PROGRAMAÇÃO LINEAR

Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

❈✉rs♦ ❞❡ ▼❡str❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛

Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r

♣♦r

P❡❞r♦ P❛✉❧♦ ❙♦❛r❡s ❞❡ ❆♥❞r❛❞❡

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ❈❛r❧♦s ❇♦❝❦❡r ◆❡t♦

❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡✲ s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ Pr♦✲ ❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠á✲ t✐❝❛ ✲ ❈❈❊◆ ✲ ❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡s✲ tr❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❆❣♦st♦✴✷✵✶✹ ❏♦ã♦ P❡ss♦❛ ✲ P❇

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛

Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

❈✉rs♦ ❞❡ ▼❡str❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛

Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r

♣♦r

P❡❞r♦ P❛✉❧♦ ❙♦❛r❡s ❞❡ ❆♥❞r❛❞❡

❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ✲ ❈❈❊◆ ✲ ❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛✳ ❆♣r♦✈❛❞❛ ♣♦r✿

Pr♦❢✳ ❉r✳ ❈❛r❧♦s ❇♦❝❦❡r ◆❡t♦ ✲❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮

Pr♦❢✳ ❉r✳ ❋❧❛♥❦ ❉❛✈✐❞ ▼♦r❛✐s ❇❡③❡rr❛ ✲ ❯❋P❇

Pr♦❢✳ ❉r✳ ❙❡✈❡r✐♥♦ ❍♦rá❝✐♦ ❞❛ ❙✐❧✈❛ ✲ ❯❋❈●

❉❡❞✐❝❛tór✐❛

❊st❡ tr❛❜❛❧❤♦ é ✉♠ t❡①t♦ s♦❜r❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✳ ■♥✐❝✐❛❧♠❡♥t❡ tr❛t❛ ✉♠ ♣♦✉❝♦ ❞♦ ❝♦♥t❡①t♦ ❤✐stór✐❝♦✱ ❞❡ s✉❛ ✈❛st❛ ❛♣❧✐❝❛çã♦ ♥♦s ♠❛✐s ❞✐✈❡rs♦s r❛♠♦s ❞❛ ❝✐ê♥❝✐❛✱ ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ ❛ ♦t✐♠✐③❛r ✈❛r✐á✈❡✐s ❞❡ ❞❡❝✐sã♦✱ ❡①♣❧✐❝❛♥❞♦ ❞❡t❛❧❤❛❞❛✲ ♠❡♥t❡ ♦s ♣❛ss♦s ❛ s❡r❡♠ ❞❛❞♦s ♣❛r❛ r❡s♦❧✈❡r t❛✐s ♣r♦❜❧❡♠❛s✳ ❊①❡♠♣❧✐✜❝❛♠♦s t❛✐s s✐t✉❛çõ❡s ❝♦♠ r❡s♦❧✉çõ❡s ❛❧❣é❜r✐❝❛s ❡ ❣rá✜❝❛s ❞❡ ❛❧❣✉♥s ♣r♦❜❧❡♠❛s ♣❛r❛ ✉♠ ♠❡❧❤♦r ❡♥t❡♥❞✐♠❡♥t♦ ❞♦ ❧❡✐t♦r✳ ❉❡s❡♥✈♦❧✈❡♠♦s ♦ ▼ét♦❞♦ ❙✐♠♣❧❡① q✉❡ ♣❡r♠✐t❡ ❛ r❡s♦❧✉çã♦ ❞❡ss❡s ♣r♦❜❧❡♠❛s ❞❡ ♣r♦❣r❛♠❛çã♦ ❧✐♥❡❛r✳ ❚❛❧ ♠ét♦❞♦ ❢♦✐ ❝r✐❛❞♦ ♣❡❧♦ ❛♠❡r✐❝❛♥♦ ●❡✲ ♦r❣❡ ❉❛♥t③✐❣ ♣♦r ✈♦❧t❛ ❞❡ ✶✾✹✼✳ ❖ ♠ét♦❞♦ s✐♠♣❧❡① é ✉♠❛ té❝♥✐❝❛ ✉t✐❧✐③❛❞❛ ♣❛r❛ s❡ ❞❡t❡r♠✐♥❛r✱ ♥✉♠❡r✐❝❛♠❡♥t❡✱ ❛ s♦❧✉çã♦ ót✐♠❛ ❞❡ ✉♠ ♠♦❞❡❧♦ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✳

P❛❧❛✈r❛s✲❈❤❛✈❡✿ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✱ ▼ét♦❞♦ ❙✐♠♣❧❡①✱ ❖t✐♠✐③❛çã♦✱ ▼❛①✐♠✐✲ ③❛r ❡ ▼✐♥✐♠✐③❛r✳

❆❜str❛❝t

❚❤✐s ✇♦r❦ ✐s ❛ t❡①t ♦♥ ▲✐♥❡❛r Pr♦❣r❛♠♠✐♥❣✳ ❋✐rst ❝♦♠❡s ❛ ❜✐t ♦❢ ❤✐st♦r✐❝❛❧ ❝♦♥t❡①t✱ ✐ts ✇✐❞❡ ❛♣♣❧✐❝❛t✐♦♥ ✐♥ ✈❛r✐♦✉s ❜r❛♥❝❤❡s ♦❢ s❝✐❡♥❝❡✱ ✇✐t❤ r❡❣❛r❞ t♦ ♦♣t✐♠✐③✐♥❣ ❞❡❝✐s✐♦♥ ✈❛r✐❛❜❧❡s✱ ❡①♣❧❛✐♥✐♥❣ ✐♥ ❞❡t❛✐❧ t❤❡ st❡♣s t♦ ❜❡ t❛❦❡♥ t♦ s♦❧✈❡ s✉❝❤ ♣r♦❜❧❡♠s✳ ❲❡ ❡①❡♠♣❧✐❢② s✉❝❤ s✐t✉❛t✐♦♥s ✇✐t❤ ❛❧❣❡❜r❛✐❝ ❛♥❞ ❣r❛♣❤✐❝❛❧ r❡s♦❧✉t✐♦♥s ♦❢ s♦♠❡ ✐ss✉❡s ❢♦r ❛ ❜❡tt❡r ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ r❡❛❞❡r✳ ❉❡✈❡❧♦♣❡❞ t❤❡ ❙✐♠♣❧❡① ♠❡t❤♦❞ t❤❛t ❛❧❧♦✇s t❤❡ r❡s♦❧✉t✐♦♥ ♦❢ t❤❡s❡ ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ♣r♦❜❧❡♠s✳ ❚❤✐s ♠❡t❤♦❞ ✇❛s ❝r❡❛t❡❞ ❜② ❆♠❡r✐❝❛♥ ●❡♦r❣❡ ❉❛♥t③✐❣ ✐♥ ✶✾✹✼ ❚❤❡ s✐♠♣❧❡① ♠❡t❤♦❞ ✐s ❛ t❡❝❤♥✐q✉❡ ✉s❡❞ t♦ ❞❡t❡r♠✐♥❡ ♥✉♠❡r✐❝❛❧❧② t❤❡ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ♦❢ ❛ ▲✐♥❡❛r Pr♦❣r❛♠♠✐♥❣ ♠♦❞❡❧✳

❑❡②✇♦r❞s✿ ▲✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣✱ ❙✐♠♣❧❡① ▼❡t❤♦❞✱ ❖♣t✐♠✐③❛t✐♦♥✱ ▼❛①✐♠✐③❡✱ ▼✐♥✐♠✐③❡✳

✶ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ✶ ✶✳✶ ❉❡✜♥✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❋♦r♠✉❧❛çã♦ ▼❛t❡♠át✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✸ ❖♣❡r❛çõ❡s ❊❧❡♠❡♥t❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✹ ❊①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✹✳✶ Pr♦❜❧❡♠❛ ❞❡ ❊❝♦♥♦♠✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✹✳✷ Pr♦❜❧❡♠❛ ❞❡ ❚r❛♥s♣♦rt❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✺ ❈♦♥❝❧✉sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✷ ▼ét♦❞♦ ❙✐♠♣❧❡① ✾

✷✳✶ ❋✉♥❞❛♠❡♥t♦s ❚❡ór✐❝♦s ♣❛r❛ ♦ ▼ét♦❞♦ ❙✐♠♣❧❡① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✷ ❉❡✜♥✐çõ❡s ❡ ❚❡♦r❡♠❛s ❋✉♥❞❛♠❡♥t❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✸ ❉❡s❝r✐çã♦ ❞♦ ▼ét♦❞♦ ❙✐♠♣❧❡① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✸✳✶ ▼✉❞❛♥ç❛ ❞❡ ❇❛s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✸✳✷ ❊①❡♠♣❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✹ ▼ét♦❞♦ ❞❡ ❉✉❛s ❋❛s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✺ ❆♣❧✐❝❛çã♦ ❞♦ ▼ét♦❞♦ ❙✐♠♣❧❡① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

■♥tr♦❞✉çã♦

❖ ♦❜❥❡t✐✈♦ ❞❛ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r é ♦t✐♠✐③❛r ♣r♦❜❧❡♠❛s ❞❡ ❞❡❝✐sã♦✱ ✉s❛♥❞♦ ♣❛r❛ ✐ss♦ ♠♦❞❡❧♦s q✉❡ ❝❛r❛❝t❡r✐③❡♠ ✉♠❛ r❡❛❧✐❞❛❞❡✳ ❆ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r t♦r♥♦✉✲s❡ ❡♥✲ tã♦ ✉♠❛ ❢♦r♠❛ ❡✜❝✐❡♥t❡ ❞❡ r❡s♦❧✈❡r ✉♠❛ ✈❛st❛ ✈❛r✐❡❞❛❞❡ ❞❡ ♣r♦❜❧❡♠❛s q✉❡ ❡stã♦ ❛ss♦❝✐❛❞♦s ❛ ✐♥ú♠❡r♦s ❞♦♠í♥✐♦s✿ ◆♦ ♣❧❛♥❡❥❛♠❡♥t♦ ❞❛ ❞✐str✐❜✉✐çã♦ ❡ ♣r♦❞✉çã♦ ❞❡ ♣r♦❞✉t♦s✱ ♥❛s ❞❡❝✐sõ❡s ❧✐❣❛❞❛s às ♣♦❧ít✐❝❛s ♠✐❝r♦ ❡❝♦♥ô♠✐❝❛s ❡ ♠❛❝r♦ ❡❝♦♥ô♠✐❝❛s ♥❛ ❡str✉t✉r❛ ❣♦✈❡r♥❛♠❡♥t❛❧ ❞❡ ♣❛ís❡s ✭♣♦r ❡①❡♠♣❧♦ s✐t✉❛çõ❡s ♠✐❧✐t❛r❡s✮✱ ♥♦ ♣❧❛♥❡❥❛✲ ♠❡♥t♦ ❞❡ ❝✉rt♦ ♣r❛③♦ ❡♠ ❛♣r♦✈❡✐t❛♠❡♥t♦ ❤✐❞r♦❡❧étr✐❝♦s ❡ ✉t✐❧✐③❛çã♦ ❝♦♠♦ s✉❜✲r♦t✐♥❛s ♣❛r❛ ♦ s✉♣♦rt❡ ❞❡ t❛r❡❢❛s ❡s♣❡❝í✜❝❛s ❡♠ ❝ó❞✐❣♦s ❞❡ ♣r♦❣r❛♠❛çã♦ ❧✐♥❡❛r✳

❉❡st❛ ❢♦r♠❛ ❛ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r t❡♠✱ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ♥❛ ✐♥❞ústr✐❛✱ ♥❛ ❛❣r✐❝✉❧✲ t✉r❛✱ ♥❛ ❡❝♦♥♦♠✐❛✱ ❡♥tr❡ ♦✉tr❛s✳ ❆ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r é ✉♠❛ té❝♥✐❝❛ ❞❛ ▼❛t❡✲ ♠át✐❝❛ ❆♣❧✐❝❛❞❛ q✉❡ ❝♦♥st✐t✉✐ ✉♠ ❞♦s r❛♠♦s ❞❛ ■♥✈❡st✐❣❛çã♦ ❖♣❡r❛❝✐♦♥❛❧✱ ❡♠ q✉❡ ✧Pr♦❣r❛♠❛çã♦✧s❡ r❡❢❡r❡ à ♣r♦❣r❛♠❛çã♦ ❞❡ t❛r❡❢❛s ♦✉ ♣❧❛♥✐✜❝❛çã♦✱ ♥ã♦ s❡♥❞♦ ♣r♦✲ ❣r❛♠❛çã♦ ♥♦ s❡♥t✐❞♦ ❞❛ ■♥❢♦r♠át✐❝❛❀ ❡ ✧❧✐♥❡❛r✧❛❞✈é♠ ❞♦ ❢❛t♦ ❞❛s ❡①♣r❡ssõ❡s q✉❡ s❡ ✉t✐❧✐③❛♠ s❡r❡♠ ❧✐♥❡❛r❡s✳

❯♠ ♠♦❞❡❧♦ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r é ❝♦♥st✐t✉í❞♦ ♣♦r ✈❛r✐á✈❡✐s ❞❡ ❞❡❝✐sã♦✱ ❛s q✉❛✐s ♣r❡t❡♥❞❡♠♦s ❞❡t❡r♠✐♥❛r✱ ♣♦r ♦❜❥❡t✐✈♦✱ ✐st♦ é ♦ q✉❡ s❡ ♣r❡t❡♥❞❡ ♦t✐♠✐③❛r ❡ ♣♦r r❡str✐çõ❡s✱ q✉❡ tê♠ ❞❡ s❡r s❛t✐s❢❡✐t❛s✳ P❛r❛ s❡ ❞❡t❡r♠✐♥❛r ❛ s♦❧✉çã♦ ❞❡ ❢♦r♠❛ ❛ ❝✉♠♣r✐r ♦ ♦❜❥❡t✐✈♦ sã♦ ✉t✐❧✐③❛❞❛s ❞✐❢❡r❡♥t❡s ♣r♦❝❡❞✐♠❡♥t♦s✱ ❞♦s q✉❛✐s ♦ ♠ét♦❞♦ ❙✐♠♣❧❡①✱ q✉❡ é ♦ ♠❛✐s ❛♥t✐❣♦✱ ❡ ♦ ♠ét♦❞♦ Pr✐♠❛❧✲❉✉❛❧✳

❖ ♠ét♦❞♦ ❙✐♠♣❧❡① ❜❛s❡✐❛✲s❡ ♥✉♠ ❛❧❣♦r✐t♠♦ q✉❡ ♣❡r♠✐t❡ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✱ ❡♥q✉❛♥t♦ q✉❡ ♦ ♠ét♦❞♦ Pr✐♠❛❧✲❉✉❛❧ s❡ ❜❛s❡✐❛ ♥❛ r❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛✳

❖ ♣r♦❜❧❡♠❛ ❞❡ ♦t✐♠✐③❛r ✉♠❛ ❢✉♥çã♦ ❧✐♥❡❛r ❝♦♠❡ç♦✉ ❡♠ ✶✽✷✻ ❝♦♠ ♦s ❡st✉❞♦s ❞❡ ❋♦✉r✐❡r r❡❧❛t✐✈❛♠❡♥t❡ ❛♦s s✐st❡♠❛s ❧✐♥❡❛r❡s ❞❡ ✐♥❡q✉❛çõ❡s✱ ♠❛s só ❡♠ ✶✾✸✾ s❡ r❡✈❡❧♦✉ ❛ ✐♠♣♦rtâ♥❝✐❛ ♣rát✐❝❛ ❞❡st❡s ♣r♦❜❧❡♠❛s✱ q✉❛♥❞♦ ❑❛♥t♦r♦✈✐❝❤ ❝r✐♦✉ ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❛ s✉❛ s♦❧✉çã♦✳

●❡♦r❣❡ ❉❛♥t③✐❣ ❡ ♦✉tr♦s ❝✐❡♥t✐st❛s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❛ ❋♦rç❛ ❆ér❡❛ ❆♠❡r✐❝❛♥❛ ❛♣r❡s❡♥t❛r❛♠ ❡♠ ✶✾✹✼ ✉♠ ♠ét♦❞♦ ❞❡♥♦♠✐♥❛❞♦ ❙✐♠♣❧❡①✱ ❞❡ ❢♦r♠❛ ❛ r❡s♦❧✈❡r❡♠ ♦s ♣r♦❜❧❡♠❛s ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✱ ❝✉❥❛s ♣r✐♠❡✐r❛s ❣r❛♥❞❡s ❛♣❧✐❝❛çõ❡s ❢♦r❛♠ ♥♦ ❞♦♠í♥✐♦ ♠✐❧✐t❛r✳ ❆✐♥❞❛ ❡♠ ✶✾✹✼✱ ❑♦♦♣♠❛♥ ❞❡♠♦♥str♦✉ ❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❞❛ Pr♦❣r❛✲ ♠❛çã♦ ▲✐♥❡❛r ♣❛r❛ ❛ ❛♥á❧✐s❡ ❞❛ t❡♦r✐❛ ❡❝♦♥ô♠✐❝❛ ❝❧áss✐❝❛✳

❊♥tr❡ ✶✾✺✵ ❡ ✶✾✻✺ ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞♦s ♦s ❛❧❣♦r✐t♠♦s ♣❛r❛ ♦s ♠♦❞❡❧♦s ❞❡ Pr♦✲

❝♦♠ ❊❧❧✐s ❏♦❤♥s♦♥✱ ❡♠ r❡❧❛çã♦ ❛♦ ♠ét♦❞♦ Pr✐♠❛❧✲❉✉❛❧✱ ❡st❡ t❡✈❡ ♦r✐❣❡♠ ♥♦ ❛❧❣♦r✐t♠♦ ❞❡ ❍❛r♦❧❞ ❑✉❧♠ ❡ ❢♦✐ ✜♥❛❧✐③❛❞♦ ❝♦♠ ♦ ❛❧❣♦r✐t♠♦ ❞❛ ❝♦♥❞✐çã♦ ❞❡ ❉❡❧❜❡rt ❋✉❧❦❡rs♦♥ ❡♠ ✶✾✻✶✳

❖s ♣r♦❜❧❡♠❛s ❞❡ ❣❡stã♦ ♦r❣❛♥✐③❛❝✐♦♥❛❧ ❝♦♠❡ç❛r❛♠ ❛ s❡r r❡s♦❧✈✐❞♦s ❝♦♠ ❣r❛♥❞❡ ❡✜❝✐ê♥❝✐❛ ♣❡❧❛ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✱ ♦ q✉❡ ❧❡✈♦✉ ❛ q✉❡ ❛s ❣r❛♥❞❡s ♦r❣❛♥✐③❛çõ❡s ❝♦✲ ♠❡ç❛ss❡♠ ❛ ❞❛r ✐♠♣♦rtâ♥❝✐❛ ❛♦ tr❛❜❛❧❤♦ ❞♦s ♠❛t❡♠át✐❝♦s✱ ♦❧❤❛♥❞♦ ♣❛r❛ ❡st❡s ❝♦♠ ♦✉tr♦s ♦❧❤♦s✳

❈❛♣ít✉❧♦ ✶

Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r

✶✳✶ ❉❡✜♥✐çã♦

❆ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r é ✉♠❛ ❢❡rr❛♠❡♥t❛ ♠❛t❡♠át✐❝❛ q✉❡ ♣❡r♠✐t❡ ❡♥❝♦♥tr❛r ❛ s♦❧✉çã♦ ót✐♠❛ ♣❛r❛ ✉♠ ❝❡rt♦ t✐♣♦ ❞❡ ♣r♦❜❧❡♠❛✳ ◆❡st❡ ❝♦♥t❡①t♦✱ ♦ t❡r♠♦ ♣r♦❣r❛♠❛✲ çã♦✱ s✐❣♥✐✜❝❛ q✉❡ ❡①✐st❡ ♦ ♣❧❛♥❡❥❛♠❡♥t♦ ❞❡ ❛t✐✈✐❞❛❞❡s ❡ ♦ t❡r♠♦ ❧✐♥❡❛r s❡ r❡❢❡r❡ ❛ ❧✐♥❡❛r✐❞❛❞❡ ❞❛s ❡q✉❛çõ❡s ❞♦ ♣r♦❜❧❡♠❛✳ ❱❡r r❡❢❡rê♥❝✐❛ ❬✶❪✱ ❈❛♣✳ ✷ ♣á❣✐♥❛ ✸✶✳

P♦❞❡✲s❡ t❛♠❜é♠ ❞❡✜♥✐r Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ❝♦♠♦ ✉♠❛ sér✐❡ ❞❡ ♦♣❡r❛çõ❡s ♠❛t❡✲ ♠át✐❝❛s q✉❡ sã♦ ✉s❛❞❛s ♣❛r❛ ❞✐str✐❜✉✐r r❡❝✉rs♦s ❧✐♠✐t❛❞♦s s♦❜r❡ ♦♣❡r❛çõ❡s q✉❡ ❡①✐❣❡♠ ❛ s✉❛ ✉t✐❧✐③❛çã♦ s✐♠✉❧tâ♥❡❛ ❞❡ ✉♠❛ ❢♦r♠❛ ót✐♠❛ ♣❛r❛ ✉♠ ❞❛❞♦ ♦❜❥❡t✐✈♦ ú♥✐❝♦✳

❆ ♣r♦❣r❛♠❛çã♦ ▲✐♥❡❛r é✱ ♣♦rt❛♥t♦✱ ✉♠❛ ♣♦❞❡r♦s❛ ❢❡rr❛♠❡♥t❛ ❞❡ ♠♦❞❡❧❛çã♦ ♠❛✲ t❡♠át✐❝❛✳ ➱ ✉♠❛ té❝♥✐❝❛ ❞❡ ♦t✐♠✐③❛çã♦ ❝♦♠ ❛♣❧✐❝❛çõ❡s ❛♠♣❧❛s ❡ ❞✐✈❡rs✐✜❝❛❞❛s ❛♦ ♥í✈❡❧ ❞❡ ♣r♦❜❧❡♠❛s r❡❛✐s✳

❊♠ ❝❛s♦s r❡❛✐s✱ ♦ ❛♥❛❧✐st❛ t❡♠ ❞❡ ❢♦r♠✉❧❛r ♦ ♣r♦❜❧❡♠❛ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ❝♦♥s✐❣❛ ❛ ♦t✐♠✐③❛çã♦ ót✐♠❛ ♣❛r❛ s❡✉ ♦❜❥❡t✐✈♦✳ ◆♦ ❡♥t❛♥t♦✱ ♥♦t❛✲s❡ q✉❡✱ ❛ ♠♦❞❡❧❛çã♦ ❞❛ r❡❛❧✐❞❛❞❡ ♥ã♦ ❢á❝✐❧ ❛♣❡s❛r ❞❡ ♣❛r❡❝❡r ❡❧❡♠❡♥t❛r✳ ❖ ✈❛❧♦r ót✐♠♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ♥ã♦ é ♦❜r✐❣❛t♦r✐❛♠❡♥t❡ ♦ ✈❛❧♦r ót✐♠♦ ♥❛ r❡❛❧✐❞❛❞❡✳

P❛r❛ q✉❛❧q✉❡r q✉❡ s❡❥❛ ♦ s❡✉ ♠♦❞❡❧♦ ❡ ♦ ♣r♦❣r❛♠❛ tr❛❞✉t♦r ❞❛ r❡❛❧✐❞❛❞❡ ❛ ❛♥❛✲ ❧✐s❛r✱ ♦ ❛♥❛❧✐st❛ ❞❡✈❡ ❡st❛❜❡❧❡❝❡r ❛s três s❡❣✉✐♥t❡s ❝♦♥s✐❞❡r❛çõ❡s✿

• ❛ r❡s♣♦st❛ é ❝r❡❞✐tá✈❡❧ ♦✉ r❡❛❧✐③á✈❡❧❄

• ❛ r❡s♣♦st❛ é ót✐♠❛❄

• q✉ã♦ s❡♥sí✈❡❧ é ♦ ✈❛❧♦r ót✐♠♦ ❛♦s ♣❛râ♠❡tr♦s ❡ q✉❛❧ ❛ r❡s♣♦st❛ ❞♦ ✈❛❧♦r ót✐♠♦

às ✈❛r✐❛çõ❡s ❞❛s ✈❛r✐á✈❡✐s ✐♥❞❡♣❡♥❞❡♥t❡s❄

❉♦ ❛❝✐♠❛ ❡①♣♦st♦ ✜❝❛ ✐♠♣❧í❝✐t♦ q✉❡ ♣❛r❛ ♦t✐♠✐③❛r é ♥❡❝❡ssár✐♦ ✉♠❛ ❢❡rr❛♠❡♥t❛ ♣❛r❛ ♠♦❞❡❧❛r ❡ s♦❧✉❝✐♦♥❛r ♦ ♣r♦❜❧❡♠❛✳ ❯♠❛s ❞❡st❛s ❢❡rr❛♠❡♥t❛s é ❛ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✳

✶✳✷ ❋♦r♠✉❧❛çã♦ ▼❛t❡♠át✐❝❛

❆♦ ❢♦r♠✉❧❛r ✉♠ ♠♦❞❡❧♦ ❧✐♥❡❛r ❞❡✈❡ t❡r✲s❡ ♦ ❝✉✐❞❛❞♦ ❞❡ ❞✐st✐♥❣✉✐r s❡♠♣r❡ ❛s s❡❣✉✐♥t❡s ❢❛s❡s✿

• ■❞❡♥t✐✜❝❛çã♦ ❞❛s ✈❛r✐á✈❡✐s ❞❡ ❞❡❝✐sã♦❀

• ■❞❡♥t✐✜❝❛çã♦ ❞❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦❀

• ■❞❡♥t✐✜❝❛çã♦ ❞❛s r❡str✐çõ❡s❀

• ❋♦r♠✉❧❛çã♦ ♠❛t❡♠át✐❝❛✳

❉❡♣♦✐s ❞❡ s❡ t❡r ♦❜t✐❞♦ ❛ ❢♦r♠✉❧❛çã♦ ♠❛t❡♠át✐❝❛ é ❡♥tã♦ ♣♦ssí✈❡❧ ❞❡ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ❞❡ ♦t✐♠✐③❛çã♦✳

❖ ♠ét♦❞♦ ❞❡ ♣r♦❣r❛♠❛çã♦ ❧✐♥❡❛r ♣❡r♠✐t❡ ♦ r❡❝✉rs♦ ❞❛ ♠❡t♦❞♦❧♦❣✐❛ ❣rá✜❝❛ ❡ ❞❛ ♠❡t♦❞♦❧♦❣✐❛ ❛❧❣é❜r✐❝❛✱ ♠❛♥✉❛❧ ♦✉ ❝♦♠ r❡❝✉rs♦ ❞❡ ✉♠ ❝♦♠♣✉t❛❞♦r✳

P❛ss♦s ♣❛r❛ ❢♦r♠✉❧❛çã♦ ❞❡ ✉♠ PP▲

P♦❞❡♠♦s ❞❡❝♦♠♣♦r ♦ ♣r♦❝❡ss♦ ❞❡ ♦r❣❛♥✐③❛çã♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ❞❡ ♣r♦❣r❛♠❛çã♦ ❧✐♥❡❛r ♥❛s s❡❣✉✐♥t❡s ❡t❛♣❛s✿

• ❉❡✜♥✐çã♦ ❞❡ ❛t✐✈✐❞❛❞❡s

❆♣ós ❛ ❛♥❛❧✐s❡ ❞♦ ♣r♦❜❧❡♠❛✱ ❛s ❛t✐✈✐❞❛❞❡s q✉❡ ♦ ❝♦♠♣õ❡♠ sã♦ ❞❡✜♥✐❞❛s✳ ◆♦r✲ ♠❛❧♠❡♥t❡✱ ❛ss♦❝✐❛❞❛ ❛ ❝❛❞❛ ❛t✐✈✐❞❛❞❡ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ ❞❡✈❡ s❡r ❛❞♦✲ t❛❞❛✳

• ❉❡✜♥✐çã♦ ❞♦s r❡❝✉rs♦s

❈♦♥s✐❞❡r❛♥❞♦ ♦s ✐♥s✉♠♦s ❞✐s♣♦♥í✈❡✐s ❞❡♥tr♦ ❞❡ ❝❛❞❛ ❛t✐✈✐❞❛❞❡✱ ❞❡t❡r♠✐♥❛✲s❡ ♦s r❡❝✉rs♦s q✉❡ ❡stã♦ s❡♥❞♦ ✉s❛❞♦s ❡ ♣r♦❞✉③✐❞♦s ❡♠ ❝❛❞❛ ✉♠❛✳

• ❈á❧❝✉❧♦ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ✐♥s✉♠♦✴♣r♦❞✉çã♦

➱ ✐♥❞✐s♣❡♥sá✈❡❧ ❡st❛❜❡❧❡❝❡r ❝❧❛r❛♠❡♥t❡ ❝♦♠♦ ❛s ❛t✐✈✐❞❛❞❡s ❡ ♦s r❡❝✉rs♦s ❡s✲ tã♦ r❡❧❛❝✐♦♥❛❞♦s ❡♠ t❡r♠♦s ❞❡ r❡❝✉rs♦s ♥❡❝❡ssár✐♦s ♣♦r ✉♥✐❞❛❞❡ ❞❡ ❛t✐✈✐❞❛❞❡ ♣r♦❞✉③✐❞❛✳

• ❉❡t❡r♠✐♥❛çã♦ ❞❛s ❝♦♥❞✐çõ❡s ❡①t❡r♥❛s

❈♦♥s✐❞❡r❛♥❞♦ q✉❡ ♦s r❡❝✉rs♦s sã♦ ✐❧✐♠✐t❛❞♦s✱ ❝✉♠♣r❡ ❞❡t❡r♠✐♥❛r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝❛❞❛ r❡❝✉rs♦ ❞✐s♣♦♥í✈❡❧ ♣❛r❛ ♦ ♣r♦❝❡ss♦ ♠♦❞❡❧❛❞♦✳ ❊ss❛s sã♦ ❛s ❞❡♥♦♠✐♥❛✲ ❞❛s ❝♦♥❞✐çõ❡s ❡①t❡r♥❛s ❞♦ ♠♦❞❡❧♦✳

Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ❖♣❡r❛çõ❡s ❊❧❡♠❡♥t❛r❡s

• ❋♦r♠❛❧✐③❛çã♦ ❞♦ ♠♦❞❡❧♦

❈♦♥s✐st❡ ❡♠ ❛ss♦❝✐❛r q✉❛♥t✐❞❛❞❡s ♥ã♦ ♥❡❣❛t✐✈❛s x1, x2, ..., xn ❛ ❝❛❞❛ ✉♠❛ ❞❛s

❛t✐✈✐❞❛❞❡s✱ ❡s❝r❡✈❡r ❛s ❡q✉❛çõ❡s ❞❡ ❜❛❧❛♥❝❡❛♠❡♥t♦ ❡ ✐♥❞✐❝❛r ♦ ✉s♦ ❞❡ ❝❛❞❛ r❡❝✉rs♦✳

❆❣♦r❛✱ ♥❛ ♣♦ss❡ ❞❡ t♦❞❛ ❡ss❛ ✐♥❢♦r♠❛çã♦✱ é ♠❛✐s ❢á❝✐❧ ❞❡ t♦♠❛r ❞❡❝✐sõ❡s ❛❝❡rt❛❞❛s ♥♦ ❞♦♠í♥✐♦ ❞❛ ❣❡stã♦ ❡♠ ❞✐✈❡rs❛s ❛♣❧✐❝❛çõ❡s✱ t❛✐s ❝♦♠♦✿ ♣❧❛♥❡❥❛♠❡♥t♦ ❛❣r❡❣❛❞♦ ❞❡ ♣r♦❞✉çã♦✱ ❛♥á❧✐s❡ ❞❡ ♣r♦❞✉t✐✈✐❞❛❞❡ ❞❡ s❡r✈✐ç♦s✱ ♣❧❛♥❡❥❛♠❡♥t♦ ❞❡ ♣r♦❞✉t♦s✱ ♦t✐♠✐③❛✲ çã♦ ❞♦ ✢✉①♦ ♣r♦❞✉t✐✈♦✱ ♦t✐♠✐③❛çã♦ ❞♦ ♣r♦❝❡ss♦ ❞❡ ♣r♦❞✉çã♦✱ ❡♥tr❡ ♦✉tr❛s✳ ✱▼❛s ❛s ❛♣❧✐❝❛çõ❡s ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ♥ã♦ t❡r♠✐♥❛♠ ♣♦r ❛q✉✐✳ ❊st❛ ♣♦❞❡ t❛♠❜é♠ s❡r ❡①tr❡♠❛♠❡♥t❡ út✐❧ ❡♠ ár❡❛s t❛✐s ❝♦♠♦ ♠❡❞✐❝✐♥❛✱ ❛❣r✐❝✉❧t✉r❛✱ s❡t♦r ♠✐❧✐t❛r✱ r❡❞❡ ❞❡ tr❛♥s♣♦rt❡s✱ ❡t❝✳

✶✳✸ ❖♣❡r❛çõ❡s ❊❧❡♠❡♥t❛r❡s

❯♠ ♠❡s♠♦ ♠♦❞❡❧♦ ❞❡ ♣r♦❣r❛♠❛çã♦ ❧✐♥❡❛r✱ ♣♦❞❡✱ s❡♠ q✉❛❧q✉❡r ♣❡r❞❛ ♣❛r❛ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ♠❛t❡♠át✐❝❛s✱ s❡r r❡str✐t♦ ❡♠ ❝❛❞❛ ✉♠❛ ❞❛s ❢♦r♠❛s ❜ás✐❝❛s✳ ❊ss❡ ♣r♦✲ ❝❡ss♦ ❞❡ tr❛❞✉çã♦ é r❡❛❧✐③❛❞♦ ❛tr❛✈és ❞❛s s❡❣✉✐♥t❡s ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s✿

❖♣❡r❛çã♦ ✶✿ ♠✉❞❛♥ç❛ ♥♦ ❝r✐tér✐♦ ❞❡ ♦t✐♠✐③❛çã♦✱ ♦✉ s❡❥❛✱ tr❛♥s❢♦r♠❛çã♦ ❞❡ ♠❛✲ ①✐♠✐③❛çã♦ ♣❛r❛ ♠✐♥✐♠✐③❛çã♦ ❡ ✈✐❝❡ ✈❡rs❛✳ ❊ss❛ ♠✉❞❛♥ç❛ ♣♦❞❡ s❡r r❡❛❧✐③❛❞❛ ❛tr❛✈és ❞❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿

▼❛①✐♠✐③❛r (f(x))❝♦rr❡s♣♦♥❞❡ ❛ ▼✐♥✐♠✐③❛r (−f(x))✳

❡

▼✐♥✐♠✐③❛r(f(x))❝♦rr❡s♣♦♥❞❡ ❛ ▼❛①✐♠✐③❛r (−f(x))✳

❖♣❡r❛çã♦ ✷✿ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ❧✐✈r❡✱ ❡♠ ✈❛r✐á✈❡❧ ♥ã♦ ♥❡❣❛t✐✈❛✳ ◆❡ss❡ ❝❛s♦✱ ❛ ♠✉❞❛♥ç❛ ❡①✐❣✐rá ❛ s✉❜st✐t✉✐çã♦ ❞❛ ✈❛r✐á✈❡❧ ❡♠ tr❛♥s❢♦r♠❛çã♦ ♣♦r ❞✉❛s ✈❛r✐á✈❡✐s ❛✉①✐❧✐❛r❡s✱ ❛♠❜❛s ♠❛✐♦r❡s ♦✉ ✐❣✉❛✐s ❛ ③❡r♦✱ ♠❛s ❝✉❥❛ ❛ s♦♠❛ é ✐❣✉❛❧ ❛ ✈❛r✐á✈❡❧ ♦r✐❣✐♥❛❧✱ ♦✉ s❡❥❛✿

xn=x1n−x

2

n ❡x

1

n ≥0✱x

2

n ≥0✳

❖♣❡r❛çã♦ ✸✿ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❞❡s✐❣✉❛❧❞❛❞❡s ❡♠ ✐❣✉❛❧❞❛❞❡s ❡ ✈✐❝❡ ✈❡rs❛✳ ◆❡ss❛ s✐t✉❛çã♦✱ t❡♠♦s ❞♦✐s ❝❛s♦s ❛ ❡①❛♠✐♥❛r✿

• ❈❛s♦ ❞❡ tr❛♥s❢♦r♠❛çã♦ ❞❡ r❡str✐çõ❡s ❞❡ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❡♠ r❡str✐çõ❡s ❞❡

✐❣✉❛❧❞❛❞❡✳

❙✉♣♦♥❞♦ ❛ r❡str✐çã♦ q✉❡ s❡ s❡❣✉❡✿

x1+x2+...+xn≤b✳

P❛r❛ tr❛♥s❢♦r♠á✲❧❛ ❡♠ ✉♠❛ r❡str✐çã♦ ❞❡ ✐❣✉❛❧❞❛❞❡ ♣♦❞❡♠♦s ✐♥tr♦❞✉③✐r ✉♠❛ ✈❛r✐á✈❡❧ ❞❡ ❢♦❧❣❛ xn+1 ❝❛♣❛③ ❞❡ ✧❝♦♠♣❧❡t❛r✧❛ ❞❡s✐❣✉❛❧❞❛❞❡✱ ♦ q✉❡

♣❡r♠✐t❡ r❡♣r❡s❡♥t❛r ❛ r❡str✐çã♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

x1+x2+...+xn+xn+1 =b✱ ❝♦♠ xn+1 ≥0✳

• ❈❛s♦ ❞❡ tr❛♥s❢♦r♠❛çã♦ ❞❡ r❡str✐çõ❡s ❞❡ ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❡♠ r❡str✐çõ❡s ❞❡

✐❣✉❛❧❞❛❞❡✳

❙✉♣♦♥❞♦ ❛ r❡str✐çã♦ q✉❡ s❡ s❡❣✉❡✿

x1+x2+...+xn≥b✳

P❛r❛ tr❛♥s❢♦r♠á✲❧❛ ❡♠ ✉♠❛ r❡str✐çã♦ ❞❡ ✐❣✉❛❧❞❛❞❡ ♣♦❞❡♠♦s ✐♥tr♦❞✉③✐r ✉♠❛ ✈❛r✐á✈❡❧ ❞❡ ❢♦❧❣❛ ❝♦♠ ✈❛❧♦r ♥❡❣❛t✐✈♦ xn+1 ❝❛♣❛③ ❞❡ ✧❝♦♠♣❧❡t❛r✧❛

❞❡s✐❣✉❛❧❞❛❞❡✱ ♣❛ss❛♥❞♦ ❛ r❡♣r❡s❡♥t❛r ❛ r❡str✐çã♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

x1+x2+...+xn−xn+1 =b✱ ❝♦♠ xn+1 ≥0✳

✶✳✹ ❊①❡♠♣❧♦s

✶✳✹✳✶ Pr♦❜❧❡♠❛ ❞❡ ❊❝♦♥♦♠✐❛

❯♠ ❝♦♠❡r❝✐❛♥t❡ ✈❡♥❞❡ ❞♦✐s t✐♣♦s ❞❡ ❛rt✐❣♦s✱ ❆ ❡ ❇✳ ◆❛ ✈❡♥❞❛ ❞♦ ❛rt✐❣♦ ❆ t❡♠ ✉♠ ❧✉❝r♦ ❞❡ R$20,00 ♣♦r ✉♥✐❞❛❞❡ ❡ ♥❛ ✈❡♥❞❛ ❞♦ ❛rt✐❣♦ ❇✱ ✉♠ ❧✉❝r♦ ❞❡ R$30,00✳

❊♠ s❡✉ ❞❡♣ós✐t♦ só ❝❛❜❡♠100 ❛rt✐❣♦s ❡ s❛❜❡✲s❡ q✉❡ ♣♦r ❝♦♠♣r♦♠✐ss♦s ❥á ❛ss✉♠✐❞♦s ❡❧❡ ✈❡♥❞❡rá ♣❡❧♦ ♠❡♥♦s 15 ❛rt✐❣♦s ❞♦ t✐♣♦ ❆ ❡ 25 ❞♦ t✐♣♦ ❇✳ ❖ ❞✐str✐❜✉✐❞♦r ♣♦❞❡ ❡♥tr❡❣❛r ❛♦ ❝♦♠❡r❝✐❛♥t❡✱ ♥♦ ♠á①✐♠♦60❛rt✐❣♦s ❆ ❡50❛rt✐❣♦s ❇✳ ◗✉❛♥t♦s ❛rt✐❣♦s ❞❡ ❝❛❞❛ t✐♣♦ ❞❡✈❡rá ♦ ❝♦♠❡r❝✐❛♥t❡ ❡♥❝♦♠❡♥❞❛r ❛♦ ❞✐str✐❜✉✐❞♦r ♣❛r❛ q✉❡✱ s✉♣♦♥❞♦ q✉❡ ♦s ✈❡♥❞❛ t♦❞♦s✱ ♦❜t❡♥❤❛ ♦ ❧✉❝r♦ ♠á①✐♠♦❄

❘❡s♦❧✉çã♦✿ ❙❡❥❛ x ♦ ♥ú♠❡r♦ ❞❡ ❛rt✐❣♦s ❞♦ t✐♣♦ ❆ ❡ y ♦ ♥ú♠❡r♦ ❞❡ ❛rt✐❣♦s ❞♦

t✐♣♦ ❇ q✉❡ ❞❡✈❡♠ s❡r ❡♥❝♦♠❡♥❞❛❞♦s✳

❋✉♥çã♦ ❖❜❥❡t✐✈♦✿ ❙❡ ♣❛r❛ ❝❛❞❛ ❛rt✐❣♦ ❆ q✉❡ ✈❡♥❞❡ t❡♠ ✉♠ ❧✉❝r♦ ❞❡R$20,00 ❡ ♣❛r❛ ❝❛❞❛ ❛rt✐❣♦ ❇ t❡♠ ✉♠ ❧✉❝r♦ ❞❡ R$30,00 ✱ ♦ ❧✉❝r♦ t♦t❛❧ é ❞❛❞♦ ♣❡❧❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ L(x, y) = 20x+ 30y✳

Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ❊①❡♠♣❧♦s

❘❡str✐çõ❡s✿

❛✮ ❈♦♠♦ ❝❛❜❡♠ ♥♦ ♠á①✐♠♦ 100 ❛rt✐❣♦s✿

x+y= 100

❜✮ ❙❡rã♦ ✈❡♥❞✐❞♦s ♣❡❧♦ ♠❡♥♦s 15 ❛rt✐❣♦s ❆✿

x≥15

❝✮ ❙❡rã♦ ✈❡♥❞✐❞♦s ♣❡❧♦ ♠❡♥♦s 25 ❛rt✐❣♦s ❇✿

y≥25

❞✮ ❖ ❞✐str✐❜✉✐❞♦r ❡♥tr❡❣❛rá ♥♦ ♠á①✐♠♦ 60❛rt✐❣♦s ❆✿

x≤60

❡✮ ❖ ❞✐str✐❜✉✐❞♦r ❡♥tr❡❣❛rá ♥♦ ♠á①✐♠♦ 50❛rt✐❣♦s ❇✿

y≤50 ●rá✜❝♦✿

❆s r❡str✐çõ❡s ❞ã♦ ♦r✐❣❡♠ ❛♦ ♣♦❧í❣♦♥♦ ❝♦♥✈❡①♦ ❧✐♠✐t❛❞♦ ♣❡❧❛s r❡t❛s x+y = 100✱ x= 15✱ y= 25✱x= 60 ❡ y= 50✳

❋✐❣✉r❛ ✶✳✶✿ Pr♦❜❧❡♠❛ ❞❡ ❊❝♦♥♦♠✐❛

❆s ❝♦♦r❞❡♥❛❞❛s ❞♦s ✈ért✐❝❡s ❞♦ ♣♦❧í❣♦♥♦ r❡s✉❧t❛♥t❡ s❡ ❡♥❝♦♥tr❛♠ ❢❛❝✐❧♠❡♥t❡ r❡✲ s♦❧✈❡♥❞♦ ♦s ♣❛r❡s ❞❡ ❡q✉❛çõ❡s q✉❡ ❝♦rr❡s♣♦♥❞❡♠ ❛♦s ❧❛❞♦s q✉❡ ❞❡t❡r♠✐♥❛♠ ♦ ✈ért✐❝❡✳ ❆s ❝♦♦r❞❡♥❛❞❛s sã♦ (15,25),(15,50),(50,50),(60,40) ❡(60,25)✳

❱❛❧♦r ❞❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ ♥♦s ✈ért✐❝❡s✿

❱ért✐❝❡ ⑤ L= 20x+ 30y

✭✶✺✱✷✺✮ ⑤ 20.15 + 30.25 = 1050 ▼í♥✐♠♦ ✭✶✺✱✺✵✮ ⑤ 20.15 + 30.50 = 1800

✭✺✵✱✺✵✮ ⑤ 20.50 + 30.50 = 2500 ▼á①✐♠♦ ✭✻✵✱✹✵✮ ⑤ 20.60 + 30.40 = 2400

✭✻✵✱✷✺✮ ⑤ 20.60 + 30.25 = 1950 ❘❡s♣♦st❛ ❞♦ Pr♦❜❧❡♠❛✿

❖ ❝♦♠❡r❝✐❛♥t❡✱ ♣❛r❛ ♦❜t❡r ♦ ❧✉❝r♦ ♠á①✐♠♦ ♥❛s ❝♦♥❞✐çõ❡s ❞♦ ♣r♦❜❧❡♠❛✱ ❞❡✈❡rá ❡♥❝♦♠❡♥❞❛r50❛rt✐❣♦s ❞♦ t✐♣♦ ❆ ❡50❛rt✐❣♦s ❞♦ t✐♣♦ ❞♦ t✐♣♦ ❇✳ ❈♦♠ ✐ss♦✱ ✈❡♥❞❡♥❞♦ t♦❞♦s✱ t❡rá ✉♠ ❧✉❝r♦ ❞❡ 2500 r❡❛✐s✳

✶✳✹✳✷ Pr♦❜❧❡♠❛ ❞❡ ❚r❛♥s♣♦rt❡

❯♠❛ ✜r♠❛ ❝♦♠❡r❝✐❛❧ t❡♠ ✹✵ ✉♥✐❞❛❞❡s ❞❡ ♠❡r❝❛❞♦r✐❛ ♥♦ ❞❡♣ós✐t♦ D1 ❡ t❡♠ ✺✵

✉♥✐❞❛❞❡s ♥♦ ❞❡♣ós✐t♦ D2✳ ❉❡✈❡ ❡♥✈✐❛r ✸✵ ✉♥✐❞❛❞❡s ❛♦ ❝❧✐❡♥t❡ A ❡ ✹✵ ✉♥✐❞❛❞❡s ❛♦

❝❧✐❡♥t❡ B✳ ❖s ❣❛st♦s ❞❡ tr❛♥s♣♦rt❡ ♣♦r ✉♥✐❞❛❞❡ ❞❡ ♠❡r❝❛❞♦r✐❛ ❡stã♦ ✐♥❞✐❝❛❞♦s ♥♦

❡sq✉❡♠❛ ❛❜❛✐①♦✳ ❉❡ q✉❡ ♠❛♥❡✐r❛ ❞❡✈❡ ❡♥✈✐❛r ❡ss❛s ♠❡r❝❛❞♦r✐❛s ♣❛r❛ q✉❡ ♦ ❣❛st♦ ❝♦♠ tr❛♥s♣♦rt❡ s❡❥❛ ♠í♥✐♠♦❄

❋✐❣✉r❛ ✶✳✷✿ Pr♦❜❧❡♠❛ ❞❡ ❚r❛♥s♣♦rt❡

❘❡s♦❧✉çã♦✿

Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ❊①❡♠♣❧♦s

❙❡❥❛ x ❛ q✉❛♥t✐❞❛❞❡ q✉❡ ❞❡✈❡ ❡♥✈✐❛r ❛♦ ❝❧✐❡♥t❡ A ❞♦ ❞❡♣ós✐t♦ D1 ❡ y ❛ q✉❛♥t✐✲

❞❛❞❡ q✉❡ ❞❡✈❡ ❡♥✈✐❛r ❛♦ ❝❧✐❡♥t❡ B ❞♦ ♠❡s♠♦ ❞❡♣ós✐t♦ D1✳ ❆ss✐♠✱ (30−x) s❡rá ❛

q✉❛♥t✐❞❛❞❡ q✉❡ ❞❡✈❡ ❡♥✈✐❛r ❛♦ ❝❧✐❡♥t❡ A❞♦ ❞❡♣ós✐t♦D2 ❡(40−y)❛ q✉❡ ❞❡✈❡ ❡♥✈✐❛r

❛♦ ❝❧✐❡♥t❡ B ❞♦ ❞❡♣ós✐t♦ D2✳

❋✉♥çã♦ ❖❜❥❡t✐✈♦✿

❖ ❣❛st♦G ❞♦ tr❛♥s♣♦rt❡ s❡rá ❞❛❞♦ ♣♦r✿ G= 10x+ 14y+ 12.(30−x) + 15.(40−y)

G= 960−2x−y✭ ❢✉♥çã♦ à ♠✐♥✐♠✐③❛r ✮

❘❡str✐çõ❡s✿

❛✮ x≥0 ❜✮ y≥0 ❝✮ x≤30 ❞✮ y≤40 ❡✮ x+y≤40

❢✮ (30−x) + (40−y)≤50⇔x+y≥20 ●rá✜❝♦✿

❋✐❣✉r❛ ✶✳✸✿ Pr♦❜❧❡♠❛ ❞❡ ❚r❛♥s♣♦rt❡

❆s ❝♦♦r❞❡♥❛❞❛s ❞♦s ✈ért✐❝❡s sã♦ (0,20),(0,40),(20,0),(30,0) ❡(30,10)✳

❱❛❧♦r ❞❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ ♥♦s ✈ért✐❝❡s✿

❱ért✐❝❡ ⑤ G= 960−2x−y

✭✵✱✷✵✮ ⑤ 960−2.0−20 = 940 ▼á①✐♠♦ ✭✵✱✹✵✮ ⑤ 960−2.0−40 = 920

✭✷✵✱✵✮ ⑤ 960−2.20−0 = 920 ✭✸✵✱✵✮ ⑤ 960−2.30−0 = 940

✭✸✵✱✶✵✮ ⑤ 960−2.30−10 = 890 ▼í♥✐♠♦ ❘❡s♣♦st❛✿

❖ ❣❛st♦ ♠í♥✐♠♦ s❡ ♦❜t❡rá ❡♥✈✐❛♥❞♦ ✸✵ ✉♥✐❞❛❞❡s ❞❡ ♠❡r❝❛❞♦r✐❛ ❞❡D1 ❛A✱ ✶✵ ❞❡

D1 ❛B✱ ✸✵ ❞❡D2 ❛B ❡ ♥❡♥❤✉♠❛ ❞❡ D2 ❛A✳

✶✳✺ ❈♦♥❝❧✉sã♦

❆❞♠✐♥✐str❛r ❝♦♠ ❡✜❝✐ê♥❝✐❛ ♦s r❡❝✉rs♦s ❞✐s♣♦♥í✈❡✐s ♥❛ ❡♠♣r❡s❛✱ ❛tr❛✈és ❞♦ ♣❧❛♥❡✲ ❥❛♠❡♥t♦✱ ❝♦♥tr♦❧❡ ❡ ❡①❡❝✉çã♦ ❞❛s ❛t✐✈✐❞❛❞❡s r❡❧❛❝✐♦♥❛❞❛s ❛ ✉t✐❧✐③❛çã♦ ❞❡st❡s✱ é ❢❛t♦r ❢✉♥❞❛♠❡♥t❛❧ ♥❛ ❜✉s❝❛ ❞❛ ♦t✐♠✐③❛çã♦ ❞♦ r❡s✉❧t❛❞♦ ❣❧♦❜❛❧ ❞❛ ❡♠♣r❡s❛✳ ❆ ♣r♦❣r❛♠❛✲ çã♦ ❧✐♥❡❛r ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛s té❝♥✐❝❛s ❞❡ ♣❡sq✉✐s❛ ♦♣❡r❛❝✐♦♥❛❧✱ ♣❡r♠✐t❡♠ ✐❞❡♥t✐✜❝❛r ♦ r❡s✉❧t❛❞♦ ót✐♠♦✱ ❝♦♥s✐❞❡r❛♥❞♦ t♦❞❛s ❛s r❡str✐çõ❡s ✐♠♣♦st❛s ♥♦ ♠♦❞❡❧♦ ❛❞♦t❛❞♦✳ ❆ss✐♠✱ ♦ r❡s✉❧t❛❞♦ ót✐♠♦ ❡s♣❡r❛❞♦ é ♣♦ssí✈❡❧ ❛❝♦♥t❡❝❡r✱ ✈✐st♦ q✉❡ ♦s ✈ár✐♦s ❝❡♥ár✐♦s q✉❡ s❡rã♦ ❛♥❛❧✐s❛❞♦s tê♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❞❡✜♥✐r ❛ ♣♦❧ít✐❝❛ ❞❡ ❛çã♦ ❞❛ ♦r❣❛♥✐③❛çã♦✳

❈❛♣ít✉❧♦ ✷

▼ét♦❞♦ ❙✐♠♣❧❡①

❖ ▼ét♦❞♦ ❙✐♠♣❧❡① é ✉♠ ❛❧❣♦r✐t♠♦ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❛❝❤❛r✱ ❛❧❣❡❜r✐❝❛♠❡♥t❡✱ ❛ s♦❧✉✲ çã♦ ót✐♠❛ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✳ ❙❛❜❡✲s❡ q✉❡ ❛ s♦❧✉çã♦ ót✐♠❛ ❞❡ ✉♠ ♠♦❞❡❧♦ é ✉♠❛ s♦❧✉çã♦ ❜ás✐❝❛ ❞♦ s✐st❡♠❛✱ ♦✉ s❡❥❛✱ ✉♠ ♣♦♥t♦ ❡①tr❡♠♦ ❞♦ ♣♦❧í❣♦♥♦ ❣❡r❛❞♦ ♣❡❧❛s r❡str✐çõ❡s ❞♦ s✐st❡♠❛✳ ❖ ♠ét♦❞♦ s✐♠♣❧❡① ✈❡r✐✜❝❛ s❡ ❛ ♣r❡s❡♥t❡ s♦❧✉çã♦ é ót✐♠❛✳ ❙❡ ❢♦r ♦ ♣r♦❝❡ss♦ ❡stá ❡♥❝❡rr❛❞♦✳ ❙❡ ♥ã♦ ❢♦r ót✐♠❛✱ é ♣♦rq✉❡ ✉♠ ❞♦s ♣♦♥t♦s ❛❞❥❛❝❡♥t❡s ❢♦r♥❡❝❡ ✉♠ ✈❛❧♦r ♠❛✐♦r✭♦✉ ♠❡♥♦r✮ q✉❡ ♦ ✐♥✐❝✐❛❧✳ ◆❡st❡ ❝❛s♦✱ ♦ ♠ét♦❞♦ s✐♠♣❧❡① ❢❛③ ❡♥tã♦ ❛ ♠✉❞❛♥ç❛ ❞♦ ♣♦♥t♦ ❛ ✉♠ ♦✉tr♦ q✉❡ ♠❛✐s ❛✉♠❡♥t❡✭♦✉ ❞✐♠✐♥✉❛✮ ♦ ✈❛❧♦r ❞❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦✳ ❖ ♣r♦❝❡ss♦ ✜♥❛❧✐③❛ q✉❛♥❞♦ s❡ ♦❜té♠ ✉♠ ♣♦♥t♦ ❡①tr❡♠♦ t❛❧ q✉❡ t♦❞♦s ♦s ♦✉tr♦s ♣♦♥t♦s ❡①tr❡♠♦s ❢♦r♥❡ç❛♠ ✈❛❧♦r❡s ♠❡♥♦r❡s✭♦✉ ♠❛✐♦r❡s✮ ♣❛r❛ ❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦✳

◆♦ q✉❡ s❡❣✉❡✱ ❡st❛r❡♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ❞❡ ♣r♦❣r❛♠❛çã♦ ❧✐♥❡❛r ❞❛ ❢♦r♠❛

▼✐♥✐♠✐③❛rz =cx✱ s✉❥❡✐t♦ ❛✿ Ax=d✱ x≥0 ✭✷✳✶✮ ♦♥❞❡ ♦♥❞❡ A ⊂ _Rm×n _{é ✉♠❛ ♠❛tr✐③✱ x ∈}

Rn ❡ d ∈ _Rm _{sã♦ ✈❡t♦r❡s ❝♦❧✉♥❛s ❡}

c∈Rn _{é ✉♠ ✈❡t♦r ❧✐♥❤❛✳}

✷✳✶ ❋✉♥❞❛♠❡♥t♦s ❚❡ór✐❝♦s ♣❛r❛ ♦ ▼ét♦❞♦ ❙✐♠♣❧❡①

Rn ❡ d ∈ _Rm _{❡ A}

❢♦ss❡ ✉♠❛ ♠❛tr✐③ ✐♥✈❡rtí✈❡❧✱ ❡♥tã♦ ❛ s♦❧✉çã♦ s❡r✐❛ ❢❛❝✐❧♠❡♥t❡ ❞❡t❡r♠✐♥❛❞❛✳

P♦ré♠✱ s❡ ❞❛❞♦ ✉♠ s✐st❡♠❛Ax=d ♦♥❞❡✿

 



A∈Rm×n

d∈Rm_{, m≤n}

x∈Rn

❖✉ s❡❥❛✱ s✐st❡♠❛ é r❡t❛♥❣✉❧❛r✱ ❝♦♠♦ ❞❡t❡r♠✐♥❛r s♦❧✉çõ❡s ❞❡Ax =d❄

❖ s✐st❡♠❛ ❛❝✐♠❛ s❡♠♣r❡ t❡♠ s♦❧✉çã♦❄

❉❡✜♥✐çã♦ ✶ ❉✐③❡♠♦s q✉❡ ✉♠❛ ♠❛tr✐③ B é ✉♠❛ s✉❜♠❛tr✐③ ❜❛s❡ ❞❡As❡B é ❢♦r♠❛❞❛

♣♦r m ❝♦❧✉♥❛s ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡A✳

❖❜s❡r✈❡ q✉❡ s❡At❡♠m❝♦❧✉♥❛s ▲■✱ ❡♥tã♦ ❛ ♠❛tr✐③B ❢♦r♠❛❞❛ ♣♦r ❡ss❛s ❝♦❧✉♥❛s

é ✉♠❛ ❜❛s❡ ♣❛r❛ Rm✳

❉❡✜♥✐çã♦ ✷ ✭❱❛r✐á✈❡✐s ❜ás✐❝❛s ❡ ✈❛r✐á✈❡✐s ♥ã♦ ❜ás✐❝❛s✮ ❈♦♥s✐❞❡r❡ ♦ s✐st❡♠❛

Ax = d ❡ B ✉♠❛ s✉❜♠❛tr✐③ ❜❛s❡ ❞❡ A✳ ❆s ✈❛r✐á✈❡✐s ❛ss♦❝✐❛❞❛s à s✉❜♠❛tr✐③ B ∈

Rm×m sã♦ ❞❡♥♦♠✐♥❛❞❛s ✈❛r✐á✈❡✐s ❜ás✐❝❛s r❡❧❛t✐✈❛s ❛ ❜❛s❡ _B ❡ ❞❡♥♦t❛❞❛s ♣♦r _x

B =

(xB1, . . . , xBm)✳ ❆s ✈❛r✐á✈❡✐s r❡st❛♥t❡s sã♦ ❞❡♥♦♠✐♥❛❞❛s ❞❡ ♥ã♦ ❜ás✐❝❛s✳ ❉❡♥♦t❛♠♦s

t❛✐s ✈❛r✐á✈❡✐s ♣♦r xN ♦♥❞❡ N é ❛ s✉❜♠❛tr✐③ ❞❡ A ❢♦r♠❛❞❛ ♣❡❧❛s n −m ❝♦❧✉♥❛s

r❡st❛♥t❡s ❞❛ ♠❛tr✐③ A✳

❚❡♦r❡♠❛ ✶ ❙❡❥❛ ❛ ♠❛tr✐③ A∈Rm×n _{❝♦♠ m ≤n✳ ❙❡ ❛ ♠❛tr✐③ A ♣♦ss✉✐ ♠ ❝♦❧✉♥❛s}

a1, a2, ..., am ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ❡♥tã♦ ♣❛r❛ q✉❛❧q✉❡rd∈Rm ♦ s✐st❡♠❛Ax=

d t❡♠ ❛♦ ♠❡♥♦s ✉♠❛ s♦❧✉çã♦ ♥♦_Rn_✳

Pr♦✈❛✳

❙❡❥❛ ♦ s✐st❡♠❛Ax=d❡ s✉♣♦♥❤❛ q✉❡ ❡①tr❛✐✲s❡ ❞❡ A✉♠❛ s✉❜♠❛tr✐③ B ∈Rm×m_✳

P❡❧❛s ❞❡✜♥✐çõ❡s ❛♥t❡r✐♦r❡s ♣♦❞❡✲s❡ ❢❛③❡r ❛s s❡❣✉✐♥t❡s ♣❛rt✐çõ❡s ♥♦ s✐st❡♠❛Ax=d✿ A= [B;N]x=

xB

xN

▲♦❣♦✱ ♣♦❞❡✲s❡ ❡s❝r❡✈❡r✿

Ax=d ⇔[B;N]

xB

xN

=d⇔BxB+N xN =d✳

P♦rt❛♥t♦✱ ♦ s✐st❡♠❛Ax=d é ❡q✉✐✈❛❧❡♥t❡ ❛♦ s✐st❡♠❛✿ BxB+N xN =d

■st♦ ✐♠♣❧✐❝❛ q✉❡xB =B−1d−B−1N xN é ✉♠❛ ♣♦ssí✈❡❧ s♦❧✉çã♦ ❞❡ Ax=d✳

❉❡✜♥✐çã♦ ✸ ❙♦❧✉çã♦ ❜ás✐❝❛ ❞❡ Ax=d✿ ❙❡❥❛ ♦ s✐st❡♠❛Ax=d✳ ❯♠❛ s♦❧✉çã♦ x¯ ❞❡ Ax=d✱ é s♦❧✉çã♦ ❜ás✐❝❛✱ s❡ ❡ s♦♠❡♥t❡ s❡✱ xN = 0✱ ✐st♦ é✱ x¯B =B−1d✳

❉✐③❡♠♦s q✉❡ ✉♠ ✈❡t♦r x=

     x1 x2 ✳✳✳ xn     

∈Rn é ✉♠ ✈❡t♦r ♥ã♦✲♥❡❣❛t✐✈♦ s❡ q✉❛❧q✉❡r ✉♠❛ ❞❡

❉❡✜♥✐çõ❡s ❡ ❚❡♦r❡♠❛s ❢✉♥❞❛♠❡♥t❛✐s ▼ét♦❞♦ ❙✐♠♣❧❡①

❉❡✜♥✐çã♦ ✹ ❙♦❧✉çã♦ ❜ás✐❝❛ ❢❛❝tí✈❡❧✭✈✐á✈❡❧✮ ❉✐③❡♠♦s q✉❡ x¯ é s♦❧✉çã♦ ❜ás✐❝❛ ❢❛❝tí✈❡❧ ♣❛r❛ Ax=d s❡ x¯B =B−1d≥0 ❡ x¯N = 0✳

◆♦t❡ q✉❡✱ ♥❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✱x=

xB

xN

é ✉♠ ✈❡t♦r ♥ã♦ ♥❡❣❛t✐✈♦ ❡♠ q✉❡ ❛s ❝♦♠♣♦✲ ♥❡♥t❡s r❡❢❡r❡♥t❡s às ✈❛r✐á✈❡✐s ♥ã♦ ❜ás✐❝❛s sã♦ ♥✉❧❛s✳

✷✳✷ ❉❡✜♥✐çõ❡s ❡ ❚❡♦r❡♠❛s ❋✉♥❞❛♠❡♥t❛✐s

❙❡❥❛ ♦ ❝♦♥❥✉♥t♦S ={x∈Rn:Ax=d, x≥0}♦♥❞❡A∈Rm×n✱d∈Rm ❡x∈Rn

❝♦♠ m ≤n✳

❉❡✜♥✐çã♦ ✺ ❉✐③❡♠♦s q✉❡ x é ✉♠ ♣♦♥t♦ ❡①tr❡♠♦ ❞❡ S s❡ ♣♦ss✉✐r n−m ✈❛r✐á✈❡✐s

♥✉❧❛s✳

❚❡♦r❡♠❛ ✷ ❖ ❝♦♥❥✉♥t♦S✱ ❞❡ t♦❞❛s ❛s s♦❧✉çõ❡s ❢❛❝tí✈❡✐s ❞♦ ♠♦❞❡❧♦ ❞❡ ♣r♦❣r❛♠❛çã♦

❧✐♥❡❛r Ax=d✱ é ✉♠ ❝♦♥❥✉♥t♦ ❝♦♥✈❡①♦✳

Pr♦✈❛✳ ❙❡❥❛♠ x1

, x2

∈ ❙ ❡ λ ∈ [0,1]✳ ▼♦str❛r❡♠♦s q✉❡ A(λx1

+ (1−λ)x2

) = d ❡ q✉❡ λx1

+ (1−λ)x2

≥0✳ ❉❡ ❢❛t♦✱ s❡ x1

∈S ❡ x2

∈S ❡♥tã♦ Ax1

=d ❡ Ax2

=d✳ ❆ss✐♠✱ A(λx1

+ (1−λ)x2

) =λAx1

+ (1−λ)Ax2

=λd+ (1−λ)d =d.

❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ x1

≥0 ❡ x2

≥0✱ s❡❣✉❡ q✉❡ λx1

≥0 ❡ (1−λ)x2

≥0✳ ❈♦♥s❡q✉❡♥✲ t❡♠❡♥t❡✱ λx1

+ (1−λ)x2

≥0✳

❙❡❣✉❡✱ ♣♦rt❛♥t♦✱ q✉❡λx1

+ (1−λ)x2

∈S ❡ S é ❝♦♥✈❡①♦✳

❚❡♦r❡♠❛ ✸ ❚♦❞❛ s♦❧✉çã♦ ❜ás✐❝❛ ❞♦ s✐st❡♠❛Ax=dé ✉♠ ♣♦♥t♦ ❡①tr❡♠♦ ❞♦ ❝♦♥❥✉♥t♦

❞❡ s♦❧✉çõ❡s ❢❛❝tí✈❡✐s S✳

Pr♦✈❛✳

❙❡❥❛x¯ ✉♠❛ s♦❧✉çã♦ ❜ás✐❝❛ ❛ss♦❝✐❛❞❛ ❛ ✉♠❛ s✉❜♠❛tr✐③ ❜❛s❡ B ∈Rm×m_✳

❊♥tã♦✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ s✉♣♦♥❤❛ q✉❡ ♦s í♥❞✐❝❡s ❞❛s ✈❛r✐á✈❡✐s ❜ás✐❝❛s s❡❥❛♠ ♦sm♣r✐♠❡✐r♦s✳ ❆ss✐♠✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡rx¯=

¯ xB ¯ xN

❝♦♠xN = (xm+1, . . . , xn) =

0✳

P♦r ❝♦♥tr❛❞✐çã♦✱ s✉♣♦♥❤❛ q✉❡x¯♥ã♦ s❡❥❛ ♣♦♥t♦ ❡①tr❡♠♦ ♦✉ ✈ért✐❝❡ ❞❡ S✳ ❊♥tã♦

❡①✐st❡♠ x¯1 _❡

¯

x2 _{♣❡rt❡♥❝❡♥t❡s ❛}

S t❛❧ q✉❡ x¯ = λx¯1

+ (1−λ)¯x2 _❝♦♠

λ ∈ (0,1) ❡ ¯

x1

6

= ¯x2_✳

❈♦♠♦ x¯N = 0✱ t❡♠♦s q✉❡✱ ♣❛r❛ i = m+ 1, ..., n✱ λx¯1i = 0 ❡ (1−λ)¯x

2

i = 0 ❡✱

❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ x¯1

i = 0 ❡ x¯

2

i = 0✱ ✐st♦ é✱ x¯

1

N = ¯x

2

N = 0✳ ❊✱ ❝♦♠♦ x¯

1

6

= ¯x2✱

x1

B 6=x

2

B✳

P♦r ♦✉tr♦ ❧❛❞♦✱ ♦ ❢❛t♦ ❞❡ q✉❡x¯1

∈ S ❡ x¯2

∈S ✐♠♣❧✐❝❛ q✉❡ Ax¯1

=d ❡ Ax¯2

=d✳

❊✱ ♣♦rt❛♥t♦✱

0 =d−d=Bx¯1

B−Bx¯

2

B =B(¯x

1

B−x¯

2

B).

▼❛sx¯1

B 6= ¯x

2

B ❡ ❡♥tã♦ x¯

1

B−x¯

2

B 6= 0✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡B é ♥ã♦ ✐♥✈❡rtí✈❡❧✱ ♦ q✉❡

é ✉♠❛ ❝♦♥tr❛❞✐çã♦✱ ✉♠❛ ✈❡③ q✉❡✱ ♣♦r ❤✐♣ót❡s❡✱ B é ✉♠❛ s✉❜♠❛tr✐③ ❜❛s❡✳

▲♦❣♦✱ t♦❞❛ s♦❧✉çã♦ ❜ás✐❝❛ ❞♦ s✐st❡♠❛ Ax =d é ✉♠ ♣♦♥t♦ ❡①tr❡♠♦ ❞♦ ❝♦♥❥✉♥t♦

❞❡ s♦❧✉çõ❡s ❢❛❝tí✈❡✐s S✳

❚❡♦r❡♠❛ ✹ ❙❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ♣r♦❣r❛♠❛çã♦ ❧✐♥❡❛r

▼✐♥✐♠✐③❛r z =cx✱ s✉❥❡✐t♦ ❛✿ Ax=d✱ x∈S

❛❞♠✐t✐r s♦❧✉çã♦ ót✐♠❛✱ ❡♥tã♦ ♣❡❧♦ ♠❡♥♦s ✉♠ ♣♦♥t♦ ❡①tr❡♠♦✭✈ért✐❝❡✮ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ✈✐á✈❡✐s é ✉♠❛ s♦❧✉çã♦ ót✐♠❛ ❞♦ ♣r♦❜❧❡♠❛✳

Pr♦✈❛✳

▼♦str❛r❡♠♦s ❡st❡ t❡♦r❡♠❛ ❛❞♠✐t✐♥❞♦✲s❡ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❙ é ❧✐♠✐t❛❞♦✳ ❙❡❥❛♠ x1

, x2

, ..., xp _{♣♦♥t♦s ❡①tr❡♠♦s ❞♦ ❝♦♥❥✉♥t♦ ❙ ❧✐♠✐t❛❞♦✳}

❊♥tã♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✹✱ ♣❛r❛ t♦❞♦ x ∈ ❙✱ x ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ❝♦♠❜✐♥❛çã♦

❝♦♥✈❡①❛ ❞♦s ♣♦♥t♦s ❡①tr❡♠♦s x1

, x2

, ..., xp ❞❡ ❙✱ ♦✉ s❡❥❛✱ _x= p

X

i=1

λixi ❡ p

X

i=1

λi = 1✳

▲♦❣♦✱ cx=c(

p

X

i=1

λixi) = λ1cx 1

+λ2cx 2

+...+λpcxp✳

❙❡❥❛x∗ _{✉♠ ♣♦♥t♦ ❡①tr❡♠♦ t❛❧ q✉❡ cx}∗ _≤cxi _{(i= 1, ...,0)✳}

▼❛scx=λ1cx1+λ2cx2+...+λpcxp ≥λ1cx∗+λ2cx∗+...+λpcx∗ =cx∗✳

❊♥tã♦cx∗ _{≤cx✱ ∀x∈ ❙✳}

∴ x∗ é ✉♠ ✈ért✐❝❡ ót✐♠♦ ✭s♦❧✉çã♦ ót✐♠❛✮ ❞♦ ♣r♦❜❧❡♠❛✳

❈♦r♦❧ár✐♦ ✶ ❙❡ ❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ ♣♦ss✉✐ ✉♠ ♠á①✐♠♦✭♠í♥✐♠♦✮ ✜♥✐t♦✱ ❡♥tã♦ ♣❡❧♦ ♠❡♥♦s ✉♠❛ s♦❧✉çã♦ ót✐♠❛ é ✉♠ ♣♦♥t♦ ❡①tr❡♠♦ ❞♦ ❝♦♥❥✉♥t♦ ❝♦♥✈❡①♦ S✳

❚❡♦r❡♠❛ ✺ ❚♦❞❛ ❝♦♠❜✐♥❛çã♦ ❝♦♥✈❡①❛ ❞❡ s♦❧✉çõ❡s ót✐♠❛s ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ♣r♦✲ ❣r❛♠❛çã♦ ❧✐♥❡❛r é t❛♠❜é♠ ✉♠❛ s♦❧✉çã♦ ót✐♠❛ ❞♦ ♣r♦❜❧❡♠❛✳

❉❡✜♥✐çõ❡s ❡ ❚❡♦r❡♠❛s ❢✉♥❞❛♠❡♥t❛✐s ▼ét♦❞♦ ❙✐♠♣❧❡①

❈♦r♦❧ár✐♦ ✷ ❙❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ♣r♦❣r❛♠❛çã♦ ❧✐♥❡❛r ❛❞♠✐t✐r ♠❛✐s ❞❡ ✉♠❛ s♦❧✉çã♦ ót✐♠❛ ❡♥tã♦ ❛❞♠✐t❡ ✐♥✜♥✐t❛s s♦❧✉çõ❡s ót✐♠❛s✳

❈♦r♦❧ár✐♦ ✸ ❙❡ ❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ ❛ss✉♠❡ ♦ ♠á①✐♠♦✭♠í♥✐♠♦✮ ❡♠ ♠❛✐s ❞❡ ✉♠ ♣♦♥t♦ ❡①tr❡♠♦✱ ❡♥tã♦ ❡❧❛ t♦♠❛ ♦ ♠❡s♠♦ ✈❛❧♦r ♣❛r❛ q✉❛❧q✉❡r ❝♦♠❜✐♥❛çã♦ ❝♦♥✈❡①❛ ❞❡ss❡s ♣♦♥t♦s ❡①tr❡♠♦s✳

✷✳✸ ❉❡s❝r✐çã♦ ❞♦ ▼ét♦❞♦ ❙✐♠♣❧❡①

❈♦♥s✐❞❡r❛♥❞♦ q✉❡ ❡①✐st❛ ✉♠❛ s♦❧✉çã♦ ❜ás✐❝❛ ✈✐á✈❡❧ ♣❛r❛ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ❞❡ ♣r♦❣r❛♠❛çã♦ ❧✐♥❡❛r✿

▼✐♥✐♠✐③❛rz =cx✱ s✉❥❡✐t♦ ❛✿ Ax=d✱ x≥0 ✭✷✳✷✮ ♦♥❞❡ ❆ é ✉♠❛ ♠❛tr✐③ m×n✱ ❝♦♠ ♣♦st♦ m ✭t♦❞❛s ❛s ❧✐♥❤❛s ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥✲

❞❡♥t❡s✮✱ ♣♦❞❡♠♦s ❞❡❝♦♠♣♦r ♦ ✈❡t♦r c ❡♠ s✉❛s ❝♦♠♣♦♥❡♥t❡s ❜ás✐❝❛s ❡ ♥ã♦ ❜ás✐❝❛s✱ c= (cB cR)✱ ❡ s✉♣♦r q✉❡ ❛ s♦❧✉çã♦ ❜ás✐❝❛ ✈✐á✈❡❧ ❡①✐st❡♥t❡ s❡❥❛ r❡♣r❡s❡♥t❛❞❛ ♣♦r ✉♠

✈❡t♦r x¯=

B−1_d

0

❝✉❥♦ ✈❛❧♦r ❛ss♦❝✐❛❞♦ é ❞❛❞♦ ♣❡❧❛ s❡❣✉✐♥t❡ ❡①♣r❡ssã♦✿

z0 =c

B−1

d

0

= (cB cR)

B−1

d

0

=cBB−1d

P♦❞❡✲s❡ ❡s❝r❡✈❡r ♦ ✈❡t♦r x✱ ❡♠ ❢✉♥çã♦ ❞❛s ✈❛r✐á✈❡✐s ❜ás✐❝❛s ❡ ♥ã♦ ❜ás✐❝❛s ❞❛

s❡❣✉✐♥t❡ ❢♦r♠❛✿ ¯ x= xB xR

=cBB−1d ❡d=Ax=BxB+RxR✳

▼✉❧t✐♣❧✐❝❛♥❞♦✲s❡ ♣♦r B−1 _{❛ ❡①♣r❡ssã♦ ❞❡}

z0 t❡♠✲s❡✿

xB =B−1d−B−1RxR=B−1d−

X

j∈J

B−1ajxj

❉❡ss❛ ❢♦r♠❛ ♣♦❞❡✲s❡ r❡❡s❝r❡✈❡r ❛ ❡①♣r❡ssã♦z =cx❝♦♠♦ s❡ s❡❣✉❡✿ z =cBxB+cRxR

=cB(B−1d−

X

j∈J

B−1ajxj) +

X

j∈J

B−1cjxj

=z0 − X

j∈J

(zj −cj)xj,

♦♥❞❡ zj =cBB−1aj ♣❛r❛ ❝❛❞❛ ✈❛r✐á✈❡❧ ♥ã♦ ❜ás✐❝❛✳

❆ ❡q✉❛çã♦ ❛❝✐♠❛ ♠♦str❛ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞♦ ❡st❛❜❡❧❡❝✐♠❡♥t♦ ❞❡ ✉♠ ❝r✐tér✐♦ ♣❛r❛ ♦ ♣r♦❝❡ss♦ ❞❡ ♠❡❧❤♦r✐❛ ❞❛ s♦❧✉çã♦ ❜ás✐❝❛✳ ◗✉❛♥❞♦ ♦ ✈❛❧♦r ❞♦ t❡r♠♦ zj −cj é ❡str✐✲

t❛♠❡♥t❡ ♠❛✐♦r ❞♦ q✉❡ ③❡r♦ ❡①✐st❡ ❛ ❝❤❛♥❝❡ ❞❡✱ ❝♦♠ ❛ ❡♥tr❛❞❛ ❞❛ ✈❛r✐á✈❡❧ ❞❡ í♥❞✐❝❡

j ♥❛ ❜❛s❡✱ r❡❞✉③✐r ♦ ✈❛❧♦r ❞❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ ❡♠ (zj−cj)xj✱ ❞❡s❞❡ q✉❡ ❡ss❛ ✈❛r✐á✈❡❧

♣♦ss❛ ❛ss✉♠✐r ✉♠ ✈❛❧♦r ♣♦s✐t✐✈♦✳ ❖ t❡r♠♦ (zj −cj) t❛♠❜é♠ é ❞❡♥♦♠✐♥❛❞♦ ✧❝✉st♦

r❡❞✉③✐❞♦✧✳ ❙❡ ❞❡♥♦♠✐♥❛r♠♦s k ♦ í♥❞✐❝❡ ❞❡ss❛ ✈❛r✐á✈❡❧ ♥ã♦ ❜ás✐❝❛ t❡r❡♠♦s✿

▼ét♦❞♦ ❙✐♠♣❧❡① ❉❡s❝r✐çã♦ ❞♦ ▼ét♦❞♦

z =z0−(zk−ck)xk ✭✷✳✸✮

❊①❛♠✐♥❛♥❞♦ ❛ ❡q✉❛çã♦ (2.3)✱ é ❢á❝✐❧ ❝♦♥❝❧✉✐r q✉❡✱ ❞❡ ✉♠❛ ❢♦r♠❛ ❣❡r❛❧✱ ♣❛r❛

♦ ♣r♦❝❡ss♦ ❞❡ ♦t✐♠✐③❛çã♦ s❡rá ✐♥t❡r❡ss❛♥t❡ q✉❡ ❛ ✈❛r✐á✈❡❧ xk s❡❥❛ ✐♥❝r❡♠❡♥t❛❞❛ ❛♦

♠á①✐♠♦✳ ❈♦♠ ❝r❡s❝✐♠❡♥t♦ ❞❡ xk✱ ♦ ✈❛❧♦r ❞❡ z ❞✐♠✐♥✉✐ ♥❛ ♥♦✈❛ s♦❧✉çã♦ ❜ás✐❝❛✱

♣r♦♣♦r❝✐♦♥❛❧♠❡♥t❡ ❛♦ ✈❛❧♦r ❞♦ ❝✉st♦ r❡❞✉③✐❞♦ ❛ss♦❝✐❛❞♦✳ ❈♦♠♦ s❛❜❡♠♦s q✉❡✿

xB =B−1d−B−1akxk= ¯d−ykxk

♦♥❞❡

yk =B−1ak ❡d¯=B−1d

❉❡♥♦t❛♥❞♦ ❛s ❝♦♠♣♦♥❡♥t❡s ❞♦ ✈❡t♦r xB ❡ d¯r❡s♣❡❝t✐✈❛♠❡♥t❡ ♣♦r xB1✱ xB2✱ ✳✳✳ ✱

xBm ❡ d¯1✱ d¯2✱ ✳✳✳✱ d¯m t❡♠♦s ✜♥❛❧♠❡♥t❡✱ ❛ s❡❣✉✐♥t❡ ❡①♣r❡ssã♦✿ 

   

xB1

xB2

✳✳✳

xBm      =      ¯ d1 ¯ d2 ✳✳✳ ¯ dm      −     

y1k

y2k

✳✳✳ ymk      xk

❊ss❛ ❡①♣r❡ssã♦ ♥♦s ♠♦str❛ q✉❡ s❡ ❡①✐st✐r ❛❧❣✉♠ ❡❧❡♠❡♥t♦ ❞❡ yik✱ yik ≤ 0✱ ❡♥tã♦

♦ xBi ❛ss♦❝✐❛❞♦ ♣♦❞❡ ❝r❡s❝❡r ✐♥❞❡✜♥✐❞❛♠❡♥t❡ ❝♦♠ ♦ ❝r❡s❝✐♠❡♥t♦ ❞❡ xk✳ ❙❡ ❡①✐st✐r

yik >0✱ ❡♥tã♦ ♦ xBi ❞❡❝r❡s❝❡ ❝♦♠ ♦ ✐♥❝r❡♠❡♥t♦ ❞❡ xk✳ P❛r❛ s❛t✐s❢❛③❡r ❛s ❝♦♥❞✐çõ❡s

❞❡ ♥ã♦ ♥❡❣❛t✐✈✐❞❛❞❡ ❞❡ ✉♠ s♦❧✉çã♦ ❜ás✐❝❛ ✈✐á✈❡❧✱ ❛ ♥♦✈❛ ✈❛r✐á✈❡❧xksó ♣♦❞❡rá ❝r❡s❝❡r

❛té q✉❡ ❛ ♣r✐♠❡✐r❛ ❝♦♠♣♦♥❡♥t❡xBi s❡❥❛ r❡❞✉③✐❞❛ ❛ ③❡r♦✱ ♦ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛♦ ♠í♥✐♠♦

❡♥tr❡ t♦❞♦s ♦s _d¯ i

yik ♣❛r❛ ♦s ✈❛❧♦r❡s ♣♦s✐t✐✈♦s ❞❡ yik✱ ♦✉✿

¯

ds

ysk

= min

1≤i≤m

¯

di

yik

:yik >0

⇒xs s❛✐ ❞❛ ❜❛s❡✳

◆♦t❡ q✉❡ ♣❛r❛ ❣❛r❛♥t✐r ❛ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ❧✐♥❡❛r ❞❛ ❝♦❧✉♥❛k❝♦♠ ❛s ❞❡♠❛✐s ❝♦❧✉♥❛s

❡①✐st❡♥t❡s ♥❛ ❜❛s❡ é ✐♥❞✐s♣❡♥sá✈❡❧ q✉❡ ysk 6= 0✳ P❡❧♦ ❝r✐tér✐♦✱ ❛ ✈❛r✐á✈❡❧ xk s❡r✐❛

❛ q✉❡ ❡♥tr❛r✐❛ ♥❛ ❜❛s❡ ♠❡❧❤♦r❛♥❞♦ ♦ ✈❛❧♦r ❞❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦✱ ❡ ❛ ✈❛r✐á✈❡❧ xs✱

❧✐♥❡❛r♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡ ❞❡ xk✱ ❞❡✐①❛r✐❛ ❛ ❜❛s❡ ❛♦ t❡r ♦ s❡✉ ✈❛❧♦r ♥✉♠ér✐❝♦ ❡s❣♦t❛❞♦

❝♦♠♣❧❡t❛♠❡♥t❡ ♣❡❧♦ ❝r❡s❝✐♠❡♥t♦ ❞❡ xk✳

P♦❞❡♠♦s ❢♦r♠❛❧✐③❛r ♦ ♣r♦❝❡ss♦ ❞❡ ❡s❝♦❧❤❛ ❞❛ ❜❛s❡ ✐♥✐❝✐❛❧ ❞❡ ❝á❧❝✉❧♦✱ ❝r✐tér✐♦ ❞❡ tr♦❝❛ ❞❡ ✈❛r✐á✈❡✐s ♥❛ ❜❛s❡ ❡ r❡❣r❛ ❞❡ ♣❛r❛❞❛ ❡♠ ✉♠ ❛❧❣♦r✐t♠♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

❉❡t❡r♠✐♥❛r ✉♠❛ s♦❧✉çã♦ ❜ás✐❝❛ ✐♥✐❝✐❛❧ x¯B = ¯d = B−1d✳ ❙❡❥❛ I ♦ ❝♦♥❥✉♥t♦ ❞❡

í♥❞✐❝❡s ❞❛s ❝♦❧✉♥❛s ❞❡ A ♣❡rt❡♥❝❡♥t❡s à ❜❛s❡ ❡ J =N\I ✭♦♣❡r❛çã♦ ❞❡ ❞✐❢❡r❡♥ç❛ ❞❡

❝♦♥❥✉♥t♦s✮✳

P❛ss♦ ✶✿ ❈❛❧❝✉❧❛r ❛ ♠❛tr✐③ Y = (yj) = (ysj)✱ s ∈ I ❡ j ∈ J ❡ ♦s ✈❛❧♦r❡s

zj −cj✱ ∀j ∈ J ❝♦♠♦ s❡ s❡❣✉❡✿ ✭♦❜s❡r✈❡ q✉❡ zh −ch = 0 ♣❛r❛ t♦❞♦ h ∈ I✱ ♣♦✐s

zh =cBB−1ah =ch✮

Y =B−1_R

zj =cByj✱ j ∈J

• ❙❡ zj−cj ≤0✱ ∀j ∈J ❡♥tã♦ ❛ s♦❧✉çã♦ ❜ás✐❝❛ ✈✐á✈❡❧ x¯B é ót✐♠❛✳ P❛r❡✦

• ❈❛s♦ ❝♦♥trár✐♦ ❢❛③❡r J1 ={j ∈J/zj −cj >0}.

P❛ss♦ ✷✿

• ❙❡ yi ≤0 ♣❛r❛ ♣❡❧♦ ♠❡♥♦s ✉♠j ∈J1✱ ♥ã♦ ❡①✐st❡ s♦❧✉çã♦ ót✐♠❛ ✜♥✐t❛✳ P❛r❡✦

• ❈❛s♦ ❝♦♥trár✐♦ ❞❡t❡r♠✐♥❛r k ❞❡ ♠♦❞♦ q✉❡zj−cj = max

j∈J {zj−cj}.

◆❛ ❝♦❧✉♥❛k ❡♥❝♦♥tr❛r ❛ r❡❧❛çã♦✿ d¯s ysk

= min

1≤i≤m

¯

di

yik

:yik >0

✐♠♣❧✐❝❛ q✉❡xs s❛✐

❞❛ ❜❛s❡✳ P❛ss♦ ✸✿

❈♦♥s✐❞❡r❡ ❛ ♥♦✈❛ ❜❛s❡ _Bˆ _{❞❡❞✉③✐❞❛ ❛ ♣❛rt✐r ❞❛ ❛♥t❡r✐♦r ♣❡❧❛ s✉❜st✐t✉✐çã♦ ❞❡ as} ♣♦r ak✳

ˆ

B = (B\ {as} ∪ {ak}).

❈❛❧❝✉❧❛r ❛ ♥♦✈❛ s♦❧✉çã♦ ❜ás✐❝❛ ✈✐á✈❡❧✿ ¯ˆ

xB = ˆB−1d

ˆ

z0 =z0−(zk−ck)

¯

xBs

ysk.

❆t✉❛❧✐③❛r✿

R = (R\ {ak} ∪ {as})

I = (I\ {s} ∪ {k})

J = (J\ {k} ∪ {s}) ❱♦❧t❛r ❛♦ ♣❛ss♦ ✶✳

◆♦ q✉❡ s❡❣✉❡✱ ✈❛♠♦s ❞❡s❝r❡✈❡r ✉♠❛ ❢♦r♠❛ ❞❡ s❡ ❡♥❝♦♥tr❛r ♦s ♥♦✈♦s ✈❛❧♦r❡szj−cj

❡yij q✉❛♥❞♦ s❡ ♣❛ss❛ ❞❡ ✉♠❛ ❜❛s❡ ♣❛r❛ ♦✉tr❛✱ ♣❡❧❛ tr♦❝❛ ❞❡ ❛♣❡♥❛s ✉♠❛ ❞❛s ❝♦❧✉♥❛s

❞❛ ❜❛s❡ ✐♥✐❝✐❛❧ ♣♦r ❛❧❣✉♠❛ ❞❛s ❝♦❧✉♥❛s aj✱ j ∈ J✱ s❡♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✈♦❧t❛r ❛♦

♣♦♥t♦ ✐♥✐❝✐❛❧ ❞♦ ♣r♦❜❧❡♠❛✳

▼ét♦❞♦ ❙✐♠♣❧❡① ❉❡s❝r✐çã♦ ❞♦ ▼ét♦❞♦

✷✳✸✳✶ ▼✉❞❛♥ç❛ ❞❡ ❇❛s❡

❖ ❝á❧❝✉❧♦ ❞♦s ♥ú♠❡r♦s zh −ch ❡ yih✱ h ∈ {1,2, . . . , n} q✉❡ ❛♣❛r❡❝❡♠ ♥♦ ❞❡s❡♥✲

✈♦❧✈✐♠❡♥t♦ ❞♦ ♠ét♦❞♦ s✐♠♣❧❡① ❞❡♣❡♥❞❡♠ ❞✐r❡t❛♠❡♥t❡ ❞❛ s✉❜♠❛tr✐③ ❜❛s❡ B ❞❡ A✳

P♦ré♠✱ q✉❛♥❞♦ ✉♠❛ ♥♦✈❛ ✈❛r✐á✈❡❧ ❡♥tr❛r ♥❛ ❜❛s❡✱ é ✐♥t❡r❡ss❛♥t❡ ❝❛❧❝✉❧❛r ♦s ♥♦✈♦s ✈❛❧♦r❡s ❞❡ zh−ch ❡ ❞♦s yih s❡♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✈♦❧t❛r♠♦s ❛♦ ♣r♦❜❧❡♠❛ ✐♥✐❝✐❛❧✳

P❛r❛ ✐ss♦✱ ❞❡♥♦t❡ ♣♦r ah ❛ h✲és✐♠❛ ❝♦❧✉♥❛ ❞❛ ♠❛tr✐③ A ❡ ❛ss✉♠❛ q✉❡✿

✶✳ B é ✉♠❛ s✉❜♠❛tr✐③ ✐♥✈❡rtí✈❡❧ ❞❡ A✱ ❝✉❥❛s ❝♦❧✉♥❛s sã♦ bi =asi✱ i= 1, . . . , m❀

✷✳ _Bˆ _{é ✉♠❛ ♠❛tr✐③ ✐♥✈❡rtí✈❡❧ ♦❜t✐❞❛ ❞❡ B tr♦❝❛♥❞♦✲s❡ ❛ ❝♦❧✉♥❛ br =as ♣♦r ak❀} ✸✳ yh =B−1ah ❡yˆh = ˆB−1ah❀

✹✳ zh−ch =cBB−1ah −ch✱ ❡zˆh−ch =c_BˆBˆ−1ah−ch✳

◆♦t❡ q✉❡ ♣♦r ❞❡✜♥✐çã♦

ah =Byh = m

X

i=1

yihbi. ✭✷✳✹✮

❋❛③❡♥❞♦ h=k ❡ ✐s♦❧❛♥❞♦ br✱ t❡♠♦s

br =−

X

i6=r

yih

yrk

bi+

1

yrk

ak. ✭✷✳✺✮

◗✉❛♥❞♦ ✭✷✳✺✮ é s✉❜st✐t✉í❞❛ ❡♠ ✭✷✳✹✮✱ t❡♠♦s

ah =

X

i6=r

(yij −

yrh

yrk

yik)bi+

yrh

rkak =

m

X

i=1

ˆ

yihˆbi, ✭✷✳✻✮

♦♥❞❡ ˆ_{bi =bi✱ i6=r❀}ˆ_{br =ak✳ ❈♦♠♣❛r❛♥❞♦ ❛s ❡q✉❛çõ❡s (2.4)❡ (2.6)✱ t❡♠♦s q✉❡} ˆ

yih =yih−

yrh

yrk

yik, i6=r ❡ yˆrh =

yrh

yrk

. ✭✷✳✼✮

❆s ❡q✉❛çõ❡s ❡♠ ✭✷✳✼✮ ✐♥❞✐❝❛♠ ❝♦♠♦ ❝❛❧❝✉❧❛r ♦yˆih ❛ ♣❛rt✐r ❞❡ yih✳

P❛r❛ ❝❛❧❝✉❧❛r zˆh−ch✱ ✉s❛♠♦s ❛ ❞❡✜♥✐çã♦

ˆ

zh−ch =c_Bˆyˆh−ch = m

X

i=1

c_Bˆ

iyˆih−ch. ✭✷✳✽✮

❊♥tr❡t❛♥t♦✱ c_Bˆ

i =cBi✱ i6=r❀ c_Bˆr =ck✳ ❯s❛♥❞♦ ✭✷✳✼✮✱ ♦❜t❡♠♦s

ˆ

zh−ch =

X

i6=r

cBi(yih−

yrh

yrk

yik) +

yrh

yrk

ck−ch ✭✷✳✾✮

❈♦♠♦ cBr(yrh−

yrh

yrk

yrk) = 0✱ t❡♠♦s✿

ˆ

zh−ch = m

X

i=1

cBi(yih−

yrh

yrk

yik) +

yrh

yrk

ck−ch ✭✷✳✶✵✮

❊✱ ♣♦rt❛♥t♦✱ ❞❡s❡♥✈♦❧✈❡♥❞♦ ✭✷✳✶✵✮✱ ♦❜t❡♠♦s✿

ˆ

zh−ch = m

X

i=1

cBiyih−ch−

yrh yrk ( m X i=1

cBiyik−ck), ✭✷✳✶✶✮

♦✉ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱

ˆ

zh−ch =zh−ch−

yrh

yrk

(zk−ck). ✭✷✳✶✷✮

❆ ❡q✉❛çã♦ ✭✷✳✶✷✮ ♠♦str❛ ❝♦♠♦ ❝❛❧❝✉❧❛r zˆh−ch ❛ ♣❛rt✐r ❞❡ zj−cj✱ zk−ck ❡yrj✱ yrk✳

❆♥❛❧♦❣❛♠❡♥t❡✱ ✈❡♠♦s q✉❡ ♦ ♥♦✈♦ ✈❛❧♦r ❞❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ é ❞❛❞♦ ♣♦r

ˆ

z0 =z0 −

¯

dr

yrk

(zk−ck). ✭✷✳✶✸✮

P♦❞❡♠♦s s✐♥t❡t✐③❛r ♦ ♠ét♦❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

❈♦♥s✐❞❡r❡ ♦ ♣r♦❜❧❡♠❛ ✭✷✳✷✮ ❡ ❡s❝♦❧❤❛ ✉♠❛ ♠❛tr✐③ ❜❛s❡ B ✐♥✐❝✐❛❧ ❡ ♦❜t❡♥❤❛ ♦

s❡❣✉✐♥t❡ q✉❛❞r♦ s✐♠♣❧❡① ✐♥✐❝✐❛❧✿

❚❛❜❡❧❛ ✷✳✶✿ ❚❛❜❡❧❛ s✐♠♣❧❡① ✐♥✐❝✐❛❧

c1 c2 . . . ck . . . cn−1 cn

x1 x2 . . . xk . . . xn−1 xn d¯

cB1 xB1 y11 y12 . . . y1k . . . y1,n−1 y1n d¯1

cB2 xB2 y21 y22 . . . y2k . . . y2,n−1 y2n d¯2

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

cBr xBr yr1 yr2 . . . yrk . . . yr,n−1 yrn d¯r

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

cBm xBm ym1 ym2 . . . ymk . . . ym,n−1 ymn d¯m

z1−c1 z2−c2 . . . zk−ck . . . zn−1−cn−1 zn−cn z0 =cBB−1d¯

❙❡zj−cj ≤ 0✱ ♣❛r❛ t♦❞♦ j ∈J✱ ❡♥tã♦ ❛ s♦❧✉çã♦ xB é ót✐♠❛ ❡ z0 =cBB−1b é ♦

✈❛❧♦r ót✐♠♦ ❞❡ z✳ ❈❛s♦ ❝♦♥trár✐♦✱ ❝♦♥s✐❞❡r❡ ♦ ❝♦♥❥✉♥t♦ J1 ={j ∈J :zj −cj >0} ❡

♥❡st❡ ❝❛s♦✱ t❡♠♦s ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✿ ✶✳ yj ≤0 ♣❛r❛ ♣❡❧♦ ♠❡♥♦s ✉♠j ∈J1❀

▼ét♦❞♦ ❙✐♠♣❧❡① ❉❡s❝r✐çã♦ ❞♦ ▼ét♦❞♦

✷✳ P❛r❛ ❝❛❞❛ j ∈J1✱ yij >0♣❛r❛ ❛❧❣✉♠ i= 1, . . . , m✳

◆♦ ♣r✐♠❡✐r♦ ❝❛s♦✱ ♦ ♣r♦❜❧❡♠❛ ♥ã♦ t❡♠ s♦❧✉çã♦ ✜♥✐t❛✱ ❡♥tã♦ ♣❛r❡✦ ❊✱ ♥♦ s❡❣✉♥❞♦ ❝❛s♦✱ ❞❡t❡r♠✐♥❛♠♦s ❛ ✈❛r✐á✈❡❧ xk ❛ ❡♥tr❛r ♥❛ ❜❛s❡✱ ♦♥❞❡ k é ❡s❝♦❧❤✐❞♦ ❞❡ ♠♦❞♦ q✉❡

zk−ck= max j∈J1

{zj−cj}✳

◆❛ ❝♦❧✉♥❛ k ❡♥❝♦♥tr❛r ❛ r❡❧❛çã♦✿ ¯bs ysk

= min

1≤i≤m

¯

bi

yik

:yik >0

♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡

xs =xBr s❛✐ ❞❛ ❜❛s❡✳

❆❣♦r❛✱ ❞❡♥♦t❛♥❞♦ zj −cj ♣♦r ym+1,j ❡ ✉s❛♥❞♦ ❛s r❡❧❛çõ❡s ✭✷✳✼✮✱ ✭✷✳✶✷✮ ❡ ✭✷✳✶✸✮✱

♦❜t❡♠♦s ♦ ♥♦✈♦ q✉❛❞r♦ s✐♠♣❧❡① ❝♦♠ ♦s ✈❛❧♦r❡s ❛t✉❛❧✐③❛❞♦s ❞♦syij✱zh−ch ❡z✱ ❝♦♠♦

s❡ s❡❣✉❡✿

❚❛❜❡❧❛ ✷✳✷✿ ❚❛❜❡❧❛ s✐♠♣❧❡① ❛t✉❛❧✐③❛❞❛ ❛♣ós ♣r✐♠❡✐r❛ ✐t❡r❛çã♦

c1 . . . ck . . . cn

x1 . . . xk . . . xn ¯b

cB1 xB1 y11− y1k

yrkyr1 . . . 0 . . . y1n−

y1k

yrkyrn

¯

d1 −y1 k

yrk

¯

dr

cB2 xB2 y21− y₂k

yrkyr1 . . . 0 . . . y2n−

y₂k

yrkyrn

¯

d2 −y_y2k rk

¯

dr

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

ck xk y

r₁

yrk . . . 1 . . .

yrn yrk ¯ dr yrk ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

cBm xBm ym1−

ymk

yrkyr1 . . . 0 . . . ymn−

ymk

yrkyrn

¯

dm− y

mk

yrk

¯

dr

ym+1,1− y r₁

yrkym+1,k . . . 0 . . . ym+1,n−

yrn

yrkym+1,k z0− ¯

dr

yrkym+1,k

◆♦t❡ q✉❡ s❡ ❞❡♥♦t❛r♠♦s _d¯_{i ♣♦r yi,n+1✱ ♦ ❝á❧❝✉❧♦ ❞❛ ♥♦✈❛ t❛❜❡❧❛ s✐♠♣❧❡① ❝♦rr❡s✲} ♣♦♥❞❡ ❛♦ ♣✐✈♦t❡❛♠❡♥t♦ ❣❛✉ss✐❛♥♦ ❞❛ ♠❛tr✐③ (yij) ❞❡ ♦r❞❡♠ (m+ 1)×(n+ 1) ❡♠

t♦r♥♦ ❞❛ ❡♥tr❛❞❛ yrk✳

❆ss✐♠✱ ♦ ♣r♦❝❡ss♦ ✐t❡r❛t✐✈♦ ❞♦ ♠ét♦❞♦ s✐♠♣❧❡① ❡stá ❢♦r♠❛❞♦✳ ❊♥tã♦✱ ❢❛③✲s❡ ♥♦✲ ✈❛♠❡♥t❡ ❛ ❛♥á❧✐s❡ ♣❛r❛ s❛❜❡r s❡ ❛ s♦❧✉çã♦ ♦❜t✐❞❛ é ót✐♠❛ ♦✉ s❡ ♥ã♦ ❤á s♦❧✉çã♦ ót✐♠❛ ✜♥✐t❛ ❡✱ ❝❛s♦ ❝♦♥trár✐♦✱ ✐t❡r❛✲s❡ ♠❛✐s ✉♠❛ ✈❡③ ♦ ♣r♦❝❡ss♦ ❝♦♠ ✉♠❛ ♥♦✈❛ ❜❛s❡✳

✷✳✸✳✷ ❊①❡♠♣❧♦

❙❡❥❛ ♦ ♣r♦❜❧❡♠❛✿

▼❛①✐♠✐③❛r z =x1 +x2

❙✉❥❡✐t♦ ❛✿       

2x1+x2 ≤8

x1+ 2x2 ≤7

x2 ≤3

x1 ❡x2 ≥0

P❛ss❛♥❞♦ ♦ ♣r♦❜❧❡♠❛ ♣❛r❛ ❛ ❢♦r♠❛ ♣❛❞rã♦✱ t❡♠♦s✿

▼✐♥✐♠✐③❛r−z =−x1−x2+ 0x3+ 0x4+ 0x5

❙✉❥❡✐t♦ ❛✿       

2x1+x2+x3+ 0x4+ 0x5 = 8

x1+ 2x2+ 0x3+x4+ 0x5 = 7

0x1+x2+ 0x3+ 0x4+x5 = 3

x1, x2, x3, x4, x5 ≥0

❯s❛♥❞♦ ❛s ✈❛r✐á✈❡✐sx3, x4, x5❝♦♠♦ ❜❛s❡✱ t❡♠♦s ✐♥✐❝✐❛❧♠❡♥t❡B =♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡✱

cB = (0 0 0)✱ zh −ch = −ch ♣❛r❛ t♦❞♦ h = 1,2,3,4,5 ❡ −z0 = 0✳ ❯s❛r❡♠♦s ✉♠❛

t❛❜❡❧❛ s✐♠♣❧✐✜❝❛❞❛ ♦♠✐t✐♥❞♦ ♦s cj ❡ ♦s cBr✳ ❆ss✐♠✱ ❛ t❛❜❡❧❛ ✐♥✐❝✐❛❧ t♦♠❛ ❛ ❢♦r♠❛✿

❚❛❜❡❧❛ ✷✳✸✿ ❚❛❜❡❧❛ ✐♥✐❝✐❛❧

x1 x2 x3 x4 x5 d

x3 ✷ ✶ ✶ ✵ ✵ ✽

x4 ✶ ✷ ✵ ✶ ✵ ✼

x5 ✵ ✶ ✵ ✵ ✶ ✸

✶ ✶ ✵ ✵ ✵ ✵

❆ s♦❧✉çã♦ ♥ã♦ é ót✐♠❛ ♣♦✐s ❤á ✈❛❧♦r❡s zj −cj > 0 ❡ yj ♥ã♦ é ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛

③❡r♦ ♣❛r❛ j = 1,2✳ ❊♥tã♦ ✈❛♠♦s ♣❛r❛ ❛ ♣ró①✐♠❛ ✐t❡r❛çã♦✳

❖❜s❡r✈❡ q✉❡ ❤á ✉♠ ❡♠♣❛t❡ ♥❛ ❡s❝♦❧❤❛ ❞❛ ✈❛r✐á✈❡❧ q✉❡ ❞❡✈❡ ❡♥tr❛r z1 −c1 =

z2−c2 = 1✱ ❡♥tã♦ ❡s❝♦❧❤❡♠♦sx1✳ ❆ ✈❛r✐á✈❡❧ ❛ s❛✐r ❞❛ ❜❛s❡ éx3✱ ♣♦✐s

8

2 = min{ 8 2,

7 1}✳

▼ét♦❞♦ ❙✐♠♣❧❡① ❉❡s❝r✐çã♦ ❞♦ ▼ét♦❞♦

❆ss✐♠ ❛ ♥♦✈❛ t❛❜❡❧❛ é

❚❛❜❡❧❛ ✷✳✹✿

x1 x2 x3 x4 x5 d

x1 ✶ 0,5 0,5 ✵ ✵ ✹

x4 ✵ 1,5 −0,5 ✶ ✵ ✸

x5 ✵ ✶ ✵ ✵ ✶ ✸

✵ 0,5 −0,5 ✵ ✵ −4

❈♦♠♦ ❛ s♦❧✉çã♦ ❛✐♥❞❛ ♥ã♦ é ót✐♠❛✱ ♣♦✐s z2 −c2 > 0✱ ✐t❡r❛♠♦s ♠❛✐s ✉♠❛ ✈❡③✳

◆♦t❡ q✉❡ x2 ❡♥tr❛ ♥❛ ❜❛s❡ ❡ x3 s❛✐ ❞❛ ❜❛s❡ ❡ ❛ ♥♦✈❛ t❛❜❡❧❛ é

x1 x2 x3 x4 x5 d

x1 ✶ 0 2 3 −

1

3 ✵ ✸

x2 ✵ 1 − 1 3

2

3 ✵ ✷

x5 ✵ ✵ 1₃ −2₃ ✶ ✶

✵ 0 −1

3 − 1

3 ✵ −5

❈♦♠♦ ♥ã♦ ❤á ♠❛✐szj−cj ♣♦s✐t✐✈♦s✱ ❛ s♦❧✉çã♦ é ót✐♠❛✳ ❚❡♠♦s q✉❡ ♦ ♠❛✐♦r ✈❛❧♦r

❞❡ z é ❛t✐♥❣✐❞♦ ♥♦ ♣♦♥t♦ (x1, x2) = (3,2) ❡ s❡✉ ✈❛❧♦r ót✐♠♦ é z = 5 ✭❡q✉✐✈❛❧❡♥t❡ ❛

−z =−5✮✳

✷✳✹ ▼ét♦❞♦ ❞❡ ❉✉❛s ❋❛s❡s

◆♦ ❛❧❣♦r✐t♠♦ ❞❡s❝r✐t♦✱ ❛ ❢❛s❡ ❞❡ ♦❜t❡♥çã♦ ❞❡ ✉♠❛ ❜❛s❡ ✈✐á✈❡❧ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❢♦✐ ❝♦♥s✐❞❡r❛❞❛ s✉♣❡r❛❞❛✳ ❘❡❛❧♠❡♥t❡✱ q✉❛♥❞♦ ♣♦r ♠❡✐♦ ❞❛ ✐♥❝❧✉sã♦ ❞❡ ✈❛r✐á✈❡✐s ❞❡ ❢♦❧❣❛ ♥❛ ♣❛ss❛❣❡♠ ❞❛ ❞❡s❝r✐çã♦ ❞❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛ ♣❛r❛ ❛ ♣❛❞rã♦ ♦❜t❡♠♦s ✉♠❛ ❜❛s❡ ✐♥✐❝✐❛❧ ✭❡ ❛✐♥❞❛ ♣♦r ❝✐♠❛ ❝❛♥ô♥✐❝❛✮✱ ♦ ❛❧❣♦r✐t♠♦ ❞❡s❝r✐t♦ s♦❧✉❝✐♦♥❛ ❜❡♠ ♦ ♣r♦❜❧❡♠❛✳ ▼❛s ❡ q✉❛♥❞♦ ✐ss♦ ♥ã♦ ❛❝♦♥t❡❝❡❄ ❙❡❥❛ ♦ ❡①❡♠♣❧♦ q✉❡ s❡ s❡❣✉❡✿

▼✐♥✐♠✐③❛rZ =−3x1−5x2

❙✉❥❡✐t♦ ❛✿

x1 ≤4

x2 ≤6

3x1+ 2x2 ≥18

x1 ≥0✱ x2 ≥0

◗✉❡ ❛♦ s❡r ❝♦♥✈❡rt✐❞♦ ♣❛r❛ ❛ ❢♦r♠❛ ♣❛❞rã♦ s❡rá✿

▼✐♥✐♠✐③❛rZ =−3x1−5x2+ 0x3+ 0x4+ 0x5

❙✉❥❡✐t♦ ❛✿

x1+x3 = 4

x2+x4 = 6

3x1+ 2x2−x5 = 18

x1 ≥0✱ x2 ≥0✱ x3 ≥0✱ x4 ≥0✱ x5 ≥0

❖❜✈✐❛♠❡♥t❡ ❛ ♠❛tr✐③ ❢♦r♠❛❞❛ ♣❡❧❛s ✈❛r✐á✈❡✐s ❞❡ ❢♦❧❣❛ ♥ã♦ é ✉♠❛ ❜❛s❡ ✈✐á✈❡❧✿

❇ ❂





1 0 0 0 1 0 0 0 −1





❆ ✜❣✉r❛ ❛❜❛✐①♦ ♠♦str❛ ❣r❛✜❝❛♠❡♥t❡ q✉❡ ♦ ♣♦♥t♦ ♦r✐❣❡♠ ✭t♦❞❛s ❛s ✈❛r✐á✈❡✐s r❡❛✐s ✐❣✉❛✐s ❛ ③❡r♦✮ ♥ã♦ ♣❡rt❡♥❝❡ ❛♦ ❡s♣❛ç♦ ❞❛s s♦❧✉çõ❡s ✈✐á✈❡✐s ❞❡ss❡ ♣r♦❜❧❡♠❛✱ ♦✉ s❡❥❛✱ ❛ ❜❛s❡ tr❛❞✐❝✐♦♥❛❧♠❡♥t❡ ❝♦♥s✐❞❡r❛❞❛ ❝♦♠♦ ✐♥✐❝✐❛❧ ❢❛❧❤❛ ❡♠ s❡r ✈✐á✈❡❧✳