Readings in fuzzy sets for intelligent systems

922 Pages·1993·42.276 MB·English

by  Didier Dubois, Henri M Prade, Ronald R Yager

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Ackwnloedgments
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M.S ugen(o1 977)F uzzMye asureasn dF uzzIyn tegralSsu-rA­ Whale(n1 984)D ecisiMoank inugn deUrn certaiwnittyhV ari­
T
veyI.n M .M.G uptaG,N Saridainsd.B .RG.a ine(sE ds)F,u zzy ousA ssumptioanbso uAtv ailaIbnlfeo rmatIiEoEnET. r ansactions
AutomaatnadD ecisPiroonc esNsoerst,Hh o llanNde.w Y orkp.p 89- onS ystemMsa.n .a ndC yberne1t4i8c8s8.- 90R0e.p rintweidt ht he
102R.e printweidtt hh ep ermissoifot nh ea uthoarn dp ublisherp ermissoifot nh ea uthoarn dp ublisher
M. Schneidaenrd A .K ande(l1 988)P ropertoifet sh eF uzzy D.W illaeaynsd N .M alvach(e1 981T)h eU seo fF uzzSye tsf or
ExpecteVadl uaen dt heF uzzEyx pecteIdn terivnaF lu zzEyn vi­ theT reatmeonftF uzzIyn formatbiyoC no mputeFru.z zSye tasn d
ronmenFtu.z zSye tasn dS yste2m6s3. 7 3-38R5e.p rintweidt ht he System5s 3.2 3-328R.e printweidt ht hep ermissioofn t he
permissoifot nh ea uthoarnsd p ublisher authoarnsd p ublisher.
W Silve(1r 9t7 9S)y mmetrSiucm matioAn C lasosf O perationsR .RY.a ge(r1 988)O n OrdereWde ightAevde ragiAnggg regation
on FuzzSye tsI.E EET ransacotniS oynsst emMsa.n .a ndC ybernetiOcpse.r atoirnsM ulticriDteecriisai onmakIiEnEgET. r ansacotni ons
9(I0 )6 57-65R9e.p rintweidt thh ep ermissoifot nh ea uthoarn d SysteMmasn,, a ndC yberne1t8i1 c8s3.- 19R0e.p rintweidt thh ep er­
publisher missioonft hea uthoarn dp ublisher
R Slowinsakndi Teghem.,(1 988)F uzzVye rsuSst ochastiRc .RY.a ge(r1 992)O n theS pecifiocfia t Pyo ssibiDliisttyr ibu­
I Ir
Appraochetso M ulticriLtienreiaaPr r ogrammiunngd erU ncer­ tionFu.z yz Setasn dS yste5m0s2 ,7 9-29R2e.p rintweidt ht hep er­
taintNya vaRle seaLrocghi st3i5c,ps p. 673-69R5e.p rinted wimtihs sioonft hea uthoarn dp ublisher.
thep ermissiofot nh ea uthoarnsd W iley-Lai dsisv.i sioofJ no hn
R.R.Y age(r1 987)U sinAgp proximaRteea sonitnogR epresent
WileyS onsI.n c.
& DefauKlnto wled.gA ertifiIcnitaell li3g19e 9n-c1e1.R2 e.p rintweidt h
Takagain dM .S ugen(o1 985)F uzzIyd entificoafSt yisotne ms thep ermissoifot nh ea uthoarn dp ublisher
T
andi tAs pplaitciontsoM odelinagn dC ontroIlE.E ET ransactions
R.RY.a ge(r1 98)1. AN ew MethodolofgoyrO rdinMaul ltiobjec­
onS ystemMsa.n a ndC yberne1t5i1:c1 s6.- 13R2e.p rintweidt ht he
tivDee cisioBnass edon FuzzSye tsD.e cisSicoine n1c2e:s5.8 9-600.
permissoifot nh ea uthoarnsd p ublisher
Reprintweidt thh ep ermissiooftn h ea uthoarn dp ublishTehre
H.T anaka,T sukiyamaan.dK .A sai(1 982)A FuzzSyy stem DecisSicoine njcoeusr niaslp ublishbeyd t heD ecisiSocni ences
T
ModelB aseodn t heL ogicSatlr uctuIrnRe ..R Y.a ge(rEd ).F uzzy Institluotcea.t aetdG eorgiSat atUen iversity
Seta ndP ossibility TheDoervye-lRoepcmPeeennrttg sa,m oPnr ess,
S YasunobSu .M iyamotaon.d H .l har(1a 9 83)F uzzCyo ntrol
New Yorkp.p 257-27R4e.p rintweidt ht hep ermissiooftn h e
forA utomatTirca iOnp eratiSoyns temIn Proceedoiftn hge4s t h
authoarnsd p ublisher.
IFAC/IFIP/IFORS ICnotnefrenraoetnniC cooenn atlri onTl r ansportation
R.M.T ong(1 980S)o meP ropertoifeF su zzFye edbaScyks tems SystemBsa.d en-BadGeenr.m an,yp p.3 3-39R.e printweidt ht he
IEEET ransactoino Snyss temMsa.n andC yberneIt i30c2 s7.- 331.p ermissoifot nh ea uthoarnsd p ublisher
Reprintweidtt hh ep ermissoifot nh ea uthoarn dp ublisher
LA Zadeh(1 981)P ossibiTlhietoyra yn dS ofDta taA nalysIins
R.M.T onga nd Bonison(n1 e9 80A) LinguisAtpipcr oactho L.C obba ndR .T hral(lE ds)M.a thematFircoanlt oifeS rosc iaanld
PA
DecisionmawkiitnhgF uzzSye tsI.E EET ransacotniS oynsst ems.P oliSccyi encBeosu.l deWre stviePwr esps.p .6 9-129R.e printed
Man.a ndC yberneIt0 i7c1s6.- 72R3e.p rintweidt ht hep ermissionw itthh ep ermissoifot nh ea uthor
oft he authaonrdp su blisher
LA Zadeh(1 977)F uzzSye tsI.n Belzer.H olzmana,n d
J. A A
E.T rillaansdL .V alver(1d 9e8 5)O n Modea ndI mplicatiino n Kent( Ed)s. E ncycloopfeC doimap utSecri enacnedT echnolMoagryc,e l
ApproximaRteea soniInngM .M. GuptaK.a ndeWl .B andier DekkeNre.w Y orkR.e printweidt ht hep ermissiooftn h ea uthor
A
and B Kiszk(aE d)s .A pproximRaetaes oniinnE gx peSryts tems.a ndp ublisher.
I
North-HollAamnsdt.e rdapmp,. 1 57-16R6e.p rintweidt ht he M. Zemankov(1a 9 89)F ILIPA FuzzIyn telliIgnefnotr mation
permissoifot nh ea uthoarnsd p ublisher
Systemw ith LearninCga pabilitiIensf ormatSiyosnt ems.
E.T rillaansd Valver(1d 9e8 5)O n Implicatainodnl ndistin1­4(6)473-4P8e6,r gamonP ressL td, London.E ngland.
L.
guishabiilnti hteyS ettionfgF uzzLyo giIcn. Kacprzaynkd R .R Reprintweidt thh ep ermissoifot nh ea uthoarn dp ublisher
I
Yage(rE.d )s M.a nagemDenetc isSiuopnp oSryts teUmssi nFgu zzSye ts HJ Zimmerma(n1 985)A pplicatioofnF su zzSye tT heoryt o
andP ossibilitVye rTlhaTegUo Vr,Ry h,e inlaKnadi.n p,p .1 98-212.
MathematiPcraolg rammiInngf.o rmaStciioenn 3c6e:s2.9 -C5o8p.y ­
Reprintweidt thh ep ermissoifot nh e authaonrdps u blisher
righ1t9 85b y ElseviSecri encPeu blishCion,g I ncR.e printed
witthh ep ermissoifot nh ea uthoarn dp ublisher
Chapter
1
Introcdituon
byD idiDeurb oiHse,n rPir adaen,d R onalRd.Y ager
Fora lontgi mcel,a ssliocghaialcbs e efnle tt oo poionutta ,n dt or egrtehta",tL ogichiaavtneos mo u ch
liemdi tafrsa mae wofromkro delailfnlag c oefth su man negletchtseet du odfy nostu psecttihneg
vagueness,
reaisnoagnn,d mroeercnett lhyue n iquenpersosbi alboift iym protapnatri ttp laiynsm athematthiocuagTlhh ti.s"
theoarsym oad elc afpotruu rnicnegr taanvidan gtuyse sn,e poionftv iewwa sa lseox pressosmeedt liamtee r by
habse eqnu estiaoswn eeldMl a.n ayt mtpethsa vbee en BertrRaunsds (e1l239l) I.n s pei otft hceo nesriadble
madee,s peciinat lhlicyse ntufroayru, g mentthien gi ntefroemrsu til ptle-vlaolgurieacdis si entd h 1e39 0bsy
reprteasteincoanpaalb iolfli otgioiercsf , op rr oposiJnagLn u kasiicaeznw hdi ssc hoowlh od evelloopgewidic tsh
non-admdodietloisfvu en caeirntOtnye. t ohmfeo srta dicailn terrmyet driuavtahl u,e i(tsw )ast heA merican
anfdur itofftu hle astemt petwsa isn iiattbeyLd o tZfia dehp hilosMoapxBh lear(c 1k39 7w)h o pfriorpstos soe-dc alled
in1 96w5i tthhp eu blicoafht iipsoa np "eFru zSzeyt s.""c onsipsrtoefni(cltyeah sne"c esotffo zurzsmy e mbership
Satrtfrionmg itdheeoa fg radmueamlb erishtha ibspe ,e n functiionon rsd)toe "rc haravcatgesuryeim zbeo Clasl.c"u li
thbea sfoirs baol tohg oifgc r adiutaypl riornpt eiaensd af ovra guper edichaavtbeee se snu ggeedbs ytH .W ey!
newp,ai rctulsairmlpayln ed evffeeu,cn tciertcaailnctuyl (us14,90 )b,yA .K apl,a naHn.Fd.S chto( t19,5w 1he)re the
cal"lPeods siTbhieloirtyZy"a dbe(yh1 89)7fo,rh andlibnags fuizcz syec to nnecatlirveaeapsdpe yaS re.Oe s tasiewicz
thneo tioofnp so ssibainldci etryt a(ionnrte yc essi(t1y19)1,9 9 2)9f or daentrdae ifelrcseet nsor elawtokersdT. h e
gradually. Frenclhi ngYuvieGsset ni tlhom(m16e98 c)o intehtdee rm
Together, afnudpz ozsys isbteihtlesoio rtfyyfa e r "ensefmlbolue" F r(etnhcceoh ut nerpfaorfrtu zszeyt ),
framewtoord ke awli tphr ediwchaotsseeas t isfaicsat iwoni thboeuitn agw aroefZ adeh('16s95 )fu zzsye ts.
matotfedr e graenewd i tmho daleixtepisressui nncge rtaiGnetnyt ilhuosmetmdhe te e rfmo nra mianp ga iorfn ested
gruadalPloys.s imbeialsiutarynend se cesmseiatsyua rrees s ubs(eEt1.s E2,)w itEhl E2,i no rdteord istinguish
no-nadditsievtfe un ctiotnhse;f omrer are �
betwe"ecne nitnretarlpr teitoan(siE"nl) tohfce o ncept
max-decompfoodrs iajsbulnec tainodtn h lea tatreer
midne-compofsoacrbo jlnnuec tiBoenrfo.ei ndictahtei ntgh us desc"rpieberdi,po hnaee(nrsidaE n 2l- "El )·
Interesetnionuiggtlh y t ihsme a thematKiacrila n
maifna coeftf su zszeyat nspd o ssibiliatntdyr ytihnego ry,
Mengwerh oi,n 1 915,w ast hfei rsuts ett hote e rm
toc layr tihfe miasisnur eaiss etdh ifisrna mewiotr k,
"ensemfblloue" t hteii nto lfae p apeirnF recnhH.e
seemasp pprroiattome e ntisoonmf eo ruenrnewrhso
consiwdheariteks dn owinnt hfuez zsyel tei rtatausr e-m ax
discuisdseoearfods r madle finivteircoyln osts ote h bea sic
prodtuarcnts iftuizvrzeey il oatnbsu,wt i tahp or babilistic
notions ibnyZt aredodhTu.hc eeidr cwaobenr vkise waesd
inteetraptrHiiowsno .r iknp roabbilgiesotmiecdt erayl s
testimotnhiceeo sn cdeoernfv elopreedsc ehbaeyrwr isth
witthr ianignuelqaura litthyle o cwahotefipn oo ni anrtes
diffebraecnoktug nrdess,p elcliiynta hsee cohnadl ft hoifs
assocwiiattpher dob bialdiittsryi buftuitnoicnoI ntws a.s
centufroyr, mocdaeplasob flhe a ndltihnkegi nodf
motivbaytt ehdne o nr-astniti(viiant s yt rsiecnts e) of
knowltehdhagutem amnasnp uil(aite.i ele.lf,-i dcnlead sses,
equailnaip thyy sciocnatli nwuhueamrne,e lemmeanybt e
impreocrui nscee rptaiienoc fei sn fornmr,au tlpieeosr vaded
indistinfrgoumti wsoh aobttlhhecae atrn s d ibes tinguished
witvha gueonree xscsie opnts).
fromo nea nothaesdr i,s cubsysP eodi ncMaorree.o ver
Mengewro'rosk n p robbialimsettirci c splaectdoe s also
thienr toducotfis oon- catlrlieadn ngourlmaasnr dc o­
TheE mergencoef F uzzyS etsa nd
normwsh,i chha vbee eenx tenssitvuedlbiyye B d.
PossibilTihteyo ry
SchewizaenrAd . S kl(a18r93 a)n,wd h ilcahth earvt eu rned
outtob eb asoipce raftoorr sfs ueztzsyC h(aspe2t)ee .r
Aboauh tu ndyreeadar gso ,A mtehrei pchainl osopher
Menge(r91 66a)c knloewdgseodm ree lationship
CharPleeisrw caeso noef t hfei risntt h meo deamg teo
betweheiinsd eoaf" cloudp-oliinkoter"s" h, a szeyt "
2 ChapIt eIrno tdruction
(Mengetrra'nss lafotri" oenn semfblloeua "n)dt, h ied ea,B enven(u1t9i6 o9f)g, e neirzaeildn formation measures
recenetxlpyr eastst ehdat ti moefa, f uzzy Isnfe atct.th e whicahr e lnoon gneerc essaardidliytb iuvteo ,nl y
ideoafg radmeedm berswhaissp u ggesbtyMe edn gerd ecomposwaibtlrhee spetcoat t riangcuol-anro rm
himsetlotf h (eE asGte)r mamna thematDi.cK ilaanu a (incluthdeci ansogef m axo peration).
whob,e twe1e9n6 a5n d1 972d,e veloapnee xdt ensiveA sw ec an botht hied eoaffu zzsye tfosrd escribing
see
theoorfmy a ny-vasleutesd h tamhsua ctih n c ommowni th vaguper edicoartc elsa swsietsih m precilsoeclayt ed
fuzzsye tth eoarlyb,e wiitta h m ormea thematicbaolulnyd­aarnidet hsne,e efdo arn on-additive calculus of
orientrteeadt m(een.tKg l.a,u1 a9,6 6S)e.Ge o ttw(a1l9d8 4)u ncertbaaisnoetndmy a xa nmdi no peratiwoenrbser ewing
foarn e xtenascicvoeuo nftt h oef tfoerng otwtoernkb sy beforet hien trodoufctt hnieoo nt ioofan f su zszeyat n dof
Klau(aa nfdo rt heigree nsisa)l,o nwgi tah fulla possibmielaistuybry eZ adeihn 1 96a5n di n1 978
bibliography. respectiHvoewleyv.Ze ard,e whh,o h adm adiem portant
TheE ngliescho nomGi.sL t. S .S hackilsea contributtosi yosnst eamnsdc ontrol enbgeifnoereer ing
particnuoltaarpblrlyee c uorfsp oors sibtihleiotIrynya . 196h5a,bs e etnh fier sttob ec onvinced
serioefws o rksst aritnit nhlgea t1e9 40Ssh,a ck(l1e9 49, ". t.h.at thc eo nventqiuoannatli tteacthinvieoq fu es
195159,6 119,7 h9a)ps r opostoe rde plparcoeb abwiiltiht y systaenma lysiinstr insuincsaulifoltryde eda ling
are
an ewc oncienpo tr dteohr a vaem orsea taicstfomroyd eolf withhu manissytsitceo mrsf, o trh amta ttaenry,
uncertianie nctoyn omHiceis n.t rod"udceegdr oefe s systwehmo sceo mpleixsci otmyp artaobt lheao tf
potenstuiraplr wihsiec,eh"s timtaowt hea t eexvteenntts humanissytsitce Tmhse. basisc ofnotre ntthiiosn
ariem possainbdlu es,et dh mea ximuamn dm inimum resotnsw hat mbiegc hatl ltehdpe r ipnlceoif
operatfioocrno sm bintihndege greaensda, p otential inocmpatibSitaltiietndyf o.r mtahleels ys,e onfc e
surprciusrevw eh icuhp t oa c omplementtoa1 t iiso n thipsr inciistp hlaaets t hceo mpleoxfai s tyys tem
nothibnugat p ossibdiilsittryi ibnuZ taidoenhs 'esn se. increaosueras b,i ltiotm ya kep reciansde yet
Moreovheerd ,i scusastel de ng"tuhn certaasi nty signifsitcaatnetma ebnotiusttbe s h avidoirm inishes
probabvisl".iu tnyc"e ratsap ionstsyi bIinls iptioytf.te "h e untait lh resihsro eladc behyeodn wdh icphr ecision
inteorfess otm leo giciinah nisws o r(ke .Hga.m,b lin, ands ignifi(coarrn eclee vabnecceo)ma el most
1959t)h,me a themaotfSi hcasc ktlhee'oswr eyr nee ver mutuaelxlcyl ucshiavrea cte(Zraidset1hi9,c7 s3.)"
complewtoerlkyoe udt . ThenZ adeh (1973)a naa ldtveocrnaaattipevpsero ach
Independtehnept hliyl,o soLp.hJCe.or h e(n1 973) ". .b. aseodnt hper emitsheat th kee ye lements in
advocattheemd i nimuomp eratfiooren s timatthien g humatnh inkairneng o tn umbebrustl, a beolfs
so-caglrlaeddoe fi nductive supptootr hte attafcuhzezsdye ttsh,ai tsc ,l asosfeo sb jeicntw sh icthh e
conjunocftt wioporn o posfritoimto hngesr adaesss ociated transfriotmim oenm bersthoni opn -membeirss hip
witeha cohf t hem. Some ytehaepr hsi lloastoeprh er gradruaatlth ehraa nb rupt. tIhnpede erevda siveness
NichoRleassc h(e1r9 7p6r)o poast ehde oorfpy l ausible off uzziinneh susm atnh oupgrhotc essusgegse sts
reasownhienrgpe r oposiatrioeor ngsa niiznle ady eorfs thamtu coh ft hleo gic behind huimsa n reasoning
vari"opulsa usidbeiglrieTtehyse".sd ee grewehsi,ci hno ur notth trea dititownoa-lv aolreu veedmn u ltivalued
terminaorlreoa gtyh" ecre rtdaeignrteyae rspe,r "o pagated logibcu,ta logiwci tfhu zztyr utfhusz,z y
bym eanosfi nferernuclewe hsi cahrs ei mitlota hroe n es connecatnifudvze zsry,u loefis n ference."
appeariintn hgse y ntapcatriotcfp ossibilloigsiatc i,c A si tc anb es eefrno mt hese qutowot atZiaodnesh,' s
logwihco ssee manctiocmsp leatgerlweyie t pho ssibilmiottyi vaatinobdna scg krounwde rqeu idtieff erfreonmtt h ose
theo(rsyeC eh apt5e)Tr.h ec onsequpernicnect ihpailtse oft haeb ove-tmieonnaeudt hHoeri sn.t rodutchceeod n cept
usegdoe sb actkoT heophraad sitsucsoi,fpA lrei staontdl eo,fa fu zzsye cto mpleitnedleyp enodfet nhtepliryro posals
expretshstaehtsce e rtaaitnttayct hoae c do nclucsainonno ti no rdteorp rovait doeo flo rre preseanntdri enags oning
bew eaktehra tnh aotft hwee akest premitsos weiststheh anevt aiiallia nbfloer mianta im oann nseirm itlota hre
drawitnhcgeo ncluFsoiromna.tl hlciyas,bn e e xpreisns edw ayh umanesx preksnso wleadngdse u mmardiaztea .
termosfa necesmseiatsyu Nr aes N,( B)m in(N(A)Z,a dehc'aslf lo nro n-convetnetcihonniahqlau sbe ese n
<::
N(A B)w)h erAe B ist hmea teriimapll ic(ait.ieo.ns, o metimmeiss understood,a tte hsbeepg eicninaialnnlgdy,
� �
"noAt o rB " )T.h iisn equaclainbt eyo btaiansea d interparsae nte exda mpolfpe e rmissivetr ytihntigon king
consequoefnt cheem in-decompoosfan beicleistsyie tsyc apes ttrhiec otfnm eastsh emartaitchse,r a sta h an
measurfoerst hceo njunction. serioautst emtpoto verttahkeel imitpeodw eorf
Thes pecciaasloe fa S hafe(r1'9s7 6ba)c, o nsonanetx pressiovfee xniessmtsoi dnegal nsmd e thodIsns. p iotfe
plausifbuinlcit(tiwyoh net nh feoc aell emewnhtisc h gitvhee e arleyn courageamnedsn utp poorfts ome
birttoht hissef tu nctaironene steidsm) a,t hematicmaaltlhye matliickiReai ncsh ard Boerll olgmiacnli iaknes
identtioZc aadle h('1s9 7p8o)s sibmielaistuSyrh ea.f er'Gsr igoMroei stihlwe,o rkosfZ adeahn,dh ifso llowers,
contour f(ucnocrtrieosnpsto ont dhiepn lga usiobfi lihtayvo ef teennc ounstkeerpetdi mciissimn,t erparnedt ations
singletobnesv )i ecwaaensda fuzzsye mte mbershiupn warracrnitteidfrc oimms amn syc holwahrisfl,ue z szeyt
functaipo oinn,fit r smta dbey K ampdee F eri(e1t9 80)a.n dp ossibtihleioatrnyyd t heaiprp licartaiiostnehsde
Shilk(r1e9t7m'1as)x itive amreaean soutrheesr examipnlteeo rfooe fts hte SresZe.a deh (fo1r9h 9io0sw) n a ccoounn t
possibilmietays-ulriienkste,r odiuntc heicdsa sfreo ma theev oluotfifu oznz yl ogiAcf.t aelr motshti ryteya rs,
puremlayt hemapotiincotafv l i eAwl.s woo rth-mentiZoandienhgo 'rsi giindaela hsa veb lossomiendt ao
ist hei ntrodubcytK iaomnp,ed eF eriFeotr,ta en d comprehecnosripovufems e thoadnsdt oolfosr h andling
ChapIt Ietnrr otdiuocn3
botchl assweistg hr adumaelm bersahnidnp o n­knowle(deg.lega .co,kf p ercissiloonp,p ilniemsist,a tion
probbialiusntcritencat iAyb.o uttw toh ousarnedsse archoeftr hse nlaatnugrau,lab guertsa )tahsea r woafye xpsriensg
havceo ntriabttu htee dt hoero ratephtlpeii cc alteivoenl sgr.a duiatlIynt. h saetsn ef,u zszeydts o n ohta veex actthley
Thep apecrosl leicntt heedrs eeai dngcsa bne s eeansa samceo nrcnea so thaeprp roatcohv easg uenFeosrs .
rperesentasatmipvleio nftg h eciornb turtiions. instaKn.Fc ien(,e1 97p5r)oo psetsh astta temeabnotaus t
vagpuree dibec attaekt eobe n t ruiaef n odn liytf h ehyo ld
fora lplo ssiwbalyoesf m akitnhgpe r edipceractieIs te .
What FuzzyS etsA reA bout enabtlhecesl assliocgairlce allas thiiobepntsw eeavn a gue
prediAc aantdie t nse gatnioonn-t Aob ep erservBeyd .
Thceo ncoefapf tu zzy sweittt hhdr eee apslresen taticoonn trfuazsztsy e atpp licattaikaoedn vsa notaftg hele i mited
ofc laswsheossbo eu ndaraireneso t qdueitteiern meIdt. overilnabegptp weenA annodn -aAn dmo rgee nelryoa fl
useasc haractfeurnicstttiaiiokcnnv g,a luuessu ailntl hye this okfoi vnedr lnagip nfp zuizpya rttiioinnos r dteohr a ve
inte[r0v,aI1lmt) a .ya lsboea f inoirdt een urmaebsleet oaf s moottrhsa iintofnr oomn ec latsosa nothPearr.i kh
lineoarrdleylr eevdeH losw.e vtehrve a luasteniteo enbd e (198q3u)e sttihopeno ss siboifpl rietcyi sely assessing
neitlhienrea orrldye (rael da ttiisec neo uagsh pooiuntt edde groeeftrs u tfohra v ague preIdnpi rctaacitech.eo ,w ever
earilnty h dee vleopmoefnfu tz zsye tnso)bor u nde(de .g.m,e mbersdheigpr eesm ahianavlneoy r dimneaaln iInng .
thsee mi-roepaeilnn te[r0v,a+lhoo a)bs e euns ed aw itoht hweorr disit s t hoer deriinndgu bcye dt he membership
cositn teetrpart.i oFno)r simpalsiscutimhteuey s ,oe f wed egrbeetewse etnhe el emetnhtaistsm eaninrgaftuhle,r
thien te[r0v,ai1lnt) h foel lowIinna fg u:z czlya ssosm,e thatnh eex avcatl oufet hdee egresT.h iissi na gerement
elemetnhtaastr ec ondseireads" mraginoarl" "l essw itthhq eu alintaattiuvorefte h bea soipecr iaotntsh aarte
accepet,aa"br glei vaed ne groefem embertshhiaipts useodnt hedseeg r(emeismn a,x ancdo mtphlee mentation
intermebdeitawre0ye ( nn on-bmeermshainpd1) ( fullt o1 a sa no rderr-srieonvpgee raitn[i 0o,n1O ]b)v.i oau sly
membehri.spF )uzzya rsgeee tnasel rulsye fodrre presentfiunzgmz eym bershipw ifldulenpe cntodin to hnce o ntienx t
linguliasbetlTishc.i c sa nb ev iewaesde xpresasni nvga riwoauysFs i.r tshtue,n vierosfed isrcso(eui .teh.e,
uncertaabionuttcy l tehaerm -ecauntio nfg lthaebB eult. domaoifnt hmee mbersfhuinpc thiaotsno b) e d efined
thiem portpoainntit st hatthv ea luasteiitoss n u ppotsoe d( e.tagl.l,ni esnso stt h sea emt hifonrga m ano rfo ra tr ee).
bec ommotnot heva riloiuns guliasbteitlchsa a tr e Secoinmtda ,yd epenodnt hoet hcelra swsheisac rhue s ed
involivnaeg di vperno ebmlT.h iisst h ceo mmensurabtiolc iotyvte hrde o maFionir.n stawnictrehepe, sc tt oa g iven
assump,wt hiioucnnhd eralltilhef eszu zsye ltei rtataunrde doma"isnm,a dloelsn" o t meexaanc tthlseya mteh ing
habse efonu ndde batabbysl oe moep ponetnott hste h eory.t hree maivnocianbgu lianrcyl oundle"yslr ageo"ri rsi cher
Iti st ob ec onesriedda caosn venierneta saonnda balnecd o ntaboitnh" sm ediaunmd""a l rg.eL"a syt,alf uifz zy
assumpitnmi aonnsy i tuaDteiloentsti.hn aigss s umptimoenm bersfhuinpc tmiaoyvn a ry fromp erosntoeo n
woullde atdod iifcfulwthieecnso mbinmienmgb ershainpo tHhoewre.v er,i rse wlahyila mtpo rtainnpt r acittsio c e
functionnasm,e ilty uissu ailmlpyo sstiomb elreg e cortrlecrye epsretnhtpe i ecoefks n owlepdrgoev ibdyae nd
complperteeor ridneagsns o,t beydA rro(w1 96i3ns) o ciaelx pearntcd a pttuhrmeee anihneig n tetnod sg tiohv ies
choitchee ory. ownw ordWsh.e thtehrei rsaeu nivecrosnasle onns utsh e
A fuzszeymt e mberfsuhnicpet nicoonad p erse fceer enmeanionfag l insgtuieicx preslsiik"oetn a mlaln i"s
amontgh peo ssiibnltee rtpiroeontftsa h ceo rrespondainnogt mhaetrt er.
labe.l A fuszecztay an l sboed efibnyee dx amplificatFiuoznz,sy e ts be ciannt erpirnde itfeefdne tr wTahyes .
rankeilnegm eanctcso rtod tihnegi r twyiptirhce ascpltei tmyo scto mmovni eiwst os emee mbersfhuinpc taisao ns
tot hceo nceupntd erltyhifenu zgzy s e(tp rotioctaylpw ayo fa ssesdseignrgoe fes sa tisfoarpc rteifeornefo nrc e
elemernetcset ihvgeer eamteemsbte rgsrhaisdp)Fe .u zzy thed iffeernte lemse notr intperarettiso.nT he
sentast aulralpyp eiannr o n-sstpreiccitf siI.cmt aa tybi eo ncommensurhaybpioltiohtnetys h diees g roefsae tiss faction
sofcto nstraoirfn ltesx riebqlueie rnetfomsr whsilcihg hwti trhe stpt eocv arifouuzssz eyt s mapkoesss itibotl e
violactaibeno tnosl er(aet.teghd.de ,ea dliintseo dabyu,t combifnuez zsye tisna meaninwgafyuG.le neral
tomorirsso twai clcle patalbtlheol ueggshoos d )o,re lastipcr incfiopfrlu ezssz eymt e mberfsnuhcitpi eolni citation
clasosfoe bjse catpsp,r oxidmeastceri opftt iyopnoesfs restoore tx ismteitnhgoi dnms e asuremen(tKr atnhtezo ry
situattowi hoinca hg i ven prcocaebnde ua rpep loire d,e ta l1.9,7 1A) p.o iwnotr tmhe ntioanbionfugut z szeyt
evepnr oceduwrietfshu zzsitlayti-ends trouncsIt.nie ach combinaetsipoenc,ii nat lhlneyu merseitctailin stg h,e
casfezu zsye ts praeg sreardavuneads l m ootthar nsitieoxni stoefan p caen oopfld yif freenfta miolfio epesr ators,
from coaneteg oriyn taon othaenrda voiadbsr upti ncludviarnigo euxst ensoifol nosg iocnaells i ke
disconttihnwauotiu tlbideec sa usbeytd h aes signomfe nctoj nunctainoddni jssu ncttoigoentswh,ie tra hv egrianagn d
precbiosuen dafroirte hse cdoenrsesdiu bsets. coTmhpee onrsyoa petriaotnTsh.i vsa rioeftc yo mbination
specifictahtubiseo cnosmm oer e raonbadud sattpi vIen. operatmiaoyhn esls po ldvief uflitcliiektseh f elo lowing:
thciass feu zzyp rsoevtaist d oeo folrb ridgtihngeag p Smitahn dO sher(s1o9n8 g4i)v teh eex amptlhea" ta
betwetehpener ceicvoendt ionfut ihwteo yr alndhd u man peercftblryo wanp plsee"e mtsob em oret ypiocfta hle
disccroegtnei rteipvrete asteinIonpn a.r tifczuuzlsyae rt,s c ojnunicotbnr owanpl pet haonfe itbhrero wonra plpe.
helwpi tihn tceirnfngau merdiactaa lsa ynmdbo lliacb els.H owevtehrei,xs ampallesp oo inotutsth awth ewne
are
Isth oubel edm phastihztaehtfde u zzionfaep srso pertiyn teriesnat celda Cs,sw hicihsm adoef b orderline
isn ovti ewaesad d efecitn ltihneg ueixsptriecso sfi oenl emeonftt wsoo t hcelra ssasyeA s a,n Bd, w em ayh ave
4 ChapIt eIrno tdruction
tor enoarlmitzhere e suolftt h ce omnbaitiooftn h e advocbaytA erdi staonhtdil ssec hooiln;d eseodml ei nkage
memebrshfuicnpt ioofn saA n Bd ino rdteoor b tatihne betwpeoesn sitbhieloairntysdy o mmeo dal slyosgtiecm s
onoef C. havbee en repcoeinnottule(tyde .Fga.r,id neaC!se rarnod
Fuzszeyct asan l sor ebleat tore adn dsoemta ssf, i rsHt erzi1g9,9 G1i)v.ae c no lleocft jioeobcnfot rws h iwceh
notibcyeF do rtaentdK ambou(z91i7a6b )y,v iewai ngh avseo m(ep ossiinbcloeymt pekl)n olwedagbeo tuhte
fuzzy( iasnfi e nti steet taisn cgoa)n vceoxm baitnioofn valoufe aatnt riXb,ur teep reesdbe ynm teanosfa
nestoerdd insaurbys eFtusz.zs ye cto unnjctiaonnds p ossidbsiitlriitbyfou rte iaoconhbj ectths,ee to bjoefc ts
disjunccatbnie oa nlssi one trpreitnte hdif rsa meworsku.ct hh a"tXi sA "( Am ayb ei tselfi ntf huez zgye neral
Someau thaolrssa od vocatiene trtphree toafta i ocna sieas)n i -lklnoswendt e scrbiyb peaad io rff uzszeyt s,
memberfusnhcitpai sol nia k elfinuhcotod(i soCeenh aptenra metlhyfe u zzy osbjeet cwtohsfi cmho roer l ess
8). possisbaltyit shrfeey q uir"eXimeAsn" at n d ftzuhzesy e t
of joebcwthsi mcohr oerl ensesc ess(air.cieela.ryi,t n ly)
satiistBf.ey c aIlu(sAe;;:) N: ( Afo)r e acohb jtet,ch lea tter
Fuzzy SeVtesr susP ossibility fuzszeyt iniucsdl eidn ftohre mTehre.p osssitbiicl i
Distributions framewotrhkue sn abulset soc apttuhrieed eoafs ub­
definsietcteos n dseirbeydv ariaouutsh( oer.Nsga .nr,'i y,a ni
Ane xepsrsisounca hs" Xi Fs" w heXr ei sav ariab1l89e0 ).
anFd afu zzsye (te ."ga.gi,eys o unmga"yb) e u seidnt wo
dfifenrtte ypeosfs itusaw,th iiocbnho ttha kaed vanotfa ge
thfeu zzinoefsF s.O nt hoen hea ntdh eex pre"sXsi iso n Degreeosf T rutha nd Incomplete
F"c atna kpel aicnae sitwuhaetrtiehov ena loufXe is Information
preciksneolwaynn wde e stitmhaeetx et enwth ittcohh i s
valiusec ompatwiibttlhhe le a bFe (lw homseaen ing At tphoiiisnts te emtsi melayn stwtoeh rqe u estion
obvioduespleyon dnts h ceo nteIxntt h)ci.as s weea re raibsyeH da ac(9k17 9a)tt heen d toh1fe9 70tsh,ia std, o
interients htgeer da dusfaotln a,t uorfte h sep ecificawtein oenef dzu zlyo gHiaca?c okb'jse crteihloyen asv ial y on
stated bFy"w h"iXcm hai yes x praec sosnr satint odre sac riopfft uizolznoy g tihcae tm phasfiuzzetzsry u th
qualifiFcoiarn tsitoanon.nc ee ,l ooskosm efbwoohrdo y valuaensdl inguaipsptriiomcxa toifo n membership
shoubleyd o uanngwd e e stitmoaw theae tx taep netr sonf ucntions biyna g liavvbeoencl asb uMloarrteyh .at ne n
satitshfirieessq eumieronrt caqnu ableia fs"i yeodu ngy"e alrast tehrpe,e pe rcstihvaers a tchhearn gfeudzl;zo yg ic
(e.Jgo.h,wn h,o saeg ei sk nowtnob e3 2,c anb e inZ aedhs'esn assew ,i tnebsyps aepde grast heirnte hde se
coindseraes"d y outnogt "hd ee gr0e.ei8 n coan dseiredr edaingissb, o tah l ogoifcg radpuraolp eratnide s
cont.e xt) incompilnefotrem aWthieonn .o nlpyer dfuizczaayrt ee s
Ont hoet hhearn" dXi s aFl"sm oa ym ea"na lli st hiantv olovnelmdyu, il ptle-vlaolguaiercdnes e ed(ewdh at
knowanb otuhtve a loufeX ,i s tXh iasFt " ( whiotut Haaccka ltlhs'e f isrtsatog fef uzzificWahteino n').
knowitnhgve a loufeX priecseilnty h icsa seT)h.i s incompilneftoer mcaotmieosnd eignr,oe fet sr uatrhe
corproensdtsoa situaotfii nocnpo elmtei nformatpieornv awdietudhn certaanidtn rtu-ytv halbueecso imlel ­
(pervwaidtiehmd p recisuinocanei rnattansy di) t,u ianti konn own aarnmedo delblyfe udz zy souftb hsee tusn it
whiwceh cano rodtnehlrpey o ssviabllouefe X sa ccordiinngt eWrhvaaatlr c.ea l'lfuezdtz ryhu -tvalbyuZ easd'te urnh
tot heliervo efpl l ausiobrpi olsistiybW ihleanifu tzyz.y outt ob ei -lklnowtnr u-tvhal,u aesis nducbeyd
seitus s edr ertpeos wehnatt k niosw n abouotf t ahi en cvoamlpulee atneadnr neeos stso m, u cahn eamtpttto grasp
single-vvaarliuatebhdlde ee ,g raetet acthoae v da luet hlei ngusiusbttilocef tt hiweeo sr d i'nttr huneea 't ural
expretshsee slo efpv oesls itbhitalhtivi tasyl iusien deedl anguIangdeef.eu dz,tz ryu- tvhalrueepsr esbeyfun ztzeyd
thvea loufte h vea rlieTa.hb feu zszeyt s t Fheiinne trpretseudb soeftt hsue n iitn tearnvmdao ld elulnicnegr toani nty
asap ossidbiisiltbirutty(i Zoand,1 e 9h7w8h)i,ec xhp resstersu th-vanloute sn edcoce osrsraersitploloy in ndgi uci st
prefereonntc hepeso ssibloeft hvieal llu-ekvsna orbwilnae exepsrsiionnn ast ulraanlg uHaogweesv.ae ttr h,te i moef
X. Ntohtadeti stvianlcumtea sys imaunletouhsalvaye Haakc'csr itpioqsusei,tb hieloirtyty h deai nmde nosfi on
degreoefp ossiebqiultaio1lt . y incompilnfoertmea htaindoo bnte elna id bare in a striking
In ctahsoeef i ncompilnefotrem awteci aocnno mputfaes hi(oanl thtohuewgyeh r ael rebardoyio ndignZ adeh,
tow haetx ttehnietn forma"tXii Fso" ni sco nsisten1t79 5w)i".tF hu ztzryu- tvhalumeigsh"tt h eanp peaasr a
as tatelmiek"nett hv ea loufeX isi n suAb"st;eh ti s siosp hisetxitcrgaatavenadcw eh,it lheen yo wc an vbiee wed
estimbaymt eeadn s poofs siamb eialsiIlut(ryAe W) h.e n asa natwuaryoa fl encompassing ao fs ingle framework
A ianso rdisnaurbysIl e(tAi,)sd e fianste hdme a ximumd egroeftre ust ahn dde groefue nsc aeirnt(tSyeC.eh ap5t er
ofµF o veAr.M oreowvees rh,ab leal l tlh meo re cerotfta hienrs eea gdsiS.nmi)i latrhl'eyl ocalfiutztyzr'yu ­ tohf
thaXt t akietsvs a liunes ome sAu baasslet lth vea luevsa lueemsp hasbiyzB eedl lmaannd Za(9d17e7hw) a s
outsAi hdaeva es maldleegrro efpe o ssibtihcliaisbnt e y ;s ometiimlnelds-e urstoogde,n ekarisan lod fac ontext­
estiemdba ymt eanosf d aua mle asoufrn ee cesNs(iAt)y depeenndcwyh,i ilrtee l almye atnhsia ntf uzlzoyg wihca,t
1- Il(A()A i st hceo mpleomfeA n)t. Tdhuia=sl itiys ctarluliteeshdv aluwaittrehed s pteoac d te scriopft ion
ana vailabolf(e v asgtiuanetc,peo l mektneo)w leadngde ,
betwepeons siabnidnl eicteyys m seiatsuirsae g sr aded
nowti trhe spteotc htae c t,uo abjlectsitvaoetft e h weo rld.
versoifto hnoe n eex isitnmi ondga l wlhoigecixhcp ,r esses
Lalsytf,uz zlyo goircs ,u bcatlhceurlaeiron efo wb eing
ar elabteitownte henene cesasnatdrh ype o ssiabllreea,d y