Neutrosophic Parametrized Soft Set Theory and Its Decision Making

: In this work, we present definition of neutrosophic parameterized (NP) soft set and its operations. Then we define NP-aggregation operator to form NP-soft decision making method which allows constructing more efficient decision processes. We also dive an example which shows that they can be successfully applied to problem that contain indeterminacy.

In 1999 a Russian researcher [27] firstly gave the soft set theory as a general mathematical tool for dealing with uncertainty and vagueness and how soft set theory is free from the parameterization inadequacy syndrome of fuzzy set theory, rough set theory, probability theory. Then, many interesting results of soft set theory have been studied on fuzzy soft sets [8,12,23], on intuitionistic fuzzy soft set theory [14,25], on possibility fuzzy soft set [7], on generalized fuzzy soft sets [26,29], on generalized intuitionistic fuzzy soft [6], on interval-valued intuitionistic fuzzy soft sets [20], on intuitionistic neutrosophic soft set [9], on generalized neutrosophic soft set [10], on fuzzy parameterized soft set theory [17,18], on fuzzy parameterized fuzzy soft set theory [13], on intuitionistic fuzzy parameterized soft set theory [15], on IFP−fuzzy soft set theory [16],on neutrosophic soft set [24].interval-valued neutrosophic soft set [11,19].
In this paper our main objective is to introduce the notion of neutrosophic parameterized soft set which is a generalization of fuzzy parameterized soft set and intuitionistic fuzzy parameterized soft set.The paper is structured as follows. In section 2, we first recall the necessary background on neutrosophic and soft set. In section 3, we give neutrosophic parameterized soft set theoryand their respective properties. In section 4, we present a neutrosophic parameterized aggregation operator. In section 5,a neutrosophic parameterized decision methods is presented with example. Finally we conclude the paper.

Preliminaries
Throughout this paper, let U be a universal set and E be the set of all possible parameters under consideration with respect to U, usually, parameters are attributes, characteristics, or properties of objects in U.
We now recall some basic notions of neutrosophic set and soft set. For more details, the reader could refer to [33,37]. Definition 1.
[37] Let U be a universe of discourse then the neutrosophic set A is an object having the form where the functions , , : U→] − 0,1 + [ define respectively the degree of membership, the degree of indeterminacy, and the degree of non-membership of the element x ∈ X to the set A with the condition. Let U be an initial universe set and E be a set of parameters. Let P(U) denotes the power set of U. Consider a nonempty set A, A ⊂ E. A pair (K, A) is called a soft set over U, where K is a mapping given by K: A → P(U).
As an illustration, let us consider the following example.
Example 2.Suppose that U is the set of houses under consideration, say U= { , , }. Let E be the set of some attributes of such houses, say E = { , , , }, where , , , stand for the attributes "beautiful", "costly", "in the green surroundings", "moderate" and technically, respectively. In this case, to define a soft set means to point out expensive houses, beautiful houses, and so on. For example, the soft set (K, A) that describes the "attractiveness of the houses" in the opinion of a buyer, says Thomas, and may be defined like this:

Neutrosophic Parameterized Soft Set Theory
In this section, we define neutrosophic parameterized soft set and their operations.

NP-aggregation operator
In this section, we define NP-aggregation operator of an NP-soft set to construct a decisionmethod by which approximate functions of a soft set are combined to produce a single neutrosophic set that can be used to evaluate each alternative.

NP-Decision Methods
Inspired by the decision making methods regard in [12][13][14][15][16][17][18][19].In this section, we also present NPdecision method to neutrosophic parameterized soft set. Based on definition 4.1 and 4.2 we construct an NP-decision making method by the following algorithm. Now, we construct a NP-soft decision making method by the following algorithm to produce a decision fuzzy set from a crisp set of the alternatives.
According to the problem, decision maker Step v .Finally, DM chooses for the position from since it has the maximum degree 0.125 among the others.

Conclusion
In this work, we have introduced the concept of neutrosophic parameterized soft set and studied some of its properties. The complement, union and intersection operations have been defined on the neutrosophic parameterized soft set. The definition of NP-aggregation operator is introduced with application of this operation in decision making problems.