Binary Relations as a Basis for Rule Induction in Presence of Quantitative Attributes

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1 440 JOUNAL OF COMPUTES, VOL. 5, NO. 3, MACH 200 ry eo for e Idco Preece of Qve Are Lpg A e Schoo, Nk Uvery, Tj, 30007, Ch Em: pg2000@.com Lgy Tog Schoo of Mgeme, Hee Uvery of Techoogy, Tj, 30030, Ch Em: oggy2008@.com Arc I org rogh e heory, he oo of e ppromo h ee rodced y g dcery reo defed o he e of ojec. I ome ce, ecery o geerze dcery reo y g ome oher ry reo. I h pper, we coder mry reo d oerce reo mog ojec. Thee ry reo re defed from ome mry mere he eve of ve of y qve re. The reo defed y ge re re ggreged o go reo he eve of he e of re. The, we corc he ower ppromo opero d he pper ppromo opero geered y ry reo d vere reo. I order o dce he mm deco re ed o ppor he deco k, he omry mr of deco e wh repec o he ower ppromo d odry defed o corc he omry fco whch re ooe fco. The e of f he deco re decoded from prme mpc of he ooe fco. A empe red o demore he ppco of h pproch. Ide Term ry reo, mry, oerce, rogh e, ower d pper ppromo I. INTODUCTION The cocep of rogh e w orgy propoed y Pwk [] mhemc pproch o hde mpreco, vgee d cery d y. A rogh e defed y me of wo ordry e ced ower d pper ppromo. The dcery ce defed y dcery reo re he dg ock for he corco of he ower d pper ppromo. The reqreme of dcery reo eem o e rge codo h my m he ppco dom of he Pwk rogh e mode [2]. For empe, h kd of reo mpe h re re om, whch hve eempfed reoe re for hm o whe ome re re ord or qve [3]. Ad, compee formo yem [4, 5] d re-ved formo yem [6, 7] c o e hded wh Pwk rogh e. To overcome he reoee, ever ereg d megf eeo o dcery reo hve ee propoed o ove hee proem. Sepk e [8] propoed oerce reo, whch refeve d ymmerc, o re ve formo yem. Skowro d Sepk [9] preeed more geer defo of ppromo pce whch c e ed for empe for mry ed rogh e mode d vre preco rogh e mode. Sepk [0] frher dced rogh e mode ed o ppromo pce wh cery fco d rogh co. The eeme of ppromo pce re prmerzed. Ad, he rege of prmeer opmzo were dced. Kryzkewcz [4] defed oerce reo compee formo yem d propoed ype of re redco h oy eme he formo, whch o ee from he po of vew of cfco or deco mkg. Eporg he de of dcery fco, deco re re fod drecy from he compee deco e. Greco e. [] rodced he rogh ppromo ed o domce reo d propoed rogh e mehodoogy o yze m-crer choce d rkg deco proem. Sowk d Vderpooe [2] propoed he o of eqvece reo wh geer reo whch oy refevy reqred. Do d Prde [3] comed fzzy e wh rogh e frf wy y defg rogh fzzy e d fzzy rogh e. Che e. [4] repced he dcery reo y fzzy mry reo d geerzed he crp rogh e o fzzy rogh e. I [5, 6], Yo veged rogh e ed o geer ry reo. The hor red from he propere of ry reo o vege he ee propere of he ower d pper ppromo opero geered y ch reo. H de re my cocered o he corcve d omc pproche of ppromo operor. I [7, 8], he cocep of cover of vere w propoed o corc he pper d ower ppromo of rrry e. Zh [9] ded coverg-ed rogh e from he opoogc po of vew. 200 ACADEMY PULISHE do:0.4304/jcp

2 JOUNAL OF COMPUTES, VOL. 5, NO. 3, MACH I rey, de o he mpreco of d decrg he ojec, m dfferece re ofe o codered gfc for he prpoe of dcrmo. Th o my e formy modeed y coderg mry or oerce reo [2, 20-23]. Eedg dcery o mry or oerce mpoe o wekeg of ome of he propere of he ry reo erm of ymmery d rvy. I h pper, he ry reo re defed from ome mry mere he eve of ve of y ge re. The he reo defed y ge re re ggreged o go reo he eve of he e of re. Se ppromo c e defed ed o he go reo. I order o dce he mm deco re ed o ppor he deco k, he omry mr of deco e wh repec o he ower ppromo d odry defed o corc he omry fco whch re ooe fco. The e of f he deco re decoded from prme mpc of he ooe fco. The oher pr of h pper re orgzed foow. Seco 2 revew ome c oo of ry reo d vere reo. I Seco 3, we dc he ry reo o qve re ve formo e d he rogh ppromo ed o he reo. Seco 4 pree mehod for deco re dco. The re dco gorhm red wh empe Seco 5. The f Seco coe coco. II. INAY ELATION AND ITS INVESE ELATION I h eco, we pree ome c cocep d propere of ry reo [2, 6, 24, 25] o e ed h pper. Defo. Le U e o-empy fe e. U U he prodc e of U d U. Ay e of U U ced ry reo o U. For y (, k ) U U, f (, k ), we y h reo wh k, d deoe h reohp k,.e., ={(, k ): k }. Defo 2. Le U U e ry reo o U. The vere reo of, deoed y, defed ={( k, ): k }. Defo 3. For y U, we c he e ( )={ k U: k } he ef eghorhood of. Defo 4. For y U, we c he e ( )={ k U: k } he rgh eghorhood of. From he Defo ~4, he foowg propoo c ey e oed. Propoo. k k ( k ) k ( ) (, k ) ( k, ) Defo 5. Le U U e ry reo o U. The reo d o e er f here e k U ch h (, k ) for U; d o e refeve f (, ), or, or ( ) for U; d o e ymmerc f for, k U, (, k ) ( k, ), or k k, or k ( ) ( k ); d o e rve f for, k, U, (, k ) ( k, ) (, ), or k k, or k ( ) ( k ) ( ). Defo 6. Le e reo o U. If refeve d ymmerc, we y oerce reo o U. Defo 7. Le e reo o U. If refeve, we y mry reo o U. Propoo 2. Le e ry reo o U. If refeve, he refeve. Proof. Le U. Sce refeve, y Propoo,, refeve. Propoo 3. If oerce reo, he =. Proof. Sppoe (, k ), he ( k, ). Sce ymmerc, we hve (, k ) ( k, ). Hece =. III. INAY ELATIONS AND SIMILAITY MEASUES IN INFOMATION TALES The rogh e phoophy ed o he mpo h wh every ojec of he vere here oced cer mo of formo, epreed y me of ome re ed for ojec decrpo. My reword ppco hve oh qve d qve re. A mpor fere of kowedge dcovery yem how re dffere ype of re. I h eco, we corc he ry reo o qve re ve. A. Smry Mere d Dce A formo e 4-pe S={U, A, V, f}, where U o-empy fe e of ojec, A o-empy fe e of re, V he dom of he re, V = d f: U A V o fco ch h U V A f(, ) V for ech A, U, ced formo fco. A re A c e qve or qve. A deco e y formo e of he form S=(U, C {d}, V, f), where C e of codo re, d qe deco re d C {d}=. Sppoe d oced dcery reo o U, deoed y I d : I d ={(, k ) U U: f(, d)=f( k, d)}. The fmy of he dcery ce of he reo I d deoed y U/I d ={U : =,2,, }, where U ced -h deco c. The drd rogh e mode c e geerzed y coderg y ype of ry reo o codo re ve ed of he dcery reo. I order o corc ry reo, oe c r from eg reo ewee re ve for ech re. Depedg o ddo formo o re, mry mere c e creed. eohp ewee he ry reo d mry mere S cod e decred foow [26]. k S (, k ) (), 200 ACADEMY PULISHE

3 442 JOUNAL OF COMPUTES, VOL. 5, NO. 3, MACH 200 where A,, k U. ry reo ewee re ve of re. () mry hrehod for ve of re d () [0, ]. Smry mere d dce re coey reed. Ug he dce mere d o re ve, wo ve f(, ) d f( k, ) re codered o e mr f d oy f d (, k ) ε(), where ε() hrehod for ve of re d ε() [0, ]. The, ry reo c e oed o re ve y g dce mere. Ad he foowg eme hod: k S (, k ) () d (, k ) ε().. ry eo o Qve Are Ve ry reo o qve re ve re corced o he of h m dfferece o ome re ve my e jdged mporce. Toerce reo d mry reo re codered for he qve re. Le S={U, A, V, f} e deco e. A qve re. The dce ewee ve f(, ) d f( k, ) c e defed : f (, ) f ( k, )) d (, k )=. () m( f (, )) m( f (, )) Defo 8. Le S={U, A, V, f} e formo e. A qve re. The ry reo of o U, deoed y, defed ={(, k ) U U: d (, k ) ε()}. (2) Whe (, k ), we y h d k re mr o re, deoed y k. Ovoy, oerce reo. For revew of dffere dce mere defed o re ve ee [27]. Sowk d Vderpooe [2, 22] hve propoed mry reo whch oy refeve. Defo 9. Le S={U, A, V, f} e formo e. A qve re. The ry reo of o U, deoed y, defed ={(, k ) U U: f(, ) f( k, ) ε (f( k, ))}, (3) where jec ojec d k refere ojec. ε fco of he ve of re for he refere ojec k. I prcce, he foowg er form of ε ffce o chrcerze gfc dfferece [22]: ε (f( k, ))=α f( k, )+β. (4) Ecep for co hrehod correpodg o α =0, defed ccordg o (3) re o ymmerc. Whe α =β =0, dcery reo. Accordg o (3) d (4), mr o k o re f.e., f(, ) f( k, ) ε (f( k, )), f( k, ) ε (f( k, )) f(, ) f( k, )+ε (f( k, )). Defo 0. For y k U, we c he erv [f( k, ) ε (f( k, )), f( k, )+ε (f( k, ))] he mry c erv of k o re, deoed y [ (), The, he foowg reo hod: k k f f(, ) [ (), k ()]. (5) k k ()]. Whe (, k ), we y h d k re mr o re, deoed y k. [22]. mry reo C. ry eo d ogh Appromo The reo defed y ge re re ggreged o go reo he eve of he e of re. Defo. Le S={U, A, V, f} e formo e. The ry reo ewee ojec wh repec o he e of re A, deoed y, defed ={(, k ): k, }. (6) If (, k ), d h he ojec mr o k wh repec o, deoed y k. The foowg reo hod: = I. (7) Defo 2. The vere reo of, deoed y, defed ={( k, ): k, }. ( )={ k U: k } he e of jec ojec k whch re mr o wh repec o. ( )={ k U: k } he e of refere ojec k o whch mr wh repec o. Defo 3. Le S={U, A, V, f} e formo e. X U d A. ry reo defed o U. The -ower ppromo of X S wh repec o, deoed y (X), d he -pper ppromo of X S wh repec o, deoed y (X), re repecvey defed foow: (X)={ U: () X}, (8) (X)={ U: () X }. (9) The -odry of X S wh repec o (X) (X), deoed y (X). 200 ACADEMY PULISHE

4 JOUNAL OF COMPUTES, VOL. 5, NO. 3, MACH Defo 4. The -ower ppromo of X S wh repec o, deoed y (X), d he -pper ppromo of X S wh repec o (X), re repecvey defed foow:, deoed y (X)={ U: () X}, (0) (X)={ U: () X }. () The -odry of X S wh repec o (X) (X), deoed y (X). If ry reo whch ymmerc, ch oerce reo, he (X)= (X), (X)= (X). IV. DECISION ULE INDUCTION Deco re re ogc eme of he ype f he, where he ecede pecfe ve med y oe or more codo re d he coeqece pecfe gme o oe or more deco ce. Ec re re ppored oy y ojec from he ower ppromo of he correpodg deco c. Approme re re ppored oy y ojec from he odre of he correpodg deco ce. Procedre for geero of deco re from deco e e dcve erg prcpe. The ojec re codered empe of deco. I order o dce deco re wh qe coeqe gme o eemery e, he empe eogg o he eemery e re ced pove d he oher egve. A deco re dcrm f dghe pove empe from egve oe, d mm,.e. removg y re from codo pr gve re coverg o egve ojec []. A. Nomry Mr d Fco I cc rogh e heory, mm deco re c e geered g dcery mr d ooe fco [28]. For he rogh ppromo y me of mry reo, he cocep of omry mr d omry fco re preeed h pper o dce he mm deco re. Defo 5. Le S=(U, C {d}, V, f) e deco e, C. U q he q-h deco c. The omry mr M S ((U q ) (U q )) of S wh repec o he ower ppromo of U q defed M S ((U q ) (U q ))=(m(, y j )) I J, (2) where =, 2,, I, j=, 2,, J, m(, y j )={ : o y j, d (U q ) (U q ), y j U ((U q ) (U q ))}. Defo 6. The omry fco from M S ((U q ) (U q )) defed y: ( U q ) f ( U q ) ( ) of * f ( )= { : m(, y j ) d m(, y j ) }, (3) y j * where he ooe vre correpodg o he re. d re repecvey geerzed cojco d djco operor. Trg f ( ) o djcve orm form d ( U q ) g he orpo w of ooe ger o mpfy, he cojc,.e., he prme mpc of he mpfed deco fco correpod o he mm ec deco re. Defo 7. Le S=(U, C {d}, V, f) e deco e, C. U q he q-h deco c. The omry mr M S ( (U q )) of S wh repec o he -odry of U q defed M S ( (U q ))=(m(, y j )) I J, (4) where =, 2,, I, j=, 2,, J, m(, y j )={ : o y j, d (U q ), y j U (U q )}. Defo 8. The omry fco f from M S ( (U q )) defed ( U q ) ( U q ) ( ) of * f ( )= { : m(, y j ) d m(, y j ) }, (5) y j Trg f ( ( U q ) ) o djcve orm form d g he orpo w of ooe ger o mpfy, he prme mpc of he mpfed deco fco correpod o he mm ppromo deco re. Whe we r he omry fco (3) or (5) o djcve orm form, he foowg ype of re c e oed:.e., (o y j ) U q, (f(, ) [ y j ( ), Sce, we hve f(, ) [ ( ), j q y ( )]). (6) U q q ( )]. (7) y (6) d (7), we ge he foowg eme. (f(, ) [ y j ( ), y ( )]) (f(, ) [ ( ), j ( )]). (8) U q q Smry, we c defe he M S ( (U q )) of S wh repec o he -odry of U q d f ( ). - Uq (. Agorhm for e Idco The gorhm of re dco from deco e preeed he foowg wy. Ipe: A deco e S=(U, C {d}, V, f). Op: Se of mm re h dgh ojec eogg o C(U q ) C (U q ), C (U q ) d C (U q ). ) 200 ACADEMY PULISHE

5 444 JOUNAL OF COMPUTES, VOL. 5, NO. 3, MACH 200 Agorhm: Sep. For ech qve re m C, compe he dcery c of ech ojec he deco e,.e., I m ( )={ U: m I }. Sep 2. For ech qve re c k C, compe he mry c erv of ech ojec he deco e ccordg o (5). Sep 3. For ech ojec, compe he mry c for ech qve re c k : c k ( )={ U: ck }. Sep 4. For of he qve d qve re, ggrege he dcery ce I ( ) d he mry ce ereco operor,.e., C ( )=( I I C m c k m ( ) of ojec g he ( m ) ) ( c I k C ck ( ) ). Sep 5. Compe he e of ojec o whch mr ccordg o C,.e., C ( )={ U: }. Sep 6. For ech deco c U q, compe he C(U q ), C (U q ) d C (U q ); C (U q ), C (U q ) d C (U q ). Sep 7. Corc he omry mr M S (C(U q ) C (U q )) of S wh repec o he ower ppromo of U q. Sep 8. For ech ojec U q, =, 2,, I, corc he omry fco f ( C( Uq ) ) of from M S (C(U q ) C (U q )). Sep 9. Corc he omry mr M S ( C (U q )) of S wh repec o he C-odry of U q. Sep 0. For ech ojec U q, =, 2,, I, corc he omry fco f ( C ( U q ) ) of from M S ( C (U q )). Sep. Smry, corc he omry mr M S ( C (U q )) of S wh repec o he C -odry of U q d f ( ). - C ( U q ) Sep 2. Cce he djcve orm form of f ( C( Uq ) ), f ( C ( U q ) ) d f ( ) where ech - C ( U q ) cojc correpod o mm deco re defed (8). Sep 3. epe he Sep 6 o 2 o o he deco re of he ojec d mpfy he re e. V. AN EXAMPLE I order o re he re geero gorhm, e coder mpe empe of hypohec deco Some oher ry reo c e defed o he qve re, ch oerce reo d orkg reo. e, how Te I. The ojec re chrcerzed y wo codo re d. qve re, whe re qve oe wh hree poe ve. Le C={, }. Deco re d mke dchoomc pro of he ojec. The deco ce U d U 2 re, repecvey: U ={ U: d()=}={, 3, 7, 9, }, U 2 ={ U: d()=2}={ 2, 4, 5, 6, 8, 0, 2 }. Le e he mry reo defed ccordg o (3) d (4) for he codo re. Sppoe h α =0.2, β =0. The mry reo o re defed ={(, j ) U U: f(, ) f( j, ) 0.2f( j, )}. Sppoe h he ry reo o re dcery reo: I ={(, j ) U U: f(, )=f( j, )}. For re, he mry c erv of re comped foow. [ (), TALE I. A DECISION TALE Frm d ()]=[43 ( ), 43+( )] =[34.4, 5.6]. The mry c erv of, =, 2,, 2, o re re how he ecod com of Te II. The, he mry c of for re how he hrd com. The dcery ce of for qve re re how he forh com. The mry c C ( ) of for re d c e ggreged g ereco operor,.e., C ( )= ( ) I ( ), how he ffh com. The c of ojec o whch mr,.e., C ( ), re how he h com of Te II. The, he ppromo of deco ce d he odry rego of U d U 2 re, repecvey: C(U )={, 7, 9 }, C (U )={, 2, 3, 4, 7, 8, 9,, 2 }, C (U )={ 2, 3, 4, 8,, 2 }. C(U 2 )={ 5, 6, 0 }, 200 ACADEMY PULISHE

6 JOUNAL OF COMPUTES, VOL. 5, NO. 3, MACH TALE II. SIMILAITY CLASS INTEVALS AND SIMILAITY CLASS Frm [ (), ()] ( ) I ( ) C ( ) C ( ) [34.4, 5.6] { } {, 2, 7, } { } { } 2 [43.2, 64.8] { 2, } {, 2, 7, } { 2, } { 2, } 3 [99.2, 48.8] { 3, 4, 7, 8, 0, 2 } { 3, 4, 8, 9, 2 } { 3, 4, 8, 2 } { 3, 8, 2 } 4 [8.6, 22.4] { 4, 5, 6, 9 } { 3, 4, 8, 9, 2 } { 4, 9 } { 3, 4, 2 } 5 [78.4, 7.6] { 4, 5, 6, 9 } { 5, 6, 0 } { 5, 6 } { 5, 6 } 6 [70.4, 05.6] { 4, 5, 6, 9 } { 5, 6, 0 } { 5, 6 } { 5, 6 } 7 [04, 56] { 3, 7, 8, 0, 2 } {, 2, 7, } { 7 } { 7 } 8 [02.4, 53.6] { 3, 7, 8, 0, 2 } { 3, 4, 8, 9, 2 } { 3, 8, 2 } { 3, 8, 2 } 9 [65.6, 98.4] { 5, 6, 9 } { 3, 4, 8, 9, 2 } { 9 } { 4, 9 } 0 [07.2, 60.8] { 3, 7, 8, 0, 2 } { 5, 6, 0 } { 0 } { 0 } [46.4, 69.6] { 2, } {, 2, 7, } { 2, } { 2, } 2 [00.8, 5.2] { 3, 4, 7, 8, 0, 2 } { 3, 4, 8, 9, 2 } { 3, 4, 8, 2 } { 3, 8, 2 } C (U 2 )={ 2, 3, 4, 5, 6, 8, 0,, 2 }, C (U 2 )={ 2, 3, 4, 8,, 2 }. C (U )={, 7 }, C (U )={, 2, 3, 4, 7, 8, 9,, 2 }, C (U )={ 2, 3, 4, 8, 9,, 2 }. C (U 2 )={ 5, 6, 0 }, C (U 2 )={ 2, 3, 4, 5, 6, 8, 9, 0,, 2 }, C (U 2 )={ 2, 3, 4, 8, 9,, 2 }. The omry mr M S (C(U ) C (U )) he foowg: c M S (C(U ) C (U ))= y M S (C(U ) C (U )), he omry fco f ( C( U ) re corced o o he mm re. For ) empe, f ( C( U )= ( ) ( ) ( ) ( ) ( ) ( ) ) ( ) ( ). Tr f ( C( U ) o djcve orm form, we hve ) f ( C( U ) )=,.e., he prme mpc of f ( C( U ). The ) correpodg mm deco re he foowg: f(,) [43.2,64.8] f(,) [99.2,48.8] f(,) [8.6, 22.4] f(,) [78.4,7.6] f(,) [70.4,05.6] f(,) [02.4,53.6] f(,) [65.6,98.4] f(,) [07.2,60.8 ] f(,) [46.4,69.6] f(,) [00.8,5.2] f(,) [34. 4,5.6] f(, d)=. 2 Afer mpfco, he foowg re c e oed: Smry, f(, ) [34.4, 43.2) f(, d)=. f( 7, ) (22.4, 56] f( 7, )=0 f( 7,d)=. The omry mr M S (C(U 2 ) C (U 2 )) comped foow: M S (C(U 2 ) C (U 2 ))= Sce f ( C( U2 ) )=, where =5, 6, 0, he foowg re c e oed. f(, )=2 f(,d)=2. Therefore, hree ec re ppored y ojec from he ower ppromo of he correpodg deco c re foow: f(, ) [34.4, 43.2) f(, d)=. ( ) f(, ) (22.4, 56] f(, )=0 f(,d)=. ( 7 ) f(, )=2 f(,d)=2. ( 5, 6, 0 ) Smry, M S ( C (U q )) d M S ( C (U q )), q=, 2, re comped foow, repecvey: M S ( C (U ))= ACADEMY PULISHE

7 446 JOUNAL OF COMPUTES, VOL. 5, NO. 3, MACH 200 M S ( C (U ))= Sce C (U )= C (U 2 )={ 2, 3, 4, 8,, 2 } d C (U )= C (U 2 )={ 2, 3, 4, 8, 9,, 2 }, we hve d M S ( C (U 2 ))=M S ( C (U )) M S ( C (U 2 ))=M S ( C (U )). The, he pprome re ppored y ojec from he odre of he correpodg deco ce re foow: f(, ) (5.6, 64.8] f(,d)= f(,d)=2, ( 2 ) f(, ) (7.6,48.8] f(, )= f(,d)= f(,d)=2, ( 3 ) f(, )= f(,d)= f(,d)=2, ( 3, 4, 8, 9, 2 ) f(, ) (5.6, 65.6) f(,d)= f(,d)=2, ( ) Some of he re, dced from he dffere ojec, my e mpfed, e.g., d f(, ) (5.6, 64.8] f(,d)= f(,d)=2 f(, ) (5.6, 65.6) f(,d)= f(,d)=2. We c e hem o oe re of f(, ) (5.6, 64.8] f(,d)= f(,d)=2. The f re e he foowg: f(, ) [34.4, 43.2) f(, d)=. ( ) f(, ) (22.4, 56] f(, )=0 f(,d)=. ( 7 ) f(, )=2 f(,d)=2. ( 5, 6, 0 ) f(, ) (5.6, 64.8] f(,d)= f(,d)=2, ( 2, ) f(, )= f(,d)= f(,d)=2, ( 3, 4, 8, 9, 2 ) VI. CONCLUSIONS Whe eedg he rogh e heory o he e of mry reo or oerce reo whch re o ecery ymmerc d rve, he cocep of defy doe o correcy cpre he preece or ece of mgy. Therefore, we propoed he ew defo of ower d pper ppromo ed o he ry reo d vere reo. Thee ew defo propery chrcerze he e of pove ojec d he e of mgo ojec whe he ry reo o ecery ymmerc or rve. I order o dce re e from deco e vovg qve d qve re, he omry mr d he omry fco re preeed ed o he cocep of he ry reo defed from ome mry mere. I compro wh cc pproche, he eeded rogh e pproch o re dco h ome m dvge. Fr, he mehod of corco of mry mere overcome he rc cojcve operor rdoy ed rogh e heory ed o dcery reo. Secody, he deco re dced from mry reo re more ro h hoe dced from dcery reo ce he former re e eve o m dfferece of re ve. Ly, he mehod doe o reqre y dcrezo of he dom of qve re. ACKNOWLEDGEMENT Th work w ppored y No Nr Scece Fodo of Ch (No ). EFEENCES [] Z. Pwk, ogh e, Iero Jor of Comper d Iformo Scece, vo., o. 5, pp , 982. [2] T. Y. L, K. J. Hg, Q. L, d W. Che. ogh Se, Neghorhood Syem d Appromo, Proceedg of he Ffh Iero Sympom o Mehodooge of Iege Syem, Kove, Teeee, pp. 30-4, 990. [3] S. Greco,. Mrzzo,. Sowk, The e of rogh e d fzzy e MCDM, : T. G, T. J. Sewr, T. He, Ed. Mcrer Deco Mkg: Advce MCDM Mode, Agorhm, Theory, d Appco, Kwer Acdemc Pher, Dordrech, pp , 999. [4] M. Kryzkewcz, ogh e pproch o compee formo yem, Iformo Scece, vo. 2, o. -4, pp , 998. [5] M. Kryzkewcz, e compee formo yem, Iformo Scece, vo. 3, o. 3-4, pp , 999. [6] Q. H. H, D.. Y, Z.X. Xe, Neghorhood cfer, Eper Syem wh Appco, vo. 34, o. 2, pp , [7] Q.H. H, D.. Y, Z.X. Xe, Iformo-preervg hyrd d redco ed o fzzy-rogh echqe, Per ecogo Leer, vo. 27, o. 5, pp , [8] J. Sepk, M. Kręowk, Deco yem ed o oerce rogh e, Proceedg of he Forh Iero Workhop o Iege Iformo Syem, Agow, Pod, pp , 995. [9] A. Skowro, J. Sepk, Toerce ppromo pce, Fdme Iformce, vo. 27, o. 2-3, pp , 996. [0] J. Sepk, Opmzo of rogh e mode, Fdme Iformce, vo. 36, o. 2-3, pp , 998. [] S. Greco,. Mrzzo,. Sowk, ogh e heory for mcrer deco y, Erope Jor of Opero eerch, vo. 29, o., pp. -47, 200. [2]. Sowk, D. Vderpooe, A geerzed defo of rogh ppromo ed o mry, IEEE Trco o Kowedge d D Egeerg, vo. 2, o. 2, pp , ACADEMY PULISHE

8 JOUNAL OF COMPUTES, VOL. 5, NO. 3, MACH [3] D. Do, H. Prde, ogh fzzy e d fzzy rogh e, Iero Jor of Geer Syem, vo. 7, o. 2-3, pp , 990. [4] D. G. Che, W. X. Zhg, D. Yeg, E.C.C. Tg, ogh ppromo o compee compeey drve ce wh ppco o geerzed rogh e, Iformo Scece, vo. 76, o. 3, pp , [5] Y. Y. Yo, Corcve d gerc mehod of he heory of rogh e, Iformo Scece, vo. 09, o. -4, pp. 2-47, 998. [6] Y. Y. Yo, eo erpreo of eghorhood operor d rogh e ppromo operor, Iformo Scece, vo., o. -4, pp , 998. [7] W. Zkowk, Appromo he pce (U,Π), Demoro Mhemc, vo. XVI, pp , 983. [8] W. Zh, F. Y. Wg, edco d omzo of coverg geerzed rogh e, Iformo Scece, vo. 52, pp , [9] W. Zh. Topoogc pproche o coverg rogh e, Iformo Scece, vo. 77, o. 6, pp , [20] S. Mrc. Toerce ogh Se, Cech Topooge, Lerg Procee,. of he Poh Acdemy of Scece Techc Scece, vo. 42, o. 3, pp , 994. [2] A. Skowro, J. Sepk. Geerzed Appromo Spce, I: T.Y. L, A. Wderger, Ed. Sof Compg: ogh Se, Fzzy Logc, Ner Nework, Ucery Mgeme, Kowedge Dcovery. Smo Coc, S Dego, CA, pp. 8-2, 995. [22]. Sowk, D. Vderpooe. Smry eo for ogh Appromo, I: P. P. Wg, Ed. Advce Mche Iegece d Sof Compg, Drhm, NC: Dke Uvery Pre, vo. IV, pp. 7-33, 997. [23] W. Zh. Geerzed ogh Se ed o eo, Iformo Scece, vo. 77, pp , [24] P. jgop, J. Moe. Dcree Mhemc for Comper Scece. Sder Coege, Cd, 992. [25] W. Zh, F. Y. Wg. ry reo ed rogh e. Lecre Noe Comper Scece, Sprger er, Hedeerg, vo. 4223/2006, pp , [26] M. Kreowk, J. Sepk, Seeco of ojec d re: oerce rogh e pproch, Proceedg of he Poer Seo of Nh Iero Sympom o Mehodooge for Iege Syem, Zkope, Pod, pp , 996. [27] D.. Wo, T.. Mrez, Improved heerogeeo dce fco, Jor of Arfc Iegece eerch, vo. 6, pp. -34, 997. [28] A. Skowro, C. zer. The Dcery Mrce d Fco Iformo Te, I:. Sowk, Ed. Iege Deco Sppor: Hdook of Appco d Advce of he ogh Se Theory, Dordrech: Kwer Acdemc Pher, pp , 99. Lpg A, or 97, receved he S degree mgeme cece d egeerg from Hee Uvery of Techoogy, Tj, Ch, 993, he MS degree mgeme cece d egeerg from Hee Uvery of Techoogy, Tj, Ch, 999, d he PhD degree mgeme cece d egeerg from Tj Uvery, Tj, Ch, Crrey, he oce profeor he Deprme of Mgeme Scece d Egeerg Nk Uvery. He o he vce drecor of he deprme. H reerch ere cde e egece, rogh e, d formo yem deveopme. Lgy Tog, or 97, receved he S degree mgeme cece d egeerg from Hee Uvery of Techoogy, Tj, Ch, 993, he MS degree mgeme cece d egeerg from Hee Uvery of Techoogy, Tj, Ch, 996, d he PhD degree coro heory d coro egeerg from Nk Uvery, Tj, Ch, Crrey, he oce profeor he Deprme of Iformo Syem d Iformo Mgeme Hee Uvery of Techoogy. Her reerch ere cde rfc egece, rogh e, d ppy ch mgeme. 200 ACADEMY PULISHE