IJCSNS Intnatonal Jounal of Coput Scnc and Ntwo Scuty VO.6 No.9A Sptb 2006 67 Doubl-lay odfcatoy ngustc Tuth-alu attc-alud Ealuaton thod Dan ng Xu Huang Zaqang Zhang and Yang Xu School of Econocs Infoaton Engnng School of Busnss and Adnstaton Southwstn Unsty of Fnanc and Econocs Chngdu Schuan Chna School of Econocs and anagnt Dpatnt of athatcs Southwst Jaotong Unsty Chngdu Schuan Chna Suay In ths pap a doubl-lay odfcatoy lngustc tuth-alu lattc-alud aluaton thod s psntd. Its logc basc s basd on odfcatoy lngustc tuth-alu lattc-alud poposton logc syst TVP(X and opato odfcatoy lngustc tuth-alu lattc-alud poposton logc OTVP(X. TVP(X and OTVP(X a sla to TVP(X and TVP(X whch ha bn gn n ou pous sach pap. But thy ha dffnt aluaton fld.. dffnt typ of attc Iplcaton Algba. It s an attpt to sol th doubl-lay aluaton pobl wth ncopaabl lngustc qualtat nfoaton. In ths aluaton thod y aluato can spcfy a odfcatoy lngustc tuth-alu fo y sub-cton. ws dffnt aluato can b ndowd wth dffnt potanc accodng to th dffnt bacgound nowldg o xpnc. oo dffnt aluato can spcfy dffnt wght fo y sub-cton accodng to th xpnc o pfnc. All dffnt potanc s agggatd to th fnal aluaton sult. It s shown that ncopaabl lngustc nfoaton can b dalt wth n ths aluaton thod. Ky wods: Atfcal Intllgnc attc Iplcaton Algba attcalud ogc syst Doubl-lay Ealuaton thod wth ngustc Tuth-alu. Intoducton Th a any aluaton stuatons n whch th nfoaton cannot b aluatd n a quanttat fo but ay b n a qualtat on so t s ncssay to us qualtat alu n an aluaton syst. Fo xapl whn w ty to aluat cofot of a ca sat w tnd to us natual languag such as {ost cofotabl o cofotabl uncofotabl} tc. oth than pcs nucal alu. Consd anoth xapl whn w aluat on s ablty w tnd to us {coptnt abl ncapabl unabl} tc. oth than nucal alu. On of th qualtat nfoaton pocssng thods s lngustc aabl and appoachs basd on lngustc aabl. Th fuzzy lngustc aabl and latd appoach s psntd by. A. Zadh n 975[6 7]. Aft. A. Zadh's wo n 975 a lot of fuzzy lngustc appoachs ha bn poposd and appld wth y good sults to dffnt pobls such as ducaton[] softwa usablty-aluaton[4 20] nfoaton tal[] s aluaton n softwa dlopnt[3] aluaton of us ntfac[24] and dcson-ang syst[2 5 6 7 8 9 2 25 26 28] tc.. Whth n aluaton syst o n dcson-ang syst nfoaton fuson plays an potant ol. So solutons to ths pobl a gn n [5 0 27]. In addton th a so sach wo whch ty to us hdg algba to dscb fuzzy lngustc aabl n fuzzy logc n [2 22 23]. How all ths thods a basd on lna odd st. In oth wods ncopaabl nfoaton can t b dalt wth n ths lngustc aabl fawo. In fact ncopaabl nfoaton dos xst n natual languag. Fo xapl t s dffcult to dstngush btwn alost Tu and o o lss Tu. In od to dscb and dal wth ncopaabl nfoaton Xu poposd attc Iplcaton Algba n 993 [30]. Xu tc. ha stablshd lattc-alud poposton logc syst P(X [29] and lattc-alud fst-od logc syst F(X [29] basd on attc Iplcaton Algba snc 993. It pods a ncssay foundaton fo th pocssng of ncopaabl nfoaton. In addton th a so sach wo on ncopaabl nfoaton pocssng. An aluaton thod wth ncopaabl nfoaton s psntd n [3]. attc-alud lngustc-basd dcsonang thod s dscussd n [4]. A odl fo handlng lngustc ts n th fawo of lattc-alud logc F(X s psntd n [5]. A fawo of lngustc tuth-alud popostonal logc basd on attc Iplcaton Algba s gn n [8]. In addton ncopaabl hdg s poposd basd on lattc Iplcaton Algba n [9]. Slaly ncopaabl nfoaton dos xst n aluaton syst. In ths pap anuscpt cd Sptb 5 2006. anuscpt sd Sptb 25 2006.
68 IJCSNS Intnatonal Jounal of Coput Scnc and Ntwo Scuty VO.6 No.9A Sptb 2006 w poposd odfcatoy lngustc tuth-alu attc Iplcaton Algba abb. to TVIA. Basd on TVIA w stablsh odfcatoy lngustc tuth-alu lattc-alud poposton logc syst TVP(X and opato odfcatoy lngustc tuth-alu lattc-alud poposton logc OTVP(X to psnt lngustc tuth-alu of a poposton and ta th tuth-alu as th aluaton alu of ach sub-cton. Basd on ths psntaton w ty to constuct an aluaton thod to sol doubl-lay aluaton pobl wth ncopaabl nfoaton by usng odfcatoy lngustc tuth-alu lattc-alud logc Syst and opato lngustc tuthalu lattc-alud logc syst. Th pap s oganzd as follows. W ntoduc so ncssay dfntons fst. Thn w g odfcatoy lngustc tuth-alu lattc-alud poposton logc syst TVP(X and opato odfcatoy lngustc tuth-alu lattc-alud poposton logc OTVP(X. Aft that w popos th doubl-lay odfcatoy lngustc tuth-alu lattc-alud aluaton thod. It s an attpt to sol ncopaabl nfoaton n a doubl-lay aluaton syst. 2. Plnas In ths scton w wll g so ncssay dfntons fo th adablty and ntllgblty of ths pap. Dfnton 2. [30]. t ( O I b a boundd lattc wth an od-sng noluton I and O b th gatst and th last lnt of spctly and : b a appng. If th followng condtons hold fo any x y z : ( x ( y z = y ( x z; (2 x x = I; (3 x y = y' x' ; (4 x y = y x = I pls x = y; (5 ( x y y = ( y x x; (6 ( x y z = ( x z ( y z; (7 ( x y z = ( x z ( y z; thn ( ' O I s calld a lattc plcaton algba. Dfnton 2.2 [3]. t T= { Tu (abb. to T Fals (abb. to F } th followng lattc plcaton algba ( T ' F T s calld a ta lngustc tuth-alu lattc plcaton algba wh ' s dfnd th sa as that of n classcal logc. In Classcal ogc th tuth-alu of a poposton s Tu o Fals. How w oftn us so odfcatoy wod to dcas o ncas th anng of a wod n daly lf and wo. Natually a odl ust b bult to dscb ths cas. In ths pap w ty to stablsh ths odl nad odfcatoy lngustc tuth-alu lattc-alu logc syst. W ntoduc so odfcatoy wods (abb. To W and cobn th odfcatoy wod (W and tuth-alu {Tu Fals} to a odfcatoy lngustc tuth-alu n odfcatoy lngustc tuth-alu attc-alu logc syst. Exapl 2.3 [29] Consd th st = { a = 2 n} dfn od on as follows: f ff a a wh {2 n}. Fo any n dfn: ( a a = aax{ } ; (2 a a = an{ } ; ' (3 a = a n + ; (4 a a = an{ n + n} thn ( ' O I s a attc Iplcaton Algba. W wll us ths attc Iplcaton Algba to dscb th hdg st. Exapl 2.4 ( t W = {Slghtly (Abb. to Sl Sowhat (Abb. to So Rath (Abb. to Ra Alost (Abb. to Al Exactly (Abb. to Ex Qut (Abb. to Qu Vy (Abb. to V Hghly (Abb. to H Absolutly (Abb. to Ab.} b a odfactoy wod st thn chan Sl So Ra Al Ex Qu V H Ab s a lattc plcaton algba wth opaton as gn n xapl 2.3 and t s a nn odfactoy wod st; (2 t W = {last lss o o lss o ost} b a odfcatoy wod st thn chan last lss o o lss o ost s a lattc plcaton algba wth opaton as gn n xapl 2.3 and t s a f odfcatoy wod st. Dfnton 2.5. t = { a a2 as} b a odfcatoy wod st ({ a a2 as} ' O I b a lattc plcaton algba as dfnd n Exapl 2.3 2 ={Tu Fals} b a tuth-alu st ({ Tu Fals} 2 2' 2 2 O2 I2 b a ta lngustc tuth-alu lattc plcaton algba dfn th poduct of and 2 b as follows: 2 = {( a b a b 2} dfn th opaton ' on as follows: 2 fo any ( a b ( a2 2 ( a b ( a2 = ( a a2 b 2 ( a b ( a2 = ( a a2 b 2 a b ' = ( a ( ' ( b 2
IJCSNS Intnatonal Jounal of Coput Scnc and Ntwo Scuty VO.6 No.9A Sptb 2006 69 ( a T ( a F = ( a ( a F ( a T = ( a ( a T ( a T = ( a ( a F ( a F = ( a ax{0 + s} n{ s + } n{ s s + } n{ s s + } F T T T thn ( 2 ' ( an F ( an T s a lattc plcaton algba [8] ( 2 ' ( an F ( an T s calld a odfactoy lngustc tuth-alu lattc plcaton algba abb. to TVIA ts Hass Gaph s as n fg.. Fg. Hass Gaph of odfcatoy ngustc Tuth-alu attc Iplcaton Algba. Not 2.6 Fo Dscats poduct of two lattc plcaton algbas w can dfn dffnt ngaton dsuncton conuncton plcaton opatons on t o than on nd of lattc plcaton algba can b dfnd as a sult of dffnt way of dfnton o dtals wll b dscussd n ou oth pap. 3. TVP(X and OTVP(X Basd on th odfcatoy lngustc tuth-alu lattc plcaton algba TVIA odfcatoy lngustc tuth-alu lattc-alu poposton logc TVIA can b dfnd as th followng. Dfnton 3. t ( 2 ' ( as F( as T b a odfactoy lngustc Tuth-alu attc Iplcaton algba th dfnton of odfcatoy lngustc Tuthalu lattc-alud poposton logc syst TVP(X s dfnd as follows: Syntax: Th sybols n TVP(X a: ( th st of popostonal aabl: X = {p q }; (2 th st of constants: TVIA; (3 logcal conncts: ' ; (4 auxlay sybols: (. Th st F of foula of TVP(X s th last st Y satsfyng th followng condtons: ( X Y ; (2 TVIA Y; (3 f p q Y thn p' p q p q p q Y. A appng : TVP(X TVIA s calld a odfcatoy lngustc lattc-alud aluaton of P(X f t s a T-hooophs. Basd on TVP(X w can dscb so odfcatoy lngustc tuth-alu but w can t us TVP(X to dscb th odfcatoy lngustc tuth-alu wth potanc coffcnt. In th followng w wll ntoduc opato odfcatoy lngustc tuth-alu lattc-alud poposton logc syst λtvp(x. Fo any λ [0] w dfn λ TVIA = { λα λ [0] α TVIA } λtvp ( X = { λf F TVP ( X λ [0]}. Dfnton 3.2 t ( 2 ' ( as F( as T b a odfcatoy lngustc tuth-alu attc Iplcaton Algba λ [0] th dfnton of OTVP(X s dfnd as follows: Syntax: Th sybols n OTVP(X a: ( th st of popostonal aabl: X = {λ p λq λ }; (2 th st of constants: λtvia; (3 logcal conncts: ' ; auxlay sybols: (. Th st F of foula of OTVP(X s th last st Y satsfyng th followng condtons: ( (λx λy; (2 λotvia λy; (3 f λ p λ 2 q λy; thn ( λp' λ p λ2q λ p λ2q λ p λ2q λy. A appng : OTVP(X λtvia s calld a lngustc lattc-alud aluaton of P(X f t s a T- hooophs. 4. Doubl-lay odfcatoy ngustc Tuth-alu attc-alud Ealuaton thod 4. Foulaton of Doubl-ay Ealuaton thod t th aluaton pobl nol n altnats (obcts { A = 2 n} to b aluatd and aluatos (xpts { = 2 }. t ach altnat wll b aluatd basd on a st of cta (attbuts { C = 2 } and ach cton C can b ddd to ( = 2 sub-cton C C C } and ach p { 2 p
70 IJCSNS Intnatonal Jounal of Coput Scnc and Ntwo Scuty VO.6 No.9A Sptb 2006 sub-cton can't b ddd futh t s a doubl-lay aluaton pobl. Th doubl-lay aluaton pobl can b shown n fgu 2. In ths pap ach cton C ( = 2 s also calld upp-lay cton. Each sub-cton C t ( = 2 t = 2 p s also calld low-lay sub-cton. Ealuatd Altnat C C 2... C C C 2... C... p C 2 C 22... C 2p2 C C 2... C p Fg. 2 Donstaton gaph of doubl-lay aluaton pobl. 4.2 Slcton of Ealuaton T St and Dsgnaton of Ealuaton Valu In ou appoach w focus on pocssng th nfoaton wth lngustc alu spcally nfoaton wth ncopaabl lngustc alu oth than nucal nfoaton. Actually th aluatd t st s usually dffnt fo dffnt aluatd cta n an aluaton pobl n ost cass. W adopt th followng thod to dnot and dsgnat th aluaton alu n ou aluaton thod. ( Slcton of Ealuaton T st. W us th followng odfcatoy lattc plcaton algba H {Tu Fals} = {(a s Fals (a s- Fals (a Fals (a Tu (a 2 Tu (a s- Tu (a s Tu} as dfnd n dfnton 2.5 to dscb th dg of tuth-alu of ach sub-cton. And slct H {Tu Fals} = {(a s Fals (a s- Fals (a Fals (a Tu (a 2 Tu (a s- Tu (a s Tu} as th aluaton t st n ths aluaton pobl. (2 Dsgnaton th Ealuaton Valu All aluatos (xpts co to an agnt on th xclus standad aluaton adct wthout any odfcatoy adb fo y sub-cton n post w. Fo xapl satsfactoy wll b usd as th xclus adct whn w ty to aluat sc qualty oth than dssatsfactoy. Basd on th xclus adct w wll ta th tuth-alu of ach poposton as th aluaton alu. W dnot th xclus adct by A wthout loss of gnalty. Fo th altnat x thn th standad poposton s as x s A. Th aluaton alu wll b gn basd on th tuth-alu of poposton x s A. Fo xapl a standad poposton s gn as n x s A. On aluato (xpt ay thn poposton p: x s y A thn th tuth-alu of ths poposton p gn by ths aluato s y Tu ; Othws f h thns poposton p x s slghtly A thn th tuth-alu of ths poposton p s slghtly Tu. Slaly f a standad poposton s gn as n y s B. If aluato (xpt thns poposton q y s xactly B thn th tuth-alu of poposton q s xactly Tu ; Othws f h thns poposton q y s slghtly not B thn th tuth-alu of poposton q s slghtly Fals. By usng ths thod th aluaton alu can b gn by ach aluato (xpt accodng to th agnt gadlss of th standad adct s A o B and ach aluaton alu blongs to th aluaton t st whch s a odfcatoy lngustc lattc plcaton algba. 4.3 Doubl-ay Ealuaton Pocss Basd on odfcatoy ngustc Tuth-alu attc-alud ogc Syst W g th followng doubl-lay aluaton pocss on odfcatoy lngustc tuth-alu lattc-alud logc syst. ( Dsgnat th aluaton alu of ach altnat fo ach low-lay sub-cton and gt sub-aluaton alu atx. Fo doubl-lay aluaton pobl ach aluato (xpt ( 2 should g aluaton alu basd on th dscpton n 4.2 fo ach sub-cton C t wh = 2 t = 2 p. Th followng sub-cton aluaton alu cto 2 dnots th aluaton alu of th th altnat p fo th sub-cton C C 2 C of th th cton p C gn by th th aluato wh = 2 = 2 n = 2. Fo = 2 n ta ach sub-cton aluaton alu cto as a colun cto w can gt th sub-aluaton alu atx fo p ( = 2 sub-cton on n altnats gn by th th aluato (xpt as n (:
IJCSNS Intnatonal Jounal of Coput Scnc and Ntwo Scuty VO.6 No.9A Sptb 2006 7 (( (2( (( (2( 2 2 2 ( (( (2( p p p Thn fo ach of n aluatd altnats { A A2 An } aluatos (xpts wll g sub-cton aluaton alu atcs as n ( wh = 2 = 2. (2 Spcfy Ipotanc Coffcnt In doubl-lay aluaton pobl th followng th nds of potant coffcnt should b consdd and ncludd n th fnal agggaton pocss. Th fst on s potanc coffcnt of ach low-lay sub-cton fo ach upp-lay cton; th scond on s potanc coffcnt of ach xpt fo ach upplay cton.. fo ach goup of low-lay subcton of ach upp-lay cton; th last on s th potanc coffcnt of ach cton. A Fo th th cton C ( = 2 th aluato (xpt ay g dffnt potanc coffcnt fo ach sub-cton C C C } of th th cton { 2 p C accodng to th dffnt bacgound pfnc o xpnc wh = 2. W us th followng sub-cton wght cto ( ( ( ( 2 p to dnot th potanc fo ach goup of sub-cta C C C } of th th cton C { 2 p gn by th th aluato (xpt wh = 2 = 2. Th a sub-cton wght ctos gn by aluatos (xpts nddually fo ach goup sub-cta of cta as follows: ( ( ( ( wh = 2 = 2. B In addton dffnt xpt ha dffnt potanc fo dffnt cta C accodng to th own nowldg o bacgound so th potanc coffcnt of th th aluato (xpt fo th cton C s ( wh ( = 2 = 2. Th potanc coffcnt of aluato can b gn by a thd-paty o oth wght gttng thod. C Th potanc coffcnt of ach upp-lay cton s dscbd as = 2. Th wght can b gn by a consnsus of all aluatos (xpts o usng oth xstng wght coputaton thods. (3 Agggat and gt fnal aluaton sult. 2 p I. Agggat th potanc coffcnt of th th aluato (xpt fo th th cton C and th subcton wght cto ( ( ( (. Th potanc coffcnt of th th aluato (xpt fo th ( cton C s wh ( = 2 = 2. Th potanc of ach sub-cton of th th cton C gn by th th aluato (xpt.. sub-cton wght cto s ( ( ( ( 2 p thn th doubl wght cto s as n (2 wh s ultplcaton on R. ( ( = ( > ( ( ( ( ( ( ( ( ( ( 2 ( ( ( ( ( 2 ( 2 ( p ( ( ( ( ( p 2 Th agggaton ncluds two an stps: low-lay agggaton and upp-lay basd on low-lay agggaton sults. W wll dscb th spctly as n II and III. II. ( p p (2 Agggat of sub-cton aluaton alu atcs and doubl wght cto. Fo th connnc of th followng dpcton w ha to ndx th odfcatoy lngustc tuth-alu st. Th a a lot of thods to ndx ths tuth-alu wthout loss of gnalty w ndx th tuth-alu st dscbd n fg. as follows: V = ( as T V2 = ( as T V4 = ( a2 F V2s 2 = ( as T = ( a F 2s s V = ( a F V 3 5 2s V = ( a s 2 T = ( a T Thn t s obous that appng f f :{ V V2 Vn} N : s V a a - appng. Fo ths n altnats th th aluato (xpt g th followng sub-cton aluaton alu atx fo ach sub-cton { C C 2 C p } of th th cton as n ( th a sub-cton C aluaton alu atcs. Th agggaton s dscbs as follows. (3
72 IJCSNS Intnatonal Jounal of Coput Scnc and Ntwo Scuty VO.6 No.9A Sptb 2006 (( (2( (( (2( 2 2 2 ( ( ( ( ( ( ( ( ( 2 ( p o (( (2( p p p ( ( (( ( ( (2( ( ( ( ( p ( ( ( (( ( ( ( (2( ( ( ( 2 2 ( 2 p ( 2 2 = (4 ( ( (( ( ( (2( ( ( ( p p ( p p ( p p wh = 2 = 2 n = 2 th atcs n (4 s calld sub-cton aluaton alu atcs wth doubl potanc coffcnt. Thn agggat ths sub-cton aluaton alu atcs wth doubl potanc coffcnt as th follows: Dfn ( ( ( q as a al nub and alu ( ( ( q = 0.0 fst and follow up th pocss to agggat as follows: Fo = 2 ; Fo = 2 n; Fo = 2 ; {Fo q = 2 2s; {Fo t = 2 p ; { ( ( ( q = 0; If ( ( = V t q ( ( q = ( q + ( ( ( ( t ls ( q = ( q ; ( ( tun ( q. } } } and fo th th cton C constuct th odfcatoy lngustc tuth-alu dstbuton cto of th th altnat basd on th alu gn by th th aluato (xpt as n (5: ( ( (2 (2s (5 thn all th ctos as n (6 and gt th odfcatoy lngustc tuth-alu st dstbuton cto of th th altnat fo th th cton C as n (6 ; ( = ( = (2 (2s (6 = constuct th odfcatoy lngustc tuth-alu dstbuton atcs basd on low-lay agggaton as n (7 (( (( (( ( (2 (2s (2( (2( (2( ( (2 (2s wh ach colun (8 ( (7 (2 (2s ( = (2 = (2s = of (7 s th odfcatoy lngustc tuth-alu dstbuton cto of th th altnat fo th th cton C. III. Agggat th upp-lay lngustc tuth-alu dstbuton atcs and gt fnal aluaton sult Th a odfcatoy lngustc tuth-alu dstbuton atcs basd on low-lay agggaton as n (7 and th odfcatoy lngustc dstbuton cto of th th altnat fo th th cton C s as n (8 thn w should agggat th upp-lay nfoaton basd on low-lay agggaton sult as n (7 and potanc coffcnt of ach cta = 2 as follows: Agggat th potant coffcnt of ach cton as n (9 and (0 = = = (( (( (( = = = = (( (( (( ( (2 (2s ( (2 = (2s = = = (2( (2( (2( ( (2 (2s (2( (2( (2( ( (2 (2s thn gt th fnal aluaton sult as n ( ( (2 = (2s = ( = (2 = (2s = (9 (0
IJCSNS Intnatonal Jounal of Coput Scnc and Ntwo Scuty VO.6 No.9A Sptb 2006 73 = = = = = = = = = = = (( (( (( (( (( (( ( (2 (2s (2s ( (2 = = = = = = = = = (2( (2( (2( ( (2 (2s (2( (2( (2( ( (2 = = (2s = = = = = = = ( (2 (2s ( ( ( ( ( ( ( (2 (2s ( ach colun n ( s th fnal aluaton odfcatoy lngustc tuth-alud dstbuton cto. Basd on th pous ndx of odfcatoy lngustc tuth-alu w can dscb th fnal aluaton alu by usng th alu n atx ( and th cospondng lngustc tuth-alu to dnot th fnal aluaton alu cto as n (2. Fo th th altnat ts fnal aluaton alu s: 2 ( ( V (2 V2 (2s V2s Th fnal aluaton sult s a cto. Each t n th cto s a constant lnt n opato odfcatoy lngustc tuth-alu lattc-alud logc syst. In addton w now that th appng laton dfnd n (3 btwn th ndx t st and th odfcatoy lngustc tuth-alu lattc plcaton algba s a - appng and t s soophc appng. oo th odfcatoy lngustc tuth-alu lattc plcaton algba has a patal od as dfnd n fg.. So th patal od can b nducd to th ndx t st slaly th nducd patal od s notd as R h. Bas on R w can dfn a patal laton on fnal aluaton alu cto st {( λv λ2v2 λ2sv2s λ [0]} as follows: Fo (2s; ; - {J: f a < b thn ( av a2v2 a2s V2 s a2sv2s ( bv V2 s V2 s sv2s ; ls f f a > b thn ( bv V2 s V 2s sv2s ( av a2v2 a2s V 2s a2sv2s ; ls f a = b and > go to J; f a = b and = thn ( av a2v2 a2s V2 s a2sv2s = ( bv V2 s V2 s sv2s ; }. By usng th patal latonshp dfnd as th abo w can gt a patal od fo y fnal aluaton alu cto st {( λ V λ V 2 2 λ V 2s 2s λ [0]} dscbd n (2. 5. Conclusons In ths pap a doubl-lay odfactoy lngustc tuthalu lattc-alud aluaton thod s psntd. It s an xtnson of ou pous sach sult n [3]. Th thod s basd on odfcatoy lngustc tuth-alu lattc-alud poposton logc syst TVP(X and opato odfcatoy lngustc tuth-alu lattc-alud poposton logc OTVP(X. It s an attpt to sol th aluaton pobl wth ncopaabl lngustc qualtat nfoaton. In ths aluaton fawo y aluato can spcfy a odfcatoy lngustc tuthalu fo ach sub-cton. oo all dffnt wghts whch co fo aluato (xpt potanc cton potanc and sub-cton potanc a synthszd nto th fnal aluaton sults. Slaly ths thod can b xtndd to ult-lay o xd-lay aluaton syst wth ncopaabl nfoaton. Acnowldgnts W would l to xpss ou thans to th suppot of th Scntfc Rsach Fund of Southwstn Unsty of Fnanc and Econocs (Gant No. 06Q76 and th Natonal Natual Scnc Fund of Chna (Gant No. 60474022. Rfncs [] C. K. aw Usng Fuzzy Nubs n Educatonal Gadng Syst Fuzzy Sts and Systs Vol. 83 pp. 26-27 996. [2] C. Kahaan D. Ruan &. Dogan. Fuzzy goup dcsonaltng fo faclty locaton slcton Infoaton Scnc Vol. 57 pp. 35-53 2003. [3] D. ng H. D. Ja Z. Q. Zhang &Y. Xu ngustc Tuth-alu attc-alud ogc Syst wth Ipotant Coffcnt and Its Applcaton to Ealuaton Syst Intnatonal Jounal of Coput Scnc and Ntwo Scuty Vol. 6 No. 6 pp. -6 2006. [4] E. Chang & T. S. Dllon. A Usablty-Ealuaton tc Basd on a Soft-Coputng Appoach IEEE Tansactons on Systs an and Cybntcs Pat A: Systs and Huans Vol. 36 No. 2 ach 2006. [5] F. Ha E. Ha-Vda &. atnz A Fuson Appoach fo anagng ult-ganulaty ngustc T
74 IJCSNS Intnatonal Jounal of Coput Scnc and Ntwo Scuty VO.6 No.9A Sptb 2006 St n Dcson ang Fuzzy Sts and Systs Vol. 4 pp. 43-58 2000. [6] F. Ha & J.. Vdgay A ngustc Dcson Pocss n Goup Dcson ang Goup Dcson Ngotaton ol. 5 pp. 65-76 996. [7] F. Ha E. Ha-Vda & J.. Vdgay Dct Appoach Pocsss n Goup Dcson ang Usng ngustc OWA Opatos Fuzzy Sts and Systs ol. 79 pp. 75-90 996. [8] F. Ha. atnz & P. J. Sanchz anagng nonhoognous nfoaton n goup dcson ang Euopan Jounal of Opatonal Rsach 66 pp. 5-32 2005. [9] F. Ha & E. Ha-Vda ngustc Dcson Analyss: Stps fo Solng Dcson Pobls und ngustc Infoaton Fuzzy Sts and Systs. ol. 5 pp. 67-82 2000. [0] F. Ha E. Ha-Vda. atnz & P.J. Sanchz Cobng ult-ganula ngustc Infoaton wth Unbalancd ngustc T Sts Tchncal Rpot Dpatnt of Coput Scnc and Atfcal Intllgnc Unsty of Ganada Span 2002. [] G. Bodogna & G.Pass A Fuzzy ngustc Appoach Gnalzng Boolan Infoaton Rtal: A odl and Its Ealuaton J. A. Soc. Info. Sc. Vol. 44 pp. 26-32 993. [2] G. Kl & B. Yuan Fuzzy Sts and Fuzzy ogc: Thoy and Applcatons Upp Saddl R NJ: Pntc Hall 995. [3] H.. Applyng Fuzzy Sts Thoy to Ealuat th Rat of Agggaton Rs n Softwa Dlopnt Fuzzy sts and Systs Vol. 80 pp. 323-326 996. [4] J. u Y. Xu D. Ruan &. atnz A attc-alud ngustc-basd Dcson-ang thod Pocdng of IEEE Intnatonal Confnc on Ganula Coputng (GC 2005 Bng Chna July 25-27 pp. 99-203 2005. [5] J. a W. J. Y. Xu & Z.. Song A odl fo handlng lngustc ts n th fawo of lattc-alud logc F(X Systs 2004 IEEE Intnatonal Confnc on an and Cybntcs Vol. 2 pp: 504 509 2004. [6]. A. Zadh Th Concpt of ngustc Vaabl and ts Applcaton to Appoxat Rasonng I II III Infoaton Scnc Vol. 8 No. 3 pp. 99-25 No. 4 pp. 30-357 Vol. 9 pp. 43-80 975. [7]. A. Zadh Fuzzy Sts and Infoaton Ganulaty n:.. Gupta R. K. Ragad & R. R. Yag ds. Adancs n Fuzzy St Thoy and Applcatons Noth-holland Nw Yo pp. 3-8 979. [8]. Zou J. a & Y. Xu A Fawo of ngustc Tuthalud Popostonal ogc Basd on attc Iplcaton Algba IEEE Intnatonal Confnc on Ganula Coputng USA ay 2006. [9]. D. Coc & E. E. K A Contxt-basd Appoach to ngustc Hdgs Intnatonal Jounal of athatcs Coputng Scnc Vol. 2 pp. 37-382 2002. [20] N. Ban &. aclod Usablty asunt n contxt Bha. Inf. Tchnol. ol. 3 no. /2 pp. 32 45 994. [2] N. C. Ho & W. Wchl Hdg Algbas: An Algbac Appoach to Stuctu of Sts of ngustc Tuth Valus Vol. 35 pp. 28-293 990. [22] N. C. Ho & W. Wchl Extndd Hdg Algbas and Th Applcaton to Fuzzy ogc Fuzzy Sts and Systs Vol. 52 pp 259-28 992. [23] N. C. Ho & H.V. Na An Algbac Appoach to ngustc Hdgs n Zadh's Fuzzy ogc" Fuzzy st and Systs Vol. 29 pp. 229-254 2002. [24] R. olch & J. Nlsn Hustc aluaton of us ntfacs n Poc. AC Huan Factos Coputng Systs (CHI Nw Yo pp. 249 256 990. [25] R. R. Yag An Appoach to Odnal Dcson ang Intnatonal Jounal of Appoxat Rasonng Vol. 2 pp. 237-26 995. [26] R. R. Yag A Nw thodology fo Odnal ultobct Dcsons Basd on Fuzzy Sts n: D. Dubos H. Pad and R. R. Yag Eds. Fuzzy Sts fo Intllgnt Systs San ato CA: ogan Kaufann Publshs 995. [27] R. R. Yag Fuson of ult-agnt Pfnc Odng Fuzzy Sts and Systs Vol. 7 No. pp. -2 200. [28] V. Huynh & Y. Naano A satsfactoy-ontd Appoach to ult-xpt Dcson-ang wth ngustc Assssnts IEEE Tans. On Systs an and Cybntcs-Pat B; Cybntcs Vol. 35 No. 2 pp. 84-96 2005. [29] Y. Xu D. Ruan K. Y. Qn & J. u attc-alud ogc: An Altnat Appoach to Tat Fuzznss and Incopaablty Gany : Spng-Vlag 2003. [30] Y. Xu attc plcaton algbas Jounal of Southwst Jaotong Un. ol. 89- pp. 20-27 993 (n Chns. Dan ng cd th Bachlo dg n Pu athatcs and ast dg n athatcal ogc and Its Applcaton to Coput Scnc fo aonng Noal Unsty n 998 and 200 spctly. Sh cd th Ph. D. dg n Tanspotaton Infoaton Engnng fo Southwst Jaotong Unsty n 2004. Sh now wth School of Econocs Infoaton Engnng Southwstn Unsty of Fnanc and Econocs Chngdu Schuan Chna as an assocat pofsso. Xu Huang cd th Bachlo dg n Industy Econoy and a ast dg n Busnss anagnt and a Ph.D dg n Busnss anagnt n 985 988 2004 fo School of Busnss and Adnstaton spctly. Sh now wth School of Busnss and Adnstaton Southwstn Unsty of Fnanc and Econocs Chngdu Schuan Chna as a pofsso. Zaqang Zhang cd th Bachlo dg n Fnanc fo Zhongnan Unsty of Econocs and aw n 999 h s a BA studnt of School of anagnt and Econoy Southwst Jaotong Unsty Chngdu Schuan Chna. Yang Xu H s a pofsso and ph. D supso n Dpatnt of athatcs and School of Econocs and anagnt of Southwst Jaotong Unsty Chngdu Schuan Chna.