F06_Simplex_1 - Sigrn B. Gunnhildardttir Tkni- og...

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Unformatted text preview: Sigrn B. Gunnhildardttir Tkni- og verkfrideild | T-403 Ageragreining Fyrirlestur 6: Simplex aferin Simplex aferin Almenn afer til a leysa lnuleg bestunarlkn ru af George Dantzig ri 1947 Er yfirleitt tfr me hjlp tlvu nema egar um mjg einfld lkn er a ra Lnulegar skorur G.r.f. a allar skorur su forminu: Hver skora skiptir lausnasvinu tvennt. Lglegt lausnasvi er skilgreint me safni af skorum. Lausnasvi lnulegum lknum eru v alltaf margfltungar (polyhedron). a 1 x 1 + a 2 x 2 + ... + a n x n b Lausnasvi 2 breytur (tvvtt lausnasvi): 3 breytur (rvtt lausnasvi): n breytur (n-vtt lausnasvi): Kpt lausnasvi (convex solution space) Skilgreining: Lausnasvi er kpt ef hgt er a taka hvaa punkta sem er innan svisins, draga beina lnu milli eirra og veri viss um a allir punktar eirri lnu eru einnig innan lausnasvisins. Punktarnir eirri lnu eru kpt, lnuleg samantekt punktunum tveimur. kpt, lnulegt lausnasvi ekki kpt lausnasvi kpt lausnasvi (en ekki lnulegt) tpunktar (extreme point) Skilgreining: Punktur lausnasvi er tpunktur ef ekki er hgt a skrifa hann sem kpta lnulega samantekt af rum punktum r lausnasvinu. Fullyringar: Ef lausnasvi inniheldur ekki endanlega lnu hefur lausnasvi amk einn tpunkt. Ef lausnasvi fyrir lnulegt lkan hefur tpunkt er hgt a finna bestu lausn tpunkti. Ef a er aeins ein besta lausn lnulegu lkani er s lausn tpunkti. Ef lnulegt lkan er me m har skorur og n breytur eru til mismunandi tpunktar. ( ) n m Jfnuhneppi & Gauss-Jordan eying Dmi: 2x + 3y = 4 4x + y = 5 x = 1.1 y = 0.6 Lausn: Dmi: 2x + 3y + 2z = 4 4x + y + 3z = 5 x = 1.1 y = 0.6 z = 0 Lausnir: x = -1 y = z = 3 x = 0 y = 0.2857 z = 1.5714 Frjlsar breytur: Setjum frjlsar breytur = 0 og leysum fyrir hinar breyturnar til a finna lausn. Eftir Gauss-Jordan eyingu: 1x + 0y = 1.1 0x + 1y = 0.6 Hugmyndin bak vi Simplex reikniriti max z = 3x 1 + 5x 2 mtt. x 1 4 2x 2 12 3x 1 + 2x 2 18 x 1 ,x 2 0 Hvar finnum vi bestu lausn lnulegu bestunarlkani? Ef a er til besta lausn lnulegu lkani getum vi fundi bestu lausn hornpunkti gjaldgenga lausnasvinu. Hugmyndin bak vi Simplex reikniriti max z = 3x 1 + 5x 2 mtt. x 1 4 2x 2 12 3x 1 + 2x 2 18 x 1 ,x 2 0 (0,0) (0,6) (2,6) (4,3) (4,0) Gjaldgengir hornpunktar og ngrannar: Tveir hornpunktar lnulegu bestunarlkani me n breytum eru ngrannar ef eir liggja smu (n-1) skorum....
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This note was uploaded on 12/08/2011 for the course ENGINEERIN 101 taught by Professor Siggabeinteins during the Spring '11 term at Uni. Reykjavik.

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F06_Simplex_1 - Sigrn B. Gunnhildardttir Tkni- og...

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