Abstract
This paper is introducing a new concept of triangular linear Diophantine fuzzy numbers (TLDFNs) in a generic way. We first introduce the concept of TLDFNs and then study the arithmetic operations on these numbers. We find a method for the ranking of these TLDFNs. At the end, we formulate the linear and quadratic equations of the types , and where the elements , and are TLDFNs. We provide a procedure for the solution of these equations using -cut and also provide the examples.
1. Introduction
In 1965, Zadeh [1] introduced a new notion of fuzzy set theory. Fuzzy set (FS) theory has been widely acclaimed as offering greater richness in applications than ordinary set theory. Zadeh popularized the concept of fuzzy sets for the first time. There is an area of FS theory, in which the arithmetic operations on FNs play an essential part known as fuzzy equations (FEQs). Fuzzy equations were studied by Sanchez [2], by using extended operations. Accordingly, a profuse number of researchers like Biacino and Lettieri [3], Buckley [4], and Wasowski [5] have studied several approaches to solve FEQs. In [6–9], Buckley and Qu introduced several techniques to evaluate the fuzzy equations of the type and where , and are fuzzy numbers (FNs). Jiang [10] studied an approach to solve simultaneous linear equations that coefficients are fuzzy numbers.
Intuitionistic fuzzy sets [11, 12], neutrosophic sets [13, 14], and bipolar fuzzy sets [15] are the generalizations of the fuzzy sets. There are several mathematicians who solved linear and quadratic equations based on intuitionistic fuzzy sets, neutrosophic sets, and bipolar fuzzy sets. Banerjee and Roy [16] studied the intuitionistic fuzzy linear and quadratic equations, Chakraborty et al. [17] studied arithmetic operations on generalized intuitionistic fuzzy number and its applications to transportation problem, Rahaman et al. [18] introduced the solution techniques for linear and quadratic equations with coefficients as Cauchy neutrosophic numbers, and Akram et al. [19–23] introduced some methods for solving the bipolar fuzzy system of linear equations, also see [24–26].
Linear Diophantine fuzzy set [27] is a new generalization of fuzzy set, intuitionistic fuzzy set, neutrosophic set, and bipolar fuzzy set which was introduced by Riaz and Hashmi in 2019. After the introduction of this concept, several mathematicians were attracted towards this concept and worked in this area. Riaz and others studied the decision-making problems related to linear Diophantine fuzzy Einstein aggregation operators [28], spherical linear Diophantine fuzzy sets [29], and linear Diophantine fuzzy relations [30]. Almagrabi et al. [31] introduced a new approach to -linear Diophantine fuzzy emergency decision support system for COVID-19. Kamac [32] studied linear Diophantine fuzzy algebraic structures.
Motivated by the work of Buckley and Qu [7], we solve the linear and quadratic equations with more generalized fuzzy numbers. As the linear Diophantine fuzzy set, [27] is the more generalized form of fuzzy sets so we studied the linear and quadratic equations based on linear Diophantine fuzzy numbers. In linear Diophantine fuzzy sets, we use the reference parameters, which allow us to choose the grades without any limitation; this helps us in obtaining better results.
In Section 2, we provided the fundamental definitions related to fuzzy sets and linear Diophantine fuzzy sets. In Section 3, we define linear Diophantine fuzzy numbers, in particular, triangular linear Diophantine fuzzy number. Also defined some basic operations on LDF numbers. In Section 4, we provide the ranking of LDF numbers, and in Section 5, we solved linear and quadratic equations based on LDF numbers.
2. Preliminaries and Basic Definitions
This section is devoted to review some indispensable concepts, which are very beneficial to develop the understanding of the prevalent model.
Definition 1 (see [1]). Letbe a classical set,be a function fromtoThe MF (membership function)of a FS (fuzzy set)is defined by
Definition 2 (see [33]). Letbe a fuzzy subset of universal set. Then,is called convex FS ifandwe have
Definition 3 (see [1]). A fuzzy setis said to be normalized if
Definition 4. An-level set of a FSis defined as
Definition 5 (see [33]). A fuzzy subsetdefined on a set(of real numbers) is said to be a FN (fuzzy number) ifsatisfies the following axioms:(a) is continuous: is a continuous function from (b) is normalized: there exists such that (c)Convexity of : i.e., , if , then (d)Boundness of support: i.e., and if , then
We denote the set of all FNs by .
Now, we study the idea of LDFSs (linear Diophantine fuzzy sets) and their fundamental operations.
Definition 6 (see [27]). Letbe the universe. A LDFSonis defined as follows:where such that
The hesitation part can be written as
where is the reference parameter.
We write in short or for
Definition 7 (see [27]). An absolute LDFS oncan be written asand empty or null LDFS can be expressed as
Definition 8 (see [27]). Letandbe two LDFSs on the reference setand. Then,(i)(ii) (iii) (iv)(v)where
Definition 9 (see [27]). Letbe an LDFS. For any constants,,,such thatwithdefine the-cut ofas follows:
3. Triangular LDF Numbers
Here, in this section, we provide definitions and arithmetic operations on LDF numbers (LDFNs).
Definition 10. A LDF numberis(i)a LDF fuzzy subset of the real line (ii)normal, i.e., there is any such that (iii)convex for the membership functions and i.e.,(iv)concave for the nonmembership functions and i.e.,
We now provide the 4 types of triangular LDF numbers.
Definition 11. Letbe a LDFS onwith the following membership functions (and) and nonmembership functions (and)where for all Then, is called (i)a triangular LDFN of type-1 if and (ii)a triangular LDFN of type-2 if and (iii)a triangular LDFN of type-3 if and (iv)a triangular LDFN of type-4 if and
Throughout the paper, we consider only triangular LDFN of type-1 and we write this type as triangular LDFN (TLDFN). This TLDFN is denoted by
The figure of is shown in Figure 1.
The figure of is shown in Figure 2.
The figure of is shown in Figure 3.
Remark 12. If we takeandthen both type-1 and type-3 become the same.
Definition 13. Consider a TLDFNThen,(i)-cut set of is a crisp subset of which is defined as follows:(ii)-cut set of is a crisp subset of which is defined as follows:(iii)-cut set of is a crisp subset of which is defined as follows:(iv)-cut set of is a crisp subset of which is defined as follows:
We can denote the -cut of by
We denote the set of all TLDFN on by . The arithmetic operations based on extension principle are defined as follows.
Definition 14. Letandbe two TLDFN on. Then,(i)(ii)(iii)(iv)
Definition 15. A TLDFNis said to be positive if and only ifand
Definition 16. Two TLDFNsandare said to be equal if and only if and
We now define the arithmetic operations on TLDFNs using the concept of interval arithmetic.
Definition 17. Consider two positive TLDFNsandthen,(i)(ii)(iii)(iv)(v)
4. Ranking Function of TLDFNs
There are many methods for defuzzification such as the centroid method, mean of interval method, and removal area method. In this paper, we have used the concept of the mean of interval method to find the value of the membership and nonmembership function of TLDFN.
Consider a TLDFN
The -cut of is
where
Now, by the mean of -cut method, the representation of membership functions is
Now, by the mean of -cut method, the representation of nonmembership functions is
Now,
Consider two positive TLDFNs and then (i) iff (ii) iff (iii) iff
5. Solution of LDF Equations
5.1. Solution of by Using the Method of -Cut
Let , and be the LDFNs and let and Then, is a LDF equation (LDFE). Let Then, in general is not the solution of Equation (27).
Let represent the -cuts of and , respectively, in the given (27). Substituting these into Equation (27), we get
By comparing the -cuts of and we get
Now,
Then, the solution of the equation exists iff (1) is monotonically increasing in (2) is monotonically decreasing in (3) is monotonically decreasing in (4) is monotonically increasing in (5) is monotonically increasing in (6) is monotonically decreasing in (7) is monotonically decreasing in (8) is monotonically increasing in (9)
Example 1. Consider the equationwhereThe -cuts of , and are respectively. The -cut equation is By comparing the -cuts of , and we get It is easy to see that and are increasing and , and are decreasing in Also, This shows that the solution of exists with -cut. The solution is The solution in continuous form is The graph of the solution is given in Figure 4.
5.2. Solution of by Using the Method of -Cut
Let , and be the LDFNs and let , and Then, is a LDF equation (LDFE). Let Then, in general is not the solution of Equation (39).
Let represent the -cuts of , and , respectively, in the given (39). Substituting these into Equation (39), we get
By comparing the -cuts of and we get
Now,
Then, the solution of the equation exists iff (1) is monotonically increasing in (2) is monotonically decreasing in (3) is monotonically decreasing in (4) is monotonically increasing in (5) is monotonically increasing in (6) is monotonically decreasing in (7) is monotonically decreasing in (8) is monotonically increasing in
Example 2. Consider the equationwhereThe -cuts of and are respectively. The -cut equation is By comparing the -cuts of and we get It is easy to see that , and are increasing and and are decreasing in Also, This shows that the solution of exists with -cut. The solution is The solution in continuous form is The graph of the solution is given in Figure 5.
5.3. Solution of by Using the Method of -Cut.
Let , and be the LDFNs and let
Then, is a LDF equation (LDFE). Let Let represent the -cuts of and , respectively, in the given (53). Substituting these into Equation (53), we get
By comparing the -cuts of and we get
Now,
Then, the solution of the equation exists iff (1) is monotonically increasing in (2) is monotonically decreasing in (3) is monotonically decreasing in (4) is monotonically increasing in (5) is monotonically increasing in (6) is monotonically decreasing in (7) is monotonically decreasing in (8) is monotonically increasing in (9)
Example 3. Consider the equationwhere
The -cuts of , and are given in Table 1.
By comparing the -cuts of , and we get
The graph obtained by -cut is shown in Figure 6.
The -cuts of , and are given in Table 2.
By comparing the -cuts of and we get
The graph obtained by -cut is shown in Figure 7.
It is easy to see that , and are increasing and and are decreasing in Also,
This shows that the solution of exists with -cut. The solution is
The solution in continuous form is
The graph of the solution is given in Figure 8.
6. Conclusion
In this paper, we have defined the linear Diophantine fuzzy numbers, in particular triangular linear Diophantine fuzzy number, and present some properties related to them. After finding the ranking function of triangular linear Diophantine fuzzy number, our study has focussed on the linear Diophantine fuzzy equations. We used the more general approach to solve LDF equations that is the method of -cut. In LDF sets, there is no limitation to take the grades like in intuitionistic fuzzy sets, Pythagorean fuzzy sets, and -rung orthopair fuzzy sets. The linear Diophantine fuzzy numbers may have several applications, like in linear programming, transportation problems, assignment problems, and shortest route problems. Our future work may be on the following topics: (i)LDF linear programming problems(ii)LDF assignment problems and transportation problems(iii)LDF shortest path problems(iv)Numerical solutions of linear and nonlinear LDF equations
Data Availability
No data were used to support this study.
Disclosure
The statements made and views expressed are solely the responsibility of the author.
Conflicts of Interest
The authors of this paper declare that they have no conflict of interest.
Acknowledgments
The fourth author (YUG) would like to acknowledge that this publication was made possible by a grant from the Carnegie Corporation of New York.