Research Article Rough Atanassov s Intuitionistic Fuzzy Sets Model over Two Universes and Its Applications

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1 e Scietific World Joural Article ID pages Research Article Rough Ataassov s Ituitioistic Fuzzy Sets Model over Two Uiverses ad Its Applicatios Shuqu Luo ad Weihua Xu School of Mathematics ad Statistics Chogqig Uiversity of Techology Chogqig Chia Correspodece should be addressed to Weihua Xu; chxuwh@gmailcom Received 26 August 2013; Accepted 18 November 2013; Published 13 May 2014 Academic Editors: E Haghverdi M Masour ad H Xu Copyright 2014 S Luo ad W Xu This is a ope access article distributed uder the Creative Commos Attributio Licese which permits urestricted use distributio ad reproductio i ay medium provided the origial work is properly cited Recetly much attetio has bee give to the rough set models based o two uiverses Ad may rough set models based o two uiverses have bee developed from differet poits of view I this paper a ovel model that is rough Ataassov s ituitioistic fuzzy sets model over two differet uiverses is firstly proposed from Ataassov s ituitioistic poit of view We study some importat properties of approximatio operators ad ivestigate the rough degree i the ovel model Furthermore a illustrated example is employed to demostrate the coceptual argumets of the model Fially rough Ataassov s ituitioistic fuzzy sets approach to decisio is preseted i the geeralized approximatio space over two uiverses by cosiderig the problem about how to arrage patiets to see the doctor reasoably from which it ca be foud that the method is valuable ad useful i real life 1 Itroductio Rough set theory origially proposed by Pawlak i the early 1980s [1 3] as a useful tool for treatig with ucertaity or imprecisio iformatio has bee successfully applied i the fields of artificial itelligece patter recogitio medical diagosis data miig coflict aalysis algebra [4 12] ad so o I recet years the rough set theory has aroused a great deal of iterest amog more ad more researchers It is widely accepted that the theory of rough sets which is very importat to costruct a pair of upper ad lower approximatio operators is based o available iformatio I the Pawlak approximatio space the lower ad upper approximatios of arbitrary subset of the uiverse of discourse ca be represeted directly The lower approximatio is the uio of all equivalece classes which are classes iducedbytheequivalecerelatiootheuiversead icluded i the give set The upper approximatio is the uio of all equivalece classes which are classes iduced by theequivalecerelatiootheuiversehavigaoempty itersectio with the give set So the equivalece relatio is a key ad primitive otio i Pawlak s rough set model I the Pawlak approximatio space the equivalece relatio is a very restrictive coditio ad the sets used are classical sets so the applicatio domai of rough set model is arrowed I 1986 Ataassov itroduced the cocept of Ataassov s ituitioistic fuzzy sets Ataassov s ituitioistic fuzzy set was cosidered as a geeralizatio of fuzzy set ad had bee foud to be very useful to deal with vagueess Ataassov thought that Ataassov s ituitioistic fuzzy set was characterized by a pair of fuctios the membership fuctio ad the omembership fuctio valued i [0 1] The degrees of membership ad omembership are idepedet So Ataassov s ituitioistic fuzzy set is more suitable ad precise to represet the essece of vagueess adismoreusefulthafuzzysetidealigwithimperfect iformatio Nowadays may excellet works over sigle uiverse have bee achieved For example Zhou ad Wu [13] preseted the rough approximatios of Ataassov s ituitioistic fuzzy sets i crisp ad fuzzy approximatio spaces over sigle uiverse i which both costructive ad axiomatic approaches are used Zhag [14] geeralized a itervalvalued rough Ataassov s ituitioistic fuzzy (IF) sets model by meas of itegratig the classical Pawlak rough set theory with iterval-valued Ataassov s ituitioistic fuzzy set theory More excellet results ca be foud i [15 16] Moreover the study o the rough set model over two uiverses was doe ad it has become oe of the hottest

2 2 The Scietific World Joural researches i recet years for authors She ad Wag [17] researched the variable precisio rough set model over two uiverses ad ivestigated the properties Ya et al [18] established the model of rough set over dual uiverses Su et al [19] proposed Ataassov s ituitioistic fuzzy rough set model over two uiverses with a costructive approach ad discussed the basic properties of this model i fuzzy approximatio space More details about recet advacemets of rough set model over two uiverses ca be foud i the literature [20 25] I this paper we will discuss rough Ataassov s ituitioistic fuzzy sets model over two uiverses i the geeralized approximatio space ivestigate itsmeasuresadstudyhowtouseitforservigourlife The rest of the paper is orgaized as follows The ext sectio reviews the basic cocepts of Ataassov s ituitioistic fuzzy sets rough sets ad rough fuzzy sets I the ext sectio rough Ataassov s ituitioistic fuzzy sets model is costructed i geeralized approximatio space over two uiverses Moreover rough Ataassov s ituitioistic fuzzy sets cut sets ad some importat properties are preseted I Sectio 4w ystudiedhowtomeasuretheucertaity of rough Ataassov s ituitioistic fuzzy set over two uiverses What is more a geeral approach to decisio makig is established based o rough Ataassov s ituitioistic fuzzy sets over two uiverses with a problem of how to order the patiets to see the doctor reasoably as the backgroud for the applicatio i Sectio 5 Fially we draw brief coclusios ad set further research directios i Sectio 6 2 Prelimiaries I this sectio we will review some ecessary defiitios ad cocepts required i the sequel of this paper 21 Ataassov s Ituitioistic Fuzzy Sets Defiitio 1 (see [26 27]) Let U be a oempty fiite set Ataassov s ituitioistic fuzzy set A o U is a object havig the form A = { x μ A (x) ] A (x) x U} (1) where μ A : U [0 1] ad ] A : U [0 1] satisfy 0 μ A (x) + ] A (x) 1 for all x U μ A (x) ad ] A (x) are respectively called the degrees of membership ad omembership of the elemet x Uto A The complemet of Ataassov s ituitioistic fuzzy set A is deoted by A ={ x] A (x) μ A (x) x U} Let IF(U) deote the family of all Ataassov s ituitioistic fuzzy sets o U For ay A B IF(U) some basic operatios o IF(U) are defied as follows: (1) A B μ A (x) μ B (x) ] A (x) ] B (x) for all x U (2) A = B μ A (x) = μ B (x) ] A (x) = ] B (x) for all x U (3) A B ={ xμ A (x) μ B (x) ] A (x) ] B (x) x U} (4) A B = { x μ A (x) μ B (x) ] A (x) ] B (x) x U} (5) if λ>0theλ A = { x 1 (1 μ A (x))λ (] A (x))λ x U} Defiitio 2 Let A be Ataassov s ituitioistic fuzzy set over U ad (α β) L L = {(α β) α [0 1] β [0 1] α+β 1} ad the (α β)-level cut set of A deoted by A β α is defied as follows: A β α = {x U μ A (x) α] A (x) β} (2) A α ={x U μ A (x) α} ad A α+ ={x U μ A (x) > α} arerespectivelycalledtheα-level cut set ad the strog αlevel cut set of membership geerated by A Ad A β ={x U ] A (x) β} ad A β+ = {x U ] A (x) < β} are respectively referred to as the β-level cut set ad the strog β-level cut set of omembership geerated by A What is more other types of cut sets ad strog cut sets of Ataassov s ituitioistic fuzzy set A are deoted for example A β+ α+ ={x U μ A (x) > α ] A (x) < β} whichis called the (α+ β+)-level cut set of A 22 Fuzzy Rough Sets Model Defiitio 3 (see [28]) Let (U R) be a fuzzy approximatio space ad A be a fuzzy subset of UThelowerapproximatio ad upper approximatio are deoted by R( A) ad R( A) respectively The memberships of x to A are defied as R ( A) (x) = { A(y) (1 R(x y)) y U} R( A) (x) = { A(y) (1 R(x y)) y U} x U x U (3) where ad mea mi ad max operatorsrespectively ad A(y) is the membership of y with respect to AThe pair ( R( A) R( A)) is called a fuzzy rough set 23 Rough Ataassov s Ituitioistic Fuzzy Sets Model over Oe Uiverse Defiitio 4 (see [13]) Let (U R) be a geeralized approximatio space for ay A ={ xμ A (x) ] A (x) x U} IF(U) The upper ad lower approximatios of A deoted by R( A) ad R( A) are respectively defied as follows: R( A) = { x μ R( A) (x) ] R( A) (x) x U} R ( A) = { x μ R( A) (x) ] R( A) (x) x U} (4)

3 The Scietific World Joural 3 where μ R( A) (x) = μ R( A) (x) = μ A (y) ] R( A) (x) = ] A (y) μ A (y) ] R( A) (x) = ] A (y) (5) where ad mea mi ad max operatorsrespectively ad μ A (y) ] A (y) are the membership ad omembership of y with respect to AThepair(R( A) R( A)) is called a fuzzy rough set i a geeralized approximatio space 24 Rough Fuzzy Sets Model over Two Uiverses Defiitio 5 (see [29]) Let (U V R) be a geeralized approximatio space over two uiverses ad for ay A F(U) B F(V) deote ( A) (y) = mi { A (x) } ( A) (y) = max { A (x) } Defiitio 6 (see [22]) Let R be a crisp biary relatio o the uiverses U ad VThe (1) R is serial if for ay x U there exists y Vst R(x y) = 1 (2) R is reverse serial if for ay y V there exists x U st R(x y) = 1 Defiitio 7 Let (U V R) be a geeralized approximatio space over two uiverses ad for ay A IF(U) B IF(V) deote where ( A) = { y μ RU ( A) (y) ] ( A) (y) y V} ( A) = { y μ RU ( A) (y) ] ( A) (y) y V} ( B) = { x μ RV ( B) (x) ] ( B) (x) x U} ( B) = { x μ RV ( B) (x) ] ( B) (x) x U} μ RU ( A) (y) = μ A (x) ] ( A) (y) = ] A (x) (7) y V; ( B) (x) = mi { B(y) } ( B) (x) = max { B(y) } (6) μ RU ( A) (y) = μ RV ( B) (x) = μ A (x) ] ( A) (y) = ] A (x) μ B (y) ] ( B) (x) = ] B (y) x U The ( A) ad ( A) are called the lower ad upper approximatios of fuzzy set A i F(U)ad ( B) ad ( B) are called the lower ad upper approximatios of fuzzy set B i F(V)respectively( ( A) ( A))ad( ( B) ( B))are called rough fuzzy sets i geeralized approximatio space over two uiverses 3 Rough Ataassov s Ituitioistic Fuzzy Sets Model over Two Uiverses I this sectio we will itroduce rough Ataassov s ituitioistic fuzzy sets model over the differet uiverses ad discuss some importat properties of rough Ataassov s ituitioistic fuzzy sets 31 Costructio of Rough Ataassov s Ituitioistic Fuzzy Sets Let U V be oempty fiite uiverses ad a subset R P(U V)(ie R: U V {01})iscalledabiaryrelatio from U to V I geeral if U=V R is called the biary relatio over sigle uiverse If R satisfies reflexivity symmetry ad trasitivity the we say R is a equivalece relatio But i geeralized approximatio space over two uiverses R is a biary relatio from U to V ad the R must ot be equivalece relatio μ RV ( B) (x) = μ B (y) ] ( B) (x) = ] B (y) (8) The ( A) ad ( A) are called the lower ad upper approximatios of Ataassov s ituitioistic fuzzy set A i IF(U) ad ( B) ad ( B) are called the lower ad upper approximatios of Ataassov s ituitioistic fuzzy set B i IF(V) ( ( A) ( A)) ad ( ( B) ( B)) are called rough Ataassov s ituitioistic fuzzy sets over the uiverses U ad V Furthermore we also defie the positive regio pos RU ( A) pos RV ( B) egative regio eg RU ( A)eg RV ( B)adboudary regio b RU ( A)b RV ( B) of A B about o the uiverse U V as follows respectively: pos RU ( A) = ( A) eg RU ( A) = ( A) b RU ( A) = ( A) ( A) ; pos RV ( B) = ( B) eg RV ( B) = ( B) b RV ( B) = ( B) ( B) If for ay y V(resp x U) ( A) = ( A) (resp ( B) = ( B)) the Ataassov s ituitioistic fuzzy set A (9)

4 4 The Scietific World Joural (resp B) is Ataassov s ituitioistic fuzzy defiable set about the geeralized approximatio space (UVR)Otherwise Ataassov s ituitioistic fuzzy set A (resp B) isaroughset about the geeralized approximatio space (UVR)over two uiverses Example 8 Let (U V R) be a geeralized approximatio space over two uiverses Let U V be two oempty fiite uiverses ad let U deote the symptom set ad let V deote the disease set Suppose U={x 1 x 2 x 3 x 4 x 5 } V= {y 1 y 2 y 3 y 4 y 5 }whereeachx i (i = 125) stads for oe symptom but each y i (25)stads for a disease Assume R be a biary relatio o U Vforayx i Uif there exists y Vso the relatioca be uderstood that if a perso has a symptom x i sohehadpossiblysufferedfroma disease y i Now we ca defie the relatio R as follows: R={(x 1 y 1 )(x 1 y 2 )(x 1 y 3 )(x 2 y 3 )(x 3 y 4 )(x 4 y 1 ) belog to IF(V) adtheloweradupperapproximatios of Ataassov s ituitioistic fuzzy set B IF(V) belog to IF(U) This property is differet from the lower ad upper approximatios over sigle uiverse What is more we ca obtai other properties i the followig 32 Correspodig Properties of Rough Ataassov s Ituitioistic Fuzzy Sets Theorem 10 Let (U V R) be a geeralized approximatio space over two uiverses ad for ay A A IF(U) B B IF(V) oe has the followig properties: (1) ( A) = ( A) ( A) = ( A); ( B) = ( B) ( B) = ( B); (2) ( A A )= ( A) ( A ) ( A A )= ( A) ( A ); (x 4 y 2 )(x 5 y 5 )} (10) ( B B )= ( B) ( B ) ( B B )= ( B) ( B ); From RwecaseethatR is serial ad reverse serial ad we ca obtai R P (y 1 )={x 1 x 4 } R P (y 2 )={x 1 x 4 } R P (y 3 )={x 1 x 2 } R P (y 4 )={x 3 } R P (y 5 )={x 5 } (11) Suppose a perso A has the symptom coditios ad we ca describe by Ataassov s ituitioistic fuzzy set A = { x x x x x } By Defiitio 7wecaobtai (12) (3) A A ( A ) ( A) ( A ) ( A); B B ( B ) ( B) ( B ) ( B); (4) ( A A ) ( A) ( A ) ( A A ) ( A) ( A ); ( B B ) ( B) ( B ) ( B B ) ( B) ( B ); (5) ( A) ( A) ( B) ( B) Proof We oly eed to prove the first part of each property as the similarity of the above properties (1) Accordig to Defiitio 7wecaobtai ( A) = { y y y y y } ( A) = { y y y y y } (13) ( A) = { { y μ A (x) ] A (x) y V} } { x R p(y) x R p(y) } From the lower ad upper approximatios of A we ca draw a coclusio that the perso must have had the diseases y 1 y 2 y 3 y 4 y 5 at the degrees (06 01) (06 01) (06 01) (02 08)ad(01 06) respectively Ad the perso may be have had the diseases y 1 y 2 y 3 y 4 y 5 at the degrees (08 01) (08 01) (08 01) (02 08) ad(01 06) respectively = { { y ] A (x) μ A (x) y V} } { x R p(y) x R p(y) } = ( A) (14) Remark 9 I a geeralized approximatio space over two uiverseswecafidoutthattheloweradupperapproximatios of Ataassov s ituitioistic fuzzy set A IF(U) So we ca have ( A) = ( A) The property ( A) = ( A) ca be proved similarly

5 The Scietific World Joural 5 (2) We ca have ( A A ) Example 12 (cotiued from Example 8) Let α=08ad let β = 01; thewehave A ={x 1} so the lower ad upper approximatios of A cabepresetedasfollows: = { { y { x R p(y) μ A A (x) ] A A (x) y V} } x R p(y) } ( A )=0 ( A )={y 1y 2 y 3 } (18) = { { y { x R p(y) (μ A (x) μ A (x)) (μ A (x) μ A (x)) y V} } x R p(y) } = { { y μ A (x) { x R p(y) x R p(y) μ A (x) ] A (x) ] A (x) y V} } x R p(y) x R p(y) } = ( A) ( A ) (15) Hece we ca obtai ( A A )= ( A) ( A ) (3) Accordig to the defiitios of Ataassov s ituitioistic fuzzy lower ad fuzzy upper approximatios (3) holds (4) It is easy to prove it by property (3) (5) For ay y ( A)wecahave A (x) μ RU ( A) (y) = μ A (x) μ A (x) ] RU ( A) (y) = ] A (x) ] (16) Therefore ( A) ( A) Defiitio 11 Let (U V R) be a geeralized approximatio space over two uiverses ad for ay A IF(U) B IF(V) deote ( A β α )={y R p (y) A β α } ( A β α )={y R p (y) A β α =0}; ( B β α )={x R s (x) B β α } ( B β α )={x R s (x) B β α =0} (17) where α β [0 1] ( A β α )ad( A β α ) are called the lower ad upper approximatio of A β α o the uiverse U ad ( B β α ) ad ( B β α ) arecalledtheloweradupper approximatios of B β α o the uiverse V Remark 13 I Defiitio 7wegivethecoceptsofthelower approximatio ad upper approximatio of A β α about R from the uiverse U to V Similarly we ca also defie the lower approximatio ad upper approximatio of sets A α A α+ A β A β+ A β α+ A β+ α ad A β+ α+ about R from the uiverse U to V For example we defie the lower approximatio ad upper approximatio of A β+ B β+ about R from the uiverse U to V as follows: ( A β+ )={y R p (y) A β+ } ( A β+ )={y R p (y) A β+ =0}; ( B β+ )={x R s (x) B β+ } ( B β+ )={x R s (x) B β+ =0} (19) I the followig discussio without loss of geerality we oly ivestigate the properties of ( A β α ) ( A β α ) ad ( B β α ) ( B β α ) The correspodig properties ca be exteded to the other lower approximatios ad upper approximatios; here we will ot list them oe by oe Theorem 14 Let (UVR) be a geeralized approximatio space over two uiverses; if > β 1 <β 2 oe ca obtai ( A β 1 ) ( A β 2 ) ( A β 1 ) ( A β 2 ); ( B β 1 ) ( B β 2 ) ( B β 1 ) ( B β 2 ) (20) Proof Sice > β 1 <β 2 so A β 1 A β 2 Forayy ( A β 1 )wecahaver p (y) A β 1 ThusR p (y) A β 2 y ( A β 2 ) that is ( A β 1 ) ( A β 2 ) The properties ( A β 1 ) ( A β 2 ) ( B β 1 ) ( B β 2 ) ad ( B β 1 ) ( B β 2 ) cabeprovedsimilarly Example 15 (cotiued from Example 12) Let = 08 = 06 β 1 = 01 adβ 2 = 02; thewehave A ={x 1x 2 x 4 }sotheloweradupperapproximatiosof A ca be obtaied as follows: ( A )={y 1y 2 y 3 } ( A )={y 1y 2 y 3 } (21) The ( A ) ( A ) ad ( A ) ( A 02 hold 06 )

6 6 The Scietific World Joural Accordig to Defiitio 7 we ca defie two pairs of Ataassov s ituitioistic fuzzy sets as follows: where R U ( A) = { y μ R U ( A) (y) ] R U ( A) (y) y V} R U ( A) = { y μ R U ( A) (y) ] R (y) y V} U ( A) R V ( B) = { x μ R V ( B) (x) ] R V ( B) (x) x U} R V ( B) = { x μ R V ( B) (x) ] R V ( B) (x) x U} μ R U ( A) (y) = {α y ( A β α )}= {α R p (y) A β α } ] R U ( A) (y)= {β y ( A β α )}= {β R p (y) A β α } μ R ] R μ R U ( A) (y) = {α y ( A β α )} = {α R p (y) A β α =φ} ] R U ( A) (y)= {β y ( A β α )} = {β R p (y) A β α =φ} μ R V ( B) (x) = {α x ( B β α )}= {α R s (x) B β α } ] R V ( B) (x) = {β x ( B β α )}= {β R s (x) B β α } (22) V ( B) (x) = {α x ( B β α )}= {α R s (x) B β α =φ} V ( B) (x) = {β x ( B β α )}= {β R s (x) B β α =φ} (23) The we ca obtai the followig properties Theorem 16 Let (U V R) be a geeralized approximatio space over two uiverses ad for ay A IF(U) B IF(V) the ( A) = R U ( A) ( A) = R U ( A) ; ( B) = R V ( B) ( B) = R V ( B) Proof For ay y V deote =μ RU (A) (y) = β 1 = ] RU (A) (y) = μ A (x) ] A (x) ; (24) = {α R p (y) A β α } β 2 = {β R p (y) A β α } (25) Let α β satisfy R p (y) A β α ifx R p(y)theμ A(x) α ] A(x) β ad x Rp (y) μ A (x) α ] A (x) β So α β 1 β; therefore; β 1 β 2 O the other had for ay α> β<β 2 accordig to the defiitios of β 2 we ca kow that there exists x R p (y) stx A β α ;thatis μ A (x) < α β 1 ] A (x) > β; thus α> β<β 1 by the arbitrary of α> ad β<β 2 ad we ca obtai β 2 β 1 Hece ( A) = R U ( A) The properties ( A) = R U ( A) ( B) = R V ( B) ad ( B) = R V ( B) cabeprovedsimilarly Defiitio 17 Let (U V R) be a geeralized approximatio space over two uiverses A 1 A 2 IF(U) B 1 B 2 IF(V) If ( A 1 )= ( A 2 )the A 1 ad A 2 are called lower rough equivaleces of U deoted by A 1 U A 2 If ( A 1 )= ( A 2 )the A 1 ad A 2 are called upper rough equivaleces of U deoted by A 1 U A 2 If ( A 1 )= ( A 2 ) ad ( A 1 )= ( A 2 )the A 1 ad A 2 are called rough equivaleces of U deoted by A 1 U A 2 If ( B 1 )= ( B 2 )the B 1 ad B 2 are called lower rough equivaleces of V deoted by B 1 V B 2 If ( B 1 )= ( B 2 )the B 1 ad B 2 are called upper rough equivaleces of V deoted by B 1 V B 2 If ( B 1 )= ( B 2 ) ad ( B 1 )= ( B 2 )the B 1 ad B 2 are called rough equivaleces of V deoted by B 1 V B 2 Propositio 18 Let (U V R) be a geeralized approximatio space over two uiverses A 1 A 2 A 1 A 2 IF(U) B 1 B 2 B 1 B 2 IF(V);the (1) A 1 U A 2 ( A 1 A 2 ) U A 2 ( A 1 A 2 ) U A 1 ; B 1 V B 2 ( B 1 B 2 ) V B 2 ( B 1 B 2 ) V B 1 (2) A 1 U A 2 ( A 1 A 2 ) U A 2 ( A 1 A 2 ) U A 1 ; B 1 V B 2 ( B 1 B 2 ) V B 2 ( B 1 B 2 ) V B 1 (3) If A 1 U A 1 A 2 U A 2 the( A 1 A 2 ) U A 1 A 2 ; if A 1 U A 1 A 2 U A 2 the( A 1 A 2 ) U A 1 A 2 ; if B 1 V B 1 B 2 V B 2 the( B 1 B 2 ) V B 1 B 2 ; if B 1 V B 1 B 2 V B 2 the( B 1 B 2 ) V B 1 B 2 (4) If A 1 U 0 or A 1 U0the A 1 A 1 U0; if B 1 V 0 or B 1 V0the B 1 B 1 V0 (5) If A 1 U U or A 1 UUthe A 1 A 1 UU; if B 1 V V or B 1 VVthe B 1 B 1 VV (6) If A 1 A 1 ad A 1 U0the A 1 U 0; if B 1 B 1 ad B 1 V0the B 1 V 0 (7) If A 1 A 1 ad A 1 U Uthe A 1 UU; if B 1 B 1 ad B 1 V Vthe B 1 VV Proof Straightforward

7 The Scietific World Joural 7 Theorem 19 Let (U V R) be a geeralized approximatio space over two uiverses A IF(U) B IF(V);the (1) ( A) = { A IF(U) A U A } ( B) = { B IF(V) B V B }; (2) ( A) = { A IF(U) A U A } ( B) = { B IF(V) B V B } Proof We ca obtai them accordig to Propositio 18 Theorem 20 Let (U V R) be a geeralized approximatio space over two uiverses A IF(U) foray0 α β β 1 β 2 1adifR is a reverse serial relatio o U V deote ( ( A)) β ={y V μ α ( A) (y) α ] ( A) (y) β} ( ( A)) β α ={y Vμ ( A) (y) α ] ( A) (y) β} (26) The (1) ( ( A)) β α (( A) β α ) (( A)) β α (( A) β α ) (2) β 1 β 2 ( ( A)) β 1 ( ( A)) β 1 ( ( A)) β 2 Proof (1) For ay y (( A) β α ) φ=r p(y) ( A) β α for ay ( A) β α for all x R p(y) μ A (x) α ] A (x) β μ A (x) α ] A (x) β μ RU ( A) (y) αad] ( A) (y) β y ( ( A)) β α Thuswe ca have ( ( A)) β α (( A) β α ) The properties ( ( A)) β α (( A) β α ) ca be proved similarly (2) Sice β 1 β 2 wecahavethatforay y ( ( A)) β 1 μ RU ( A) (y) = μ A (x) ] RU ( A) (y) = ] A (x) β 1 x Rp (y) μ A (x) ad x Rp (y) ] A (x) β 1 β 2 y ( ( A)) β 2 Therefore ( ( A)) β 1 ( ( A)) β 1 ( ( A)) β 2 Theorem 21 Let (U V R) be a geeralized approximatio space over two uiverses A IF(U) for ay 0 α ββ 1 β 2 1adifR is a serial relatio o U V deote ( ( B)) β ={x V μ α ( B) (x) α] ( B) (x) β} (27) ( ( B)) β ={x Vμ α ( B) (x) α] ( B) (x) β} The (1) ( ( B)) β α (( B) β α ) (( B)) β α (( B) β α ) (2) β 1 β 2 ( ( B)) β 1 ( ( B)) β 1 ( ( B)) β 2 Proof The proof is similar to Theorem 20 4 The Measures of Rough Ataassov s Ituitioistic Fuzzy Sets Model over Two Uiverses I this sectio we will research some measures of rough Ataassov s ituitioistic fuzzy set over differet uiverses Defiitio 22 Let (U V R) be a geeralized approximatio space over two uiverses A IF(U)foray0< 1 0<β 1 β 2 1adtheapproximateprecisioof A about R ca be defied as follows: α U ( A) = ( ( A)) β 1 (28) β 1 )( β 2 )] ( ( A)) β 2 where A =0 ad the otatio deotes the cardiality of set Let ρ U ( A) β 1 )( β 2 )] = 1 α U ( A) β 1 )( β 2 )] ad ρ U ( A) β 1 )( β 2 )] is called the rough degree of A about the uiverse U Theorem 23 Let (U V R) be a geeralized approximatio space over two uiverses A IF(U) foray0 < 1 0 < β 1 β 2 1adtheapproximateprecisio α U ( A) β 1 )( β 2 )] ad the rough degree ρ U ( A) β 1 )( β 2 )] satisfy the properties as follows: 0 α U ( A) 1 β 1 )( β 2 )] (29) 0 ρ U ( A) 1 β 1 )( β 2 )] Proof Accordig to Defiitio 22 this theorem ca be proved easily Example 24 (cotiued from Example 8) We ca fid the (08 01)-level cut set of ( A) ad the (06 01)-level cut set of R( A) as follows: ( ( A)) 01 =0 (R 08 U ( A)) 01 = {y 06 1y 2 y 3 } (30) So we ca compute the approximatio precisio ad rough degree as follows: α ( A) [(0801)(0601)] =0 ρ ( A) [(0801)(0601)] =1 (31) Theorem 25 Let (U V R) be a geeralized approximatio space over two uiverses A A 1 IF(U) A A 1 ad ( A) β 2 =( A 1 ) β 2 foray0< 1 0<β 1 β 2 1;the α U ( A) α β 1 )( β 2 )] U( A 1 ) β 1 )( β 2 )] (32) ρ U ( A 1 ) ρ β 1 )( β 2 )] U( A) β 1 )( β 2 )]

8 8 The Scietific World Joural Proof Sice A A 1 wecahave( ( A)) β 1 ( ( A 1 )) β 1 O the other had ( ( A)) β 2 =( ( A 1 )) β 2 Therefore the theorem ca be proved by Defiitio 22 Theorem 26 Let (UVR) be a geeralized approximatio space over two uiverses A A 1 IF(U) A A 1 ( ( A)) β 1 = ( ( A 1 )) β 2 foray0< 1 0<β 1 β 2 1;the α U ( A 1 ) α β 1 )( β 2 )] U( A) β 1 )( β 2 )] (33) ρ U ( A) ρ β 1 )( β 2 )] U( A 1 ) β 1 )( β 2 )] Proof The proof is similar to Theorem 25 Theorem 27 Let A A 1 F(U) adif A 1 U A foray0< 1 0<β 1 β 2 1 oe ca have α U ( A 1 ) =α β 1 )( β 2 )] U( A) β 1 )( β 2 )] (34) ρ U ( A) =ρ β 1 )( β 2 )] U( A 1 ) β 1 )( β 2 )] Proof ItcabeprovedbyTheorem 25 ad Defiitio 22 Theorem 28 Let U V be two oempty fiite uiverses ad let R be the relatio of U VForay A A 1 IF(U)The rough degrees ad precisios of A A 1 A A 1 ad A A 1 have the followig relatios Cosider Proof Accordig to Theorem 10we ca obtai ρ U ( A A 1 ) β 1 )( β 2 )] =1 O the other had =1 1 ρ U ( A A 1 ) β 1 )( β 2 )] =1 =1 1 ( ( A A 1 )) β 1 ( ( A A 1 )) β 2 ( ( A A 1 )) β 1 ( ( A)) β 2 ( ( A 1 )) β 2 ( ( A)) β 1 (R U ( A 1 )) β 1 ( ( A)) β 2 (R α U ( A 1 )) β 2 2 (36) ( ( A A 1 )) β 1 ( ( A A 1 )) α2 β 2 ( ( A)) β 1 ( ( A 1 )) β 1 ( ( A A 1 )) β 2 ( ( A)) β 1 (R U ( A 1 )) β 1 ( ( A)) β 2 (R α U ( A 1 )) β 2 2 (37) ρ U ( A A 1 ) β 1 )( β 2 )] ( ( A)) β 2 (R α U ( A 1 )) β 2 2 ρ U ( A) β 1 )( β 2 )] ( ( A)) β 2 +ρ U ( A 1 ) β 1 )( β 2 )] ( ( A 1 )) β 2 ρ U ( A A 1 ) β 1 )( β 2 )] For classical sets X ad Ywehave Hece X Y = X + Y X Y (38) ρ U ( A A 1 ) β 1 )( β 2 )] ( ( A)) β 2 (R α U ( A 1 )) β 2 2 ( ( A)) β 2 ( ( A 1 )) β 2 α U ( A A 1 ) β 1 )( β 2 )] ( ( A)) β 2 (R α U ( A 1 )) β 2 2 α U ( A) β 1 )( β 2 )] ( ( A)) β 2 +α U ( A 1 ) β 1 )( β 2 )] ( ( A 1 )) β 2 α U ( A A 1 ) β 1 )( β 2 )] ( ( A)) β 2 ( ( A 1 )) β 2 (35) ( ( A)) β 2 ( ( A 1 )) β 2 ( ( A)) β 1 ( ( A 1 )) β 1 = ( ( A)) β 2 + ( ( A 1 )) β 2 ( ( A)) β 2 ( ( A 1 )) β 2 ( ( A)) β 1 ( ( A 1 )) β 1 + ( ( A)) β 1 ( ( A 1 )) β 2

9 The Scietific World Joural 9 ( ( A)) β 2 + ( ( A 1 )) β 2 ( ( A)) β 1 ( ( A 1 )) β 1 ρ U ( A A 1 ) β 1 )( β 2 )] ( ( A)) β 2 (R α U ( A 1 )) β 2 2 =ρ U ( A) β 1 )( β 2 )] ( ( A)) β 2 +ρ U ( A 1 ) β 1 )( β 2 )] ( ( A 1 )) β 2 ρ U ( A A 1 ) β 1 )( β 2 )] ( ( A)) β 2 (R α U ( A 1 )) β 2 2 (39) The other iequality ca be proved similarly Propositio 29 Let (U V R) (U V S) be two geeralized approximatio spaces over two uiverses S R The we ca obtai α ( A) β 1 )( β 2 )] αs U ( A) β 1 )( β 2 )] ρ S U ( A) β 1 )( β 2 )] ρ ( A) β 1 )( β 2 )] α ( B) β 1 )( β 2 )] αs V ( B) β 1 )( β 2 )] ρ S V ( B) β 1 )( β 2 )] ρ ( B) β 1 )( β 2 )] (41) Example 30 (cotiued from Examples 12 ad 24) I the followigwe give aother biary relatio S over the uiverses U ad V S = {(x 1 y 2 )(x 2 y 3 )(x 3 y 4 )(x 4 y 1 )(x 5 y 5 )} (42) Obviously S RadbyDefiitio 7wecaobtai S U ( A) = { y y y y y } α ( A) = (R A) β 1 U β 1 )( β 2 )] (A) β 2 α S U ( A) = (R A) β 1 U β 1 )( β 2 )] (A) β 2 ρ ( A) =1 (R A) β 1 U β 1 )( β 2 )] (A) β 2 ρ S U ( A) =1 (R A) β 1 U β 1 )( β 2 )] (A) β 2 α ( B) = (R B) β 1 V β 1 )( β 2 )] (B) β 2 α S V ( B) = (R B) β 1 V β 1 )( β 2 )] (B) β 2 ρ ( B) =1 (R B) β 1 V β 1 )( β 2 )] (B) β 2 ρ S V ( B) =1 (R B) β 1 V β 1 )( β 2 )] (B) β 2 (40) S U ( A) = { y y y At this time y y } (43) (S U ( A)) } (S U ( A)) y 2 y 3 } (44) The α S U ( A) [(0801)(0601)] = 1 3 ρs U ( A) [(0801)(0601)] = 2 3 (45) So α ( A) [(0801)(0601)] α S U ( A) [(0801)(0601)] ρ S U ( A) [(0801)(0601)] ρ ( A) [(0801)(0601)] (46) Similarly we ca discuss that the approximated set is Ataassov s ituitioistic fuzzy set over the uiverse V 5 Case Study I medical diagosis sometimes we eed to arrage patiets order accordig to the state of their illesses If the patiet is iabadwayheorsheshouldbearragedtoseeadoctor firstiftheillessislighterwecalethimorherwaitfor the time beig So a good orderig system is eeded Whe the patiets symptoms are Ataassov s ituitioistic fuzzy sets ad the relatioship betwee the symptoms ad diseases is a geeral biary relatio decided by medical specialists who have abudat experiece ow we will desig the order system i view of the actual situatio of the hospital by usig rough Ataassov s ituitioistic fuzzy sets theory i order to

10 10 The Scietific World Joural cosider most of the patiets This scheme ca be used i both primary diagosis ad further diagosis Assume triple (U V R) is a geeralized approximatio space where U is a symptom set V is a disease set ad R is a relatio from U to V I geeral the relatio R ca be give by medical specialist Detailed steps are devised as follows Step 1 Weight Ataassov s ituitioistic fuzzy sets of patiets symptoms Because some symptoms are severe ad some symptoms are ot severe we ca weigh the symptoms i order to kow the overall situatio of the patiet Let A = { x 1 μ A (x 1) ] A (x 1) x 2 μ A (x 2) ] A (x 2) x μ A (x ) ] A (x ) } be Ataassov s ituitioistic fuzzy set of a patiet ad ley ω=(ω 1 ω 2 ω ) be a weightig vector satisfyig ω 1 +ω 2 + +ω =1 The weighted ituitioistic fuzzy set ofapatietcabeobtaiedasfollows: A ω ={ x 1 1 (1 μ A (x 1)) ω 1 (] A (x 1)) ω 1 x 2 1 (1 μ A (x 2)) ω 2 (] A (x 2)) ω 1 x 1 (1 μ A (x )) ω (] A (x )) ω } (47) Step 2 Compute the lower ad upper approximatios of weighted Ataassov s ituitioistic fuzzy sets of patiets symptoms accordig to the model we proposed i Defiitio 7 Step 3 Weigh the lower ad upper approximatios of Ataassov s ituitioistic fuzzy sets Step 4 By usig Ataassov s ituitioistic fuzzy hybrid operator [30] calculate weighted lower ad upper approximatios of Ataassov s ituitioistic fuzzy sets comprehesive attribute value d ad the weightig vector w = (w 1 w 2 w ) that satisfied w 1 +w 2 + +w =1is usually give by medical specialist i advace because objects of the lower ad upper approximatios of weighted Ataassov s ituitioistic fuzzy sets are subsets of disease sets If the patiets kow the coditios of themselves well we ca cosider the attribute value i terms of the lower approximatios If the patiets are ot familiar with coditios of themselves well we ca cosider the attribute value i terms of the upper approximatios The lower ad upper approximatio comprehesive attribute values are defied as follows respectively Cosider d( ( Aω)) = IFHw ((μ RU ( A ω ) (y 1)] RU ( A ω ) (y 1)) =(1 (μ RU ( A ω ) (y 2)] RU ( A ω ) (y 2)) (μ RU ( A ω ) (y )] RU ( A ω ) (y ))) (1 μ RU ( A ω ) (y i)) w i (] RU ( A ω ) (y i)) w i) d( ( Aω)) =IFHw((μ RU ( A ω ) (y 1)] RU ( A ω ) (y 1)) =(1 (μ RU ( A ω ) (y 2)] RU ( A ω ) (y 2)) (μ RU ( A ω ) (y )] RU ( A ω ) (y ))) (1 μ RU ( A ω ) (y i)) w i (] RU ( A ω ) (y i)) w i) (48) Step 5 Compute the comprehesive score value sthe higher the score the more serious the patiet The we ca order the patiets i the view of comprehesive score values Accordig to Step 4wealsocagaitwotypesofcomprehesivescore values the first type of the comprehesive score value is about the lower approximatio ad the secod type of the comprehesive score value is about the upper approximatio The results of comprehesive score values are show as follows: s( ( Aω)) = (1 ( s( ( Aω)) = (1 ( (1 μ RU ( A ω ) (y i)) w i) (] RU ( A ω ) (y i)) w i) (1 μ RU ( A ω ) (y i)) w i) (] RU ( A ω ) (y i)) w i) (49) Fiallywecaorderthepatietstoseethedoctoritur Example 31 Let us cosider the problem about how to arrage patiets for a hospital These patiets have eough time ad they agree with the project which maily cosiders the coditios of patiets That is to say the more severe the patiet is the earlier he ca go to see the doctor The uiverse U = {fever (x 1 )headache(x 2 ) stomachache (x 3 ) cough (x 4 ) ad chest pai (x 5 )} is asymptomsetadtheuiversev = {viral fever (y 1 ) dysetery (y 2 ) typhoid fever (y 3 ) gastritis (y 4 ) ad peumoia (y 5 )} is a disease set Table 1 is the coditios of five patiets ad Table 2 is the relatio from U to V determied by medical specialist i advace The the triple (U V R) formed a geeralized approximatio space over two uiverses Ithefollowigwewillrakthepatietsaccordigthe method proposed above i order to cure the severe patiets earlier (1) Weigh Ataassov s ituitioistic fuzzy sets of patiets symptomstheresultsarepresetedi Table 3 (2) Compute the lower ad upper approximatios of Ataassov s ituitioistic fuzzy sets i Table 3 The

11 The Scietific World Joural 11 Table 1: The coditios of five patiets x i μ A i (x i ) ] A i (x i ) x 1 x 2 x 3 x 4 x 5 A 1 (03 05) (07 01) (04 02) (08 01) (02 06) A 2 (06 02) (05 03) (07 02) (07 01) (03 06) A 3 (04 04) (08 01) (05 02) (06 03) (04 05) A 4 (03 04) (06 02) (09 00) (08 01) (03 05) A 5 (05 03) (03 06) (06 02) (07 02) (06 03) Table 2: The relatios betwee symptoms ad diseases (UVR) y 1 y 2 y 3 y 4 y 5 x x x x x Ataassov s ituitioistic fuzzy sets o uiverse U are trasformed ito Ataassov s ituitioistic fuzzy sets o uiverse V ad Table 4 is the calculatio of lower approximatios ad Table 5 is the calculatio of upper approximatios (3) Because some diseases are very serious they must be cured at oce while some diseases are chroic ad they ca be cured earlier or later So the hospital ca weigh diseases so that the doctors kow the patiets coditios itegrally This weight ca be decided by mathematical methods such as statistical method ad differetial method For coveiece we ivite experts who have rich experiece to score the weighig value Assume weighig vector of the diseases is w = ( ) (4) Now we ca compute the comprehesive attribute value of five patiets Accordig to the project we proposed above obviously we have d( ( A 1ω )) =(1 (1 μ RU ( A 1ω ) (y i)) w i (] RU ( A 1ω ) (y i)) w i) Thecomprehesiveattributevalueaboutupperapproximatio ca be obtaied as follows: d( ( A 1ω )) = (1 Similarly ( (1 μ RU ( A 1ω ) (y i)) w i) (] RU ( A 1ω ) (y i)) w i) = ( ) d( ( A 2ω )) = ( ) d( ( A 3ω )) = ( ) d( ( A 4ω )) = ( ) d( ( A 5ω )) = ( ) (52) (53) (5) We calculate the comprehesive score value i the followig: = ( ) Similarly d( ( A 2ω )) = ( ) d( ( A 3ω )) = ( ) d( ( A 4ω )) = ( ) d( ( A 5ω )) = ( ) (50) (51) s( ( A 1ω )) = (1 Similarly ( = (1 μ RU ( A 1ω ) (y i)) w i) (] RU ( A 1ω ) (y i)) w i) s( ( A 2ω )) = s ( ( A 3ω )) = s( ( A 4ω )) = (54) s ( ( A 5ω )) = (55)

12 12 The Scietific World Joural Table 3: The weighted coditios of five patiets x i μ A iω (x i ) ] A iω (x i ) x 1 x 2 x 3 x 4 x 5 A 1ω ( ) ( ) ( ) ( ) ( ) A 2ω ( ) ( ) ( ) ( ) ( ) A 3ω ( ) ( ) ( ) ( ) ( ) A 4ω ( ) ( ) (0929 0) ( ) ( ) A 5ω ( ) ( ) ( ) ( ) ( ) Table 4: The lower approximatios of weighted Ataassov s ituitioistic fuzzy sets y i μ RU ( A iω ) (y i) ] RU ( A iω ) (y i) y 1 y 2 y 3 y 4 y 5 ( A 1ω ) ( ) ( ) ( ) ( ) ( ) ( A 2ω ) ( ) ( ) ( ) ( ) ( ) ( A 3ω ) ( ) ( ) ( ) ( ) ( ) ( A 4ω ) ( ) ( ) ( ) ( ) ( ) ( A 5ω ) ( ) ( ) ( ) ( ) ( ) Table 5: The upper approximatios of weighted Ataassov s ituitioistic fuzzy sets y i μ RU ( A iω ) (y i) ] RU ( A iω ) (y i) y 1 y 2 y 3 y 4 y 5 ( A 1ω ) ( ) ( ) ( ) ( ) ( ) ( A 2ω ) ( ) ( ) ( ) ( ) ( ) ( A 3ω ) ( ) ( ) ( ) ( ) ( ) ( A 4ω ) ( ) ( ) (0929 0) (0929 0) ( ) ( A 5ω ) ( ) ( ) ( ) ( ) ( ) Accordig to the lower approximatio that is to say the first type of projects the patiets order should be A 3 A 2 A 4 A 1 A 5 because of > > > > This project is very strict with patiets they must kow themselves well The secod type of project is about upper approximatio This project is suitable for the situatio i which the patiets do ot kow their coditio well By the similar way we ca calculate the comprehesive score value as follows: s( ( A 1ω )) = (1 Similarly ( = (1 μ RU ( A 1ω ) (y i)) w i) (] RU ( A 1ω ) (y i)) w i) s( ( A 2ω )) = s ( ( A 3ω )) = s( ( A 4ω )) = s ( ( A 5ω )) = So the patiets order should be A 4 A 2 A 5 A 1 A 3 (56) (57) 6 Coclusios I this paper we have itroduced rough Ataassov s ituitioistic fuzzy sets model over two differet uiverses i the geeralized approximatio space (U V R) The the properties of lower ad upper approximatio operators are discussed Meatime we gave the defiitios ad properties about rough Ataassov s ituitioistic fuzzy cut sets i the geeralized approximatio space (UVR)Afterwardswe represeted how to measure rough Ataassov s ituitioistic fuzzy sets over two differet uiverses I additio we also studied a example to illustrate our results Fially we desiged a ew method about how to arrage the patiets reasoablysothatitcatakemostofpatietsitoaccout; this issue is meaigful ad useful i our real life Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper Ackowledgmets This work is supported by Natioal Natural Sciece Foudatio of Chia (o ) Natioal Natural Sciece Foudatio of CQ CSTC (os cstc2011jja40037 ad cstc2013jcyja40051) Sciece ad Techology Program of

13 The Scietific World Joural 13 Board of Educatio of Chogqig (KJ120805) ad Graduate Iovatio Foudatio of Chogqig Uiversity of Techology (o YCX ) Refereces [1] Z Pawlak Rough sets Iteratioal Joural of Computer & Iformatio Scieces vol 11 o 5 pp [2] Z Pawlak RoughSet:TheoreticalAspectsofReasoigabout Data Kluwer Academic Publishers Dordrecht The Netherlads 1991 [3]ZPawlakadRSowiski Roughsetapproachtomultiattribute decisio aalysis Europea Joural of Operatioal Researchvol72o3pp [4] V S Aathaarayaa M N Murty ad D K Subramaia Tree structure for efficiet data miig usig rough sets Patter Recogitio Lettersvol24o6pp [5] W Cheg Z Mo ad J Wag Notes o the lower ad upper approximatios i a fuzzy group ad rough ideals i semigroups Iformatio Scieces vol177o22pp [6] A P Dempster Upper ad lower probabilities iduced by a multivalued mappig The Aals of Mathematical Statistics vol 38 pp [7] I Dütsch ad G Gediga Ucertaity measures of rough set predictio Artificial Itelligece vol 106 o 1 pp [8] R Jese ad Q She Fuzzy-rough sets assisted attribute selectio IEEE Trasactios o Fuzzy Systemsvol15o1pp [9] J Mi ad W Zhag A axiomatic characterizatio of a fuzzy geeralizatio of rough sets Iformatio Scieces vol 160 o 1 4 pp [10] Z Pawlak Rough sets decisio algorithms ad Bayes theorem Europea Joural of Operatioal Research vol 136 o 1 pp [11] Z Pawlak Some remarks o coflict aalysis Europea Joural of Operatioal Research vol166o3pp [12] Z Pawlak ad A Skowro Rudimets of rough sets Iformatio Sciecesvol177o1pp [13] L Zhou ad W Wu Characterizatio of rough set approximatios i Ataassov ituitioistic fuzzy set theory Computers ad Mathematics with Applicatios vol 62 o 1 pp [14] Z Zhag A iterval-valued rough ituitioistic fuzzy set model Iteratioal Joural of Geeral Systems vol39o2 pp [15] W Xu Q Wag ad X Zhag Multi-graulatio fuzzy rough sets i a fuzzy tolerace approximatio space Iteratioal Joural of Fuzzy Systemsvol13o4pp [16]WHXuYFLiuadWXSu Ucertaitymeasureof ituitioistic fuzzy T equivalece iformatio systems Joural of Itelliget ad Fuzzy SystemsIpress [17] Y She ad F Wag Variable precisio rough set model over two uiverses ad its properties Soft Computigvol15o3 pp [18] R Ya J Zheg J Liu ad Y Zhai Research o the model of rough set over dual-uiverses Kowledge-Based Systems vol 23o8pp [19] B Z Su W M Ma ad Q Liu A approach to decisio makig based o ituitioistic fuzzy rough sets over two uiverses Joural of the Operatioal Research Society vol 64 pp [20] T Li ad W Zhag Rough fuzzy approximatios o two uiverses of discourse Iformatio Scieces vol 178 o 3 pp [21] G Liu Rough set theory based o two uiversal sets ad its applicatios Kowledge-Based Systems vol 23 o 2 pp [22] W H Xu Y F Liu ad W X Su Ucertaity measure of Ataassov s ituitioistic fuzzy T equivalece iformatio systems Joural of Itelliget ad Fuzzy Systemsvol26 o4pp [23] L Shu ad X Z He Rough set model with double uiverse of discourse i Proceedigs of the IEEE Iteratioal Coferece o Iformatio Reuse ad Itegratio (IRI 07) pp Las Vegas Nev USA August 2007 [24] B Su ad W Ma Fuzzy rough set model o two differet uiverses ad its applicatio Applied Mathematical Modellig vol35o4pp [25] H Zhag W Zhag ad W Wu O characterizatio of geeralized iterval-valued fuzzy rough sets o two uiverses of discourse Iteratioal Joural of Approximate Reasoig vol51o1pp [26] K T Ataassov Ituitioistic fuzzy sets Fuzzy Sets ad Systemsvol20o1pp [27] K Ataassov Ituitioistic Fuzzy Sets: Theory ad Applicatio Physica Heidelberg Germay 1999 [28] D Dubois ad H Prade Rough fuzzy sets ad fuzzy rough sets Iteratioal Joural of Geeral Systems vol17pp [29] WHXuWXSuYFLiuadWXZhag Fuzzyrough set models over two uiverses Iteratioal Joural of Machie Learig ad Cybereticsvol4o6pp [30] Z Xu Ituitioistic fuzzy aggregatio operators IEEE Trasactios o Fuzzy Systemsvol15o6pp

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