On Fuzzy Soft Matrix Based on Reference Function

I.J. Information Engineering and Electronic Business, 2013, 2, 52-59 Published Online August 2013 in M ECS (http://www.mecs-press.org/) DOI: 10.5815/ ijieeb.2013.02.08

On Fuzzy Soft Matrix Based on Reference Function Said Brou mi Facu lt y o f A rts an d Hu man it ies, Hay El Baraka Ben M's ik Casab lanca B.P. 7951, HassanII Un ivers ity Moh ammed ia-Casab lanca, Mo ro cco 1 broumisaid78@g mail.co m Florentin Smarandache Depart ment of Mathemat ics, University of New Mexico, 705 Gurley Avenue, Gallup, NM 87301, USA fsmarandache@gmail.co m Mamoni Dhar Depart ment of Mathemat ics, Science College, Kokrajhar-783370, Assam, India mamon idhar@g mail.co m

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

Index Terms—Soft set, fuzzy soft set, fuzzy soft set based on reference function, fuzzy soft matrix based on reference function.

I.

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 Copyright © 2012 M ECS

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-max 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

I.J. In formation Engineering and Electronic Business, 2013, 2, 52-59

On Fu zzy Soft Matrix Based on Reference Function

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.

2.1.Definition (Soft Set [13]) 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.

2.2. 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.

(F, E) = {{result = s1, s3, s4} {conduct = s1,s2 } {sports performances = s2,s3,s4 }} 2.3. Definition (FuzzySoft Set [9, 12] ) 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. 2.4.Example.

Consider the example2.2,in soft set(F,E),if s1 is mediu m in studies, we cannot expressed with only the two numbers 0 and 1,we can characterize it by a membership function instead of the crisp number 0 and 1,which associates with each element a real number in the interval [0,1].Then fuzzy soft set can describe as (F, A)={F(e1) = {(s1,0.9), (s2,0.3), (s3,0.8), (s4,0.9)}, F(e2) = {(s1,0.8), (s2,0.9), (s3,0.4), (s4,0.3)}},where A={ e1,e2}.

In the following, Neog et al. [29] showed by an example Copyright Š 2012 M ECS

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that this definition sometimes gives degenerate cases and revised the above definition as follows: 2.5 .Definition [29] Let A (Îź1 ,Îź2 )={x,Îź1 (x) , Îź2 (x) ; x ∈ U } and B (Îź3 ,Îź4 ) ={x, Îź3 (x), Îź4 (x) ; x ∈ U } be two fuzzy sets defined over the same universe U. Then the operations intersection and union are defined asA ( Îź1 , Îź2 ) â‹‚ B ( Îź3 , Îź4 )= {x, min(Îź1 (x),Îź3 (x)) ,max( Îź2 (x) ,Îź4 (x)) ; x ∈ U } and A {x, ( Îź1 , Îź2 ) ⋃ B ( Îź3 , Îź4 )= max(Îź1 (x) ,Îź3 (x)) ,min( Îź2 (x),Îź4 (x)) ; x ∈ U } 2.6.Definition [29] Let A ( đ?œ‡đ?œ‡ 1 ,đ?œ‡đ?œ‡ 2 )={x, đ?œ‡đ?œ‡ 1 (đ?‘Ľđ?‘Ľ) , đ?œ‡đ?œ‡ 2 (đ?‘Ľđ?‘Ľ) ; x ∈ U } and B (đ?œ‡đ?œ‡ 3,đ?œ‡đ?œ‡ 4 )={x, đ?œ‡đ?œ‡ 3 (đ?‘Ľđ?‘Ľ) , đ?œ‡đ?œ‡ 4 (đ?‘Ľđ?‘Ľ) ; x ∈ U } be two fuzzy sets defined over the same universe U.To avoid degenerate cases we assume that min( đ?œ‡đ?œ‡ 1 (đ?‘Ľđ?‘Ľ) , đ?œ‡đ?œ‡ 3 (đ?‘Ľđ?‘Ľ) ) ≼ max( đ?œ‡đ?œ‡ 2 (đ?‘Ľđ?‘Ľ) ,đ?œ‡đ?œ‡ 4 (đ?‘Ľđ?‘Ľ)) for all x∈ U. Then the operations intersection and union are defined as A ( Îź1 , Îź2 ) â‹‚ B ( Îź3 , Îź4 )= {x, min(Îź1 (x),Îź3 (x)) ,max( Îź2 (x) ,Îź4 (x)) ; x ∈ U } and A ( đ?œ‡đ?œ‡ 1 , đ?œ‡đ?œ‡ 2 ) ⋃ B ( đ?œ‡đ?œ‡ 3 , đ?œ‡đ?œ‡ 4 )= {x, max(đ?œ‡đ?œ‡ 1 (đ?‘Ľđ?‘Ľ) ,đ?œ‡đ?œ‡ 3 (đ?‘Ľđ?‘Ľ)) ,min( đ?œ‡đ?œ‡ 2 (đ?‘Ľđ?‘Ľ) ,đ?œ‡đ?œ‡ 4 (đ?‘Ľđ?‘Ľ)) ; x ∈ U }

2.7. Definition[29] For usual fuzzy setsA (đ?œ‡đ?œ‡, 0)={x, đ?œ‡đ?œ‡(đ?‘Ľđ?‘Ľ),0 ; x ∈ U }and B (1, đ?œ‡đ?œ‡)={x, 1 , đ?œ‡đ?œ‡(đ?‘Ľđ?‘Ľ) ; x ∈ U } defined over the same universe U, we have A ( đ?œ‡đ?œ‡ , 0 ) â‹‚ B (1, đ?œ‡đ?œ‡ )= {x, min((đ?‘Ľđ?‘Ľ), 1 ) ,max(0 ,đ?œ‡đ?œ‡(đ?‘Ľđ?‘Ľ)) ; x ∈ U }= {x, đ?œ‡đ?œ‡(đ?‘Ľđ?‘Ľ), đ?œ‡đ?œ‡(đ?‘Ľđ?‘Ľ) ; x ∈ U }, which is nothing but the null setđ?œ‘đ?œ‘ and A (đ?œ‡đ?œ‡, 0 ) ⋃ B (1, đ?œ‡đ?œ‡)= {x, max(đ?œ‡đ?œ‡(đ?‘Ľđ?‘Ľ), 1) ,min( 0, đ?œ‡đ?œ‡(đ?‘Ľđ?‘Ľ)) ; x ∈ U } = {x,1, 0 ; x ∈ U }, which is nothing but the universal set U. This means if we define a fuzzy set(A (đ?œ‡đ?œ‡, 0) ) đ?’„đ?’„ ={x, 1, đ?œ‡đ?œ‡(đ?‘Ľđ?‘Ľ) ; x ∈ U } it is nothing but the complement ofA (đ?œ‡đ?œ‡, 0)={x, đ?œ‡đ?œ‡(đ?‘Ľđ?‘Ľ) ,0 ; x ∈ U }.

2.8. Definition[29] Let A (đ?œ‡đ?œ‡ 1,đ?œ‡đ?œ‡ 2 )={x, đ?œ‡đ?œ‡ 1 (đ?‘Ľđ?‘Ľ) , đ?œ‡đ?œ‡ 2 (đ?‘Ľđ?‘Ľ) ; x ∈ U } and B (đ?œ‡đ?œ‡ 3,đ?œ‡đ?œ‡ 4)={x, đ?œ‡đ?œ‡ 3 (đ?‘Ľđ?‘Ľ) , đ?œ‡đ?œ‡ 4 (đ?‘Ľđ?‘Ľ) ; x ∈ U } be two fuzzy sets defined over the same universe U.The fuzzy setA (đ?œ‡đ?œ‡ 1,đ?œ‡đ?œ‡ 2)is a subset of the fuzzy set B (đ?œ‡đ?œ‡ 3,đ?œ‡đ?œ‡ 4)if for all x ∈ U , đ?œ‡đ?œ‡ 1 (đ?‘Ľđ?‘Ľ) ≤ đ?œ‡đ?œ‡ 3 (đ?‘Ľđ?‘Ľ) and đ?œ‡đ?œ‡ 4 (đ?‘Ľđ?‘Ľ) ≤ đ?œ‡đ?œ‡ 2 (đ?‘Ľđ?‘Ľ).

Two fuzzy setsC={x, đ?œ‡đ?œ‡ đ??śđ??ś (đ?‘Ľđ?‘Ľ)x ∈ U }and D={x, đ?œ‡đ?œ‡ đ??ˇđ??ˇ (đ?‘Ľđ?‘Ľ); x ∈ U } in the usual definition would be expressed asC(đ?œ‡đ?œ‡ đ??śđ??ś , 0)={x, đ?œ‡đ?œ‡ đ??śđ??ś (đ?‘Ľđ?‘Ľ) , 0; x ∈ U }and D (đ?œ‡đ?œ‡ đ??ˇđ??ˇ , 0) ={x, đ?œ‡đ?œ‡ đ??ˇđ??ˇ (đ?‘Ľđ?‘Ľ),0; x ∈ U } Accordingly, we have C(đ?œ‡đ?œ‡ đ??śđ??ś , 0) ⊆ D(đ?œ‡đ?œ‡ đ??ˇđ??ˇ , 0) if for all x ∈ U, đ?œ‡đ?œ‡ đ??śđ??ś (đ?‘Ľđ?‘Ľ) ≤ đ?œ‡đ?œ‡ đ??ˇđ??ˇ (đ?‘Ľđ?‘Ľ) , wh ich can be obtained by puttingđ?œ‡đ?œ‡ 2 (đ?‘Ľđ?‘Ľ) = đ?œ‡đ?œ‡ 4 (đ?‘Ľđ?‘Ľ)=0 in the new definition.

2.9 .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 (đ?‘“đ?‘“đ??´đ??´ , E) be fu zzy soft set (FS) over U. Then a subset of U Ă—E is uniquely defined by đ?‘…đ?‘…đ??´đ??´ = {(u, e); e ∈ A, u∈ đ?‘“đ?‘“đ??´đ??´ (đ?‘’đ?‘’)} which is called a relation form of (đ?‘“đ?‘“đ??´đ??´ , E).

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On Fu zzy Soft Matrix Based on Reference Function

2.10. Example Assume that U ={u1 ,u2 ,u3 ,u4 } is a universal setand E ={ e1 , e ,e3 ,e4 ,e5 } be the set of parameters and A={e1 ,e2 ,e3 } ⊆ E and

đ?‘“đ?‘“đ??´đ??´ (đ?‘’đ?‘’1 ) ={ đ?‘˘đ?‘˘ 1â „(0.7 , 0) , đ?‘˘đ?‘˘ 2 â „(0.1 , 0) , đ?‘˘đ?‘˘ 3 â „(0.2 , 0) , đ?‘˘đ?‘˘ 4 â „(0.6 , 0) } đ?‘“đ?‘“đ??´đ??´ (đ?‘’đ?‘’2 ) ={ đ?‘˘đ?‘˘ 1â „(0.8 , 0) , đ?‘˘đ?‘˘ 2 â „(0.6 , 0) , đ?‘˘đ?‘˘ 3 â „(0.1 , 0) , đ?‘˘đ?‘˘ 4 â „(0.5 , 0) } đ?‘“đ?‘“đ??´đ??´ (đ?‘’đ?‘’3 ) ={ đ?‘˘đ?‘˘ 1â „(0.1 , 0) , đ?‘˘đ?‘˘ 2 â „(0.2 , 0) , đ?‘˘đ?‘˘ 3 â „(0.7 , 0) , đ?‘˘đ?‘˘ 4 â „(0.3 , 0) } Then the fuzzy soft set (đ?‘“đ?‘“đ??´đ??´ , E) is a parameterized family { đ?‘“đ?‘“đ??´đ??´ (đ?‘’đ?‘’1 ) , đ?‘“đ?‘“đ??´đ??´ (đ?‘’đ?‘’2 ) ,đ?‘“đ?‘“đ??´đ??´ (đ?‘’đ?‘’3 ) } of all fuzzy soft sets over U. Then the relation form of (đ?‘“đ?‘“đ??´đ??´ , E) is written as đ?‘…đ?‘…đ??´đ??´ u1 u2 u3 u4

TABLE 1.The relation form of (đ?‘“đ?‘“đ??´đ??´ , E)

e1 (0.7, 0) (0.1, 0) (0.2, 0) (0.6, 0)

e2 (0.8, 0) (0.6, 0) (0.1, 0) (0.5, 0)

e3 (0.1, 0) (0.2, 0) (0.7, 0) (0.3, 0)

e4 (0, 0) (0, 0) (0, 0) (0, 0)

Hence,the fuzzy soft matrix representing this fuzzy soft set would be represented as (đ?&#x;Žđ?&#x;Ž. đ?&#x;•đ?&#x;•, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;–đ?&#x;–, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž, đ?&#x;Žđ?&#x;Ž) (đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;”đ?&#x;”, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž, đ?&#x;Žđ?&#x;Ž) ďż˝ ďż˝ (đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;•đ?&#x;•, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž, đ?&#x;Žđ?&#x;Ž) (đ?&#x;Žđ?&#x;Ž. đ?&#x;”đ?&#x;”, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž, đ?&#x;Žđ?&#x;Ž)

2.14. Example

Let U={đ?‘?đ?‘?1 ,đ?‘?đ?‘?2 ,đ?‘?đ?‘?3 ,đ?‘?đ?‘?4 } be the un iversal set and Eb e the set of parameters g iven by E={đ?‘’đ?‘’ 1,đ?‘’đ?‘’ 2 ,đ?‘’đ?‘’ 3 } We cons ider th e fu zzy soft sets based on reference funct ion.

(F,E)={F(đ?‘’đ?‘’ 1)={(đ?‘?đ?‘?1 ,0.3, 0) ),(đ?‘?đ?‘?2 ,0.5, 0), (đ?‘?đ?‘?3 , 0.6, 0), (đ?‘?đ?‘?4 ,0.5,0)},F(đ?‘’đ?‘’ 2 )={(đ?‘?đ?‘?1 ,0.7, 0) ),(đ?‘?đ?‘?2 ,0.9, 0), (đ?‘?đ?‘?3 , 0.7, 0), (đ?‘?đ?‘?4 ,0.8,0)},F(đ?‘’đ?‘’ 3 )={(đ?‘?đ?‘?1 ,0.6, 0) ),(đ?‘?đ?‘?2 ,0.7, 0), (đ?‘?đ?‘?3 , 0.7, 0), (đ?‘?đ?‘?4 ,0.3,0)}}.

(G,E)={G( đ?‘’đ?‘’ 1)={(đ?‘?đ?‘?1 ,0.8, 0) ),(đ?‘?đ?‘?2 ,0.7, 0), (đ?‘?đ?‘?3 , 0.5, 0), (đ?‘?đ?‘?4 ,0.4,0)}, G(đ?‘’đ?‘’ 2 )={(đ?‘?đ?‘?1 ,0.9, 0) ),(đ?‘?đ?‘?2 ,0.9, 0), (đ?‘?đ?‘?3 , 0.8, 0), ( đ?‘?đ?‘?4 ,0.7,0)},G( đ?‘’đ?‘’ 3 )={( đ?‘?đ?‘?1 ,0.5, 0) ),( đ?‘?đ?‘?2 ,0.9, 0), ( đ?‘?đ?‘?3 , 0.6, 0), (đ?‘?đ?‘?4 ,0.8,0)}}.

The fu zzy soft mat rices based on reference funct ion rep resenting these t wo fu zzy soft sets are respect ively

A

A=

2.11. Definition[29 ] We define the membership value matrix corresponding to the matrix A as MV(A) =[ đ?›żđ?›żđ?‘—đ?‘—đ?‘—đ?‘— (đ?‘?đ?‘?đ?‘–đ?‘– )] Where đ?›żđ?›ż(đ??´đ??´)đ?‘–đ?‘–đ?‘–đ?‘– = đ?œ‡đ?œ‡ đ?‘—đ?‘—1 (đ?‘?đ?‘?đ?‘–đ?‘– )- đ?œ‡đ?œ‡ đ?‘—đ?‘—2 (đ?‘?đ?‘?đ?‘–đ?‘– ) i= 1,2,3,‌,m and j =1,2,3‌..,n ,where đ?œ‡đ?œ‡ đ?‘—đ?‘—1 (đ?‘?đ?‘?đ?‘–đ?‘– ) and đ?œ‡đ?œ‡ đ?‘—đ?‘—2 (đ?‘?đ?‘?đ?‘–đ?‘– ) represent the fuzzy membership function and fuzzy reference function respectively ofđ?‘?đ?‘?đ?‘–đ?‘– in the fuzzy set F( đ?‘’đ?‘’đ?‘—đ?‘— ).

2.12. Definition [29] Let the fuzzy soft matrices corresponding to the fuzzy soft sets (F,E), and (G,E) beA=[ aij ] ∈ FSM m Ă—n , and bij B=[ bij ]where aij = ( Îźj1 (ci ) , Îźj2 (ci ) ) =( χj1 (ci ) , χj2 (ci ) ),i =1,2,3,‌, m ;j =1,2,3,‌,n ;Then Aand Bare called fuzzy soft equal matrices denoted byA=B, if đ?œ‡đ?œ‡ đ?‘—đ?‘—1 (đ?‘?đ?‘?đ?‘–đ?‘– ) = đ?œ’đ?œ’ đ?‘—đ?‘—1 (đ?‘?đ?‘?đ?‘–đ?‘– ) and đ?œ‡đ?œ‡ đ?‘—đ?‘—1 (đ?‘?đ?‘?đ?‘–đ?‘– ) = đ?œ’đ?œ’ đ?‘—đ?‘—1 (đ?‘?đ?‘?đ?‘–đ?‘– ) for all i,j.

In [13], the ‘add it ion (+)’ op eration bet ween t wo fu zzy soft mat rices is defined as fo llo ws 2.13. Definition [29] Let U={đ?‘?đ?‘?1 ,đ?‘?đ?‘?2 ,đ?‘?đ?‘?3 ,‌ . . , đ?‘?đ?‘?đ?‘šđ?‘š } be the un iversal set and Ebe th e set of parameters g iven by E={đ?‘’đ?‘’ 1,đ?‘?đ?‘?2 ,đ?‘’đ?‘’ 3 ,‌ . . , đ?‘’đ?‘’ đ?‘›đ?‘› }.Let the set of all mĂ— n fu zzy soft matrices ov er U b e FSM đ?‘šđ?‘š Ă—đ?‘›đ?‘› .

Let A , B ∈ FSM m Ă—n ,where A= [đ?‘Žđ?‘Žđ?‘–đ?‘–đ?‘–đ?‘– ] đ?‘šđ?‘š Ă—đ?‘›đ?‘› , đ?‘Žđ?‘Žđ?‘–đ?‘–đ?‘–đ?‘– = and B= [đ?‘?đ?‘?đ?‘–đ?‘–đ?‘–đ?‘– ] đ?‘šđ?‘š Ă—đ?‘›đ?‘› , đ?‘?đ?‘?đ?‘–đ?‘–đ?‘–đ?‘– ( đ?œ‡đ?œ‡ đ?‘—đ?‘—1 (đ?‘?đ?‘?đ?‘–đ?‘– ) , đ?œ‡đ?œ‡ đ?‘—đ?‘—2 (đ?‘?đ?‘?đ?‘–đ?‘– ) ) =(đ?œ’đ?œ’ đ?‘—đ?‘—1 (đ?‘?đ?‘?đ?‘–đ?‘– ) ,đ?œ’đ?œ’ đ?‘—đ?‘—2 (đ?‘?đ?‘?đ?‘–đ?‘– )).To avoid degenerate cases we assume Copyright Š 2012 M ECS

thatmi n (đ?œ‡đ?œ‡ đ?‘—đ?‘—1 (đ?‘?đ?‘?đ?‘–đ?‘– ) ,đ?œ’đ?œ’ đ?‘—đ?‘—1 (đ?‘?đ?‘?đ?‘–đ?‘– )) ≼max (đ?œ‡đ?œ‡ đ?‘—đ?‘—2 (đ?‘?đ?‘?đ?‘–đ?‘– ),đ?œ’đ?œ’ đ?‘—đ?‘—2 (đ?‘?đ?‘?đ?‘–đ?‘– )) for all i and j .Th e op erat ion o f ‘add it ion (+)’ b et ween A and B is d efined as A+ B =C ,where C= [đ?‘?đ?‘?đ?‘–đ?‘–đ?‘–đ?‘– ] đ?‘šđ?‘š Ă—đ?‘›đ?‘› ,đ?‘?đ?‘?đ?‘–đ?‘–đ?‘–đ?‘– =(max (đ?œ‡đ?œ‡ đ?‘—đ?‘—1 (đ?‘?đ?‘?đ?‘–đ?‘– ) ,đ?œ’đ?œ’ đ?‘—đ?‘—1 (đ?‘?đ?‘?đ?‘–đ?‘– )), min (đ?œ‡đ?œ‡ đ?‘—đ?‘—2 (đ?‘?đ?‘?đ?‘–đ?‘– ) , đ?œ’đ?œ’ đ?‘—đ?‘—2 (đ?‘?đ?‘?đ?‘–đ?‘– )) )

(đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;•đ?&#x;•, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;”đ?&#x;”, đ?&#x;Žđ?&#x;Ž) (đ?&#x;Žđ?&#x;Ž. đ?&#x;–đ?&#x;–, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;—đ?&#x;—,đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“,đ?&#x;Žđ?&#x;Ž) ⎥(đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;—đ?&#x;—, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;•đ?&#x;•, đ?&#x;Žđ?&#x;Ž)⎤ ⎥(đ?&#x;Žđ?&#x;Ž. đ?&#x;•đ?&#x;•, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;—đ?&#x;—,đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;—đ?&#x;—,đ?&#x;Žđ?&#x;Ž) ⎤ ⎢ ⎼ ⎢ =⎢(đ?&#x;Žđ?&#x;Ž. đ?&#x;”đ?&#x;”, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;•đ?&#x;•, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;•đ?&#x;•, đ?&#x;Žđ?&#x;Ž)⎼,B =⎢(đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;–đ?&#x;–,đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;”đ?&#x;”,đ?&#x;Žđ?&#x;Ž) ⎼⎼ ⎢(đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;–đ?&#x;–, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘, đ?&#x;Žđ?&#x;Ž)⎼ ⎢(đ?&#x;Žđ?&#x;Ž. đ?&#x;’đ?&#x;’, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;•đ?&#x;•,đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;–đ?&#x;–,đ?&#x;Žđ?&#x;Ž) ⎼ ⎢ ⎼ ⎢ ⎼ (đ?&#x;Žđ?&#x;Ž. đ?&#x;–đ?&#x;–, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;—đ?&#x;—, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;”đ?&#x;”, đ?&#x;Žđ?&#x;Ž)

đ?&#x;•đ?&#x;•, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;—đ?&#x;—, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;—đ?&#x;—, đ?&#x;Žđ?&#x;Ž) HereA+B = ďż˝(đ?&#x;Žđ?&#x;Ž. ďż˝ (đ?&#x;Žđ?&#x;Ž. đ?&#x;”đ?&#x;”, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;–đ?&#x;–, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;•đ?&#x;•, đ?&#x;Žđ?&#x;Ž) (đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;–đ?&#x;–, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;–đ?&#x;–, đ?&#x;Žđ?&#x;Ž)

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.

3.1. Definition Let A= ďż˝(aij , 0)ďż˝

m Ă—n

∈ FSM m Ă—n

according to the

c

definition in [26], then A is calledfuzzy soft complement for all aij ∈ [0 , 1]. matrix if Ac = ďż˝(1 , aij )ďż˝ 3.2 .Example

m Ă—n

(0.7, 0)(0.8,0)

ďż˝ be fuzzy soft matrix based on Let A = ďż˝( 0.1, 0)(0.6,0) reference function, then the complement of this matrix is (1, 0.7)(1, 0.8)

ďż˝. đ??´đ??´đ?‘?đ?‘? = ďż˝ ( 1, 0.1)(1, 0.6)

3.3. Proposition

Let A, B be two fuzzy soft matrix based on fuzzy reference function .Then (i)(đ??´đ??´đ?‘?đ?‘? ) đ?‘‡đ?‘‡ = (đ??´đ??´đ?‘‡đ?‘‡ )đ?‘?đ?‘?

(1)

I.J. In formation Engineering and Electronic Business, 2013, 2, 52-59

On Fu zzy Soft Matrix Based on Reference Function

(ii)(đ??´đ??´đ?‘?đ?‘? + đ??ľđ??ľ đ?‘?đ?‘? )đ?‘‡đ?‘‡ = (đ??´đ??´đ?‘‡đ?‘‡ )đ?‘?đ?‘? +(đ??ľđ??ľ đ?‘‡đ?‘‡ )đ?‘?đ?‘?

(2)

Proof: To show (i) (đ??´đ??´đ?‘?đ?‘? )đ?‘‡đ?‘‡ = (đ??´đ??´đ?‘‡đ?‘‡ )đ?‘?đ?‘?

we have

A+B= C where C=[đ?‘?đ?‘?đ?‘–đ?‘–đ?‘–đ?‘– ]

Following the defin it ion o f add it ion of t wo fu zzy soft mat rices, we hav e đ??śđ??śđ?‘–đ?‘–đ?‘–đ?‘– = (max(đ?‘Žđ?‘Žđ?‘–đ?‘–đ?‘–đ?‘– , đ?‘?đ?‘?đ?‘–đ?‘–đ?‘–đ?‘– ),min(đ?‘&#x;đ?‘&#x;đ?‘–đ?‘–đ?‘–đ?‘– ,đ?‘&#x;đ?‘&#x;đ?‘–đ?‘–đ?‘–đ?‘–′ ))

(đ??´đ??´đ?‘‡đ?‘‡ )đ?‘?đ?‘? = [1 ,đ?œ‡đ?œ‡ đ?‘–đ?‘–1 (đ?‘?đ?‘?đ?‘—đ?‘— )] Hen ce (đ??´đ??´đ?‘?đ?‘? )đ?‘‡đ?‘‡ = (đ??´đ??´đ?‘‡đ?‘‡ ) đ?‘?đ?‘?

According to definition 4.1 the trace of fuzzy soft matrixbased on reference function would be:

tr (C) = [ max {max( đ?‘Žđ?‘Žđ?‘–đ?‘–đ?‘–đ?‘– , đ?‘?đ?‘?đ?‘–đ?‘–đ?‘–đ?‘– ) }, min {min( đ?‘&#x;đ?‘&#x;đ?‘–đ?‘–đ?‘–đ?‘– ,đ?‘&#x;đ?‘&#x;đ?‘–đ?‘–đ?‘–đ?‘–′ )}] = [ max {max(đ?‘Žđ?‘Žđ?‘–đ?‘–đ?‘–đ?‘– ) ,max( đ?‘?đ?‘?đ?‘–đ?‘–đ?‘–đ?‘– )} , min {min(đ?‘&#x;đ?‘&#x;đ?‘–đ?‘–đ?‘–đ?‘– ), min(đ?‘&#x;đ?‘&#x;đ?‘–đ?‘–đ?‘–đ?‘–′ )}] = trA+trB, Conversely, trA+trB = [ max {max(đ?‘Žđ?‘Žđ?‘–đ?‘–đ?‘–đ?‘– ) ,max( đ?‘?đ?‘?đ?‘–đ?‘–đ?‘–đ?‘– )} , min {(min(đ?‘&#x;đ?‘&#x;đ?‘–đ?‘–đ?‘–đ?‘– ), min(đ?‘&#x;đ?‘&#x;đ?‘–đ?‘–đ?‘–đ?‘–′ ))}] = [ max {max(đ?‘Žđ?‘Žđ?‘–đ?‘–đ?‘–đ?‘– , đ?‘?đ?‘?đ?‘–đ?‘–đ?‘–đ?‘– ), min (min(đ?‘&#x;đ?‘&#x;đ?‘–đ?‘–đ?‘–đ?‘– ,đ?‘&#x;đ?‘&#x;đ?‘–đ?‘–đ?‘–đ?‘–′ )}] =tr(A+B) hence the resulttrA+trB =tr(A+B)

3.4. Example đ?&#x;?đ?&#x;?,đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘,đ?&#x;Žđ?&#x;Ž) (đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;’đ?&#x;’, đ?&#x;Žđ?&#x;Ž) Let A=ďż˝(đ?&#x;Žđ?&#x;Ž. ďż˝, B=ďż˝ ďż˝ (đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?,đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;’đ?&#x;’,đ?&#x;Žđ?&#x;Ž) (đ?&#x;Žđ?&#x;Ž. đ?&#x;”đ?&#x;”, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž) đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘) đ?‘?đ?‘? (đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;’đ?&#x;’) đ??´đ??´đ?‘?đ?‘? =ďż˝(đ?&#x;?đ?&#x;?, ďż˝, đ??ľđ??ľ =ďż˝ ďż˝ (đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;’đ?&#x;’) (đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;”đ?&#x;”)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?)

0.2)(1,0.1) (1, 0.5)(1, 0.6) đ?‘‡đ?‘‡ đ?‘?đ?‘? đ?‘‡đ?‘‡ đ?‘?đ?‘? (đ??´đ??´đ?‘?đ?‘? )đ?‘‡đ?‘‡ =ďż˝(1, ďż˝, (đ??ľđ??ľđ?‘?đ?‘? )đ?‘‡đ?‘‡ = ďż˝ ďż˝, (đ??´đ??´ ) +(đ??ľđ??ľ ) = (1, 0.3)(1,0.4) (1, 0.4)(1, 0.2)

đ?‘?đ?‘?

đ??´đ??´ + đ??ľđ??ľ =

Then

(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?) ďż˝ ďż˝ (đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?)

(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘) ďż˝ ďż˝ (đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?)

đ?‘?đ?‘?

đ?‘?đ?‘? đ?‘‡đ?‘‡

, (đ??´đ??´ + đ??ľđ??ľ ) =

trB= (max đ?‘?đ?‘?đ?‘–đ?‘–đ?‘–đ?‘– ,min đ?‘&#x;đ?‘&#x;đ?‘–đ?‘–đ?‘–đ?‘–′ )

then

The proof of (ii) follows similar lines as above.

đ?‘?đ?‘?

fu zzy so ft matrices trA= (max đ?‘Žđ?‘Žđ?‘–đ?‘–đ?‘–đ?‘– ,min đ?‘&#x;đ?‘&#x;đ?‘–đ?‘–đ?‘–đ?‘– )

and

We have, let A ∈ đ??šđ??šđ??šđ??šđ??šđ??šđ?‘šđ?‘š Ă—đ?‘›đ?‘› , then A= [(đ?œ‡đ?œ‡ đ?‘—đ?‘—1 (đ?‘?đ?‘?đ?‘–đ?‘– ) ,đ?œ‡đ?œ‡ đ?‘—đ?‘—2 (đ?‘?đ?‘?đ?‘–đ?‘– )] đ??´đ??´đ?‘?đ?‘? = [1 ,đ?œ‡đ?œ‡ đ?‘—đ?‘—1 (đ?‘?đ?‘?đ?‘–đ?‘– )] (đ??´đ??´đ?‘?đ?‘? )đ?‘‡đ?‘‡ = [1 ,đ?œ‡đ?œ‡ đ?‘–đ?‘–1 (đ?‘?đ?‘?đ?‘—đ?‘— )] For đ??´đ??´đ?‘‡đ?‘‡ = [(đ?œ‡đ?œ‡ đ?‘–đ?‘–1 (đ?‘?đ?‘?đ?‘—đ?‘— ) ,đ?œ‡đ?œ‡ đ?‘–đ?‘–2 (đ?‘?đ?‘?đ?‘—đ?‘— )],

55

4.3. Example:

(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?) ďż˝ ďż˝ (đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?)

Let us consider the following two fuzzy soft matrices A and Bbased on reference function for illustration purposes

(đ??´đ??´đ?‘?đ?‘? + đ??ľđ??ľ đ?‘?đ?‘? )đ?‘‡đ?‘‡ = (đ??´đ??´đ?‘‡đ?‘‡ )đ?‘?đ?‘? +(đ??ľđ??ľ đ?‘‡đ?‘‡ )đ?‘?đ?‘?.

(đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;•đ?&#x;•, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;–đ?&#x;–, đ?&#x;Žđ?&#x;Ž)

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.

4.1. Definition

(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?,đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž . đ?&#x;‘đ?&#x;‘,đ?&#x;Žđ?&#x;Ž)

A =ďż˝(đ?&#x;Žđ?&#x;Ž. đ?&#x;’đ?&#x;’, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘, đ?&#x;Žđ?&#x;Ž)ďż˝andB= ďż˝(đ?&#x;Žđ?&#x;Ž . đ?&#x;–đ?&#x;–,đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž)ďż˝ (đ?&#x;Žđ?&#x;Ž. đ?&#x;”đ?&#x;”, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;’đ?&#x;’, đ?&#x;Žđ?&#x;Ž)

(đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“, đ?&#x;Žđ?&#x;Ž)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž . đ?&#x;–đ?&#x;–,đ?&#x;Žđ?&#x;Ž)

The addition of two soft matrices would be: (đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;•đ?&#x;•, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;–đ?&#x;–, đ?&#x;Žđ?&#x;Ž)

A+B = ďż˝(đ?&#x;Žđ?&#x;Ž. đ?&#x;–đ?&#x;–, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘, đ?&#x;Žđ?&#x;Ž)ďż˝ (đ?&#x;Žđ?&#x;Ž. đ?&#x;”đ?&#x;”, đ?&#x;Žđ?&#x;Ž)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;–đ?&#x;–, đ?&#x;Žđ?&#x;Ž)

Using the defin it ion of trace of fu zzy soft matrices, we see the following results:

Let A be a square matrix. Then the trace ofthe matrix A is denoted by tr A and is defined as:

tr A = { max(0.3, 0.5, 0.4), min (0, 0, 0)}=(0.5 , 0) tr B = { max(1, 0.5, 0.8), min (0, 0, 0)}=(1 , 0)

trA= (max(đ?œ‡đ?œ‡ đ?‘–đ?‘–đ?‘–đ?‘– ),min(đ?‘&#x;đ?‘&#x;đ?‘–đ?‘–đ?‘–đ?‘– ))

Thus we have

(3)

where Îźii stands for the membership functions lying along the principal diagonal and rii refers to the reference function of the corresponding membership functions.

4.2. Proposition Let A and B be t wo fu zzy softsquare matrices each of order n.

Then tr (A+B) =t rA+ trB (4) proof.

We h ave fro m th e p roposed defin it ion o f trace o f Copyright Š 2012 M ECS

trA+trB = { max(1, 0.5, 0.8) , min (0, 0, 0)}=(1 , 0) And tr (A+B) = { max(1, 0.5, 0.8) , min (0, 0, 0)}= (1, 0) Hence the result trA+trB =tr(A+B) 4.4.Proposition Let A = [đ?‘Žđ?‘Žđ?‘–đ?‘–đ?‘–đ?‘– , đ?‘&#x;đ?‘&#x;đ?‘–đ?‘–đ?‘–đ?‘– ] ∈ FSM đ?‘šđ?‘š Ă—đ?‘›đ?‘› be fuzzy soft squarematrix of order n, if đ?œ†đ?œ† is a scalar such that 0 ≤ đ?œ†đ?œ† ≤ 1 . Then

I.J. In formation Engineering and Electronic Business, 2013, 2, 52-59

56

On Fu zzy Soft Matrix Based on Reference Function

tr(đ?œ†đ?œ† A)= đ?œ†đ?œ† tr(A)

(5)

(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘)

proof. to prove tr(đ?œ†đ?œ† A) = đ?œ†đ?œ†trA0 ≤ đ?œ†đ?œ† ≤ 1 we have tr(đ?œ†đ?œ† A) ={ max(đ?œ†đ?œ†đ?‘Žđ?‘Žđ?‘–đ?‘–đ?‘–đ?‘– ), min (đ?œ†đ?œ†đ?‘&#x;đ?‘&#x;đ?‘–đ?‘–đ?‘–đ?‘– )} =đ?œ†đ?œ†{ max(đ?‘Žđ?‘Žđ?‘–đ?‘–đ?‘–đ?‘– ), min(đ?‘&#x;đ?‘&#x;đ?‘–đ?‘–đ?‘–đ?‘– )} = đ?œ†đ?œ†tr (A)

đ??´đ??´đ?‘?đ?‘? +đ??ľđ??ľ đ?‘?đ?‘? = ďż˝(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;’đ?&#x;’)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?)ďż˝ (đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;’đ?&#x;’)

tr (đ??´đ??´đ?‘?đ?‘? +đ??ľđ??ľ đ?‘?đ?‘? ) ={max(1,1,1),min (0.3, 0.5, 0.4)} = (1, 0.3)

V.

4.5. Example

(đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;•đ?&#x;•, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;–đ?&#x;–, đ?&#x;Žđ?&#x;Ž)

Let A = ďż˝(đ?&#x;Žđ?&#x;Ž. đ?&#x;’đ?&#x;’, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘, đ?&#x;Žđ?&#x;Ž)ďż˝and đ?œ†đ?œ† =0.5

NEW FUZZY SOFT MATRIX THEORY IN DECISION MAKING

In th is section we adopted the defin it ion o f fu zzy soft mat rix decis ion method p roposed by P. Rajarajes wari,P. Dh analaks h mi in [30] to the case of fu zzy soft mat rix based on referen ce funct ion in o rder to define a new fu zzy soft mat rix decis ion method based on reference funct ion .

(đ?&#x;Žđ?&#x;Ž. đ?&#x;”đ?&#x;”, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;’đ?&#x;’, đ?&#x;Žđ?&#x;Ž)

Then

soft matrices based on reference function, we have.

(đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?đ?&#x;?đ?&#x;?,đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘đ?&#x;‘đ?&#x;‘, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;’đ?&#x;’đ?&#x;’đ?&#x;’, đ?&#x;Žđ?&#x;Ž)

đ?œ†đ?œ†A =ďż˝ (đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?đ?&#x;?đ?&#x;?,đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž)ďż˝ (đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘đ?&#x;‘đ?&#x;‘,đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;Žđ?&#x;Žđ?&#x;Žđ?&#x;Ž, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž)

5.1. Definition: (Value Matrix) Let A =[ a ij , 0 ] ∈ [ FSM ] m Ă—n .Then we defin e the value mat rix of fu zzy soft matrix A based on reference funct ion as V(A )=[ a ij ] =[ a ij - r ij ], i=1, 2, ,‌.,m , j= 1,2,3,‌, n, where r ij = [ 0] m Ă—n .

tr(đ?œ†đ?œ† A) = { max(0.15, 0.25, 0.20), min ( (0, 0, 0)} = (0.25,0) Again tr A=(0.5,0) and hence tr(đ?œ†đ?œ† A) =0.5 (0.5, 0) = (0.25,0)

5.2. Definition:(Score Matrix)

4.6. Proposition: Let A =[ đ?‘Žđ?‘Žđ?‘–đ?‘–đ?‘–đ?‘– , đ?‘&#x;đ?‘&#x;đ?‘–đ?‘–đ?‘–đ?‘– ] ∈ FSM đ?‘šđ?‘š Ă—đ?‘›đ?‘› be fuzzy soft square matrices each of order n .

Then trA=tr(đ??´đ??´đ?‘Ąđ?‘Ą ),wheređ??´đ??´đ?‘Ąđ?‘Ą is the transpose of A 4.7.Example (đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;’đ?&#x;’, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;”đ?&#x;”, đ?&#x;Žđ?&#x;Ž)

Let đ??´đ??´đ?‘Ąđ?‘Ą =ďż˝(đ?&#x;Žđ?&#x;Ž. đ?&#x;•đ?&#x;•, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž)ďż˝ (đ?&#x;Žđ?&#x;Ž. đ?&#x;–đ?&#x;–, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;’đ?&#x;’, đ?&#x;Žđ?&#x;Ž)

Then tr(đ??´đ??´đ?‘Ąđ?‘Ą )= { max(0.3, 0.5, 0.4) , min (0, 0, 0)} =(0.5, 0) HencetrA=tr( đ??´đ??´đ?‘Ąđ?‘Ą )

The same result will hold if we consider the complements of fuzzy soft square matrices. (đ?&#x;?đ?&#x;? , đ?&#x;Žđ?&#x;Ž.đ?&#x;‘đ?&#x;‘)(đ?&#x;?đ?&#x;? ,đ?&#x;Žđ?&#x;Ž. đ?&#x;•đ?&#x;•)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;–đ?&#x;–)

đ??´đ??´đ?‘?đ?‘? = ďż˝(đ?&#x;?đ?&#x;? , đ?&#x;Žđ?&#x;Ž.đ?&#x;’đ?&#x;’)(đ?&#x;?đ?&#x;? ,đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘)ďż˝ (đ?&#x;?đ?&#x;? , đ?&#x;Žđ?&#x;Ž.đ?&#x;”đ?&#x;”)(đ?&#x;?đ?&#x;? ,đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;’đ?&#x;’)

tr đ??´đ??´đ?‘?đ?‘? ={max (1, 1, 1), min (0.3, 0.5, 0.4)}=(1, 0.3) If we consider another fuzzy soft matriceB: (đ?&#x;?đ?&#x;? , đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘,đ?&#x;Žđ?&#x;Ž)

B = ďż˝ (đ?&#x;Žđ?&#x;Ž. đ?&#x;–đ?&#x;–,đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“,đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž . đ?&#x;?đ?&#x;?,đ?&#x;Žđ?&#x;Ž) ďż˝ (đ?&#x;Žđ?&#x;Ž . đ?&#x;“đ?&#x;“,đ?&#x;Žđ?&#x;Ž)(đ?&#x;?đ?&#x;? , đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;–đ?&#x;–,đ?&#x;Žđ?&#x;Ž)

(đ?&#x;?đ?&#x;?, đ?&#x;?đ?&#x;?)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘)

đ??ľđ??ľ đ?‘?đ?‘? = ďż˝ (đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;–đ?&#x;–)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž.đ?&#x;?đ?&#x;?) ďż˝ (đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž.đ?&#x;“đ?&#x;“)(đ?&#x;?đ?&#x;?, đ?&#x;?đ?&#x;?)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;–đ?&#x;–)

Then the trace of đ??ľđ??ľ đ?‘?đ?‘? will be the following: tr(đ??ľđ??ľ đ?‘?đ?‘? )={max(1,1,1),min (1, 0.5, 0.8)}=(1, 0.5) Following the definition 2.13 of addition of two fuzzy Copyright Š 2012 M ECS

If A =[ a ij ] ∈ FSM,B=[ bij ] ∈ [ FSM ] m Ă—n .Th en we define sco re matrix of A and B as : s A ,B =[dij ]mxn where [dij ]=V(A )-V(B)

5.3. Definition:(Total Score) If A =[ a ij , 0 ] ∈ [ FSM ] m Ă—n ,B=[ bij , 0 ] ∈ [ FSM ] m Ă—n .Let the correspond ing valu e mat rices be V(A ),V(B) and their score matrix is s A,B =[dij ] mxn then we define total score fo r each ci in U as s i = ∑ nj=1 dij .

Methodology and algorith m Assume th at there is a set of candid ates( p rog rammer), U={c1 , c2 , ,‌,cn } is a set of cand idates to be recru ited by soft ware develop ment org anizat ion in p rogrammer post.Let E is a set of paramet ers related to innovat ive att itude o f the prog rammer. We construct fu zzy soft set (F,E)over U rep resent the select ion o f cand idate by field expert X,where F is a mapp ing F:E→ F u ,F u is the collection o f all fu zzy subsets of U. We further construct another fu zzy soft set (G,E)over Urepresent the select ion o f cand idat e by field expert Y,where G is a mapp ing G:E → đ??šđ??šđ?‘˘đ?‘˘ , đ??šđ??šđ?‘˘đ?‘˘ is the co llection of all fu zzy subsets of U.The mat ricesA and B corresponding to the fu zzy softsets (F,E) and (G,E) are constructed,we co mpute the co mp lementsand theirmatrices đ??´đ??´ đ?‘?đ?‘? and đ??ľđ??ľ đ?‘?đ?‘? corresponding to (F, E) đ?‘?đ?‘? and (G, E) đ?‘?đ?‘? respectively . Co mpute A +B wh ich is themaxi mu m memb ersh ip o f select ion of cand idat es by the judges.

Co mpute đ??´đ??´ đ?‘?đ?‘? + đ??ľđ??ľ đ?‘?đ?‘? wh ich is the maxi mu m membership o f non selection o f cand idat es by the judges. us ingdef (5.1) ,Co mpute V(A +B),V( đ??´đ??´ đ?‘?đ?‘? + đ??ľđ??ľ đ?‘?đ?‘? ) đ?‘ đ?‘ ((đ??´đ??´ +đ??ľđ??ľ ),( đ??´đ??´đ?‘?đ?‘? +đ??ľđ??ľđ?‘?đ?‘? )) and the total

I.J. In formation Engineering and Electronic Business, 2013, 2, 52-59

On Fu zzy Soft Matrix Based on Reference Function

score đ?‘†đ?‘†đ?‘–đ?‘– for each cand idate in U.Finally find đ?‘†đ?‘†đ?‘—đ?‘— = max( đ?‘†đ?‘†đ?‘–đ?‘– ),then conclud e that the cand id ate đ?‘?đ?‘?đ?‘—đ?‘— has selected by the judg es.If đ?‘†đ?‘†đ?‘—đ?‘— has mo re than on e v alu e the process is repeated by reassessing the parameters.

No w, us ing defin it ions 5-1, 5-2 and 5-3 we can construct a fu zzy soft matrix decis ion making method based on reference funct ion by the fo llo wing algo rith m.

Then the add it ion mat rices are (0.2, 0)(0.6, 0)(0.5, 0) ⎥(0.6, 0)(0.5, 0)(0.8, 0) ⎤

Then we con clud e that the cand idate đ?‘?đ?‘?đ?‘–đ?‘– is selected for the post.

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

Case S tudy

(1, 0.1)(1, 0.6)(1, 0.5) ⎥(1, 0.5)(1, 0.4)(1, 0.7) ⎤

A+B= ⎢⎢(0.2, 0)(0.6, 0)(0.7, 0) ⎼⎼,đ??´đ??´ đ?‘?đ?‘? +đ??ľđ??ľ đ?‘?đ?‘? = ⎢⎢(1, 0.1)(1, 0.5)(1, 0.6) ⎼⎼ ⎢(0.4, 0)(0.8, 0)(0.5, 0) ⎼ ⎢ ⎼

V(A +B) =

Alg ori thm Step1 :Input th e fu zzy soft set (F, E), (G,E) and obtain the fu zzy soft mat rices A ,B correspond ing to (F,E) and (G, E) respectiv ely.

Step2 :W rite the fu zzy soft co mplement set (F, E) c , (G, E) c and obtain the fu zzy soft mat rices Ac , B c corresponding to (F, E) c ,and (G, E) c respect ively.

Step3 :Co mpute (A +B),(Ac +B c ), V(A +B),V( Ac +B c ) and s ((A +B),( A c + B c )) .

Step4 :Co mpute the total score S i for each ci in U. Step5 :Find ci fo r wh ich max ( S i ) .

57

đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;? đ?&#x;Žđ?&#x;Ž.đ?&#x;”đ?&#x;” đ?&#x;Žđ?&#x;Ž.đ?&#x;“đ?&#x;“ ⎥ đ?&#x;Žđ?&#x;Ž. đ?&#x;”đ?&#x;” đ?&#x;Žđ?&#x;Ž.đ?&#x;“đ?&#x;“ đ?&#x;Žđ?&#x;Ž.đ?&#x;–đ?&#x;– ⎤ ⎢ ⎼ ⎢ đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;? đ?&#x;Žđ?&#x;Ž.đ?&#x;”đ?&#x;” đ?&#x;Žđ?&#x;Ž.đ?&#x;•đ?&#x;• ⎼ ⎢ đ?&#x;Žđ?&#x;Ž. đ?&#x;’đ?&#x;’ đ?&#x;Žđ?&#x;Ž.đ?&#x;–đ?&#x;– đ?&#x;Žđ?&#x;Ž.đ?&#x;“đ?&#x;“ ⎼ ⎢ ⎼

⎢(1, 0.3)(1, 0.7)(1, 0.5) ⎼ ⎢ ⎼

đ?‘?đ?‘?

đ?‘?đ?‘?

,V(đ??´đ??´ +đ??ľđ??ľ ) =

đ?&#x;Žđ?&#x;Ž. đ?&#x;—đ?&#x;— đ?&#x;Žđ?&#x;Ž.đ?&#x;’đ?&#x;’ đ?&#x;Žđ?&#x;Ž.đ?&#x;“đ?&#x;“ ⎥đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“ đ?&#x;Žđ?&#x;Ž.đ?&#x;”đ?&#x;” đ?&#x;Žđ?&#x;Ž.đ?&#x;‘đ?&#x;‘ ⎤ ⎢ ⎼ ⎢ đ?&#x;Žđ?&#x;Ž. đ?&#x;—đ?&#x;— đ?&#x;Žđ?&#x;Ž.đ?&#x;“đ?&#x;“ đ?&#x;Žđ?&#x;Ž.đ?&#x;’đ?&#x;’ ⎼ ⎢ đ?&#x;Žđ?&#x;Ž.đ?&#x;•đ?&#x;• đ?&#x;Žđ?&#x;Ž.đ?&#x;‘đ?&#x;‘ đ?&#x;Žđ?&#x;Ž.đ?&#x;“đ?&#x;“ ⎼ ⎢ ⎼

Calcu late the score matrix and the total sco re for select ion đ?‘ đ?‘

( (đ??´đ??´+đ??ľđ??ľ ),( đ??´đ??´ đ?‘?đ?‘? +đ??ľđ??ľđ?‘?đ?‘? ))

=

−đ?&#x;Žđ?&#x;Ž. đ?&#x;•đ?&#x;• đ?&#x;Žđ?&#x;Ž.đ?&#x;?đ?&#x;? đ?&#x;Žđ?&#x;Ž ⎥đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;? − đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;? đ?&#x;Žđ?&#x;Ž.đ?&#x;“đ?&#x;“ ⎤ ⎢ −đ?&#x;Žđ?&#x;Ž.đ?&#x;•đ?&#x;• đ?&#x;Žđ?&#x;Ž.đ?&#x;?đ?&#x;? đ?&#x;Žđ?&#x;Ž.đ?&#x;‘đ?&#x;‘ ⎼⎼ ⎢ đ?&#x;Žđ?&#x;Ž.đ?&#x;“đ?&#x;“ đ?&#x;Žđ?&#x;Ž ⎼ ⎢ −đ?&#x;Žđ?&#x;Ž. đ?&#x;‘đ?&#x;‘ ⎢ ⎼

Total score =

⎥ ⎢ ⎢ ⎢ ⎢

−đ?&#x;Žđ?&#x;Ž.đ?&#x;“đ?&#x;“ đ?&#x;Žđ?&#x;Ž.đ?&#x;“đ?&#x;“ ⎤ ⎼ −đ?&#x;Žđ?&#x;Ž.đ?&#x;‘đ?&#x;‘ ⎼ đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;? ⎼ ⎼

We see that the second candidate has the maximu m value and thus conclude that from both the expert’s opinion, candidateđ?‘?đ?‘?2 is selected for the post.

VI. CONCLUSIONS

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= {c1 ,c2 ,c3 ,c4 } 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 .

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.

(F,E)={F(đ?‘’đ?‘’ 1)={(đ?‘?đ?‘?1 ,0.1, 0) ),(đ?‘?đ?‘?2 ,0.5, 0), (đ?‘?đ?‘?3 , 0.1, 0), (đ?‘?đ?‘?4 ,0.4,0)},F(đ?‘’đ?‘’ 2 )={(đ?‘?đ?‘?1 ,0.6, 0) ),(đ?‘?đ?‘?2 ,0.4, 0), (đ?‘?đ?‘?3 , 0.5, 0), (đ?‘?đ?‘?4 ,0.7,0)},F(đ?‘’đ?‘’ 3 )={(đ?‘?đ?‘?1 ,0.5, 0) ),(đ?‘?đ?‘?2 ,0.7, 0), (đ?‘?đ?‘?3 , 0.6, 0), (đ?‘?đ?‘?4 , 0.5, 0)}}.

VII. ACKNOW LEDGM ENTS

(G,E)={G( đ?‘’đ?‘’ 1)={(đ?‘?đ?‘?1 ,0.2, 0) ),(đ?‘?đ?‘?2 ,0.6, 0), (đ?‘?đ?‘?3 , 0.2, 0), (đ?‘?đ?‘?4 ,0.3,0)}, G(đ?‘’đ?‘’ 2 )={(đ?‘?đ?‘?1 ,0.6, 0) ),(đ?‘?đ?‘?2 ,0.5, 0), (đ?‘?đ?‘?3 , 0.6, 0), (đ?‘?đ?‘?4 ,0.8,0)}, ,G( đ?‘’đ?‘’ 3 )={(đ?‘?đ?‘?1 ,0.5, 0) ),(đ?‘?đ?‘?2 ,0.8, 0), (đ?‘?đ?‘?3 , 0.7, 0), (đ?‘?đ?‘?4 ,0.5,0)}}.

These t wo fu zzy soft sets based on reference funct ion are rep resented by th e follo wing fu zzy soft mat rices based on referen ce funct ion respect ively (đ?&#x;Žđ?&#x;Ž .đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;”đ?&#x;”, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“, đ?&#x;Žđ?&#x;Ž) ⎥(đ?&#x;Žđ?&#x;Ž .đ?&#x;“đ?&#x;“, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;’đ?&#x;’, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;•đ?&#x;•, đ?&#x;Žđ?&#x;Ž)⎤ ⎢ ⎼ ⎢(đ?&#x;Žđ?&#x;Ž .đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;”đ?&#x;”, đ?&#x;Žđ?&#x;Ž)⎼ ⎢(đ?&#x;Žđ?&#x;Ž .đ?&#x;’đ?&#x;’, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;•đ?&#x;•, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“, đ?&#x;Žđ?&#x;Ž)⎼ ⎢ ⎼

A=

B=

(đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. )(đ?&#x;Žđ?&#x;Ž. đ?&#x;”đ?&#x;”, đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“, đ?&#x;Žđ?&#x;Ž) ⎥ (đ?&#x;Žđ?&#x;Ž . đ?&#x;”đ?&#x;”,đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž . đ?&#x;“đ?&#x;“,đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;–đ?&#x;–, đ?&#x;Žđ?&#x;Ž) ⎤ ⎢ ⎼ ⎢ (đ?&#x;Žđ?&#x;Ž . đ?&#x;?đ?&#x;?,đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž . đ?&#x;”đ?&#x;”,đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;•đ?&#x;•, đ?&#x;Žđ?&#x;Ž) ⎼ ⎢ (đ?&#x;Žđ?&#x;Ž . đ?&#x;‘đ?&#x;‘,đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž . đ?&#x;–đ?&#x;–,đ?&#x;Žđ?&#x;Ž)(đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“, đ?&#x;Žđ?&#x;Ž) ⎼ ⎢ ⎼

Then, th e fu zzy soft co mp lement matricesbased on reference funct ion are (đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;”đ?&#x;”)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“) ⎥(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;’đ?&#x;’)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;•đ?&#x;•)⎤

(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž.đ?&#x;?đ?&#x;?)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž.đ?&#x;”đ?&#x;”)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž.đ?&#x;“đ?&#x;“) ⎥(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž.đ?&#x;”đ?&#x;”)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž.đ?&#x;“đ?&#x;“)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž.đ?&#x;–đ?&#x;–)⎤

đ??´đ??´ đ?‘?đ?‘? =⎢⎢(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;?đ?&#x;?)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;”đ?&#x;”)⎼⎼,đ??ľđ??ľ đ?‘?đ?‘? =⎢⎢(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž.đ?&#x;?đ?&#x;?)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž.đ?&#x;”đ?&#x;”)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž.đ?&#x;•đ?&#x;•)⎼⎼ ⎢(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;’đ?&#x;’)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;•đ?&#x;•)(đ?&#x;?đ?&#x;?, đ?&#x;Žđ?&#x;Ž. đ?&#x;“đ?&#x;“)⎼ ⎢ ⎼

Copyright Š 2012 M ECS

The authors wou ld like to than k the anony mous rev iewers fo r their carefu l read ing o f th is paper and for their help fu l co mments.

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Dr. FlorentinS marandacheis a Professor of Mathematics at the University of New Mexico in USA. He published over 75 books and 200 articles and notes in mathematics, physics, philosophy, psychology, rebus, literature. In mathematics his research is in number theory, non-Euclidean geomet ry, synthetic geometry, algebraic structures, statistics, neutrosophic logic and set (generalizations of fuzzy logic and set respectively), neutrosophic probability (generalizat ion of classical and imp recise probability).A lso, small contributions to nuclear and particle physics, informat ion fusion, neutrosophy (a generalization of dialectics), law of sensations and stimu li, etc. He got the 2010 Telesio-Galilei Academy of Science Go ld Medal,

I.J. In formation Engineering and Electronic Business, 2013, 2, 52-59

On Fu zzy Soft Matrix Based on Reference Function

59

Adjunct Professor (equivalent to Doctor HonorisCausa) of Beijing Jiaotong University in 2011, and 2011 Ro manian Academy Award for Technical Science (the highest in the country). Dr. W. B. VasanthaKandasamy and Dr.Flo rentinSmarandache got the 2012 and 2011 New Mexico-Arizona Book Award for A lgebraic Structures.

Said Broumi is an Admin istrator of Hassan II university Mohammedia- Casablanca. He worked in University for five years. He received his M.Sc in Industrial Automatic fro m Hassan II University Ainchok- Casablanca.

His research concentrates on soft set theory, fuzzy theory, intuition istic fuzzy theory ,neutrosophic theory, control systems.

Mamoni Dharis an Assistant Professor in the department of Mathematics, Science College, Kokrajhar, Assam, India. She received M.Sc degree fro m Gauhati University, M.Ph il degree from Madurai Kamraj University, B.Ed fro m Gauhati University and PGDIM fro m India Gandhi National Open University. Her research interest is in Fuzzy Mathematics. She has published eighteen articles in different national and international journals .

Copyright Š 2012 M ECS

I.J. In formation Engineering and Electronic Business, 2013, 2, 52-59