Some Results of Intuitionistic Fuzzy Soft Sets and. its Application in Decision Making

Transcription

1 pplied Mathematial Sienes, Vol. 7, 2013, no. 95, HIKRI Ltd, Some Results of Intuitionisti Fuzzy Soft Sets and its ppliation in Deision Making. hetia Department of Mathematis, rahmaputra Valley ademy North Lakhimpur ,ssam, India P. K. Das Department of Mathematis, NERIST, Nirjuli , Itanagar, runahal Pradesh, India opyright hetia and P. K. Das. This is an open aess artile distributed under the reative ommons ttribution Liense, whih permits unrestrited use, distribution, and reprodution in any medium, provided the original work is properly ited. bstrat The soft set is one of the reent topis developed for dealing with the unertainties present in most of our real life situations. The parametrization tool of soft set theory enhanes the flexibility of its appliations. In this paper, we establish ( some results in ontinuation to agman et al.[3]. Keywords: Soft sets, Intuitionisti Fuzzy soft sets, and Deision making 1. Introdution Zadeh[10] introdued fuzzy set theory to represent unertainty and vagueness mathematially with formalized logial tools for dealing with the impreision inherent in many real world problems. gain tanassov[1] generalised the fuzzy set by defining the intuitionisti fuzzy to many situations where fuzzy set theory

2 4694. hetia and P. K. Das annot be suitably applied but intuitionisti fuzzy set theory an be used for a fair analysis. Maji et al.[ 6,7,8] have further studied the theory of soft sets introdued by Molodtsov[9]and introdued the onept of fuzzy soft sets[7] and intuitionisti ( fuzzy soft sets[8]. agman et al. [2] redefined the operations of Molodtsov s soft sets to make them more funtional for obtaining several new results. lso,they[3] further redefined the onept of intuitionisti fuzzy soft sets introdued by Maji et al.[8] and studied some of its operations. The purpose of present paper is to ontinue his study by defining a few new operations of intuitionisti fuzzy soft(ifs) sets and reprove various results on intuitionisti fuzzy soft set theory along with an appliation in deision making problem. 2. brief review of soft sets theory and intuitionisti fuzzy soft sets theory: In order to make our disussion self-ontained, we summarize below some of the basi definitions and results on the theory of soft sets and intuitionisti fuzzy soft sets introdued by Molodtsov [9] and subsequently redefined by ( agman et al. [2,3]. Throughout this work, U refers to an initial universe, E a set of parameters, P(U) the power set of U and F(U) denotes the set of all intuitionisti fuzzy sets of U. Definition 2.1[ 2] For E, a soft set (f, E) or F on the universe U is defined by the set of ordered pairs (f, E )= F = {(e, f (e)) : e E, f (e) P (U )}, where f : E P (U ) suh that f (e) = φ if e. Here, f is alled an approximate funtion of the soft set F.The set f (e) is alled e-approximate value set or e-approximate set whih onsists of related objets of the parameter e E. Let S(U) be the set of all soft sets over U. Example 2.1 Let U={ 1, 2, 3 } be the set of three ars and E ={ostly(e 1 ), metalli olour (e 2 ), heap (e 3 )} be the set of parameters, where ={e 1, e 2 } E. Then f (e 1 )={ 1, 2, 3 }, f (e 2 )={ 1, 3 }, and we write a risp soft set (f, E)= F ={( e 1, { 1, 2, 3 }), (e 2,{ 1, 3 )}over U whih desribes the attrativeness of the ars whih Mr. S(say) is going to buy. Definition 2.2[2] Let F S(U). If f (x) =φ for all x E, then F is alled an empty soft set, denoted by F φ.

3 Intuitionisti fuzzy soft sets 4695 Definition 2.3[ 2] Let F S(U). If f (x) =U for all x, then F is alled -universal soft set, denoted by F %. If = E, then the -universal soft set is alled universal soft set denoted by F E %. Definition 2.4[2 ] Let F, F S(U). Then F is a soft subset of F, denoted by F % F, if f ( x) f ( x) for all x E. Remark 1.[2] F % Fdoes not imply that every element of F is an element of F. Therefore the definition of lassial subset is not valid for the soft subset. For example, Let U = { u1, u2, u3, u4} be a universal set of objets and E = { x1, x2, x3} be the set of all parameters. If = { x1} and = { x1, x3}, and F = {( x1,{ u2, u4})}, and F = {( x1,{ u2, u3, u4})}, ( x3{ u1, u5})}, for all e E then f( x) f( x) is valid. Hene F F. ut ( x, f ( x )) F where as ( x, f ( x )) F. % Definition 2.5[2] Let F,F S(U).Then, F and F, are soft equal, denoted by F =F only if f ( x) = f ( x) for all x E. Definition 2.6[2 ] Let F S(U). Then the omplement of F, denoted by if and F %, is a soft set defined by the approximate funtion f % ( x) = f ( x) for all x E, where f ( ) x is the omplement of the set f ( x ), that is, f ( x ) =U\ f ( x ) for all x E. Definition 2.7[2 ] Let F,F S(U). Then, union of F and F, denoted by F % F, is a soft set defined by the approximate funtion f % ( x) = f( x) f( x) for all x E. Definition 2.8[ 2] Let F,F S(U). Then, intersetion of F and F, denoted by F % F, is a soft set defined by the approximate funtion f % ( x) = f( x) f( x) for all x E. Definition 2.9[ 2] Let F,F S(U). Then, differene of two soft sets F and F, denoted by F \ F, is a soft set defined by the approximate funtion f \ ( x) = f( x)\ f( x) for all x E. Definition 2.10[3] n IFS set (f, E) or IF on the universe U is defined by the set of ordered pairs (f, E )= IF = {(e, f (e)) : e E, f (e) F(U) }, where f : E F(U) suh that f (e) = null intuitionisti fuzzy set of U, if e.

4 4696. hetia and P. K. Das Example 2.2 Let U={ 1, 2, 3 } be the set of three ars, E ={ostly(e 1 ), metalli olour(e 2 ), getup(e 3 )} be the set of parameters and ={e 1,e 2 } E f ( e) = / (.6,.3), / (.4,.5), / (.3,.6), and 1 { } f ( e ) { / (.5,.3), / (.7,.2), / (.8,.2)} 2 = Then, IF = {( e1, f ( e1)),( e2, f ( e2))} is the IFS set over U desribing the attrativeness of the ars whih Mr. S(say) is going to buy. Definition 2.11[3] Let IF be an IFS set. If f ( x) = φ, the null intuitionisti fuzzy set of U, for all x E, then IF is alled a empty IFS set, denoted by IF φ. Definition 2.12[3] Let IF be an IFS set. If f ( x) = U, for all x then IF is alled -universal IFS If IF, IF be two IFS sets,then IF and IF are intuitionisti fuzzy soft equal, denoted by IF = IF, if and only if f ( x) = f ( x) for all x E. % Let IF be an IFS set, then the omplement of IF,denoted by IF is an IFS set set, denoted by IF. If = E then -universal IFS set is alled universal IFS set, denoted by IF. E% % Definition 2.13[ 3 ] If IF, IF be two IFS sets, then IF is an IFS subset of IF, denoted by IF IF, if f ( x) is an intuitionisti fuzzy subset of f ( x) for all x E. Definition 2.14[ 3] Definition 2.15[3] defined by the approximate funtion f % ( x) = f ( x), for all x E, where f ( x) is omplement of the intuitionisti fuzzy set f ( x). Remark: We use two different symbols ' % ' and ' ' whih represent the omplement of IFS set and intuitionisti fuzzy set respetively. ut, in the subsripts, the symbol ' % ' indiates that f % is the approximate funtion of IF % whih is not a set operation. We will write < μ f ( ), ( ) for ( ) e ν f e > f e and IF(U) for the set of all IFS sets over U and prove the results using this notation..

5 Intuitionisti fuzzy soft sets 4697 Definition 2.16[3] If IF, IF IF( U), then, union of IF and IF, denoted by IF U IF, is an IFS set defined by the approximate funtion f ( x) = < μ, ν > = < max{ μ, μ }, min{ ν, ν } >, % f ( x) f ( x) f ( x) f ( x) f ( x) f ( x) % % for all x E. Definition2.17[3] If IF, IF IF( U),then, intersetion of IF and IF, denoted by IF %I IF, is an IFS set defined by the approximate funtion f ( x) = < μ, ν > = < min{ μ, μ }, max{ ν, ν } >, % f ( x) f ( x) f ( x) f ( x) f ( x) f ( x) % % for all x E. Proposition 2.1 [3] If IF, IF IF( U), then % % % () i ( IF U% IF ) = IF I% IF. % % % ( ii) ( IF I% IF ) = IF U% IF. Proof. For all x E, () i f % ( x) =< μ, ν > ( %U ) (ii) Similar to (i). f ( ) ( ) ( % x f ) ( % ) % x U % U = < 1 μ, 1-ν > f ( x) f ( x) U% U% = < 1 [max( μ, μ )],1 [min( ν, ν )] > = < min{1-μ = < μ, ν > = f % %( x). %I f( x) f( x) f( x) f( x) f ( x) f( x) f( x) f( x) f ( ) ( ) % x f x I% % % I% %,1 μ },max{1- ν,1 ν } > Proposition 2.2[ 3] If IF, IF, IF IF( U), then () i IF U% ( IF I% IF ) = ( IF U% IF ) I% ( IF U% IF ). ( ii) IF I% (IF U% IF ) = ( IF I% IF ) U% ( IF I% IF ).

6 4698. hetia and P. K. Das Proof. For all x E () i f ( x) =< μ, ν > U% ( I% ) f ( x) f ( x) U% ( I% ) U% ( I% ) = < max( μ, μ ),min( ν, ν ) > f( x) f ( x) f( x) f ( x) I% I% = < max( μ,min( μ, μ )),min( ν,max( ν, ν )) f( x) f( x) f( x) f( x) f( x) f( x) = < min(max(( μ, μ ),max( μ, μ )),max(min(( ν, ν ),min( ν, ν )) > ( U% ) I% ( U% ) f( x) f( x) f( x) f( x) f( x) f( x) f( x) f( x) = < min( μ, μ ),max( ν, ν ) > = f ( x). f ( x) f ( x) f ( x) f ( x) U% U% U% U% (ii) Similar to (i). 3. Some new operations Definition 3.1 If IF, IF IF( U ), then, '+' operation of IF and IF, denoted by IF + IF is an IFS set defined by the approximate funtion f ( x) = < μ, ν > = < μ + μ μ. μ, ν. ν > + f ( x) f ( x) f ( x) f ( x) f ( x) f ( x) f ( x) f ( x) + + for all x E. Definition 3.2 If IF, IF IF ( U ), then, Dot of IF and IF, denoted by IF. IF is an IFS set defined by the approximate funtion f ( x) = < μ, ν > = < μ. μ, ν + ν ν. ν >. f ( x) f ( x) f ( x) f ( x) f ( x) f ( x) f ( x) f ( x).. for all x E. Definition 3.3 If IF, IF IF( U), then, '@' operation of IF and IF, denoted by IF is an IFS set defined by the approximate funtion μf ( x) + μf ( x) ν f ( x) + ν f ( x) f@ ( x) = < μf ( x), ν f ( x) > = 2 2 for all x E.

7 Intuitionisti fuzzy soft sets 4699 Definition 3.4 If IF, IF IF( U),then, '$' operation of IF and IF, denoted by IF$ IF is an IFS set defined by the approximate funtion f () x = < μ, ν > = < μ. μ, ν. ν > $ f ( x) f ( x) f ( x) f ( x) f ( ) ( ) $ $ x f x for all x E. Example 3.1 Let U= {h 1, h 2, h 3, h 4 } be a universal set and E = {e 1, e 2, e 3, e 4 } be set of parameters. Let = { e 2, e 3, e 4 }, ={e 1, e 2, e 4 }and = { e 2, e 3 }. Suppose, f (e 2 ) = {h 1 /(.4,.1), h 2 /(1,0), h 3 /(.3,.5), h 4 /(.5,.2)}, f (e 3 )={h 1 /(.2,.5), h 2 /(.3,.4), h 3 /(1,0), h 4 /(.4,.1)}and f (e 4 )={h 1 /(.6,.1),h 2 /(.4,.3),h 3 /(.3,.5),h 4 /(.6,.1)}, f (e 1 )={h 1 /(.3,.2),h 2 /(.2,.4), h 3 /(.5,.1),h 4 /(.6,.2)},f (e 2 )={h 2 /(.4,.1),h 2 /(.3,.3),h 3 /(.5,.2),h 4 /(.7,.1)}and f (e 4 )={h 1 /(.2,.5), h 2 /(.3,.4), h 3 /(.6,.1), h 4 /(.7,.2)}, f (e 2 )={h 1 /(.2,.4), h 2 /(.5,.1), h 3 /(.6,.2), h 4 /(.4,.3)} and f (e 3 )={h 1 /(.6,.1), h 2 /(.5,.2), h 3 /(.4,.3), h 4 /(.1,.5)}. Therefore IF = {( e h /.3,.2, h /.2,.4, h /.5,.1, h /.6,.2 ), IF = {( e { h /.4,.1, h / 1,0, h /.3,.5, h /.5,.2 }), ( e h /.2,.5, h /.3,.4, h / 1,0, h /.4,.1 ), ( e h /.6,.1, h /.4,.3, h /.3,.5, h /.6,.1 )} 1, ( e h /.4,.1, h /.3,.3, h /.5,.2, h /.7,.1 ), ( e h /.2,.2, h /.3,.4, h /.6,.1, h /.7,.2 )} IF = {( e, h /.2,.4, h /.5,.1, h /.6,.2, h /.4,.3 ), { } ( e, h /.6,.1, h /.5,. 2, h /.4,.3, h /.1,.5 )}. Then, IF. IF = {( e h /.0,.2, h / 0,.4, h / 0,.1, h /.0,.2 ), 1, { 1 ( ) 2 ( ) 3 ( ) 4 ( )} ( e h /.16,.19, h /.3,.3, h /.15,.6, h /.35,.28 ), ( e h / 0,.5, h / 0,.4, h / 0,0, h /.0,.1 ), ( e h /.12,.55, h /.12,.58, h /.18,.55, h /.42,.28 )} Similarly, IF + IF, IF and IF $ IF an be derived.

8 4700. hetia and P. K. Das Proposition 3.1 If IF, IF, IF IF( U), then () i IF. IF = IF. IF ( ii) IF + IF = IF + IF ( iii) IF = IF ( iv) IF $ IF = IF $ IF ( v)( IF. IF ). IF = IF.( IF. IF ) ( vi)( IF + IF ) + IF = IF + ( IF + IF ). On the other hand, it is seen that ( IF )@ IF IF ) ( IF $ IF )$ IF IF $( IF $ IF ) i.e., though the all four operations mentioned above are ommutative but only + and. operations are assoiative and other two operations don t satisfy assoiative property. Example 3.2 From example 3.1, we have ( IF. IF ). IF = {( e h /.0,.2, h / 0,.4, h / 0,.1, h /.0,.2 ), 1, { 1 ( ) 2 ( ) 3 ( ) 4 ( )} (,.55), / ( 0,.58 ), / (.0,.55), / ( 0,.28) )} 1, ( e h /.032,.514, h /.15,.37, h /.09,.68, h /.14,.49 ), ( e h / 0,.55, h / 0,.52, h / 0,.3, h /.0,.55 ), ( e h / 0,.55, h / 0,.58, h /.0,.55, h / 0,.28 )} and IF.( IF. IF ) = {( e h /.0,.2, h / 0,.4, h / 0,.1, h /.0,.2 ), ( e h /.032,.514, h /.15,.37, h /.09,.68, h /.14,.49 ), ( e h / 0,.55, h / 0,.52, h / 0,.3, h /.0,.55 ), ( e h / 0 h h h Therefore, ( IF. IF ). IF = IF.( IF. IF ) is verified.

9 Intuitionisti fuzzy soft sets 4701 Proposition 3.2 If IF, IF, IF IF( U), then () i ( IFU% IF) + IF = ( IF + IF) U% ( IF + IF), ( ii) ( IFI% IF) + IF = ( IF + IF) I% ( IF + IF), ( iii) ( IFU% IF). IF = ( IF. IF) U% ( IF. IF), ( iv) ( IF I% IF ). IF = ( IF. IF ) I% ( IF. IF ), ( v) ( IF U% IF ). IF = ( IF. IF ) U% ( IF. IF ), ( vi) ( IF I% IF ). IF = ( IF. IF ) I% ( IF. IF ), ( vii) ( IF I% IF )@ IF = ( IF ) I% ( IF ), ( viii) ( IF U% IF )@ IF = ( IF ) I% ( IF ), ( ix) ( IF ) + IF = ( IF + IF )@( IF + IF ), ( x) ( IF ). IF = ( IF. IF )@( IF. IF ), ( xi) ( IF + IF ). IF % ( IF. IF ) + ( IF. IF ), ( xii) ( IF + IF )@ IF % ( IF ) + ( IF ), ( xiii) ( IF. IF ) + IF % ( IF + IF ).( IF + IF ), ( xiv) ( IF. IF )@ IF % ( IF ).( IF ). Definition 3.5 f ( x) f ( x) f( x) f( x) If IF IF( U),then, the neessity operation of IF denoted by IF is an IFS set defined by the approximate funtion f ( x) = < μ, ν > = < μ, 1-μ > for all x E. Definition 3.6 If IF IF( U),then, the possibility operation of IF denoted by IF is an IFS set defined by the approximate funtion f ( x) = < μ, ν > = < 1 ν, ν > for all x E. f ( x) f ( x) f( x) f( x) Example 3.3 From example 3.1 we have

10 4702. hetia and P. K. Das ( ) ( ) ( ) IF = {( e { h / 0,1, h / 0,1, h / 0,1, h / 0,1}), 1, ( e h /.4,.6, h / 1,0, h /.3,.7, h /.5,.5 ), ( e h /.2,.8, h /.3,.7, h / 1,.0, h /.4,.6 ), ( e h /.6,.4, h /.4,.6, h /.3,.7, h /.6,.4 )} and IF = {( e { h / 1, 0, h / 1, 0, h / 1, 0, h / (1, 0}), 1, ( e h /.9,.1, h / 1,0, h /.5,.5, h /.8,.2 ), ( e h /.5,.5, h /.6,.4, h / 1,.0, h /.9,.1 ), ( e h /.9,.1, h /.7,.3, h /.5,.5, h /.9,.1 )}. Proposition 3.3 If IF, IF IF( U ), then ( i) ( IF % IF) = IF % IF, (ii) ( IF % IF ) = IF % IF, ( iii) IF = IF, ( iv) ( IF % IF) = IF % IF, ( v) ( IF % IF ) = IF % IF, ( vi) IF = IF, ( vii) ( IF ) = ( x) ( IF$ IF) % IF$ IF. Definition 3.7 ( viii) ( IF ) = IF, ( ix) ( IF $ IF ) % IF $ IF, IF, If IF IF( U), then, the'!' operation of IF denoted by!( IF )is an IFS set defined by the approximate funtion 1 1 f!( ) ( x) = < μf ( x), ν f ( x) > = < max{, μf ( )},min{, ( )} for all.!( )!( ) x ν f x > x E 2 2 Definition 3.8 If IF IF( U), then, the '?' operation of IF denoted by?( IF )is an IFS set defined by the approximate funtion 1 1 f?( ) ( x) = < μf ( x), ν f ( x) > = < min{, μf ( )},max{, ( )} for all.?( )?( ) x ν f x > x E 2 2

11 Intuitionisti fuzzy soft sets 4703 Example 3.4 From example 3.1 we have! IF = {( e { h /.5,0, h /.5,0, h /.5,0, h /.5,0 }), and 1, ( e h /.5,.1, h / 1,0, h /.5,.5, h /.5,.2 ), ( e h /.5,.5, h /.5,.4, h / 1,0, h /.5,.1 ), ( e h /.6,.1, h /.5,.3, h /.5,.5, h /.6,.1 )}? IF = {( e { h / 0,.5, h / 0,.5, h / 0,.5, h / 0,.5 }), 1, ( e h /.4,.5, h /.5,.5, h /.3,.5, h /.5,.5 ), ( e h /.2,.5, h /.3,.5, h /.5,.5, h /.4,.5 ), ( e h /.5,.5, h /.4,.5, h /.3,.5, h /.5,.5 )} Proposition 3.4 If IF, IF IF( U),then ( i)!( IF % IF) =!( IF) %!( IF) (ii)!( IF % IF) =!( IF) %!( IF) ( iii)?( IF % IF) =?( IF) %?( IF) ( iv)?( IF % IF ) =?( IF ) %?( IF ) Proposition 3.5 If IF IF( U), then, () i! IF =! IF. ( ii)? IF =? IF. ( iii)! IF =! IF. ( iv)? IF =? IF. Definition 3.9 If IF IF( U), then, the ' ' operation of IF denoted by ( IF )is an IFS set defined by the approximate funtion μ ν + 1 f( x) f( x) f ( ) ( x) = < f ( x), f ( x) > = <, > for all x E. ( ) ( ) μ ν 2 2

12 4704. hetia and P. K. Das Definition 3.10 If IF IF( U). Then, the ' ' operation of IF denoted by ( IF )is an IFS set defined by the approximate funtion + 1 ν f( x) f( x) f ( ) ( x) = < μf ( x), ν f ( x) > = <, > for all x E. ( ) ( ) Proposition 3.6 If IF, IF IF( U ), then ( i) ( IF % IF ) = ( IF ) % ( IF ). (ii) ( IF % IF ) = ( IF ) % ( IF ). ( iii) ( IF % IF ) = ( IF ) % ( IF ). ( iv) ( IF % IF ) = ( IF ) % ( IF ). ( v) ( IF + IF ) % ( IF ) + ( IF ). ( vi) ( IF + IF ) % ( IF ) + ( IF ). ( viii) ( IF. IF ) % ( IF ) + ( IF ). ( ix) ( IF $ IF ) % ( IF )$ ( IF ). ( x) ( IF $ IF ) % ( IF )$ ( IF ). Proposition 3.7 μ If IF IF( U), then () i IF = IF ( ii) IF = IF ( iii) IF = IF ( iv) IF = IF Example 3.5 From example 3.1, we have 2 2 ( ) ( ) ( ) h ( ) IF = {( e { h / 0,.5, h / 0,.5, h / 0,.5, h / 0,.5 }), ( vii) ( IF. IF ) % ( IF ). ( IF ). 1, { 4 } ( e h /.2,.55, h /.5,.5, h /.15,.75, h /.25,.6 ), ( e h /.1,.75, h /.15,.7, h /.5,.5, h /.2,.55 ), ( e h /.3,.55, h /.2,.65, h /.15,.75, /.3,.55 )} and 4, 1 2 3

13 Intuitionisti fuzzy soft sets 4705 ( ) ( ) ( ) h ( ) IF = {( e { h /.15,.6, h /.1,.7, h /.25,.55, h /.3,.6 }), 1, { 4 } ( e h /.2,.55, h /.15,.65, h /.25,.6, h /.35,.55 ), ( e h / 0,.5, h / 0,.5, h / 0,.5, h / 0,.5 ), ( e h /.1,.75, h /.15,.7, h /.3,. 55, /.35,.6 )}.Then 4, ( IF % IF ) = ( IF ) % ( IF ) = {( e { h /.15,.5, h /.1,.5, h /.25,.5, h /.3,.5 }), 1, ( e h /.2,.55, h /.5,.5, h /.25,.6, h /.35,.55 ), ( e h /.1,.5, h /.15,.5, h /.5,.5, h /.2,.5 ), ( e h /.3,.55, h /.2,.65, h /.3,.55, h /.35,.55 )} Definition 3.11 If IF, IF IF( U ),then, ' a ' operation of IF and IF, denoted by IF a IF is an IFS set defined by the approximate funtion f ( x) =< μ, ν >=< max{ ν, μ },min{ μ, ν } > a f ( x) f ( x) f( x) f( x) f( x) f( x) a a for all x E. Proposition 3.8 If IF, IF, IF IF( U), then % % () i IF a IF = IF ( ii) ( IFI% IF) a IF = ( IF a IF) U% ( IF a IF) ( iii) ( IF U% IF ) a IF = ( IF a IF ) I% ( IF a IF ) Example 3.6 From example 3.1 we have ( IF I% IF ) a IF = ( IF a IF ) U% ( IF a IF ) ( ) ( ) = ( e { h /(.2,0), h /.4,0, h /(.1,0), h /.2,0 }), 1, ( e h /.2,.4, h /.5,.1, h /.6,.2, h /.4,.3 ), ( e h /.6,0, h /.5,0, h /.4,0, h /.1,0 ), ( e h /.5,0, h /.4,0, h /.5,0, h /.2,0 )}.

14 4706. hetia and P. K. Das Definition 3.12[3 ] If IF, IF IF( U), then, ' ' operation of IF and IF, denoted by IF IF is an IFS set defined by the approximate funtion f : E E IF( U) where f ( x, y) =< μ, ν >=< min{ μ, μ },max{ ν, ν } > f ( x, y) f ( x, y) f ( x) f ( y) f ( x) f ( y) for all x E. Definition 4.13[ 3 ] If IF, IF IF( U),then, ' ' operation of IF and IF, denoted by IF IF is an IFS set defined by the approximate funtion f : E E IF( U) where f ( x, y) =< μ, ν >=< max{ μ, μ }, f ( x, y) f ( x, y) f ( x) f ( y) min{ νf x, ν f y} > for all x E. ( ) ( ) Proposition: 3.9[ 3] Let IF, IF IF U, then ( ) () i ( IF IF ) = IF IF % % % ( ii) ( IF IF ) = IF IF % % % Proof: With the help of approximation funtions, these proofs an be done. For all x,y E () i f % ( x, y) f ( x, y) = ( ) (ii) Similar to (i). ( ) = < μ, ν > f ( x, y) f ( x, y) = < (max{ μ, μ }),(min{ ν, ν }) > = < min{ μ, μ ( ) f ( x) f ( y) f ( x) f ( y) f ( x) f ( y) = < μ ( xy, ), ν ( xy, ) > f f = f ( x, y). },max{ ν, ν } > f ( x) f ( y) 4. n ppliation Of Intuitionisti Fuzzy Soft Sets to Deision Making Problems agman et al.[5 ]applied the theory of fuzzy soft sets to solve a deision making problem. In this setion, we present an appliation of intuitionisti fuzzy soft sets to a different type of deision making problem by generalising the onept of agman et al.[5] to make it ompatible with our work. For this, we start with a few definitions.

15 Intuitionisti fuzzy soft sets 4707 Definition 4. 1 Let IF IF( U), U = { u1, u2, u3,..., um}, E = { e1, e2, e3,..., en} and and E, then IF an be represented by the following table, IF e e... e 1 2 u μ ( u ), ν ( u ) μ ( u ), ν ( u )... μ ( u ), ν ( u ) 1 f( e1) 1 f( e1) 1 f( e1) 1 f( e1) 1 f( en) 1 f( en) 1 u μ ( u ), ν ( u ) μ ( u ), ν ( u )... μ ( u ), ν ( u ) 2 f( e1) 2 f( e1) 2 f( e2) 2 f( e2) 2 f( en) 2 f( en) u μ ( u ), ν ( u ) μ ( u ), ν ( u )... μ ( u ), ν ( u ) m f( e1) m f( e1) m f( e2) m f( e2) m f( en) m f( en) m where μ f ( ) and e ν f( e) are the membership and non membership funtions of f respetively. ( μ, ν ) = μ ( u ), ν ( u ) where i=1, 2,.m and j=1, 2, n then IFS If ij ij f ( e j) i f ( e j) i set IF is uniquely determined by a matrix n [( μ, ν )] ij ij m n ( μ11, ν11) ( μ12, ν12)... ( μ1n, ν1n) ( μ, ν ) ( μ, ν )... ( μ, ν ) n 2n = ( μm 1, νm 1) ( μm2, νm2)... ( μmn, νmn) is alled an m n IFS matrix of IF over U. Definition 4. 2 Let IF IF( U). Then the ardinal set of IF, denoted by ( IF ) and is defined by ( IF ) = { μ ( x), ν ( x) } x: x E} is an intuitionisti fuzzy set over E ( IF) ( IF) where the membership and non-membership funtions μ μ ( x) and ν ( x) of ( IF ) are respetively defined by ( IF) ( IF) : E [0,1], μ = ( IF) ( IF) μ ( x) f ν f ( x) and ν( IF ) : E [0,1], ν ( IF) = U U

16 4708. hetia and P. K. Das where U is the ardinality of the universe U and μf ( x) and ν ( ) are the salar f x ardinilities of the intuitionisti fuzzy set f ( x) respetively. Remarks: The set of all ardinal sets of the IFS sets over U will be denoted by IF ( ) U. Definition 4. 3 Let IF IF( U) and ( IF ) IF ( U ). Suppose that 1 2 E = { x, x,..., x } and E, n then ( IF ) an be presented by the following table E x1 x2... xn μ, ν μ ( x ), ν ( x) μ ( x), ν ( x)... μ ( x), ν ( x) ( IF ) ( IF ) ( IF ) 1 ( IF ) 1 ( IF ) 2 ( IF ) 2 ( IF ) n ( IF ) n Note that if ( μ, ν ) = μ ( x ), ν ( x ) for j= 1,2,..., nthen the ardinal set ( IF ) 1j 1 j ( IF) j ( IF) j is uniquely haraterized by a matrix, [ μ, ν ] = [( μ, ν ),( μ, ν )...( μ, ν )] whih is alled the ardinal matrix of the 1j 1j 1 n n 1n ardinal set ( IF ) over E. Definition 4. 4 Let IF IF( U) and ( IF ) IF ( U). Then the intuitionisti fuzzy soft aggregation operator, denoted by (IF),is defined by g (IF) : IF ( U ) IF( U ) F( U ) suh that ( IF ) (( IF ), IF ) = IF g g where IF = { μ ( u), ν ( u) / u : u U} IF IF is an intuitionisti fuzzy set over U.Here IF is alled the aggregate intuitionisti fuzzy set of the IFS set IF.The membership funtion μ and and non membership funtion ν of IF are defined as follows: IF 1 μ : U [0,1] suh that μ ( u) : ( ) ( ). ( )( ) and IF IF μ IF u μ f x u E x E 1 ν : U [0,1] suh that ν ( u) : IF ( ) ( ). ( )( ) where is the IF ν IF u ν f x u E E x E ardinality of E. IF

17 Intuitionisti fuzzy soft sets 4709 Definition 4.5 Let IF IF( U) and IF be its aggregate intuitionisti fuzzy set and U={u 1,u 2,.,u m }, then IF an be presented by the following table IF ( μ, ν ) IF IF u μ ( u ), ν ( u ) 1 1 IF 1 IF u μ ( u ), ν ( u ) 2 2 IF 2 IF u μ ( u ), ν ( u ) m m IF m IF If ( μ, ν ) = μ ( u ), ν ( u ) for i = 1,2,..., m then the IF i1 i1 IF i ( IF) i is uniquely haraterized by a matrix, ( μ11, ν11) ( μ21, ν21). [ μi1, νi1] m 1 =,.. ( μm 1, νm 1) whih is alled the aggregate matrix of IF over U. Theorem 4. 1 If IM, IM and IM are representation matries of IF ( IF) IF IF,( IF ) and IF resetively where IF IF( U) and E,then E IM = IM IM, IF IF T where IM is the transposition of IM. ( IF ) Definition 4. 6 ( Sore value) ( IF ) The sore value S for u U is defined as S = ( μ ν. π ) where π = 1 μ ν. T ( IF ) i i i i j j j i j ui U

18 4710. hetia and P. K. Das Note that we are applying this theorem for omputing the aggregate intuitionisti fuzzy set of an IFS-set. lgorithm: 1.onstrut an IFS set IF over U. 2. ompute the ardinal set ( IF) of IF. 3.ompute the aggregate intuitionisti fuzzy set IF of IF. 4. ompute the sore value S i for eah u U from the set IF and arrange S i s in desending order of magnitude, say Si і S... 1 i і і 2 Si.Then S i is the 3 maximum. 5. Stop. ase Study: Suppose an investor wants to invest a part of his money in a ompany in order to get high profits.if all the ompanies proess are almost same, it is rather very diffiult for the investor to selet a ompany right way. For this ase, let the set of parameters be E={onstant growth rate(e 1 ),negative growth rate (e 2 ),low growth rate(e 3 ),medium growth rate(e 4 ),high growth rate(e 5 )}. Suppose the investor wants to selet a ompany from the view points of parameters e 1,e 4 and e 5, i.e. ={e 1,e 4,e 5 }. There are five ompanies who form the set of universe U={ 1, 2, 3, 4, 5 } where 1 is a mediine ompany 2 is a software ompany 3 is a ar manufaturing ompany 4 is a food proessing ompany and 5 is an online shopping ompany. Step 1: Then the investor onstruts an IFS set IF to selet a ompany whih an provide more profit. ie.., IF = {( e,{ / (.5,.3), / (.6,.2), / (.4,.4), / (.7,.2), / (.8,.2)}), ( e,{ / (.7,.1), / (.3,.5), / (.5,.4), / (.3,.6), / (.6,.4)}) ( e,{ / (.5,.5), / (.4,.5), / (.6,.3), / (.7,.1), / (.4,.5)})} The tabular representation of the IFS set IF is 4 5

19 Intuitionisti fuzzy soft sets e e e (.5,.3) (.7,.1) (.5,.5) (.6,.2) (.3,.5) (.4,.5) (.4,.4) (.5,.4) (.6,.3) (.7,.2) (.3,.6) (.7,.1) (.8,.2) (.6,.4) (.4,.5) Step2: Then the ardinal of the IFS set IF is ( IF ) = {(.6,.26) / e,(.48,.4) / e,(.52,.38) / e } Step3: Now from the theorem 4.3.1, we have E IM = IM IM IF IF T ( IF ) 1 T i.. e IM = ( IM ) IF IM IF ( IF ) E (.5,.3) 0 0 (.7,.1) (.5,.5) (.6,.26) (.6,.2) 0 0 (.3,.5) (.4,.5) 0 1 = (.4,.4) 0 0 (.5,.4) (.6,.3) 0 5 (.7,.2) 0 0 (.3,.6) (.6,. 3) (.48,.4) (.8,.2) 0 0 (.6,.4) (.4,.5) (.52,.38) (.179,.06) (.14,.088) = (.158,.074) (.185,.066) (.194,.08)) i.e. IF = {(.179,.06) / u,(.14,.088) / u,(.158,.074) / u,(.185,.066) / u,(.194,.08)) / u } Step4: ompute the sore fution S i for eah i

20 4712. hetia and P. K. Das i.e. S 1 =0.618, S 2 =0.423, S 3 =0.513, S 4 =0.648 and S 5 =0.693 and hene S S S S S i.e. 5 is the best alternative for the investor onlusion Molodtsov introdued the onept of soft sets, whih is a generalised mathematial tool for dealing with unertain onepts. The parametrization tool of soft set theory provides an additional advantage for its appliations. In this paper, we have introdued some new definitions and results of intuitionisti fuzzy soft ( sets that are established in ontinuation to the work of agman et al.[ 3]. & Referenes [1] K.T., tanassov, Intuitionisti Fuzzy Sets- Theory and ppliations, Physia-Verlag, Springer-Verlag ompany, New York (1999). ( ( [2] N., agman and S., Enginoglu, Soft set theory and uni-int deision & making, European Journal of Operational Researh, 207(2)(2010) ( [3] N., agman and S., Karatas, Intuitionisti Fuzzy Soft set theory and its & deision making, Journal of Intelligent and fuzzy Systems, 24(2013) ( [4] N., agman and I., Dali, Similarity measures of Intuitionisti fuzy soft sets & and their deision making,arxiv: v1[math.lo]3 Jan ( ( [5] N., agman and S., Enginoglu, Fuzzy Soft Sets theory and its & ppliations, Iranian Journal of Fuzzy Systems, 8(3)(2011) [6] P.K., Maji., R., iswas, and.r., Roy, Intuitionisti Fuzzy Soft Sets., The Journal of Fuzzy Mathematis, 9(3) (2001) [7] P.K., Maji., R., iswas, and.r., Roy, On Intuitionisti Fuzzy Soft Sets., The Journal of Fuzzy Mathematis, 12(3) (2004) [8] P.K., Maji., R., iswas, and.r., Roy, Soft Set Theory, omputers & Mathematis with ppliations, 45(2003) [9] D.,Molodtsov, Soft Set Theory-First Results, omputers and Mathematis with ppliation, 37(1999), [10] L..,Zadeh, Fuzzy sets,inform. ontrol 8(1965) Reeived: June 16, 2013