A new approach to bipolar soft sets and its applications
aa r X i v : . [ m a t h . G M ] J un A new approach to bipolar soft setsand its applications
Faruk Karaaslan a ∗ and Serkan Karata¸s b a Department of Mathematics, Faculty of Sciences, Karatekin University, 18100 C¸ ankırı,Turkey, [email protected] b Department of Mathematics, Faculty of Arts and Sciences, Ordu University, 52200Ordu, Turkey [email protected]
Abstract
Molodtsov [25] proposed the concept of soft set theory in 1999,which can be used as a mathematical tool for dealing with problemsthat contain uncertainty. Sabir and Naz [30] defined notion of bipolarsoft set in 2013. In this paper, we redefine concept of bipolar soft setand bipolar soft set operations as more functional than former. Alsowe study on their basic properties and we present a decision makingmethod with application.
Keyword 0.1
Soft set, bipolar soft set, decision making, bipolar softset operations
In the real life, problems in economy, engineering, environmental scienceand social science and many fields involve data that contain uncertainties.These problems may not be successfully modeled by existing methods inclassical mathematics. There are some well known mathematical theories fordealing with uncertainties such as; fuzzy set theory [35], intuitionistic fuzzyset theory [4], rough set [27] and so on. But all of these theories have theirown difficulties which are pointed out in [25]. To cope with these difficulties,Molodtsov proposed the concept of soft set as a new mathematical tool fordealing with uncertainties. Then Maji et al. [22], equality of two soft sets, ∗ corresponding author In this section, we will remind the definition of soft set which is definedby Molodtsov [25] and present contributions to soft set which is made byC¸ a˘gman [10]. Moreover, we will give some properties of these topics. Through-out this paper, we will denote initial universe, set of parameters and powerset of U by U , E and P ( U ), respectively. Definition 2.1 [25] Let consider a nonempty set A , A ⊆ E . A pair ( F, A ) is called a soft set over U , where F is a mapping given by F : A → P ( U ) .
2n this paper, we will use following definition which was defined by C¸ a˘gman[10] and we will do our work based on this definition.
Definition 2.2 [10] A soft set F over U is a set valued function from E to P ( U ) . It can be written a set of ordered pairs F = (cid:8) ( e, F ( e )) : e ∈ E (cid:9) . Note that if F ( e ) = ∅ , then the element ( e, F ( e )) is not appeared in F . Setof all soft sets over U is denoted by S . Definition 2.3 [10] Let
F, G ∈ S . Then,i. If F ( e ) = ∅ for all e ∈ E , F is said to be a null soft set, denoted by Φ .ii. If F ( e ) = U for all e ∈ E , F is said to be absolute soft set, denoted by ˆ U .iii. F is soft subset of G , denoted by F ˜ ⊆ G , if F ( e ) ⊆ G ( e ) for all e ∈ E .iv. F = G , if F ˜ ⊆ G and G ˜ ⊆ F .v. Soft union of F and G , denoted by F ˜ ∪ G , is a soft set over U and definedby F ˜ ∪ G : E → P ( U ) such that ( F ˜ ∪ G )( e ) = F ( e ) ∪ G ( e ) for all e ∈ E .vi. Soft intersection of F and G , denoted by F ˜ ∩ G , is a soft set over U anddefined by F ˜ ∩ G : E → P ( U ) such that ( F ˜ ∩ G )( e ) = F ( e ) ∩ G ( e ) for all e ∈ E .vii. Soft complement of F is denoted by F ˜ c and defined by F ˜ c : E → P ( U ) such that F ˜ c ( e ) = U \ F ( e ) for all e ∈ E . In this section, we will redefine bipolar soft set and introduce some basicproperties.
Definition 3.1
Let E be a parameter set and E and E be two non emptysubsets of E such that E ∪ E = E and E ∩ E = ∅ . If F : E → P ( U ) and G : E → P ( U ) are two mappings such that F ( e ) ∩ G ( f ( e )) = ∅ , then triple ( F, G, E ) is called bipolar soft set, where f : E → E is a bijective function.Set of all bipolar soft sets over U is denoted by BS . We can represent abipolar soft set ( F, G, E ) as following form: ( F, G, E ) = n(cid:10) ( e, F ( e )) , ( f ( e ) , G ( f ( e ))) (cid:11) : e ∈ E and F ( e ) ∩ G ( f ( e )) = ∅ o . f F ( e ) = ∅ and G ( f ( e )) = ∅ for e ∈ E , then (cid:10) ( e, ∅ ) , ( f ( e ) , ∅ ) (cid:11) is no writtenin the bipolar soft set ( F, G, E ) . Remark 3.2
According to above form of the bipolar soft set ( F, G, E ) , wecan construct a tabular form every bipolar soft set for a finite universal set U and finite parameter set E . Therefore, let U = { u , u , . . . , u m } and E = { e , e , . . . , e n } . Then, ( F, G, E ) ( F ( e ) , G ( f ( e ))) ( F ( e ) , G ( f ( e ))) . . . ( F ( e n ) , G ( f ( e n ))) u ( a , b ) ( a , b ) . . . ( a n , b n ) u ( a , b ) ( a , b ) . . . ( a n , b n ) ... ... ... . . . ... u m ( a m , b m ) ( a m , b m ) . . . ( a mn , b mn ) where, a ij = ( , u i ∈ F ( e j )0 , u i / ∈ F ( e j ) and b ij = ( , u i ∈ G ( f ( e j ))0 , u i / ∈ G ( f ( e j )) . Note that, a ij and b ij must not be in same time. So, there are three casesfor ( a ij , b ij ) : (1 , , (0 , or (0 , . Example 3.3
Let U = { u l , u , u , u , u , u , u , u } be the universe whichare eight houses and E = { e , e , e , e , e , e , e , e } be the set of parameters.Here, e i ( i = 1 , , , , , , , stand for the parameters “ large ”, “ small ”,“ modern ”, “ standard ”, “ cheap ”, “ expensive ” “ with parking ”, and “ noparking area ” respectively. Therefore, we can chose E and E sets as E = { e , e , e , e } and E = { e , e , e , e } . Now, we define the bijective function f as f ( e i ) = ¬ e i ( i = 1 , , , . Here, the notion ¬ e i means that “ not e i ”for all i = 1 , , , . Thus, we have following results. f ( e ) = ¬ e = e f ( e ) = ¬ e = e f ( e ) = ¬ e = e f ( e ) = ¬ e = e So, we can describe following t bipolar soft sets ( F , G , E ) and ( F , G , E ) to buy a house. ( F , G , E ) = n(cid:10) ( e , { u , u , u } ) , ( e , { u , u } ) (cid:11) , (cid:10) ( e , { u , u , u } ) , ( e , { u , u , u } ) (cid:11) , (cid:10) ( e , { u , u } ) , ( e , { u , u , u , u } ) (cid:11) , (cid:10) ( e , { u , u , u , u } ) , ( e , { u , u } ) (cid:11)o F , G , E ) = n(cid:10) ( e , { u , u , u } ) , ( e , { u , u , u , u } ) (cid:11) , (cid:10) ( e , { u , u } ) , ( e , { u , u , u , u } ) (cid:11) , (cid:10) ( e , { u , u , u } ) , ( e , { u , u , u , u } ) (cid:11) , (cid:10) ( e , { u } ) , ( e , { u , u , u } ) (cid:11)o Tabular representations of bipolar soft sets ( F , G , E ) and ( F , G , E ) are intable .... ( F , G , E ) ( F ( e ) , G ( f ( e ))) ( F ( e ) , G ( f ( e ))) ( F ( e ) , G ( f ( e ))) ( F ( e ) , G ( f ( e ))) u (1 ,
0) (0 ,
1) (0 ,
1) (0 , u (0 ,
1) (1 ,
0) (0 ,
1) (0 , u (1 ,
0) (0 ,
1) (1 ,
0) (0 , u (1 ,
0) (0 ,
0) (1 ,
0) (0 , u (0 ,
0) (1 ,
0) (0 ,
1) (1 , u (0 ,
1) (0 ,
0) (0 ,
0) (1 , u (0 ,
0) (1 ,
0) (0 ,
0) (1 , u (0 ,
0) (0 ,
1) (0 ,
1) (1 , and ( F , G , E ) ( F ( e ) , G ( f ( e ))) ( F ( e ) , G ( f ( e ))) ( F ( e ) , G ( f ( e ))) ( F ( e ) , G ( f ( e ))) u (1 ,
0) (0 ,
1) (1 ,
0) (0 , u (1 ,
0) (1 ,
0) (0 ,
1) (0 , u (0 ,
1) (0 ,
1) (1 ,
0) (0 , u (1 ,
0) (0 ,
1) (1 ,
0) (0 , u (0 ,
1) (0 ,
0) (0 ,
1) (1 , u (0 ,
1) (0 ,
0) (0 ,
0) (0 , u (0 ,
1) (0 ,
0) (0 ,
1) (0 , u (0 ,
0) (0 ,
1) (0 ,
1) (0 , Definition 3.4
Let ( F , G , E ) , ( F , G , E ) ∈ BS . Then, ( F , G , E ) is bipo-lar soft subset of ( F , G , E ) , denoted by ( F , G , E ) ⊑ ( F , G , E ) , if F ( e ) ⊆ F ( e ) and G ( f ( e )) ⊆ G ( f ( e )) for all e ∈ E . Definition 3.5
Let ( F , G , E ) , ( F , G , E ) ∈ BS . Then, ( F , G , E ) and ( F , G , E ) equal, denoted by ( F , G , E ) = ( F , G , E ) , if ( F , G , E ) ⊑ ( F , G , E ) and ( F , G , E ) ⊑ ( F , G , E ) . Definition 3.6
Let ( F , G , E ) , ( F , G , E ) ∈ BS . Then, bipolar soft unionof ( F , G , E ) and ( F , G , E ) is, denoted by ( F , G , E ) ⊔ ( F , G , E ) , definedby ( F ⊔ F )( e ) = F ( e ) ∪ F ( e ) and ( G ⊔ G )( f ( e )) = G ( f ( e )) ∩ G ( f ( e )) for all e ∈ E . Definition 3.7
Let ( F , G , E ) , ( F , G , E ) ∈ BS . Then, bipolar soft inter-section of ( F , G , E ) and ( F , G , E ) is, denoted by ( F , G , E ) ⊓ ( F , G , E ) ,defined by ( F ⊓ F )( e ) = F ( e ) ∩ F ( e ) and ( G ⊓ G )( f ( e )) = G ( f ( e )) ∪ G ( f ( e )) for all e ∈ E . efinition 3.8 Let ( F, G, E ) ∈ BS . For all e ∈ E , F ( e ) = ∅ and G ( f ( e )) = U , then ( F, G, E ) is called null bipolar soft set and denoted by (Φ , ˜ U , E ) . Definition 3.9
Let ( F, G, E ) ∈ BS . For all e ∈ E , F ( e ) = U and G ( f ( e )) = ∅ , then ( F, G, E ) is called absolute bipolar soft set and denoted by ( ˜ U , Φ , E ) . Definition 3.10
Let ( F, G, E ) ∈ BS . Then, complement of ( F, G, E ) , de-noted by ( F, G, E ) ˜ c , is a bipolar soft set over U such that ( F, G, E ) ˜ c =( H, K, E ) , where H ( e ) = G ( f ( e )) and K ( f ( e )) = F ( e ) for all e ∈ E . Example 3.11
Let U = { u l , u , u , u , u , u , u , u } be an initial universeand E = { e , e , e , e , e , e , e , e } be a set of parameters. If we chose E and E sets as E = { e , e , e , e } and E = { e , e , e , e } . Now, we definethe bijective function f as f ( e i ) = ¬ e i ( i = 1 , , , . Here, the notion ¬ e i means that “ not e i ” for all i = 1 , , , . Thus, we have following results. f ( e ) = ¬ e = e f ( e ) = ¬ e = e f ( e ) = ¬ e = e f ( e ) = ¬ e = e So, we can describe following the bipolar soft sets ( F , G , E ) , ( F , G , E ) and ( F , G , E ) . ( F , G , E ) = n(cid:10) ( e , { u , u , u } ) , ( e , { u , u } ) (cid:11) , (cid:10) ( e , { u , u , u } ) , ( e , { u , u , u } ) (cid:11) , (cid:10) ( e , { u , u } ) , ( e , { u , u , u , u } ) (cid:11) , (cid:10) ( e , { u , u , u , u } ) , ( e , { u , u } ) (cid:11)o ( F , G , E ) = n(cid:10) ( e , { u , u , u } ) , ( e , { u , u , u , u } ) (cid:11) , (cid:10) ( e , { u , u } ) , ( e , { u , u , u , u } ) (cid:11) , (cid:10) ( e , { u , u , u } ) , ( e , { u , u , u , u } ) (cid:11) , (cid:10) ( e , { u } ) , ( e , { u , u , u } ) (cid:11)o ( F , G , E ) = n(cid:10) ( e , { u , u } ) , ( e , { u , u , u , u } ) (cid:11) , (cid:10) ( e , { u , u } ) , ( e , { u , u , u , u } ) (cid:11) , (cid:10) ( e , { u } ) , ( e , { u , u , u , u , u } ) (cid:11) , (cid:10) ( e , { u , u } ) , ( e , { u , u , u , u } ) (cid:11)o e ∈ E ,since F ( e ) ⊆ F ( e ) and G ( f ( e )) ⊇ G ( f ( e )), ( F , G , E ) ⊑ ( F , G , E )( F , G , E ) ⊔ ( F , G , E ) = { ( e , { u , u } , { u , u , u , u , u } ) , ( e , { u , u } , { u , u , u , u } ) , ( e , { u , u } , { u , u , u , u , u } ) , ( e , { u } , { u , u , u } ) } and( F , G , E ) ⊓ ( F , G , E ) = { ( e , { u , u , u , u } , { u } ) , ( e , { u , u , u } , { u , u , u } ) , ( e , { u , u , u } , { u , u , u } ) , ( e , { u , u , u , u } , { u , u } ) } Theorem 3.12
Let ( F, G, E ) , ( F , G , E ) , ( F , G , E ) ∈ BS . Then,i. ( F, G, E ) ⊑ ( ˜ U , Φ , E ) ii. (Φ , ˜ U , E ) ⊑ ( F, G, E ) iii. ( F, G, E ) ⊑ ( F, G, E ) iv. If ( F, G, E ) ⊑ ( F , G , E ) and ( F , G , E ) ⊑ ( F , G , E ) , then ( F, G, E ) ⊑ ( F , G , E ) Proof.
It is clear from Definition 3.4.
Theorem 3.13
Let ( F, G, E ) , ( F , G , E ) , ( F , G , E ) ∈ BS . Then,i. ( F, G, E ) ⊔ ( F, G, E ) = (
F, G, E ) ii. ( F, G, E ) ⊔ (Φ , ˜ U , E ) = (
F, G, E ) iii. ( F, G, E ) ⊔ ( ˜ U , Φ , E ) = ( ˜ U , Φ , E ) iv. ( F, G, E ) ⊔ ( F, G, E ) ˜ c = ( ˜ U , Φ , E ) v. ( F , G , E ) ⊔ ( F , G , E ) = ( F , G , E ) ⊔ ( F , G , E ) vi. ( F, G, E ) ⊔ (cid:2) ( F , G , E ) ⊔ ( F , G , E ) (cid:3) = (cid:2) ( F, G, E ) ⊔ ( F , G , E ) (cid:3) ⊔ ( F , G , E ) Proof.
It can be proved by Definition 3.6.7 heorem 3.14
Let ( F, G, E ) , ( F , G , E ) , ( F , G , E ) ∈ BS . Then,i. ( F, G, E ) ⊓ ( F, G, E ) = (
F, G, E ) ii. ( F, G, E ) ⊓ (Φ , ˜ U , E ) = (Φ , ˜ U , E ) iii. ( F, G, E ) ⊓ ( ˜ U , Φ , E ) = ( F, G, E ) iv. ( F, G, E ) ⊓ ( F, G, E ) ˜ c = (Φ , ˜ U , E ) v. ( F , G , E ) ⊓ ( F , G , E ) = ( F , G , E ) ⊓ ( F , G , E ) vi. ( F, G, E ) ⊓ (cid:2) ( F , G , E ) ⊓ ( F , G , E ) (cid:3) = (cid:2) ( F, G, E ) ⊓ ( F , G , E ) (cid:3) ⊓ ( F , G , E ) Proof.
It can be proved by Definition 3.7.
Theorem 3.15
Let ( F, G, E ) , ( F , G , E ) , ( F , G , E ) ∈ BS . Then,i. ( F, G, E ) ⊓ (cid:2) ( F , G , E ) ⊔ ( F , G , E ) (cid:3) = (cid:2) ( F, G, E ) ⊓ ( F , G , E ) (cid:3) ⊔ (cid:2) ( F, G, E ) ⊓ ( F , G , E ) (cid:3) ii. ( F, G, E ) ⊔ (cid:2) ( F , G , E ) ⊓ ( F , G , E ) (cid:3) = (cid:2) ( F, G, E ) ⊔ ( F , G , E ) (cid:3) ⊓ (cid:2) ( F, G, E ) ⊔ ( F , G , E ) (cid:3) Proof.
It can be proved simply.
Theorem 3.16
Let ( F, G, E ) , ( F , G , E ) , ( F , G , E ) ∈ BS . Then,i. (cid:0) ( F, G, E ) ˜ c (cid:1) ˜ c = ( F, G, E ) ii. (Φ , ˜ U , E ) ˜ c = ( ˜ U , Φ , E ) iii. ( ˜ U , Φ , E ) ˜ c = (Φ , ˜ U , E ) Proof.
It is obvious from Definition 3.10.
Theorem 3.17
Let ( F , G , E ) , ( F , G , E ) ∈ BS . Then, De Morgan’s lawis valid.i. (cid:2) ( F , G , E ) ⊔ ( F , G , E ) (cid:3) ˜ c = ( F , G , E ) ˜ c ⊓ ( F , G , E ) ˜ c ii. (cid:2) ( F , G , E ) ⊓ ( F , G , E ) (cid:3) ˜ c = ( F , G , E ) ˜ c ⊔ ( F , G , E ) ˜ c roof.i. Let ( F , G , E ) ⊔ ( F , G , E ) = ( H, K, E ) and (
H, K, E ) ˜ c = ( S, T, E ).Then, H ( e ) = F ( e ) ∪ F ( e ) and K ( f ( e )) = G ( f ( e )) ∩ G ( f ( e )) for all e ∈ E . Thus, we have S ( e ) = K ( f ( e )) = G ( f ( e )) ∩ G ( f ( e )) (1)and T ( f ( e )) = H ( e ) = F ( e ) ∪ F ( e ) (2)for all e ∈ E . Moreover, ( F , G , E ) ˜ c = ( H , K , E ) and ( F , G , E ) ˜ c =( H , K , E ). Then, H ( e ) = G ( f ( e )), H ( e ) = G ( f ( e )), K ( f ( e )) = F ( e ) and K ( f ( e )) = F ( e ). Therefore,( H ⊓ H )( e ) = G ( f ( e )) ∩ G ( f ( e )) (3)and ( K ⊓ K )( e ) = F ( e ) ∪ F ( e ) (4)for all e ∈ E . On the other hand, it can be seen clearly that right handof (1) is equal to right hand of (3) and right hand of (2) is equal to righthand of (4). So, the proof is completed. ii. It can be proved similar way ( i. ). Definition 3.18
Let ( F , G , E ) , ( F , G , E ) ∈ BS . Then, and -product ofbipolar soft sets ( F , G , E ) and ( F , G , E ) is, denoted by ( F , G , E ) ∧ ( F , G , E ) ,defined by ( F ∧ F )( e, e ′ ) = F ( e ) ∩ F ( e ′ ) and ( G ∧ G )( f ( e )) = G ( f ( e )) ∪ G ( f ( e ) , f ( e ′ )) for all e, e ′ ∈ E . Definition 3.19
Let ( F , G , E ) , ( F , G , E ) ∈ BS . Then, or -product ofbipolar soft sets ( F , G , E ) and ( F , G , E ) , denoted by ( F , G , E ) ∨ ( F , G , E ) ,is defined by ( F ∨ F )( e, e ′ ) = F ( e ) ∪ F ( e ′ ) and ( G ∨ G )( f ( e ) , f ( e ′ )) = G ( f ( e )) ∩ G ( f ( e ′ )) for all e, e ′ ∈ E . Example 3.20
Let U = { u l , u , u , u , u , u , u , u } be the universe whichare eight houses and E = { e , e , e , e , e , e } be the set of parameters. Here, e i ( i = 1 , , , stand for the parameters “ large ”, “ small ”, “ cheap ”,“ modern ”,“ standard ”, “ expensive ”, respectively. Therefore, we can chose E and E sets as E = { e , e , e } and E = { e , e , e } . Now, we define the bijective unction f as f ( e i ) = ¬ e i ( i = 1 , , . Here, the notion ¬ e i means that “ not e i ” for all i = 1 , , . Thus, we have following results. f ( e ) = ¬ e = e f ( e ) = ¬ e = e f ( e ) = ¬ e = e So, we can describe following the bipolar soft sets ( F , G , E ) , ( F , G , E ) and ( F , G , E ) . ( F , G , E ) = n(cid:10) ( e , { u , u , u } ) , ( e , { u , u } ) (cid:11) , (cid:10) ( e , { u , u , u } ) , ( e , { u , u , u } ) (cid:11) , (cid:10) ( e , { u , u } ) , ( e , { u , u , u , u } ) (cid:11)o ( F , G , E ) = n(cid:10) ( e , { u , u , u } ) , ( e , { u , u , u , u } ) (cid:11) , (cid:10) ( e , { u , u } ) , ( e , { u , u , u , u } ) (cid:11) , (cid:10) ( e , { u , u , u } ) , ( e , { u , u , u , u } ) (cid:11)o . Then, ( F , G , E ) ∧ ( F , G , E ) = (cid:8)(cid:10) (( e , e ) , { u , u } ) , (( e , e ) , { u , u , u , u , u } ) (cid:11) , (cid:10) (( e , e ) , ∅ ) , (( e , e ) , { u , u , u , u , u , u } ) (cid:11) , (cid:10) (( e , e ) , { u , u , u } ) , (( e , e ) , { u , u , u , u , u } ) (cid:11) , (cid:10) (( e , e ) , ∅ ) , (( e , e ) , { u , u , u , u , u , u } ) (cid:11) , (cid:10) (( e , e ) , { u , u } ) , (( e , e ) , { u , u , u , u } ) (cid:11) , (cid:10) (( e , e ) , ∅ ) , (( e , e ) , { u , u , u , u , u , u } ) (cid:11) , (cid:10) (( e , e ) , { u , u , u } ) , (( e , e ) , { u , u , u , u , u } ) (cid:11) , (cid:10) (( e , e ) , ∅ ) , (( e , e ) , { u , u , u , u , u , u } ) (cid:11) , (cid:10) (( e , e ) , { u , u } ) , (( e , e ) , { u , u , u , u , u } ) (cid:11)(cid:9) nd ( F , G , E ) ∨ ( F , G , E ) = (cid:8)(cid:10) (( e , e ) , { u , u , u , u } ) , (( e , e ) , { u } ) (cid:11) , (cid:10) (( e , e ) , { u , u , u , u } ) , (( e , e ) , ∅} ) (cid:11) , (cid:10) (( e , e ) , { u , u , u } ) , (( e , e ) , { u } ) (cid:11) , (cid:10) (( e , e ) , { u , u , u , u , u } ) , (( e , e ) , { u } ) (cid:11) , (cid:10) (( e , e ) , { u , u , u } ) , (( e , e ) , { u , u , u } ) (cid:11) , (cid:10) (( e , e ) , { u , u , u , u , u , u } ) , (( e , e ) , { u } ) (cid:11) , (cid:10) (( e , e ) , { u , u , u } ) , (( e , e ) , { u } ) (cid:11) , (cid:10) (( e , e ) , { u , u , u , u } ) , (( e , e ) , { u , u } ) (cid:11) , (cid:10) (( e , e ) , { u , u , u } ) , (( e , e ) , { u , u , u } ) (cid:11)(cid:9) Theorem 3.21
Let ( F , G , E ) , ( F , G , E ) ∈ BS . Then,i. (cid:2) ( F , G , E ) ∨ ( F , G , E ) (cid:3) ˜ c = ( F , G , E ) ˜ c ∧ ( F , G , E ) ˜ c ii. (cid:2) ( F , G , E ) ∧ ( F , G , E ) (cid:3) ˜ c = ( F , G , E ) ˜ c ∨ ( F , G , E ) ˜ c Proof.
The proof is clear.
In this section we will construct a decision making method over the bipolarsoft set. Firstly, we will define some notions that necessary to constructalgorithm of decision making method.
Definition 4.1
Let E = { e , e , ..., e n } be a parameter set, U = { u , u , ..., u m } be initial universe and ( F, G, E ) be a BSS over U .Then, score of an object,denoted by s i , is computed as s i = c + i − c − i . Here, c + i and c − i are computedform c + i = P nj =1 a ij and c − i = P nj =1 b ij Now we present an algorithm for most appropriate selection of an object.
Algorithm
Step 1:
Input the bipolar soft Set (
F, G, E ) Step 2:
Consider the bipolar soft set (
F, G, E ) and write it in tabular form
Step 3:
Compute the score s i of h i ∀ i tep 4: Find s k = maxs i Step 5:
If k has more than one value then any one of h i could be the prefer-able choice.Let us use the algorithm to solve the problem. Example 4.2
We consider the problem to select the most suitable housewhich Mr. X is going to choose on the basis of his number of parametersout of m number of houses. Let E = { e = beautif ul, e = cheap, e = in good repairing,e = moderate, e = wooden } and E = { e = not beautif ul, e = not cheap,e = not in good repairing, e = not moderate, e = not wooden } and U be ainitial universe. Here, we consider E = E ∪ E . For the sake of shortness,we will denote the ( F ( e i ) , G ( f ( e i ))) with F G ( e i ) Step 1:
Let us consider BSS ( F, G, E ) over U defined as follow, ( F, G, E ) = n(cid:10) ( e , { u , u } ) , ( e , { u , u , u , u } ) (cid:11) , (cid:10) ( e , { u } ) , ( e , { u , u , u , u } ) (cid:11) , (cid:10) ( e , { u , u , u } ) , ( e , { u , u , u , u } ) (cid:11) , (cid:10) ( e , { u } ) , ( e , { u , u , u } ) (cid:11)(cid:10) ( e , { u , u , u } ) , ( e , { u , u } ) (cid:11)o Step 2:
Tabular representation of the BSS ( F, G, E ) is as below: ( F, G, E ) F G ( e ) F G ( e ) F G ( e ) F G ( e ) F G ( e ) u (1 ,
0) (0 ,
1) (1 ,
0) (0 ,
0) (1 , u (1 ,
0) (1 ,
0) (0 ,
1) (0 ,
1) (0 , u (0 ,
1) (0 ,
1) (1 ,
0) (0 ,
1) (1 , u (1 ,
0) (0 ,
1) (1 ,
0) (0 ,
1) (0 , u (0 ,
1) (0 ,
0) (0 ,
1) (1 ,
0) (0 , u (0 ,
1) (0 ,
0) (0 ,
0) (0 ,
0) (0 , u (0 ,
1) (0 ,
0) (0 ,
1) (0 ,
0) (0 , u (0 ,
0) (0 ,
1) (0 ,
1) (0 ,
0) (1 , Tabular form of the BSS ( F, G, E ) Step 3:
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