Solution of Linear and Quadratic Equations Based on Triangular Linear Diophantine Fuzzy Numbers

Department of Mathematics and Statistics, Riphah International University, I-14 Islamabad, Pakistan Quantum Leap Africa (QLA), AIMS Rwanda Centre, Remera Sector KN 3, Kigali, Rwanda Institut de Mathematiques et de Sciences Physiques (IMSP/UAC), Laboratoire de Topologie Fondamentale, Computationnelle et leurs Applications (Lab-ToFoCApp), BP 613, Porto-Novo, Benin African Center for Advanced Studies, P.O. Box 4477, Yaounde, Cameroon Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan


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][7][8][9], Buckley and Qu introduced several techniques to evaluate the fuzzy equations of the type A · X + B = C and A · X 2 + B · X + C = D, where A, B, C, D, and X 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][20][21][22][23] introduced some methods for solving the bipolar fuzzy system of linear equations, also see [24][25][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 q-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.

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]). Let X be a classical set, μ M be a function from X to ½0, 1: The MF (membership function) μ M ðϑÞ of a FS (fuzzy set) M is defined by Definition 2 (see [33]). Let M be a fuzzy subset of universal set X . Then, M is called convex FS if ∀r, s ∈ X and λ ∈ ½0, 1 we have Definition 3 (see [1]). A fuzzy set M is said to be normalized if hðMÞ = 1: Definition 5 (see [33] We denote the set of all FNs by F ns ðℝÞ. Now, we study the idea of LDFSs (linear Diophantine fuzzy sets) and their fundamental operations.
Definition 6 (see [27]). Let X be the universe. A LDFS £ R on X is defined as follows: The hesitation part can be written as where ξ is the reference parameter. We write in short Definition 7 (see [27]). An absolute LDFS on X can be written as and empty or null LDFS can be expressed as Definition 8 (see [27]). Let £ R = ðhM τ R , N ν R i, hα, βiÞ and £ P = ðhM τ P , N ν P i, hγ, δiÞ be two LDFSs on the reference set X and ϑ ∈ X . Then, Journal of Function Spaces Definition 9 (see [27]). Let £ R = fðϑ, hM τ R ðϑÞ, N ν R ðϑÞi, hαðϑ Þ, βðϑÞiÞ: ϑ ∈ Xg be an LDFS. For any constants s,t,u ,v ∈ ½0, 1 such that 0 ≤ su + tv ≤ 1 with 0 ≤ u + v ≤ 1, define the ðhs, ti, hu, viÞ -cut of £ R as follows:

Triangular LDF Numbers
Here, in this section, we provide definitions and arithmetic operations on LDF numbers (LDFNs).
Definition 11. Let £ R be a LDFS on ℝ with the following membership functions (M τ R and α) and nonmembership functions (N ν R and β ) where 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 ðϑ 1 , ϑ 2 , ϑ 3 , ϑ 4 , ϑ 5 Þ is shown in Figure 1.
The figure of £ R TLDFN is shown in Figure 3.
Definition 13. Consider a TLDFN £ R TLDFN = f ðϑ 1 ,ϑ 2 ,ϑ 3 ,ϑ 4 ,ϑ 5 Þ ðϑ 1 ′ ,ϑ 2 ′ ,ϑ 3 ,ϑ 4 ′ ,ϑ 5 ′ Þ : Then, (i) s-cut set of £ R TLDFN is a crisp subset of ℝ, which is defined as follows: (ii) t-cut set of £ R TLDFN is a crisp subset of ℝ, which is defined as follows: (iii) u-cut set of £ R TLDFN is a crisp subset of ℝ, which is defined as follows: (iv) v-cut set of £ R TLDFN is a crisp subset of ℝ, which is defined as follows: We can denote the ðhs, ti, hu, viÞ-cut of £ R TLDFN = We denote the set of all TLDFN on ℝ by £ R TLDFN ðℝÞ. The arithmetic operations based on extension principle are defined as follows.
to be positive if and only if ϑ 1 ≥ 0 and ϑ 1 ′ ≥ 0: We now define the arithmetic operations on TLDFNs using the concept of interval arithmetic.

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 ðhs, ti, hu, viÞ-cut of £ R TLDFN is where 5 Journal of Function Spaces Now, by the mean of ðhs, ti, hu, viÞ -cut method, the representation of membership functions is Now, by the mean of ðhs, ti, hu, viÞ -cut method, the representation of nonmembership functions is Now, Consider two positive TLDFNs

Solution of LDF Equations
represent the ðhs, ti, hu, viÞ-cuts of A, B, and X, respectively, in the given (27). Substituting these into Equation (27), we get Journal of Function Spaces By comparing the ðhs, ti, hu, viÞ -cuts of A, B, and X, we get Now, Then, the solution of the equation respectively. The ðhs, ti, hu, viÞ-cut equation is By comparing the ðhs, ti, hu, viÞ-cuts of A, B, and X, we get It is easy to see that M τ X ðsÞ , N ν X ðtÞ,α X ðuÞ , and β X ðvÞ are increasing and M τ X ðsÞ,N ν X ðtÞ , α X ðuÞ, and β X ðvÞ are decreasing in 0 ≤ s, t, u, v ≤ 1: Also, This shows that the solution of A + X = B exists with ðhs, ti, hu, viÞ -cut. The solution is The solution in continuous form is The graph of the solution is given in Figure 4.

Solution of A · X + B = C by Using the Method of ðhs, ti
, hu, viÞ-Cut. Let A, B, C, and X be the LDFNs and let A = .
represent the ðhs, ti, hu, viÞ-cuts of A, B, C, and X, respectively, in the given (39). Substituting these into Equation By comparing the ðhs, ti, hu, viÞ-cuts of A, B, C, and X, we get  Journal of Function Spaces Now, Then, the solution of the equation The ðhs, ti, hu, viÞ-cuts of A, B, C, and X are respectively. The ðhs, ti, hu, viÞ-cut equation is By comparing the ðhs, ti, hu, viÞ-cuts of A, B, C, and X, we get It is easy to see that M τ X ðsÞ , N ν X ðtÞ,α X ðuÞ , and β X ðvÞ are increasing and M τ X ðsÞ,N ν X ðtÞ , α X ðuÞ, and β X ðvÞ are decreasing in 0 ≤ s, t, u, v ≤ 1: Also, This shows that the solution of A · X + B = C exists with ðhs, ti, hu, viÞ -cut. The solution is The solution in continuous form is The graph of the solution is given in Figure 5.

Solution of
A · X 2 + B · X + C = D by Using the Method of α-Cut. Let A, B, C, D, and X be the LDFNs and let Then, is a LDF equation (LDFE). Let X ≈ ðx 1 , represent the ðhs, ti, hu, viÞ-cuts of A, B, C, D, and X, respectively, in the given (53). Substituting these into Equation By comparing the ðhs, ti, hu, viÞ -cuts of A, B, C, D, and X, we get Journal of Function Spaces Now, Then, the solution of the equation A · X 2 + B · X + C = D exists iff (1) M τ X ðsÞ is monotonically increasing in 0 ≤ s ≤ 1 The hs, ti-cuts of A, B, C, D, and X are given in Table 1.
The solution in continuous form is  Figure 6: Solution by hs, ti-cut.
The graph of the solution is given in Figure 8.

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 ðhs, ti, hu, viÞ-cut. In LDF sets, there is no limitation to take the grades like in intuitionistic fuzzy sets, Pythagorean fuzzy sets, and q-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.