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Ically mathematical programuvannya - Nakonechny S.І.

2.4. Geometric іnterpretatsіya zadachі lіnіynogo programuvannya

Rozglyanemo on ploschinі H10x2 sumіsnu system lіnіynih nerіvnostey:

(2.9)

Cutaneous nerіvnіst tsієї Sistemi geometrically viznachaє pіvploschinu s boundary line ai 1 ai + 1 x 2 x 2 = bi (i = 1, 2, ..., m). Minds nevіd'єmnostі zmіnnih viznachayut pіvploschini s boundary line x 1 = 0 that x 2 = 0. sumіsna system to pіvploschini yak opuklі mnozhini, peretinayuchis, utvoryuyut spіlnu Chastain, scho Yea i opukloyu mnozhinoyu yavlyaє him sukupnіst tochok coordinates kozhnoї s yakih Je rozv'yazkom danoї system (Fig. 2.1).

Sukupnіst Tsikh tochok (rozv'yazkіv) nazivayut bagatokutnikom rozv'yazkіv, the range of permissible abo planіv (rozv'yazkіv) zadachі lіnynogo programuvannya. Tse Mauger Buti point (єdiny rozv'yazok) vіdrіzok, Promin, bagatokutnik, neobmezhena bagatokutna area.

Yakscho in sistemі obmezhen (2.9) bude three zmіnnih, the cutaneous nerіvnіst geometrically viznachatime pіvprostіr trivimіrnogo expanse, Boundary ploschinami MDM will be 1 x 1 ai ai + 2 x 2 + 3 x 3 ai = bi (i = 1, 2, ... , t), and the minds nevіd'єmnostі - pіvprostori s boundary ploschinami x j   = 0 (j   = 1, 2, 3), de i - number obmezhennya and j - zmіnnoї number. Yakscho system obmezhen sumіsna then tsі pіvprostori yak opuklі mnozhini, peretinayuchis, utvoryat in trivimіrnomu prostorі spіlnu Chastain, scho nazivaєtsya bagatogrannikom rozv'yazkіv. Vіn Mauger Buti point vіdrіzkom, Promen, bagatokutnikom, bagatogrannikom, bagatogrannoyu neobmezhenoyu area.

Let him have sistemі obmezhen (2.9) Quantity zmіnnih bіlsha, nіzh three: x 1, x 2, ... x n; todі cutaneous nerіvnіst viznachaє pіvprostіr n -vimіrnogo expanse s boundary gіperploschinoyu a i 1 x 1 x 2 + ai ai 2 + 3 x 3 + ... + ainxn = bi (i = 1, 2, ..., m). Obmezhennyu cutaneous form (2.9) vіdpovіdayut gіperploschina that napіvprostіr, yaky lezhit s one side tsієї gіperploschini and minds nevіd'єmnostі - pіvprostori s boundary gіperploschinami x j   = 0 (j   = 1, 2, 3, ..., n).

Yakscho system obmezhen sumіsna, over analogієyu s trivimіrnim spacious Won utvoryuє spіlnu Chastain in n -vimіrnomu prostorі - opukly bagatogrannik admissibility rozv'yazkіv.

Otzhe, geometric problem lіnіynogo programuvannya yavlyaє him vіdshukannya coordinate takoї point bagatogrannika rozv'yazkіv at pіdstanovtsі yakih at tsіlovu lіnіynu funktsіyu Stop nabiraє maximum (mіnіmalnogo) values, and the allowable rozv'yazkami Je usі point bagatogrannika rozv'yazkіv.

Tsіlovu funktsіyu

in paragraph -vimіrnomu prostorі mainly zmіnnih mozhna geometrically іnterpretuvati yak sіm'yu gіperploschin parallel, the provisions kozhnoї s yakih viznachaєtsya parameter value Z.

Rozglyanemo geometric іnterpretatsіyu zadachі lіnіynogo programuvannya on prikladі. Nekhay farmer priynyav rіshennya viroschuvati winter wheat i tsukrovі Buriak ploschі on 20 hectares, vіdvіvshi pid tsukrovі Buriak not less then yak 5 hectares. Tehnіko-ekonomіchnі pokazniki viroschuvannya Tsikh cultures maєmo in Table. 2.3:

table 2.3

POKAZNIKI VIROSCHUVANNYA SІLSKOGOSPODARSKIH CULTURES

Kriterієm optimalnostі Je maksimіzatsіya pributku.

Zapishemo ekonomіko-ically mathematical model of the structure virobnitstva ozimoї pshenitsі that tsukrovih buryakіv, vvіvshi takі poznachennya:

x 1 - Suka Ploscha posіvu ozimoї pshenitsі, ha;

x 2 - Suka Ploscha posіvu tsukrovih buryakіv, ha.

Task lіnіynogo programuvannya Got Taqiy viglyad:

max Z = 0,7 x 1 + x 2 (2.10)

of minds:

x 1 + x 2 ≤ 20; (2.11)

5 x 1 + 25 x 2 ≤ 270; (2.12)

1 x 2 + 8 x 2 ≤ 80; (2.13)

2 x ≥ 5; (2.14)

x 1 ≥ 0, x 2 ≥ 0. (2.15)

Geometric іnterpretatsіyu zadachі zobrazheno in Fig. 2.2.

Fig. 2.2. The range of permissible rozv'yazkіv zadachі

The range of permissible rozv'yazkіv tsієї zadachі dіstaєmo so. Cutaneous obmezhennya, napriklad x 1 + x 2 20 zadaє pіvploschinu s boundary lines x 1 + x 2 = 20. Buduєmo її i viznachaєmo pіvploschinu, yak opisuєtsya nerіvnіstyu x 1 + x 2 20. W tsієyu metoyu in nerіvnіst x 1 + x 2 20 pіdstavlyaєmo coordinates harakternoї point skazhіmo, x 1 = 0 i = 0 x 2 Perekonuєmosya, scho tsya point nalezhit pіvploschinі x 1 + x 2 20. Tsei fact in Fig. 2.2 іlyustruєmo vіdpovіdnoyu napryamlenoyu strіlkoyu. Analogіchno buduєmo pіvploschini, SSMSC vіdpovіdayut nerіvnostyam (2.11) - (2.15). In rezultatі peretinu Tsikh pіvploschin utvoryuєtsya the range of permissible rozv'yazkіv zadachі (Figure 2.2 -. Chotirikutnik ABCD). Tsіlova funktsіya Z = 0,7 x 1 + x 2 yavlyaє him sіm'yu parallel straight lines, cutaneous s yakih vіdpovіdaє Pevnyi values Z. Zokrema, Yakscho Z = 0, maєmo 0.7 x 1 + x 2 = 0. Tsya line passes through the ear of the coordinate system. Koli Z = 3,5, then a straight maєmo 0.7 x 1 + x 2 = 3.5.