Linear Diophantine Fuzzy Relations and Their Algebraic Properties with Decision Making

: Binary relations are most important in various ﬁelds of pure and applied sciences. The concept of linear Diophantine fuzzy sets (LDFSs) proposed by Riaz and Hashmi is a novel mathematical approach to model vagueness and uncertainty in decision-making problems. In LDFS theory, the use of reference or control parameters corresponding to membership and non-membership grades makes it most accommodating towards modeling uncertainties in real-life problems. The main purpose of this paper is to establish a robust fusion of binary relations and LDFSs, and to introduce the concept of linear Diophantine fuzzy relation (LDF-relation) by making the use of reference parameters corresponding to the membership and non-membership fuzzy relations. The novel concept of LDF-relation is more ﬂexible to discuss the symmetry between two or more objects that is superior to the prevailing notion of intuitionistic fuzzy relation (IF-relation). Certain basic operations are deﬁned to investigate some signiﬁcant results which are very useful in solving real-life problems. Based on these operations and their related results, it is analyzed that the collection of all LDF-relations gives rise to some algebraic structures such as semi-group, semi-ring and hemi-ring. Furthermore, the notion of score function of LDF-relations is introduced to analyze the symmetry of the optimal decision and ranking of feasible alternatives. Additionally, a new algorithm for modeling uncertainty in decision-making problems is proposed based on LDFSs and LDF-relations. A practical application of proposed decision-making approach is illustrated by a numerical example. Proposed LDF-relations, their operations, and related results may serve as a foundation for computational intelligence and modeling uncertainties in decision-making problems.


Introduction
In this modern age of technology, modeling uncertainties in engineering, computer sciences, social sciences, medical sciences and economics is growing widely. To tackle such types of problems, classical mathematical methods are not always useful. In 1965, Zadeh [1], coined the notion of fuzzy set (FS) to handel the uncertainties in every day language. In FS theory, a person who is very sick could have the degree of sickness near to 0.89. On contrary, a person who is having degree of sickness 0.12 indicates that he has nearly recovered from illness. In decision making (DM) and operational research the concept of FS theory was broadly studied since 1965 (see [2][3][4]).
However, only the membership function is not always sufficient to describe the complexities in real life problems. In [5][6][7], Atanassov proposed the concept of intuitionistic fuzzy set (IFS) as an extension of FS. Atanassov's IFS enhanced the idea of FS by allowing uncertain real-life problems with the help of parameterizations. Riaz et al., introduced the idea of m-polar neutrosophic sets and m-polar neutrosophic topology with applications to MADM [36].
The main objective of this paper is to introduce the concept of linear Diophantine fuzzy relation (LDF-relation) as an extension of IF-relation. A second objective of LDF-relation is to address modeling uncertainties in MCDM. Because LDF-relation is more efficient to relax the strict restrictions of IF-relation regarding membership and non-membership grades. Some operations on LDF-relation and their properties are investigated. Additionally, algebraic structures such as semigroup, semiring and hemiring are studied in the set of all LDF-relations. A third objective is to introduce notion of score function of LDF-relation and to analyze the symmetry of the optimal decision and ranking of feasible alternatives. A fourth objective is to develop a new algorithm and present its practical application to MCDM problems based on LDFSs and LDF-relations.
This manuscript is composed in the following order: Section 2 contains some basic concepts of FSs, IFSs, LDFSs, F-relation, IF-relations, semigroup, semiring and hemiring. Section 3 introduces the concept of LDF-relation and some fundamental operations with some significant properties. With the help of these operations and properties, some algebraic structures such as semigroup, semiring and hemiring, in the set of all LDF-relations are introduced. Section 4 is devoted to constructing an algorithm for DM with a numerical example. Finally, Section 5 presents the conclusion of this research paper.

Preliminaries
This section includes some essential concepts which are useful in the remaining sections of the manuscript. For detailed study, we refer the reader to [1,2,8,26,32,37]. In the whole manuscript, Q will be supposed to be a universal set. Definition 1. [1] A FS δ on Q is a mapping δ : Q → [0, 1] known as membership function which assigns the grade of membership to each object υ ∈ Q in δ. The set of all FSs on Q is denoted by F (Q).
A binary relation from Q 1 to Q 2 is a subset of the cartesian product Q 1 × Q 2 , where Q 1 and Q 2 are two universes. In 1971, Zadeh [2] fuzzified the structure of binary relation and introduced a new concept, known as F-relation.

Definition 2. [2]
A F-relation R from Q 1 to Q 2 is simply a F-subset of Q 1 × Q 2 . That is, a F-relation or a F-binary relation from Q 1 to Q 2 is a membership function which assigns the grade of membership to each pair (υ 1 , υ 2 ) ∈ Q 1 × Q 2 in R. The set of all F-relations from Q 1 to Q 2 is represented by F (Q 1 × Q 2 ). Definition 3. [5] An IFS in Q is an object of the following form: represent the membership and non-membership functions, respectively, satisfying the following condition: for all υ ∈ Q. Hesitation part is defined by λ(υ) = 1 − (δ M (υ) + δ N (υ)) for each υ ∈ Q. The set of all IFSs is denoted by IF (Q).
In 1984, Atanassov [8] also generalized the concept of F-relation [2] and introduced the concept of IF-relation. Definition 4. [8] An IF-relation from Q 1 to Q 2 is an IF-subset of Q 1 × Q 2 , that is an expression of the following form: where the membership and non-membership F-relations The set of all IF-relations from Q 1 to Q 2 is denoted by IF R(Q 1 × Q 2 ). Definition 5. [26] A LDFS on Q is an object defined as follows: where π D (υ) is known to be the degree of indeterminacy of υ to £ D , and ξ(υ) is reference parameter related to the degree of indeterminacy. We shall denote the collection of all LDFSs on Q by LDF (Q).
In the following of this section, we shall recall some definitions of semigroup, semiring and hemiring. Definition 6. A non-empty set S together with an associative binary operation * defined on S is called a semigroup. It is usually denoted by the pair (S, * ).

Definition 7.
A semigroup (S, * ) is called: (1) monoid, if there exists an element e ∈ S such that e * a = a * e = a for all a ∈ S.
(2) idempotent, if a * a = a for all a ∈ S. Definition 8. A non-empty set R with two binary operations +, and · is called a semiring, if (1) (3) Multiplication is distributive over addition from both sides, that is, for all a, b, c ∈ R. We shall denote a semiring with two binary operations +, · by (R, +, ·). (4) A semiring (R, +, ·) is called commutative, if (R, ·) is commutative semigroup, that is, a · b = b · a for all a, b ∈ R. (5) A semiring (R, +, ·) is said to have an identity element e, if for any a ∈ R a · e = e · a for some e ∈ R. (6) A semiring (R, +, ·) is said to have a zero element 0, if for any a ∈ R (i) a + 0 = 0 + a = a.

Linear Diophantine Fuzzy Relation (Ldf-Relation)
We know that the binary relations are just the subsets of the cartesian product of two universes and they play a vital role in both pure and applied sciences. To extend the existing notion of IF-relation, we applied the notion of LDFS [26] to binary relations which removes the restrictions of IF-relations on membership and non-membership F-relations. In this regard, a new concept of LDF-relation is introduced in the motivation of Riaz and Hashmi's work [26] only with the addition of reference parameters corresponding to membership and non-membership F-relations respectively.

Definition 10.
A LDF-relation R D from Q 1 to Q 2 is an expression of the following form: are denoting the membership, and non-membership F-relations from Q 1 to Q 2 , respectively, and α(υ 1 , υ 2 ), β(υ 1 , υ 2 ) ∈ [0, 1] are the corresponding reference parameters to δ M R D (υ 1 , υ 2 ) and δ N R D (υ 1 , υ 2 ) respectively. These membership and non-membership F-relations satisfy the condition For an LDF-relation from Q 1 to Q 2 , we shall use for the sake of simplicity. The F-relation π D : Q 1 × Q 2 → [0, 1] associated with each LDF-relation 1, where The number π D (υ 1 , υ 2 ) is an index (a degree) of hesitation wether υ 1 and υ 2 are the relation R D or not, and γ D (υ 1 , υ 2 ) is the reference parameter of degree of hesitation. We shall denote the set of all LDF-relations from Q 1 to Q 2 by LDF R(Q 1 × Q 2 ).
By Definition 10, an LDF-relation R D is simply an LDFS on Q 1 × Q 2 .

Remark 1.
(i) Since every binary relation is a F-relation and every F-relation is an IF-relation with non-zero membership grade and zero non-membership grade. For parametric values α(υ 1 , Hence, every IF-relation is also an LDF-relation. However, the converse is not true in general as it is proved in case of LDFSs [26], page 5423. (ii) If the reference parameters α(υ 1 , υ 2 ), β(υ 1 , υ 2 ) ∈ [0, 1] do not satisfy the condition The Definition 10 of LDF-relation can be extended to n−universal sets Q 1 × Q 2 × ... × Q n in similar manners.
In the motivation of the matrix notation of F-relations defined in [30], the matrix notation of LDF-relations is defined below.
Then, an LDF-relation R D can be represented in the form of four matrices as follows: Or in the form of one matrix as follows: Since an LDF-relation is an LDFS on Q 1 × Q 2 , they have the same set-theoretic operations as LDFSs.
Proof. The proof is straightforward in view of Definition 12.
As an illustration of the Definition 12, we present the following example.
The LDF-relations R D and P D from Q 1 to Q 2 are defined in Table 1 and Table 2, respectively.   After simple calculations, the union R D ∪ P D is obtained in the Table 3.
Their intersection R D ∩ P D is given in Table 4. Table 4.
Further, LDF-relation P −1 D from Q 2 to Q 1 is calculated in Table 5.  In addition, R c D is presented in Table 6.
Proof. The proof is very easy in view of Definition 12.
Definition 13. In LDF R(Q 1 × Q 2 ), we denote and define full LDF-relation, and null LDFrelation as follows: As a direct consequence of the Definition 12 (2) and (3), and Definition 13, we get the following result.
. Then, the following properties hold: The above Proposition 3 is very important which yields to the following algebraic structure (see Corollary 1).
The next result is very important which gives rise to some other algebraic structures.
From Proposition 4, we have the following corollary.
The above Corollary 2 gives rise to the following result.
In the motivation of the composition of F-relations [2,30], we define the composition of two LDF-relations and study some of its important properties in the sequel of this manuscript.
be an LDF-relation from Q 2 to Q 3 . We denote and define their composition as follows: for all (υ 1 , υ 3 ) ∈ Q 1 × Q 3 .

Proposition 5.
With the same notations as in Definition 14, we have R D• P D ∈ LDF R(Q 1 × Q 3 ).

Theorem 1.
With the same assumptions as in the above Proposition 5, the following assertion hold: According to the Definition 12 (4), and Definition 14, υ 1 ). This completes the proof.
Proof. (1)The proof is straightforward in view of Definition 12 (1) and 14.
The following Theorem 4, informs us that LDF-relations satisfies the associative laws with respect to the composition defined in Definition 14.
Proof. Let υ 1 ∈ Q 1 , υ 4 ∈ Q 4 . Then, by Definition 14 According to the Definition 14, In the following two results, the distributive laws of union and intersection over composition are proved.
. Then, the following properties hold: Proof. The proof is similar to the proof of Theorem 5.
Theorem 4, 5, and 5 giving rise the following algebraic structures. (1) semiring with identity element1 D ∈ LDF R(Q 1 × Q 1 ) and zero element 0 D ∈ LDF R(Q 1 × Q 1 ). (2) hemiring with zero element0 D ∈ LDF R(Q 1 × Q 1 ). Now we define the concept of an equivalence LDF-relation. Let us assume that is an LDF-relation on a Q. If |Q| = n, where |.| denotes the number of elements, and where i, j = 1, 2, ..., n. Then LDF −relation R D is reflexive, if: Since a relation is symmetric, if and only if its matrix is the same as its transpose. So, R D is symmetric, if and only if, The LDF-relation R D is said to be an equivalence LDF-relation, if R D is reflexive, symmetric and transitive.
For illustration, we construct the following Example. Then, it can be easily seen that R D is an equivalence LDF-relation.

Application of LDF-Relations in Decision Making (DM)
Since LDF-relations are LDFSs, so its applications can be found in the field of AI, engineering, medical, DM and MADM [26]. DM as an abstract technique results best alternative among various choices. In this section, an algorithm is produced to solve some DM problems by utilizing the concept of LDF-relation, in the motivation of Naeem et al. [19], which is supported by a numerical example.
First, we define the score function on LDF-relations, in the motivation of Riaz et al. [26]. υ 2 ) >) be a LDF-relation from Q 1 to Q 2 . Define the score function on R D by a map given as follows: Now, we propose an Algorithm 1 to DM approach in view of LDF-relations as follows:
(2) Compute the LDF-relations R D from Q 1 to Q 2 , and P D from Q 2 to Q 3 .
(3) Perform the composition operation• among R D and P D , that is, R D• P D .
(4) Compute the error or hesitation values of according to Definition 10, that is, , and θ ik = α ij• α jk , θ ik = β ij• β jk , and 1 ≤ i ≤ |Q 1 |, 1 ≤ j ≤ |Q 2 |, and 1 ≤ k ≤ |Q 3 |, where |Q l |, l = 1, 2, 3, represents the number of elements of |Q l |. (5) Compute the association grades among the elements of the sets Q 1 and Q 3 by using Find out the pair (q i , q k ), where q i ∈ Q 1 , q k ∈ Q 3 having the maximum association grade valueÄ ik . (7) Decision: The pair (q i , q k ) is the optimal choice.
To explain the above algorithm, the following example is elaborated.

Example 3.
Suppose that a person Mr. X wants to purchase a new brand one canal double story bungalow and the property dealer visited four bungalows Q 1 = {u 1 , u 2 , u 3 , u 4 } as per his requirement Q 2 = {l 1 = near to play ground, l 2 = near to park , l 3 = near to main service road } in reasonable price, where the set of prices is Q 3 = {p 1 = low , p 2 = medium , p 3 = high }. Now, we consider an LDF-relation R D from Q 1 to Q 2 which describes the location of bungalows in a certain membership and non-membership degree functions δ M R D , and δ M R D together with the parametric values α = good location and β =not good location, to the locations, respectively, in the Table 7.  In addition, we consider the LDF-relation P D from Q 2 to Q 3 which describes the relationship among the locations of bungalows and their prices by the membership and non-membership Frelations δ M P D , δ N P D together with parametric values α = reasonable price, β = not reasonable price in Table 8.  By simple calculations of the composition 14, LDF-relation R D• P D from Q 1 to Q 3 given in Table 9 describes the relationship among the bungalows and their prices according to the locations.  Now, by using the Definition 19, hesitation degrees η ik = 1 − (γ ik θ M ik + γ ik θ N ik ) of R D• P D are given in Table 10. Next, the association grades among objects of Q 1 and Q 3 by using the formulaë A ik = θ M ik − θ N ik η ik are given in Table 11. Clearly, the pair (u 4 , p 2 ) have the highest association grade. Thus, u 4 is the optimal choice for Mr. X to purchase property in good location and reasonable price. For confirmation of our result, we calculate the score values among the objects of Q 1 and Q 3 by using the Definition 19 are computed in Table 12. It can be easily seen in the last row the pair (u 4 , p 2 ) has the highest score value. Thus, our decision is true. Hence, our results are valid, and thus our proposed algorithm is a reliable method.

Conclusions
Binary relations play an important role in various fields of pure and applied sciences. This manuscript is devoted to studying the concept of LDF-relation in the motivation of Riaz and Hashmi's work. This new concept of LDF-relation removes the limitations of IF-relation and enhances the space of membership and non-membership grades by adding the reference or control parameters. Some primary operations are defined and certain important results are established. With the help of these operations, it is investigated that the set of all LDF-relations give rise to some algebraic structures namely, semigroup, semiring and hemiring. Moreover, the concept of score function on an LDF-relation is introduced. Moreover, the notion of score function of LDF-relations is introduced to analyze the symmetry of the optimal decision and ranking of feasible alternatives. As an application of proposed LFD-relations in DM, an algorithm is rendered together with a numerical example. In future studies, this new work may be applied to various directions of MCDM and rough set theory using different hybrid techniques, for further research work. LDF-relation comes up with a rigorous mathematical model for modeling uncertainties in decision-making problems, including AI, robotics, machine learning, medical analysis, medicine, economics, and many other real life problems. We hope that the proposed model of LDF-relations and all the ideas in this paper shall exist as an establishment for LDFS theory and will lead to new fruitful results.