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e ooo-yaap cy C.., .A.yoa Moe eo p pee ycox eopeeeoc L1={b} a21 a22 a2m 2 ai1 ai2 aim i ak1 ak 2

akm k e 2000 C.. , .A. yoa Moe eo p pee ycox eopeeeoc e 2001 22.18 519.816 71 C.., yoa .A.

Moe eo p pee ycox eopeeeoc. e: , 2001. 138 c.

Moopa ocea aay aa p pee ycox e opeeeoc;

paccapac oe eo pee poex cya op ye aepa, papoae, pyooe yopo ee aepa. aac aeaece oco eop p pee, poc pep cooa eoo pa pae cx aa.

ceaco oac cceoa oepa, oep p pee, cce cycceoo eea, a ae cyeo acpao cooecyx ceaoce.

Ta. 12. . 25. op. 124 a.

ISBN 5-900037-19- C.. , .A. yoa, e ooo-yaap cy, Cco coyex copae B epooca ( apo) BC eea cpyypa ( apo) APOC Aye Poeyp y Oopx Cya (eo p pee) P aaa p pee P apaoc pace (coco) C* ec coacoaoc C opaoa ccea C** apaoc cy (coco) C opaoo-ypaa ccea ococepeca ( apo) P o, paee peee MA eo aaa epapx .o.. eeoe ooee ecpooo peoe OPACC OPaa ACCa (eo p pee) OC ooee coacoaoc AP APa oeca (eo p pee) P pe pee C poca cpyypa ( apo) P paoa pya C ceea ( apo) CP ccea oep p pee T* pacopyeoc (coco) T yppa ( apo) C cepa ccea cepa occ Beee poe p pee, oope poo ae oo pac capa a poe aaa cox cce, aa ce oee eco copeeo aye [69].

Bo o cee cce oep p pee pcy cy o opaoo-ypae ccee. o epe pa pep, yopoe cpyyp opaa aaa eop opax ce, poea papao epe cce oep p pee (CP) caoc ocoeo ayao.

C poeoo pep oa oaa oooc MIS (Manager Information System opepoaa a oeceee poecca ypae eoxoo opae o poo, acoe yye ypaeo cce), DSS (Decision Support System opepoaa a eeyaoe oeceee poecca p pee ca coe e oepy aoo poecca) cce, a a pyooe pep eoxoa e oo opea opa p ypae ecoo pee, o oooc oep pee. Booo pea ae aoo oxoa ec papaoa epee cyecyy C pep CP a ee occe.

O ax opoco papao CP ec op aea ecx oee eoo p pee, cocax ocoy ee yopoa. pe pee ccee ypae poe pep cao co cooc cce, pacpeeeoc ee occe, eopeeeoc eyeo coco, eoxooc ya ooe co pax aopo pepe, xapaepy x apa pee. ooy p papaoe CP poeoo pep oae poea opa aeax aeaecx eo o, oox opaa cpyypy coo cce, oopo p aec peee, oeppoa cye oea cepo, p a o ae aece (epa) xapaep oe cea ca apao pee poe, ya ecoc, eooc a x cpeca eeo o.

ypo paaac oceee pe aeaeca eop o a coaa cooyoc eoo, ooax p oepo oepe eo pa pee p cpoax ecx apaepax, xapaepyx cceye poecc, a ae o cyae, oa apaep cyae e. Oao ocoe pyoc o a o cyae, oa apaep oaac eopeee oa o o e pe co a peya pee. Tae cya oy oa a cece eocaoo yeoc poecco, oopx paec peee, a -a yac ypaee ecox , peceyx pae e.

pee, o o e pe ee coco aaa co x, oxo opeeex cce, e oaxc ooy aeaecoy oca, opac a cooae cecx epeex e ex aopo. Ocoe poe aoo oxoa oocc o ac oo, ypae pooco, cycceoy eey, cxoo, c, opao opa, e, oo.

oepa oeca aa cpoe pacpee cep c ooa oecex eoo aaa a ce x pee aaa ooecx, ypacecx, coax, ooecx pyx cce. oco eoo, coyex acoee pe aaa yacecx cce, .e. cce, oopx yacye eoe, peca coo oa eoo, oope eee eoo pee coaac exacecx cce. aeaee ycex, ocye c oo x eoo, oo oc oe p poe e coaa ce oee oee coepee ycpoca. Ho e ycex ce poo pacpocpaeoe yeee o, o e e eoa oo oo cpaeo eo cceo a yacece cce. Ta, apep, ycex, ocye c o o eop ypae ocpypoa ocecx aaox cce coo ooc, cypoa peee o eop aa a ooecx ooecx cce. cex apocoecoo aaa ecx cce c oo oepoa a BM pe oy, o ooepecoe oepoae c oo BM ca p e pee aa poopoa, ooecoo apoa ypae pooco.

o yoo yopeec pa ayoo e oae e ooec c oooc eo oeceoo aaa. O ao, o coe cy oe oecee eo aaa cce ep o yacecx cce ooe x cce, cpax o cooc c yacec ccea. B ocoe oo eca e o, o oo o aa po ecoecoc. Cy oo p a oo pa pepo a: e coee ccea, e eee coco a oe o e pe ee paecoe aee cy e o ee oee. e ye aapye peay aay, e eopeeeee caoc ee peee. cce, cooc oopx peocxo eoop opoo ypoe, ooc paec cc caoc o ca py pya xapaepca. eo o cce o oece aa oee yacecx cce e ee, o-oy, ooo paecoo ae pea x coax, ooecx pyx aaax, cax c yace ooo eoea py e.

pecaea paoa opaec a peocy o o, o eea e eoea c e ca, a ee eoopx eex oec acco oeo [61-64, 69], oopx epexo o p aeoc accy epaeoc e caoopae, a epep e. cao ee, eeoc, pcya poeccy e eoea, ao a c o o, o ocoe oo poecca e e paoa yaa ae ooaa oa, a oa c eeo co c, ee c ee paa oa. eo aa e ea oa pae ocoy po o, o oe oaac oo aoee ax copo eoeecoo e cocooc oea opa, .e. pa paoopa cee e oo e, oo pe e ooee aapyeo poee.

o coe ppoe oea ec pee. Bo ox cyax ocaoa eca pea xapaepa aopa ax, ocoy oce ocox aa, peaex eoeo, e peyec coa ooc. eoe coye oycoc ao eooc, opy opa, ocaoy pee aa eea eex o ec, oope peo oca cxoe ae. oo opa, ocyae o epe opa pe, cyxa, oca p., cyaec, a opao, oy cpyy opa, eoxoo pee ocaeo aa c ao cee ooc. Co cooc oeppoa ee oeca eaa ee co cooc oea opa ec o aoee ex a ec eoeecoo paya, oopoe yaea opao oae eoeec pay o a aaeoo aoo paya, pcaeoo ce aa.

Tpaoe eo aaa cce eocaoo po aa a yacecx cce eo ooy, o o e coco oxa eeoc eoeecoo e oee. ooy ec eoo aaa yacecx cce y oxo, oopx o oc, cpooc aeaec opa e c e-o aco o eoxo oopx coyec eoooeca cxea, oyc aa eeoc ace c. B paoe, c oo copo, poo c cpae aa eoo p pee ycox eope eeoc;

c pyo copo, paccapaec oooc coa e oo oxoa poeccy p pee, o oo ceacy o peo peeo oac e pa cpe exc eoo ae a ao cya, a ooac yepca aopo, aeo ypoa poeypy p pee.

1. Ocoe o eop p pee 1.1. Cce oep p pee x eco opaoo-ypax cceax Bce poecc yopoa copeeoo poeoo pe p, o poepoa e o eo poa, eco aoca pey eoo epaoaoo ypae. Ocoe pee, p aee a ypoe pyooe pep, eooo peaoa e pao opaoo pacpyyp. aeco opaooo oecee ypae o ax aopo, opeex ec eoc paex ypaeecx pee. Ocyce caeo cce opaooo oecee ypae po epooc oy xapaepy paex ypaeecx pee, ypoa cope opa, oep yo opa , a cece, e coo eoc ypae. Coae C pep ooe opoa coec aa copa opa oece oee ooe yoeopee opaox opeoce pyooee oea eo. Cyecye e acoee pe cceo exece pacpyyp oca pep oecea o o epe oo oee pooceo-xoceo, acoo-ooeco eeoc ypae;

eo ypoe C e cooecye copeeoy ypo opaox exoo eo peecx papaoo o ao poee.

C pecae coo coy ooypoey opaoy ccey, apapyy aoapoaoe ypaee ce occe a ypae cce a eeoc pep. Ha y ypyey oe C oo peca e aoec pex occe (pc. 1.1). popaa coa C peycapae p aa [84]: coepecoae pae cyecye cce copa opao opa o pep acaoo oepaoo oec ee ypax cpyyp pyooca pep ce eoxo o ocoepo opae eoxoe cpo;

pae C e x aoaa oep p ypaeecx pee;

ocpoe e cpaeeco opaoo-ypae cce pep.

Ocoa e opoo aa coae cce oep p pe e a occe C, oae aeco oepax, ae cx cpaeecx pee.

Aaeca CP ccea ( occe a oe (Cce a ooo e e oec oc yoocy C ( o a o a occe a Xpae ax o o ece ooce e ec o e o o e oaae oaae oaae oaae ep apx pacpeeee a Pc 1.1. Moe opaoo-ypae cce poeoo pep ep a oo paccapa a poecc pa C, oop acoee pe ye ao-o epe peaoa oco pep . Bo e pe, opoc papao CP o cx op c o aoee ocyaex ayax [52, 66, 101, 121], o cao co ce oee opacae po apoao px pee poec ce oaoo ypae, c oo copo, eocaoo pa, pea oa a ceo e, CP, c pyo copo.

Coae epee CP C pep peye oao papao pa cooyoc cex oeceax occe CP: execoo, aeaecoo, popaoo, opaooo, opaaooo oecee. B oopa paccapac opoc papao aeaecoo popaoo oecee a cooyoc aeaecx eoo, oee, aopo popa peaa ee aa oep p pee.

epaoe eco coo, oopao pyoeo eeo c o opaaooy ypae poe pepe a a poecc p pee. ypae opaae aa ec c aa noyeu o opao aaa aau o eco eo ea, yce ocaex aa oe oooc x o e cooec c ec pecypca yco [111]. Ha opo ae pooc eonouu oe aau ynpaeu a aa o ex eeo, .e. pacpeec pao ey opaee opaa coaco x yoaoy peaae ooo c. a eeoc cocae poecc p pee a opo ae a ypae. Cey a aaec c nepeau no uop auo aaa opauauu npumx peeu. oe e ypae epeaec opooe aae, a ecoey ey cooa ec ooea xapaepca poo pee. aee ceye a onepamuoo ynpaeu u ompo, o peyaa oopoo pac pee. cxo opeeex a cocoo ao, ypaee po e pepe oo peca a ec poecc opaooo oea p pee ex epapx.

Aa pee poecce poeoo pooca xapaepa eeoc o x oepe ooe e 10 ocox oace pee, paex a pep (oee) xoe yca e [66] paumue: ycaoee oecex aecex oaa ee pa pep (oee), op aaooe, pe ocpy ooe cpoeco, ce e c poeoo po oca;

peopauau: ce c py pep, coae co ecx pep, eee ypee opaaoo cpyyp;

ynpaeue: op pooceo popa, peypoae poo ca, epee oo poy pooco;

npoemupoaue: papa oa ooo e, papaoa oo pecypcocepeae exoo pooca poy;

mexoou: execoe exooecoe epe oopyee pooca, exooeca oooa yca ooo e;

caeue: opeeee pya ocao cp, aepao, o eyx e;

peauau: op epcex po ca po y, opeeee pya opeee e, aao;

ocyua ue: ope cocex ocyax epo, epecop exe cx yco peoa e;

ap: aeca, oyee, coae aopx poocex yco;

couao-moe ycyu: oe cpoeco, yypo-poceee epop, opaa pao oeocex oopoex ypee.

Oaae oo pyooe o cex epecex apaex yopoa pep, ocyeceoe cceaec, o ope ee poeypa, yao ope ycox oe o pao, o a pa opepoaoe a paoy opex oex pee cox ecpyyppoax poe, a pe oopx oeae pyooe, o ocoa y CP, oopa ec occeo opaoo-ypae cce poeoo pep.

CP o ccea, ea opaaoy cpey oa aa oo pyooe oye peex pee ecpy yppoax poe, aa ce ceye a: aa c ya ocaoa poe, oppoae op apao pee, opaa oe pee, opo oe pee (pc. 1.2).

Aa poo coco coco Bee poeo cya oppoae ee Aau cumyau u nocmaoa npoe ocaoa aa opo Coacoae, opypoae oe yepee apao Aapa pee pee e oex Pyooco pyooca pee. Aa Bop pee oppe ompo opupoaue u op noeu peeu apuamo peeu Ceac cep Papaoa popa oe pee e px pee Opauau noeu peeu Pc 1.2. Cxea yopoa CP Cyecy pae CP. B acoc o ypo poec co ypaeecx pee yaoo, pyooo, opaa ooo eopaaooo, e cooecye CP.

uuyaa CP ocyae oeo oe o, paee peee pyooe oee, pep, opaa. Boo oc ao cce ac o x aec pyooe, eo a, ao, oa. Ha cpyypy oypa cce eocpeceoe e oaa c e pyooca opeoo a ooae cce. pynnoa CP opepoaa a ocyae py , aoecyx ey coo p pee ao-o poe. oepa poecca pao pyox pee ocyec ec a ce ycpae oyaox apepo ey ea py , pee oecex eoo aaa pee pyo , paoao opaae cax poeyp pao py. Opaao e eopaaoe CP pec p aae cox po e oecoo, ecapoo xapaepa, pee oopx y a o cax paoopax oacx.

B acoc o a paex pee opae pae ypo CP: oepa, aec cpaeec.

Oepa ypoe oeceae peee oopao oop xc aa oepa a opoo peeo epae (ee, eaa, ec ..). Ha o ypoe e a oe oex oepa, a aa p ypaeecx pee. Oepae pee, a pao, pac p aae poe ox ee opaa, ee yaco, paox ec. Taec ypoe oeceae peee a a, peyx peapeoo aaa opa, oooeo a epo ypoe. Taece pee pac a oee eo poeye pee (apa, oyoe ..). Ha o ypoe oe pe aex aa yeaec, o opacae x cooc. Taece pee xapaep occe C. Cpaeec ypoe oeceae paoy pee, apaex a ocee oocpox cpae ecx ee opaa. Tao pee xapaepye e peeo epa (o, ecoo e ..), cepa ec ec ypae oe eo (pepe, eopaao oec ..). acca CP peea a pc. 1.3.

Cpaeec Meoa a o eC P ype CP Oa a o eC P (Cpae ec ypoe ypae) yo eC P Taec ype ya eC P CP (yoa (a ec) ypoe ypae) Oepa ype CP (Oepa ypoe ypae) Pc 1.3. acca CP 1.2. oepa oepa p pee oep p pee cyecy eoe pe, o c o oee ceo ex oac opaoa exoo oep p pee, ao ocoeoc oopo ec oe ea aa aa peaex p oeceoc, pee cooc opaca Cee aeceo o eo opaa aoec eoea o epa. epoaao oep paccapac a cpeco a a eoo p pee, a cpeco cpo peaa ce. B acoee pe oep ec apepo eoea p pee. CP coaec oo opeeeoo acca a a oeceae oepy P p aae poe. P apaa e eoxoe ae, yae poe, oyae coe CP, oo cec oy pee oox poe, poye pe pee pae eo, a cepo [64]. Tao yo aa a c, pee ceo, o xopoe peapeo ooo CP, o oa ee yx ax a, eoxox eoo. o aa ooae P o poey, yo co peoe paoa ay apa ee pee. oco cyecyx CP opepoa a cpaeo y py aa. B acoee pe CP paac ceyx apaex:

oeee CP c aoapoa opao ccea ccea c;

cee CP c cep ccea oee e eyax CP;

coepecoae exooeco a CP.

B yye oc cce, oope coy ocpaac o c e eoea, poa pe eo pao, oope cay a pooee ߻ pyooe. Ho ec eoope pa e pa CP caa o cee e oe opo aeceo o apa pee. Oao ec aea, o ao apa oe oy o poecce aoa eoea c CP, o a oaa, oopo co cocoa o ao.

Bec pae oepo oep p pe e: CP, cepe cce, coeye cce .. [64].

Coemyue cucme [71] peoaa oceoaeoe epa oe aoece c oepaopo c e e apaepo ey e poeo cya a oax peoea oepaopy o pee oe poe.

cnepme cucme [63, 64, 101] papaaac oepo o pecae xpae a cooapoax cepo c e, o o aee ocooac ceac c oee o aae. cepe cce apae a acc aa c oopc pee, p o o y cepa opac ae c oa. peoaaec, o poe, oeae pee, c cao cpyyppoa. cepe cce oy pec pax o eeoc, oope oo cpypoa o ce y aeop: eppea, poo, aoca, poepoae, apoae, aee, oaa, peo, oyee, ypaee. eo p pee cao cpyyppoax poe eoeeca y ee ocoy eoc. oa cepa, ocoae a eo poo o e, a ye, oo ey pea poe a coo ypoe. B c c oa e o epeae x ye oepy. aao p ee C oe po, o aa ax cce oe paoa oo oo opaeo oac. o pacp peey oac, ae peeax oo oac a, oae oce cya e eyce.

onmepa CP [64, 66, 101] o epaa aoapo aa ccea, coya oe pao pee, oecea a ooae e ocy pacpeeeo ae ax peocaa paoopae oooc o oopae opa. Bao oa ccea oep p pee pe cae coo cooyoc ceyx occe: oeca pacpee ex execx cpec;

oeca aeaecx oee;

aaa coco pao pee;

a ax;

cce ypae oe , o oepoa, opao oopae opa. B coca CP xo p ax ooea: aa ax, aa oee po paa occea (pc. 1.4).

co ax popaa occea ypae aa oee aaeoo C oepaooo ypo oee aa ax C CM cpaeecx cpaeecx oye aecx aecx Bee co oepax oepax Ccea ypae poe ypee co aeaecx aeaecx epeco eoe, pa pee Pc 1.4. Apxeypa CP oepe CP, a pao, opepoa a opee a eaece eo p pee [13, 32, 38, 52, 64, 71, 76, 87, 101, 106, 117, 121]. B c c ec aya aa aeaecoo oecee CP: aa, oee, eoo, cxe .. c e, o cpypoa x, ae y yopoeoy, oocoaoy cce oa cooa [8].

1.3. ac poecca p pee B oooe p pee yacye ea pya cea co, oeaa a o o a poecca oep pee, pyo oe, a oopoo oc aaa ooaeoo opa ayeo apaa, a ae p aepecoax . B cao cpyyppoax aaax p pee (P) caa poea opa eco caa c e oeo ee aee [64]. Baee poe ec eoe, oo p (o e opyax) oe ee pea ece oeceoc a pe pee. pee oy oaa eocpeceoe oe ce a eo aococoe oeceoo ooee. Ho o aeo e cea oaae, o aee poe ec ae uo, npuua u peeue (P). oeo, P oe ao, o a cya , oa aee poe ec o ecox eoe, pax yace ee pee. O oe pecaee o eoo opaa, paeo pee, e oopoo ye a opocc, o oc coac. Tpe oo cya P aee poe pae . ec pep, oa pyoo e cpec epeo a pyx pe pee: aa p oaaec a aece, ocae oooee py ( oa poopee) pacope. Ta opao, aee poe P oy a oo, a pa oc. poe pee oo ceac, a oop, aapa P. Ec P oepe co o oa, o oe ae e a peee, a poco oca eo. Ho oeceoc ce pao e a P, a e a ex, o yacoa o ooe pee.

Ha pe pee amue pynn ( [93, 94] x aa amopau) py e, ex oe epec o ooe poee, peye pee. Ta, p p pee o ocpoe AC aopa c: copy cepca epe, aepeco ae ppoce epoep;

copy cpoeo opaa, ocyece ocpoy;

pecae pox paa;

pecae ao opyae cpe. B ao cyae aee poe ( oa P) c ece ac, oope o a papeee a ocpoy AC a coe eppop.

Payoe P cea pae o ae epec ax py, ya x o x pep p oee aepax a pao pee.

p paeco paoe ao eo oe a cycc, oa paccapac pae apa pee, o aa p pee, oce oopoo ao peee o, a e ocya. p paccope apao pee ay po pa cnepm , oope poec coao (ye, e P) a oee ace paccapaeo poe. o-ac expert o ceac, pycco e a coa e ecoo paac cc: o cepo oa ooo cooapoaoo ceaca, yeeo cooa c y p pee [80]. oo opaac a oea, a pooa cxoo ex x pee. aa ae oe , cep caa coe cyeoe ee. Ho ec cep ec pcpace ec poeccoao coe ee, eo oe oe. p p cox (oo cpaeecx) pee x oooe pae yace ocymam no npumu peeu.

Eo po coc payo opaa poecca p pee: o o P aey poe pao ocaoe aa;

e e poe o aopo;

opaa pao c cepa. ocy a ( aa) oo e ae cocex oeo p p pe e, o ooae py yc peoe, ec ce a po paoa pay opocc.

aco oo ocapoa o ey eeepa oo o e opaa o ooy cep oeceoc o a o oeae, o ae pee pae. ooy oe a peae, opee e opo pao. Heapo oe copae po aa c y epe peceaecyeo oec acea, a paoy oo pep oeceoo oee c yepe eo ycaa.

paec o peee oo ca ya (poe cyae p pee p occe, oopx op ocyec ec oocoae e ocye). Ta a ae cyae oeoo opaa, paeo ae pee, oo aaec cpeee eo oo opaa oc co o, a oy oy, oee peee. B o cyae ao opa cyae a P, oaaee ope eeo oo.

1.4. e pecypc p oe pax y eeea oae eoxo oc pa pee [80]. Hapep, poecc apoa oe aepc peee o yepe aa, poecc opo peee o ope a ooe o oppepoe aa.

aoe peee apaeo a ocee oo ecox e e. Hapep, p cpoece AC eaeo o ceye aa: oy ppoc epoep;

oy acay p o yopoa AC. e e oo oc oopeeo.

Oao a ae e cea.

aco cpeaac opypoa acy p p y e apa ypee poopea. My apa pae 0, oa paoa e pooc, o p oa oe paa 0. Ec e p ea, o apa e, ocoy o, pyoe cao c oeo pooca.

Moo o acpoa p p cpoax apaax, o poa apa p aao p.

aoe peee peoaae cooae ex x pecypco.

Ta, cepco epe cxo cyecoa eoxooo a epaoo apooo oecee pao AC. Ec aoo oecee e o , o cycc e ea cca.

B oeo ae ceo pae pee, oya oa p ycy. y coepeo co, o aoe pecypc o oeco exc ee. Bcya a opee, eoe oe pe, ae oap ey ceye oya o ao ee. Acya a pooe a o payo opa co yc. B aax cpaax eop p pee caec paeo oo. B paax ooeco eo p ocyac oe opoc o coooe oeoc oapa ( opee) ooe oeoc ee. aoo popeaeoo oapa ec co oeoc opee. ao peeo oeoc ac, o peea oeoc yae. ae oop, oceye ap oapa eee e opee e epe. Ec ec eoxo oc oy ecox oapo, o opee cpec pacpee co e a, o ooee oeoc oo oapa oe ee epe o oco. ae oop, ec oeoc oapa oe, o cpeca, apaee a eo, o oe. Too a e ee ce eoe p pee aa o aaooe: o a ae oe cpeca oee oee apae eeoc. oo c ca, o aoe oeee eoea ec eceo pa , aa eoea, ocyeceo a opao co op, paoa eoeo.

Ta opao, p paeco paoe a poeo pee ao poaapoa: eo xo oc? ae pecypc oo c ooa oo?.

1.5. Aepa pep Amepamuau [64] aa apa paex pee. o poeo cyae oa cya, oopa ee e eee yx apao pee. B pepe co cpoeco AC a y oo e e aepa: papeee cpoeca eo ape. Ceo aeo, cyecoa cao aa p pee eoxoo e xo e aepa. Heaucuu c e aepa, e ec c oop (yaee paccope, eee ae ce ye ..) e oaa a aeco pyx aepa.

p aucux aepaax pee o o x oaa e a aeco pyx. pepo ec pyoa acoc ec pee o paccapa xo oy aepay py, o ao paccap a c pyy. py o acoc ec acoc o a epa, caex paccope. Hapep, acoc pae x pecopae o ex, oope e ce e.

Be ae acoc o ecyecyx aepa. Ta, opa eao aepa, coaaeo eoeo o pe opa, oe oaa e a op peax aepa, ocoeo ec ec aea a peayeoc eaoo apaa.

aa p pee oy cyeceo oac o cy a epa x a a oe pao o p pee.

Bcpeac aa, oa ce aepa ye aa eoxo op oo oeca. Ta, oe ca pao opa yx e ye exc, opee aoee ey opaa , y yepce .. Ocoeoc x aa ec ay oe epacpeec oeco aepa. Ho cyecye oeco aa pyoo a, e ce aepa x aea ac e coppoa a oe p pee. B ax aaax, a op aa pa opoa, op acoa oe .. ocox aepa, c paccope oopx aaec op, cpaeo eoo. Ho o e c eceo oo. aco a ocoe x aepa poecce opa oa o oe aepa, o cooyoc peoa eoca aepaa. o acc aa oo aam aaau c ocmpyupyeu amepamuau. a, aepa, p cycye P, oy cey:

eac;

ac;

apaee aa;

oc oce pao paa p pee;

ocpypye poecce p pee.

pep o coco oca aepax apao pee, coco pae pa ey c o pe peoe P.

oeco pepe pax eopeecx ocpoex pax eoax p pee oo peae ey. Copeee eo p pee opepoa a ye cex oex ocoe oce aec aepa, o cyeceo pae opae cxe peaoy py. ooy acoee pe oopepaoe o cae aepa caoc ce oee p. a pao, pep oe e aa a aao ae aaa poe, a o e aoe Pcep.

pep oy ac eac. pep aa aucuu, oa oea aepa o ooy x opeee (eeppoao o c oo cee epooc) oey o py oy pep. P eo x pee ac o ca pepe. p eoo oece pepe (2-5) aaa coocae yx aepa ocaoo poca P. p oe ce pepe aaa ca oc aooopo. B o cyae pep oec py, oope oo ca eac oec epapx pepe.

Ta opao, ee epe aepa cpyyp pepe ec eoxo ep ao P.

1.6. Pc eopeeeoc Moe pee pac ycox pca, .e. p ooo oacoc oep [80]. Cao o c paoopa eopeeeoc, opya ac. oa poo ocoa a xopoo yex aoo epocx ocyecec aepa. Hapep, eo poopoa e ocecx aapao papaoa acoo, o ooa a oaeca coa opae. Oao cae epe eeepo p poe poopoa oo e oo a ooa oocoa poo. o eopeeeoc [78] oac e, e oaec aay epe co co yoo oo ooc.

B [80] poc acca pax o eopeeeoce, ac oopx caa c eocaooc a o ppox ex poeccax, apep:

- eopeeeoc, cae c eocao a o ppoe (apep, a eece o oe oex coaex ope o ecopoe, a ooy e oe oo pecaa pae o ae poeoc oe aoox ocye o ee pep );

- eopeeeoc ppox e, ax, a ooa, a a ypoaoc, a apa a ooee, a yp, a apyy pacop x ye p.;

- eopeeeoc, cae c ocyecee ecyx (e oae aap) poepyex (ooe o papaoo eca eoooc ocyece poecca, oopy apaee e yaoc pecaa) exooecx poecco.

Moe ooe eopeeeoc ca c a opye e p, eeep oopo aaec poopoae:

- eopeeeoc, cae c eeoc yaco ooe co (pee ceo apepo oypeo p), acoc, c x eoo aoc, aco ooee, coee oa ec;

- eopeeeoc, cae c coa acpa aopa opex peoax, oopx pa ee eoe epec.

ooe aee e eopeeeoc a ypoe cpa, a coc:

- eopeeeoc yye poo cya cpae, o ce ocyce ocoepo opa o yyx ecx ocao c c ec peoe opeee;

- eopeeeoc, cae c oea e (ao ), op poea, ax ypco pyx apoooecx o aaee;

- eopeeeoc, opoee ecaoc aooaeca eye ooeco o (.e. c eeoc pyooca cpa, cepc eoc), cae c oeco cyae, ec ap, pocoo, ooecx pyx opaa acae cpa.

aco pxoc ya eeooece eopeeeo c, cae c cyae apyex cpaax eyapox opa ax, c oop oepaee eoe ooe.

Ta opao, eeepy pxoc poopoa yyee, p a pee ecoa, yao yac oeae eopeeeo ce. oeo ec x acca a CT-aop (o ep y a o co - coae, exooece, ooece, oece) aop oypeoo opye. CT-aop ecy eaco o eeepa, a o oype o a e epa. Booo, o yy opoc c a, cpec ece ae p c pa.

Ho oo epeoop, eye ooooo ooopeoc.

aa epecex o eopeeeoc oe cpy yppoaa aee. Ta, ec pye papao o aay eopee eoce p exooecx aapx, acoc, a xecx po ocax a aox epocax.

e pco eopeeeoce poecce p pee ope ee op aeaecx eoo, oox ya ae aop.

1.7. cepoe oeae p oocoax pee eoxoo opac a o, a y ceaco [80]. oce opo poo o pa ax eop ypae (eeea) caa paac caocoea ca cepe oe.

Meo cepx oeo o eo opaa pao co ce aca-cepa opao e cepo, paex o eceo / aeceo ope c e ooo opa p pee P. poee pao o eoy cepx oe o coa paoy pyy (P), oopa opaye o opye P eeoc cepo, oeex (opao o cyecy) c epy occ ().

Cyecye pae eo oye cepx oeo. B ox c a cepo paoa oeo, o ae e ae, o ee ec cepo, a ooy caae coe ee eaco o aopeo.

B pyx cepo copa ece ooo aepao P, p o cep ocya poey py c pyo, yac py y pya, eepe e opacac. B ox eoax co cepo cpoao aoo, o cacece eo poep coacoaoc e ae x ycpee oo pa oocoae pee.

B pyx co cepo pace poecce poee cep, a pep, p cooa eoa ceoo oa (o e ae). He ee cyecye eoo opao oeo cepo, o ce eca aeapoax oeppoax.

o oa peca cepa occ peyae coe pao opa p pee P poe caoo pee?

O oea a o eoooec opoc ac opaa pao occ.

Ec e cepoo coea cop opa P, o oa pa oa pya oa copa ooo oe oocec ey op a, apyeo a po opeeex apao pee. o ee eo oceeoo yee ca cepo: caaa ep c ep po co coopae o paccapaeoy opocy;

cocae aepa epeaec opoy cepy, oop oae co ap ye;

aoe aepa ocyae ceyey peey c epy... poeypa aaaec, oa ccae oo ox coopae.

Oe, o cep paccapaeo eoe oo oca opa, apye a po, o e paaa coacoa oo poea pee. oee oo, aoy oy poc cep c ee, ooc o accooo, ocoy eo o x ce ye oa aoee opax apyeo.

Ec e oooa poea pee P, o pec e o pao eoo e cepo.

oa coacoaocmu. Caec, o peee oe po a ocoe coacoax e cepo. ooy ca cepo py ex, e ee oaec o e oca. p o oceac a eapoae a, oae coca c epo occ o eopaye o coopae, e e ooe x poeccoaoy ypo, a aoee opae ce.

ae a, o cep ec a e oee py, ex ee pyoe o pe.

Meu uccuemo. C e cycceo oc coacoaoc capac ye e e cepo-cceo. ec coco op c ccea coco x ce cocaa cep o occ. Opaoa cepo, a opaoa peo exc peyao ae, po poeypa, e oxe e ece cacece coca.

M coco op c ccea coco pee poac x (ycox) cacecx poeyp. poce pep: ec o e cepa eceoe co, o peo eeec ee cc ea co e a cpeee apeecoe oeo cepo e e a x eay. ooy payo aece coacoaoo e paccapa eay. Oao p o oppyc (e oca P) apye cceo.

Ocoe cmauu cnepmoo onpoca. a oaae o poee cepx cceoa, eecoopao e ceye ca cepoo opoca:

1) opypoa P e cepoo opoca;

2) oop P ocooo cocaa P, oo pyooe cepe ap;

3) papaoa P yepee y P execoo aa a poe ee cepoo opoca;

4) papaoa P opooo ceap poee copa aaa c epx e (oeo), a a ope cepo op a (coa, ycoe paa, ca, papo, pae e oeo ecoo ppo) opee eo aaa o opa;

5) oop cepo cooec c x oeeoc;

6) oppoae cepo occ (eecoopao aee o oopo c cepa o ycox x pao ee oa, yepee P cocaa cepo occ);

7) poeee copa cepo opa;

8) aa cepo opa;

9) p pee poeyp ecox ypo oopee yx peyx ao;

10) eppea oyex peyao oooa ae P;

11) oaoe ooae eeoc P ( o ce oooa yepee ayoo acooo oeo o poee cepoo cceoa, oaa pya cepo copyo P).

oop cnepmo. poea oopa cepo ec oo aoee cox. Oeo, aece cepo eoxoo coo a ex e, cye aoee ooy p aeaoo pe e. Ho a e, a, oopa ax e? Hao po ca a, o e eoo oopa cepo, aepa oeceax ycex cep. aco peaa cooa eo aooe cao oe oeeoc cepo. C oo copo, o ye oe a oooc cepa, e o ca? C pyo copo, p caooee o eeoc copee oeaec cee caoyepeoc cepa, e eo peaa oeeoc. Te oee, o cao oe oeeoc cpoo e opeeeo. Moo eo yo, e cocae, o p o ycoec peapea ac eeoc cepo occ.

p cooa eoa aooe, oo oooc po e ocx pyox ca aa, pae po eoce oeoc cepo o ooocx py pya. B copeex ycox ocaoo xopoee aoco c paoa oooc py pya oe y ceaco, oo e paoax coeco. Oao peee ax ap ceaco e oe-o eecoopao, ocoy o co oxo py a pya.

cooae opax oaaee (ooc, yee cee ae, ca, co ya...), oeo, oe oc cooae xapaep. ceoc yac peyx cepax xopo pep eeoc eycaopa, paa, cy copx copeo ax, .e. ax cepo, oope yacy x cepx oo x cep.

Ec oe eo ceoo oa, p oopo o aoo ce aca, peaeoo aece cepa, oya ecoo a ex, o oe cepo o paccapaeo eae. Oeo, eoope x a cpeac paee eeoc P, a eoo pe oe. poecc pacpe cca ocaaaec, oa oe a epeca cpeac. B peyae oyaec ocaoo op cco oox cepo. co, o ec a epo ae ce c ep ooo aa, o eo ceoo oa ac, copee ce o, oo aa, e apye pyx ao yy yye.

Heoxoo oepy, o oop cepo oeo cee y paoe py, ae eo oopa e ca c ee oeceoc. py coa, eo a paoe pye e oe ceoc a oeeoc cepo, a x pay coco oc pe ocaey aay. Ba ec peoae P o yepe cca cepo.

2. Maeaec aapa, coye eoax p pee 2.1. Bcee aoo coceoo eopa px ap Oco eoa aaa epapx apyc a acceco eop ap, oeo [9, 33, 60, 93, 94].

Map apx cpae MA peca coo ooee opaocepe epoe ap, oop peec peoae coacoaoc.

apae ap A = (aij), oopx aij > 0, i, j = 1,2,...,n, aij = 1/ aji, i,j = 1, 2, , n, aac nooumeu opamocuempu u ampuau.

ooee opaocepe ap A = (aij), ee o oopx oec coooee aik = aijajk, i,j,k = 1, 2, , n, c coacoau.

apaa apa enpuoua, ec oa e oe peca A1 ea e, e A1 A3 apae ap, 0 yea ap A A a. B poo cyae apy aa npuouo. ae, o eo opx paoax ae ap aa epaouu [33].

- 2 0 puep. Mapa A = 1 3 4 poa.

3 0 pa, cooecy o ape, ee yy epo epy pe ep aaoo pee epy pe ep, o epexo o opy epy eooe (pc. 2.1). opo ep oo epe o ce p ep.

3 Pc. 2.1. pa, cppy apy Ta opao, epa pe ep opay epoy ooey, a opa caa c .

ae, o oeca apa A epoa o oo o cyae, ec ee apae pa D(A) co c.

Teopea. apaa apa epoa, e oe peea ye epecaoo eco y:

A1 0 0 0 0 A2 0 0 0 0 Ak 0 Ak +1,1 Ak +1,2 Ak +1,k Ak +1 Am1 Am2 Amk Am k +1 Am coepaey o - aoay apy c epo apa Ai a aoa. p o, o pae epe, oa ap c o e co ao cpoe, oopo o oc, yea.

MA ooe c pope aepa U = {u1, u2, , un}. popea aepa, oye a ocoe a p x apx cpae, cya opaoae ae aoo coceoo eopa ap.

Beop x, oopx Ax = x (x 0), aac cocmeu emopau, a cooecye ca xapamepucmuecuu, cocmeu ucau ampu A.

Cocee ca ap A c op xapaepcecoo ypae ap |A I| = 0, e I ea apa. a oo aoe ypaee ooceo, oo ee N ope. aoy coce oy ae cac cooece coce eop, opeee c ooc o capoo oe.

B eopee poeyca [33] yepaec, o epoa eopa ea apa A ( eopee eppoa, o ooea apa A) cea ee eceoe ooeoe pocoe coceoe aee.

max pe oy cex pyx xapaepcecx ce e peocxo.

max Teopea (eppo -poeuyca) yc A 0 epoa apa. Toa:

1. A ee eceoe ooeo pocoe (.e. epaoe) coc eoe aee, oopoe o oy e ee oo pyoo co max ceoo ae ap A (eoope oopx oy o ec ca).

2. Coce eop A, cooecy coceoy ae, max ee ooee ooe , o cyecy (c ooc o o cooo oe), ecee.

3. co (oa aaeoe oe eppoa ap A) yoeope max yco ( Ax )i ( Ax )i = max min = min max ;

x 0 pooo.

max x0 1in xi x0 1in xi Cecmue. yc A 0 epoa yc x 0 pooo. Toa o pe eppoa yoeope yco ( Ax )i ( Ax )i min max.

1in xi max 1in xi aa a cocme aeue ( eoopx co max ax ope eppoa) ap A, a cooecy ey coce e op w a cocme emopo.

Ta opao, c pope eeo (axo a coce eop) ooo cey opao: cocaec xapae pcecoe ypaee ap A, cpe ope aoo ypae pa ec aoee, cec cooecy coce eop w, a e eopa opayc. Cocaee peee aoo poa ypae ao ap eoe aaa epapx ocaoo pyoe co poecc.

B MA peoeo cooa pye coco oye aoo coceoo eopa ap, o oopx opaec a cece eopee eppoa-poeyca, p o ap o yoeop yco poc.

Ec epoa eopaea apa ee ceo h xapaep cecx ce:,,, c aca oye r ( = | | = = | | = 1 2 h 1 2 h = r), o aa apa aaec npuumuo p h = 1 (aoe coce oe aee o oy ec ece = ), p h > 1 ap 1 max a aaec unpuumuo (cyecy xapaepcece op a p, coaae o oy c a coce aee).

T. Caa coye aece opeee po ap yep ee eope: epoa apa A 0 ec po o oo o cyae, oa m 1, aoe, o Am > 0 (eoopa cee ap A ooea).

Ocoa eopea MA opypyec px ap.

Ak e Teopea. po ap A lim = cw, Ak = eT Ake, Ak k e c ocoa, a w coce eop, cooecy =.

max coy cece eope eppoa, oyae eee pyoe (p oepo oepe) coco ce eopa popeo ap apx cpae: apa ooc pooo oe cee, cc cy eeo cpo ap-peyaa, oy ee cy opayc. Tao oxo peocae oooc axo a coce eop ap A e cooa xapa epcecoo ypae, o aoo oeae poeypy eo ce . pep ce aoo coceoo eopa accec eoo, c cooae ocoo eope MA, paccope a paoe [20].

2.2. epapx pope ep ao MA ec ocpoee epapx, opaae po ecc p pee. epapx paccapac [93] a cea yopoeoo oeca ac cya paa. epa ep pea paa aece oco opaoo opeee, a opa aece cpa. o eoope aeaece oco e papx.

acmuo ynopoe oecmo aaec oeco S c ap ooee, oopoe yoeope aoa peecoc, a cepoc paoc:

Peecuocm: cex x, x x.

Amucuempuocm: ecu x y u y x, mo x = y.

Tpaumuocm: ecu x y u y z, mo x z.

oo ooe x y aoo a oo opee x < y, o oaae x y x y.

oop, o y nopaem (opye) x, ec x < y ec x < t < y e ooo aoo t.

opoee oeca c oe co eeo oy yoo pecae apae pao. a ee cce pecae epo a, o ya apaea o a b, ec a > b.

Bnoe ynopoeoe oecmo (ae aaeoe en) ec yo poeoe oeco co cey ooe coco: ec x, y S, o x y y x.

Boc ooaee x = {y | x opae y} x+= {y | y opae x} oo eea x yopoeo oece.

yc H oeoe aco yopoeoe oeco c ao eeo b. H ec uepapxu, ec oc ceye yco:

1. Cyecye paee H a ooeca Lk, k = 1,2,...,h, e L1 = {b}.

2. x Lk ceye, o x- Lk +1, k = 1,...,h - 1.

3. x Lk ceye, o x+ Lk-1, k = 2,...,h.

aoo x H cyecye ecoa y (cyoc ee ac o e, oopoo cpoc epapx) wx : x- [0;

1], o wx ( y ) = 1.

y x Moeca Lk c ypo epapx, a y wx ec y popea eea ooo ypo ooceo e x.

puep. Paccop epapx H, ocpoey aa opa py ooe pex aao. eea oeca H ao cyae c e aa aop, a ee e, a ae aa a ooc pyooe.

Pyooe 1 2 Pc. 2.2. epapx aa Bop pyooe H = {pyooe, opaaoe cocooc, poeccoa, a aoc, oyaeoc, ae oe, aop e cpe oex, aa 1, aa 2, aa 3}.

epapx oc ce yco opeee.

1. Cyecye paee oeca H a ooeca L1, L2, L3 (h = 3), e L1 = {Pyooe}, L2 = {opaaoe cocooc, poeccoa , a aoc, oyaeoc, ae oe, aope cpe oex}, L3 = {aa 1, aa 2, aa 3}.

2. Paccop x = Pyooe L1, o cyae x = L2. (aao e o cpae cex pyx x).

3. Paccop x = aa 1 L3, o cyae x+ = L2. (aaoe o cpae cex pyx x).

Opee ecoy y eea x = Pyooe. a y ca cooece aeca pyooe (eea ypo L2) ae opea [0, 1] opeee pope x aec ooc aope cpe oex a aoc ae oe poeccoa opaaoe cocooc oyaeoc eo e Pyooe. Becoa y aaec cyeo c epa. pepy, oa oe ao:

wpyooe (op. coco.) = 0,3;

wpyooe(poec.) = 0,2;

wpyooe (. a.) = 0,1;

wpyooe(oyaeoc) = 0,1;

wpyooe (ae o.) = 0,1;

wpyooe(aope) = 0,2.

coe ( y) = 1 oec.

wx yx Ocoa aaa MA aaec ceye: a opee oo aaoo eea x L ooeca S L ( < ) y wx,S : S [0;

1], o oa opaaa coca y popeo wx a ypox Lk+1, k =,..., - 1. B acoc, o o a y wb,L : Lh [0;

1].

h B [93, 94] peaaec cey eo pee ocoo aa.

peoo, o Y = Lk, X = Lk+1. yc ae cyecye ee z Lk -1. Paccop y popeo wz :Y [0;

1], wy : X [0;

1], j = 1,..., nk, j e nk oeco oeo a k-o ypoe epapx.

Ooa w(xi) y popea eea xi ooceo e z.

nk B [93] aa y aaec e w(xi ) = (xi )wz (y ), i = 1,...,nk+1.

wy j j j= To ec, pocxo poecc ea oaae eea yj a pope eea xi ye yoe oo oaae a aoc ee a yi ooceo z. Ec wy (xi ) opaoa apy B, oo j / bij = wy (xi ), W=w(xi), W =wz(yj), o ooaea opya pe :

j W = BW. Taoe eae pooc aoo ypo epapx.

Paccapa poecc ea a ce epapx, oyae opy y eopa popeo caoo oo ypo ooceo e b:

W = BhBh-1BW, e h oeco ypoe epapx, W pope eeo epoo ypo epapx ( oce cyae ep ypoe coco ooo eea e poecca p pee, o cy ae W cap).

2.3. oe. Heee oeca. Heee ooe ee eee oeca e cya, peye p pee, coepa, a pa o, ooe oeco eopeeeoce. B o cyae opa, a ocoe oopo paec peee, oe paea eeo. Haa o pa o eex oec cao c ee .A. ae. Copee e o eeo o pecae paoax A.H. Mexoa [70, 71], A.H. opcoa [23, 24, 25, 26], C.A. Opocoo [81], X. Paa [89], A. oaa [56], B.. ya [57, 58], paoax [74, 78, 79, 87].

Moeco A eoe oeco, ec A ac eoopoo yep caoo ao pao aa oeca U, xapaepyeoc yco:

- ce ee oeca eo pa ey coo, oece e ecox epo eoopx eeo;

- ooceo aoo eea u U oo eo opee, pa e o aoy oecy e.

yco oo oxapaepoa eoe oeco eo xapa epceco ye, aao a yepcao oece U p ae ae oece {0, 1}:

0, u A, (u) = u U.

A 1, u A Oa o epoo yco po oee oey, e oeco, o onema, oycaeo ae ecox epo eo opx eeo. oe xapaepyec ye epoc, a ao a yepcao oece U pae ae oe ce eopaex ex ce: (u) {0, 1, 2, .} co e A po eea u U oee A.

Oa o opoo yco po oee oey, e oeco, o eemoo oecma, oycaeo opeeee eoo po cee paeoc eeo aoy oecy.

~ Hee ooeco A oeca X aaec cooyoc ~ ap a A = {< x, (x) >}, e xX, a (x) y paeoc, A A caa cooece oecy X opeo [0;

1].

~ X aooe oeco, aoa aa. B oeco A e ac ee, oopx (x) = 0. Heeoe oeco - ycoe, A ec (x) = 0 aoo xX. Heeoe oeco X yepcaoe, ec (x) = 1 aoo xX.

X y paeoc paec cyeo, ac o cye a, eo acpoe, e ocpoe oec, peaeo aa ..

puep.

yc X oeco oeecex a. X = {Boa, Oa, Mo ~ c, y}. Toa oo opee eeoe oeco A xopo ~ x a a: A = {(Boa;

1), (aopoe;

0,4), (Moc;

0,6), (y;

0,8)}.

puep.

y paeoc eoy oecy B = {x| 0 x 2} p ae aee 1, ec 0 x 2 aee 0 poo cyae. Ee pa pee a pcye.

B(x) 0 x Pc. 2.3. pa y paeoc oecy B = {x| 0 x 2} pa e y paeoc oy eeoy oecy (a cee ycoo yepcaoo oec) ye peca co o ey py. Paccop, apep, eeoe oeco C = { x| aee x o 1}. pa eo y paeoc o e e a, a apep a pc. 2.4.

C(x) 0 1 X Pc. 2.4. pa y paeoc oecy C = { x| aee x o 1} ~ Hocee eeoo oeca A aaec ooeco A o eca X, coepaee e ee X, oopx ae y paeoc (x) > 0. Ceye ae, o oce eeoo o A eca o oeco oo cce.

puep.

yc X oeco aypax ce. Toa eo eeoe oo ~ ~ eco M oe ax ce oe a: M = {(1;

1), (2;

0,8), (3;

0,7), (4;

0,6), (5;

0,5), (6;

0,3), (7;

0,1)}.

~ Hocee eeoo oeca M ec oeco M = {1, 2, 3, 4, 5, 6, 7}. o ooe eoe ooeco oeca X.

Heee caa oepa a Onpeeeue. Heemoe caaue peoee, ooceo o opoo oo cy o cee eo coc ooc acoee ~ pe. Cee coc ooc d( A ) pae ae [0;

1].

e 0, 1 peee ae cee coc coaa c o c ex caa.

Heee caa co cee c 0,5 aac uue pemocm, ocoy oo co o e epe, o oo.

puep.

2 aeoe co eeoe caae, cee coc oopoo 0,9.

~ Onpeeeue. Opae eeoo caa ec ca A ~ ae, cee coc oopoo opeeec paee A ~ ~ ~ d( )= 1 d( A ). oo opeee ceye, o cee ooc A A ~ coaae co cee coc .

A ~ ~ Onpeeeue. oe eex caa A B, aaec ~ ~ eeoe caae A & B, cee coc oopoo coaae co cee coc eee coo caa.

~ ~ ~ ~ d(A & B) = min(d(A),d(B)).

~ ~ Onpeeeue. e eex caa A B, aaec ~ ~ eeoe caae A B, cee coc oopoo coaae co cee coc oee coo caa ~ ~ ~ ~ d(A B) = max(d(A),d(B))..

~ ~ Onpeeeue. ae eex caa B, aaec A ~ ~ ~ ~ eeoe caae A B cee coc oopoo d(A B ) =, ~ ~ = max(1 d( A ), d( B )). coc a e ee e cee ooc ee oc cee coc ee cec.

puep.

~ yc eeoe caae A ee cee coc 0,3;

ee ~ ~ ~ oe caae B 0,6. a x caa A B ye ~ ~ e cee coc d(A B ) = max(0,7;

0,6) = 0,7.

Cee a e e, e ee cee coc o c oe cee coc cec.

~ ~ Onpeeeue. aeoc eex caa A B, a ~ ~.

aec eeoe caae A B ~ ~ ~ ~ ~ ~ d( B ) = min((max(1 d( A ), d( B )), (max(1 d( B ), d( ))).

A A coc aeoc coaae co cee coc eee ~ ~ ~ ~ co a B B.

A A Ec cee coc caa 0 1, o ce opeee co oecy oec oepa a e caa.

~ ~ Onpeeeue. a caa B aac eeo , A ~ ~ ec cee coc B oe paa 0,5. B ocee cyae A ~ ~ ye aa A B ao eeo epe.

opo oe oepa a ee caa - Co.

- Opae.

- o.

- .

- a.

- aeoc.

puep.

Bc cee coc cocaoo eeoo caa ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ D = ( & B & B ) ( & C );

ec A = 0,7;

B = 0,4;

C = 0,9.

A A A ~ ~ ~ ~ ~ ~ ~ D = max((1- d( A & B & B )), d( ( A & C ))) = A ~ ~ ~ ~ ~ ~ = max ((1 max(d( & B ), d ( & B )), d(1 ( & C ))) = A A A ~ ~ ~ ~ = max ((1 max(min(d( A ), 1 d( B )), min (1 d( A ), d( B )))), ~ ~ 1 min (d( A ),d( C )))= = max((1 max(min 0,7;

0,6), min (0,3;

0,4))), 1 min (0,7;

0,9))= = max ((1 max(0,6;

0,3)), 0,3)= max (0,4;

0,3)= 0,4.

Heee oece opy x coca ~ Onpeeeue. Heea caaea epeea xi o eeoe caae, cee coc oopoo oe pa poooe aee opea [0;

1].

~ Onpeeeue. Heeo oeco opyo A ~, x2, x3,...,~n, n x1 ~ ~ x aaec:

a) a eea caaea epeea ocaa [0;

1], ~ x1 ~ ~ x ) paee A ~, x2, x3,...,~n, oyeoe eex oecx ~ ~ opy ~, x2, x3,...,~n ~, x2, x3,...,~n peee o x1 ~ ~ x x1 ~ ~ x A1 A o oeoo ca oecx oepa.

B acoc cocae eee caa ae c ee oec opya, ec paccope opaye x poce eee caa a eee caaee epeee.

~ Onpeeeue. Cee paococ opy ~, x2, x3,...,~n x1 ~ ~ x A ~ ~ ~ ~ ~ ~ ~ ~ ~ ( x1, x2, x3,..., xn ) ooaaec (, ) opeeec (, ) A2 A A2 A1 A ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ = & ( ( x1, x2, x3,..., xn ) ( x1, x2, x3,..., xn ).

A1 A ~ ~ x1, x2,....~n x Ec cee paococ eex oecx opy ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ( x1, x2, x3,..., xn ) ( x1, x2, x3,..., xn ) a cex opeeex aopax A1 A ceee coc caaex epeex oe paa 0,5, o ae opy ye aa eeo a x aopax oo ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ aa ( x1, x2, x3,..., xn ) ( x1, x2, x3,..., xn ).

A A ~ ~ Ec (, ) 0,5, o opy e c eeo :

A1 A ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ( x1, x2, x3,..., xn ) ( x1, x2, x3,..., xn ).

A1 A ~ ~ ~ ~ ~ ~ ~ ae, o p (, ) = 0,5 opy ( x1, x2, x3,..., xn ) A1 A2 A ~ ~ ~ ~ ~ ( x1, x2, x3,..., xn ) oopeeo c e c eeo A . x aa ao epe ooaa ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ( x1, x2, x3,..., xn ) ( x1, x2, x3,..., xn ).

A1 A Paococ ex oecx opy ec ac cyae eeo oc.

~ ~ Ec a ox ex e aopax ceee coc epee A1 A x pa o e e ae cee coc, o aee ce e x paococ cea oe pao 0,5, o ec ac cyae eeo oc. To ec, eee oece opy, ee a ox ex e aopax epeex oaoe cee coc e pa oc, a e eoopy cee paococ 0,5, o cea 1.

puep.

~ ~ ~, ~ ~, ~ Opee cee paococ opy ( x y ) ( x y ), e A1 A ~ ~ ~, ~ ~ ~ ~, ~ ~ ~ ( x y ) = x y, ( x y ) = x & ~, x pae aee cee y A1 A ~ coc oeca cpex ae {0,8;

0,6;

0,7}, a y {0,3;

0,4}.

Peee.

~ ~ &~ ~ ~ ~ y (, ) = ( x y ) ( x & ~ )). Bpa ce ooe a A1 A2 ~,.y x ~ ~ op ceee coc x y, ae ~ ~ (, ) = ((0,80,3) (0,8& 0,3)) & (0,80,4) (0,8& 0,4)) A1 A & (0,60,3) (0,6& 0,3)) & (0,60,4) (0,6& 0,4)) & (0, 0,3) (0,7&0,3)) & (0,70,4) (0,7& 0,4)) & = (0,80,7) & (0,80,6) & (0,60,6) & (0,60,6) & (0,70,7) & (0,70,6) = 0,7 & 0,6& 0,6& 0,6& 0,7& 0,6 = 0,6.

~ ~ Oya ceye, opy, eeo p aax aopax A1 A ceee coc.

~ Ec cea ay e poepy, oaa, o x pae aee ~ ~ ~ aopa {0,2;

0,4}, a y {0,6;

0,7;

0,8}, o (, ) = 0,2. o cy A1 A ae opy e c eeo .

Onpeeeue. Ec p cex opeeex aex cee co ~ ~ ~ ~ c eex epeex x1, x2, x3,..., xn, aee cee coc e ~ ~ ~ ~ ~ eo oeco opy A ( x1, x2, x3,..., xn ) oe pao 0,5, o opya ec eeo co a ax aopax epeex oo ~ aaec epe . Ec aee cee coc ee pao 0,5, o ay opyy ye aa eeo oo a ax aopax ~ epeex ooa .

~ ~ ~ ~ yc 1, 2, 1, 2 eoope eeo ce eeo o e opy a ox ex e aopax epeex, oa cpae ceye coooe.

~ ~ ~ ~ ~ ~ 1 2 1 2 1 & ~ ~ ~ ~ ~ ~ 1 2 1 2 1 & ~ ~ ~ 1 & 1 ~ ~ ~ 1 1 1.

~ ~ Ec, pooe eee oece opy, o cpae A1 A coooe:

~ ~ ~ ~ A A 1 1 ~ ~ ~ ~ A & 1 & 2, A 1 ~ ~ ~ ~ ~ ~ e,, 1, 2, 1, 2 opeee a ox ex e aopax epe A1 A ex.

puep.

pee poce pep eeo cx eeo ox opy.

~ ~&~ = x x ~ ~~ = x x.

o ceye opeee oepa opa, o ~&~ ~~ , .. x x 0,5, x x 0,5.

Toeca oo opee acc eeo x opy, e ex aaoo eeo oe.

Coomoeu, cnpaeue x aopo aeu ucmuocmu eemux nepeex.

~, ~, ~ yc x y z eee oece opy.

~ ( ~ ) x ;

(1) x ~ ~ ~ x & x x ;

(2) ~ ~ ~ x x x ~ ~ ~ ~ x & y y & x ;

(3) ~ ~ ~ ~ x y y x ~ ~ ~ ~ ~ ~ ~ ~ ~ x & ( y & z ) ( x & y ) & z x & y & z ;

(4) ~ ~ ~ ~ ~ ~ ~ ~ ~ x ( y z ) ( x y ) z x y z ~ ~ ~ ~ ~ ~ ~ x & ( y z ) ( x & y ) ( x & z ) ;

(5) ~ ~ ~ ~ ~ ~ ~ x ( y & z ) ( x y ) & ( x z ) ~ ~ ( x & y ) ~~ x y ;

(6) ~ ~ ~ ( x y ) ~& y x ~ ~ ~ ~ x & ( x y ) x ;

(7) ~ ~ ~ ~ x ( x & y ) x ~ ~ ~ ~ ~ ~ ( x y ) ( x & y ) x y ;

(8) ~ ~ ~ ~ ~ ~ ( x y ) & ( x & y ) x & y ;

(9) ~&~ ~&~ x x y y ;

(10) ~~ ~ ~~ ~ x x y y y x ;

(11) ~&~ ~~ ~&~ ( x x ) & ( y y ) x x ;

(12) ~~ ~&~ ~~ ( x x ) ( y y ) x x ~ ~ x y ~ ~ ;

(13) y x ~ ~ ~ ~ ~ y ~ x x y ;

(14) x y ~ ~~ ~&~ ~ x ( y y ) ( x x ) y ;

(15) ~&~ ~ ~ ~& ~ ~ ~ ( x x ) ( y z ) ( y y ) x z. (16) poe oo, yc 0, c, 1 oca 0 < c < 1, oa ~ ~ ~ x & 0 0, x & 1 x.

~ ~, ~ x 0 x x 1 x, ecu x c, ~ ~ ~ x & c ~ c, ecu x c.

~ c, ecu x c, ~ x c ~ ~ x, ecu x c.

oaaeca aoo pae eoxoo oaa, o ~ ~ ~ ~ cee paococ (A1, A2 ) opayx eo opy A1, A2 oe pao 0,5.

~ ~ o ooo oa, oa opy A1, A2 pa o e e ae cee coc a oaox aopax epeex o e cee coc oopeeo ee pao 0,5 oe pao 0,5 a oaox aopax epeex.

~) oae opyy (6) (~ & y ~~, ooa x x y ~ ~) ~) ~ x ~) x x x y A1 (~, y = (~ & y, A2 (~, y = ~~.

~ ~ ~, ~ ~ ~ ~, ~ ~ ~, d( A1( x y )) = 1 - min( d( x ),d( y )) d( A2( x y )) = max( 1 - d( x ), 1 - d( y )). Ec ~ ~ ~,~ ~,~ d (~) < d (~), oa d( A1 ( x y )) = 1 - d (~) d( A2 ( x y )) = 1 - d (~), .e.

x y x x ~ ~ ~ ~) ~) p cex x cee coc opy A1 (~, y A2 (~, y coaa, x x ~ ~ oya ceye (A1, A2 ) 0,5.

~ ~ ~,~ ~,~ Ec d (~) > d(~), o d( A1 ( x y )) = 1 - d (~), d( A2 ( x y )) = 1 - d (~), x y y y ~ ~ oya ceye (A1, A2 ) 0,5.

~ ~ ~,~ ~,~ Ec d (~) = d(~), o d( A1 ( x y )) = 1 - d (~), d( A2 ( x y )) = 1 - d (~), x y x y ~ ~ oya ceye (A1, A2 ) 0,5.

Heee pea aop Onpeeeue. Heee oece opy, oope opeee a ao-o oece X pa co ae ayoo ep aa [0, 1] aa ee peao.

Hapep, y paeoc ec ooec e A e peao.

puep.

X = {1, 2, 3, , 10}, oa ee pea eo c ~ ~ ~ o pae ceye ae: A (1) = 1, A (2) = 0,9, A (3) = 0,7, ~ ~ ~ ~ ~ ~ ~ A (4) = 0,5, A (5) = 0,1, A (6) = 0, A (7) = 0, A (8) = 0, A (9) = 0, A (10) = 0 ~ aec aae eeoe oeco A = {(1;

1), (2;

0,9), (3;

0,7), (4;

0,3), (5;

0,1)} oece X.

~ yc oac opeee eeoo peaa A ec oeco X = {x1, x2, x3, ., xn}, oa aoo x X oe ceo a ~ ee (x) peaa A (x).

A ~ Onpeeeue. Bea (A) = A (x1)& A (x2)& A (x3)& & A ~ (xn) = = & A( x ) aaec cee ooc coc A (x) ee xX o oeca X.

~ ~ Ec ( A ) 0,5, o a oecy opyy A (x) oe aea ~ aop eeo ooc, oop aec cex oo.

~ Onpeeeue. Bea (A) = A (x1) A (x2) A (x3) A (xn) = ~ = A( x ) aaec cee cyecoa coca A (x) ee xX o oeca X.

~ ~ Ec (A) 0,5, o a oecy opyy A (x) oe aea ~ aop eeoo cyecoa, oop aec cyecye ao eec ao.

~ yc A (x) eea oeca opya o oo epeeo, p ~ ~ ae ae X. Bpaee ( xX) A (x) ec eeo c ~ o opyo aec oo xX cee coc A (x) oe pao 0,5.

Oepa a ee oeca.

Heeoe ee eeoe paeco oec Tae a a e oeca opeec oece oepa e, paeca, oee, epecee, ooe .., opeec o a ee oeca, oo eaec o p oo y paeoc.

~ ~ Onpeeeue. yc aa eee ooeca A, B oeca X.

~ ~ ~ Cee e ( A, B ) eeoo oeca A eeoe oeco ~ ~ ~ B axoc o opye ( A, B ) = ( A( x ) B( x )), e ( x ), & A xX ( x ) oac a eee caaee epeee, B a, & oepa o, oopa epec o ce xX.

~ ~ ~ ~ Ec ( A, B ) 0,5, o A eeo aec oeco B ooa ~ ~ ~ ~ ~ aec A B. Ec ( A, B ) 0,5, o A eeo e aec oec ~ ~ ~ o B ooaaec A B. o oe ec ooee o e ex oec. eceo, yc A B ee oe ca AB, oca ceye (A,B) = 1. Ec e AB, o (A,B) = 0.

puep.

X ={x1, x2, ,xn}.

~ ~ A = {(x2;

0,3), (x3;

0,6), (x5;

0,4)}, B = {(x1;

0,8), (x2;

0,5), (x3;

0,7), (x5;

0,6)}, o ~ ~ a ( A, B ) = (00,8)& (0,30,5)& (0,60,7)& (00)& (0,40,6) = = 1&0,7&0,7&1&0,6 = 0,6.

~ ~ ~ ~ Aaoo oo c ( B, A ) = 0,2, oya ceye A B, ~ ~ o B A.

~ ~ Onpeeeue. Moeco A aec o oeco B ~ ~ A B ec x X, (x) (x).

A B ~ Cpaeo ceyee yepee: ec eeoe oeco A ~ aec eeoe oeco B, o oec eeoe ee ~ ~ A B.

eceo, yc oec x X, (x) (x), oae, A B ~ ~ o ( A, B ) 0,5.

Ec ~ ~ (x) (x) 0,5, o ( A, B ) = ( (x1) (x1))&( (x2) (x2)) & A B A B A B ( (xn) (xn)) = (max(1 - (x1), (x1)))& (max(1 - (x2), (x2)))& A B A B A B &(max(1 - (xn), (xn))) = (1 - (x1))& (1 - (x2))& & (1 - (xn)).

A B A A A opeee oepa o ceye, o peya ye a cex (1 - (xi)), i = 1...n. A ocoy x X (x) 0,5, A A ~ ~ o ( A, B ) 0,5.

~ ~ Ec 0,5 < (x) (x), o ( A, B ) = ( (x1) (x1))& A B A B & ( (x2) (x2))&( (xn) (xn)) = (max(1- (x1), (x1)))& A B A B A B (max(1- (x2), (x2)))&&(max(1- (xn), (xn))) = ( (x1))& ( (x2)) A B A B B B ~ ~ & & ( (xn)). Ta a x X (x) > 0,5, o ( A, B ) > 0,5.

B B ~ ~ To ec, x ( A, B ) 0,5 x ae y p aeoc (x) (x), x X.

A B ~ ~ Ec e oec ( A, B ) 0,5, o oo e ceye, o ~ ~ xX, (x) (x). eceo, ( A, B ) = ( (x1) (x1))& A B A B ( (x2) (x2))& ( (xn) (xn)) = (max(1 - (x1), (x1)))& A B A B A B (max(1 - (x2), (x2)))& &(max(1 - (xn), (xn))), a a A B A B ~ ~ ( A, B ) 0,5, o o opeee oepa o aoe, a a ce ocae ae pae max(1 - (xi), (xi)) 0,5.

A B Oao ae, ec, apep (xi) = 0,3, a (xi) = 0,2, o A B max(1 - (xi), (xi)) 0,5, o (xi) (xi). To ec, ee oec A B A B ~ ~ a o oeco B e apapye eeoo e, a ec A ocao ycoe eeoo e.

~ ~ Onpeeeue. Cee paeca yx eex ooec A, B ~ ~ oeca X opeeec a ( A, B ) = ( (x) (x)). Ec A B & xX ~ ~ ~ ~ ~ ~ ( A, B ) 0,5, o oeca eeo pa A B. Ec ( A, B ) 0,5, o ~ ~ ~ ~ oeca eeo e pa A B. Ec ( A, B ) = 0,5, o oeca ~ ~ ao epe A B.

o eeoo paeca epaeca, epeoc c ooee o paeca epaeca ex oec. ec eo, yc A B ee oeca, oa cyae A = B, (A, B) = 1, ec e A B (A,B) = 0.

puep.

X = {x1, x2, x3, ., x5}, ~ A = {(x2;

0,8), (x3;

0,6), (x5;

0,1)}, ~ B = {(x1;

0,3), (x2;

0,6), (x3;

0,7), (x4;

0,2), (x5;

0,3)}.

~ ~ ( A, B ) = (0 0,3)&(0,8 0,6)&(0,6 0,7)&(0 0,2)&(0,1 0,3) = ~ ~ = 0,7&0,6&0,6&0,8&0,7 = 0,6, oya ceye A B.

~ ~ peopaye cee paeca ( A, B )= ( (x) (x)) = A B & xX = (( (x) (x))& ( (x) (x))), y oyaoc A B B A & xX ~ ~ o ( A, B ) = ( ( (x) (x)))&( ( (x) (x))), A B B A & & xX xX ~ ~ ~ ~ ~ ~ oca ceye ( A, B ) = ( A, B )&( B, A ), .e. cee paeca ee x oec paa ao ceee x aoo e.

~ ~ ~ ~ ~ ~ Ec ( A, B ) 0,5, .e. oeca A, B eeo pa, o ( A, B ) 0, ~ ~ ~ ~ ~ ~ ( B, A ) 0,5, A B B A. Oca ceye eo oaae ca eeoo paeca eex oec, ocoa a oaaece aoo eeoo e.

~ ~ Onpeeeue. Heeoe oeco pao eeoy oecy B A ~ ~ A = B, ec x X, (x) = (x).

B A ~ ~ Hepyo ae, ec oec paeco oec A = B, o ~ ~ oeca c eeo pa A B. eceo, ec (x) = B ~ ~ ~ ~ ~ ~ = (x) x X, o ( A, B ) = ( (x) (x)) = ( A, B )&( B, A )0,5.

A & A B xX Teopeo-oecee oepa ~ ~ yc aa eee ooeca A, B oeca X.

~ ~ A = {}, B = {}, x X.

A B ~ ~ Onpeeeue. Oeee eex oec A B ec ~ ~ oeco A B = {x, (x)}, x X, y paeoc ee A o oopoy opeeec a B(x) = max{ (x), (x)} = (x) (x) A A B A B (pc. 2.5).

AB(x) A(x) B(x) X Pc. 2.5. Oeee eex oec ~ ~ ~ ~ ~ T.e. A B o eeoe oeco, aoe, o A A B ~ ~ ~ B A B.

Onpeeeue. epeceee eex oec A B aaec o ~ ~ eco A B = {x, (x)}, x X, y paeoc eeo A oopoy opeeec a B(x) = min{ (x), (x)} = (x)& (x) A A B A B (pc. 2.6).

AB(x) A(x) B(x) X Pc. 2.6. epeceee eex oec ~ ~ ~ ~ ~ T.e. A B o eeoe oeco, aoe, o A B A ~ ~ ~ A B B.

~ Onpeeeue. ooee eeoo oeca A aaec o ~ eco A ={}, x X, aoe, o A (x) = 1 (x), x X.

A puep.

~ Paccop eeoe oeco B ce, opao ox y. o ~ oee oy oecy ye c oeco A ce, opao ex y (pc. 2.7.).

A(x) B(x) X Pc. 2.7. ooee eeoo oeca Onpeeeue. Paoc eex oec aaec oeco ~ ~ A \ B = {x, (x)}, x X, y paeoc eeo oopoy A\ B & (x).

opeeec a (x) = A\ B A B ~ ~ Onpeeeue. Cepeco paoc B aaec oeco A ~ ~ A B = {

(x)}, e (x) = (x) (x).

A A A\ B B\ A puep.

~ A = {(x1;

0,3), (x3;

0,8), (x6;

0,4)} ~ B = {(x1;

0,9), (x2;

0,2), (x3;

0,4), (x4;

0,5)}.

~ ~ A B = {(x1;

0,9), (x2;

0,2), (x3;

0,4), (x4;

0,5), (x6;

0,4),}.

~ ~ A B = {(x1;

0,3), (x3;

0,4)}.

~ A = {(x1;

0,7), (x2;

1), (x3;

0,2), (x4;

1), (x5;

1), (x6;

0,6), (x7;

1)}.

~ ~ A \ B = {(x1;

0,1), (x3;

0,6), (x6;

0,4)}.

~ ~ A B = {(x1;

0,7), (x2;

0,2), (x3;

0,2), (x4;

0,5), (x6;

0,6)}.

Onpeeeue. Byo oae oec A1, A2, An aaec eeoe oeco A c ye paeoc (x) = (x), e 0, i=1, 2, 3, . n, = 1.

A i i i i ~ Onpeeeue. Moeco ypo eeoo oeca X, a A aec oeco oo cce, cocaeoe eeo xX, cee paeoc oopx eeoy oecy A oe pa . A = {x | x X, (x) }.

A ~ ~ Onpeeeue. p poeee eex oec A B a ~ ~ aec epe A x B ooaaec eeoe ooeco XxY, oopoe ~ ~ opeeec paee x B = { < ( x, y ), ( x, y ) > }, x X, y Y, A AxB e ( x, y ) = x )& ( y ).

AxB A( B ~ ~ Onpeeeue. ooe eex oec B aaec A oeco ~ ~ F P = { < ( x,z ), ( x,z ) > },( x z ) X Z, F P ( x,z ) = ( ( x, y ) & ( y,z )).

F P F P yY Ocoe coca eex oec:

~ ~ 1. ( A ) A o.

~ ~ ~ 2. A A A, ~ ~ ~ A A A eoeoc.

~ ~ ~ ~ 3. A B B A, ~ ~ ~ ~ A B B A oyaoc.

~ ~ ~ ~ ~ ~ ~ ~ ~ ( 4. A B C ) ( A B )C A B C, ~ ~ ~ ~ ~ ~ ~ ~ ~ A ( B C )( A B )C A B C accoaoc.

~ ~ ~ ~ ~ ~ ~ 5. A ( B C )( A B )( A C ), ~ ~ ~ ~ ~ ~ ~ A ( B C )( A B ) ( A C ) cpyoc.

~ ~ ~ ~ 6. ( A B ) A B, ~ ~ ~ ~ ( A B ) A B ao e Mopaa.

~ ~ ~ ~ 7. A A B B, ~ ~ ~ ~ A A B B.

~ ~ ~ ~ ~ ~ 8. A A B B B A, ~ ~ ~ ~ ~ ~ A A B B B A.

~ ~ ~ ~ ~ ~ 9. ( A A ) ( B B ) A A, ~ ~ ~ ~ ~ ~ ( A A ) ( B B ) A A.

~ ~ ~ ~ 10. A \ B A B.

~ ~ ~ ~ 11. A B B A.

~ ~ ~ ~ ~ ~ ~ ~ ~ 12. A ( B C ) ( A B ) C A B C.

~ ~ ~ ~ ~ ~ 13. A B ( A \ B )( B \ A ).

~ ~ ~ ~ 14. ( A B )( B A ).

~ ~ ~ ~ 15. ( A B )( B A ).

~ ~ ~ ~ ~ ~ 16. ( A ( B B ))(( A A ) B ).

~ ~ ~ ~ ~ ~ ~ ~ 17. (( A A ) ( B C ))(( B B ) ( A C ).

~ ~ ~ 18. A A, A.

~ ~ ~ 19. A X X, A X A.

epecee e ocoe coca eex oec ee e c cceo aco. Cpoa e ccea aco, aeaa, aco c, aepe eex oec a copypoaa a 7 e o x o oe. Cooecya aepaeca cpyypa opeeec a oece X c y ccea oepa: , e x y = (x+ y)*y, x y = (x* y) + y (co oepa pa poco opypoo, x e ceye ocpa y ao, a cooecye apeece oece oepa).

Ccea aco 1. x + y = y + x 1'. x * y = y * x 2. x + (y + z) = (x + y) + z 2'. x * (y * z) = (x * y) * z 3. x+ x = 1 3'. x * x = 4. x + 1 = 1 4'. x * 0 = 5. x + 0 = x 5'. x * 1 = x 6. (x + y) = x * y 6'. (x * y) = x + y 7. x = ( x) 8. 0 = 9'. x y = y x 9. x y = y x 10'. x (y z) = (x y) z 10. x (y z)= (x y) z 11'. x * (y z) = (x*y) (x * z) 11. x + (y z) = (x + y) (x + z) a ccea aco oa. oaepa BX ex eeo X, o opx x + x = x (, o paoco x*x = x), ec yeo ae po, oopo x + y = x y, x*y = x y.

C c ee oeca caoc co oce paccope acca pepo ax aep, opaoaoo oeca S ece x ce ey 0 1, yoeop yco (oe ++, -- oo aa oe apeece oepa):

1. 0 S 1 S;

2. Ec x, y S, o min(1, x++y) S;

3. Ec x, y S, o max(0, x++y - 1) S;

4. Ec x S, o 1 - - x S.

Oepa S opeec cey opao:

x + y = min(1, x++y), x*y = max(0, x++y - 1), x = 1 - x, x y = max(x,y), x y = min(x,y).

e pya poepec, o a opeeea cpyypa yoeope pee acoa, o e acoa yeo aep. Copypoa yco 1-4 yoeop pae opee oeca, ap ep, S = {0,1};

S = [0,1];

S = {ce paoae ca ey 0 1};

S(m) = = {ce paoae ca a n/m eoopoo cpoaoo ay paoo m ex 0 n m}, c oepa, +, *.

Heeoe ooeco A yepcaoo oeca U oe opeeeo ye paeoc (x) X, e X yoeope pe A ye acoa (paoo X = S = [0,1]);

(u) = 1. Oepa a e U e oeca opeec epax x y paeo c coc (ooeo) oepa a ae ocex, o ec oepa X.

Oepa, +, * cyae X = S = [0,1] c ec eop eex oec ooee, pa cyo poeee, eee oyp, e,, o axo coe oocoae oo oece. Be oo oeca ( acoc, paax yeo aep) e ocpecey c ey oepa, a ee oec a ycao apyeo.

Heee cooec ooe B eoe p pee p eeo cxoo opa [81] aecex eoax p pee [64] cyeceo coy c o cooec, ooe, eex cooec ooe, oepa a .

Heemu coomemcmue ey oeca X Y aaec e ~ ~ pe = (X, Y, F ) ooaaec poa oec, oopo X, Y po ~ oe ee oeca, F eeoe oeco XxY. ooo aa eeo eoo cooec oeco X aa oac o ~ pae, oeco Y oac p, a F ee pao eeoo cooec.

~ ~ Haoe ocee eeoo cooec = (X, Y, F ) cooece ~ = (X, Y, F), y oopoo pa F ec ocee eeoo paa F.

Heeoe cooece oe aao eopeo-oeceo, paec apo e.

eopeo-oeceoo aa eeoo cooec eo xoo epec ee oec X Y aa eeoe oeco ~ F XxY.

~ ~ F B apo e eeoe cooece = (X, Y, ) aaec c o o ap e R, cpo oopo oee eea xi X (i I = {1, 2,..., n}), co eea yi Y (j J = {1, 2,...,m}), a a epecee cpo xi coa yj cac ee rij= mF < xi, yj>, e mF y paeoc eeo XxY eeoy pay.

Heeoe cooece oo aa e opepoaoo paa c oeco ep X Y, ao ye oopoo pcao a ee y paeoc F puep.

~ ~ aa eoopoe eeoe cooece = (X, Y, F ), opee X ~ F Y a X = {x1, x2,...,x5} Y = {y1, y2, y, y4}, = {<(x1, y2);

0,2>, <(x3, y1);

1>, <(x3, y3);

0,4>, <(x4, y2);

0,3>, <(x5, y2);

0,7>, <(x5, y3);

0,8>} Mapa e R pa eeoo cooec opae a pc. 2.8.

y1 y2 y3 y x1 x2 x3 x4 x x1 0 0,2 0 0,8 x2 0 0 0 0, 0,2 1 0, ~ R = x3 1 0 0,4 0, x4 0 0,3 0 x5 0 0,7 0,8 y1 y2 y3 y Pc. 2.8. paecoe apoe aae eeoo cooec ~ ~ = (X,Y,F) Heemu omoeue a eyco oece X aaec epe ~ ~ ~( X,F ) ooaaec apa oec, oopo F ec ee o oeco X.

~ Moeco X aaec oac aa, a F -ee pao ooe. Heee ooe oo aaa eopeo-oeceo, paec apo e. B eoe p pee coyec ~ apoe aae eex ooe: ooee ~(X, F) aaec c oo ap ceoc R, a epecee ao i-o cpo j-o coa cac ee rij = F < xi,x >, e F y pa j ~ eoc eeo X eeoy pay F.

Cocma eemux omoeu [64, 81].

Peecuocm. Heeoe ooee R a oece X aaec peec, ec oo x X oec paeco ( x, X ) = 1.

R Amupeecuocm. Heeoe ooee R a oece X aa ec apeec, ec oo x X oec paeco R( x, X ) = 0.

Cocm. Heeoe ooee R a oece X aaec c, ec oo x,y X oec epaeco R ( x, y ) > 0 R( y,x ) > 0.

Cuempuocm. Heeoe ooee R a oece X aaec cep, ec oo x, y X R ( x, y ) > 0 R( y,x ) > 0.

Amucuempuocm. Ec x x, y X R ( x, y ) > R ( y,x ) = 0, o aoe ooee ye c acep.

Tpaumuocm. Ec x x, y X y paeoc eeoo ooe R a oece X yoeope epaecy R( x, y ) sup min R( x,z ),R( z, y )}, o aoe ooee aaec pa zX .

Ooee R aaec pa, ec yi, y, yk Y ax, o j ( yi, y ) R ( y, yk ) R, ceye ( yi, yk ) R.

j j aunopo. Ooee R aa aopo, ec oo pe eco pao.

Ooee R aa omoeue cmpooo npenomeu, ec oo acepo pao.

Ooee R aa omoeue epauu, ec oo cep o pao.

oye ap oo ooe coyec a coe poeee cooecyx ap. Ec T S eee o oe, o x acoe poeee M = T S = [ mki ][ mij ] = max{min( mki,mij )}.

i 2.4. oe aepaece pee Oe coca peeo paccapac paoax .A. Copoa, C.. Caaeo, M.. aeo, Yi-Jia Tan [79, 95, 110, 124].

aco yopoeoe oeco L aaec ue [epxe] no ypeemo, ec aoe yxeeoe eo ooeco ee oy [epx] pa. Ec aco yopoeoe oeco e c e epxe oypeeo oopeeo, o oo aaec pe emo.

Ec L peea, o x eeo a b oo ec oepa ab = inf{ a,b }, a + b = sup{ a,b }.

Peea aaec noo, ec e cyecy oee e pecee x oec eeo. Bca oea peea oa.

Oopaee pee L peey L aaec epxu [uu] ooopuo, ec ( a + b ) = ( a ) + ( b ) [ ( ab ) = ( a ) ( b ) ] x a, b L. ooopu pee L peey L opeeec a oo paee, eec epx ooopo oopeeo.

Bao-ooa ooop aaec uoopuo. Bepxe e ooop c oo oopae.

Heycoe ooeco H pee L aaec nopeemo, ec a,b H ceye a + b H, ab H.

yc L peea. a,b L. Hao ee x L, ao, o ax b, aaec omocume nceoonoeue eea a ooc eo eea b peee L. Ec ao ee cyecye, o opee ec ooao aae a b ooaaec epe a b. Peea, oopo x a,b L opeeea oepa a b, aaec pe emo c omocumeu nceoonoeuu ( paypoo peem o).

Hae ee x L, ao, o a + x b, aaec ya omocume nceoonoeue eema a ooceo eea b.

Peea, oopo x a,b L opeeea oepa a b, aa ec yao peemo c omocumeu nceoonoeuu ( y ao paypoo peemo).

Peea L aaec ucmpuymuo peemo, ec e o c oeca: x(y + z) = xy + xz, x + yz = (x +y )(x + z), aaee cpy aoa. Peea cpya ye oa, oa e ee eco ye o yaax aoo.

Peea L aaec ecoeo ucmpuymuo peeo, ec oo x L x ceec eeo {yi | i I }, e I oeco eco eeo, oec:

x yi = x yi, (17) iI iI x yi = x yi. (18) iI iI oa peea ec paypoo, ec oa ecoeo -cpya, .e. oec (17). oa peea ec yao paypoo peeo, ec oa ecoeo -cpya, .e. o ec (18). Ceoaeo, oa peea L o paypoa peea y aa paypoa peea, ec L ecoeo cpya ecoeo -cpyo, a a, paypoo, pee ec a oe a cpya peea, opeo [0;

1].

Peea L aaec peemo c omocumeu onoeuu, ec coo eea x oo ee epaa [a;

b] aec ao e e d, o c + d = b cd = a, p o d [ a;

b ]. Peea L c ye ee 1 aaec peeo c ooe, ec a ee ee ee ooee epae [0;

1]. ooe epae [0;

1] aa c poco ooe.

yeo peemo aaec cpya peea c ooe , .e. cpya peea c 0 1. B yeo peee B a e e x ee ooc oo ooee x/, aoe, o x x/.

3. Moe eo p pee, ocoae a apo cpae aepa p p ypaeecx pee pyooe pep oe e oo oaac a co o y, o opaac xopoo papaoa acoee pe aeaec oe o ep p pee, oox oppeo pa aoee ye aepa exc. O oo, acoo paoo, a poao ocyecec oepa p ypaeecx pee , ac yceoc pa ceo pep eo [36, 39, 56, 65, 66, 75, 80, 84, 113].

Moocee cceoa oaa, o a, pae pee e ooeo aaeco oep, coy ypo ee, a oa poopee peae paa [1, 2, 12, 14, 29, 53, 59, 67, 74, 92]. oepa p pee peyec o cex e ce oacx pao eeoc eoea [4, 13, 15, 19, 50, 56, 58, 59, 62, 64, 65, 68, 74, 85, 91, 101, 108, 115, 118], o cao c yeac oeo opa, eoxooc ya ooe oeco po opex aopo, oex cyex cocax p p pee.

3.1. acca oee eoo p pee pee acca oee eoo p pee. Mo e aa p pee [25] pecaec e:

, e t ocaoa aa (apep, pa oy ayy eoopo cce aepay yopo c oe co aepa);

X oeco oycx aepa;

R oeco pepe oe cee oce ocaex ee;

A oeco a epe o pep (a aeoa, opoe, ep ae, ooe);

F oopaee oeca oycx aepa oeco pepax oeo;

G ccea peoe peaeo eea;

D peaee pao, opaaee ccey peoe.

acca oee aa p pee [25] pooc coo ec co cey paa:

1) o y oopae F eeppoaoe, epoocoe e opeeeoe, oo e cooeceo: P ycox opeee oc, P ycox pca, P ycox eopeeeoc. Aao opao, o ooe oca cceyeoo oea accpy c P [80];

2) o ooc oeca R ooeeoe oeco coco ee ecox pepe, ec cooeceo: P co cap pepe, P c eop pepe (oopepae aa);

3) o y cce G opaae peoe ooo a oe a eo, ec aa yaoo P, aa pyooo P.

B [27] o oe opa oaec apa (X, R), cocoa o eca aepa X apoo ooe R a e. B [8] p opeee oe P peoaaec, o paccapaec eoopoe oeco cxox cpyyp peoe cceyec opeeea P, poecc pee oopo oaec a oa op eoa opao c xoo cpyyp eoopoo aooo acca eoo. p o oo ca, o a oece cxox cpyyp aaa oe pee o caeo P, ec yaa e p pao, coaco oopo y poooy ooe cac cooece eoop aop eoo. opee oe opepoa a cooece ex x eoo p pee opeee ao cpyypa.

B [25] peea acca eoo P o a paa, a coepae cepo opa, oyaeo opa, a oco e oopo oo opee pyy eoo P ycox eopeee oc (pc. 3.1).

B oopa paccapaec oooc aaa pax opo co ypae eoa p pee ycox eopeeeoc.

o cao c e, o p cceoa ooecx, coax py x cce, yopoa oopx yacye eoe, aeoe oeco opa oe oyeo o e, ex o pao c ao cceo ax ee ocoeoc, o e, ex pecaee o ex yopoa cce. a opa oc cye xapaep, ee pecaee ececeo e coep eopeeeoc, oope e e aaoo e paoo ae a. B o cyae ye paccapa aa oaoo ypae c o eoo, yax eopeeeoc oca oe cceyeoo oea.

Ta opao, oae poecce ypae pepe poe, oope oaa paa ecpyyppoax aa P, ooo cecopoe poaapoa eoa, ya eope eeoc [10, 25, 26, 31, 36, 37, 49, 51, 56, 57, 58, 68, 70, 77, 78, 81, 86, 89, 96, 113].

Cepae pa py e T pa 1. c e 1.1.1. e e 1.1.2. e c e e e c x e e 2.1.1. e c ec e e 2.1. ec e 2.1.2. e e c e ( ) e x e 2.1.3. e " c " 2. 2.2. ec e e e x e 2.2.1. e ec e e e e e c "c c - e c " e e 2.2.2. e e e e c e 2.2.3. e ec c 2.2.4. e e 2.3. ec e e e x 2.3.1. e x e 2.3.2. e e e c e 3. ec e e e 3.1. e c e e e c 3.1.1. e..

e e 3.1.2. e e e c e e e e 3.1.3. e e c e Me p pee ycx 3.1.4. e epeeec 3.1.5. e e e ec e 4.1. c c e 4.1.1. e c c e 4. e e e e c e e x e e x;

ec e e e ec e c e c x c e.

4.2.1. x c ec e e e 4.2.2. e c x c 4.2. ec e e e e e c c e e e x x e e ec e c e c x 4.3.1. e ec 4.3. ec e 4.3.2. e c c ec e e x e e x e e c e c x 4.4. ec e 4.4.1. e x e e e x c x c ec e x e e e e c c e c x 4.4.2. e ee e e e 4.4.3. e e e e ( e e c ) Pc 3.1. acca eoo P a ocoe coepa cepo opa p o o eopeeeoc ye oa e, e o aec aay epe co co yoo oo ooc [78].

B [80] poc acca eopeeeoce, oopo, aco c, ec eopeeeoc, cae c a opyee p, eeep oopo aaec poopoae: eopeeeo c, cae c eeoc yaco ooeco (pee ceo apepo oypeo p), acoc, c x eoo ao c, aco ooee, coee oaec;

eopeeeo c, cae c coa acpa aopa ope x peoax, oopx pa ee eoe epec.

3.2. Moe eoo yopoa coyee pae oe eoo yopoa pa oo paec a e oe py, paaec co oxoo pee aa yopoa oeo [8].

B oex nepo pynn, coyx cmamucmuecue emo, aoy oey xi coocaec opeee epa oaae, oea o eo cpae c oca oea, a aee i oe poco yopoac o ya ae oo papy eo aopa. B oex mopo pynn, coyx ouamopo ouecue u meopemuo-paoe emo, oeac oaae e o ex oeo, a ceo yopoa eo paec yopo ae, acpyee eoop yoa aeca. Oeo a oc p o e eaec. Paccop eoope oe epo py.

yc aao eoopoe cpoaoe oeco oeo X={x1, x2,..., xn}, oope cpaac oapo c o pe x peo eoc, eaeoc, aoc .., a peya acac e ap apx cpae A = ||aij||nxn, opaae oaee apoe ooee peoe/epa a oece X. Ce pe ee ap apx cpae aij aji o pac pa, ec cooecye oe paoe ecpa (xi xj);

ec e xi > xj, o oo aij > aji. poe oo, a ee a p A oo aaac ooee apooe opae , ooao cae oapo cepe ee aij, aji. p ee ocoe apoo.

1) poca cpyypa (C).

1, ecu xi > xj ;

(19) i, j, i j aij = ecu xj > xi ;

0, 1 / 2, ecu x ~ xj.

i eppea: aij aop aa peocxoca ooo oea a py x paoeoc (ecpaoc).

2) Typpa apoa (T).

(20) i, j aij 0;

aij + aji = c.

eppea: aij co oo, apax poo xi o cex cpeax c poo xj;

co c = const p o oe eppepoac a oeco ax cpe. Hepeo ooeo ocypyec eo ceoc ap.

3) ococepeca apoa ().

(21) + = 0.

i, j aij a ji eppea: oe xi peocxo apo cpae oe xj a aij.

4) Ceea apoa (C).

. (22), > 0;

= i j aij aij a ji eppea: oe xi peocxo apo cpae oe xj aij pa.

5) Bepooca apoa (B).

(23), 0 aij ji i j aij 1;

+ a = eppea: aij epooc peocxoca xi a.

xj oo apoo (19)-(23) oo aaa oo ec ee pooy eey cpyypy (BC), paax oopo peoaa ec oo oo eopaeoc ap A, ca e ee ee o y eppepoac o-paoy.

epexo o ap A, aao eoopo apoe, oapo ao o-oy ape B ooe e cea, o p coe eoopx ooex coepaex yco epeo cope c oepe ao opa. Bopoc o oooc epexoa ape c pyo apoo o yx aoo epexoa c pa oe pacca pac c yeo coepaex ocoeoce aa. Cxe apae oox epexoo pee a. 3.1.

Taa 3. Peaa peopaoa apo T epexoa Booe coco peaa BC T = + bij aij ( max sij - sij ) / 2 ;

i, j = ;

= + ;

bij ( aij max sij ) / sij sij aij a ji i, j T = ;

= bij aij - a bij ( aij - a ) / 2 ;

ji ji T = + + bij aij( signaij 1) / 2 ( max aij - | aij |) / 2 ;

i, j = + > bij c( aij max aij );

c 0 ;

i, j aij C = > ;

bij r ;

r C = > ;

bij logr aij ;

r B, T C = ;

bij aij / a ji C B, T = + ;

bij aij /( 1 a ) ji T B = + ;

bij aij /( aij a ) ji B T = > ;

bij c aij ;

c BC, T, B, , C C = +.

bij [ sign( aij - a ) 1] / ji aa oee eoo yopoa peye ap ap x cpae opeeex apoox opae.

1) Moeu cnopmuoo muna: i = 1,n si = aij.

i j Taoe aae copec yopeoc a eo pyo cxox oee, oopx aece papyeo aopa coyec a paa oeo cya oo. Opaaaea apa A oe e apoy a T, C .

Haae Maeaeca ooee Heoca oe P oe xapaepc oe Typpa Oe yopoac pocoa oye- Bcee o oe peyaa. ecex o ya si.

Boc oeo oe e coca a Moe B aece yeo (e peoaae.

paoc oceoa- pa) paec oe c cy (C**), eoo max si, epaec i- i apaoc ee cpoa i- coe, o pace e paec ep ..

(P), oo po ea pea (P).

Moe Moeco X paaec pocoa oye- He oaae oceoa- a a co co c o- peyaa. coca c eo ooc si co c e xoo ao (M), si .. ec P.

a co.

2) Moe umepao cmeneu npeocxocma ec o oe copoo a;

peec ococepecx ap.

Boc oe epao cee peocxoca n ~ ( xi, x ) = ( ait - a ), oeae peocxoco xi a xj cpae j t jt t = c po oea. oa epaa cee peocxoca aa ~ a, ee oo peca e ( xi,x ) = f ( xi ) - f ( x ), p o j j n ~ i f ( xi ) = ( xi,x ), e f y oeoc a X, oe j j j= peaaec yopoa o ya ee ae.

eocaa oe oo oec eee oeo t eco oeo, oao ax eco apaee aao e oe. Moe coc yppo caocoeoo epeca e ee.

3) Moe yuu ouupyeocmu opepoaa a opaoy e ex ooe peoe, .e. aij 0;

1. peec apo o T, .

y opyeoc l( xi ) = max a xapaepye acay ji ji cy, c oopo oe xi opyec oca oea oeca X.

p l( xi ) = 0 acoo e opyec, p l( xi ) = 1 acoo o pyec, p 0 < l( xi ) < 1 cao opyec. Oe yopo ac o ya cooecyx ae y m( xi ) = 1 - l( xi ).

B [81] coyc pye coco oye l(xi).

Booo ucnooaue aeu m( xi ) aecme ouecmex oeo aocmu oemo. Moe oaae coca M, P, C**, P.

4) Moe pu-Teppu poa pocx cpyyp e paoe x eeo eocex yppx ap.

aoy oey coocaec eo ca, pe peoaae i c, o epooc peocxoca apo cpae P( xi > x ) po j P( xi x ) poopoaa : > = /( + ) = 1 - P( x > xi ). a i j i i j j o ap (i,j) pooc k eacx ao apx cpae. Ooa n si / = k ( + )-1 ;

i = 1,n;

i i j j= eo oyaec:

n = 1.

i i= Ccea oe peea epaoo. oce ce cex i oe yopoac o x ya.

oyaee ooe opaoaoo eopa oy cy oece oea aoc oeo. Boc coca C**, P.

5) Moe pa-pya-ypoa peec opao pocx cpyyp, ap c yppo ceeo apoa.

aoy oey xi cac cooece eoa a aaex eppoax c, oopo ca k-o opa pik opeeec a cy a eeo i- cpo ape Ak :

n i = 1,n pi k = aij k ;

aij k = Ak ;

k = 1,2,....

nxn j= k k p k ee eco lim( pi / pi ) =, i = 1,n, e opao k i a coce eop = (,..., ) ap A oeae ac 1 n aoy o oy coceoy cy eope eppoa-poeyca.

Moe oaae coca P, M;

T* (p aii = 1). oyaee a eu onoem cocmeoo emopa oym cyum oeo aocmu oemo.

6) Cmoxacmueca oe aoa peoea opao ap, aax ceeo epooco apoax.

Mapa A peopayec epoocy apy P, e pij epo oc peocxoca xj a xi.

p apoe (23) P = AT.

- p apoe (22) pij = a /(1 + a ), i j.

ji ji ~ ~ Cpoc coxaceca apa.

P = pij nxn ~ ~ ~ pii = 1 - pij, i, j = 1,n, pij = pij /( n - 1), i j;

i j n pi = /, i = 1,n, ii jj j= ~ e op oyae - P ) epae i-o coa ii det( E i-o cpo.

opoae xi pooc o pi.

Oaae coca M, P. oyaee ooe aoo pacpeee oy cooac aece oecex oeo aoc oeo.

He oaae coca C**, P.

7) Moe paoepoo cauau peec ooex ap c apoo T C.

o acoe ca apo T ee eco: i, j / = aij / a i j ji n n n (*) p o = / = ( / )-1 = ( bji )-1.

i i j j i j=1 j =1 j = Ooaa i, j zij = lnbij, oyae i, j zij = zik + zkj = -zki + zkj, a o apa Z = zij nxn oe occaoea o o cpoe.

B ao oe o cxoo ap A eoxoo epe ap e Z ocpo n pax ap Z(1), , Z(n), oaa, o apa Z(k), opoaec k-o cpoo ap Z o opye (*).

k = 1,n n Z = ( 1 / n ) Zi ( i ), pe i, j zij = ( 1 / n ) ( zik + zkj ). poea peopao i k = n a i, j bij = exp zij, aij = cbij ( 1 + bij )-1, oe oy = ( b )-1.

i ji j= Oaae coca P, T, M, P. oyaee oe o o cooa oeceo oe aoc oeo.

Bop oe yopoa c e coca, oope ocoeo eae ao opeo cyae, pecaec eca oe cceax oep p pee [8]. Moe a: oe y opyeoc, p-Tepp, epa-pya-ypoa, coxaceca o e aoa, oe paoepoo caa peaa opao oee yeee oo oy cooecyx oax yopo a. Moe p-Tepp poa pocx cpyyp eoce x yppx ap, oope e ya eooc, eopeee oc oeax cepo. Coxaceca oe aoa copee ope poaa a epooc acc eopeeeoce, oe o ee o e epa-pya-ypoa oe y opyeoc oo ya eopeeeoc e, e oaxc epe co co yoo oo ooc c yeo eeoc cooeceo. Moe paoepoo caa e oaae coco C**, o e ooe cepa poo eoe cpae. B [8] yepaec, o oe epa-pya-ypoa ae e oaae coco C**, oao [93] ya a oxo e popeo eoo ap a ocoe a o oe. Ta opao, eoo yopoa ycox e opeeeoc peoeee ooac oe y o pyeoc epa-pya-ypoa. Paccop eo P, opepoa e a eee oe. poe x, oopa paccapac ae aecee eo P, poeyp cpae oeo oopx opepoa a aecee oe cepo, o oeae opoc c epo, oo oeppoa epa, coce opeo pe eo oac (pc. 3.2).

P Moe np pee ycox eonpeeeoc cxoe ae: Meo p A Opt( A ) Q ape ooe pee Q Opt(A)= |Opt(A)|>= opee oe oe oeo Moe eoo yopoa Moe pyooo yopoa I=(i,...,i3), I* - oeco epecaoo =(j,...,j2), ji - cpa m AI*opt(A) = X i = m i j = i i i j i= Moe y Moe epa-pya-ypoa Moe eopo oe aepa n opyeoc A = { ai ) - amepamu,i =,...,n;

Y = X X... X - oecmo ( k ) Q i =,...,n pi( k ) = ;

aij mx( xi ) = - max aij j= emopx oeo A = { ai( yij )} {( kij )} ji ( emopeoeu, yumaue npuopumem pumepu ) aij( k ) = Ak ;

k =,,..

nxn Meo p MA (Meo Meo APOC (Aye Meo AP Meo OPACC pee a ae Aaa Poeyp y Oopx (APa (OPaa eeo o epapx) Cya) oeca) ACCa) aecee eo p pee opoa eo apx ooe Pc. 3.2. acca oee eoo p pee ycox eopeeeoc Meo aaa epapx [93, 94] p p ypaeecx pee poopoa oo x peyao o, paee peee, oo caaec co coo cceo aoacx ooe (pecypc, eaee cxo e, a pya ..), oopy yo poaapo a [93, 94]. MA paae oe epa-pya-ypoa [8].

pa peee, pya cepo poo eoo coo poe opeee ee ooe ooe ey .

oyaec oe peao eceoc, ocpoea e epap x. Bepa epapx oa e, aee pacoaac oe, ae c, oope a oe, , x e, o, cpae , , aoe, cxo, ec peyaa cpae. Ha cey e ae pee cpaac ye oee ooe epapx ey coo. B peyae oe paea oocea cee ecoc aoec eeo epapx. ae cye paac ceo. B aepe aaa poe MA ae poeyp cea oecex cye, oye popeoc pepe axoe aepax pee. Ta opao, oco e a p pee c oo MA ceye:

ocpoee epapx paccapaeo poe;

apoe cpaee ooe epapx;

aeaeca opaoa oyex cye.

B aoee eeapo e epapx cpoc c ep (ee c o pe ypae), epe poeyoe ypo (pep, o oopx ac oceye ypo) caoy oy ypo (oo p oo ec epee aepa). Cyecy ecoo o epapx: oae, xoapx, ac .. Haoee aco peec ep epapx.

ape cpae pooc epax opoa ooo eeo a py. cye ae paac ex cax.

Ec ee A opye a eeo , o ea ap, cooe cya cpoe A coy , aoec e co, a ea, co oecya cpoe coy A, aoec opa ey c o (po). B MA peoea aa ooceo aoc e eo epapx (a. 3.2).

Bce ap MA o opao cep, .e.

aij =1/aji.o ao aoa ap apaee cac e, ..

aepaa paoea cao cee. aoe ao ap paepo nxn ocaoo poec oo n(n-1)/2 cye.

Cocaee ax ap pooc cex ypoe py e papx. pe oyee ap o coacoa oc oepoo pee. Coacoaoc poec coo (apa o coacoaoc aij a = aik ) pao (opoo coacoa jk oc). Coacoaoc ap oo poep.

Taa 3. aa ooceo aoc ecoc Opeeee ooceo aoc 1 paa aoc 3 yepeoe peocxoco ooo a py 5 cyeceoe coe peocxoco 7 aeoe peocxoco 9 oe coe peocxoco 2, 4, 6, 8 poeyoe pee ey y co ce cye Opae e p- ec p cpae ooo apaepa eex e ce c py oyeo oo eyaax ce, o p cpae opoo apaepa c ep oy opay ey Bc eop popea (coce eop) ao a p apx cpae oo pa cocoa [94]. B acoc o paoo cocoa aae oe aac oa ea opeoc. Haoee oocoa peya oyaec p pe e eope eppoa-poeyca.

Ha ocee ae opao oyee eop popeo c epyc, aa co opoo ypo . oae pope e peoac a pope cooecyeo pep a ecoe ypoe cypyc o aoy eey cooec c pep, a oope oecye o ee (a ee opoo ypo yoaec a ey, .e. a ec eceo e caoo epxeo ypo.) o ae cocao, oa, pope oo eea, oop ae coyec ea oax popeo eeo, cpaaex o ooe ey a pep paco oex ypoe e. poeypa pooaec o caoo eo ypo.

Meo p pee p eeo cxoo opa [81, 8, 48] B paoe C.A. Opocoo [81] paccapac eo p pee, ocoae a apx cpaex aepa, oope pa ac e eex ooe. Meo coyc oe y eopyeoc [8]. B paoe [48] poeea cpyypa ax eoo, peyae oopo e ceye eo eop p pee p eeo cxoo opa:

- eo p pee c o cepo;

- eo p pee c pyo cepo, xapaepye x eco oea;

- eo p pee c pyo cepo, xapaepye x ee ooee ecpooo peoe.

aaa npumu peeu c ou cnepmo aao oeco oox pee aepa U = {u1, u2,, un } eeoe ooee ecpooo peoe (.o..) R a oece U c ye paeoc (ui, uj) [0, 1] oe R peecoe eeoe ooee a U, a o (ui, ui)=1, ui U.

R H.o.. aaec oo P peyae opoca cepo, oaa x a pecae o coepa cyece aa, oope e opaoa cy peepo cooc ao opaa o py pa.

o ap aepa ui, uj U aee (ui, uj) oaec R a cee peoe ui, e xye uj ac ui uj. Paeco (ui, uj) = 0 oe oaa a o, o (uj, ui) > 0, o ec c oo R R eo cee oeo opaoe peoee uj ui, a o, o (uj, ui) = 0, o ec aepa uj ui ecpa. Peec R oc .o.. opaae o ecece a, o a aepaa e xye cao ce.

aaa p pee aaec paoao ope a oee peoex aepa oeca U, a oopo aao eeoe ooee peoe R.

Aopum peeu aau S 1. Cpoc eeoe ooee cpooo peoe R, acco poaoe c R, opeeeoe ye paeoc (ui,u )- (uj,ui ), (ui,u )> (uj,ui ), j j R R R R ( ui,u ) = R j 0, ( ui,u ) ( u,ui ).

R j R j S T T o ooee oe pecaeo e R = R\R, e R T opaoe ooee (apa ooe R oyaec pacopo ae ap ooe R).

nd 2. Cpoc eeoe ooeco UR U eopyex a epa, accopoaoe c R aee e aepa, oope e opyc a py, opeeeoe ye pae oc nd S S (ui ) = min1 - ( u,ui ) = 1 - max ( u,ui ), ui U.

R R j R j u U u U j j nd o aepa uj U aee ( ui ) oaec a R cee eopyeoc o aepa, o ec cee, c oopo nd ui e opyec oo aepa oeca U;

( ui ) = R oaae, o aa aepaa uj e oe ye ui co cee opoa oe ;

ae oop, ui oe opoac py aepaa, o co cee e e 1 -. Paoa ecec eo ca op aepa, ex o oooc oy nd cee paeoc oecy UR.

nd 3. Bpaec a aepaa u*, oopo aee (u*), a R cao:

nd u* = arg max ( ui ).

R uiU Oa ae peee aa. Ec aoy cee eopye oc ee e oa, a ecoo aepa, o P oe o ca pa oy x, cxo ax-o ooex coopae, o pacp py cepo p oppoa cxox ax a a oop ee peee.

aaa npumu peeu c pynno cnepmo, xapamepuyex e cou ouuemau Ha oece ceoox pee (aepa) U = {u1, u2,,un} aao ecoo .o.. Heee ooe ecpooo peoe Rk oye peyae opoca aoo cepa aoe ap eeoo ooe ecpooo peoe (.o..) Rk, a ee oopo ec aee y paeoc (ui, uj), paaee R cee peoeoc aepa ui o cpae c uj. p (ui, uj) > 0 ui peoeee, e uj;

ec e (ui, uj) = 0, o o R R epa aepaa xye opo, o o ecpa. o, p aee peee, o-paoy oocc cepa, o axo opae e ecox oeax k, (e 0 1, = 1), cooecy k k x aoy x.

e ao aa ec yopoee cooyoc aepa U = {u1,..., un}.

Aopum peeu aau 1. Cpoc cepa P ooe a epeceee eex ooe ecpooo peoe cepo P = Rk (ui, uj) = min { (ui, uj)};

Rk a opao, oyaec ooe eeoe ooee ecpooo pe oe. aee c .o.. accopyec ooee cpooo peoe S PS = P\PT c ye paeoc.

P P j j P j j ( ui,u ) - T ( ui,u ), ecu ( ui,u ) > T ( ui,u );

S P P ( ui,u ) = P j 0, ecu P ( ui,u j ) P T ( ui,u j ).

nd aee opeeec oeco eopyex aepa U c y P e paeoc nd S (ui ) = 1 - max ( u,ui ), ui U.

P P j u P j 2. Cpoc ya cepa Q ooe Rk, oopa opeeec a Q = Rk, (ui,u ) = ( ui,u ). Oa ec o .o.., k Q j k k j k c oop accopyc eo ooee cpooo peoe QS nd nd nd oeco eopyex aepa UQ. Moeca U UQ e P cy ooy py pya opa o eopyeoc aepa .

3. Paccapaec epeceee oyex oec UPnd UQnd:

nd nd Und= UPnd UQnd c ye paeoc (ui)=min{ (ui), P nd (ui)}.

Q nd 4. Bpaec a aepaa u*, oopo aee (u*) a nd cao: u*=arg max (ui), ui U.

aaa npumu peeu c pynno cnepmo, xapamepuyex e emu omoeue ecmpoo npenomeu ey uu.

Moo paccope aay p pee c pyo cepo, xapaepyex e eco oea, a p oo ee ooo .o.. N, aaoo a oece E cepo c ye paeoc (ek, el), ek, el E, ae oopo oaa cee peoe c N epa ek o cpae c cepo el.

Aopum peeu aau nd 1. C a Rk accopyc RkS Uk, oc ooaee nd (ui) = (ek, ui), i = 1,..., n, k = 1,..., m. Te ca aaec eeoe k cooece ey oeca E U.

2. Cpoc cepa e oo cooec = T N .

pe, peypyee ooee opeeec a acoe po eee ap T, N, . To ec, oyaec eoe peypyee ooee, oyeoe c yeo opa o ooceo aoc nd .o.. Rk. C ooee accopyec ooee S oeco U.

nd nd 3. oppepyec oeco U o oeca U c ye nd nd paeoc (ui)= min { (ui), (ui, ui)}.

4. Bpaec a aepaa, oopo aee y p nd aeoc coppepoaoo eeoo ooeca U eo pyex aepa acao.

aecee eo p pee [61-64] aecee eo p pee papaoa O.. ape ocac paoe [64]. Meo oy cooac oex eoo yopoa oeo a ocoe x eopo peoe.

B aecex eoax p pee coyc oo ae coco oye opa o cepo oece poeyp ocpoe oo, oope, coaco a cxooecx cce oa, cooecy oooc eoeeco cce epepao opa. O ax cocoo, oop oe c ycexo pee pee ecpyyppoax poe c aece epee apoa ayx cceoa, oypcoo oopa poeo, poe oo opa ec eo yopoa o opepax aepa APOC (aye poeyp y oopx cya).

Paccop peee oo eoa a pepe papoa e eoeox e, a eo, eoopx o poyapx o, c o pe opeecoo cpoca. peoo, o co o e p: 1 (a eeco), 2 (a aya), 3 ( a aoa). Be pep x oea: e , co oc, yooc ya (oeco aepa pepe oe cyae oe poo). pepaoe ocae aepa x e oe ceeo ay:

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