On Fuzzy Soft Matrix Based on Reference Function

— In this paper we study fuzzy soft matrix based on reference function.Firstly, we define some new operations such as fuzzy soft complement matrix and trace of fuzzy soft matrix based on reference function.Then, we introduced some related properties, and some examples are given. Lastly, we define a new fuzzy soft matrix decision method based on reference function.


INTRODUCTION
Fuzzy set theory was proposed by LotfiA.Zadeh [1] in 1965,where each element ( real valued ) [ 0, 1] had a degree of membership defined on the universe of discourse X, the theory has been found extensive application in various field to handle uncertainty.Therefore,several researches were conducted on the generalization on the notions of fu zzy sets such as intuitionistic fuzzy set proposed by Atanassov [2,3] , interval valued fuzzy set [5 ] .In the literature we found many well -known theories to describe uncertainty: rough set theory [6].. etc, but all of these theories have their inherit difficulties as pointed by Molodtsov in his pioneer wo rk [7] .The concept introduced by Molodtsov is called "soft set theory" which is set valued mapping.This new mathematical model is free fro m the difficult ies mentioned above.Since its introduction, the concept of soft set has gained considerable attention and this concept has resulted in a series of wo rk [8,9,10,11,12,13,14] .Also as we know, mat rices play an important ro le in science and technology.However, the classical mat rix theory sometimes fails to solve the problems involving uncertainties,occurring in an imprecise environ ment.In [4] Thomason, introduced the fuzzy mat rices to represent fuzzy relat ion in a system based on fuzzy set theory and discussed about the convergence of powers of fuzzy matrix.In [15,16,17] ,some important results on determinant of a square fuzzy mat rices are discussed .Also,Ragab et al. [18,19] presented some properties of the min-ma x composition of fuzzy matrices.Later on, several studies and some applications of fuzzy matrices are defined in [20,21].
In 2010,Cagmanet al [13] defined soft matrix wh ich is representation of soft set, to make operations in theoretical studies in soft set more functional.Th is representation has several advantages, it's easy to store and manipulate matrices and hence the soft sets represented by them in a co mputer.
Recently severalresearch have been studied the connection between soft set and soft matrices [ 13,14,22] .Later,Maji et al [9 ] introduced the theory of fuzzy soft set and applied it to decision making problem.In 2011, Yang and C.Ji [22] ,defined fuzzy soft matrix (FSM) which is very useful in representing and computing the data involving fuzzy soft sets.
The concept of fuzzy set based on reference function was first introduced by Baruah [23,24,25] in the following manner -According to him, to define a fuzzy set, two functions namely fuzzy membership function and fuzzy reference function are necessary.Fuzzy membership value is the difference between fuzzy membership function and reference function.Fuzzy membership function and fuzzy membership value are two different things.In [26,27] M.Dhar applied this concept to fuzzy square matrix and developed some interesting properties as determinant, trace and so on.Thereafter, in [28] , T.J.Neog, D. K.Sutwere extended this new concept to soft set theory, introducing a new concept called "fuzzy soft set based on fuzzy reference function".Recently,Neog .T.J, Sut D. K,M .Bora [29] combinedfu zzy soft set based on reference function with soft matrices.The paper unfolds as follows.The next section briefly introduces some definit ions related tosoft set,fuzzy soft set, and fuzzy soft setbased on reference function.Section 3 presents fuzzy soft complement matrix based on reference function.Sect ion 4presentstrace of fuzzy soft matrix based on reference function..Section5presentsnew fuzzy soft mat rix theory in decision making.Conclusions appear in the last section.

II. PRELIMINA RIES
In this section first we review some concepts and definitions of soft set,fuzzy soft set, and fuzzy soft set based on reference functionfrom [9,12,13,29] , which will be needed in the sequel.

Remark:
For the sake of simplicity we adopt the following notation of fuzzy soft set based on reference function defined in our way as: Fuzzy soft set based on reference =(F, A) rf To make the difference between the notation (F, A) defined for classical soft set or its variants as fuzzy soft set. [13])

2.1.Definition (Soft Set
Suppose that U is an initial universe set and E is a set of parameters, let P(U) denotes the power set of U.A pair(F,E) is called a soft set over U where F is a mapping given by F: E→P(U).Clearly, a soft set is a mapping from parameters to P(U),and it is not a set, but a parameterized family of subsets of the universe.

Example.
Suppose that U={s1,s2,s3,s4} is a set of students and E={e1,e2,e3} is a set of parameters, which stand for result, conduct and sports performances respectively.Consider the mapping from parameters set E to the set of all subsets of power set U.Then soft set (F,E) describes the character of the students with respect to the given parameters, for finding the best student of an academic year.

Definition (FuzzySoft Set
Let U be an initial universe set and E be the set of parameters.Let A⊆E .A pair (F,A) is called fuzzy soft set over U where F is a mapping given by F: A→F u ,whereF u denotes the collection of all fu zzy subsets of U.
Then the operations intersection and union are defined asA (

.Defintion [29] (Fuzzy soft matrices (FSMs) based on reference function)
Let U be an initial universe, E be the set of parameters and A ⊆ E. Let (  , E) be fu zzy soft set (FS) over U. Then a subset of U ×E is uniquely defined by   = {(u, e); e ∈ A, u∈   ()} which is called a relation form of (  , E).

Example
Let U={ 1 , 2 , 3 , 4 } be the un iversal set and Eb e the set of parameters g iven by E={ 1 , 2 , 3 } We cons ider th e fu zzy soft sets based on reference funct ion.
The fu zzy soft mat rices based on reference funct ion rep resenting these t wo fu zzy soft sets are respect ively

III. FUZZY SOFT COMPLEM ENT MATRIX BASED ON REFERENCE FUNCTION
In this section ,westart by introducing the notion of the fuzzy soft complement matrix based on reference function,and we prove some formal properties.

Definition
Let A= �(a ij , 0)� m ×n ∈ FSM m ×n according to the definition in [26], then A c is calledfuzzy soft complement matrix if

Proof:
To show (i) we have The proof of (ii) follows similar lines as above.

IV. TRA CE OF FUZZY SOFT MATRIXBASED ON REFERENCE FUNCTION
In this section we extend the concept of trace of fuzzy square matrix proposed M. Dhar [26] to fuzzy soft square matrix based on reference function, and we prove some formal properties.

Definition
Let A be a square matrix.Then the trace ofthe matrix A is denoted by tr A and is defined as: where μ ii stands for the membership functions lying along the principal diagonal and r ii refers to the reference function of the corresponding membership functions.Then we con clud e that the cand idate   is selected for the post.

Proposition
Incase max   occurs for more than one v alu e, then repeatthe pro cess by reassessing the parameters.

Case S tudy
Let (F,E) and (G,E) be t wo fu zzy so ft set based on reference funct ion rep resenting the select ion o f four cand idat es fro m the un iversal set U= {c 1 ,c 2 ,c 3 ,c 4 } by the experts X,and Y.Let E = {e 1 , e 2 ,e 3 } be the set of parameters wh ich stand fo r intelligence,innovat ive and analysis .

VI. CONCLUSIONS
In ou r work, we hav e put fo rward so me new concepts such as co mp lement , t race o f fu zzy soft mat rix based on reference fun ction .So me related properties have been established with examp le.Finally an applicat ion of fu zzy soft mat rix based on reference funct ion in decis ion making p roblem is given .It 's hoped that ou r work will enhance th is study in fu zzy soft mat rix.