Programacion Lineal

 

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]rfiroaoh`fl k`lgok

OWTFRGX5

Zgkordg, Horkfs H.@. L½ Z-20.0:=.207

 

]rfiroaoh`fl k`lgok, ?.- Wlo polodgráo gs joafso pfr sus dfs gspgh`ok`dodgs dg tortos5 ko torto dg hnfhfkotg y ko torto dg K`aøl. Ko torto dg hnfhfkotg rgqu`grg poro su gkobfroh`øl agd`f c`kf dg ozõhor y > nugvfs y glg ul prgh`f dg vglto dg > $. Ko torto dg K`aøl lghgs`to ? c`kf dg ozõhor y > nugvfs, y glg ul prgh`f dg vglto dg ?= $. Gl gk okaohàl kgs qugdobol ?= c`kfs dg ozõhor y ?2= nugvfs. o) ¶_uà hfab`loh`flgs dg gspgh`ok`dodgs pugdgl nohgr6 ]koltgo gk prfbkgao y rgprgsglto irïhoagltg gk hfleultf dg sfkuh`flgs. b) ¶Huïltos ul`dodgs dg hodo gspgh`ok`dod nol dg prfduh`rsg poro fbtglgr gk aoyfr `lirgsf pfr vgltos6

O) Xg tfao gl gl huglto huglto qug ^1‑luagr ^1‑luagrf f dg tortos tortos dg hnfhfkotg‑ hnfhfkotg‑ g P1‑luag P1‑luagrf rf dg tortos tortos dg k`aøl‑ Xg nohg ko tobko poro gstobkghgr kos rgstr`hh`flgs

Hnfhfkot g K`aøl

{

Ozõhor

Nugvfs

 =.  =.7 c

>

?c

>

?=

?2=

{

  ^ ≧= ,P ≧ =  ^ ≧ = , y ≧ = =.7 ^ + y ≤ ?=  1<  x + 2  y ≤ 2= > x + > y ≤ ?2=  x + y ≤ ?7

Ko julh`øl fbegvf qug rgprgsglto kfs `lirgsfs pfr vgltos y qug hfls`dgroldf kos rgstr`hh`flgs oltgr`fgrgs noy qug aox`a`zor5 z1 j(x,y)1>x+?=y

Xg rgprgsglto gk hfleultf dg rgstr`hh`flgs y ko rghto 4x+7y1=, qug do ko d`rghh`øl dg kos rghtos z1j(x.y)1>x+?=y B)

{

  x + y 1?7 1x+?=y (=,?=)5 j(=,?=)1?=.?=1?== (?=,7)5 j(?=,7)1>.?=+?=.71?8= (?7,=)5 j(?=,7)1>.?71?2=

Gk aoyfr `lirgsf sg fbglg hfl ?= tortos dg hnfhfkotg y 7 tortos dg k`aøl

2.- Wl hfagrh`oltg ohudg o h`grtf agrhodf o hfapror porhn`to hfl 7== $. Kg fjrghgl dfs pfs dg porhn`to5 kos dg pf O o =,7 $ gk ci y kos dg pf B o =,> $ gk ci. Xobgafs qug sfkf d`spflg gl su juriflgto dg gspoh`f poro trolspfrtor 9== ci dg porhn`tos hfaf aïx`af y qug p`glso vgldgr gk c`kf dg porhn`tos dg pf O o =,7> $ y gk dg pf B o =,0 $. ¶Huïltfs c`kfiroafs dg porhn`tos dg hodo pf dgbgrï hfapror poro fbtglgr bglgh`f aïx`af6 Xgol ^1‑c`kfs dg porhn`tos dg pf O‑ G P1‑c`kfs dg porhn`tos dg pf B‑ Kos rgstr`hh`flgs dgk prfbkgao sfl 5

{

{

  ^ ≧= ,P ≧ =   ^ ≧=, y≧=  ^ + y ≤ 9==  1<  x + y ≤ 9== =.7 x + =.>  y ≤ 7== 7 x + >  y ≤ 7===

Ko julh`øl qug do gk bglg`h`f, suegto o kos rgstr`hh`flgs oltgr`frgs oltgr`frgs gs5

 

z1 j(x,y)1(=.7>-=.7)^ j(x,y)1(=.7>-=.7)^ +(=.0-=.>)P1=.=>x+ +(=.0-=.>)P1=.=>x+=.?y =.?y

Xg rgprgsglto ko rghto =.=>x+=.?y1= 1< >x+?=y1= >x+?=y 1= 1< 4x+7y1= Gk aïx`af sg fbt`glg gl gk pultf dg `ltgrsghh`øl dg kos rghtos5

{

  x + y 1 9==  1  y 1 7===

Xg dgbgl hfapror 2==c dg t`pf O y 7==ci dg t`pf B 8.- Xg vo o friol`zor ulo pkolto dg ul tokkgr dg outfaøv`kgs dfldg vol o troboeor gkghtr`h`stos y aghïl`hfs. ]fr lghgs`dodgs dgk agrhodf, gs lghgsor`f qug noyo aoyfr f `iuok lõagrf dg aghïl`hfs qug dg gkghtr`h`stos y qug gk lõagrf dg aghïl`hfs lf supgrg ok dfbkg qug dg gkghtr`h`stos. Gl tftok noy d`spfl`bkgs 8= gkghtr`h`stos y 2= aghïl`hfs. Gk bglgh`f dg ko gaprgso pfr efrlodo gs dg ?7= $ pfr gkghtr`h`sto y ?2= $ pfr aghïl`hf. ¶Huïltfs troboeodfrgs dg hodo hkosg dgbgl gkgi`rsg poro fbtglgr gk aïx`af bglgh`f6

Dglftoldf pfr x1‑luagrf dg gkghtr`h`stos‑ g y 1‑luagrf dg aghïl`hfs‑ aghïl`hfs‑

{

 ^ gP sflgltgrfs = ≤ x ≤ 8= , = ≤ y ≤ 2= Kos rgstr`hh`flgs gstobkgh`dos sfl P ≧ ^  P ≤ 2 x Ko julh`øl bglgh`f qug noy qug aox`a`zor5 z1J(x,y)1?7=x+?2=y Xg rgprgsglto gk hfleultf dg rgstr`hh`flgs y ko rghto ?7=x+?2=y1= 1< 7x+4y1= qug do do ko d`rghh`øl dg kos rghtos z1j(x,y)1?7=x+?2=y z1j(x,y)1?7=x+?2=y

 

z1j(x,y)1?7=x+?2=y gl gk hfleultf dg rgstr`hh`flgs sfkf noy 4 pultfs (=,=) , (?=,?=) , (?=,2=) (?=,2=) , (2=,2=) Gk aïx`af sg okholzo gl gk (2=,2=) Xg gk`igl 2= gkghtr`h`stos y 2= aghïl`hfs

4. Trgs prfduhtfs sfl jobr`hodfs gl ulo aïqu`lo. Gk gapf dg prgporoh`øl dg hodo prfduhtf gs dg 2, 8 y 4 a`lutfs rgspghvoagltg, y gk gapf dg prfhgsf dg 8, 2 y ? a`lutfs. Gk bglgh`f opfrtodf pfr hodo prfduhtf gs rgspghvoagltg dg ?2, ?= y ?7 dfkorgo. Xg d`spflg dg ?== a`lutfs dg aïqu`lo y 2== poro ko prgporoh`øl dg ko a`sao. Dgtgra`lg gk lõagrf øpaf dg ul`dodgs o jobr`hor dg hodo orhukf.

]koltgoa`gltf dgk afdgkf5 ^`1 Wl`dodgs o jobr`hor dgk prfduhtf `(?, 2, 8) aox z1 ?2^?+?=^2+?7^8 Xuegtf o5 2^?+8^2+4^8;12== 8^?+2^2+?^8;1?== ^`4=

VVVV

?

8/?=

-?/7

4=

=

-?/?=

2/7

2=

]fdgafs fbsgrvor qug gk Ue-He gs pfs`vf gl su tftok`dod, kf qug qu`grg dgh`r qug, ngafs kkgiodf o ko sfkuh`øl øpao. U1>4=

 

^?12= ^21= ^814= ^41= ^71=

Xustuygldf kfs vokfrgs gl kos rgstr`hh`flgs5 2(2=)+8(=)+4(4=)12== qug gs ;12== 8(2=)+2(=)+?(4=)1?== qug gs ;1?== Gstf s`il`ho qug ko gaprgso uk`zo tfdfs kfs rghursfs d`spfl`bkgs, gl gstg hosf gk gapf(a`lutfs) dg prgporoh`øl y prfhgsf, pfr kf qug kos vor`obkgs dg nfkiuro tfaol ul vokfr dg hgrf gl ko sfkuh`øl øpao. Xustuygldf kfs vokfrgs gl ko julh`øl fbegvf5 aox z1 ?2(2=)+?=(=)+?7(4=) U1>4=

Gstf qu`grg dgh`r qug ko gaprgso poro fbtglgr gk aïx`af bglgh`f dgbg prfduh`r 2= ul`dodgs dgk prfduhtf ?, = ul`dodgs dgk prfduhtf 2 y 4= ul`dodgs dgk prfduhtf 8, dg gsto jfrao fbglg ul bglgh`f `iuok o >4= ul`dodgs aflgtor`os, gl gstg hosf gurfs.

7.- Wl gstud`oltg dg `ligl`gráo dg Norvord lghgs`to hfapkgtor ul tftok dg :7 hursfs poro iroduorsg. Gk lõagrf dg hursfs dg `ligl`gráo tgldrï qug sgr aoyfr qug f `iuok o 28. 2 8. Gk lõagrf dg hursfs oeglfs ok ïrgo dg `ligl`gráo dgbgrï sgr aoyfr qug f `iuok o 2=. Gk hursf dg `ligl`gráo prfagd`f rgqu`grg ul k`brf dg tgxtf qug hugsto $:= g `apk`ho ?2= nfros dg gstud`f. Kfs hursfs oeglfs ok ïrgo dg `ligl`gráo rgqu`grgl ul k`brf dg tgxtf qug hugsto $24 g `apk`hol 2== nfros dg gstud`f. Gk gstud`oltg d`spflg dg ul prgsupugstf dg $8,=== poro k`brfs. Jfraukg ul hfleultf dg ghuoh`flgs k`lgokgs poro dgshr`b`r ko julh`øl fbegvf y kos rgstr`hh`flgs. Wk`hg gk olïk`s`s irïhf poro glhfltror ko sfkuh`øl v`suok. ¶Hfl quà hfab`loh`øl dg hursfs dg `ligl`gráo y ftrfs oeglfs o gsto ïrgo sg a`l`a`zoráo gk lõagrf tftok dg nfros dg gstud`f6  @dglqug vor`obkgs dg nfkiuro f supgrïv`t O) Xg dglfto qug x1‑Hursfs dg `ligl`gráo‑ y1‑Hursfs oeglfs ok ïrgo dg `ligl`gráo‑ Julh`fl fbegvf a`l (?2=x+2==y)

 

Rgstr`hh`flgs Rgstr`hh`flgs5

Hursfs Lghgsor`fs poro iroduorsg5

^ + P 1 :7

Holdod dg Hursfs dg `ligl`gráo5

^ ≧ 28

Holdod dg Hursfs oeglfs o `ligl`gráo5

P ≧ 2=

]rgsupugstf dgk gstud`oltg5

B)

:=^ + 24P ≤ 8===

 

Ko sfkuh`øl v`suok sg glhfltroráo gl gk pultf G5 ^ 1 4=

P 1 27

H) Hfl Hfl kfs kfs vokf vokfrg rgss fbtg fbtgl` l`df dfss dg dg ^ 1 4=, 4=, P 1 27, 27, sg a`l` a`l`a` a`zo zorï rï kos kos nfr nfros os dg gst gstud ud`f `f,, tgl`gldf hfaf rgsuktodf 0>== nfros. D) Xg glg kos vor`obkgs dg supgrïv`t poro kos rgstr`hh`flgs rgspghtf o ko holdod dg hursfs dg `ligl`gráo (s?) y hursfs oeglfs o ko `ligl`gráo (s2). s? 1 4= ‘ 28 1 ?9 s2 1 27 ‘ 2= 1 7 :.- Wl jobr`holtg dg hfkfroltgs poro tgkos pugdg uk`zor dfs rutos dg prfhgsoa`gltf d`jgrgltgs poro gkobfror ul pf porhukor dg hfkfroltg. Ko ruto ? uk`zo ko prglso sghodfro O y ko ruto 2 uso ko prglso sghodfro B. Oabos rutos rgqu`grgl ko uk`zoh`øl dg ko a`sao lo dg agzhkodf poro rgvfkvgr kfs `lirgd`gltgs quáa`hfs dgk hfkfroltg oltgs dgk sghodf. Ko s`iu`gltg tobko augstro kfs rgqu`s`tfs dg gapf y kos hopoh`dodgs dg gstfs prfhgsfs5

 

Hodo c`kfiroaf dg hfkfroltg prfhgsodf gl ko ruto ? rgqu`grg 2= k`trfs dg prfduhtfs quáa`hfs, gl toltf qug hodo c`kfiroaf dg ltg prfhgsodf gl ko ruto 2 uk`zo sfkoagltg ?7 k`trfs. Ko d`jgrglh`o sg dgbg o kos d`sltos tosos dg prfduhh`øl dg kos prglsos sghodfros. ]fr hfls`iu`gltg, ko uk`dod pfr hodo c`kfiroaf prfhgsodf gl ko ruto ? gs dg $7= y gl ko ruto 2 gs dg $:7. Xg d`spflg dg ul tftok dg 47= k`trfs dg `lirgd`gltgs quáa`hfs. Jfraukg kos rgstr`hh`flgs y ko julh`øl fbegvf poro aox`a`zor kos uk`dodgs. Opk`qug gk aàtfdf irïhf dg prfiroaoh`øl k`lgok poro glhfltror ko sfkuh`øl øpao. @dglqug kos rgstr`hh`flgs hfl nfkiuro f supgrïv`t.

O) Zor`obkgs5 x 1 C`kfiroafs dg hfkfroltg prfhgsodos gl ko ruto ? y 1 Lõagrf tgkos prfhgsodos gl ko ruto 2 Julh`øl Fbegvf5 Aox (7=x + :7y) Rgstr`hh`flgs5 Agzhko

2x + 2y ≤ 74

Xghodfro O

:x ≤ ?2=

Xghodfro B

>y ≤ ?>=

]rfduhtfs _uáa`hfs

2=x + ?7y ≤ 47=

 

B)

Ko sfkuh`øl v`suok sg glhfltroráo gl gk pultf D5 x 1 4.7

y 1 22.7

Wk`dod Aïx`ao 1 ?:>9.7 h) Xg glg nfkiuro dg 08n gl ko rgstr`hh`øl dg ko sghodfro O y nfkiuro dg 22.7 k`trfs gl ko rgstr`hh`øl dg prfduhtfs quáa`hfs.