Nguyên lý phân rã trong quy hoạch tuyến tính

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ì ì ị P ❚❯❨➌◆ ❚➑◆❍ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✸ Số hóa trung tâm học liệu http://www.lrc.tnu.edu.vn/ ❈ỉ♥❣ tr➻♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ❍å❝ ❑❤♦❛ ữớ ữợ ❤å❝✿ ●❙ ✲ ❚❙✳ ❚r➛♥ ❱ô ❚❤✐➺✉ P❤↔♥ ❜✐➺♥ ✶✿ ❚❙✿ ◆❣✉②➵♥ ❱➠♥ ◆❣å❝ ✲ ❱✐➺♥ ❚♦→♥ ❤å❝ P❤↔♥ ❜✐➺♥ ✷✿ ●❙✳ ❚❙✿ ◆❣✉②➵♥ ❇÷í♥❣ ✲ ❱✐➺♥ ❈◆❚❚ ▲✉➟♥ ✈➠♥ ữủ trữợ ỗ t↕✐✿ ❚r÷í♥❣ ✣↕✐ ❍å❝ ❑❤♦❛ ❍å❝ ✲ ✣↕✐ ❍å❝ ❚❤→✐ ◆❣✉②➯♥ ◆❣➔② ✶✷ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✶✸ ❈â t❤➸ t➻♠ ❤✐➸✉ t↕✐ ❚❤÷ ❱✐➺♥ ✣↕✐ ❍å❝ ❚❤→✐ ◆❣✉②➯♥ Số hóa trung tâm học liệu http://www.lrc.tnu.edu.vn/ ✶ ▼ư❝ ❧ư❝ ▼ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶✳ ✶✳✷✳ ✶✳✸✳ ỗ t ỗ ✳ ✳ ✳ ✳ ✳ ❇➔✐ t♦→♥ q✉② ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✣è✐ ♥❣➝✉ tr♦♥❣ q✉② ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ❚❤✉➟t t♦→♥ ✤ì♥ ❤➻♥❤ ❝↔✐ ❜✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ữỡ ỵ r t ị tữ ♣❤➙♥ r➣ ❝õ❛ ❉❛♥t③✐❣ ✲ ❲♦❧❢❡ ✳ ✳ ❚r÷í♥❣ ❤đ♣ ● ❦❤æ♥❣ ❜à ❝❤➦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❳➙② ❞ü♥❣ ♣❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ ❜❛♥ ✤➛✉ ✳ ❱➼ ❞ö ♠✐♥❤ ❤å❛ ♣❤➙♥ r➣ ❉❛♥t③✐❣ ✲ ❲♦❧❢❡ ◗✉② ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ❝➜✉ tró❝ ❦❤è✐ ✳ ✳ ✳ ❈❤÷ì♥❣ ỵ r rs ị tữ r ❝õ❛ ❇❡♥❞❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇➔✐ t♦→♥ ❝❤õ ✈➔ ❜➔✐ t♦→♥ ❝❤õ t❤✉ ❤➭♣ ✳ ✳ ✳ ❱➼ ❞ö ♠✐♥❤ ❤å❛ ♣❤➙♥ r➣ ❇❡♥❞❡rs ✳ ✳ ✳ ✳ ✳ ◗✉② ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ❝➜✉ tró❝ ❜➟❝ t❤❛♥❣ ✳ ❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ Số hóa trung tâm học lieäu ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✼ ✳ ✼ ✳ ✶✷ ✳ ✶✹ ✳ ✶✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ http://www.lrc.tnu.edu.vn/ ✷✶ ✷✶ ✷✼ ✸✵ ✸✷ ✸✺ ✸✽ ✸✽ ✹✶ ✹✹ ✹✻ ✺✶ ✺✷ ✷ ▼ð ✤➛✉ ◗✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ✭▲✐♥❡❛r Pr♦❣r❛♠♠✐♥❣✮ ❧➔ ❜➔✐ t♦→♥ tè✐ ÷✉ ✤ì♥ ❣✐↔♥ ♥❤➜t✳ ✣â ❧➔ ❜➔✐ t♦→♥ t➻♠ ❝ü❝ t✐➸✉ ✭❤❛② ❝ü❝ ✤↕✐✮ ❝õ❛ ♠ët ❤➔♠ t✉②➳♥ t➼♥❤ ✈ỵ✐ ❝→❝ r➔♥❣ ❜✉ë❝ ✤➥♥❣ t❤ù❝ ❤❛② ❜➜t ✤➥♥❣ t❤ù❝ t✉②➳♥ t➼♥❤✳ ◗✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ❝â ♥❤✐➲✉ ù♥❣ rở r tr ỵ tt tỹ t Pữỡ ♣❤→♣ ✤ì♥ ❤➻♥❤ ✭❞♦ ●✳ ❇✳ ❉❛♥t③✐❣ ✤➲ ①✉➜t ♥➠♠ ✶✾✹✼✮ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ q✉❡♥ t❤✉ë❝✱ ❝â ❤✐➺✉ q✉↔ ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ✈➔ ❝→❝ ❜➔✐ t♦→♥ ✤÷❛ ✤÷đ❝ ✈➲ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤✳ P❤÷ì♥❣ ♣❤→♣ ✤ì♥ ❤➻♥❤ ❝â ♥❤✐➲✉ ❜✐➳♥ t❤➸ ❦❤→❝ ♥❤❛✉✳ ❈❤➥♥❣ ❤↕♥✱ ✈ỵ✐ ❜➔✐ t♦→♥ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ❝â ♥❤✐➲✉ ❤➺ sè ❜➡♥❣ ❦❤ỉ♥❣✱ t❛ t❤÷í♥❣ ❞ị♥❣ t❤✉➟t t♦→♥ ✤ì♥ ❤➻♥❤ ❝↔✐ ❜✐➯♥ ✤➸ t➟♥ ❞ư♥❣ ✤ë t❤÷❛ ❝õ❛ ❝→❝ ❤➺ sè ❦❤→❝ ❦❤ỉ♥❣✳ ❱ỵ✐ ❜➔✐ t♦→♥ ❝â ❝➜✉ tró❝ ✤➦❝ ❜✐➺t✱ t❛ ❝➛♥ sû ❞ö♥❣ ❝→❝ t❤✉➟t t♦→♥ r✐➯♥❣✱ ❤✐➺✉ q tợ ỵ r t ỵ r rs ✤➸ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ❝â tữợ ợ õ trú t tró❝ ♠❛ tr➟♥ ❦❤è✐ ❤❛② ❝➜✉ tró❝ ♠❛ tr➟♥ ❜➟❝ t ị tữ ỡ tt r ❧➔ ✤÷❛ ❜➔✐ t♦→♥ ❜❛♥ ✤➛✉ ✈➲ ❝→❝ ❜➔✐ t♦→♥ ❝♦♥ ♥❤ä ❤ì♥ ✭➼t ❜✐➳♥ sè ❤ì♥ ❤♦➦❝ ➼t r➔♥❣ ❜✉ë❝ ❤ì♥✮ ❤♦➦❝ ❝â t❤➸ t➟♥ ❞ư♥❣ ❝➜✉ tró❝ r✐➯♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❝♦♥ ✭❝❤➥♥❣ ❤↕♥ ❝➜✉ tró❝ ❜➔✐ t♦→♥ ✈➟♥ t↔✐✮ ✤➸ ❣✐↔✐ ❤✐➺✉ q✉↔ ❤ì♥✳ ❚❛ ❜➢t ✤➛✉ tứ trữớ ủ ỡ õ t ỗ ❤❛✐ ❦❤è✐ r➔♥❣ ❜✉ë❝ ✤ë❝ ❧➟♣ ♥❤❛✉ ✭t✇♦ ✐♥❞❡♣❡♥❞❡♥t s✉❜s②st❡♠s✮ ❦❤æ♥❣ ❝â ❜✐➳♥ sè ❝❤✉♥❣✳ ❈❤➥♥❣ ❤↕♥✱ n1 z= n cj xj → M in cj xj + j=1 Soá hóa trung tâm học liệu j=n1 +1 http://www.lrc.tnu.edu.vn/ ✸ ❱ỵ✐ ✤✐➲✉ ❦✐➺♥✳ n1 aij xj = bi , i = 1, , m1 ✭✵✳✶✮ j=1 n1 aij xj = bi , i = m1 + 1, , m j=n1 +1 xj ≥ 0, j = 1, 2, , n ❉♦ ❦❤æ♥❣ ❝â ♠è✐ ❧✐➯♥ ❤➺ ♥➔♦ ❣✐ú❛ ❤❛✐ ❦❤è✐ ♥➔② ✭trø ❤➔♠ ♠ö❝ t✐➯✉✮ ♥➯♥ rã r➔♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ✭✵✳✶✮ ❝â t❤➸ t➻♠ ❜➡♥❣ ❝→❝❤ ❣✐↔✐ ❤❛✐ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ❝♦♥ ✭♠é✐ q✉✐ ❤♦↕❝❤ ù♥❣ ✈ỵ✐ ♠ët ❦❤è✐✮ t t rỗ tr tố ữ ❧↕✐ ✤➸ ♥❤➟♥ ✤÷đ❝ ❣✐→ trà tè✐ ÷✉ ❝❤✉♥❣ ③✳ ▼ët ♠ð rë♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✭✵✳✶✮ ❧➔ ❜➔✐ t♦→♥ ❝â ❝➜✉ tró❝ ❣â❝ ✲ ❦❤è✐ ✭❜❧♦❝❦ ✲ ❛♥❣✉❧❛r s②st❡♠✮ ✭✵✳✷✮ s❛✉ ✤➙②✳ ❇➔✐ t♦→♥ ♥➔② ❝â ❑ ❦❤è✐ ✤ë❝ ❧➟♣ ♥❤❛✉ ✈➔ ❝→❝ r➔♥❣ ❜✉ë❝ ❧✐➯♥ ❦➳t✿ z = (c0 )T x0 + (c1 )T x1 + + (ck )T xk → M in ❱ỵ✐ ✤✐➲✉ ❦✐➺♥ A0 x0 + A1 x1 + + AK xK = b B x1 = b1 ✳✳ ✳✳✳ ✭✵✳✷✮ B K xK = bK x0 ≥ 0, x1 ≥ 0, , xK ≥ ❈â t❤➸ ❣✐↔✐ t ỵ tỹ t ổ ố ữ s sỷ ởt ổ t ỗ ♠→② ❤♦↕t ✤ë♥❣ ❣➛♥ ♥❤÷ ✤ë❝ ❧➟♣ ❦ ❂ ✶✱ ✳✳✳ ✱ ❑✳ ▼é✐ ♥❤➔ ♠→② ❝â ❝→❝ r➔♥❣ ❜✉ë❝ r✐➯♥❣✱ ✤ë❝ ❧➟♣ ✈ỵ✐ r➔♥❣ ❜✉ë❝ ❝õ❛ ❝→❝ ♥❤➔ ♠→② ❦❤→❝✳ ❚✉② ♥❤✐➯♥✱ ❣✐ú❛ ❝→❝ ♥❤➔ ♠→② ❝â ♠ët sè r➔♥❣ ❜✉ë❝ ❝❤✉♥❣✱ ♥❤÷ ❤↕♥ ❝❤➳ ✈➲ t➔✐ ❝❤➼♥❤✱ ✈➲ ❧❛♦ ✤ë♥❣ ❧➔♥❤ ♥❣❤➲ ✈➔ ❝â ♠ët ❤➔♠ ❧ñ✐ ➼❝❤ ❝❤✉♥❣ ✤↕✐ ❞✐➺♥ ❝❤♦ q✉②➲♥ ❧đ✐ ❝õ❛ ❝ỉ♥❣ t② ✈➔ ❑ ♥❤➔ ♠→②✳ ❚r♦♥❣ ❜➔✐ t♦→♥ ✭✵✳✷✮ xk ❧➔ ✈➨❝tì ❜✐➸✉ t❤à ❝÷í♥❣ ✤ë ❤♦↕t ✤ë♥❣ ❝õ❛ ♥❤➔ ♠→② t❤ù ❦ ✈➔ x0 ❧➔ ❝÷í♥❣ ✤ë ❤♦↕t ✤ë♥❣ ❝õ❛ ❜ë q ỵ ổ t ổ t ❜➜t ❝ù ♥❤➔ ♠→② ♥➔♦✳ ❉á♥❣ t❤ù ♥❤➜t ❧➔ ❤➔♠ ♠ư❝ t✐➯✉ ❝❤✉♥❣ ❝❤♦ t♦➔♥ ❝ỉ♥❣ t②❀ ❞á♥❣ t❤ù ❤❛✐ ❧➔ ♠ r➔♥❣ ❜✉ë❝ ✈➲ sû ❞ö♥❣ ❝❤✉♥❣ ♠ t➔✐ Số hóa trung tâm học liệu http://www.lrc.tnu.edu.vn/ ✹ ♥❣✉②➯♥ ❦❤❛♥ ❤✐➳♠ ✭t➔✐ ❝❤➼♥❤✱ ❧❛♦ ✤ë♥❣ ❧➔♥❤ ♥❣❤➲ ✳✳✳✮❀ ❞á♥❣ t❤ù ❜❛ ❧➔ x1 r➔♥❣ ❜✉ë❝ ❝❤➾ r✐➯♥❣ ❝õ❛ ♥❤➔ ♠→② t❤ù ♥❤➜t ✈➔ ❞á♥❣ ❝✉è✐ ❝ò♥❣ ❧➔ ♠❑ r➔♥❣ ❜✉ë❝ ❝❤➾ r✐➯♥❣ ❝õ❛ ♥❤➔ ♠→② t❤ù ❑✳ ❈➜✉ tró❝ õ ố ủ ỵ t ọ t♦→♥ ❜❛♥ ✤➛✉ t❤➔♥❤ ❑ ❜➔✐ t♦→♥ ❝♦♥ ✤ë❝ ❧➟♣ rỗ s õ t♦→♥ ♥➔② ✤➸ t➼♥❤ ✤➳♥ ❝→❝ r➔♥❣ ❜✉ë❝ ❧✐➯♥ ❦➳t ởt ỷ ỵ ố ợ ❝→❝ ♥❤➔ ❦✐♥❤ t➳ ❧➔ ❜➢t ✤➛✉ tø ✤à♥❤ ❣✐→ tý ỵ t ❝→❝ ♥❤➔ ♠→② tü tè✐ ÷✉ ❤♦→ ❤♦↕t ✤ë♥❣ ❝õ❛ ♠➻♥❤ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❧➔ ♥❤➔ ♠→② ♣❤↔✐ tr↔ t✐➲♥ ❝❤♦ ❝→❝ t➔✐ ♥❣✉②➯♥ ❤✐➳♠ t❤❡♦ ❣✐→ ✤➣ q✉✐ ✤à♥❤✳ t q ỵ ❝æ♥❣ t② ✈➔ ❝→❝ ♥❤➔ ♠→② ❝➛♥ ✤➳♥ ♥â✐ ❝❤✉♥❣ ữủt q ự ỗ t ổ t ❝â✮✳ ❇➔✐ t♦→♥ trð t❤➔♥❤ t➻♠ ♠ët t❤✉➟t t♦→♥ ❤✐➺✉ qõ❛ ✤➸ ✤✐➲✉ ❝❤➾♥❤ ❝→❝ ❣✐→ ✭❝→❝ ♥❤➙♥ tû ▲❛❣r❛♥❣❡✮✳ ỵ r t ♥❤í ♠ët sè ❤ú✉ ❤↕♥ ❧➛♥ ❧➦♣✳ P❤➙♥ r➣ ❇❡♥❞❡rs t❤ü❝ ❝❤➜t ❧➔ ♣❤➙♥ r➣ ❉❛♥t③✐❣ ✲ ❲♦❧❢❡ →♣ ❞ö♥❣ ✈➔♦ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉✳ ❚❤❡♦ ❝→❝❤ ♥➔②✱ sè ❜✐➳♥ t ữủ ợt t r➔♥❣ ❜✉ë❝ ♠ỵ✐ ✈➔♦ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉✳ ❚r♦♥❣ ♣❤➙♥ r➣ ❉❛♥t③✐❣ ✲ ❲♦❧❢❡ ❝→❝ ❝ët ❝õ❛ ✧❜➔✐ t♦→♥ ❝❤õ✧ ✤÷đ❝ s✐♥❤ r❛ ❦❤✐ ❝➛♥✳ ❚r♦♥❣ ♣❤➙♥ r➣ ❇❡♥❞❡rs ❝ơ♥❣ ✈➟②✱ ❝→❝ r➔♥❣ ❜✉ë❝ ❝õ❛ ❜➔✐ t♦→♥ ❝❤õ ❇❡♥❞❡rs ❝❤➾ ữủ s r ởt ợ t ❦❤→❝ ❝ô♥❣ ❤❛② ❣➦♣ tr♦♥❣ t❤ü❝ t✐➵♥ ✈➔ ❝â t❤➸ ỵ r tố ❜➟❝ t❤❛♥❣ ✭st❛✐r❝❛s❡ s②st❡♠s✮✳ ❈→❝ ❤➺ t❤è♥❣ ♥➔② ❦❤→❝ ✈ỵ✐ ❤➺ t❤è♥❣ ❣â❝ ✲ ❦❤è✐ ✭✵✳✷✮ ð ❝❤é✿ ❝→❝ ❤♦↕t ộ ữợ t s ✤➛✉ ✈➔♦ ✈➔ ❝→❝ s↔♥ ♣❤➞♠ ✤➛✉ r❛ ✈ỵ✐ ❤❛✐✳ ữợ t trữợ s õ ✭✵✳✸✮ ♠æ t↔ ♠ët ❤➺ t❤è♥❣ ❜➟❝ t❤❛♥❣ ❝â ❜è♥ ❝➜♣✳ z = (c1 )T x1 + (c2 )T x2 + (c3 )T x3 + (c4 )T x4 → ❱ỵ✐ ✤✐➲✉ ❦✐➺♥ A11 x1 A21 x1 + A22 x2 A32 x2 + A33 x3 A43 x3 + A44 x4 = = = = b1 , b2 , b3 , b4 , x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ Số hóa trung tâm học liệu http://www.lrc.tnu.edu.vn/ ✭✵✳✸✮ ✺ ❍➺ t❤è♥❣ ❜➟❝ t❤❛♥❣ t❤÷í♥❣ ♥→② s✐♥❤ ❦❤✐ ①➨t ❝→❝ q✉→ tr➻♥❤ ❞✐➵♥ r❛ t❤❡♦ t❤í✐ ❣✐❛♥✱ tr♦♥❣ ✤â ❝→❝ ❤♦↕t ✤ë♥❣ ð ♠ët t❤í✐ ❦ý ✭❤❛② ❣✐❛✐ ✤♦↕♥✮ trü❝ t✐➳♣ t→❝ ✤ë♥❣ tỵ✐ ✭❤❛② ❝❤à✉ ↔♥❤ ❤÷ð♥❣ ❜ð✐✮ ❝→❝ ❤♦↕t ✤ë♥❣ ð t❤í✐ ❦ý t✐➳♣ s trữợ ổ õ t tợ ✭❤❛② ❝❤à✉ ↔♥❤ ❤÷ð♥❣ ❜ð✐✮ ❝→❝ ❤♦↕t ✤ë♥❣ ð ❝→❝ t❤í✐ ❦ý ✭❣✐❛✐ ✤♦↕♥✮ ❦❤→❝✳ ❈→❝ ❤➺ t❤è♥❣ ♥❤÷ t❤➳ ♥↔② s✐♥❤ tr♦♥❣ s↔♥ ①✉➜t ❦❤✐ ❤♦↕t ✤ë♥❣ s↔♥ ①✉➜t ð ♠é✐ ❣✐❛✐ ✤♦↕♥ ❝❤➾ ❝❤à✉ ↔♥❤ ❤÷ð♥❣ ❜ð✐ s↔♥ t trữợ õ t ✤ë♥❣ tỵ✐ s↔♥ ①✉➜t ð ❣✐❛✐ ✤♦↕♥ ❧✐➲♥ s❛✉✳ ❚r♦♥❣ ❝→❝ ❜➔✐ t♦→♥ ♥❤÷ t❤➳✱ t❤÷í♥❣ ♠ët sè ♠❛ tr➟♥ ❝♦♥ tr➯♥ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤ Aii ❧➔ ❣✐è♥❣ ♥❤❛✉ ✈➔ ♠ët sè ♠❛ tr➟♥ ❝♦♥ tr➯♥ ✤÷í♥❣ ❝❤➨♦ ♣❤ư Ai,i−1 ❝ô♥❣ ❣✐è♥❣ ♥❤❛✉✳ ❑❤✐ ✤â ❝â t❤➸ ❦❤❛✐ t❤→❝ ❝➜✉ tró❝ ✤➦❝ ❜✐➺t ♥➔② ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥✳ ◆ë✐ ❞✉♥❣ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ t❤➔♥❤ ❜❛ ❝❤÷ì♥❣ ❝❤➼♥❤✳ ❈❤÷ì♥❣ ✶ ✈ỵ✐ t✐➯✉ ✤➲ ✧❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✧ tr➻♥❤ ❜➔② ♠ët số tự ỡ s t ỗ t ỗ ✤❛ ❞✐➺♥✱ ❝→❝❤ ①→❝ ✤à♥❤ t➟♣ ✤➾♥❤ ✈➔ t➟♣ ❝↕♥❤ ổ ởt t ỗ ỵ t ỗ q ✈➔ ❝↕♥❤ ✈æ ❤↕♥ ❝õ❛ ♥â✳ ◆➯✉ ❝→❝ ❦✐➳♥ t❤ù❝ ❝➛♥ t❤✐➳t ✈➲ ❜➔✐ t♦→♥ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ✈➔ t➼♥❤ ❝❤➜t✱ ❝→❝❤ ①➙② ❞ü♥❣ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ✈➔ ❝→❝ q✉❛♥ ❤➺ ✤è✐ ♥❣➝✉ tr♦♥❣ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤✳ ❈✉è✐ ❝❤÷ì♥❣ ♥❤➢❝ ❧↕✐ t❤✉➟t t♦→♥ ✤ì♥ ❤➻♥❤ ❝↔✐ ❜✐➯♥ ❣✐↔✐ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤✳ ❚❤✉➟t t♦→♥ ♥➔② s➩ ✤÷đ❝ ❞ị♥❣ ✤➳♥ ð ❝❤÷ì♥❣ s❛✉ ✤➸ ❣✐↔✐ ✧❜➔✐ t♦→♥ ❝❤õ✧ t r t ữỡ ợ t ỵ r t tr tt t t ỵ r t q t t õ tữợ ợ õ trú tr ố t ỵ t÷ð♥❣ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✈➔ ❝→❝ ❦ÿ t❤✉➟t t➼♥❤ t♦→♥ t ị tữ r ữ t ợ r t ➼t r➔♥❣ ❜✉ë❝ ❤ì♥ ✭❣å✐ ❧➔ ❜➔✐ t♦→♥ ❝❤õ✮ ♥❤÷♥❣ ✈ỵ✐ r➜t ♥❤✐➲✉ ❜✐➳♥✳ ❈ët ❤➺ sè ❝õ❛ ❝→❝ ❜✐➳♥ ♥➔② s➩ ✤÷đ❝ t➻♠ ❞➛♥ ❦❤✐ ❝➛♥✱ ♥❤í ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ♣❤ư✳ ❚r÷í♥❣ ❤đ♣ ♠✐➲♥ r➔♥❣ ❜✉ë❝ ❝õ❛ ❜➔✐ t♦→♥ ❝♦♥ ❦❤ỉ♥❣ ❜à ❝❤➦♥ ✈➔ ✈➜♥ ✤➲ t➻♠ ♣❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ ❜❛♥ ✤➛✉ ❝❤♦ ❜➔✐ t♦→♥ ❝❤õ ❝ô♥❣ ữủ t tợ s õ ởt sè ✤➸ ♠✐♥❤ ❤♦↕✳ ❈✉è✐ ❝❤÷ì♥❣ ♥➯✉ →♣ ❞ư♥❣ ♣❤➙♥ r➣ ❉❛♥t③✐❣ ✲ ❲♦❧❢❡ ✈➔♦ ❜➔✐ t♦→♥ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ❝â ❝➜✉ tró❝ ♠❛ tr➟♥ ✤÷í♥❣ ❝❤➨♦ ❦❤è✐✳ ❈❤÷ì♥❣ ợ t ỵ r rs tr tt t t ỵ r rs q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤✱ tr♦♥❣ ✤â ❝→❝ ✈➨❝tì ✤✐➲✉ ❦✐➺♥ t ữủ t t õ ỗ tớ Số hóa trung tâm học liệu http://www.lrc.tnu.edu.vn/ ✻ ❝→❝ ✈➨❝tì t❤✉ë❝ ♥❤â♠ t❤ù ♥❤➜t ❝â ❝➜✉ tró❝ ✤➦❝ ❜✐➺t ✭❝❤➥♥❣ ❤↕♥ ❝➜✉ tró❝ ❜➔✐ t♦→♥ ✈➟♥ t↔✐✱ ❝➜✉ tró❝ ♠❛ tr➟♥ ❦❤è✐ ❤❛② ❝➜✉ tró❝ ♠❛ tr➟♥ ❜➟❝ t❤❛♥❣ ✳✳✳✮✱ ❝❤♦ ♣❤➨♣ ❦❤✐ ❣✐↔✐ ❜➔✐ t♦→♥ ❝❤➾ ✈ỵ✐ ❝→❝ ✈➨❝tì ✤✐➲✉ ❦✐➺♥ t❤✉ë❝ ♥❤â♠ ♥➔② ❝â t❤➸ sû ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ r✐➯♥❣ ❝â ❤✐➺✉ qõ❛ ❤ì♥ s♦ ợ ữỡ ỏ tỡ tở õ tự õ tờ qt ị tữ r ữ t ợ ✈➲ ❜➔✐ t♦→♥ ➼t ❜✐➳♥ ❤ì♥ ✭❣å✐ ❧➔ ❜➔✐ t♦→♥ ữ ợ rt r r ♥➔② ✤÷đ❝ t➻♠ ❞➛♥ ❦❤✐ ❝➛♥✱ ♥❤í ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ♣❤ö✳ ❙❛✉ ✤â ♥➯✉ ♠ët ✈➼ ❞ö sè ✤➸ ♠✐♥❤ ❤♦↕✳ ❈✉è✐ ❝❤÷ì♥❣ ♥➯✉ →♣ ❞✉♥❣ ♣❤➙♥ r➣ ❇❡♥❞❡rs ✈➔♦ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ❝â ❝➜✉ tró❝ ♠❛ tr➟♥ t ữủ t ợ sỹ ữợ t t ❚r➛♥ ❱ô ❚❤✐➺✉ ✲ ❱✐➺♥ t♦→♥ ❤å❝ ❱✐➺t ◆❛♠✳ ❚ø ✤→② ❧á♥❣ ♠➻♥❤✱ ❡♠ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤è✐ ✈ỵ✐ sü q✉❛♥ t➙♠✱ ✤ë♥❣ ✈✐➯♥ sỹ ữợ t tr trồ ỡ tợ ổ tr rữớ ✣↕✐ ❍å❝ ❑❤♦❛ ❍å❝ ✲ ✣↕✐ ❍å❝ ❚❤→✐ ◆❣✉②➯♥✱ ♣❤á♥❣ rữớ ỗ tớ tổ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ t➟♣ t❤➸ ❧ỵ♣ ❝❛♦ ❤å❝ t♦→♥ ❑✺❈✱ ❑✺❆ ❚r÷í♥❣ ✣↕✐ ❍å❝ ❑❤♦❛ ❍å❝ ✤➣ ✤ë♥❣ ✈✐➯♥ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❧➔♠ ❧✉➙♥ ✈➠♥ ♥➔②✳ ❚ỉ✐ ①✐♥ ❝↔♠ ì♥ tỵ✐ ❯❇◆❉ ❤✉②➺♥ ◆❛ ❍❛♥❣ ✲ P●❉ ✣➔♦ t↕♦ ◆❛ ❍❛♥❣ ỗ trữớ t❤ỉ♥❣ ❉➙♥ ❚ë❝ ❇→♥ ❚ró ❚❍❈❙ ❙✐♥❤ ▲♦♥❣ ✲ ◆❛ ❍❛♥❣ ✲❚✉②➯♥ ◗✉❛♥❣ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤➔♥❤ ❦➳ ❤♦↕❝❤ ❤å❝ t➟♣✳ ❉♦ ✤➙② ❧➔ ❧➛♥ ✤➛✉ t✐➯♥ t❤ü❝ ❤✐➺♥ ❝æ♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉✱ ♥➯♥ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✱ rt ữủ sỹ õ õ ỵ ❝→❝ ❚❤➛② ❈æ ✈➔ ❝→❝ ❜↕♥ ✤➸ ❜↔♥ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✵✽ t❤→♥❣ ✵✽ ♥➠♠ ✷✵✶✸ ◆❣÷í✐ t❤ü❝ ❤✐➺♥ ❱÷ì♥❣ ▼↕♥❤ ❈÷í♥❣ Số hóa trung tâm học liệu http://www.lrc.tnu.edu.vn/ ✼ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ tự ỡ s t ỗ t ỗ ỵ t ỗ q ❝→❝ ✤➾♥❤ ✈➔ ❝↕♥❤ ✈æ ❤↕♥ ❝õ❛ ♥â✳ ◆➯✉ ❝→❝ ❦✐➳♥ t❤ù❝ ❝➛♥ t❤✐➳t ✈➲ ❜➔✐ t♦→♥ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ✈➔ t➼♥❤ ❝❤➜t✱ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ✈➔ ❝→❝ q✉❛♥ ❤➺ ✤è✐ ♥❣➝✉ tr♦♥❣ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤✳ ❈✉è✐ ữỡ tợ tt t ỡ ❣✐↔✐ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ t❤❛♠ ❦❤↔♦ ❝❤õ ②➳✉ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✷❪✱ ❬✸❪ ỗ t ỗ ỗ ởt q trồ ữủ ũ rở r tr tố ữ ỗ õ t t ú ỵ t t ỗ ✤❛ ❞✐➺♥✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❚➟♣ ❝♦♥ ❈ tr♦♥❣ Rn ữủ ởt t ỗ õ ự trồ ✤♦↕♥ t❤➥♥❣ ♥è✐ ❤❛✐ ✤✐➸♠ ❜➜t ❦ý t❤✉ë❝ ♥â✳ ◆â✐ t ỗ a + (1 − α)b ∈ C ✈ỵ✐ ♠å✐ ❛✱ ❜ ∈ C ✈➔ ♠å✐ ≤ α ≤ ◆â✐ r✐➯♥❣✱ t➟♣ , t ỗ t ởt tỷ t ổ Rn t ỗ ã ởt số t ỗ ú ỵ tự t➟♣ ❝❤ù❛ trå♥ ✤÷í♥❣ t❤➥♥❣ ✤✐ q✉❛ ❤❛✐ ✤✐➸♠ ❜➜t ❦ý t❤✉ë❝ ♥â✳ ❜✱ ❙✐➯✉ ♣❤➥♥❣✱ tù❝ ❧➔ t➟♣ ❝â ❞↕♥❣ H = x ∈ Rn : aT x = α, a ∈ Rn \ {0} , α ∈ R } ❝✱ ❈→❝ ♥û❛ ❦❤æ♥❣ ❣✐❛♥ ✤â♥❣ H + = x ∈ Rn : aT x ≤ α , H − = x ∈ Rn : aT x ≥ α} ❞✱ ❈→❝ ♥û❛ ❦❤æ♥❣ ❣✐❛♥ ♠ð H + = x ∈ Rn : aT x < α , H − = x ∈ Rn : aT x > α} ❡✱ ❍➻♥❤ ❝➛✉ ✤â♥❣ B(a, r) = {x ∈ Rn : r > trữợ.) Soỏ hoựa bụỷi trung taõm học liệu x − a ≤ r} (a ∈ Rn http://www.lrc.tnu.edu.vn/ ã ứ t ỗ trỹ t s✉② r❛ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ s❛✉ ✤➙②✿ ởt t ý t ỗ ởt t ỗ ữ ủ ổ ú t ỗ t ỗ ụ t ỗ C ⊂ Rm, D ⊂ Rn t❤➻ t➼❝❤ C × D = {(x, y) : x ∈ C, y ∈ D} ởt t ỗ tr Rm+n õ t rở t ỗ M ởt t ❛❢✐♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ M = a + L ✈ỵ✐ a ∈ M ✈➔ ▲ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥✱ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ s♦♥❣ s♦♥❣ ✈ỵ✐ M ✱ ❤❛② t÷ì♥❣ ✤÷ì♥❣✿ M ❧➔ ♠ët t➟♣ ❛❢✐♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ M ❧➔ t➟♣ ♥❣❤✐➺♠ ❝õ❛ ♠ët ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤✱ tù❝ ❝â ❜✐➸✉ ❞✐➵♥ M = {x ∈ Rn : Ax = b, A ∈ Rm+n , b ∈ Rm } ●✐❛♦ ❝õ❛ ❜➜t ❦ý ❝→❝ t➟♣ ❛❢✐♥ ❧➔ t➟♣ ❛❢✐♥✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ❛✮ ✣✐➸♠ x ∈ Rn ❝â ❞↕♥❣ x = λ1a1+λ2a2+ +λk ak ✈ỵ✐ ∈ Rn , λi ≥ 0, λ1 + λ2 + + λk = ✱ ❣å✐ ❧➔ ởt tờ ủ ỗ a1 , a2 , , ak ❜✮ ✣✐➸♠ x ∈ Rn ❝â ❞↕♥❣ x = λ1a1 + λ2a2 + + λk ak ✈ỵ✐ ∈ Rn, λi ≥ 0, λ1 + λ2 + + λk = ✱ ❣å✐ ❧➔ ♠ët ❛❢✐♥ ❝õ❛ ❝→❝ ✤✐➸♠ a1 , a2 , , ak ❝✮ ✣✐➸♠ x ∈ Rn ❝â ❞↕♥❣ x = λ1a1 + λ2a2 + + λk ak ✈ỵ✐ ∈ Rn, λi ≥ ✱ ❣å✐ ❧➔ ♠ët tê ❤ì♣ t✉②➳♥ t➼♥❤ ❦❤ỉ♥❣ ➙♠ ❤❛② ❧➔ tê ❤đ♣ ♥â♥ ❝õ❛ ❝→❝ ✤✐➸♠ a1 , a2 , , ak ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳ ❈❤♦ E ❧➔ ♠ët t➟♣ ❜➜t ❦ý tr♦♥❣Rn ✱ ✳ ✳ ✳ ✱ ❛✮ ●✐❛♦ ❝õ❛ t➜t ❝↔ ❝→❝ t➟♣ ❛❢✐♥ ❝❤ù❛ E ❣å✐ ❧➔ ❜❛♦ E, ỵ E õ t➟♣ ❛❢✐♥ ♥❤ä ♥❤➜t ❝❤ù❛ E ✳ ❜✮ ●✐❛♦ ❝õ❛ tt t ỗ ự E ỗ E, ỵ E õ t ỗ ọ t ự E ự số ởt t M, ỵ ❞✐♠M, ❧➔ t❤ù ♥❣✉②➯♥ ✭sè ❝❤✐➲✉✮ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ s s ợ õ ữợ dim = ự số ởt t ỗ C, ỵ E, tự số ❝❤✐➲✉ ❝õ❛ ❜❛♦ ❛❢✐♥ ❛❢❢ C ❝õ❛ ♥â✳ ▼ët t➟♣ ỗ C tr Rn õ tự dimC = n ỗ ởt t ỗ õ trú ỡ rt tr ỵ tt tố ữ t t ởt t ỗ ♠ët sè ❤ú✉ ❤↕♥ ❝→❝ ♥û❛ ❦❤æ♥❣ ❣✐❛♥ ✤â♥❣ ❣å✐ ởt t ỗ õ õ ❧➔ t➟♣ Số hóa trung tâm học liệu http://www.lrc.tnu.edu.vn/ ữỡ ỵ r rs ữỡ ♥➔② tr➻♥❤ ❜➔② t❤✉➟t t♦→♥ ♣❤➙♥ r➣ ❇❡♥❞❡rs ❣✐↔✐ q✉✐ t t ỵ tữ ữỡ tt t t ữ t ợ ♥❤✐➲✉ ❜✐➳♥ ✈➲ ❝→❝ ❜➔✐ t♦→♥ ➼t ❜✐➳♥ ❤ì♥ ✈➔ ✤÷❛ r❛ ✈➼ ❞ư sè ♠✐♥❤ ❤♦↕✳ ❈✉è✐ ❝❤÷ì♥❣ ♥➯✉ →♣ ❞ö♥❣ ♣❤➙♥ r➣ ❇❡♥❞❡rs ✈➔♦ ❜➔✐ t♦→♥ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ✈ỵ✐ ❝➜✉ tró❝ ♠❛ tr➟♥ ❜➟❝ t❤❛♥❣✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ t❤❛♠ ❦❤↔♦ ❝❤õ ②➳✉ tø ❝→❝ t ị tữ r ❝õ❛ ❇❡♥❞❡rs ▼ö❝ ♥➔② t❛ ①➨t ❜➔✐ t♦→♥ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ s❛✉ ✤➙②✿ ❚➻♠ ❝→❝ ❜✐➳♥ sè x1, x2, , xn ✈➔ y1, y2, , yp t❤♦↔ ♠➣♥ p n (P0 ) f (x, y) = dk yk → ci xj + j=1 ✭✸✳✶✮ k=1 ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ p n aij xj + j=1 dik yk = bi , i = 1, 2, , m, k=1 x ∈ X, yk ≥ 0, k = 1, 2, , p ✭✸✳✷✮ ✭✸✳✸✮ ❚r♦♥❣ ✤â aij , bi, cj , dik , dk ỳ số trữợ t n X ⊂ Rn ✭❈❤➥♥❣ ❤↕♥ X = R+ ❤♦➦❝ X = {0, 1}n) ỵ A = (aij)mìn, D = (dik )m×p , b = (b1 , b2 , , bm )T , d = (d1 , d2 , , dp )T , c = (c1 , c2 , , cn )T ❚❛ ✈✐➳t ❧↕✐ ❜➔✐ t♦→♥ ✭✸✳✶✮ ✲ ữợ (P0 ) cT x + dT y : Ax + Dy ≥ b, x ∈ X, y ≥ ❚❤❡♦ ❝→❝❤ t✐➳♣ ❝➟♥ ❇❡♥❞❡rs✱ số t ữủ rút ợt t r t tự ợ ữỡ tỹ ♥❤÷ ❦ÿ t❤✉➟t Số hóa trung tâm học liệu http://www.lrc.tnu.edu.vn/ ✸✾ s✐♥❤ ❝ët tr♦♥❣ ❜➔✐ t♦→♥ ❝❤õ ❉❛♥t③✐❣ ✲ ❲♦❧❢❡✱ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ r➔♥❣ ❜✉ë❝ tr♦♥❣ ❜➔✐ t♦→♥ ❝❤õ ❇❡♥❞❡rs ❝❤➾ ✤÷đ❝ s✐♥❤ r❛ ❦❤✐ ❝➛♥✱ ♥❤í ❞ị♥❣ ❝→✐ ❣å✐ ❧➔ ❦ÿ t❤✉➟t s✐♥❤ ❞á♥❣ ❝â tr➵ ✭❉❡❧❛②❡❞ ❘♦✇ ✲ ●❡♥❡r❛ t✐♦♥ Pr♦❝❡❞✉r❡✮✳ P❤➙♥ r➣ ❇❡♥❞❡rs ❣✐ú ✈❛✐ trá tr✉♥❣ t➙♠ tr♦♥❣ ✈✐➺❝ ❣✐↔✐ ❜➔✐ t♦→♥ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ♥❣➝✉ ♥❤✐➯♥ ♥❤✐➲✉ ❣✐❛✐ ✤♦↕♥✳ ▼ö❝ ♥➔② tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ r➣ ❇❡♥❞❡rs →♣ ❞ư♥❣ ❣✐↔✐ ❜➔✐ t♦→♥ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ (P0) P❤➙♥ r➣ ❇❡♥❞❡rs ❜✐➳♥ ✤ê✐ ❤➺ ♠ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝õ❛ (n + p) ➞♥ sè (x, y) t❤➔♥❤ ❤➺ p + ➞♥ số tữớ ợ t ữỡ tr ỡ ổ ❦❤✐ ♣❤➙♥ r➣ ❇❡♥❞❡rs ❝á♥ ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ❦❤û ❜✐➳♥ ✭❡❧✐♠✐♥❛t❡ ✈❛r✐❛❜❧❡s✮✳ ❑❤✐ ❝è ✤à♥❤ x ∈ X, (P0) trð t❤➔♥❤ ❜➔✐ t♦→♥ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ t❤❡♦ ❜✐➳♥ ②✿ ▲P✭ ①✮ dT y + cT x : Dy ≥ b − Ax, y ≥ ✭✸✳✹✮ ✣➸ ✤ì♥ ❣✐↔♥ ❧➟♣ ❧✉➟♥✱ t❛ ❣✐↔ t❤✐➯t r➡♥❣ ✈ỵ✐ ộ x X ố ổ tỗ t y ≥ t❤ä❛ ♠➣♥ Dy ≥ b − Ax, tù❝ (x, y) ữủ ố ợ t♦→♥ ❜❛♥ ✤➛✉ (P0) ❱ỵ✐ ❣✐↔ t❤✐➯t ♥➔②✱ ❜➙② ❣✐í t❛ s➩ ❝❤➾ r❛ ❝→❝❤ t➻♠ ♥❣❤✐➺♠ tè✐ ÷✉ ❝õ❛ ❜➔✐ t♦→♥ LP (x) ①→❝ ✤à♥❤ t❤❡♦ ✭✸✳✹✮✳ ▲➜② ✤è✐ ♥❣➝✉ ❝õ❛ LP (x) t❛ ♥❤➟♥ ✤÷đ❝ ❜➔✐ t♦→♥ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ✭t❤❡♦ ❜✐➳♥ ✤è✐ ♥❣➝✉ u ∈ Rm ỵ LD(x) max (b Ax)T u + cT x : DT y ≤ d, u = (x) ỵ C = u ∈ Rm : DT y ≤ d, u ≥ ❧➔ ♠✐➲♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ▲❉✭ ①✮✳ ó r ởt t ỗ sû u1, u2, , uL ❧➔ ❝→❝ ✤➾♥❤ ✈➔ v1, v2, , vM ❧➔ ♣❤÷ì♥❣ ❝ü❝ ❜✐➯♥ ✭❝❤✉➞♥ ❤â❛✮ ❝õ❛ ❈ tù❝ ❧➔ DT ui ≤ d, ui ≥ 0, ∀i, ✈➔ DT v j ≤ 0, v j ≥ 0, ∀j ❚❛ ♥❤➢❝ ❧↕✐ ❦❤→✐ ♥✐➺♠ ✤✐➸♠ ❝ü❝ ❜✐➯♥ ữỡ ỹ t ỗ C Rn : • x0 ∈ ❈ ❧➔ ✤✐➸♠ ❝ü❝ ❜✐➯♥ ❝õ❛ C ổ tỗ t x1 , x2 C x1 = x0 , x2 = x0 ✈➔ λ ∈ (0, 1) s❛♦ ❝❤♦ x0 = λx1 + (1 − λ) x2 ◆â✐ ❝→❝❤ ❦❤→❝✱ ✤✐➸♠ ❝ü❝ ❜✐➯♥ ❝õ❛ ởt t ỗ ỳ ổ tr ✤♦↕♥ t❤➥♥❣ ♥è✐ ❤❛✐ ✤✐➸♠ ❦❤→❝ ❜➜t ❦ý t❤✉ë❝ t➟♣ õ ã ởt ữỡ ổ C ♠ët ♣❤÷ì♥❣ ❝ü❝ ❜✐➯♥ ❝õ❛ C ♥➳✉ ♥â ❦❤ỉ♥❣ t❤➸ ữợ tờ ủ t t ữỡ ❤❛✐ ♣❤÷ì♥❣ ✈ỉ ❤↕♥ ❦❤→❝ ❝õ❛ C ✳ ◆❤÷ ✈➟②✱ ♥➳✉ d0 ∈ Rn ❧➔ ♣❤÷ì♥❣ ❝ü❝ ❜✐➯♥ ❝õ❛ C t❤➻ Số hóa trung tâm học liệu http://www.lrc.tnu.edu.vn/ ✹✵ ❦❤ỉ♥❣ t❤➸ ❝â d0 = λ1d1 + λ2d2 ✈ỵ✐ λ1, λ2 ≥ ✈➔ d1 = d0, d2 = d0 ❧➔ ❤❛✐ ♣❤÷ì♥❣ ✈ỉ ❤↕♥ ✭♣❤÷ì♥❣ ❧ị✐ ①❛✮ ❝õ❛ C C ởt t ỗ t ❝ü❝ ❜✐➯♥ ❝õ❛ C ❣å✐ ❧➔ ♠ët ✤➾♥❤ ❝õ❛ C ✈➔ ✈➨❝tì ❝❤➾ ♣❤÷ì♥❣ ❝õ❛ ♠ët ❝↕♥❤ ✈ỉ ❤↕♥ ❝õ❛ C ❧➔ ♠ët ♣❤÷ì♥❣ ❝ü❝ ❜✐➯♥ ❝õ❛ C ❚❤❡♦ ✣à♥❤ ỵ ộ u C õ L K i u= βj v j , αi u + i=1 ✭✸✳✺✮ j=1 ✈ỵ✐ αi ≥ 0(i = 1, 2, , L), α1+α2+ +αL = ✈➔ βj ≥ 0(j = 1, 2, , M ) ❚ø ✤â L T K i D u= βj Dv j ≤ d,u ≥ αi Du + i=1 j=1 ❈→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✤ó♥❣ ❧➔ ❞♦ ❝→❝ ✤➾♥❤ ui ✈➔ ♣❤÷ì♥❣ ❝ü❝ ❜✐➯♥ vi t❤ä❛ ♠➣♥ DT ui ≤ d, ui ≥ (i = 1, 2, , L) ✈➔ DT v j ≤ 0, v j ≥ (j = 1, 2, , M ) ❱ỵ✐ ❜✐➸✉ ❞✐➵♥ ✉ t❤❡♦ ✭✸✳✺✮✱ ❜➔✐ t♦→♥ LD(x) ❝â t❤➸ ✈✐➳t ❧↕✐ t❤❡♦ ❜✐➳♥ αi , βj t❤➔♥❤✿ L K T i (b − Ax)T v j βj + cT x → max (b − Ax) u αi + i=1 ✭✸✳✻✮ j=1 ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ α1 + α2 + + αL = 1, αi ≥ (i = 1, 2, , L) , βj ≥ (j = 1, 2, , M ) ❚❛ ❝â ❝→❝ ♥❤➟♥ ①➨t s❛✉ ✈➲ ❜➔✐ t♦→♥ ✭✸✳✻✮✿ • ❉♦ ❣✐↔ t❤✐➯t ❜➔✐ t♦→♥ (P0 ) ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ♥➯♥ ❜➔✐ t♦→♥ ✭✸✳✻✮ ❝â ♥❣❤✐➺♠ tè✐ ÷✉ ❤ú✉ ❤↕♥ ✈ỵ✐ ♠é✐ ① ✤➣ ❝❤å♥✱ ❜ð✐ ✈➻ ♥➳✉ tr→✐ ❧↕✐ ❜➔✐ t♦→♥ LP (x) s➩ ❦❤ỉ♥❣ ❝â ♣❤÷ì♥❣ →♥ ✭♠✐➲♥ r➔♥❣ ❜✉ë❝ ré♥❣✮✳ ◆❤÷ ✈➟②✱ t➼♥❤ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ LP (x) ❦➨♦ t❤❡♦ ① t❤ä❛ ♠➣♥ (b − Ax)T v j ≤ 0, ✈ỵ✐ ♠å✐ (j = 1, 2, , M ) ✭✸✳✼✮ Số hóa trung tâm học liệu http://www.lrc.tnu.edu.vn/ ✹✶ ❈❤♦ ① t❤ä❛ ♠➣♥ ✭✸✳✼✮✱ ❦❤✐ ✤â ♥❣❤✐➺♠ tè✐ ÷✉ ❝õ❛ ▲❉✭①✮ ♣❤↔✐ ❧➔ ♠ët tr♦♥❣ ❝→❝ ✤➾♥❤ ❝õ❛ ♠✐➲♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ▲❉✭①✮✱ ♥❣❤➽❛ ❧➔ • ϕ (x) = max (b − Ax)T ui 1≤i≤L ◆❤÷ ✈➟②✱ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ❜❛♥ tữỡ ữỡ ợ t cT x + max (b − Ax)T ui x∈X 1≤i≤L ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✭✸✳✼✮✳ ❉➵ ❞➔♥❣ t❤➜② r➡♥❣ ❜➔✐ t♦→♥ ♥➔② ❝â t❤➸ ✈✐➳t ❧↕✐ t❤➔♥❤ cT x + γ → ✭✸✳✽✮ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✈ỵ✐ ♠å✐ i = 1, 2, , L (b − Ax)T v j − γ ≤ ✈ỵ✐ ♠å✐ j = 1, 2, , M ✭✸✳✾✮ ✭✸✳✶✵✮ x∈X ✭✸✳✶✶✮ ❝→❝ r➔♥❣ ❜✉ë❝ ✭✸✳✶✵✮ ✤÷đ❝ ✤÷❛ ✈➔♦ ✤➸ ✤↔♠ ❜↔♦ r➡♥❣ ✈✐➺❝ ❝❤å♥ x ∈ X s tỗ t s (x, y) ữủ ố ợ t (P0) (b − Ax)T ui − γ ≤ ✸✳✷✳ ❇➔✐ t♦→♥ ❝❤õ ✈➔ ❜➔✐ t♦→♥ ❝❤õ t❤✉ ❤➭♣ ✣➸ ❝❤♦ ❣å♥✱ t❛ ✤➦t Gi = Aui , H j = Av j , gi = bT ui , hj = bT v j ❚❤➳ ❝→❝ ✤↕✐ ❧÷đ♥❣ ♥➔② ✈➔♦ ❜➔✐ t♦→♥ ✭✸✳✽✮ ✲ ✭✸✳✶✶✮ t❛ ♥❤➟♥ ✤÷đ❝ ❜➔✐ t♦→♥ tũ ỵ s ❝❤♦ β = cT x + γ → ✭✸✳✶✷✮ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ xT Gi − γ ≤ gi , ✈ỵ✐ ♠å✐ ✐ = 1, 2, , L ✭✸✳✶✸✮ xT H j ≤ 0, ✈ỵ✐ ♠å✐ ❥ = 1, 2, , M ✭✸✳✶✹✮ x ∈ X ✭✸✳✶✺✮ Số hóa trung tâm học liệu http://www.lrc.tnu.edu.vn/ ✹✷ ❇➔✐ t♦→♥ ✭✸✳✶✷✮ ✲ ✭✸✳✶✺✮ ❣å✐ ❧➔ ❜➔✐ t♦→♥ ❝❤õ ❇❡♥❞❡rs✳ ❙♦ ✈ỵ✐ (P0)✱ ❜➔✐ t♦→♥ ♥➔② ❝â sè ❜✐➳♥ ➼t ❤ì♥ ♥❤÷♥❣ sè r➔♥❣ ❜✉ë❝ ❝â t❤➸ ❤ì♥ r➜t ♥❤✐➲✉✱ ❝ư t❤➸ ❧➔ ✭▲ ✰ ▼✮×✭♥ ✰ ✶✮✱ tr♦♥❣ ✤â ▲ ❧➔ sè ✤➾♥❤ ✈➔ ▼ ❧➔ sè ♣❤÷ì♥❣ ❝ü❝ ❜✐➯♥ ❝õ❛ t➟♣ ỗ C = {u Rm : DT u d, u 0} ỵ ✤ê✐ ❇❡♥❞❡rs✮✳ ❇➔✐ t♦→♥ ❝❤õ ❇❡♥❞❡rs ✭✸✳✶✷✮ ✲ ✭✸✳✶✺✮ t÷ì♥❣ ữỡ ợ t q t t tữợ m ì (n + p) ❈→❝ r➔♥❣ ❜✉ë❝ ❝õ❛ ❜➔✐ t♦→♥ ❝❤õ s➩ ✤÷đ❝ ①→❝ ✤à♥❤ ❞➛♥ tr♦♥❣ q✉→ tr➻♥❤ ❣✐↔✐✳ ●✐è♥❣ ♥❤÷ tr♦♥❣ ♣❤➙♥ r➣ ❉❛♥t③✐❣ ✲ ❲♦❧❢❡✱ t❤÷í♥❣ ❦❤ỉ♥❣ t❤➸ ❜✐➸✉ ❞✐➵♥ t÷í♥❣ ♠✐♥❤ t➜t ❝↔ ❝→❝ ✤➾♥❤ ✈➔ ♣❤÷ì♥❣ ❝ü❝ ❜✐➯♥ ❝õ❛ t ỗ r r rs ụ t❤÷í♥❣ ❦❤ỉ♥❣ t❤➸ ❜✐➸✉ ❞✐➵♥ ❤✐➸♥ t♦➔♥ ❜ë ❝→❝ r➔♥❣ ❜✉ë❝ ❝õ❛ ❜➔✐ t♦→♥ ❝❤õ ✭✸✳✶✷✮ ✲ ✭✸✳✶✺✮✳ ✣➸ ❜➢t ✤➛✉ ♠ët ✈á♥❣ ❧➦♣ ❦✱ ❣✐↔ t❤✐➳t r➡♥❣ t❛ ✤➣ tứ tứ ỏ trữợ ởt t Lk ≥ ❜➜t ✤➥♥❣ t❤ù❝ ❝õ❛ t➟♣ ▲ ❜➜t ✤➥♥❣ t❤ù❝ ✤➛✉ ✈➔ t➟♣ ❝♦♥ Mk ≥ ❜➜t ✤➥♥❣ t❤ù❝ ❝õ❛ M ❜➜t ✤➥♥❣ t❤ù❝ t❤ù ❤❛✐ tr♦♥❣ ❜➔✐ t♦→♥ ❝❤õ✳ ❇➔✐ t♦→♥ t✉②➳♥ t➼♥❤ ✈ỵ✐ Lk + Mk ❜➜t ✤➥♥❣ t❤ù❝ ✤÷đ❝ ❣å✐ ❧➔ ❜➔✐ t♦→♥ ❝❤õ ❇❡♥❞❡rs t❤✉ ❤➭♣ ✭❇❡♥❞❡rs ❘❡str✐❝t❡❞ ▼❛st❡r Pr♦❣r❛♠✮✳ ❈→❝ ❜➜t ✤➥♥❣ t❤ù❝ ♥➔② ❣å✐ ❧➔ ❝→❝ ❧→t ❝➢t ✭❝✉ts✮✱ ♠ët t❤✉➟t ♥❣ú ❧➜② tø q✉✐ ❤♦↕❝❤ ♥❣✉②➯♥✱ ð ✤â ♠é✐ ❜➜t ✤➥♥❣ t❤ù❝ s✐♥❤ r❛ s➩ ✧❝➢t ❜ä✧ ♠ët ♣❤➛♥ ♠✐➲♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❦❤ỉ♥❣ ❝❤ù❛ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❝➛♥ t➻♠✳ ●✐↔ sû ❜➔✐ t♦→♥ ❝❤õ ❇❡♥❞❡rs t❤✉ ❤➭♣ ❝â ♥❣❤✐➺♠ tè✐ ÷✉ xk ✈➔ γk , t❛ ①➙② ❞ü♥❣ ❜➜t ✤➥♥❣ tự ợ ữ ρk = b − Axk ✈➔ ❣✐↔✐ ❜➔✐ t♦→♥ ❝♦♥ ρk T u : DT u ≤ d, u ≥ ❑❤✐ ❣✐↔✐ ✭✸✳✶✻✮ ❝â ❤❛✐ ❦❤↔ ♥➠♥❣ r ã t t tố ữ ữủ ♥❣❤✐➺♠ tè✐ ÷✉ ❤ú✉ ❤↕♥ ♥❣❤✐➺♠ ♥➔② s✐♥❤ r❛ ❜➜t ✤➥♥❣ t❤ù❝ ♠ỵ✐ ✭✸✳✶✻✮ uk xT GLk +1 + γ ≥ gLk +1 , tr♦♥❣ ✤â GL +1 = Auk , gL +1 = bT uk ❇➜t ✤➥♥❣ t❤ù❝ ♥➔② ❣å✐ ❧➔ ❧→t ❝➢t tè✐ ÷✉ ✭❖♣t✐♠❛❧✐t② ❈✉t✮✳ k k Số hóa trung tâm học liệu http://www.lrc.tnu.edu.vn/ t❤➻ ã t t ữủ t ♥❤➟♥ ✤÷đ❝ ♥❣❤✐➺♠ tè✐ ÷✉ ✈ỉ ❝ü❝ ❧➔ ♠ët ♣❤÷ì♥❣ ❝ü❝ ❜✐➯♥ ✭❝❤✉➞♥ ❤â❛✮ vk t❤➻ ♥❣❤✐➺♠ ♥➔② s✐♥❤ r❛ ❜➜t ✤➥♥❣ t❤ù❝ ♠ỵ✐ xT H Mk +1 ≥ hMk +1 , tr♦♥❣ ✤â H M +1 = Avk , hM +1 = bT vk ❇➜t ✤➥♥❣ t❤ù❝ ♥➔② ❣å✐ ❧➔ ❧→t ❝➢t ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ✭❋❡❛s✐❜✐❧✐t② ❈✉t✮✳ ❙❛✉ ❦❤✐ t❤➯♠ ✈➔♦ ❜➔✐ t♦→♥ ❝❤õ ❇❡♥❞❡rs t❤✉ ❤➭♣ ❧→t t tố ữ t t ữủ ợ ❝❤➾ sè i = Lk+1 = Lk +1 ❤♦➦❝ j = Mk+1 = Mk +1, t❛ ❣✐↔✐ ❧↕✐ ❜➔✐ t♦→♥ ❝❤õ t❤✉ ❤➭♣ ♠ỵ✐✳ ●✐↔✐ ❜➔✐ t♦→♥ ♠ỵ✐ ♥➔②✳ ✣à♥❤ ỵ (P0) ổ õ ữỡ s❛✉ ❦❤✐ t❤➯♠ ✈➔♦ ❜➔✐ t♦→♥ ❝❤õ t❤✉ ❤➭♣ ❧→t t ữủ ợ t ❜➔✐ t♦→♥ ❦❤ỉ♥❣ ❝â ♣❤÷ì♥❣ →♥ t❤➻ ❜➔✐ t♦→♥ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ❜❛♥ ✤➛✉ (P0 ) s➩ ❦❤æ♥❣ ❝â ữỡ ỵ tố ữ t t tố ữ ữủ ố ợ x = xk ✈➔ γ = γk t❤➻ (x, y) = xk , yk ❧➔ ♠ët ♥❣❤✐➺♠ tè✐ ÷✉ ❝õ❛ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ❜❛♥ ✤➛✉ (P0), tr♦♥❣ ✤â yk ❧➔ ♣❤÷ì♥❣ →♥ tè✐ ÷✉ ❝õ❛ ❜➔✐ t♦→♥ LP (xk ) ✲ ❜➔✐ t♦→♥ ✭✸✳✹✮✳ k k ❚❤✉➟t t♦→♥ ♣❤➙♥ r rs ữợ sỹ t = , ❝❤➾ sè ✈á♥❣ ❧➦♣ ❦ ❂ ✶✱ t➟♣ ✤➾♥❤ t➟♣ ♣❤÷ì♥❣ ❝ü❝ ❜✐➯♥ V0 = ∅, L1 = |U0| = 0, M1 = |V0| = ✈➔ ❝❤å♥ ❜➜t ❦ý x1 X ữợ ỷ ỵ t ❉ị♥❣ t❤✉➟t t♦→♥ ✤ì♥ ❤➻♥❤ ❣✐↔✐ ❜➔✐ t♦→♥ LD(xk )✿ • ◆➳✉ LD(xk ) ❝â ♥❣❤✐➺♠ tè✐ ÷✉ ✈ỉ ❝ü❝ t❤➻ ❣✐↔ sû v k ❧➔ ♣❤÷ì♥❣ ❝ü❝ ❜✐➯♥ ♥❤➟♥ ữủ tứ tt t ỡ rỗ s ữợ ✷✳ • ❚r→✐ ❧↕✐ ♥➳✉ LD(xk ) ❝â ♥❣❤✐➺♠ tè✐ ÷✉ ❤ú✉ ❤↕♥✱ ❣✐↔ sû uk ❧➔ ✤➾♥❤ tè✐ ÷✉ s ữợ ữợ ỷ ỵ ♣❤÷ì♥❣ ❝ü❝ ❜✐➯♥✳ ❚❤➯♠ vk ✈➔♦ t➟♣ ♣❤÷ì♥❣ ❝ü❝ ❜✐➯♥ ❜➡♥❣ ❝→❝❤ ✤➦t Vk ← Vk−1 ∪ vk (Mk = Mk + 1) , Uk ← Uk−1(Lk = Lk−1 ) ❚➼♥❤ H k = Av k , hk = bT v k , t❤➯♠ ✈➔♦ ❜➔✐ t♦→♥ ❝❤õ t❤✉ ❤➭♣ ❧→t ❝➢t ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ xT H k ≥ hk s ữợ ữợ ỷ ỵ ✤✐➸♠ ❝ü❝ ❜✐➯♥✳ ❚➼♥❤ Gk = Auk , gk = bT uk ◆➳✉ U0 = ∅, xk Số hóa trung tâm học liệu T Gk + γk ≥ gk http://www.lrc.tnu.edu.vn/ ✹✹ t❤➻ ❞ø♥❣ t❤✉➟t t♦→♥✿ ❣✐↔✐ LP xk ✤➸ t➻♠ yk ✈➔ xk , yk ❧➔ tè✐ ÷✉ (P0 ) ỵ r t uk ✈➔♦ t➟♣ ✤➾♥❤ ❜➡♥❣ ❝→❝❤ ✤➦t Uk ← Uk−1 ∪ uk (Lk = Lk + 1) , Vk ← Vk−1 (Mk = Mk−1 ) ❚❤➯♠ ✈➔♦ ❜➔✐ t♦→♥ ❝❤õ t❤✉ ❤➭♣ ❧→t ❝➢t tè✐ ÷✉ xk T Gk + γk gk s ữợ ữợ ❇➔✐ t♦→♥ ❝❤õ (Mk ) ●✐↔✐ βk = cT x + γ → ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✈ỵ✐ ♠å✐ ui ∈ Uk−1, ✈ỵ✐ ♠å✐ vj ∈ Vk , xT Gi − γk ≤ gi , xT H j ≤ hj , x ∈ X ◆➳✉ Mk ❦❤ỉ♥❣ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ t ứ tt t ỵ (P0) ổ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ✭♠✐➲♥ r➔♥❣ ❜✉ë❝ ❝õ❛ (P0) ré♥❣ ✮✳ • ◆➳✉ Mk ❝â ❧í✐ ❣✐↔✐ tè✐ ÷✉ ❤ú✉ ❤↕♥ t❤➻ ❝❤å♥ βk+1 , xk+1 ❧➔ ❧í✐ ❣✐↔✐ tè✐ ÷✉ t ý ã Mk õ tố ữ ✈æ ❤↕♥ t❤➻ ❝❤å♥ βk+1 = βk ✈➔ xk+1 ❧➔ ① ❜➜t ❦ý t❤ä❛ ♠➣♥ xT H j ≥ hj ợ v j Vk ã k k + tr ữợ ã ỵ tử tt t rs ỵ ✸✳✷✳✹✳ ●✐↔ sû (P0) ❧➔ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝✳ ❑❤✐ ✤â✱ s ởt số ỳ ữợ tt t tr s t➻♠ ✤÷đ❝ ♥❣❤✐➺♠ tè✐ ÷✉ ❝❤♦ ❜➔✐ t♦→♥ (P0) ❤♦➦❝ r r ổ tỗ t tố ữ ❱➼ ❞ö ♠✐♥❤ ❤å❛ ♣❤➙♥ r➣ ❇❡♥❞❡rs ❱➼ ❞ö ✸✳✸✳✶✳ ❳➨t ❜➔✐ t♦→♥ q✉✐ ❤♦↕❝❤ ♥❣✉②➯♥ ❤é♥ ❤ñ♣ ✵ ✲ ✶✳ 42x1 + 18x2 + 33x3 − 8y1 − 6y2 + 2y3 → 10x1 + 8x2 − 2y1 − y2 + y3 ≥ 4, + 8x3 − y1 − y2 − y3 ≥ 3, 5x1 x1 , x2 , x3 ∈ {0, 1} , y1 , y2 , y3 ≥ Số hóa trung tâm học liệu http://www.lrc.tnu.edu.vn/ ✹✺ ❈è ✤à♥❤ ❝→❝ ❜✐➳♥ x1, x2, x3 ✈➔ ❝❤✉②➸♥ ❝❤ó♥❣ s❛♥❣ ✈➳ ♣❤↔✐ r➔♥❣ ❜✉ë❝ t❛ ♥❤➟♥ ✤÷đ❝ ❜➔✐ t♦→♥ t❤❡♦ ❝→❝ ❜✐➳♥ y1, y2, y3 −8y1 − 6y2 + 2y3 + (42x1 + 18x2 + 33x3 ) → −2y1 − y2 + y3 ≥ − 10x1 − 8x2 , −y1 − y2 − y3 ≥ − 5x1 − 8x3 y1 , y2 , y3 ≥ ❇➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ❝õ❛ ❜➔✐ t♦→♥ ♥➔② ❝â ❞↕♥❣✿ (4 − 10x1 − 8x2 )u1 + (3 − 5x1 − 8x3 )u2 + (42x1 − 18x2 − 33x3 )u → max ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ −2u1 − u2 ≤ −8, −u1 − u2 ≤ −6, u1 − u2 ≤ 2, u1 ≥ 0, u2 ≥ ▼✐➲♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ C ❝õ❛ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ✈➩ ð ❍➻♥❤ ✸✳✶✳ ❍➻♥❤ ✸✳✶✳ ▼✐➲♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ C ❝õ❛ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ k T k k T k ❱ỵ✐ k = 5, β ≥ c x + b − Ax u ✤÷đ❝ t❤ä❛ ♠➣♥✳ ❚❛ ❦➳t ❧✉➟♥ x∗ = x5 = (1, 0, 0) , β = 26, tữỡ ự ợ y = (2, 0, 0) t➻♠ ✤÷đ❝ ♥❤í ❣✐↔✐ P1 y5 Số hóa trung tâm học liệu http://www.lrc.tnu.edu.vn/ ✹✻ ✸✳✹✳ ◗✉② ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ❝➜✉ tró❝ ❜➟❝ t❤❛♥❣ ◗✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ❝➜✉ tró❝ ❜➟❝ t❤❛♥❣ ❝â ♠❛ tr➟♥ r➔♥❣ ❜✉ë❝ ❝❤✐❛ t→❝❤ t❤➔♥❤ K × K ỉ ✈✉ỉ♥❣ ❤❛② ❝❤ú ♥❤➟t✱ ♠é✐ ỉ ❧➔ ♠ët ♠❛ tr➟♥ ❝♦♥ ♠➔ ♣❤➛♥ ❧ỵ♥ ✤➲✉ ❧➔ ❝→❝ ♠❛ tr➟♥ ❦❤æ♥❣✱ trø ❝→❝ ♠❛ tr➟♥ ❝♦♥ tr ữợ ữớ ❤↕♥✿ z = c1 T x + c2 T x + c3 T x + c4 T x → ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ A11 x1 A21 x1 + A22 x2 A32 x2 + A33 x3 A43 x3 + A44 x4 = = = = b1 , b2 , b3 , b4 , ð ✤➙② K = ❚❛ s➩ ❣å✐ ❝→❝ ❜➟❝ t❤❛♥❣ ❧✐➯♥ t✐➳♣ tø ✤➛✉ tỵ✐ ❝❤➙♥ t❤❛♥❣ ❧➔ ❝→❝ t❤í✐ ❦ý t ❂ ✶✱ t ❂ ✷✱ t ❂ ✸ ✈➔ t ❂ ✹✱ ♠➦❝ ❞ị tr♦♥❣ ♥❤✐➲✉ ù♥❣ ❞ư♥❣ ❝→❝ ❜➟❝ t❤❛♥❣ ❝â t❤➸ ❧➔ ❝→❝ ❣✐❛✐ ✤♦↕♥ ❝õ❛ q✉→ tr➻♥❤ s↔♥ t trú t ỵ P❤÷ì♥❣ ♣❤→♣ ♠ỉ t↔ s❛✉ ✤➙② ❝â t➼♥❤ ❦❤→✐ q✉→t✱ ❝â t❤➸ ❞ị♥❣ ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥ ❝â ❝➜✉ tró❝ ❜➟❝ t❤❛♥❣ ✈ỵ✐ ❑ ❜➜t ❦ý✳ ❚r♦♥❣ ♠ư❝ ♥➔② t❛ s ỵ r rs t ủ ợ sỷ tr ỗ t r ỗ st st r ởt t ỗ t t t❤í✐ ❣✐❛♥✱ ❜➢t ✤➛✉ tø ♣❤➙♥ ❤♦↕❝❤ ① t❤➔♥❤ {x1}, {x2, x3, x4} ❑➳t q✉↔ ❧➔ ❜➔✐ t♦→♥ ❝♦♥ xk ≥ 0, k = 1, , K, Số hóa trung tâm học liệu http://www.lrc.tnu.edu.vn/ ✹✼ ❇❡♥❞❡rs ❝➛♥ ❣✐↔✐ t❤❡♦ x1 ❝è ✤à♥❤✿ γ1 = c2 T x + c3 T x + c4 T x → ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ A22 x2 = b2 − A21 x1 , A32 x2 + A33 x3 = b3 , A43 x3 + A44 x4 = b4 , xk ≥ 0, k = 2, 3, ❇➔✐ t♦→♥ ❝♦♥ ♥➔② ✤÷đ❝ ❣✐↔✐ ❜➡♥❣ ❝→❝❤ t✐➳♣ tư❝ ♣❤➙♥ ❤♦↕❝❤ {x2, x3, x4} t❤➔♥❤ {x2}, {x3, x4}✱ ✈✳✈ ✳✳✳ ▼é✐ ♣❤➙♥ ❤♦↕❝❤ ❧➔♠ ❣✐↔♠ ♠ët ❜➟❝ t❤❛♥❣ ❝❤♦ ✤➳♥ ❦❤✐ ❜➔✐ t♦→♥ ỏ ỗ t ởt t t rs t tữỡ ự ợ ❤♦↕❝❤ t❤ù ♥❤➜t✮ ❝â ❞↕♥❣ t÷ì♥❣ tü ♥❤÷ ❜➔✐ t♦→♥ ✭✸✳✽✮ ✲ ✭✸✳✶✶✮✱ ❝ư t❤➸ ❧➔ ❝â ❞↕♥❣ ✈➨❝tì ♥❤÷ s❛✉✳ c1 T x + γ1 = z → ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ GT x1 + eγ1 ≥ g, e = (1, 1, , 1)T , H T x1 x1 ≥ h, ≥ tr♦♥❣ ✤â G = G1, G2, , g = (g1, g2, )T , H = H 1, H 2, ✈➔ h = (h1 , h2 , ) T ❇➔✐ t♦→♥ ❝❤õ t❤✉ ❤➭♣ ❝❤♦ ❝→❝ ❜➔✐ t♦→♥ ❝♦♥ t✐➳♣ t❤❡♦ ✤÷đ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝→❝❤ t÷ì♥❣ tü✳ ❱➼ ❞ư ✸✳✹✳✶✳ ✭①❡♠ ❬✺❪✮✳ ❳➨t q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ s❛✉✳ tr♦♥❣ ✤â xj ≥ 0✱ ❥ ❂ ✶✱ ✷✱ ✳✳✳ ✱ ✻✳ ❇➔✐ t♦→♥ ❝â ❞↕♥❣ ❜➟❝ t❤❛♥❣ ✈ỵ✐ ❑❂✷✿ cT x + dT y = z (min) Ax = b1 , Bx + Dy = b2 , x ≥ 0, y ≥ Số hóa trung tâm học liệu http://www.lrc.tnu.edu.vn/ ✹✽ ✈ỵ✐ x = (x1, x2, x3)T , y = (y1, y2, y3)T = (x4, x5, x6)T , c = (1, 1, 1)T d = (3, 2, 1)T , b1 = (6, 6)T , b2 = (9, 15, 9)T ✈➔  A= 3  1 ,B =   −1 ✈➔   −1 D =   1 ✣➸ ❣✐↔✐ ❜➔✐ t♦→♥ t❤❡♦ t❤✉➟t t♦→♥ ♣❤➙♥ r➣ ❇❡♥❞❡rs t❛ ①✉➜t ♣❤→t tø ✈✐➺❝ t↕♦ r❛ ❜➔✐ t♦→♥ ❝❤õ ✤➛✉ t✐➯♥ β = cT x + δθ = z (min) , Ax =b1 , x ≥ tr♦♥❣ ✤â δ = ♥➳✉ ❦❤ỉ♥❣ ❝â ❧→t ❝➢t tè✐ ÷✉ ♥➔♦ ✈➔ δ = ♥➳✉ ❝â ➼t ♥❤➜t ♠ët ❧→t ❝➢t tè✐ ÷✉✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② δ = ✈➻ ❧ó❝ ♥➔② ❝❤÷❛ ❝â ❧→t ❝➢t tè✐ ÷✉✳ ●✐↔ sû ❜➔✐ t♦→♥ ❝❤õ ❝â ♥❣❤✐➺♠ tè✐ ÷✉ x = x0, t❛ ❣✐↔✐ ❜➔✐ t♦→♥ ❝♦♥ dT y = w (min) , Dy = b2 − Bx0 , y ≥ ✭✸✳✶✼✮ ❙❛✉ ✤â ♥❣❤✐➺♠ tè✐ ÷✉ ❝õ❛ ❜➔✐ t♦→♥ ♥➔② ✤÷đ❝ ❞ị♥❣ ✤➸ ①➙② ❞ü♥❣ ❧→t ❝➢t ❝❤♦ ❜➔✐ t♦→♥ ❝❤õ✳ ●✐↔✐ ❜➔✐ t♦→♥ ❝❤õ ✤➛✉ t✐➯♥ z = x1 + x2 + x3 → Số hóa trung tâm học liệu http://www.lrc.tnu.edu.vn/ ✹✾ x1 + 2x2 + 3x3 = 3x1 + 2x2 + x3 = x1 ≥ 0, x2 ≥ 0, x3 ≥ t❛ ♥❤➟♥ ✤÷đ❝ ♥❣❤✐➺♠ tè✐ ÷✉ x0 = t♦→♥ ❝♦♥ ✭✸✳✶✼✮✿ 1, 1, T ✳ ❚➼♥❤ ✈➳ ♣❤↔✐ ❝õ❛ ❜➔✐         1 1, b2 − Bx0 =  15  −   ×   =   −1 1, 1, ✈➔ s❛✉ ✤â ❣✐↔✐ ❜➔✐ t♦→♥ ❝♦♥ z = 3y1 + 2y2 + 1y3 →  4y1 − y2 + y3 =    3y1 + 2y2 + y3 = y + y2 + y3 = 1,    y1 ≥ 0, y2 ≥ 0, y3 ≥ ❇➔✐ t♦→♥ ♥➔② ❦❤ỉ♥❣ ❝â ♣❤÷ì♥❣ →♥ ✈➔ ✈➻ t❤➳ t❛ ❞ị♥❣ ❝→❝ ♥❤➙♥ tû ❦❤ỉ♥❣ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ❝õ❛ ♥â ✤➸ t↕♦ r❛ ❧→t ❝➢t ❝❤➜♣ ♥❤➟♥ ✤÷đ❝✳ ❈→❝ ♥❤➙♥ tû T ❦❤ỉ♥❣ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ♥➔② ❧➔ u = −0, −2, ❚✐➳♣ t❤❡♦✱ t❛ t➼♥❤ ❧→t ❝➢t ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ H 1x ≥ h1 ✈ỵ✐ T T H = u1 B = −6 −1, ✈➔ h1 = u1 b2 = −6, ❇➔✐ t♦→♥ ❝❤õ t❤✉ ❤➭♣ ♠ỵ✐ ❧➔ z = x1 + x2 + δγ →     x1 + 2x2 + 3x3 = 6, 3x1 + 2x2 + x3 = 6, −6x1 + 4x2 − 1, 4x3 = −6, 6,    x1 ≥ 0, x2 ≥ 0, x3 ≥ tr♦♥❣ ✤â δ = ✈➻ ✈➝♥ ❝❤÷❛ ❝â ❧→t ❝➢t tè✐ ÷✉✳ ▲í✐ ❣✐↔✐ tè✐T÷✉ ❝õ❛ ❜➔✐ t♦→♥ ♥➔② ❧➔ z1 = 3, x1 = 1, 207792; 0, 584416; 1, 207792 ✳ ❉ò♥❣ ♥❣❤✐➺♠ ♥➔② t❛ t➼♥❤ ✈➳ ♣❤↔✐ ❝õ❛ ❜➔✐ t♦→♥ ❝♦♥ ♠ỵ✐✿         1 1, 207792 6,  b2 −Bx1 =  15  −   ×  0, 584416  =  9, −1 1, 207792 3, 545456 Số hóa trung tâm học liệu http://www.lrc.tnu.edu.vn/ ✺✵ ✈➔ ❣✐↔✐ ❜➔✐ t♦→♥ ❝♦♥ ♠ỵ✐ w = 3y1 + 2y2 + 1y3 →  4y1 − y2 + y3 = 6,    3y1 + 2y2 + y3 = 9, y + y2 + y3 = 3, 545456    y1 ≥ 0, y2 ≥ 0, y3 ≥ ❚❛ ♥❤➟♥ ✤÷đ❝ ❧í✐ ❣✐↔✐ tè✐ ÷✉ w∗ = 9, ✈➔ y ∗ = 1, 9091 1, 6364 T 0, T ợ tỷ tố ữ u2 = ✤â✱ t❛ t➼♥❤ ❧→t ❝➢t tè✐ ÷✉ G1x + γ ≥ g1 ✈ỵ✐ T G1 = u2 B = , ✈➔ g1 = u2 ❇➔✐ t♦→♥ ❝❤õ t❤✉ ❤➭♣ ♠ỵ✐ ❧➔ T b = 15 z = x1 + x2 + γ →            x1 + 2x2 + 3x3 = 6, 3x1 + 2x2 + x3 = 6, −6x1 + 4x2 − 1, 4x3 = −6, 6, 3x1 + 2x2 + x3 + γ = 6, x1 ≥ 0, x2 ≥ 0, x3 ≥ ▲í✐ ❣✐↔✐ tè✐ ÷✉ ❝õ❛ ❜➔✐ t♦→♥T ❝❤õ ♥➔② ❧➔ 1, 207792; 0, 584416; 1, 207792 z2 = 12, x2 , γ2 = 9, γ2 = w∗ = ♥➯♥ t❛ tè✐ ÷✉ ✈➔ ❞ø♥❣ t❤✉➟t t♦→♥✳ ❑➳t q✉↔✱ t❛ ♥❤➟♥ ✤÷đ❝ ❧í✐ ❣✐↔✐ tè✐ ÷✉ ❝õ❛ ❜➔✐ t♦→♥ ❜❛♥ ✤➛✉ ❧➔ ✈➔ x∗ = 1, 207792; 0, 584416; 1, 207792 y∗ = 1, 9091 1, 6364 0, T = ✤↕t T , fmin = 12 ❚â♠ ❧↕✐✱ ❝❤÷ì♥❣ ♥➔② ✤➣ tr➻♥❤ ❜➔② t❤✉➟t t♦→♥ r rs q t t tữợ ợ õ trú t ữ ❜➔✐ t♦→♥ ❜❛♥ ✤➛✉ ✈ỵ✐ ♥❤✐➲✉ ❜✐➳♥ ✈➲ ❝→❝ ❜➔✐ t♦→♥ ✐t ❜✐➳♥ ❤ì♥✳ ❑❤→❝ ✈ỵ✐ ♣❤➙♥ r➣ ❉❛♥③✐❣✲❲♦❧❢❡✱ ❜➔✐ t♦→♥ ❝❤õ ❇❡♥❞❡rs ❝â ♥❤✐➲✉ r➔♥❣ ❜✉ë❝ ➞♥✱ ❝→❝ r➔♥❣ ❜✉ë❝ ♥➔② s➩ ✤÷đ❝ t➻♠ ✈➔ ✤÷❛ ✈➔♦ ❜➔✐ t♦→♥ ❝❤õ t❤✉ ❤➭♣ tr♦♥❣ q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ t❤✉➟t t♦→♥✳ P❤➙♥ r➣ ❇❡♥❞❡rs t❤➼❝❤ ❤ñ♣ ❝❤♦ ❝→❝ ❜➔✐ t♦→♥ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ❝â ❝➜✉ tró❝ ♠❛ tr➟♥ ❜➟❝ t❤❛♥❣✳ Số hóa trung tâm học liệu http://www.lrc.tnu.edu.vn/ ✺✶ ❑➳t ❧✉➟♥ ❑❤✐ ❣✐↔✐ ❜➔✐ t♦→♥ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ❝â tữợ ợ õ trú t ữớ t❛ t❤÷í♥❣ t➻♠ ❝→❝❤ ❝❤✐❛ ♥❤ä ♥â t❤➔♥❤ ♥❤✐➲✉ ❜➔✐ t tữợ ọ ỡ ỷ ỵ ❤ì♥ ♥❤í t➟♥ ❞ư♥❣ ❝➜✉ tró❝ r✐➯♥❣ ❝õ❛ ❝→❝ ❜➔✐ t õ ỵ tữ r tr q ❤♦↕❝❤ t✉②➳♥ t➼♥❤✳ ❈â ❤❛✐ ❝→❝❤ t✐➳♣ ❝➟♥ ✤→♥❣ ❝❤ó þ ❧➔ ♥❣✉②➯♥ ❧þ ♣❤➙♥ r➣ ❉❛♥t③✐❣ ✲ ❲♦❧❢❡ ✈➔ ỵ r rs tr ✈➔ ✤↕t ✤÷đ❝ ♠ët sè ❦➳t q✉↔ s❛✉✿ ✶✳ ❈→❝ tự ỡ s t ỗ t ỗ ❞✐➺♥✱ ❜➔✐ t♦→♥ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤✱ ❜➔✐ t♦→♥ ✤è✐ ♥❣➝✉ ✈➔ t❤✉➟t t♦→♥ ✤ì♥ ❤➻♥❤ ❝↔✐ ❜✐➯♥✳ ✷✳ ◆❣✉②➯♥ ỵ r t tr q t t ỵ tữ tt t t t ✈➔ ✈➼ ❞ö sè ♠✐♥❤ ❤♦↕✳ ⑩♣ ❞ö♥❣ ♣❤➙♥ r➣ ❉❛♥t③✐❣ ✲ ❲♦❧❢❡ ✈➔♦ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ❝â ❝➜✉ trú tr ữớ ố ỵ r rs tr q t t ỵ tữ t❤✉➟t t➼♥❤ t♦→♥ ✈➔ ❝→❝ ✈➼ ❞ö sè✳ ⑩♣ ❞✉♥❣ ♣❤➙♥ r➣ ❇❡♥❞❡rs ✈➔♦ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ❝â ❝➜✉ tró❝ ♠❛ tr➟♥ ❜➟❝ t❤❛♥❣✳ ❈â t❤➸ ①❡♠ ❧✉➟♥ ✈➠♥ ữ ữợ t t ỵ ♣❤➙♥ r➣ tr♦♥❣ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤✳ ❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥ ❤② ✈å♥❣ s➩ ❝â ❞à♣ ✤÷đ❝ t➻♠ ❤✐➸✉ s➙✉ ❤ì♥ ✈➲ ♠➦t ❦ÿ t❤✉➟t t➼♥❤ t♦→♥ ✈➔ ❝→❝ ♣❤÷ì♥❣ r ữ ữủ tợ tr ❧✉➟♥ ✈➠♥ ♥➔②✳ Số hóa trung tâm học liệu http://www.lrc.tnu.edu.vn/ ✺✷ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ◆✳ ❚✳ ❇✳ tr ữỡ tố ữ ỵ t❤✉②➳t ✈➔ t❤✉➟t t♦→♥✳ ◆❳❇ ❇→❝❤ ❑❤♦❛ ❍➔ ◆ë✐✳ ❬✷❪ ❇✳ ❚✳ ❚➙♠✱ ❚✳ ❱✳ ❚❤✐➺✉ ✭✶✾✾✽✮✳ ❈→❝ ♣❤÷ì♥❣ ♣❤→♣ tè✐ ÷✉ ❤♦→✳ ◆❳❇ ●✐❛♦ t❤ỉ♥❣ ✈➟♥ t↔✐✳ ❬✸❪ ❚✳ ❱✳ ❚❤✐➺✉ ✭✷✵✵✹✮ ●✐→♦ tr➻♥❤ tè✐ ÷✉ t✉②➳♥ t➼♥❤✳ ◆❳❇ ✣❍ ◗● ❍◆✳ ❬✹❪ ▼✳ ❙✳ ❇❛③❛r❛ ❡t ❛❧✳✱ ◆♦♥❧✐♥❡❛r Pr♦❣r❛♠♠✐♥❣✿ ❚❤❡♦r② ❛♥❞ ❆❧❣♦✲ r✐t❤♠s✳ 3rd ❊❞✐t✐♦♥✳ ❆ ❏♦❤♥ ❲✐❧❧❡② ❛♥❞ ❙♦♥s✱ ■♥❝✳✱ P✉❜❧✐❝❛t✐♦♥✱ ✷✵✵✻✳ ❬✺❪ ●✳ ❇✳ ❉❛♥t③✐❣✱ ▼✳ ◆✳ ❚❤❛♣❛ ✭✷✵✵✸✮ ▲✐♥❡❛r Pr♦❣r❛♠♠✐♥❣✿ ❚❤❡♦r② ❛♥❞ ❊①t❡♥s✐♦♥s✳ ❙♣r✐♥❣❡r✳ ❬✻❪ ❙✳ ●❛ss ✭✶✾✾✹✮ ▲✐♥❡❛r Pr♦❣r❛♠♠✐♥❣ ❛♥❞ ✐ts ❊①t❡♥s✐♦♥s✳ ■t❡r♥❛t✐♦♥❛❧ ❊❞✐t✐♦♥s✳ Số hóa trung tâm học liệu http://www.lrc.tnu.edu.vn/ ... +1 xj ≥ 0, j = 1, 2, , n ❉♦ ❦❤æ♥❣ ❝â ♠è✐ ❧✐➯♥ ❤➺ ♥➔♦ ❣✐ú❛ ❤❛✐ ❦❤è✐ ♥➔② ✭trø ❤➔♠ ♠ö❝ t✐➯✉✮ ♥➯♥ rã r➔♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ✭✵✳✶✮ ❝â t❤➸ t➻♠ ❜➡♥❣ ❝→❝❤ ❣✐↔✐ ❤❛✐ q✉✐ ❤♦↕❝❤