards uncertainty.Generally it is better to takeλ
=
1
/
2.The ranking of the given number can be written as follows.(57)
R
A
~
n
i
=
V
A
~
n
i
,
1
2
+
A
A
~
n
i
,
1
2Basically we cannot compare between two intuitionistic fuzzy numbers, that is, determining which one is greater than, equal to, or less than the other. But using the concept of ranking we can give some approximate relation between them. There exist different literature sources where researchers define ranking in their way. For different definition the comparison may be different.For our definition we can conclude that(1)
IfR
A
~
n
i
>
~
R
B
~
n
i then A
~
n
i
>
~
B
~
n
i
(2)
IfR
A
~
n
i
=
~
R
B
~
n
i then A
~
n
i
=
~
B
~
n
i
(3)
IfR
A
~
n
i
<
~
R
B
~
n
i then A
~
n
i
<
~
B
~
n
iWhere the symbol “>
~”, “=
~” and “<
~” defines the greater than, equal to, and less than relation in fuzzy sense.Example 18.
LetA
~
n
i
=
(
22,25,27
;
21,25,28
) and B
~
n
i
=
(
21,25,26
;
20,25,28
), n
=
1
/
2; then compare the numbers.Solution. See Table 2.Table 2
Comparison of two intuitionistic fuzzy numbers
Valuation of the membership function and the nonmembership function
V
μ
A
~
n
i
=
49.66
V
μ
B
~
n
i
=
49
V
ϑ
A
~
n
i
=
24.86
V
ϑ
B
~
n
i
=
24.73
Ambiguities of membership function and nonmembership function
G
μ
A
~
n
i
=
1.12
G
μ
B
~
n
i
=
1.12
G
ϑ
A
~
n
i
=
0.58
G
ϑ
B
~
n
i
=
0.66
Valuation index
V
A
~
n
i
,
λ
=
62.06
V
B
~
n
i
,
λ
=
61.13
Ambiguity index
A
A
~
n
i
,
λ
=
0.85
A
B
~
n
i
,
λ
=
0.89
Ranking
R
A
~
n
i
=
62.91
R
B
~
n
i
=
62.02HenceA
~
n
i
>
~
B
~
n
i, i.e., A
~
n
i is greater than B
~
n
i in fuzzy sense.


## 4. De-i-Fuzzification Based on Average of(
α
,
β
)-Cut Method

The crispification value of an intuitionistic fuzzy number is named as de-i-fuzzification value [33]. Here we tried to find the de-i-fuzzification value of NTIFN using average of (
α
,
β
)-cut method.


### 4.1. De-i-Fuzzification Based on Average of(
α
,
β
)-Cut Method of an IFN

For an IFNA
~
i, the de-i-fuzzification value of A
~
i is a crisp value which can be derived in the following way(58)
A
^
i
=
A
^
α
i
+
A
^
β
i
2where A
^
α
i is de-i-fuzzification of α-cut and A
^
β
i is de-i-fuzzification value of β-cut.That is,A
^
α
i
=
∫
0
1
A
1
α
+
A
2
α
/
2
d
α and A
^
β
i
=
∫
0
1
A
1
′
β
+
A
2
′
β
/
2
d
β.Now ifA
~
i
=
(
a
1
,
a
2
,
a
3
;
a
1
′
,
a
2
,
a
3
′
) is intuitionistic triangular fuzzy number, then its defuzzification value is as follows.(59)
A
^
i
=
a
1
+
a
1
′
+
4
a
2
+
a
3
+
a
3
′
8


### 4.2. De-i-Fuzzification Method For NTIFN

We can find the de-i-fuzzification value of the NTIFNA
~
n
i
T
I
F
N
=
(
a
1
,
a
2
,
a
3
;
a
1
′
,
a
2
,
a
3
′
) by the above said method. If p
=
q
=
r
=
s
=
n, then the de-i-fuzzification value is defined as follows.(60)
A
^
n
i
=
n
a
1
+
n
a
3
+
2
n
+
1
a
2
+
n
a
1
′
+
n
a
3
′
4
n
+
1Example 19.
Find the de-i-fuzzified value of a nonlinear intuitionistic fuzzy numberA
~
n
i
T
I
F
N
=
(
105,110,116
;
104,110,115
). Here p
=
q
=
r
=
s
=
2.Solution. The de-i-fuzzified value of the number is A
^
n
i
=
91.66.


## 4.1. De-i-Fuzzification Based on Average of(
α
,
β
)-Cut Method of an IFN

For an IFNA
~
i, the de-i-fuzzification value of A
~
i is a crisp value which can be derived in the following way(58)
A
^
i
=
A
^
α
i
+
A
^
β
i
2where A
^
α
i is de-i-fuzzification of α-cut and A
^
β
i is de-i-fuzzification value of β-cut.That is,A
^
α
i
=
∫
0
1
A
1
α
+
A
2
α
/
2
d
α and A
^
β
i
=
∫
0
1
A
1
′
β
+
A
2
′
β
/
2
d
β.Now ifA
~
i
=
(
a
1
,
a
2
,
a
3
;
a
1
′
,
a
2
,
a
3
′
) is intuitionistic triangular fuzzy number, then its defuzzification value is as follows.(59)
A
^
i
=
a
1
+
a
1
′
+
4
a
2
+
a
3
+
a
3
′
8


## 4.2. De-i-Fuzzification Method For NTIFN

We can find the de-i-fuzzification value of the NTIFNA
~
n
i
T
I
F
N
=
(
a
1
,
a
2
,
a
3
;
a
1
′
,
a
2
,
a
3
′
) by the above said method. If p
=
q
=
r
=
s
=
n, then the de-i-fuzzification value is defined as follows.(60)
A
^
n
i
=
n
a
1
+
n
a
3
+
2
n
+
1
a
2
+
n
a
1
′
+
n
a
3
′
4
n
+
1Example 19.
Find the de-i-fuzzified value of a nonlinear intuitionistic fuzzy numberA
~
n
i
T
I
F
N
=
(
105,110,116
;
104,110,115
). Here p
=
q
=
r
=
s
=
2.Solution. The de-i-fuzzified value of the number is A
^
n
i
=
91.66.


## 5. Intuitionistic Fuzzy Distance and Integrals

### 5.1. Generalized Hukuhara Distance on Intuitionistic Fuzzy-Valued Function

Definition 20.
The Hausdorff distance between intuitionistic fuzzy numbers is given byD
:
R
F
×
R
F
→
R
+
∪
{
0
} as in (61)
D
u
,
v
;
p
,
q
=
sup
α
,
β
∈
0,1
⁡
d
u
α
,
v
α
;
p
β
,
q
β
=
s
u
p
α
,
β
∈
0,1
max
⁡
u
1
α
-
v
1
α
,
u
2
α
-
v
2
α
,
p
1
β
-
q
1
β
,
p
2
β
-
q
2
βwhere d is Hausdorff metric and metric space (
R
F
,
D
) is complete, separable, and locally compact. The following substances for metric D are tenable:(1)
D
u
⨁
w
,
v
⨁
w
;
p
⨁
z
,
q
⨁
z
=
D
u
,
v
;
p
,
q
,
∀
u
,
v
,
p
,
q
,
w
,
z
∈
R
F;
(2)
D
λ
u
,
λ
v
;
λ
p
,
λ
q
=
λ
D
u
,
v
;
p
,
q
,
λ
∈
R
,
u
,
v
,
p
,
q
∈
R
F;
(3)
D
u
1
⨁
u
2
,
v
1
⨁
v
2
;
p
1
⨁
p
2
,
q
1
⨁
q
1 ≤ D
u
1
,
v
1
;
p
1
,
q
1 + D
(
u
2
,
v
2
;
p
2
,
q
1
)  ∀
u
1
,
u
2
,
v
1
,
v
2
,
p
1, p
2
,
q
1
,
q
1
∈
R
F;
(4)
D
u
1
⊖
u
2
,
v
1
⊖
v
2
;
p
1
⊖
p
2
,
q
1
⊖
q
1
≤
D
u
1
,
v
1
;
p
1
,
q
1
+
D
(
u
2
,
v
2
;
p
2
,
q
1
), as long as u
1
⊖
u
2
,
v
1
⊖
v
2
;
p
1
⊖
p
2
,
q
1
⊖
q
1 exists and u
1
,
u
2
,
v
1
,
v
2
,
p
1
,
p
2
,
q
1
,
q
1
∈
R
F.Lemma 21.
Letu
n
,
v
n
,
p
n
,
q
n
,
u
,
v
,
p
,
q
∈
R
F
,
n
∈
N. Let u
n
→
u
,
v
n
→
v
,
p
n
→
p
,
q
n
→
q, as n
→
∞. Then D
u
n
,
v
n
;
p
n
,
q
n
→
D
u
,
v
;
p
,
q, as n
→
∞. In particular we can write the following.(62)
lim
n
→
+
∞
⁡
D
u
n
,
v
n
;
p
n
,
q
n
=
D
lim
n
→
+
∞
⁡
u
n
,
lim
n
→
+
∞
⁡
v
n
;
lim
n
→
+
∞
⁡
p
n
,
lim
n
→
+
∞
⁡
q
n
=
D
u
,
v
;
p
,
qLemma 22.
Letu
n
,
u
∈
R
F and c
n
,
c
∈
R, such that u
n
→
u and c
n
→
c as n
→
+
∞. Therefore in D-metric (63)
lim
n
→
+
∞
⁡
u
n
⨀
c
n
=
lim
n
→
+
∞
⁡
u
n
⨀
lim
n
→
+
∞
⁡
c
n
=
u
⨀
c
.Definition 23.
The generalized Hukuhara difference between two intuitionistic fuzzy numbersa
,
b
∈
R
F is defined as follows.(64)
a
⊖
g
H
b
=
k
⟺
a
=
b
+
k
b
=
a
+
-
1
kThe (
α
,
β
)-cut set (65)
a
⊖
g
H
b
α
,
β
=
min
⁡
r
1
α
,
r
2
α
,
max
⁡
r
1
α
,
r
2
α
;
min
⁡
s
1
β
,
s
2
β
,
max
⁡
s
1
β
,
s
2
βwhere r
1
(
α
)
=
a
1
α
-
b
2
α, r
2
(
α
)
=
a
2
α
-
b
1
α, s
1
(
β
)
=
a
1
′
β
-
b
2
′
β and s
2
(
β
)
=
a
2
′
β
-
b
1
′
(
β
).
Lete
=
a
⊖
g
H
b.
The existence conditions for whicha
⊖
g
H
b exists are as follows:
(1)e
1
α
=
a
1
α
-
b
2
α, e
2
α
=
a
2
α
-
b
1
α, e
1
′
β
=
a
1
′
β
-
b
2
′
β, and e
2
′
β
=
a
2
′
β
-
b
1
′
(
β
), with (i) e
1
α, e
2
′
β being increasing and e
2
α, e
1
′
β being decreasing function and (ii) e
1
α
≤
e
2
α, e
2
′
β
≤
e
1
′
β for all α
,
β
∈
[
0,1
].
(2)e
2
α
=
a
1
α
-
b
2
α, e
1
α
=
a
2
α
-
b
1
α, e
2
′
β
=
a
1
′
β
-
b
2
′
β, and e
1
′
β
=
a
2
′
β
-
b
1
′
(
β
), with (i) e
1
α, e
2
′
β being increasing and e
2
α, e
1
′
β being decreasing function and (ii) e
1
α
≤
e
2
α, e
2
′
β
≤
e
1
′
β for all α
,
β
∈
[
0,1
].Remark 24.
In the whole paper, we considereda
⊖
g
H
b
∈
R
F.


## 5.1. Generalized Hukuhara Distance on Intuitionistic Fuzzy-Valued Function

Definition 20.
The Hausdorff distance between intuitionistic fuzzy numbers is given byD
:
R
F
×
R
F
→
R
+
∪
{
0
} as in (61)
D
u
,
v
;
p
,
q
=
sup
α
,
β
∈
0,1
⁡
d
u
α
,
v
α
;
p
β
,
q
β
=
s
u
p
α
,
β
∈
0,1
max
⁡
u
1
α
-
v
1
α
,
u
2
α
-
v
2
α
,
p
1
β
-
q
1
β
,
p
2
β
-
q
2
βwhere d is Hausdorff metric and metric space (
R
F
,
D
) is complete, separable, and locally compact. The following substances for metric D are tenable:(1)
D
u
⨁
w
,
v
⨁
w
;
p
⨁
z
,
q
⨁
z
=
D
u
,
v
;
p
,
q
,
∀
u
,
v
,
p
,
q
,
w
,
z
∈
R
F;
(2)
D
λ
u
,
λ
v
;
λ
p
,
λ
q
=
λ
D
u
,
v
;
p
,
q
,
λ
∈
R
,
u
,
v
,
p
,
q
∈
R
F;
(3)
D
u
1
⨁
u
2
,
v
1
⨁
v
2
;
p
1
⨁
p
2
,
q
1
⨁
q
1 ≤ D
u
1
,
v
1
;
p
1
,
q
1 + D
(
u
2
,
v
2
;
p
2
,
q
1
)  ∀
u
1
,
u
2
,
v
1
,
v
2
,
p
1, p
2
,
q
1
,
q
1
∈
R
F;
(4)
D
u
1
⊖
u
2
,
v
1
⊖
v
2
;
p
1
⊖
p
2
,
q
1
⊖
q
1
≤
D
u
1
,
v
1
;
p
1
,
q
1
+
D
(
u
2
,
v
2
;
p
2
,
q
1
), as long as u
1
⊖
u
2
,
v
1
⊖
v
2
;
p
1
⊖
p
2
,
q
1
⊖
q
1 exists and u
1
,
u
2
,
v
1
,
v
2
,
p
1
,
p
2
,
q
1
,
q
1
∈
R
F.Lemma 21.
Letu
n
,
v
n
,
p
n
,
q
n
,
u
,
v
,
p
,
q
∈
R
F
,
n
∈
N. Let u
n
→
u
,
v
n
→
v
,
p
n
→
p
,
q
n
→
q, as n
→
∞. Then D
u
n
,
v
n
;
p
n
,
q
n
→
D
u
,
v
;
p
,
q, as n
→
∞. In particular we can write the following.(62)
lim
n
→
+
∞
⁡
D
u
n
,
v
n
;
p
n
,
q
n
=
D
lim
n
→
+
∞
⁡
u
n
,
lim
n
→
+
∞
⁡
v
n
;
lim
n
→
+
∞
⁡
p
n
,
lim
n
→
+
∞
⁡
q
n
=
D
u
,
v
;
p
,
qLemma 22.
Letu
n
,
u
∈
R
F and c
n
,
c
∈
R, such that u
n
→
u and c
n
→
c as n
→
+
∞. Therefore in D-metric (63)
lim
n
→
+
∞
⁡
u
n
⨀
c
n
=
lim
n
→
+
∞
⁡
u
n
⨀
lim
n
→
+
∞
⁡
c
n
=
u
⨀
c
.Definition 23.
The generalized Hukuhara difference between two intuitionistic fuzzy numbersa
,
b
∈
R
F is defined as follows.(64)
a
⊖
g
H
b
=
k
⟺
a
=
b
+
k
b
=
a
+
-
1
kThe (
α
,
β
)-cut set (65)
a
⊖
g
H
b
α
,
β
=
min
⁡
r
1
α
,
r
2
α
,
max
⁡
r
1
α
,
r
2
α
;
min
⁡
s
1
β
,
s
2
β
,
max
⁡
s
1
β
,
s
2
βwhere r
1
(
α
)
=
a
1
α
-
b
2
α, r
2
(
α
)
=
a
2
α
-
b
1
α, s
1
(
β
)
=
a
1
′
β
-
b
2
′
β and s
2
(
β
)
=
a
2
′
β
-
b
1
′
(
β
).
Lete
=
a
⊖
g
H
b.
The existence conditions for whicha
⊖
g
H
b exists are as follows:
(1)e
1
α
=
a
1
α
-
b
2
α, e
2
α
=
a
2
α
-
b
1
α, e
1
′
β
=
a
1
′
β
-
b
2
′
β, and e
2
′
β
=
a
2
′
β
-
b
1
′
(
β
), with (i) e
1
α, e
2
′
β being increasing and e
2
α, e
1
′
β being decreasing function and (ii) e
1
α
≤
e
2
α, e
2
′
β
≤
e
1
′
β for all α
,
β
∈
[
0,1
].
(2)e
2
α
=
a
1
α
-
b
2
α, e
1
α
=
a
2
α
-
b
1
α, e
2
′
β
=
a
1
′
β
-
b
2
′
β, and e
1
′
β
=
a
2
′
β
-
b
1
′
(
β
), with (i) e
1
α, e
2
′
β being increasing and e
2
α, e
1
′
β being decreasing function and (ii) e
1
α
≤
e
2
α, e
2
′
β
≤
e
1
′
β for all α
,
β
∈
[
0,1
].Remark 24.
In the whole paper, we considereda
⊖
g
H
b
∈
R
F.


## 6. Linear Fredholm Integral Equation in Intuitionistic Fuzzy Environment

Integral equations are very important in the area of calculus theory. They appear in different application forms. Practically, when we modeled with integral equation the uncertainty parameters can arise. For that purpose we need to study imprecise integral equation. In this paper we study the intuitionistic integral equation.


### 6.1. Intuitionistic Fuzzy Integral Equation

Considering the linear Fredholm integral equation of second kind(66)
u
x
=
f
x
+
λ
∫
a
b
k
x
,
t
u
t
d
twhere x
∈
D, u
(
x
), and f
(
x
) are functions on D
=
[
a
,
b
], k
(
x
,
t
) is arbitrary kernel function over T
=
a
,
b
×
[
a
,
b
], and u is unknown on D.The upstairs integral equation is said to be intuitionistic integral equation if(1)
f
(
x
) is intuitionistic fuzzy-valued function;
(2)
onlyk
(
x
,
t
) is intuitionistic fuzzy-valued function;
(3)
bothf
(
x
) and k
(
x
,
t
) are intuitionistic fuzzy-valued functions.


### 6.2. Condition for Existence for Solution Intuitionistic Fuzzy Integral Equation

Consider the following intuitionistic fuzzy integral equation.(67)
u
x
=
f
x
+
λ
∫
a
b
k
x
,
t
u
t
d
tLet the solution of the above IFIE be u
~
x and its (
α
,
β
)-cut be u
x
α
,
β
=
[
u
1
x
,
α
,
u
2
x
,
α
;
u
1
′
x
,
β
,
u
2
′
x
,
β
].The solution is called strong solution if(i)∂
u
1
x
,
α
/
∂
α
>
0, ∂
u
2
x
,
α
/
∂
α
<
0
∀
α
∈
0,1, u
1
x
,
1
≤
u
2
x
,
1and(ii)∂
u
1
′
x
,
β
/
∂
β
<
0, ∂
u
2
′
x
,
β
/
∂
β
>
0
∀
β
∈
0,1, u
1
′
x
,
0
≤
u
2
′
x
,
0
.In the rest of the cases, the solution is a weak solution.


### 6.3. Solution of Intuitionistic Fuzzy Integral Equation

Problem 25.
Consider the integral equationu
x
=
f
~
x
+
λ
∫
a
b
k
~
x
,
t
u
(
t
)
d
t.
(In this integral equationf
~
x and k
~
x
,
t are intuitionistic fuzzy function.)Solution. The two possible cases are as follows.Case 1 (whenk
~
x
,
t is positive function).
In this case if we take fuzzy Laplace transformation, then the intuitionistic fuzzy integral equation becomes(68)
L
u
x
=
L
f
~
x
+
λ
L
k
~
x
,
t
L
u
twhich implies the following results.(69)
u
1
x
,
α
=
l
-
1
l
f
1
s
,
α
1
-
λ
l
k
1
s
,
α
u
2
x
,
α
=
l
-
1
l
f
2
s
,
α
1
-
λ
l
k
2
s
,
α
u
1
′
x
,
β
=
l
-
1
l
f
1
′
s
,
β
1
-
λ
l
k
1
′
s
,
α
u
2
′
x
,
β
=
l
-
1
l
f
2
′
s
,
β
1
-
λ
l
k
2
′
s
,
βCase 2 (whenk
~
x
,
t is negative function).
Letk
~
x
,
t
=
-
m
~
(
x
,
t
).
In this case if we take fuzzy Laplace transformation, then the intuitionistic fuzzy integral equation becomes(70)
L
u
x
=
L
f
~
x
+
λ
L
-
m
~
x
,
t
L
u
twhich implies the following results.(71)
u
1
x
,
α
=
l
-
1
λ
l
m
1
x
,
t
,
α
l
f
1
s
,
α
1
-
λ
2
l
m
1
x
,
t
,
α
l
m
2
x
,
t
,
α
u
2
x
,
α
=
l
-
1
λ
l
m
2
x
,
t
,
α
l
f
2
s
,
α
1
-
λ
2
l
m
1
x
,
t
,
α
l
m
2
x
,
t
,
α
u
1
′
x
,
β
=
l
-
1
λ
l
m
1
′
x
,
t
,
β
l
f
1
′
s
,
β
1
-
λ
2
l
m
1
′
x
,
t
,
β
l
m
2
′
x
,
t
,
β
u
2
′
x
,
β
=
l
-
1
λ
l
m
2
′
x
,
t
,
β
l
f
2
′
s
,
β
1
-
λ
2
l
m
1
′
x
,
t
,
β
l
m
2
′
x
,
t
,
βExample 26.
Consider the integral equationu
x
=
f
~
x
+
λ
∫
a
b
k
x
,
t
u
(
t
)
d
t, where f
~
x is a intuitionistic fuzzy-valued function defined as f
~
x
=
(
1,2
,
2.5
;
0.5,2
,
3
)
e
-
x and λ
=
1, a
=
0, b
=
x, k
x
,
t
=
sin
⁡
(
x
-
t
) (here p
=
2, q
=
3, r
=
4 and s
=
2).Solution. Taking intuitionistic fuzzy Laplace transform we can find the solution as follows.(72)
u
1
x
,
α
=
1
+
α
1
/
2
2
e
-
x
+
x
-
1
u
2
x
,
α
=
2.5
-
0.5
α
1
/
3
2
e
-
x
+
x
-
1
u
1
′
x
,
β
=
2
-
1.5
β
1
/
4
2
e
-
x
+
x
-
1
u
2
′
x
,
β
=
2
+
β
1
/
2
2
e
-
x
+
x
-
1Remark. Clearly from Figure 9 we see that u
1
x
,
α, u
2
′
x
,
β is increasing function and u
2
x
,
α, u
1
′
x
,
β is decreasing function at x
=
2. Hence at this particular point the solution is a strong solution.Figure 9
Plot ofu
1
x
,
α, u
2
x
,
α, u
1
′
x
,
β, and u
2
′
x
,
β at x
=
2 for α
,
β
∈
[
0,1
].Defuzzification Value. The defuzzification value of solution is given by the following.(73)
u
^
x
=
1.81
2
e
-
x
+
x
-
1


## 6.1. Intuitionistic Fuzzy Integral Equation

Considering the linear Fredholm integral equation of second kind(66)
u
x
=
f
x
+
λ
∫
a
b
k
x
,
t
u
t
d
twhere x
∈
D, u
(
x
), and f
(
x
) are functions on D
=
[
a
,
b
], k
(
x
,
t
) is arbitrary kernel function over T
=
a
,
b
×
[
a
,
b
], and u is unknown on D.The upstairs integral equation is said to be intuitionistic integral equation if(1)
f
(
x
) is intuitionistic fuzzy-valued function;
(2)
onlyk
(
x
,
t
) is intuitionistic fuzzy-valued function;
(3)
bothf
(
x
) and k
(
x
,
t
) are intuitionistic fuzzy-valued functions.


## 6.2. Condition for Existence for Solution Intuitionistic Fuzzy Integral Equation

Consider the following intuitionistic fuzzy integral equation.(67)
u
x
=
f
x
+
λ
∫
a
b
k
x
,
t
u
t
d
tLet the solution of the above IFIE be u
~
x and its (
α
,
β
)-cut be u
x
α
,
β
=
[
u
1
x
,
α
,
u
2
x
,
α
;
u
1
′
x
,
β
,
u
2
′
x
,
β
].The solution is called strong solution if(i)∂
u
1
x
,
α
/
∂
α
>
0, ∂
u
2
x
,
α
/
∂
α
<
0
∀
α
∈
0,1, u
1
x
,
1
≤
u
2
x
,
1and(ii)∂
u
1
′
x
,
β
/
∂
β
<
0, ∂
u
2
′
x
,
β
/
∂
β
>
0
∀
β
∈
0,1, u
1
′
x
,
0
≤
u
2
′
x
,
0
.In the rest of the cases, the solution is a weak solution.


## 6.3. Solution of Intuitionistic Fuzzy Integral Equation

Problem 25.
Consider the integral equationu
x
=
f
~
x
+
λ
∫
a
b
k
~
x
,
t
u
(
t
)
d
t.
(In this integral equationf
~
x and k
~
x
,
t are intuitionistic fuzzy function.)Solution. The two possible cases are as follows.Case 1 (whenk
~
x
,
t is positive function).
In this case if we take fuzzy Laplace transformation, then the intuitionistic fuzzy integral equation becomes(68)
L
u
x
=
L
f
~
x
+
λ
L
k
~
x
,
t
L
u
twhich implies the following results.(69)
u
1
x
,
α
=
l
-
1
l
f
1
s
,
α
1
-
λ
l
k
1
s
,
α
u
2
x
,
α
=
l
-
1
l
f
2
s
,
α
1
-
λ
l
k
2
s
,
α
u
1
′
x
,
β
=
l
-
1
l
f
1
′
s
,
β
1
-
λ
l
k
1
′
s
,
α
u
2
′
x
,
β
=
l
-
1
l
f
2
′
s
,
β
1
-
λ
l
k
2
′
s
,
βCase 2 (whenk
~
x
,
t is negative function).
Letk
~
x
,
t
=
-
m
~
(
x
,
t
).
In this case if we take fuzzy Laplace transformation, then the intuitionistic fuzzy integral equation becomes(70)
L
u
x
=
L
f
~
x
+
λ
L
-
m
~
x
,
t
L
u
twhich implies the following results.(71)
u
1
x
,
α
=
l
-
1
λ
l
m
1
x
,
t
,
α
l
f
1
s
,
α
1
-
λ
2
l
m
1
x
,
t
,
α
l
m
2
x
,
t
,
α
u
2
x
,
α
=
l
-
1
λ
l
m
2
x
,
t
,
α
l
f
2
s
,
α
1
-
λ
2
l
m
1
x
,
t
,
α
l
m
2
x
,
t
,
α
u
1
′
x
,
β
=
l
-
1
λ
l
m
1
′
x
,
t
,
β
l
f
1
′
s
,
β
1
-
λ
2
l
m
1
′
x
,
t
,
β
l
m
2
′
x
,
t
,
β
u
2
′
x
,
β
=
l
-
1
λ
l
m
2
′
x
,
t
,
β
l
f
2
′
s
,
β
1
-
λ
2
l
m
1
′
x
,
t
,
β
l
m
2
′
x
,
t
,
βExample 26.
Consider the integral equationu
x
=
f
~
x
+
λ
∫
a
b
k
x
,
t
u
(
t
)
d
t, where f
~
x is a intuitionistic fuzzy-valued function defined as f
~
x
=
(
1,2
,
2.5
;
0.5,2
,
3
)
e
-
x and λ
=
1, a
=
0, b
=
x, k
x
,
t
=
sin
⁡
(
x
-
t
) (here p
=
2, q
=
3, r
=
4 and s
=
2).Solution. Taking intuitionistic fuzzy Laplace transform we can find the solution as follows.(72)
u
1
x
,
α
=
1
+
α
1
/
2
2
e
-
x
+
x
-
1
u
2
x
,
α
=
2.5
-
0.5
α
1
/
3
2
e
-
x
+
x
-
1
u
1
′
x
,
β
=
2
-
1.5
β
1
/
4
2
e
-
x
+
x
-
1
u
2
′
x
,
β
=
2
+
β
1
/
2
2
e
-
x
+
x
-
1Remark. Clearly from Figure 9 we see that u
1
x
,
α, u
2
′
x
,
β is increasing function and u
2
x
,
α, u
1
′
x
,
β is decreasing function at x
=
2. Hence at this particular point the solution is a strong solution.Figure 9
Plot ofu
1
x
,
α, u
2
x
,
α, u
1
′
x
,
β, and u
2
′
x
,
β at x
=
2 for α
,
β
∈
[
0,1
].Defuzzification Value. The defuzzification value of solution is given by the following.(73)
u
^
x
=
1.81
2
e
-
x
+
x
-
1


## 7. Conclusion

It is not necessary that the membership and the nonmembership function of an intuitionistic fuzzy number be linear. They may be nonlinear. If they are nonlinear, then their order may be fraction or may not. For taking the above concept we introduce nonlinear intuitionistic fuzzy numbers (NIFN) and their arithmetic operation. We apply max-min principle for arithmetic operation on NIFN. The exact resulting number is written in parametric form. Finally we use this number in integral equation in intuitionistic fuzzy environment. For helping the readers who try to compare the fuzzy solution with crisp number, we defuzzify or crispify the result by min of(
α
,
β
)-cut method. The paper can help those who are modeled with uncertainty and integral and differential calculus. In future any one can use this concept for finding various types of fuzzy numbers and apply them in various fields of mathematics.



---

*Source: 4142382-2019-03-03.xml*