home Other disciplines books Ically mathematical programuvannya - Nakonechny S.І.

Ically mathematical programuvannya - Nakonechny S.І.

2. OBJECTIVE ROZDІL ZAGALNA LІNІYNOGO PROGRAMUVANNYA TA DEYAKІ W METODІV ЇЇ ROZV'YAZUVANNYA

"It's all there, if there is no deception:
And hell, and love, and fear, and flowers. "

Griboyedov

2.1. Butt pobudovi ekonomіko-ically mathematical models ekonomіchnih protsesіv that yavisch

In danomu rozdіlі rozglyadaєtsya nayprostіshy type tasks. Yak Bulo z'yasovano ranіshe, takі zadachі Je static. In їh modelі vikoristovuyut determіnovanі danі that lіnіynі funktsії for inventory vzaєmozv'yazkіv mіzh Some of the elements. Rozv'yazok znahoditsya on deyakіy neperervnіy mnozhinі. Navedemo kіlka rozglyanutih vische ically mathematical types of problems programuvannya, sformulovanih in termіnah lіnіynogo programuvannya.

Task viznachennya optimal plan virobnitstva: for deyakoї virobnichoї System (shop, pіdpriєmstva, the Branch) neobhіdno viznachiti Plan Key infrastructure n vidіv produktsії X = (x 1, x 2, ..., x n) for a method of minds naykraschogo vikoristannya її nayavnih resursіv. In protsesі virobnitstva zadіyanі m resursіv: sirovina, trudovі resources tehnіchne equipping toscho. Vіdomі zagalnі reserves resursіv , Normalized vitrat i-th share on virobnitstvo odinitsі j -oї produktsії that Prybutok s odinitsі j -oї realіzovanoї produktsії .

Kriterіy optimalnostі: maximum pributku.

Poznachimo by x 1, x 2, ..., x n obsyagi virobnitstva vіdpovіdno Perche, another t i. E. Vidіv produktsії.

Oskіlki odinitsyu produktsії at 1st sight vitrachaєtsya resource Perche mind, then virobnitstvo Perche mind produktsії obsyagom x 1 neobhіdno vitratiti and 11 x 1 tsogo resource. In another kind of produktsії obsyagom x 2 vitrati Perche resource dorіvnyuvatimut and 12 x 2 i, etc. At virobnitstvo vsіh vidіv produktsії bude vikoristano Taqiy obsyag Perche resource:.. And 11 x 1 + 12 x 2 + ... + a 1 n xn. Tsya magnitude Got no perevischuvati nayavnogo obsyagu Perche resource - b 1. Otzhe, obmezhennya schodo vikoristannya Perche resource matim viglyad: a 11 x 1 + x 2 + 12 ... + a 1 n xn ≤ b 1. Analogіchno zapisuyut obmezhennya stosovno vikoristannya vsіh іnshih virobnichih resursіv. Prybutok od realіzatsії vigotovlenoї produktsії vsіh vidіv stanovitime: 1 x 1 + x 2 2 + ... + c n xn.

Zagalom lіnіyna ekonomіko-ically mathematical model danoї zadachі matim viglyad:

of minds:

.

Ically mathematical model virobnichoї zadachі Mauger Buti zastosovana for rіznih ekonomіchnih tasks de vinikaє problem Vibor naykraschogo varіanta rozpodіlu obmezhenoї kіlkostі resursіv, Hoca Persha s Look zdavatisya Mauger, scho statement zadachі not stosuєtsya virobnichih protsesіv. Navedemo kіlka prikladіv virobnichih specific tasks.

FIRMA Got 1 million UAH obіgovih koshtіv. Vіdomі vitrati pennies in cutaneous mіsyatsі and takozh obov'yazkovі zalishki obіgovih koshtіv on kіnets mіsyatsya skin. Peredbachaєtsya takozh, scho for uspіshnogo funktsіonuvannya FIRMA vitrachatime uniquely Mensch scrip, nіzh 1 million UAH. Otzhe, Rasht koshtіv mozhna nadavati in credit. Neobhіdno viznachiti optimally rozpodіl obіgovih koshtіv protyagom quarter for dosyagnennya maximum pributku for interest rates, Yakscho vіdomі vitrati that require in reserves:

1.01 -31.01: vitrati - 80,000 USD; neobhіdny stock on 31.01 - 300 000 USD;

1.02 -28.02: vitrati - UAH 30 000; neobhіdny stock on 28.02 - 200 000 USD;

1.03 -31.03: vitrati - 50 000 UAH; neobhіdny stock on 31.03 - 190 000 UAH.

Credit termіnom 1 mіsyats daє 2% pributku, termіnom mіsyatsі 2 - 5% and 3 termіnom mіsyatsі - 8%.

Vvazhatimemo, scho loans nadayutsya Perche of skin mіsyatsya i pogashayutsya takozh Perche number vіdpovіdnogo mіsyatsya.

Pobudova ekonomіko-matematichnoї modelі

Loans termіnom one mіsyats mozhna nadavati skin mіsyatsya protyagom quarter, to poznachimo through x 11 scrip loan, scho Nada one mіsyats s 1.01 analogіchno x 12 x 13 - sumi odnomіsyachnih kreditіv scho nadanі vіdpovіdno to another that a third of mіsyatsyah.

Loans termіnom two mіsyatsі protyagom Persha quarter mozhna nadavati deprivation in Persha i mіsyatsyah another, to poznachimo through x 21 scrip loan, scho Nada two mіsyatsі in sіchnі x 22 - scrip credit, scho Nada in lyutomu two mіsyatsі. Nareshtі, credit for three mіsyatsі mozhna Nadachi deprivation once іz 1.01 yogo poznachimo through x 31.

Rozglyanemo situatsіyu on the cob Perche mіsyatsya quarter: Pochatkova suma 1 million UAH vitrachatimetsya on vkladennya koshtіv at All types kreditіv to be required for obіgovih Costa gospodarskoї dіyalnostі fіrmi stanovitimut 80 thousand UAH, and on kіnets mіsyatsya FIRMA bazhaє mother obsyagom reserve 300 000 UAH. Otzhe, vikoristannya koshtіv in sіchnі mozhna opisati in modelі as follows:

.

Nayavnі in Costa kіntsі mіsyatsya (okrіm reserve) viznachayutsya of the formula:

On the cob another mіsyatsya scrip S 1 Mauger nadavatisya credit, ale deprivation dvoh vidіv that Got zabezpechuvati vitrati dіyalnostі. Odnochasno on the cob another mіsyatsya povertayutsya Costa, scho Yea odnomіsyachny percent of the loan, yaky Bulo nada in sіchnі. Vrahovuyuchi neobhіdnіst reserve to another kіnets mіsyatsya, maєmo TAKE obmezhennya schodo vikoristannya koshtіv in lyutomu:

.

and Dire naprikіntsі obsyag nayavnih koshtіv stanovitime:

.

Analogіchno zapishemo vikoristannya koshtіv in bereznі:

.

Zagalna koshtіv scrip, yak otrimanih nadanі percent for loans, dorіvnyuvatime:

.

Zagalom ically mathematical model tsієї zadachі Got viglyad:

of minds:

On rinok postavlyaєtsya kartoplya s troh fermerskih Gospodarstwa for tsіnami vіdpovіdno 80, 75 that 65 kopecks. 1 kg. On zavantazhennya 1 m kartoplі in Gospodarstwa vіdpovіdno vitrachaєtsya 1, 6 is the 5 hvilin. Zamovleno kartoplі 12 t, i for svoєchasnoї delivery neobhіdno, dwellers on її zavantazhennya vitrachalosya not bіlshe forty hvilin. Potrіbno viznachiti, s yakih fermerskih Gospodarstwa i in yakіy kіlkostі neobhіdno dostavlyati kartoplyu, dwellers Total value zakupіvlі Bula mіnіmalnoyu, Yakscho farmers mozhut vidіliti for sale vіdpovіdno 10, 6 t 8 that kartoplі.

Pobudova ekonomіko-matematichnoї modelі

Poznachimo: x 1 - Quantity kartoplі scho bude purchased from Perche gospodarstvі (t); x 2, x 3 - Quantity kartoplі, zakuplenoї vіdpovіdno another that third fermerіv (t).

Supply potrіbnoї kіlkostі kartoplі opisuєtsya rіvnіstyu:

.

the following obmezhennya opisuє vitrati hour to zavantazhennya produktsії:

.

obmezhennya schodo supply mozhlivostey produktsії s skin Gospodarstwa:

Vartіst produktsії scho zakupovuєtsya, viznachaєtsya yak scrip dobutkіv tsіni on vіdpovіdnі її obsyagi. Tsіni 1 m kartoplі vіdpovіdno dorіvnyuyut 800, 750 that 650 UAH per danih troh fermerskih Gospodarstwa. Otzhe, tsіlovu funktsіyu mozhna zapisati as follows:

.

Ekonomіko-ically mathematical model zadachі Got viglyad:

of minds:

The problem about "dієtu": deyaky ratsіon skladaєtsya s n vidіv produktіv. Vіdomі vartіst odinitsі skin product - , Quantity neobhіdnih organіzmu pozhivnih rechovin m the one required in the kozhnіy i- іy rechovinі - . In odinitsі j th product mіstitsya pozhivnoї rechovini i. Neobhіdno Know optimally ratsіon Scho vrahovuє vimogi zabezpechennya organіzmu neobhіdnoyu kіlkіstyu pozhivnih rechovin.

Kriterіy optimalnostі - mіnіmalna vartіst ratsіonu.

Poznachimo through x 1, x 2, ..., xn - Quantity vіdpovіdnogo j- th product view . System obmezhen opisuvatime zabezpechennya in ratsіonі kozhnoї pozhivnoї rechovini not nizhche zaznachenogo rіvnya . Ekonomіko-ically mathematical model matim viglyad:

of minds:

Analogіchno yak at virobnichіy zadachі, ekonomіko-ically mathematical model zadachі about "dієtu" (abo about Sumish) takozh Mauger opisuvati INSHI ekonomіchnі processes. According to the type of tasks sutі Tsey daє zmogu znahoditi optimally poєdnannya deyakogo set of components in odne tsіle, and Takeo poєdnannya Got zadovolnyati pevnі minds.

Standard peredbachaєtsya scho octane number of gasoline A-76 Got Booty not nizhchim 76 and vmіst sіrki - not bіlshim, nіzh 0.3%. For this vigotovlennya gasoline on zavodі vikoristovuyutsya Vier components. Danі about obsyagi zapasіv Components Connection, SSMSC zmіshuyutsya, їh vartostі, oktanovі number that vmіst sіrki navedenі Table. 2.1

table 2.1

TEHNІKO EKONOMІCHNІ POKAZNIKI-octane gasoline components

Neobhіdno viznachiti, skіlki skin tones component potrіbno vikoristati for dwellers otrimati 1,000 tons of gasoline A-76 of mіnіmalnoyu sobіvartіstyu.

Pobudova ekonomіko-matematichnoї modelі

Poznachimo by x j j th component Quantity in sumіshі (t), j = 1,2,3,4.

Pershe obmezhennya zabezpechuє potrіbne octane values ​​in sumіshі:

.

Vmіst sіrki in sumіshі Got no perevischuvati 0.3%:

.

and zagalna masa utvorenoї sumіshі Got dorіvnyuvati 1000 tonnes:

.

Vikoristannya skin component Got no perevischuvati yogo nayavnogo obsyagu:

Sobіvartіst sumіshі viznachaєtsya of the formula:

.

Zagalom, ekonomіko-ically mathematical model zadachі Got viglyad:

of minds:

.

The participant ekspeditsії skladaє backpack, i Yomou neobhіdno rozv'yazati power about those SSMSC take products. In rozporyadzhennі Je m'yaso, Boroshnev, Suha milk, Zukor. In backpack zalishilos for produktіv lishe 45 dm3 ob'єmu before neobhіdno Well, dwellers zagalna masa produktіv not perevischuvala 35 kg. Lіkar ekspeditsії rekomenduvav, dwellers m'yasa (for masoyu) Bulo bіlshe, nіzh Boroshnev prinaymnі udvіchі, Boroshnev not less then, nіzh milk and milk Hoca used in vіsіm razіv bіlshe, nіzh tsukru. Skіlki i yakih produktіv potrіbno poklast backpack, dwellers sumarno kalorіynіst produktіv Bula naybіlshoyu? Features produktіv navedenі Table. 2.2.

table 2.2

FEATURES PRODUKTІV

Pobudova ekonomіko-matematichnoї modelі

Poznachimo by x 1, x 2, x 3, x 4 masu (in kg) m'yasa, Boroshnev, i tsukru vіdpovіdno milk.

Sumarno masa produktіv Got no perevischuvati 35 kg:

.

and ob'єm, yaky stink toil zaymati - not bіlshe 45 dm3:

.

Krіm of toil vikonuvatisya spіvvіdnoshennya stosovno proportsіy for masoyu produktіv:

a) m'yasa prinaymnі udvіchі bіlshe, nіzh Boroshnev, otzhe:

;

b) not less then Boroshnev, nіzh milk: ;

c) milk used in Hoca vіsіm razіv bіlshe, nіzh tsukru: .

Kalorіynіst vsogo set produktіv mozhna viznachiti as follows:

.

Otzhe, ekonomіko-ically mathematical model zadachі Got viglyad:

of minds:

.

Transportation problem: rozglyadaєtsya m punktіv virobnitstva that n punktіv spozhivannya deyakoї odnorіdnoї produktsії. Vіdomі obsyagi virobnitstva produktsії in cutaneous i- th punktі - that require skin j th point spozhivannya - . Takozh a given matrix rozmіrnіstyu , Elements yakoї Je vartostyami transportuvannya odinitsі produktsії h i- second paragraph virobnitstva to j- th paragraph spozhivannya. Neobhіdno viznachiti optimalnі obsyagi transported produktsії s urahuvannyam nayavnostі produktsії in virobnikіv that zabezpechennya vimog spozhivachіv.

Kriterіy optimalnostі: mіnіmalna sumarno vartіst transported.

Poznachimo by x ij obsyag produktsії scho transported od i- govirobnika to j- gospozhivacha.

Mozhna vivezti od skin virobnika produktsіyu scho Yea in nayavnostі. Tom skin for i Got vikonuvatis Umov: . Zabezpechennya skin spozhivacha potrіbnoyu kіlkіstyu produktsії daє Umov: for dermal . Total value of the transported Je sumoyu dobutkіv . Neobhіdno, dwellers vikonuvalas Umov . Otzhe, ekonomіko-ically mathematical model transportnoї zadachі Got Taqiy viglyad:

of minds:

Yak i in dvoh poperednіh problems ically mathematical model transportnoї zadachі Mauger i vikoristovuvatis todі, if in postanovtsі zadachі Absent navіt zgadki transported about produktsії toscho.

Fermerske Gospodarstwa spetsіalіzuєtsya on viroschuvannі ozimoї pshenitsі i Got three dіlyanki zemlі Area S 1 = 40 n, S 2 n = 90, S = 55 m 3. Vrahovuyuchi nayavnu Quantity posіvnogo materіalu, Je mozhlivіst zasіyati entire Area of ​​winter wheat troh sortіv. Quantity pshenitsі class "Mironіvska-808" zabezpechit posіv 80 hectares, "Security hundred-1" - 60 hectares is the "Odeska - 51" - 45 hectares. Urozhaynіst class "Mironіvska-808" on danih dіlyankah becoming vіdpovіdno 41 kg / ha, 40 kg / ha, 46 kg / ha. Analogіchno for variety "inermis-1" maєmo: 38 kg / ha, 41 kg / ha, 45 kg / ha, and for "Odeskoї-51" - 30 kg / ha, 28 kg / ha, 40 kg / ha.

Neobhіdno rozpodіliti posіvny materіal for land dіlyankami so dwellers otrimati maximum yield (gross zbіr) ozimoї pshenitsі.

Pobudova ekonomіko-matematichnoї modelі

Poznachimo by x ij Area (ha) i -oї zemelnoї dіlyanki scho bude zasіyana j th grade ozimoї pshenitsі (domovimosya scho variety "Mironіvska-808", "beardless-1", "Odeska-51" vіdpovіdatimut numbers 1, 2 3) (i = 1, 2, 3), (j = 1, 2, 3).

Todі vikoristannya land ugіd opisuvatime obmezhen such a system:

;

;

.

Vikoristannya posіvnogo materіalu opisati formally be as follows:

;

;

.

Gross grain zbіr rozrahovuєtsya yak scrip dobutkіv Yields vіdpovіdnih sortіv pshenitsі on їh posіvnі ploschі, tobto:

Otzhe, ekonomіko-ically mathematical model zadachі zagalom bude mother viglyad:

of minds:

.

Navedenі ekonomіko butt-ically mathematical models ekonomіchnih protsesіv that yavisch Je The Teaching. Adekvatnі ekonomіko-matematichnі modelі will uniquely skladnіshimi.