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Advances in Fuzzy Systems
Volume 2019, Article ID 4142382, 14 pages
https://doi.org/10.1155/2019/4142382
Research Article

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

1Department of Natural Science, Maulana Abul Kalam Azad University of Technology, West Bengal, Haringhata, Nadia, West Bengal, India
2Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India
3Department of Mathematics, Midnapore College (Autonomous), Midnapore-721101, West Bengal, India

Correspondence should be addressed to Sankar Prasad Mondal; moc.liamg@70ser.raknas

Received 22 May 2018; Revised 12 October 2018; Accepted 9 December 2018; Published 3 March 2019

Academic Editor: Ferdinando Di Martino

Copyright © 2019 Sankar Prasad Mondal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 -cut method. Finally we solve integral equation with NIFN by the help of intuitionistic fuzzy Laplace transform method.

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, [36]). 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 [1116].

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 [1823] and fuzzy Volterra integral equation [2429].

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 -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.

2. Preliminaries

2.1. Basic Concept Intuitionistic Fuzzy Set Theory

Definition 1 (intuitionistic fuzzy set: [8]). An IFS in is an object having the form , where the and define the degree of membership and degree of nonmembership, respectively, of the element to the set , which is a subset of , for every element of , .

Definition 2 (triangular intuitionistic fuzzy number: [30]). A TIFN is a subset of IFN in R with following membership function and nonmembership function as follows:where and .
The TIFN is denoted by .

Definition 3. Let us consider intuitionistic fuzzy-valued function defined in the parametric formTherefore if we consider the above said intuitionistic fuzzy number, the parametric form is as follows.

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.

Definition 4 (see [31]). The intuitionistic fuzzy Laplace transform of an intuitionistic fuzzy-valued function is defined as follows.Consider the intuitionistic fuzzy-valued function ; then the lower and upper intuitionistic fuzzy Laplace transform of this function are denoted based on the lower and upper intuitionistic fuzzy-valued function as follows.
= , ; ,
i.e., ;

Now we define the absolute value of an intuitionistic fuzzy-valued function as follows.

Basic Property(1) Linearity property: Let be two continuous intuitionistic fuzzy-valued functions; then = .

Remark 5. Let be a continuous intuitionistic fuzzy-valued function on and , then
= .(2) First Translation Theorem: Let be a continuous intuitionistic fuzzy-valued function and , then, where is real-valued function and .

Definition 6. In order to solve intuitionistic fuzzy differential equations, it is necessary to know the intuitionistic fuzzy Laplace transform of the derivative of . The virtue of is that it can be written in terms of .

Theorem 7. Suppose that is continuous intuitionistic fuzzy-valued function on and exponential order p and that is piecewise continuous intuitionistic fuzzy-valued function on ; then(a), if is (i)-differentiable,(b), if is (ii)-differentiable.

Proof. (a)  L.H.S:   
and R.H.S:   
  
  
.   
Hence, L.H.S=R.H.S.

Proof. (b)  L.H.S: and  
R.H.S:   
=   
= .   
Hence, L.H.S=R.H.S.

3. Nonlinear Triangular Intuitionistic Fuzzy Number and Its Arithmetic Operations

Definition 8 (see [32]). A NTIFN is a subset of IFN in R with the following membership function and nonmembership function:where and .
The TIFN is denoted by .

Definition 9 (-cut set). A -cut set of is a crisp subset of which is defined as follows.

Definition 10 (-cut set). A -cut set of is a crisp subset of which is defined as follows.

Definition 11 (-cut set). A -cut set of is a crisp subset of which is defined as follows.

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 (see Figure 1), membership and nonmembership function are denoted by and ; then

Figure 1: Nonlinear intuitionistic fuzzy number.

Proof. From Figure 1 we prove the theorem by splitting up the region.
Now we split up the region into different intervals and points asand first take . 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.

Note 13. The above proof is done for taking . It can be proved by all values of . 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

Let and be two fuzzy numbers and be an arbitrary operation such thatWe 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 be , be , and be .

Now the component of the resulting fuzzy number in parametric form is written aswhere , are increasing functions and , are decreasing functions.

3.2. Arithmetic Operation on Nonlinear Intuitionistic Fuzzy Number

If and are two nonlinear triangular fuzzy numbers with the membership function and the nonmembership function asandwith -cut andfor a particular case we consider that and .

3.2.1. Addition of Two Normal Fuzzy Numbers Using -Cut

The membership and the nonmembership function are defined as of and

3.2.2. Subtraction of Two Normal Fuzzy Numbers Using -Cut

The membership and the nonmembership function are defined as for and

3.2.3. Multiplication by a Scalar

If is a positive scalar, thenTherefore, .

If is a negative scalar, thenTherefore, .

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

Consider two intervals and

where may be positive or negative.

Therefore we can write = , and = , .

For intuitionistic fuzzy multiplication or division we can use this concept on the -cut interval.Multiplication:

whereDivision:

whereFor every    where = , and = , .

Example 14. If and , then find and .
SolutionNow let , , , and .
Now by interval rule base system we find the following.From Figure 2 we can conclude that the and should be chosen as follows.  From Figure 3 we can conclude that the and should be chosen as the following.   Hence by interval rule base the -cut of is given by the following.  Similarly,From Figure 5 we can conclude that the and should be chosen as follows.From Figure 6 we can conclude that the and should be chosen asandHence by interval rule base the -cut of is given by the following.Remark. We recommend seeing graphical representation Figures 2, 3, 4, 5, 6, and 7

Figure 2: Possible and as .
Figure 3: Possible and as .
Figure 4: , , , and for .
Figure 5: Possible and as .
Figure 6: Possible and as .
Figure 7: , , , and for .
3.3. Intuitionistic Fuzzy Function

Considering that , , , and are the continuous functions on the interval .

The set can be determined by membership and nonmembership functions as follows:andwhere, .

The intuitionistic fuzzy function is denoted as , and the -cut of is as follows.

Example 15. Consider the intuitionistic fuzzy-valued function .
Due to presence of the intuitionistic coefficient , the function becomes intuitionistic fuzzy function (see also the graphical representation Figure 8).
The -cut of isi.e., .