Neutrosophic soft sets and neutrosophic soft matrices based on decision making

Maji\cite{maj-13}, firstly proposed neutrosophic soft sets can handle the indeterminate information and inconsistent information which exists commonly in belief systems. In this paper, we have firstly redefined complement, union and compared our definitions of neutrosophic soft with the definitions given by Maji. Then, we have introduced the concept of neutrosophic soft matrix and their operators which are more functional to make theoretical studies in the neutrosophic soft set theory. The matrix is useful for storing an neutrosophic soft set in computer memory which are very useful and applicable. Finally, based on some of these matrix operations a efficient methodology named as NSM-decision making has been developed to solve neutrosophic soft set based group decision making problems.


Introduction
In recent years a number of theories have been proposed to deal with uncertainty, imprecision, vagueness and indeterminacy. Theory of probability, fuzzy set theory [54], intuitionistic fuzzy sets [4], interval valued intuitionistic fuzzy sets [3], vague sets [26], rough set theory [41], neutrosophic theory [46], interval neutrosophic theory [52] etc. are consistently being utilized as efficient tools for dealing with diverse types of uncertainties and imprecision embedded in a system. However, each of these theories has its inherent difficulties as pointed out by Molodtsov [39]. The reason for these difficulties is, possibly, the inadequacy of the parameterization tool of the theories. Later on, many interesting results of soft set theory have been obtained by embedding the idea of fuzzy set, intuituionstic fuzzy set, vague set, rough set, interval intuitionistic fuzzy set, intuitionistic neutrosophic set, interval neutrosophic set, neutrosophic set and so on. For example, fuzzy soft set [34], intuitionistic fuzzy soft set [17,31], rough soft set [24,25], interval valued intuitionistic fuzzy soft set [27,53,55], neutrosophic soft set [32,33], generalized neutrosophic soft set [7], intuitionstic neutrosophic soft set [8], interval valued neutrosophic soft set [20]. The theories has developed in many directions and applied to wide variety of fields such as on soft decision making [12,56], fuzzy soft decision making [18,19,30,45] ,on relation of fuzzy soft set [50,51], on relation on intuiotionstic fuzzy soft set [22,40], on relation on neutrosophic soft set [21], on relation on interval neutrosophic soft set [20] and so on.
Researchers published several papers on fuzzy soft matrices and intuitionistic fuzzy soft matrices, and it has been applying in many fields of real life scenarios(see [28,30,29,35]). Recently Cagman et al [13] introduced soft matrices and applied it in decision making problem. They also introduced introduced fuzzy soft matrices [15], Chetia and Das [11] defined intuitionistic fuzzy soft matrices with different products and properties on these products. Further Saikia et al [48] defined generalized fuzzy soft matrices with four different product of generalized intuitionstic fuzzy soft matrices and presented an application in medical diagnosis. Next, Broumi et al [9] studied fuzzy soft matrix based on reference function and defined some new operations such fuzzy soft complement matrix, trace of fuzzy soft matrix based on reference function a new fuzzy soft matrix decision method based on reference function is presented. Recently, Mondal et al [36,37,38] introduced fuzzy and intuitionstic fuzzy soft matrix and multicrita in desion making based on three basic t-norm operators. The matrices has differently developed in many directions and applied to wide variety of fields in [5,6,44,47].
Our objective is to introduce the concept of neutrosophic matrices and its applications in decision making problem. The remaining part of this paper is organized as follows. Section 2 contains basic definitions and notations that are used in the remaining parts of the paper. we investigated redefined neutrosophic soft set and some operations and compared our definitions of neutrosophic soft with the definitions given Maji [32] in section 3. In section 4, we introduce the concept of neutrosophic matrices and presente some of theirs basic properties. In section 5, we present two special products of neutrosophic matrices. In section 6, we present a soft decision making method based on and-product of neutrsophic matrice. Finally ,conclusion is made in section 7.

Preliminary
In this section, we give the basic definitions and results of neutrosophic set theory [46], soft set theory [39] and soft matrix theory [13] that are useful for subsequent discussions. 1]. It can be written as

Definition 1. [46] Let U be a space of points (objects), with a generic element in U denoted by u. A neutrosophic sets(N-sets) A in U is characterized by a truth-membership function T A , a indeterminacy-membership function I A and a falsity-membership function F
There is no restriction on the sum of T A (u); I A (u) and F A (u), so 0 ≤ supT A (u) + supI A (u) + supF A (u) ≤ 3.

Definition 2.
[39] Let U be a universe, E be a set of parameters that are describe the elements of U, and A ⊆ E. Then, a soft set F A over U is a set defined by a set valued function f A representing a mapping where f A is called approximate function of the soft set F A . In other words, the soft set is a parametrized family of subsets of the set U, and therefore it can be written a set of ordered pairs is a set called x-element of the soft set for every x ∈ E. Definition 3. [13] Let F A be a soft set over U. Then a subset of U × E is uniquely defined by which is called a relation form of F A . The characteristic function of R A is written by ., x n } and A ⊆ E, then the R A can be presented by a table as in the following form which is called an m × n s-matrix of the soft set F A over U.
From now on we shall delete the subscripts m × n of [a ij ] m×n , we use [a ij ] instead of [a ij ] m×n for i = 1, 2, ...m and j = 1, 2, ...n.
kn} for all k ∈ I = {1, 2, ..., n}. Then f s-max-min decision function, denoted Mm, is defined as follows The one column f s-matrix Mm[c ip ] is called max-min decision f s-matrix.
where f N is called approximate function of the neutrosophic soft set N. In other words, the neutrosophic soft set is a parametrized family of some elements of the set P (U), and therefore it can be written a set of ordered pairs 2. Then the union of N 1 and N 2 is denoted by N 1∪ N 2 and is defined by where the truth-membership, indeterminacy-membership and falsity-membership of N 3 are as follows: ∀u ∈ U, 3. Then the intersection of N 1 and N 2 is denoted by N 1∩ N 2 and is de- where the truth-membership, indeterminacymembership and falsity-membership of N 3 are as follows: ∀u ∈ U, Definition 12.
[23] t-norms are associative, monotonic and commutative two valued functions t that map from These properties are formulated with the following conditions:

Definition 13. [23] t-conorms (s-norm) are associative, monotonic and commutative two placed functions s which map from
These properties are formulated with the following condi- iv. s(a, s(b, c)) = s(s(a, b, c) t-norm and t-conorm are related in a sense of lojical duality. Typical dual pairs of non parametrized t-norm and t-conorm are complied below: i. Drastic product: ii. Drastic sum: iii. Bounded product: iv. Bounded sum: Einstein product: vi. Einstein sum: vii. Algebraic product: viii. Algebraic sum: ix. Hamacher product: x. Hamacher sum:

Neutrosophic soft set and some operations redefined
Notion of the neutrosophic soft set theory is first given by Maji [32]. This section, we has modified the definition of neutrosophic soft set and operations as follows. Some of it is quoted from [1,2,14,22,32,46]. Definition 14. Let U be a universe, N(U) be the set of all neutrosophic sets on U, E be a set of parameters that are describe the elements of U Then, a neutrosophic soft set N over U is a set defined by a set valued function f N representing a mapping is called x-approximation of the neutrosophic soft set N which may be arbitrary, some of them may be empty and some may have a nonempty intersection. In other words, the neutrosophic soft set is a parametrized family of some elements of the set N(U), and therefore it can be written a set of ordered pairs, Definition 15. Let N 1 and N 2 be two neutrosophic soft sets. Then, the complement of a neutrosophic soft set N 1 denoted by N c 1 and is defined by Definition 16. Let N 1 and N 2 be two neutrosophic soft sets. Then, the union of N 1 and N 2 is denoted by N 3 = N 1∪ N 2 and is defined by Definition 17. Let N 1 and N 2 be two neutrosophic soft sets. Then, the intersection of N 1 and N 2 is denoted by N 4 = N 1∩ N 2 and is defined by and here; Let us consider the t-norm min{a, b} and s-norm max{a, b}. Then, Proof. The proofs can be easily obtained since the t-norm function and s-norm functions are commutative and associative.

Comparision of the Definitions
In this subsection, we compared our definitions of neutrosophic soft with the definitions given Maji [32] by inspiring from [14].
Let us compare our definitions of neutrosophic soft with the definitions given Maji [32] in Table 1.
In this paper our approach in Maji Table 1 Let us compare our complement definitions of neutrosophic soft with the definitions given Maji [32] in Table 2.
In this paper our approach in Maji Let us compare our union definitions of neutrosophic soft with the definitions given Maji [32] in Table 2.
In this paper our approach in Maji Table 2 Let us compare our intersection definitions of neutrosophic soft with the definitions given Maji [32] in Table 2.
In this paper our approach in Maji

Neutrosophic Soft Matrices
In this section, we presented neutrosophic soft matrices which are representative of the neutrosophic soft sets. The matrix is useful for storing an neutrosophic soft set in computer memory which are very useful and applicable. Some of it is quoted from [13,15,5].
This section are an attempt to extend the concept of soft matrices [13], fuzzy soft matrices [15], intuitionistic fuzzy soft matrices [5]. (N, E). The characteristic function of R N is written by

Definition 18. Let N be an neutrosophic soft set over N(U). Then a subset of N(U) × E is uniquely defined by
, which is called an m × n neutrosophic soft matrix (or namely NS-matrix) of the neutrosophic soft set N over N(U).
According to this definition, an a neutrosophic soft set N is uniquely characterized by matrix [a ij ] m×n . Therefore, we shall identify any neutrosophic soft set with its soft NS-matrix and use these two concepts as interchangeable. The set of all m × n NS-matrix over N(U) will be denoted by N m×n . From now on we shall delete th subscripts m × n of [a ij ] m×n , we use [a ij ] instead of [a ij ] m×n , since [a ij ] ∈ N m×n means that [a ij ] is an m × n NS-matrix for i = 1, 2, . . . , m and j = 1, 2, . . . , n.
for all i and j. iii for all i and j.
for all i and j. ii.

Products of NS-Matrices
In this section, we define two special products of NS-matrices to construct soft decision making methods.
The one column f s-matrix Mm[c ip ] is called max-min decision f s-matrix.

Algorithm
The algorithm for the solution is given below Step 1: Choose feasible subset of the set of parameters.
Step 2: Construct the neutrosophic matrices for each parameter.
Step 3: Choose a product of the neutrosophic matrices , Step 4: Find the method min-max-max decision N-matrices.
Step 5: Find an optimum fuzzy set on U.
Remark 1. We can also define NS-matrices max-min-min decision making methods. One of them may be more useful than the others according to the type of problem.
Case study: Assume that , a car dealer stores three different types of cars U = {u 1 , u 2 , u 3 } which may be characterize by the set of parameters E = {e 1 , e 2 } where e 1 stands for costly , e 2 stands for fuel effciency. Then we consider the following example. Suppose a couple Mr. X and Mrs. X come to the dealer to buy a car before Dugra Puja. If each partner has to consider his/her own set of parameters, then we select the car on the basis of partner's parameters by using NS-matrices min-max-max decision making as follow.
Step 1: First Mr.X and Mrs.X have to chose the sets of their parameters A = {e 1 , e 2 } and B = {e 1 , e 2 }, respectively.
Step 2:Then we construct the NS-matrices [a ij ] and [b ij ] according to their set of parameters A and B, respectively, as follow: where s i = µ 11 − ν 11 .w 11 Step 5:Finally , we can find an optimum fuzzy set on U as: ] (U) = {u 1 /0.95, u 2 /0.93, u 3 /0.92} Thus u 1 has the maximum value. Therefore the couple may decide to buy the car u 1 .

Conclusion
In this paper we have redefine the notion of neutrosophic set in a new way and proposed the concept of neutrosophic soft matrix and after that different types of matrices in neutrosophic soft theory have been defiend.then we have introduced some new operations and properties on these matrices.