A DECISION MAKING APPROACH BASED ON WEIGHTED FUZZY SOFT SET

This paper is aimed at developing an approach of a real life decision making problem with respect to an weighted fuzzy soft set with preference. This paper introduces weighted fuzzy soft set and studies some of its properties. This paper also enquires about the relations on weighted fuzzy soft sets. Finally, a real life decision making problem in weighted fuzzy soft set is proposed.


INTRODUCTION
Many real life problems in society, economics, management and engineering are uncertain and imprecise. To dealing with uncertainty there are many theories, such as probability theory, fuzzy set theory [2], intuitionistic fuzzy set theory [3], interval mathematics, rough set theory [4] etc. But all these theories have their own difficulties due to lack of parametrization of the theories. Molodtsov [1] proposed the notion of soft set theory to deal with the intrinsic flows of the above mention theories. Maji et al. [7,8] continued the study and gave first practical example of the soft set in decision making problem by constructing comparison table. It is observed that the application of soft set theory is in various areas, such as forecasting [5], decision We write (S, A)∪(T , B) = (J,C).
Definition 2.6. [9]. For two fuzzy soft sets (S, A) and (T , B) over a common universe U, we say that (S, A) is a fuzzy soft subset of (T , B) if It is denoted by (S, A)⊂(T , B).
if * satisfies the following conditions: (i) * is commutative and associative , Some examples of continuous t-norm are: Some examples of continuous t-conorm are:

WEIGHTED FUZZY SOFT SETS
Throughout this paper, U be the initial universe, and P be the set of parameters related to the elements in U, and A, B,C ⊆ P and α, β , γ are the fuzzy subsets of A, B,C respectively. Then consider the mappingF α (p) : is called a weighted fuzzy soft set over (U, P) We now defineF α as follows: Then clearlyF α will be called weighted fuzzy soft set. (i) α is a fuzzy subset of β , Example 3.4. SupposeF β be a weighted fuzzy soft set over the same universe as in the previous example andḠ β be defined as the following manner: are same as in the previous exmple and p 4 stands for 'good academic score'. Now if we takeF α as in the previous example andḠ β then we see that F α⊆Ḡβ . i.e.F α is a weighted fuzzy soft subset ofḠ β Definition 3.5. LetF α andḠ β be two weighted fuzzy soft sets over the soft set universe (U, P). Then the intersection ofF α andḠ β is denoted byF α∩Ḡβ and is defined by a weighted Example 3.6. Consider two weighted fuzzy soft setsF α andḠ β in the previous example and Definition 3.7. LetF α andḠ β be two weighted fuzzy soft sets over the soft set universe (U, P). Then the union ofF α andḠ β is denoted byF α∪Ḡβ and is defined by a weighted fuzzy where µH (p) (x) = µF (p) (x) µḠ (p) (x) and γ(p) = α(p) β (p) Proposition 3.9. LetF α ,Ḡ β andH γ be any three weighted fuzzy soft sets over (U, P). Then Proof. The proof is obvious as t-norm function and t-conorm functions are commutative and associative.

RELATIONS ON WEIGHTED FUZZY SOFT SETS
Throughout this sectionF α : A → F(U) andḠ β : B → F(U) two weighted fuzzy soft sets over (U, P).
Definition 4.1. SupposeF α andḠ β be two weighted fuzzy soft sets over (U, P). By a weighted fuzzy soft relation R fromF α toḠ β we mean a function R : Definition 4.2. Let R be a weighted fuzzy soft relation fromF α toḠ β . Then the inverse relation of R fromḠ β toF α is denoted by R −1 and is defined by If R is a weighted fuzzy soft relation fromF α toḠ β then R −1 is also a weighted fuzzy soft relation fromḠ β toF α .  a, b).
Definition 4.4. Let R 1 and R 2 be two weighted fuzzy soft relations fromF α toḠ β andḠ β toH γ respectively. Then the composition of two weighted fuzzy soft relations fromF α toH γ is denoted by R 1 • R 2 and defined as ( is also a weighted fuzzy soft relation.
Proof. We know that Hence R 1 • R 2 is a weighted fuzzy soft relation fromF α toH γ .

MAKING PROBLEM
In our daily life based problem, we can use application of weighted fuzzy soft set. Suppose a person wants to buy a car depending on some parameters. Here we can help the person to choose the best car for the person according to his preference. Let us consider a weighted fuzzy soft setF α and it's approximation is given below. (iv) Compute the weighted score of v i , ∀ i in tabular form.

Suppose initially the person shortlisted six cars
(v) If the maximum score occurs in k-th row then that person will buy the car v k .
(vi) If k has more than one value then one of v k may be chosen.  Thus here we see that the score of the car v 6 is maximum. So the person will buy the car v 6 otherwise his next better option will be v 3 . It is obvious that for different problem we use different type of lavel soft sets, the choice depends on the nature of data under consideration and nature of problem under consideration.