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

,


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][4][5][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][12][13][14][15][16].

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][19][20][21][22][23] and fuzzy Volterra integral equation [24][25][26][27][28][29].

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 2 Advances in Fuzzy Systems 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.

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 (, )-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 (, )-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.
Definition 2 (triangular intuitionistic fuzzy number: [30]).A TIFN Ã is a subset of IFN in R with following membership function and nonmembership function as follows: where Definition 3. Let us consider intuitionistic fuzzy-valued function   () defined in the parametric form Therefore if we consider the above said intuitionistic fuzzy number, the parametric form is as follows.
[ Ã ] (,) 2.2.Intuitionistic Fuzzy Laplace Transform.Suppose that   () is an intuitionistic fuzzy-valued function and s is a real parameter.We define the intuitionistic fuzzy Laplace transform of f as follows.

Nonlinear Triangular Intuitionistic Fuzzy Number and Its Arithmetic Operations
Definition 8 (see [32]).A NTIFN Ã is a subset of IFN in R with the following membership function and nonmembership function: where ) is a crisp subset of  which is defined as follows.
) is a crisp subset of  which is defined as follows.
Definition 11 Theorem 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 number 1), membership and nonmembership function are denoted by  Ã () and  Ã ( 1 ); then 0 ≤  Ã () +  Ã ( 1 ) ≤ 1.
Proof.From Figure 1 we prove the theorem by splitting up the region.

Now we split up the region into different intervals and points as
and first take  = 1.Then in each of the above points and intervals, the values of membership function and nonmembership function are assumed as in Table 1.
Note 13.The above proof is done for taking  = 1.It can be proved by all values of .For other values other than 1, we can take numerical examples for proving the theorem.

Max-Min Principle Method for Arithmetic Operation on
Intuitionistic Fuzzy Number.Let Ã and B be two fuzzy numbers and * be an arbitrary operation such that 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 (, )-cut of C be [ Now the component of the resulting fuzzy number in parametric form is written as where  1 (),   1 () are increasing functions and  2 (),   1 () are decreasing functions.

Arithmetic Operation on
and with (, )-cut and for a particular case we consider that

Addition of Two Normal Fuzzy Numbers Using
The membership and the nonmembership function are defined as of ) and

Subtraction of Two Normal Fuzzy Numbers Using
The membership and the nonmembership function are defined as for and
From Figure 3 we can conclude that the   1 () and   2 () should be chosen as the following.
Hence by interval rule base the (, )-cut of ẽ is given by the following.
The set G can be determined by membership and nonmembership functions as follows: and where, 0 ≤  +  ≤ 1.
The intuitionistic fuzzy function is denoted as G(), and the (, )-cut of G() is as follows. and Theorem 17.The ambiguities of the membership function and the nonmembership function of nonlinear IFN Ã are calculated as follows: and Proof. and .
(54)  Where the symbol " >", "=" and " <" defines the greater than, equal to, and less than relation in fuzzy sense.
Hence Ã > B , i.e., Ã is greater than B in fuzzy sense.

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 (, )cut method.

De-i-Fuzzification
Based on Average of (, )-Cut Method of an IFN.For an IFN Ã , the de-i-fuzzification value of Ã is a crisp value which can be derived in the following way where Â  is de-i-fuzzification of -cut and Â  is de-ifuzzification value of -cut.
That is, Â  = ∫   where  is Hausdorff metric and metric space (R F , ) is complete, separable, and locally compact.The following substances for metric  are tenable: