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M oy caapy aay eoo popapoa, pee ee ocoo ope. oo ee p ooee epeee, peyae eo opae ayc e cce paec:

18x1 + 15x2 + 12x + x4 = 360, 6x1 + 4x2 + 8x + x5 = 192, 5x1 + 3x2 + 3x + x6 = 180.

ooec cc eex ooex epeex coco o, o o paa coo oeco ecoyex cao-aco oo oo a p eye ae pooca.

Cec-eo ec epao poecco, .e. ao, a ao ae pocxo yyee eyeo aa eaec o o o, o o eooo, .e. ey a ye ec oa. o aa o poecc, aae eoxoo p eoop a epoo aa. pe o a oe yoeop eoop yco.

B acoc, o oe c epoe, o ec e oo poo coo eyex ooe, coo opae aae (e ca opae xi 0, i=1, n ), .e. m. Tao a eo ocpo, ec ape A opae aa eec m eoeacx ex coo. Ta ae, ec cxoa aaa ee opy caapo. Toa oce pee e ocoo ocpeco ee m ooex epeex cea oc ae co. Hapep, ae cyae p coa ap opae, cooecye ooe epee x4, x5, x6 c a coa.

oa ec ae co, aece epoaaoo aa oo p a, p oopo e pa y oo ooee epeee, pe aa x pae aee paoe ee coe pao ac opae.

Heyee epeee aa aa ac, a x cooyoc aco.

a pae aece epoaaoo (oopoo) aca x4, x5, x6 acae ae aa ay, aaey cec-ae.

Taa 2.a 1 Cec-aa epoo aa x1 x2 x3 x4 x5 x6 P C A C x4 18 15 12 1 0 0 360 x5 6 4 8 0 1 0 192 x6 5 3 3 0 0 1 180 -9 -10 -16 0 0 B coe "C" acac oe, c oop ace epeee xo eeo yoa (ocoy ooee epeee eo e xo, o eye coe "C" ee y). B coe "P" aca ey a aa. B aco oe ecooa ocac ce pecypc, o cooecye "yeoy" ay, o ec eo e pooc. Ececeo ao a e oe c oa.

Be i ocee cpo paccac o opye i =< c,ai > -ci, e < c,ai > - capoe poeee coa "C" coa cec a ai cooecyeo epeeo xi, ci oe eeoo yoaa p epeeo xi. Ec cpe e i ec opaee, ceoaeo ey a oo yy. y ye ee ac epeeo oopo i opaea. pe ac o y epeey, y oopo i aoa o oy. Hapep, ae cyae o x3.

aee eoxoo opee epeey, oopa ye cea aca. oo axo cpoy, oopo ocaec y ooe cooecyeo eea coa "P" aa eey ooo ac coa, p o paccapac oo ooee ee oo coa. To ec axo min(pi/ai3), cex ai3 > 0. B ae cyae o y min(360/12; 192/8;180/3) = 192/8 = 24, ocaec o opo cpoe, .e. cae aca x5. ee co a epecee o ooo ac eea cpo caeoo eea aaec papeau, yac o 8.

Cey ao ec epece cec-a, oop pooc a opao. B coe acx epeex eco x5 acae x3, oa cpoa xoyaec eee cpo capo cec-a a papea ee. Ta opao oa cpoa oo a ye oyea. Ocae cpo oo ceca oyac ocpeco cooa peopaoa aaox eeap peopaoa cpo ap. p o ye paccaa oa cpoa yoea a yy ey aec cpo capo a a, o coe o eeo ac epeeo ca e, pe ea coa ye ocao oo cpoe eeo epeeo.

ae, o capo ae ce ace epeee e ee co.

eo oo oaec oo ae o eeo epeeo. p o co ocaxc ace epeex oo e epeca. poepe, o ec o oac cea coy yaae peopaoa, o o ce pao e ec.

Coe "C" o cex x peopaoax e yacye, e acac cooecye oe. B ae cyae o opo cpoe acaec oe c5 eco c3.

oce cpoy oo cec-a oo pacca y cocoa, o a e a ocae, pe eeape peopaoa a, o oc y coe oo aco epeeo, o coy paccopey paee opyy.

Taa 2.a 2 Cec-aa opo epa x1 x2 x3 x4 x5 x6 P C A C x4 9 9 0 1 -3/2 0 72 x3 3/4 1/2 1 0 1/8 0 24 x6 11/4 3/2 0 0 -3/8 1 108 3 -2 0 0 2 Ee pa oc, a oa a oyea capo. Cpoa x3 a oyea eee capo cpo x5 a papea ee 8. Hoa epa cpoa oyea capo epo ae ee cpo x3 yoeo a 12. Ta opao oye o coe x3. Tpe ae capo cpo oo cpo x3 yoeo a 3. eepa ae capo cpo cpo x3 yoeo a -16. Ta opao yy oye ce ee coa x3, poe e axoec cpoe x3.



aee coa , o ocee cpoe ec opaee ee (a o pa ao ee ece). a a oo yy e ac x2.

Haxo min{72/9,24/0,5,108/1,5}=72/9=8, a cae aca x4. Papea ee 9.

Taa 2.a 3 Cec-aa peeo aa x1 x2 x3 x4 x5 x6 P c A C x2 1 1 0 1/9 -1/6 0 8 x3 1/4 0 1 -1/18 5/24 0 20 x6 5/4 0 0 -1/6 -1/8 1 96 5 0 0 2/9 5/3 BM, B OCEHE CTPOE HET OTPATEHX CE, HAT TE AH X1 = 0, X2 = 8, X3 = 20, X4 = 0, X5 = 0, X6 = BETC OTMAHM. B COOTBETCB CHM HHO POBECT 8 E BA B, 20 - BA C, P TOM OCTAHTC HEBOCTPEOBAHHM 96 CTAHO-ACOB OBAHOO OOPOBAH. EBA A POBOT HE HHO, MT BPEMEH PEEPHOO TOAPHOO OOPOBAH ET OHOCT CEPAH. P P TOM COCTABT CMAX = 10 + 16=400.

2.6 Cec eo c eee cycceoo aca Paccope cocoo oo pe aa, oopx epoaao ce cxoe opae e epaec "". Oao, ec eoope cxoe opae e opy paec epaec a "", o oce pee aa aoecoy y, e ye e m ex coo ape cce, a peye cyae.

oce caaoo eapye cya. B opaee-paeco ooey epeey ec eooo, o pe a e oy ocycoa epeee xoe eo c oeo 1 c oeo 0 pye opae. A epaecax a "" ooee epeee oy co ao yc. oa e coe pa oca e opao cxo ac pee aa cec eoo.

oc caaoe a pepe.

pep 2.3 yc peye aae eco opae o oaoy oopyoa ( poc eo yee paepoc aa acoeo pepa) peyec, o cyap oe yca e o A B e eee 100. Taoe oooe opaee oe e eco, apep, cy ye aex pepe ooopo. o opaee aec cey opao 15x1 + 10x2 100, e 15, 10 - apep, coocae e poy.

PBEEH AHOHECOM B BBOM OOHTEH EPEMEHH X7, OAEM OPAHEHE 15x1 + 10x2 - x6 = 100.

HAT MATPA CCTEM OPAHEH TEEP MEET B:

18 15 12 1 0 A = 6 4 8 0 1 15 10 0 0 0 - e e pex (o oecy opae) ex coo.

B o aaox cyax coy a aae eo cycceoo aca. B opaee oopo e epeeo, xoe eo c oeo 1 c oeo 0 pye opae, o a aaey cyccey epeey. Hapep x8 0, 15x1 + 10x2 - x6 + x7 = 100.

oce oo cpe oc coo ap A ye oaec epe eo eacx ex.

cyccee epeee o yoa c ocaoo o o oy opae oeo. Toa oce pee aa cec-eoo oao ae ae cex cyccex epeex oaeo oayc pa y. oa oyc, o aec x eee a e eo cxoy aay. oa ae e aac ope aee yoyx oeo, a aca x e coa, o oop opayea opaeoe co, peocxoee o oy oe aepe aaoe. Bo cyae ecoo eec coco ac cec-a, ooy yoee ce e ooac ope aee. Oao p o oe oaac, o oe pee aa e ce cyccex epeex pa y. B ao cyae ceye ye oe epepea aay aoo.

pep 2.3 (npooeue) a ee aay C = 9x1 + 10x2 + 16x3 - 200x7 max, p opaex:

18x1 + 15x2 + 12x3 + x4 = 360, 6x1 + 4x2 + 8x3 + x5 = 192, 15x1 + 10x2 - x6 + x7 = 100, xi 0, (i = 1,7 ).

He poc oceoaeoc cec-a oyex p pee aa.

a Taa 2.ac x1 x2 x3 x4 x5 x6 x7 P x4 18 15 12 1 0 0 0 x5 6 4 8 0 1 0 0 x8 15 10 0 0 0 -1 1 3009 2010 -16 0 0 200 0 -Bo ac x1. cae x8. Papea ee 15.

Oe cyccex epeex coco o, oce oo a eoopa cyccea epeea cea aca ceyx cec-aax ee coe oo oe ye e ca, a a oa ac e opac. co, o o cae oeco eoxox ce. Oao cpa oca ce e ye aee epeca coe x7.

a Taa 2.ac x1 x2 x3 x4 x5 x6 x7 P x4 0 3 12 1 0 1,2 -1,2 x5 0 0 8 0 1 0,4 -0,4 2 2 2 x1 1 0 0 - 3 30 30 0 -4 -16 0 0 -0,6 200, Bo ac x3. cae x5. Papea ee 8.

a Taa 2.ac x1 x2 x3 x4 x5 x6 x7 P x4 0 3 0 1 -1,5 0,6 -0,6 x3 0 0 1 0 1/8 0,05 -0,05 x1 1 2 0 0 0 2 2 - 3 30 30 0 -4 0 0 2 0,2 199, Bo ac x2. cae x4. Papea ee 3.

a 4.

Taa 2.ac x1 x2 x3 x4 x5 x6 x7 P x2 0 1 0 1/3 -1/2 1/5 -1/5 x3 0 0 1 0 1/8 1/20 -1/20 x1 1 0 0 -2/9 1/3 -1/5 1/5 0 0 0 4/3 0 1 oye a ec oa. B cooec c yo yc e e a A, 4 - a B, 19 - a C. p p o coca 380 e.

2.5 ocea aaa eoo popapoa ooe aee eop aa eoo popapoa pae oe oceoc. Oaaec o aae oo cooca, a aaey ocey aay, c oopo o opay aoocey apy. p o cpaea ceya eopea, oopa aaec ocoo eopeo oceoc: ecu oa u aa ocmeo nap ueem onmuaoe peeue, mo pya mae ueem onmuaoe peeue, npue acuy eeo yuu ucxoo aau u uuy ocmeo uceo pa.





Pee po oceo aa ca pye ae coooe.

Ec oa aa coco aca coeo yoaa, o ocea e coco a ee yoaa. Ec oao ae cxoo aa aoe-o j-e opaee oec a cpooe epaeco, o cooecya j epeea oao ae oceo aa oa paa y. Ec e i- epeea cxoo aa e xo oa, a, o oao ae oceo aa cooecyee i-e opaee ye oc a cpooe epaeco (a cee poeoo cya).

B oe cyae paa, o oop cpoc ocea aaa ooo co e ye x paccapa. Oao paccop a eca ax opao oee pocx cya ap aoocex aa.

Bo-epx, paccop cya oa cxoa aaa acaa caapo ope (2.4) - (2.5). M po ee e aece cxoo aa.

cxoa aaa ocmea aaa C = c1x1 + c2x2 +...+ cnxn max, C* = b1y1+ b1y1+ + bmym min, n m aijxi bj, j = 1,...,m. aij y ci, i = 1,...,n.

j i = 1 j = xi 0, (i = 1,..., n). yj 0, (j = 1,..., m).

oceo opo aa ec cxoa, .e. a ye oopoc, yaae aa opay aoocey apy aa. Ha o pepe ae cye oe ey aaa aooceo ap. B acoc:

oeco epeex oo pao oecy opae pyo;

oo aa yoa oe aca, pyo - a;

eop oeo yoaa oo aa opeee opae opo aoopo.

aae aa opay a aaey cepy apy xaoocex aa.

Baece opoo pepa paccop ceyy (aaey ecepo) apy aa xc a e aooce.

cxoa aaa ocmea aaa C = c1x1+ c2x2++ cnxn max, C* = b1y1+ b1y1++bmym min, n m aijxi = bj, j = 1,...,m. aij y ci, i = 1,...,n.

j i = 1 j = xi 0, (i = 1,..., n).

a ec "cxoa" acaa ope ocoo aa . Bao a e oe, o oceo aae e opae a eopaeoc epeex.

Oaaec, ec pa aaa peaac cec-eoo, o peee oceo aa axoc cec-ae, cooecye oaoy pee. A eo ocee cpoe (cpoe ) m coax, cooecyx epee, oope coca epoaa ac (ec oo o p o e, o oo ae). p o aoy eey ocee cpo ceye pa oe, c oop epeea cooecyeo coa xo yoa. Hapep, ooex epeex pae poco o.

Baeee aee pee oceo aa, oopoe aaec oceo oeo aoc pecypco, paaec cey opya * Cmax Cmin * y* =,( j = 1,m); xi =,(i = 1,n).

j bj ci To ec a e ocex oeo oe caa, a coo npuuumeo ueumc onmuaoe aeue yuoaa, ecu ueeum euuy coomemcmyeo opaueu a euuy. a opa ee paecy aoc. a aee pee oceo aa oe cea o o o ae pecypc eoxoo pee ceo apaa, ae opae ac a o opao c.

pep 2.4 aa pepa 2.C = 9x1 + 10x2 + 16x3 max, 18x1 + 15x2 + 12x3 360, 6x1 + 4x2 + 8x3 192, 5x1 + 3x2 + 3x3 180.

BOCTBEHHO BETC CEA ~ C = 360y1 + 192y2 + 180y3 min, 18y1 + 6y2 + 5y 9, 15y1 + 4y2 + 3y3 10, 12y1 + 8y2 + 3y3 16.

* * * Ee peee axo ae 2.3: y1 = 2 / 9, y2 = 5/ 3, y3 = 0. oca opae , o a oo a pex opae oc c ooc o paeca, ae yoao coaa. o e acaec e ocex oeo, o oo caa, o acaoe aee Cmax oe yeeo a 2/9, ec epoe opaee ye c 360 o 361. - a 5/3, ec opoe - c 192 o 193. o oe cooao p p pee o o ao pecypco pee ceo ceye yea.

pep 2.5 aa pepa 2.C = 9x1 + 10x2 + 16x3 - 200x8 max, p opaex:

18x1 + 15x2 + 12x + x4 = 360, 6x1 + 4x2 + 8x + x = 192, 3 15*x1 + 10*x2 - x6 + x7 = 100, xi 0, (i=1,7 ).

oceo e ec ceya aaa C = 360y1+192y2 + 100y3 min, p opaex:

18y1 + 6y2 + 15y3 360, 15y1 + 4y2 + 10y3 192, 12y1 + 8y2 100, y1 0, y2 0, -y3 0, y3 -200.

* * * TA 2.7 HAXOM EE PEEHE y1 = 4 / 3, y2 = 0, y3 = -1.

HTEPPETA OEHHOO PETATA A BOCTBEHHX OEHO BAHOCT OPAHEH CXOHO AA AHAOHA PEEM PMEP.

2.6 Peee aa oee. cooae ocex oeo B pe cyae paec oae aa oa oaac a o, opae e epaec co ao "", a eey y peyec e acpoa, a paccopex aaax, a poa.

Paccop pep.

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