Bài toán vận tải có hạn chế khả năng lưu thông và ứng dụng

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈

P❤↕♠ ❇→ ❚➻♥❤

❇⑨■ ❚❖⑩◆ ❱❾◆ ❚❷■ ❈➶ ❍❸◆ ❈❍➌
❑❍❷ ◆❿◆● ▲×❯ ❚❍➷◆● ❱⑨ Ù◆● ❉Ư◆●
❈❤✉②➯♥ ◆❣❤➔♥❤✿ ❚❖⑩◆ Ù◆● ❉Ư◆●
▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✶✷

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✸
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✶

▼ư❝ ❧ư❝

▲í✐ ♥â✐ ✤➛✉
❈❤÷ì♥❣ ✶✳ ❇⑨■ ❚❖⑩◆ ❱❾◆ ❚❷■ ❈➶ ❑❍❷ ◆❿◆● ▲×❯
❚❍➷◆● ❍❸◆ ❈❍➌
✺
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷

✶✳✶✳

❇➔✐ t♦→♥ ✈➔ t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✶✳✷✳

P❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ ❜❛♥ ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✶✳✸✳

❚✐➯✉ ❝❤✉➞♥ tè✐ ÷✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵

✶✳✹✳

❚❤✉➟t t♦→♥ ❣✐↔✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵

✶✳✺✳

❱➼ ❞ư ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶

❈❤÷ì♥❣ ✷✳ ❇⑨■ ❚❖⑩◆ ❱❾◆ ❚❷■ ❱❰■ ▲×Đ◆● ❈❯◆● ❇➚
❈❍➄◆ ❉×❰■

✶✽
✷✳✶✳

◆ë✐ ❞✉♥❣ ❜➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✽

✷✳✷✳

❇➔✐ t♦→♥ t÷ì♥❣ ✤÷ì♥❣ ✈➔ t➼♥❤ ❝❤➜t

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✶

✷✳✸✳

❚❤✉➟t t♦→♥ ❣✐↔✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✺

✷✳✹✳

❱➼ ❞ư ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✽

❈❤÷ì♥❣ ✸✳ ❇⑨■ ❚❖⑩◆ ❱❾◆ ❚❷■ ❱❰■ ▲×Đ◆● ❈❯◆● ❇➚
❈❍➄◆
✸✷

✸✳✶✳

◆ë✐ ❞✉♥❣ ❜➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✷

✸✳✷✳

❚➼♥❤ ❝❤➜t ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✹

✸✳✸✳

❚❤✉➟t t♦→♥ ❣✐↔✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✻

✸✳✹✳

❱➼ ❞ư ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✵

✸✳✺✳

❚r÷í♥❣ ❤đ♣ tê♥❣ q✉→t

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✷

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✾

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✵

❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

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▲❮■ ◆➶■ ✣❺❯
▼ỉ ❤➻♥❤ ❜➔✐ t♦→♥ ✈➟♥ t↔✐ ❝ê ✤✐➸♥ ✈ỵ✐ ❧÷đ♥❣ ❝✉♥❣ ð ❝→❝ ♥ì✐ ❣✐❛♦ ❤➔♥❣
✭❣å✐ ❧➔ ❝→❝ tr↕♠ ♣❤→t✮ ✈➔ ❧÷đ♥❣ ❝➛✉ ð ❝→❝ ♥ì✐ ♥❤➟♥ ❤➔♥❣ ✭❣å✐ ❧➔ ❝→❝ tr↕♠
t❤✉✮ ✤à♥❤ tr÷ỵ❝ ✤➣ r➜t q✉❡♥ t❤✉ë❝ tr♦♥❣ ❧þ t❤✉②➳t tè✐ ÷✉ t✉②➳♥ t➼♥❤✳ ❇➔✐
t♦→♥ ✈➟♥ t↔✐ ❞↕♥❣ ♥➔② ❝â ♥❤✐➲✉ ù♥❣ ❞ư♥❣ rë♥❣ r➣✐ tr♦♥❣ t❤ü❝ t✐➵♥ ✈➔ ✤➣
✤÷đ❝ ♥❤✐➲✉ ♥❣÷í✐ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ù♥❣ ❞ư♥❣ ✭①❡♠ ❬✹❪✱ ❬✺❪✮✳
▼ët ♠ð rë♥❣ tü ♥❤✐➯♥ ❝õ❛ ❜➔✐ t♦→♥ ✈➟♥ t↔✐ ❧➔ ❦❤ỉ♥❣ q✉✐ ✤à♥❤ tr÷ỵ❝
❧÷đ♥❣ ❝✉♥❣ ❝õ❛ ❝→❝ tr↕♠ ♣❤→t ✈➔ ❧÷đ♥❣ ❝➛✉ ❝õ❛ tr↕♠ t❤✉ ♠➔ ❝❤♦ ♣❤➨♣
❧÷đ♥❣ ❝✉♥❣ ❝õ❛ ♠é✐ tr↕♠ ♣❤→t ❤❛②✴✈➔ ❧÷đ♥❣ ❝➛✉ ❝õ❛ ♠é✐ tr↕♠ t❤✉ ❝â

t❤➸ t❤❛② ✤ê✐ tr♦♥❣ ♠ët ❦❤♦↔♥❣ ❝❤♦ tr÷ỵ❝✳ ❑❤✐ ✤â t❛ ❣➦♣ ❜➔✐ t♦→♥ ✈➟♥
t↔✐ ✈ỵ✐ r➔♥❣ ❜✉ë❝ ❤❛✐ ♣❤➼❛ ✈➲ ❧÷đ♥❣ ❝✉♥❣ ✭❝õ❛ ❝→❝ tr↕♠ ♣❤→t✮ ✈➔ ❧÷đ♥❣
❝➛✉ ✭❝õ❛ ❝→❝ tr↕♠ t❤✉✮✳ ▼ỉ ❤➻♥❤ ♠ð rë♥❣ ♥➔② ♥↔② s✐♥❤ tø ♠ët sè ù♥❣
❞ư♥❣✱ tr♦♥❣ ✤â ❝â ✈➜♥ ✤➲ ✤✐➲✉ ❤➔♥❤ ♠↕♥❣ ①❡ ❜✉þt ð ❝→❝ t❤➔♥❤ ♣❤è✳
❇➡♥❣ ❝→❝❤ ❞ò♥❣ ❦ÿ t❤✉➟t t❤➯♠ ➞♥ sè ♣❤ư✱ t❛ ❝â t❤➸ ✤÷❛ ❜➔✐ t♦→♥ ✈➟♥
t↔✐ ✈ỵ✐ r➔♥❣ ❜✉ë❝ ❤❛✐ ♣❤➼❛ ✤è✐ ✈ỵ✐ ❧÷đ♥❣ ❝✉♥❣ ✭❝➛✉✮ ✈➲ ❜➔✐ t♦→♥ ✈➟♥ t↔✐
❝ê ✤✐➸♥ ✈ỵ✐ ♠ët sè ❜✐➳♥ ❝â r➔♥❣ ❜✉ë❝ ❝➟♥ tr➯♥✱ tù❝ ❧➔ q✉✐ ✈➲ ❜➔✐ t♦→♥
✈➟♥ t↔✐ ❝â ❤↕♥ ❝❤➳ ❦❤↔ ♥➠♥❣ ❧÷✉ t❤ỉ♥❣ ✈➔ ❞♦ ✤â ❝â t❤➸ →♣ ❞ư♥❣ ♣❤÷ì♥❣
♣❤→♣ t❤➳ ✈à ✤➣ ❜✐➳t ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥✳
▲✉➟♥ ✈➠♥ ♥➔② ✤➲ ❝➟♣ tỵ✐ ❜➔✐ t♦→♥ ✈➟♥ t↔✐ ❝â ❤↕♥ ❝❤➳ ❦❤↔ ♥➠♥❣ ❧÷✉
t❤ỉ♥❣✱ tr➻♥❤ ❜➔② t❤✉➟t t♦→♥ t❤➳ ✈à ❣✐↔✐ ❜➔✐ t♦→♥ ✈➔ ù♥❣ ❞ư♥❣ ✈➔♦ ①û ❧þ
❜➔✐ t♦→♥ ✈➟♥ t↔✐ ✈ỵ✐ r➔♥❣ ❜✉ë❝ ❤❛✐ ♣❤➼❛ tr♦♥❣ ❤❛✐ tr÷í♥❣ ❤đ♣✿ ❛✮ ❧÷đ♥❣
❝✉♥❣ ✭❤❛② ❝➛✉✮ ❜à ❝❤➦♥ ❞÷ỵ✐ ❜ð✐ ♠ët sè ❦❤ỉ♥❣ ➙♠❀ ❜✮ ❧÷đ♥❣ ❝✉♥❣ ✭❤❛②
❝➛✉✮ ❜à ❝❤➦♥ ✭❝↔ tr➯♥ ✈➔ ❞÷ỵ✐✮✳ ❚r÷í♥❣ ❤đ♣ ❦❤✐ ❝↔ ❧÷đ♥❣ ❝✉♥❣ ✈➔ ❧÷đ♥❣
❝➛✉ ✤➲✉ t❤❛② ✤ê✐ ❝ơ♥❣ s➩ ✤÷đ❝ ✤➲ ❝➟♣ tỵ✐✳
▲✉➟♥ ✈➠♥ ❣ç♠ ❧í✐ ♥â✐ ✤➛✉✱ ❜❛ ❝❤÷ì♥❣✱ ❦➳t ❧✉➟♥ ✈➔ ❞❛♥❤ ♠ư❝ t➔✐ ❧✐➺✉
t❤❛♠ ❦❤↔♦✳
❈❤÷ì♥❣ ✶ ✈ỵ✐ t✐➯✉ ✤➲ ✧❇➔✐ t♦→♥ ✈➟♥ t↔✐ ❝â ❤↕♥ ❝❤➳ ❦❤↔ ♥➠♥❣ ❧÷✉

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t❤ỉ♥❣✧ tr➻♥❤ ❜➔② ♥ë✐ ❞✉♥❣ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❜➔✐ t♦→♥ ✈➟♥ t↔✐✱ tr♦♥❣
✤â ❧÷đ♥❣ ❤➔♥❣ ✈➟♥ ❝❤✉②➸♥ tø ♠é✐ tr↕♠ ♣❤→t ✤➳♥ ♠é✐ tr↕♠ t❤✉ ❦❤ỉ♥❣
✤÷đ❝ ✈÷đt q✉→ ♠ët ♠ù❝ ❣✐ỵ✐ ❤↕♥ q✉✐ ✤à♥❤ tr÷ỵ❝ ✭❞♦ ♥➠♥❣ ❧ü❝ ♣❤÷ì♥❣ t✐➺♥
✈➟♥ ❝❤✉②➸♥ ❝â ❤↕♥ ❤♦➦❝ ❞♦ ♥➠♥❣ ❧ü❝ ❝➛✉ ✤÷í♥❣ tr➯♥ t✉②➳♥ ✈➟♥ ❝❤✉②➸♥
✤â ❜à ❤↕♥ ❝❤➳ ✳✳✳✮✳ ❚✐➳♣ ✤â✱ ✤➲ ❝➟♣ tỵ✐ ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ♣❤÷ì♥❣ →♥ ❝ü❝
❜✐➯♥ ❜❛♥ ✤➛✉ ❝õ❛ ❜➔✐ t♦→♥✳ ❙❛✉ ✤â✱ tr➻♥❤ ❜➔② ❝ì sð ❧þ ❧✉➟♥ ✈➔ ♥ë✐ ❞✉♥❣

t❤✉➟t t♦→♥ t❤➳ ✈à ✭♠ët ❜✐➳♥ t❤➸ ❝õ❛ t❤✉➟t t♦→♥ ✤ì♥ ❤➻♥❤ ①û ❧þ ❜➔✐ t♦→♥
❝â ❜✐➳♥ ❜à ❝❤➦♥ tr➯♥✮ ❣✐↔✐ ❜➔✐ t♦→♥✳ ❈✉è✐ ❝❤÷ì♥❣ ♥➯✉ r❛ ✈➼ ❞ư sè ♠✐♥❤
❤å❛ t❤✉➟t t♦→♥ ❣✐↔✐ ✤➣ tr➻♥❤ ❜➔②✳ ❈→❝ ❦✐➳♥ t❤ù❝ ð ❝❤÷ì♥❣ ♥➔② s➩ ❝➛♥ ✤➳♥
ð ❝→❝ ❝❤÷ì♥❣ s❛✉ ✤➸ ①û ❧þ ✈➔ ❣✐↔✐ ❜➔✐ t♦→♥ ✈➟♥ t↔✐ ✈ỵ✐ r➔♥❣ ❜✉ë❝ ❤❛✐ ♣❤➼❛✳
❈❤÷ì♥❣ ✷ ✈ỵ✐ t✐➯✉ ✤➲ ✧❇➔✐ t♦→♥ ✈➟♥ t↔✐ ✈ỵ✐ ❧÷đ♥❣ ❝✉♥❣ ❜à ❝❤➦♥ ❞÷ỵ✐✧
①➨t ❜➔✐ t♦→♥ ✈➟♥ t↔✐✱ tr♦♥❣ ✤â ❣✐↔ t❤✐➳t ❧÷đ♥❣ ❝✉♥❣ ❝õ❛ ❝→❝ tr↕♠ ♣❤→t
❦❤ỉ♥❣ q✉✐ ✤à♥❤ tr÷ỵ❝ ♠➔ ❝❤➾ ❜à ❝❤➦♥ ❞÷ỵ✐ ✭❧ỵ♥ ❤ì♥ ♠ët ♠ù❝ tè✐ t❤✐➸✉
♥➔♦ ✤â✱ t❤÷í♥❣ ❧➔ sè ❞÷ì♥❣✮✱ ❝á♥ ❧÷đ♥❣ ❝➛✉ ❝õ❛ ❝→❝ tr↕♠ t❤✉ ✤➣ ❜✐➳t
tr÷ỵ❝✳ ◆➯✉ ♠ỉ ❤➻♥❤ t♦→♥ ❤å❝ ❝õ❛ ❜➔✐ t♦→♥ ✈➔ ♥➯✉ ❝→❝❤ ✤÷❛ ✈➲ ❜➔✐ t♦→♥
✈➟♥ t↔✐ ❝â ❤↕♥ ❝❤➳ ❦❤↔ ♥➠♥❣ ❧÷✉ t❤ỉ♥❣✳ ❙❛✉ ✤â tr➻♥❤ ❜➔② t✐➯✉ ❝❤✉➞♥ tè✐
÷✉ ✈➔ t❤✉➟t t♦→♥ ❣✐↔✐ ❜➔✐ t♦→♥✳ ❈✉è✐ ❝❤÷ì♥❣ ♥➯✉ ✈➼ ❞ư sè ♠✐♥❤ ❤å❛ t❤✉➟t
t♦→♥ ✤➣ tr➻♥❤ ❜➔②✳
❈❤÷ì♥❣ ✸ ✈ỵ✐ t✐➯✉ ✤➲ ✧❇➔✐ t♦→♥ ✈➟♥ t↔✐ ✈ỵ✐ ❧÷đ♥❣ ❝✉♥❣ ❜à ❝❤➦♥✧ ①➨t
❜➔✐ t♦→♥ ✈➟♥ t↔✐✱ tr♦♥❣ ✤â ❣✐↔ t❤✐➳t ❧÷đ♥❣ ❝✉♥❣ ❝õ❛ ❝→❝ tr↕♠ ♣❤→t ❦❤ỉ♥❣
q✉✐ ✤à♥❤ tr÷ỵ❝ ♠➔ t❤❛② ✤ê✐ tr♦♥❣ ♠ët ❦❤♦↔♥❣ ❝❤♦ tr÷ỵ❝✱ tù❝ ❧➔ ❧÷đ♥❣
❝✉♥❣ ❜à ❝❤➦♥ ✭❝↔ tr➯♥ ✈➔ ❞÷ỵ✐✮✱ ❝á♥ ❧÷đ♥❣ ❝➛✉ ❝õ❛ ❝→❝ tr↕♠ t❤✉ ✤➣ ❜✐➳t
tr÷ỵ❝✳ ❇➔✐ t♦→♥ ①➨t ð ✤➙② ♠ð rë♥❣ ❤ì♥ ✤ỉ✐ ❝❤ót s♦ ✈ỵ✐ ❜➔✐ t♦→♥ ①➨t ð
❝❤÷ì♥❣ ✷ ✈➔ ❞♦ ✤â ❝→❝❤ ①û ❧þ ❝ơ♥❣ ♣❤ù❝ t↕♣ ❤ì♥✳ ◆➯✉ ♠ỉ ❤➻♥❤ t♦→♥
❤å❝ ❝õ❛ ❜➔✐ t♦→♥ ✈➔ ♥➯✉ ❝→❝❤ ✤÷❛ ✈➲ ❜➔✐ t♦→♥ ✈➟♥ t↔✐ ❝â ❤↕♥ ❝❤➳ ❦❤↔
♥➠♥❣ ❧÷✉ t❤ỉ♥❣✳ ❙❛✉ ✤â tr➻♥❤ ❜➔② t✐➯✉ ❝❤✉➞♥ tè✐ ÷✉ ✈➔ t❤✉➟t t♦→♥ ❣✐↔✐
❜➔✐ t♦→♥✳ ❈✉è✐ ❝❤÷ì♥❣ ♥➯✉ ✈➼ ❞ư sè ♠✐♥❤ ❤å❛ t❤✉➟t t♦→♥ ❣✐↔✐✳
❉♦ t❤í✐ ❣✐❛♥ ✈➔ ❦✐➳♥ t❤ù❝ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❝❤➢❝ ❝❤➢♥ ❧✉➟♥ ✈➠♥ ❝á♥ ❝â
♥❤ú♥❣ s❛✐ sât ♥❤➜t ✤à♥❤✱ ❦➼♥❤ ♠♦♥❣ q✉þ t❤➛② ❝ỉ ✈➔ ❝→❝ ❜↕♥ ✤â♥❣ ❣â♣ þ
❦✐➳♥ ✤➸ t→❝ ❣✐↔ t✐➳♣ tư❝ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✳
◆❤➙♥ ❞à♣ ♥➔②✱ t→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❚❤➛② ❤÷ỵ♥❣

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❞➝♥ ●❙ ✲ ❚❙ ❚r➛♥ ❱ơ ❚❤✐➺✉ ✤➣ t➟♥ t➻♥❤ ❤÷ỵ♥❣ ❞➝♥ ✈➔ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣
s✉èt q✉→ tr➻♥❤ ❧➔♠ ❧✉➟♥ ✈➠♥✳
❚→❝ ❣✐↔ ①✐♥ ❣û✐ tỵ✐ ❝→❝ ❚❤➛②✱ ❝ỉ ð ❱✐➺♥ ❚♦→♥ ❤å❝✱ ❝→❝ ❚❤➛②✱ ❝ỉ ❦❤♦❛
❚♦→♥✱ ♣❤á♥❣ ✣➔♦ t↕♦ s❛✉ ✤↕✐ ❤å❝ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✲ ✣↕✐ ❤å❝
❚❤→✐ ◆❣✉②➯♥ ❝ơ♥❣ ♥❤÷ ❝→❝ ❚❤➛② ❝ỉ t❤❛♠ ❣✐❛ ❣✐↔♥❣ ❞↕② ❦❤â❛ ❈❛♦ ❤å❝
✷✵✶✶ ✲ ✷✵✶✸ ❧í✐ ❝↔♠ ì♥ s➙✉ s➢❝ ✈➲ ❝ỉ♥❣ ❧❛♦ ❣✐↔♥❣ ❞↕② ✈➔ t↕♦ ♠å✐ ✤✐➲✉
❦✐➺♥ t❤✉➟♥ ❧đ✐ ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ tr÷í♥❣✳
❚→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❙ð ●✐→♦ ❞ư❝ ✈➔ ✣➔♦ t↕♦ t➾♥❤
❍➔ ●✐❛♥❣✱ ❇❛♥ ❣✐→♠ ❤✐➺✉✱ ❝→❝ tê ❝❤ù❝ ✣♦➔♥ t❤➸✱ tê ❝❤✉②➯♥ ♠ỉ♥✱ ♥❤â♠
❚♦→♥ tr÷í♥❣ ❚❍P❚ ▲✐➯♥ ❍✐➺♣ ❝ò♥❣ ❜↕♥ ❜➧ ✤ç♥❣ ♥❣❤✐➺♣ ✈➔ ❣✐❛ ✤➻♥❤ ✤➣
t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❣✐ó♣ ✤ï✱ ✤ë♥❣ ✈✐➯♥ t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳

❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✵✶ t❤→♥❣ ✽ ♥➠♠ ✷✵✶✸

❚→❝ ❣✐↔
P❤↕♠ ❇→ ❚➻♥❤

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❈❤÷ì♥❣ ✶
❇⑨■ ❚❖⑩◆ ❱❾◆ ❚❷■ ❈➶ ❑❍❷
◆❿◆● ▲×❯ ❚❍➷◆● ❍❸◆ ❈❍➌
❈❤÷ì♥❣ ♥➔② ✤➲ ❝➟♣ tỵ✐ ❜➔✐ t♦→♥ ✈➟♥ t↔✐ ❝â ❦❤↔ ♥➠♥❣ ❧÷✉ t❤ỉ♥❣ ❤↕♥
❝❤➳✳ ◆➯✉ ♠ỉ ❤➻♥❤ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❜➔✐ t♦→♥✳ ❚✐➳♣ ✤â✱ ♥➯✉ ♣❤÷ì♥❣
♣❤→♣ t➻♠ ♣❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ ❜❛♥ ✤➛✉ ❝õ❛ ❜➔✐ t♦→♥✳ ❙❛✉ ✤â✱ tr➻♥❤ ❜➔②

❝ì sð ❧þ ❧✉➟♥ ✈➔ ♥ë✐ ❞✉♥❣ t❤✉➟t t♦→♥ t❤➳ ✈à ✭♠ët ❜✐➳♥ t❤➸ ❝õ❛ t❤✉➟t t♦→♥
✤ì♥ ❤➻♥❤✮ ❣✐↔✐ ❜➔✐ t♦→♥✳ ❈✉è✐ ❝❤÷ì♥❣ ♥➯✉ ✈➼ ❞ư sè ♠✐♥❤ ❤å❛ t❤✉➟t t♦→♥
❣✐↔✐✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ t❤❛♠ ❦❤↔♦ ❝❤õ ②➳✉ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱
❬✹❪ ✈➔ ❬✺❪✳

✶✳✶✳ ❇➔✐ t♦→♥ ✈➔ t➼♥❤ ❝❤➜t
●✐↔ sû ❝➛♥ ✈➟♥ ❝❤✉②➸♥ ♠ët ❧♦↕✐ ❤➔♥❣ ✭①✐ ♠➠♥❣ ❝❤➥♥❣ ❤↕♥✮ tø ♠ ❦❤♦
❝❤ù❛ ❤➔♥❣ ✭❣å✐ ❧➔ ❝→❝ tr↕♠ ♣❤→t✮ tỵ✐ ♥ ❤ë t✐➯✉ t❤ư ✭❣å✐ ❧➔ ❝→❝ tr↕♠ t❤✉✮✳

ai > 0 ✭✐ ❂
t❤✉ ❥ ❧➔ bj > 0

❈❤♦ ❜✐➳t ❧÷đ♥❣ ❤➔♥❣ ❝â ✭❣å✐ ❧➔ ❧÷đ♥❣ ❝✉♥❣✮ ð tr↕♠ ♣❤→t ✐ ❧➔
✶✱ ✷✱ ✳✳✳ ✱ ♠✮ ✈➔ ❧÷đ♥❣ ❤➔♥❣ ❝➛♥ ✭❣å✐ ❧➔ ❧÷đ♥❣ ❝➛✉✮ ð tr↕♠

✭❥ ❂ ✶✱ ✷✱ ✳✳✳ ✱ ♥✮✳ ❈❤✐ ♣❤➼ ✈➟♥ ❝❤✉②➸♥ ♠ët ✤ì♥ ✈à ❤➔♥❣ tø tr↕♠ ♣❤→t ✐
tỵ✐ tr↕♠ t❤✉ ❥ ❧➔

cij ≥ 0✳

◆❣♦➔✐ r❛✱ ❞♦ ✤✐➲✉ ❦✐➺♥ ✈➲ ✤÷í♥❣ s→ ❤♦➦❝ ❤↕♥

❝❤➳ ✈➲ ♣❤÷ì♥❣ t✐➺♥ ✈➟♥ ❝❤✉②➸♥ ♥➯♥ tø ✐ tỵ✐ ❥ ❝❤➾ ✤÷đ❝ ✈➟♥ ❝❤✉②➸♥ tè✐ ✤❛

dij ≥ 0

✤ì♥ ✈à ❤➔♥❣ ✭dij ❣å✐ ❧➔ ❦❤↔ ♥➠♥❣ ❧÷✉ t❤ỉ♥❣ tr➯♥ t✉②➳♥ ✐ ✲ ❥✮✳ ❱➜♥

✤➲ ❧➔ ❝➛♥ ✈➟♥ ❝❤✉②➸♥ ❜❛♦ ♥❤✐➯✉ ✤ì♥ ✈à ❤➔♥❣ tø ♠é✐ tr↕♠ ♣❤→t tỵ✐ ♠é✐
tr↕♠ t❤✉ s❛♦ ❝❤♦ ❦❤ỉ♥❣ ✈÷đt q✉→ ❦❤↔ ♥➠♥❣ ❧÷✉ t❤ỉ♥❣ tr➯♥ ♠é✐ t✉②➳♥✱

♠å✐ tr↕♠ ♣❤→t ❣✐❛♦ ❤➳t ❤➔♥❣✱ ♠å✐ tr↕♠ t❤✉ ♥❤➟♥ ✤õ ❤➔♥❣ ✈➔ tê♥❣ ❝❤✐
♣❤➼ ✈➟♥ ❝❤✉②➸♥ ❧➔ ♥❤ä ♥❤➜t❄
▼ỉ ❤➻♥❤ t♦→♥ ❤å❝ ❝õ❛ ❜➔✐ t♦→♥ ✈➟♥ t↔✐ ✈ỵ✐ ❦❤↔ ♥➠♥❣ ❧÷✉ t❤ỉ♥❣ ❤↕♥

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❝❤➳ ❝â ❞↕♥❣ ♥❤÷ s❛✉ ✭xij ❧➔ ❧÷đ♥❣ ❤➔♥❣ ✈➟♥ ❝❤✉②➸♥ ❝➛♥ t➻♠✮✿

m

n

cij xij → min

✭❝ü❝ t✐➸✉ tê♥❣ ❝❤✐ ♣❤➼ ❝÷ỵ❝ ✈➟♥ ❝❤✉②➸♥✮

✭✶✳✶✮

i=1 j=1
✈ỵ✐ ❝→❝ ✤✐➲✉ ❦✐➺♥

n

xij = ai , i = 1, 2, ..., m ✭♠å✐

tr↕♠ ♣❤→t ❣✐❛♦ ❤➳t ❤➔♥❣✮

✭✶✳✷✮

j=1
m

xij = bj, j = 1, 2, ..., n ✭♠å✐

tr↕♠ t❤✉ ♥❤➟♥ ✤õ ❤➔♥❣✮

✭✶✳✸✮

i=1

0 ≤ xij ≤ dij , i = 1, 2, ..., m, j = 1, 2, ..., n
✭❧÷đ♥❣ ❤➔♥❣ ✈➟♥ ❝❤✉②➸♥ ❦❤ỉ♥❣ ➙♠ ✈➔ ❦❤ỉ♥❣ ✈÷đt q✉→
❦❤↔ ♥➠♥❣ ❧÷✉ t❤ỉ♥❣

✭✶✳✹✮

dij ✮

●✐↔ t❤✐➳t ❜➔✐ t♦→♥ ✭✶✳✶✮ ✲ ✭✶✳✹✮ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❝➙♥ ❜➡♥❣ ❝✉♥❣ ❝➛✉✿

m

n

ai =
i=1

bj

✭tê♥❣ ❝✉♥❣ ❜➡♥❣ tê♥❣ ❝➛✉✮✳

✭✶✳✺✮

j=1

xij ≥ 0 s✉② r❛ xij ≤ min {ai , bj } ✈ỵ✐ ♠å✐ ✐✱ ❥
♥➯♥ ❞➵ ❞➔♥❣ t❤➜② r➡♥❣ ♥➳✉ ♠å✐ ❦❤↔ ♥➠♥❣ ❧÷✉ t❤ỉ♥❣ dij ≥ min {ai , bj } t❤➻
✤÷ì♥❣ ♥❤✐➯♥ ❝â t❤➸ ❜ä ✤✐➲✉ ❦✐➺♥ xij ≤ dij ✈➔ ❦❤✐ ✤â ✭✶✳✶✮ ✲ ✭✶✳✺✮ trð t❤➔♥❤
❚ø ✤✐➲✉ ❦✐➺♥ ✭✶✳✷✮✱ ✭✶✳✸✮ ✈➔

❜➔✐ t♦→♥ ✈➟♥ t↔✐ t❤ỉ♥❣ t❤÷í♥❣ ✭❦❤ỉ♥❣ ❤↕♥ ❝❤➳ ❦❤↔ ♥➠♥❣ ❧÷✉ t❤ỉ♥❣✮✳ ❈❤➾
❣➦♣ ❦❤â ❦❤➠♥ ❦❤✐ ❝â ♥❤ú♥❣

dij ≤ min {ai , bj }✳

❈❤÷ì♥❣ ♥➔② s➩ ①➨t ❜➔✐

t♦→♥ ✈➟♥ t↔✐ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✳
❘➜t t✐➳❝ ❦❤ỉ♥❣ ❝â ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ❜➔✐ t♦→♥ ✭✶✳✶✮ ✲ ✭✶✳✺✮ ❣✐↔✐
✤÷đ❝✱ ♥❤÷ tr♦♥❣ ❜➔✐ t♦→♥ ✈➟♥ t↔✐ t❤ỉ♥❣ t❤÷í♥❣✱ ♠➔ ❝❤➾ ❝â ✤✐➲✉ ❦✐➺♥ ❝➛♥
✈➔ ✤✐➲✉ ❦✐➺♥ ✤õ r✐➯♥❣✳
❉➵ t❤➜② r➡♥❣ ♠ët ✤✐➲✉ ❦✐➺♥ ✤õ ✤➸ ❜➔✐ t♦→♥ ✭✶✳✶✮ ✲ ✭✶✳✺✮ ❣✐↔✐ ✤÷đ❝ ❧➔

dij ≥ min {ai , bj }

✈ỵ✐ ♠å✐ ✐✱❥✳

✭✶✳✻✮

✣➸ ✤÷❛ r❛ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ❝❤♦ ✭✶✳✶✮ ✲ ✭✶✳✺✮ ❣✐↔✐ ✤÷đ❝✱ ❦❤ỉ♥❣ ❣✐↔♠ tê♥❣
q✉→t t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t

dij ≤ min {ai , bj }

Số hóa bởi trung tâm học liệu

✈ỵ✐ ♠å✐ ✐ ✈➔ ❥✱ ❜ð✐ ✈➻ ✈ỵ✐

/>

✼

dij ≥ min {ai , bj }

t❤➻ r➔♥❣ ❜✉ë❝

xij ≤ dij

trð ♥➯♥ ❦❤ỉ♥❣ ❝➛♥ t❤✐➳t✳ ❇➡♥❣

❝→❝❤ ❝ë♥❣ ✭✶✳✹✮ t❤❡♦ ♠å✐ ❥ ✈➔ s♦ s→♥❤ ✈ỵ✐ ✭✶✳✷✮❀ s❛✉ ✤â ❝ë♥❣ ✭✶✳✹✮ t❤❡♦
♠å✐ ✐ ✈➔ s♦ s→♥❤ ✈ỵ✐ ✭✶✳✸✮ t❛ ♥❤➟♥ ✤÷đ❝ ♠ët

✤✐➲✉ ❦✐➺♥ ❝➛♥

✤➸ ❜➔✐ t♦→♥

✭✶✳✶✮ ✲ ✭✶✳✺✮ ❣✐↔✐ ✤÷đ❝ ❧➔

n

ai ≤

m

dij , ∀i = 1, ..., m ✈➔ bj ≤
j=1

dij , ∀j = 1, ..., n.

✭✶✳✼✮

i=1

◆➳✉ ✈✐ ♣❤↕♠ ✭✶✳✼✮ ✭tù❝

dij

❦❤ỉ♥❣ ✤õ ❧ỵ♥ s♦ ✈ỵ✐ ♥❤✉ ❝➛✉ ✈➟♥ ❝❤✉②➸♥✮

t❤➻ ❞ò ❝â ✤✐➲✉ ❦✐➺♥ ❝➙♥ ❜➡♥❣ ❝✉♥❣ ❝➛✉ ✭✶✳✺✮✱ ❜➔✐ t♦→♥ ✈➝♥ s➩ ❦❤ỉ♥❣ ❝â
♣❤÷ì♥❣ →♥ t❤ä❛ ♠➣♥ ✭✶✳✷✮ ✲ ✭✶✳✹✮✱ ❞♦ ✤â ❜➔✐ t♦→♥ s➩ ❦❤ỉ♥❣ ❣✐↔✐ ✤÷đ❝✳
❚r♦♥❣ t❤ü❝ ❤➔♥❤✱ ✤➸ ❜➔✐ t♦→♥ ❣✐↔✐ ✤÷đ❝ t❤÷í♥❣ ♥❣÷í✐ t❛ ♣❤↔✐ ✤✐➲✉ ❝❤➾♥❤

dij

❞➛♥ ❝→❝ ❤➺ sè

♠ët ❝→❝❤ t❤➼❝❤ ❤đ♣✳

❚✉② ♥❤✐➯♥✱ ✤✐➲✉ ❦✐➺♥ ✭✶✳✼✮ ❦❤ỉ♥❣ ❧➔ ✤✐➲✉ ❦✐➺♥ ✤õ ✤➸ ✭✶✳✶✮ ✲ ✭✶✳✺✮ ❣✐↔✐
✤÷đ❝✱ ♥❤÷ ❝❤➾ r❛ ð ✈➼ ❞ư ✤ì♥ ❣✐↔♥ s❛✉✳

❱➼ ❞ư ✶✳✶

✳ ❳➨t ❜➔✐ t♦→♥ ✈➟♥ t↔✐ ✈ỵ✐ ♠ ❂ ✸ tr↕♠ ♣❤→t ✈➔ ♥ ❂ ✸ tr↕♠

t❤✉✳ ▲÷đ♥❣ ❝✉♥❣ ❝õ❛ ❝→❝ tr↕♠ ♣❤→t ✈➔ ❧÷đ♥❣ ❝➛✉ ❝õ❛ ❝→❝ tr↕♠ t❤✉ ❧➛♥
❧÷đt ❧➔ ✶✱ ✸ ✈➔ ✹✳ ❑❤↔ ♥➠♥❣ ❧÷✉ t❤ỉ♥❣

dij ≤ min {ai , bj }

✭✐✱ ❥ ❂ ✶✱ ✷✱ ✸✮

✤÷đ❝ ❝❤♦ tr♦♥❣ ❇↔♥❣ ✶✳✶✳
❘ã r➔♥❣ ✤➲ ✭✶✳✷✮ t❤ä❛ ♠➣♥ ✈ỵ✐ ✐ ❂ ✷✱ ✸ ♣❤↔✐ ❝â

xij = dij

✈ỵ✐ ♠å✐ ✐ ❂ ✷✱

✸ ✈➔ ♠å✐ ❥ ❂ ✶✱ ✷✱ ✸✳ ◆❤÷♥❣ ❦❤✐ ✤â s➩ ✈✐ ♣❤↕♠ r➔♥❣ ❜✉ë❝ ✭✶✳✸✮ ✈ỵ✐ ❥ ❂ ✶✳
❱➟② ❦❤ỉ♥❣ t❤➸ ❝â

xij

t❤ä❛ ♠➣♥ ✭✶✳✶✮ ✲ ✭✶✳✹✮ ✈ỵ✐ ♠å✐ ✐✱ ❥ ❂ ✶✱ ✷✱ ✸ ✈➔

❜➔✐ t♦→♥ ✭✶✳✶✮ ✲ ✭✶✳✺✮ ❧➔ ❦❤ỉ♥❣ ❣✐↔✐ ✤÷đ❝✳

❑þ ❤✐➺✉

Aij ∈ Rm+n

❧➔ ✈➨❝tì ❤➺ sè ❝õ❛ ❜✐➳♥

xij ✳

❉➵ t❤➜② r➡♥❣ ✈➨❝tì

♥➔② ❝â ❤❛✐ t❤➔♥❤ ♣❤➛♥ ❜➡♥❣ ✶ t↕✐ ❤➔♥❣ ✐ ✈➔ ❤➔♥❣ ♠ ✰ ❥✱ ❝á♥ ♠å✐ t❤➔♥❤
♣❤➛♥ ❦❤→❝ ❜➡♥❣ ✵✳

Số hóa bởi trung tâm học liệu

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✽
✣➸ ❝❤♦ ❣å♥✱ t❛ ❣❤✐ ❧↕✐ ❞ú ❧✐➺✉ ❝õ❛ ❜➔✐ t♦→♥ ❞÷ỵ✐ ❞↕♥❣ ♠ët ❜↔♥❣ ❝❤ú
♥❤➟t✱ ❣å✐ ❧➔

❜↔♥❣ ✈➟♥ t↔✐

✭❇↔♥❣ ✶✳✷✮✳ ❇↔♥❣ ❣ç♠ ♠ ❤➔♥❣ ✭✐ ❂ ✶✱ ✳✳✳ ✱ ♠✮

✈➔ ♥ ❝ët ✭❥ ❂ ✶✱ ✳✳✳ ✱ ♥✮✳ ❈❤é ❣✐❛♦ ♥❤❛✉ ð ❤➔♥❣ ✐✱ ❝ët ❥ ❣å✐ ❧➔ ỉ ✭✐✱ ❥✮✳ ▼é✐
❤➔♥❣ t÷ì♥❣ ù♥❣ ✈ỵ✐ ♠ët tr↕♠ ♣❤→t✱ ♠é✐ ❝ët t÷ì♥❣ ù♥❣ ✈ỵ✐ ♠ët tr↕♠

t❤✉✳ ❙è ❣❤✐ ð ✤➛✉ ♠é✐ ❤➔♥❣ ❧➔ ❧÷đ♥❣ ❝✉♥❣✱ sè ❣❤✐ ð ✤➛✉ ♠é✐ ❝ët ❧➔ ❧÷đ♥❣
❝➛✉✳ ❈❤✐ ♣❤➼ ✈➟♥ ❝❤✉②➸♥

dij

cij

❣❤✐ ð ❣â❝ tr➯♥ ❜➯♥ tr→✐✱ ❦❤↔ ♥➠♥❣ ❧÷✉ t❤ỉ♥❣

❣❤✐ ð ❣â❝ tr➯♥ ❜➯♥ ♣❤↔✐✱ ❧÷đ♥❣ ❤➔♥❣ ✈➟♥ ❝❤✉②➸♥

xij

s➩ ❣❤✐ ð ❣✐ú❛ ❤➔♥❣

❞÷ỵ✐ ❝õ❛ ỉ ✭✐✱ ❥✮✳ ➷ ✭✐✱ ❥✮ ❜✐➸✉ t❤à t✉②➳♥ ✈➟♥ ❝❤✉②➸♥ tø tr↕♠ ♣❤→t ✐ tỵ✐
tr↕♠ t❤✉ ❥✳ ✣➦t

cij = ∞

❤♦➦❝

dij = 0

♥➳✉ ❦❤ỉ♥❣ t❤➸ ✈➟♥ ❝❤✉②➸♥ ❤➔♥❣ tø

✐ ✤➳♥ ❥✳

❙❛✉ ✤➙② ♥❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ q✉❡♥ ❞ò♥❣✳ ▼❛ tr➟♥

x = {xij }m×n

♣❤÷ì♥❣ →♥ ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✶✮ ✲ ✭✶✳✺✮✳ ▼ët
♣❤÷ì♥❣ →♥ ✤↕t ❝ü❝ t✐➸✉ ✭✶✳✶✮ ❣å✐ ❧➔ ♣❤÷ì♥❣ →♥ tè✐ ÷✉ ❤❛② ❧í✐ ❣✐↔✐✳ P❤÷ì♥❣
→♥ x ❣å✐ ❧➔ ♣❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ ♥➳✉ ❝→❝ ✈➨❝tì Aij ù♥❣ ✈ỵ✐ xij t❤ä❛ ♠➣♥ 0 <
xij < dij ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ❤❛② t➟♣ ❤đ♣ ỉ G = {(i, j) : 0 < xij < dij }
❦❤ỉ♥❣ t↕♦ t❤➔♥❤ ❝❤✉ tr➻♥❤✳ P❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ x ❣å✐ ❧➔ ❦❤ỉ♥❣ s✉② ❜✐➳♥
♥➳✉ ● ❝â ✤ó♥❣ ♠ ✰ ♥ ✲ ✶ ỉ✱ tr→✐ ❧↕✐ x ❣å✐ ❧➔ s✉② ❜✐➳♥✳
t❤ä❛ ♠➣♥ ✭✶✳✷✮ ✲ ✭✶✳✹✮ ❣å✐ ❧➔ ♠ët

Số hóa bởi trung tâm học liệu

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✾

✶✳✷✳ P❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ ❜❛♥ ✤➛✉
❱➼ ❞ư ✶✳✷

✳ ❚➻♠ ♣❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ ❜❛♥ ✤➛✉ ❝❤♦ ❜➔✐ t♦→♥ ✈➟♥ t↔✐ ✈ỵ✐

❞ú ❧✐➺✉ ❝❤♦ tr♦♥❣ ❇↔♥❣ ✶✳✸✳

• ❍➔♥❣ t❤ù ♥❤➜t✿ ❇➢t ✤➛✉ tø ỉ ✭✶✱ ✶✮✱ t❛ ♣❤➙♥ ✈➔♦ ỉ ♥➔② ❧÷đ♥❣ ❤➔♥❣✿
x11 = min {a1 , b1 , d11 } = min {40, 25, 15} = 15 = d11 ✭✶✺ tỉ ✤➟♠✱ ♠➔✉
✤ä✮✳
❚✐➳♣ ✤â✱ ♣❤➙♥ ✈➔♦ ỉ ✭✶✱ ✷✮ ❧÷đ♥❣ ❤➔♥❣✿

x12 = min {a1 − 15, b2 , d12 } = min {25, 30, 20} = 20 = d12

✭✷✵ tỉ ✤➟♠✱

♠➔✉ ✤ä✮✳
❈✉è✐ ❝ò♥❣✱ ♣❤➙♥ ✈➔♦ ỉ ✭✶✱ ✸✮ ❧÷đ♥❣ ❤➔♥❣✿

x13 = min {a1 − 15 − 20, b3 , d13 } = min {5, 40, 25} = 5 < d13

✭ỉ ❝ì sð

✧•✧✮✳

•
•

❍➔♥❣ t❤ù ❤❛✐✿ ▲➛♥ ❧÷đt ♣❤➙♥ ❤➔♥❣ ✈➔♦ ❝→❝ ỉ ✭✷✳ ✶✮✱ ✭✷✳ ✷✮ ✈➔ ✭✷✳ ✸✮✳
❍➔♥❣ t❤ù ❜❛✿ ▲➛♥ ❧÷đt ♣❤➙♥ ❤➔♥❣ ✈➔♦ ❝→❝ ỉ ✭✸✳ ✸✮ ✈➔ ✭✸✳ ✹✮✳

✣➦t

xij = 0

❝❤♦ t➜t ❝↔ ❝→❝ ỉ ❝á♥ ❧↕✐ ✭❦❤ỉ♥❣ ✤÷đ❝ ♣❤➙♥ ♣❤è✐ ❤➔♥❣✮✳

❑➳t q✉↔ ❧➔ t❛ ✤÷đ❝ ♣❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ ✭❣ç♠ ✻ ỉ ❝ì sð ✧•✧✮ ❣❤✐ ð ❇↔♥❣
✶✳✸✳
❚r♦♥❣ ỉ ✭✐✱ ❥✮✿ sè ❣❤✐ ð ❣â❝ tr➯♥ ❜➯♥ tr→✐ ❧➔
♣❤↔✐ ❧➔

dij

✈➔ sè ❣❤✐ ð ❤➔♥❣ ❞÷ỵ✐ ❧➔

xij

cij ✱

sè ❣❤✐ ð ❣â❝ tr➯♥ ❜➯♥

✭sè ✐♥ ✤➟♠ ♠➔✉ ✤ä ❝❤➾

➷ ❝ì sð ✤→♥❤ ❞➜✉ ✧•✧✳

Số hóa bởi trung tâm học liệu

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xij = dij ✮✳


✶✵

✶✳✸✳ ❚✐➯✉ ❝❤✉➞♥ tè✐ ÷✉
●✐↔ sû

x0 = x0ij

m×n

❧➔ ♠ët ♣❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✶✮ ✲

G = (i, j) : 0 ≤ x0ij ≤ dij ❣ç♠ ♠ ✰ ♥ ✲

✶ ỉ ❦❤ỉ♥❣ ❝❤ù❛ ❝❤✉ tr➻♥❤✳ ●✐↔ sû ui ✱ vj ❧➔ ❝→❝ t❤➳ ✈à ❤➔♥❣✱ ❝ët t❤ä❛ ♠➣♥
❤➺ ♣❤÷ì♥❣ tr➻♥❤ ui +vj = cij ✈ỵ✐ ♠å✐ (i, j) ∈ G✳ ❑þ ❤✐➺✉ ∆ij = ui +vj −cij
✈ỵ✐ ♠å✐ (i, j) ∈
/ G (∆ij = 0, ∀ (i, j) ∈ G)✳ ✣à♥❤ ❧þ s❛✉ ❝❤♦ t❛ ♠ët ❞➜✉
0
❤✐➺✉ ♥❤➟♥ ❜✐➳t ❦❤✐ ♥➔♦ x ❧➔ ♣❤÷ì♥❣ →♥ tè✐ ÷✉✳
0
✭❉➜✉ ❤✐➺✉ tè✐ ÷✉✮✳ P❤÷ì♥❣ →♥ x =
x0ij m×n ❧➔ tè✐
÷✉ ❝õ❛ ✭✶✳✶✮ ✲ ✭✶✳✺✮ ❦❤✐ ∆ij ≤ 0 ✈ỵ✐ ♠å✐ (i, j) ∈
/ G, x0ij = 0 ✈➔ ∆ij ≥
0 ✈ỵ✐ ♠å✐ (i, j) ∈
/ G, x0ij = dij ✳

✭✶✳✺✮ t÷ì♥❣ ù♥❣ ✈ỵ✐ t➟♣ ỉ ❝ì sð

✣à♥❤ ❧þ ✶✳✶

✶✳✹✳ ❚❤✉➟t t♦→♥ ❣✐↔✐
❇÷ỵ❝ ✵
✭❑❤ð✐

t↕♦✮✳

❳➙②

❞ü♥❣

♣❤÷ì♥❣

→♥

❝ü❝

❜✐➯♥

❜❛♥

✤➛✉

x0 = x0ij m×n ✳ ❚➟♣ ỉ ❝❤å♥ ❝ì sð t÷ì♥❣ ù♥❣ ✈ỵ✐ x0 ❧➔ G0 =
(i, j) : 0 ≤ x0ij ≤ dij ❣ç♠ ✭♠ ✰ ♥ ✲ ✶✮ ♣❤➛♥ tû ✈➔ ❦❤ỉ♥❣ ❝❤ù❛ ❝❤✉ tr➻♥❤✱
♥❣❤➽❛ ❧➔ ❤➺ ✈➨❝tì {Aij : (i, j) ∈ G0 } ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✳ ✣➦t ❝❤➾ sè ✈á♥❣
❧➦♣ ❦ ❂ ✵✳

❇÷ỵ❝ ✶

ù♥❣ ✈ỵ✐

✳ ❚➼♥❤ ❝→❝ t❤➳ ✈à

Gk ✱

ui

✭✐ ❂ ✶✱ ✳✳✳ ✱ ♠✮ ✈➔

vj

✭❥ ❂ ✶✱ ✳✳✳ ✱ ♥✮ t÷ì♥❣

❜➡♥❣ ❝→❝❤ ❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✭❞↕♥❣ t❛♠ ❣✐→❝✮✿

ui + vj = cij , ∀ (i, j) ∈ Gk .

❇÷ỵ❝ ✷
❝â

✳ ❚➼♥❤ ❝→❝ ÷ỵ❝ ❧÷đ♥❣

∆ij = 0, ∀ (i, j) ∈ Gk ✮

❇÷ỵ❝ ✸

∆ij = ui + vj − cij , ∀ (i, j) ∈
/ Gk

✭❚❛ ❧✉ỉ♥

✳

✳ ✭❑✐➸♠ tr❛ ✤✐➲✉ ❦✐➺♥ tè✐ ÷✉✮

∆ij ≤ 0 ✈ỵ✐ ♠å✐ (i, j) ∈
/ Gk , xkij = 0 ✈➔ ∆ij ≥ 0 ✈ỵ✐ ♠å✐ (i, j) ∈
/
Gk , xkij = dij t❤➻ ❞ø♥❣ t❤✉➟t t♦→♥✿ xk ❧➔ ♣❤÷ì♥❣ →♥ tè✐ ÷✉ ✭✣à♥❤ ❧þ ✶✳✶✮✳
◆➳✉

◆➳✉ tr→✐ ❧↕✐✱ ❝❤✉②➸♥ s❛♥❣ ❇÷ỵ❝ ✹✳

❇÷ỵ❝ ✹

✳ ✭✣✐➲✉ ❝❤➾♥❤ ♣❤÷ì♥❣ →♥✮

✹❛✮ ➷ ✤✐➲✉ ❝❤➾♥❤ ✭r✱s✮ ❧➔ ỉ ✤↕t ♠❛① tr♦♥❣ ❜✐➸✉ t❤ù❝

∆ = max ∆ij

✈ỵ✐

(i, j) ∈
/ Gk , xkij = 0, −∆ij

Số hóa bởi trung tâm học liệu

✈ỵ✐

(i, j) ∈
/ Gk , xkij = dij > 0.

/>

✶✶
✹❜✮ ▲➟♣ ❝❤✉ tr➻♥❤ ❈ t↕♦ ♥➯♥ ❜ð✐ ỉ ✭r✱ s✮ ✈ỵ✐ ❝→❝ ỉ t❤✉ë❝
❤↕♥ ❜➡♥❣ ❝→❝❤ ❧♦↕✐ ❞➛♥ ❝→❝ ỉ tr❡♦ ❝õ❛

Gk ∪ {(r, s)}

Gk

✭❝❤➥♥❣

✮✳

C1 ✭t➟♣ ỉ ❧➫✮ ✈➔ C2
(r, s) ∈ C2 ♥➳✉ xkrs =

✹❝✮ P❤➙♥ ❤♦↕❝❤ t➟♣ ❈ t❤➔♥❤ ❤❛✐ t➟♣ ❝♦♥ rí✐ ♥❤❛✉
✭t➟♣ ỉ ❝❤➤♥✮ ✈ỵ✐ q✉✐ ÷ỵ❝

(r, s) ∈ C1

♥➳✉

xkrs = 0 ✈➔

drs ✳
✹❞✮ ❳→❝ ✤à♥❤ ❧÷đ♥❣ ✤✐➲✉ ❝❤➾♥❤ ❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝

h = min dij − xkij

(i, j) ∈ C1 , xkij

✈ỵ✐

✈ỵ✐

(i, j) ∈ C2 ≥ 0.

➷ ✭♣✱ q✮ ✤↕t ♠✐♥ tr÷í♥❣ ❤đ♣ ❜✐➸✉ t❤ù❝ tr➯♥ s➩ ❜à ❧♦↕✐✳

❇÷ỵ❝ ✺

✳ ❳➙② ❞ü♥❣ ♣❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ ♠ỵ✐ t❤❡♦ ❝ỉ♥❣ t❤ù❝

xk+1
ij

 k
 xij + h ❦❤✐ (i, j) ∈ C1 ,
xkij − h ❦❤✐ (i, j) ∈ C2 ,
=
 k
xij
❦❤✐ (i, j) ∈
/ C.

✈➔ t➟♣ ỉ ❝❤å♥ t÷ì♥❣ ù♥❣
❚➠♥❣

k ←k+1

Gk+1 = (Gk \ {(p, q)}) ∪ {(r, s)}✳

✈➔ trð ❧↕✐ ❇÷ỵ❝ ✶ t❤ü❝ ❤✐➺♥ ✈á♥❣ ❧➦♣ ❦ ♠ỵ✐✳

✶✳✺✳ ❱➼ ❞ư ♠✐♥❤ ❤å❛
❱➼ ❞ư ✶✳✸

✳ ●✐↔✐ ❜➔✐ t♦→♥ ✈➟♥ t↔✐ ❝â ❦❤↔ ♥➠♥❣ ❧÷✉ t❤ỉ♥❣ ❤↕♥ ❝❤➳ ✈ỵ✐

✈➨❝tì ❝✉♥❣ ❛✱ ✈➨❝tì ❝➛✉ ❜✱ ♠❛ tr➟♥ ❝÷ỵ❝ ♣❤➼
♥➠♥❣ ❧÷✉ t❤ỉ♥❣

D = {dij }3×4

C = {cij }3×4

✈➔ ♠❛ tr➟♥ ❦❤↔

♥❤÷ s❛✉✳

a = (40, 50, 60)T , b = (25, 30, 40, 55)T




2 10 12 2
15 20 25 30
C =  5 1 2 10  , D =  20 25 30 30 
20 3 10 15
15 25 35 55
●✐↔✐✳ ❱➼ ❞ư ♥➔② ❝â ♠ ❂ ✸ tr↕♠ ♣❤→t✱ ♥ ❂ ✹ tr↕♠ t❤✉ ✈➔ t❤ä❛ ♠➣♥
✤✐➲✉ ❦✐➺♥ ❝➙♥ ❜➡♥❣ ❝✉♥❣ ❝➛✉ ✭✹✵ ✰ ✺✵ ✰ ✻✵ ❂ ✷✺ ✰ ✸✵ ✰ ✹✵ ✰ ✺✺✮✳ ❳✉➜t
♣❤→t tø ♣❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ ❜❛♥ ✤➛✉

G0

x0

❝❤♦ ð ❇↔♥❣ ✶✳✸ ✈ỵ✐ t➟♣ ỉ ❝ì sð

❂ ✭✶✱ ✸✮✱ ✭✷✱ ✶✮✱ ✭✷✱ ✷✮✱ ✭✷✱ ✸✮✳ ✭✸✱ ✸✮✱ ✭✸✱ ✹✮✱ ❣ç♠ ✻ ỉ ❦❤ỉ♥❣ ❝❤ù❛ ❝❤✉

tr➻♥❤ ✈➔ ❣✐→ trà ❤➔♠ ♠ư❝ t✐➯✉

Số hóa bởi trung tâm học liệu

f x0 = 1285✳

/>

✶✷

❱á♥❣ ❧➦♣ ✵
❇÷ỵ❝ ✶

✳

✳ ❈→❝ t❤➳ ✈à t÷ì♥❣ ù♥❣ ✈ỵ✐

x0 ✿

u0 = (0, −10, −2) , v 0 = (15, 11, 12, 17)

❇÷ỵ❝ ✷

∆ij = ui + vj − cij ❣❤✐ ð ♠❛ tr➟♥ ∆0 ❞÷ỵ✐





15 20 5 0
13 1 0 15
x0 =  10 10 30 0  , ∆0 =  0 0 0 −3 
0 0 5 55
−7 6 0 0

✳ ❈→❝ sè

❇÷ỵ❝ ✸

x0 ❝❤÷❛ tè✐ ÷✉ ✈➻ ❝á♥ ∆14 = 15 > 0 ✈ỵ✐ (1, 4) ∈
/
= 6 > 0 ✈ỵ✐ (3, 2) ∈
/ G0 , x32 = 0 ✭❤❛✐ sè ❞÷ì♥❣ ❣↕❝❤

✳ P❤÷ì♥❣ →♥

G0 , x14 = 0

✈➔

❝❤➙♥ tr♦♥❣

∆0 ✮✳

❇÷ỵ❝ ✹

✤➙②✿

∆32

✳

∆14 = max {∆14 , ∆32 } = 15 > 0✳
❜✮ ❈❤✉ tr➻♥❤ ✤✐➲✉ ❝❤➾♥❤ ❣ç♠ ✹ ỉ C = {(1, 4), (3, 4), (3, 3), (1, 3)}
❝✮ ❈→❝ ỉ ❧➫ C1 = {(1, 4), (3, 3)} ✈➔ ❝→❝ ỉ ❝❤➤♥ C2 = {(3, 4), (1, 3)}✳
d14 − x014 = 30, x034 = 55,
❞✮ ▲÷đ♥❣ ✤✐➲✉ ❝❤➾♥❤
h = min
= 5✳
d33 − x033 = 30, x013 = 5
❛✮ ➷ ✤✐➲✉ ❝❤➾♥❤ ✭r✱ s✮ ❂ ✭✶✱ ✹✮ ✈ỵ✐

➷ ❧♦↕✐ ✭♣✱ q✮ ❂ ✭✶✳ ✸✮✳

❇÷ỵ❝ ✺

x1 ✭❣❤✐ ð
G1 = {(1, 4) , (2, 1) , (2, 2) , (2, 3) . (3, 3) , (3, 4)}
f x0 = 1258✳
✳ P❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ ♠ỵ✐

❱á♥❣ ❧➦♣ ✶
❇÷ỵ❝ ✶

❞÷ỵ✐✮ ✈ỵ✐ t➟♣ ỉ ❝ì sð
✈➔

f x1

= 1210 <

✳

✳ ❈→❝ t❤➳ ✈à t÷ì♥❣ ù♥❣ ✈ỵ✐

x1 ✿

u1 = (0, 5, 13) , v 1 = (0, −4, −3, −3, 2) .

❇÷ỵ❝ ✷

∆ij = ui + vj − cij ❣❤✐ ð ♠❛ tr➟♥ ∆1 ❞÷ỵ✐ ✤➙②✿




15 20 0 5
−2 −14 −15 0
x1 =  10 10 30 0  , ∆1 =  0
0
0 −3 
0 0 10 50
−7 6
0
0
✳ ❈→❝ sè

❇÷ỵ❝ ✸

✳ P❤÷ì♥❣ →♥

G1 , x132 = 0

x1

✭sè ❞÷ì♥❣ ❣↕❝❤ ❝❤➙♥ tr♦♥❣

Số hóa bởi trung tâm học liệu

∆32 = 6 > 0
∆11 = −2 < 0

❝❤÷❛ tè✐ ÷✉ ✈➻ ❝á♥

∆1 ✮✱

/>
✈ỵ✐
✈ỵ✐

(3, 2) ∈
/
(1, 1) ∈
/

✶✸

G1 , x111 = d11

✈➔

❣↕❝❤ ♥❣❛♥❣ tr➯♥

❇÷ỵ❝ ✹

∆12 = −14 < 0
tr♦♥❣ ∆1 ✮✳

✈ỵ✐

(1, 2) ∈
/ G1 , x112 = d12

✭❤❛✐ sè ➙♠

✳

❛✮ ➷ ✤✐➲✉ ❝❤➾♥❤ ✭r✱ s✮ ❂ ✭✶✱ ✷✮ ✈ỵ✐✿

−∆12 = max {−∆11 = 2, −∆12 = 14, ∆32 = 6} = 14 > 0.
❜✮ ❈❤✉ tr➻♥❤ ✤✐➲✉ ❝❤➾♥❤ ❣ç♠ ✻ ỉ✿

C = {(1, 2), (1, 4), (3, 4), (3, 3), (2, 3), (2, 2)} .
❝✮

❈→❝

ỉ

C1 = {(1, 4), (3, 3), (2, 2)}✱

❧➫

❝→❝

ỉ

❝❤➤♥

C2

=

= 15✳

➷

{(1, 2), (3, 4), (2, 3)}✳
❞✮ ▲÷đ♥❣ ✤✐➲✉ ❝❤➾♥❤

x112 = 20, d14 − x114 = 25, x134 = 50,
d33 − x133 = 25, x123 = 30, d22 − x122 = 15

h = min
❧♦↕✐ ✭♣✱ q✮ ❂ ✭✷✳ ✷✮✳

❇÷ỵ❝ ✺

x2 ✭❣❤✐ ð ❞÷ỵ✐✮ ✈ỵ✐ t➟♣ ỉ ❝ì sð
G2 = {(1, 2), (1, 4), (2, 1), (2, 3).(3, 3), (3, 4)} ✈➔ f (x2 ) ❂ ✶✵✵✵ ❁ f (x1 ) ❂
✳ P❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ ♠ỵ✐

✶✷✶✵✳

❱á♥❣ ❧➦♣ ✷
❇÷ỵ❝ ✶

✳

✳ ❈→❝ t❤➳ ✈à t÷ì♥❣ ù♥❣ ✈ỵ✐

x2 ✿

u2 = (0, 5, 13) , v 2 = (0, 10, −3, 2) .

❇÷ỵ❝ ✷

∆ij = ui + vj − cij ❣❤✐ ð ♠❛ tr➟♥ ∆2 ❞÷ỵ✐ ✤➙②✿




15 5 0 20
−2 0 −15 0
x2 =  10 25 15 0  , ∆2 =  0 9

0 −3 
0 0 25 35
−7 20 0
0
✳ ❈→❝ sè

❇÷ỵ❝ ✸
G1 , x232
G2 , x211

x2

∆32 = 20 > 0 ✈ỵ✐ (3, 2) ∈
/
= 0 ✭sè ❞÷ì♥❣ ❣↕❝❤ ❝❤➙♥ tr♦♥❣ ∆2 ✮ ✱ ✈➔ ∆11 = −2 < 0 ✈ỵ✐ (1, 1) ∈
/
= d11 ✭sè ➙♠ ❣↕❝❤ ♥❣❛♥❣ tr➯♥ tr♦♥❣ ∆2 ✮✳
✳ P❤÷ì♥❣ →♥

Số hóa bởi trung tâm học liệu

❝❤÷❛ tè✐ ÷✉ ✈➻ ❝á♥

/>

✶✹

❇÷ỵ❝ ✹

✳

❛✮ ➷ ✤✐➲✉ ❝❤➾♥❤ ✭r✱ s✮ ❂ ✭✸✱ ✷✮ ✈ỵ✐✿

∆32 = max {−∆11 = 2, ∆32 = 20} = 20 > 0.
❜✮ ❈❤✉ tr➻♥❤ ✤✐➲✉ ❝❤➾♥❤ ❣ç♠ ✹ ỉ✿

C = {(3, 2), (1, 2), (1, 4), (3, 4)} .
❝✮ ❈→❝ ỉ ❧➫

C1 = {(3, 2), (1, 4)}✱

❝→❝ ỉ ❝❤➤♥

C2 = {(1, 2), (3, 4)}✳

❞✮ ▲÷đ♥❣ ✤✐➲✉ ❝❤➾♥❤

h = min d32 − x232 = 25, x212 = 5, d14 − x214 = 10, x234 = 35 = 5✳

➷

❧♦↕✐ ✭♣✱ q✮ ❂ ✭✶✱ ✷✮✳

❇÷ỵ❝ ✺

x3 ✭❣❤✐ ð ❞÷ỵ✐✮ ✈ỵ✐ t➟♣ ỉ ❝ì
G3 = {(1, 4), (2, 1), (2, 3), (3, 2).(3, 3), (3, 4)} ✈➔ f (x3 ) ❂ ✾✵✵ ❁ f (x2 )
✳ P❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ ♠ỵ✐

sð

❂

✶✵✵✵✳

❱á♥❣ ❧➦♣ ✸✳
❇÷ỵ❝ ✶

✳ ❈→❝ t❤➳ ✈à t÷ì♥❣ ù♥❣ ✈ỵ✐

x3 ✿

u3 = (0, 5, 13) , v 3 = (0, −10, −3, 2) .

❇÷ỵ❝ ✷

∆ij = ui + vj − cij ❣❤✐ ð ♠❛ tr➟♥ ∆3 ❞÷ỵ✐ ✤➙②✿




15 0 0 25
−2 −20 −15 0
x3 =  10 25 15 0  , ∆3 =  0 −6
0 −3  .
0 5 25 30
−7 0
0
0
✳ ❈→❝ sè

❇÷ỵ❝ ✸

x3 ❝❤÷❛ tè✐ ÷✉ ✈➻ ❝á♥ ∆11 = −2 < 0 ✈ỵ✐ (1, 2) ∈
/
3
= −6 < 0 ✈ỵ✐ (2, 2) ∈
/ G3 , x22 = d22 ✭❤❛✐ sè ➙♠ ❣↕❝❤

✳ P❤÷ì♥❣ →♥

G3 , x311 = d11
♥❣❛♥❣ tr➯♥

❇÷ỵ❝ ✹

∆22
tr♦♥❣ ∆3 ✮✳
✈➔

✳

❛✮ ➷ ✤✐➲✉ ❝❤➾♥❤ ✭r✱ s✮ ❂ ✭✷✱ ✷✮ ✈ỵ✐✿

−∆22 = max {−∆11 = 2, −∆22 = 6} = 6.
C = {(2, 2), (2, 3), (3, 3), (3, 2)}✳
C1 = {(2, 3), (3, 2)}✱ ❝→❝ ỉ ❝❤➤♥ C2 = {(2, 2), (3, 3)}✳

❜✮ ❈❤✉ tr➻♥❤ ✤✐➲✉ ❝❤➾♥❤ ❣ç♠ ✻ ỉ
❝✮ ❈→❝ ỉ ❧➫

Số hóa bởi trung tâm học liệu

/>

✶✺
❞✮ ▲÷đ♥❣ ✤✐➲✉ ❝❤➾♥❤✿

h = min x322 = 25, d23 − x323 = 15, x333 = 25, d32 − x332 = 20 = 15✳
➷ ❧♦↕✐ ✭♣✱ q✮ ❂ ✭✷✳ ✸✮✳

❇÷ỵ❝ ✺

x4 ✭❣❤✐ ð ❞÷ỵ✐✮
G4 = {(1, 4), (2, 1), (2, 2), (3, 2).(3, 3), (3, 4)} ✈➔ f (x4 ) ❂
✳ P❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ ♠ỵ✐

✈ỵ✐ t➟♣ ỉ ❝ì sð
✽✶✵ ❁

f (x3 )

❂

✾✵✵✳

❱á♥❣ ❧➦♣ ✹✳
❇÷ỵ❝ ✶

✳ ❈→❝ t❤➳ ✈à t÷ì♥❣ ù♥❣ ✈ỵ✐

x4 ✿

u4 = (0, 11, 13) , v 4 = (−6, −10, −3, 2)

❇÷ỵ❝ ✷

∆ij = ui + vj − cij ❣❤✐ ð ♠❛ tr➟♥ ∆4 ❞÷ỵ✐ ✤➙②✿




15 0 0 25
−8 −20 −15 0
x4 =  10 10 30 0  , ∆4 =  0
0
6 −3  .
0 20 10 30
−13 0
0
0
✳ ❈→❝ sè

❇÷ỵ❝ ✸

✳ P❤÷ì♥❣ →♥

G4 , x411 = d11

❇÷ỵ❝ ✹

x4

∆11 = −8 < 0 ✈ỵ✐ (1, 1) ∈
/
∆4 ✮✳

❝❤÷❛ tè✐ ÷✉ ✈➻ ❝á♥

✭sè ➙♠ ❣↕❝❤ ♥❣❛♥❣ tr➯♥ tr♦♥❣

✳

❛✮ ➷ ✤✐➲✉ ❝❤➾♥❤ ✭r✱ s✮ ❂ ✭✶✱ ✶✮ ✈ỵ✐

−∆11 = 8 > 0

❜✮ ❈❤✉ tr➻♥❤ ✤✐➲✉ ❝❤➾♥❤ ❣ç♠ ✻ ỉ✿

C = {(1, 1), (1, 4), (3, 4), (3, 2), (2, 2), (2, 1)} .
❝✮

❈→❝

ỉ

❧➫

C1 = {(1, 4), (3, 2), (2, 1)}✱

❝→❝

ỉ

❝❤➤♥

C2 = {(1, 1), (3, 4), (2, 2)}✳
❞✮ ▲÷đ♥❣ ✤✐➲✉ ❝❤➾♥❤

h = min

x411 = 15, d14 − x414 = 5, x434 = 30,
d32 − x432 = 5, x422 = 10, d21 − x421 = 10

= 5✳

➷ ❧♦↕✐

✭♣✱ q✮ ❂ ✭✶✳ ✹✮✳

❇÷ỵ❝ ✺

x5 ✭❣❤✐ ð ❞÷ỵ✐✮ ✈ỵ✐ t➟♣ ỉ ❝ì
G5 = {(1, 1), (2, 1), (2, 2), (3, 2), (3, 3), (3, 4)} ✈➔ f (x5 ) ❂ ✼✼✵ ❁ f (x4 )
✳ P❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ ♠ỵ✐

✽✶✵✳

Số hóa bởi trung tâm học liệu

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sð
❂


✶✻

❱á♥❣ ❧➦♣ ✺✳
❇÷ỵ❝ ✶

✳ ❈→❝ t❤➳ ✈à t÷ì♥❣ ù♥❣ ✈ỵ✐

x5 ✿

u5 = (0, 3, 5) , v 5 = (2, −2, 5, 10)

❇÷ỵ❝ ✷

∆ij = ui + vj − cij ❣❤✐ ð ♠❛ tr➟♥ ∆5 ❞÷ỵ✐ ✤➙②✿




10 0 0 30
0 −12 −7 8
x5 =  15 5 30 0  , ∆5 =  0
0
6 3 .
0 25 10 25
−13 0
0 0

✳ ❈→❝ sè

❇÷ỵ❝ ✸

✳ P❤÷ì♥❣ →♥

G5 , x524 = 0

❇÷ỵ❝ ✹

x5

❝❤÷❛ tè✐ ÷✉ ✈➻ ❝á♥

✭sè ❞÷ì♥❣ ❣↕❝❤ ❝❤➙♥ tr♦♥❣

∆5

∆24 = 3 > 0

✈ỵ✐

(2, 4) ∈
/

✮✳

✳

∆24 = 3 > 0✳

❜✮ ❈❤✉ tr➻♥❤ ✤✐➲✉ ❝❤➾♥❤ ❣ç♠ ✹ ỉ C = {(2, 4), (3, 4), (3, 2), (2, 2)}✳
❝✮ ❈→❝ ỉ ❧➫ C1 = {(2, 4), (3, 2)}✱ ❝→❝ ỉ ❝❤➤♥ C2 = {(3, 4), (2, 2)}✳
❛✮ ➷ ✤✐➲✉ ❝❤➾♥❤ ✭r✱ s✮ ❂ ✭✷✱ ✹✮ ✈ỵ✐

❞✮ ▲÷đ♥❣ ✤✐➲✉ ❝❤➾♥❤

h = min d24 − x524 = 30, x534 = 25, d32 − x532 = 0, x522 = 5

= 0✳

➷ ❧♦↕✐ ✭♣✱ q✮ ❂ ✭✸✱ ✷✮✳

❇÷ỵ❝ ✺

x6 ✭❣❤✐ ð ❞÷ỵ✐✮ ✈ỵ✐ t➟♣ ỉ ❝ì
G6 = {(1, 1), (2, 1), (2, 2), (2, 4).(3, 3), (3, 4)} ✈➔ f (x6 ) = f (x5 ) = 770✳
✳ P❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ ♠ỵ✐

❱á♥❣ ❧➦♣ ✻
❇÷ỵ❝ ✶

✳ ❈→❝ t❤➳ ✈à t÷ì♥❣ ù♥❣ ✈ỵ✐

sð

x6 ✿

u6 = (0, 3, 8) , v 6 = (2, −2, 2, 7)

❇÷ỵ❝ ✷

∆ij = ui + vj − cij ❣❤✐ ð ♠❛ tr➟♥ ∆6 ❞÷ỵ✐ ✤➙②✿




10 0 0 30
0 −12 −10 5
x6 =  15 5 30 0  , ∆6 =  0
0
3 0 .
0 25 10 25
−10 3
0 0
✳ ❈→❝ sè

❇÷ỵ❝ ✸
✭❝→❝
✭❝→❝

∆ij ≤ 0 ✈ỵ✐ ♠å✐ (i, j) ∈
/ G6 , x6ij = 0
sè ➙♠ ❣↕❝❤ ❝❤➙♥ tr♦♥❣ ∆6 ✮✱ ∆ij ≥ 0 ✈ỵ✐ ♠å✐ (i, j) ∈
/ G6 , x6ij = dij
sè ❞÷ì♥❣ ❣↕❝❤ ♥❣❛♥❣ tr➯♥ tr♦♥❣ ∆6 ✮✳

❱➟②

✳ P❤÷ì♥❣ →♥

x6

tè✐ ÷✉ ✈➻

xopt = x6 , fopt = f x6 = 770✳

Số hóa bởi trung tâm học liệu

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✶✼
❚â♠ ❧↕✐✱ ❝❤÷ì♥❣ ♥➔② ✤➣ tr➻♥❤ ❜➔② tâ♠ t➢t ❝→❝ ❦➳t q✉↔ ✈➲ ❜➔✐ t♦→♥
✈➟♥ t↔✐ ❝â ❦❤↔ ♥➠♥❣ ❧÷✉ t❤ỉ♥❣ ❤↕♥ ❝❤➳✿ ♠ỉ ❤➻♥❤ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛
❜➔✐ t♦→♥✱ ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ♣❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ ❜❛♥ ✤➛✉ ❝õ❛ ❜➔✐ t♦→♥✱
t✐➯✉ ❝❤✉➞♥ tè✐ ÷✉ ✈➔ t❤✉➟t t♦→♥ t❤➳ ✈à ❣✐↔✐ ❜➔✐ t♦→♥✳ ❈✉è✐ ❝❤÷ì♥❣✱ ♥➯✉
✈➼ ❞ư sè ♠✐♥❤ ❤å❛ t❤✉➟t t♦→♥ ❣✐↔✐✳ ◆â✐ ❝❤✉♥❣✱ t❤✉➟t t♦→♥ ❣✐↔✐ ❜➔✐ t♦→♥
✈➟♥ t↔✐ ❝â ❤↕♥ ❝❤➳ ❦❤↔ ♥➠♥❣ ❧÷✉ t❤ỉ♥❣ ♣❤ù❝ t↕♣ ❤ì♥ ✤ỉ✐ ❝❤ót s♦ ✈ỵ✐ ❜➔✐
t♦→♥ ✈➟♥ t↔✐ ❦❤ỉ♥❣ ❤↕♥ ❝❤➳ ❦❤↔ ♥➠♥❣ ❧÷✉ t❤ỉ♥❣✳

Số hóa bởi trung tâm học liệu

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✶✽

❈❤÷ì♥❣ ✷
❇⑨■ ❚❖⑩◆ ❱❾◆ ❚❷■ ❱❰■
▲×Đ◆● ❈❯◆● ❇➚ ❈❍➄◆ ❉×❰■
❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ỉ ❤➻♥❤ t♦→♥ ❤å❝ ❝õ❛ ❜➔✐ t♦→♥ ✈➟♥ t↔✐✱ tr♦♥❣
✤â ❧÷đ♥❣ ❝✉♥❣ ❝õ❛ ❝→❝ tr↕♠ ♣❤→t ❦❤ỉ♥❣ q✉✐ ✤à♥❤ tr÷ỵ❝ ♠➔ ❜à ❝❤➦♥ tr➯♥

✈➔ ❞÷ỵ✐✱ ❝á♥ ❧÷đ♥❣ ❝➛✉ ❝õ❛ ❝→❝ tr↕♠ t❤✉ ✤➣ ❜✐➳t tr÷ỵ❝✳ ❙❛✉ ✤â ①➨t tr÷í♥❣
❤đ♣ ✤ì♥ ❣✐↔♥ ❦❤✐ ❧÷đ♥❣ ❝✉♥❣ ❝õ❛ ❝→❝ tr↕♠ ♣❤→t ❝❤➾ ❜à ❝❤➦♥ ❞÷ỵ✐ ❜ð✐ sè
❦❤ỉ♥❣ ➙♠✳ ◆➯✉ ❝→❝❤ ✤÷❛ ✈➲ ❜➔✐ t♦→♥ ❣➛♥ ✈ỵ✐ ❜➔✐ t♦→♥ ✈➟♥ t↔✐ t❤ỉ♥❣
t❤÷í♥❣ ✈➔ ①➨t t➼♥❤ ❝❤➜t ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥✳ ❙❛✉ ✤â tr➻♥❤ ❜➔② t✐➯✉
❝❤✉➞♥ tè✐ ÷✉ ✈➔ t❤✉➟t t♦→♥ ❣✐↔✐✳ ❈✉è✐ ❝❤÷ì♥❣ ♥➯✉ ✈➼ ❞ư sè ♠✐♥❤ ❤å❛✳ ◆ë✐
❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ t❤❛♠ ❦❤↔♦ ❝❤õ ②➳✉ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✸❪ ✈➔ ❬✺❪✳

✷✳✶✳ ◆ë✐ ❞✉♥❣ ❜➔✐ t♦→♥
✣➸ t❤✉➟♥ t✐➺♥ ❝❤♦ ✈✐➺❝ ❦❤↔♦ s→t✱ ð ✤➙② t❛ s➩ ♠ỉ t↔ ❜➔✐ t♦→♥ ✈➟♥ t↔✐
✈ỵ✐ ❧÷đ♥❣ ❝✉♥❣ ❝õ❛ ❝→❝ tr↕♠ ♣❤→t ❜à ❝❤➦♥ ✭tr➯♥ ✈➔ ❞÷ỵ✐✮ ♥❤÷ s❛✉✳
●✐↔ sû ❝â ♠ët ❧♦↕✐ ❤➔♥❣ ✭❝❤➥♥❣ ❤↕♥ ❧÷ì♥❣ t❤ü❝✱ ①✐ ♠➠♥❣✱ ✈✳✈ ✳✳✳✮ ❝➛♥
✈➟♥ ❝❤✉②➸♥ tø ♠ ✤✐➸♠ ❝✉♥❣ ❝➜♣ ✭❣å✐ ❧➔ ❝→❝ tr↕♠ ♣❤→t✮✱ ❦þ ❤✐➺✉ ✐ ❂ ✶✱ ✷✱
✳✳✳ ✱ ♠✱ tỵ✐ ♥ ✤✐➸♠ t✐➯✉ t❤ư ✭❣å✐ ❧➔ ❝→❝ tr↕♠ t❤✉✮✱ ❥ ❂ ✶✱ ✷✱ ✳✳✳ ✱ ♥✳ ❈❤♦ ❜✐➳t
❦❤↔ ♥➠♥❣ ❝✉♥❣ ❝➜♣ ❤➔♥❣ ❝õ❛ tr↕♠ ♣❤→t ✐ t❤✉ë❝ ❦❤♦↔♥❣ ❝❤♦ tr÷ỵ❝

ai ❧➔ ♠ù❝
(0 ≤ ai ≤ ai )

ai

❚r♦♥❣ ✤â

❝✉♥❣ tè✐ t❤✐➸✉ ✈➔

♣❤→t ✐

✱ ♥❤✉ ❝➛✉ t✐➯✉ t❤ư ❤➔♥❣ ❝õ❛ tr↕♠ t❤✉ ❥ ❧➔

[ai , ai ]✳

❧➔ ♠ù❝ ❝✉♥❣ tè✐ ✤❛ ❝õ❛ tr↕♠

bj > 0

✭bj ❝è ✤à♥❤✮ ✈➔ ❝÷ỵ❝ ♣❤➼ ✈➟♥ ❝❤✉②➸♥ ♠ët ✤ì♥ ✈à ❤➔♥❣ tø tr↕♠ ♣❤→t ✐ tỵ✐
tr↕♠ t❤✉ ❥ ❧➔

cij ≥ 0✳

❇➔✐ t♦→♥ ✤➦t r❛ ❧➔ ❤➣② t➻♠ ♠ët ♣❤÷ì♥❣ →♥ ✈➟♥ ❝❤✉②➸♥ ❤➔♥❣ tø ❝→❝
tr↕♠ ♣❤→t tỵ✐ ❝→❝ tr↕♠ t❤✉ s❛♦ ❝❤♦ tê♥❣ ❝❤✐ ♣❤➼ ✈➟♥ ❝❤✉②➸♥ ❧➔ ♥❤ä ♥❤➜t❄

Số hóa bởi trung tâm học liệu

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✶✾
✣➸ ✤↔♠ ❜↔♦ ❝❤♦ ❜➔✐ t♦→♥ ❣✐↔✐ ✤÷đ❝✱ t❛ ❣✐↔ t❤✐➳t

m

n

ai ≤
i=1

m

bj ≤
j=1

ai

✭✷✳✶✮

i=1

❇➔✐ t♦→♥ tr➯♥ ❝â t❤➸ ❞✐➵♥ t↔ ❞÷ỵ✐ ❞↕♥❣ ♠ỉ ❤➻♥❤ t♦→♥ ❤å❝ ♥❤÷ s❛✉✿

m

n

f (x) ≡

(P )

cij xij → min,

✭✷✳✷✮

xij − xi0 = ai , i = 1, 2, ..., m,

✭✷✳✸✮

xij = bj , j = 1, 2, ...,n,

✭✷✳✹✮

i=1 j=1

n

j=1
m

i=1

tr♦♥❣ ✤â
t❤✉ ❥✱

xij

xi0

xij ≥ 0, i = 1, ..., m, j = 1, ..., n,

✭✷✳✺✮

0 ≤ xi0 ≤ ei ≡ ai − ai , i = 1, 2, ..., m,

✭✷✳✻✮

❜✐➸✉ t❤à ❧÷đ♥❣ ❤➔♥❣ ❝➛♥ ✈➟♥ ❝❤✉②➸♥ tø tr↕♠ ♣❤→t ✐ tỵ✐ tr↕♠

❧➔ tê♥❣ ❧÷đ♥❣ ❤➔♥❣ ❝❤✉②➸♥ ✤✐ tø tr↕♠ ♣❤→t ✐ ✭tỵ✐ ♠å✐ tr↕♠

ai ≤ ai + xi0 ≤ ai , i =
1, 2, ..., m✳ ❱➻ t❤➳✱ ✤✐➲✉ ❦✐➺♥ ✭✷✳✸✮ ❝â t❤➸ ✈✐➳t ❧↕✐ t❤➔♥❤ ai ≤ xi1 +...+xin ≤
ai ✈ỵ✐ ♠å✐ ✐✳ ❉♦ ✤â✱ ✭P✮ ❝á♥ ✤÷đ❝ ❣å✐ ❧➔ ❜➔✐ t♦→♥ ✈➟♥ t↔✐ ✈ỵ✐ r➔♥❣ ❜✉ë❝
❤❛✐ ♣❤➼❛ ✈➲ ❧÷đ♥❣ ❝✉♥❣ ❝õ❛ ❝→❝ tr↕♠ ♣❤→t✳

t❤✉✮ ✈÷đt ♠ù❝ ❝✉♥❣ tè✐ t❤✐➸✉

ai ✳

❚ø ✭✷✳✻✮ s✉② r❛

❑þ ❤✐➺✉

s1 = a1 + a2 + ... + am ✭tê♥❣ ❝✉♥❣
s2 = a1 + a2 + ... + am ✭tê♥❣ ❝✉♥❣
d = b1 + b2 + ... + bn ✭tê♥❣ ❝➛✉✮✳

tè✐ t❤✐➸✉✮✱
tè✐ ✤❛✮✱

❚r♦♥❣ ❬✷❪ ❝→❝ t→❝ ❣✐↔ ✤➣ sû ❞ư♥❣ ♠ỉ ❤➻♥❤ tr➯♥ ✤➸ ♠ỉ t↔ ❜➔✐ t♦→♥ ♣❤➙♥
❜ê tè✐ ÷✉ ❝→❝ tr✉♥❣ t➙♠ ✤✐➲✉ ❤➔♥❤ ✭❞❡♣♦t✮✱ ✐ ❂ ✶✱ ✷✱ ✳✳✳ ✱ ♠✱ q✉↔♥ ❧þ
❝→❝ t✉②➳♥ ①❡ ❜✉þt✱ ❥ ❂ ✶✱ ✷✱ ✳✳✳ ✱ ♥✱ ♥❤➡♠ ❣✐↔♠ ✤➳♥ ♠ù❝ t❤➜♣ ♥❤➜t tê♥❣
q✉➣♥❣ ✤÷í♥❣ ①❡ ❝❤↕② ❦❤ỉ♥❣ t↔✐ ✭❜✐➳♥
t➙♠ ✐ q✉↔♥ ❧þ ✈➔

xij = 0

xij = 1

♥➳✉ t✉②➳♥ ①❡ ❥ ❞♦ tr✉♥❣

♥➳✉ tr→✐ ❧↕✐✮✱ ♥❤í ✤â ❝❤♦ ♣❤➨♣ t✐➳t ❦✐➺♠ ❦❤♦↔♥❣

10% ❝❤✐ ♣❤➼ ✈➟♥ ❤➔♥❤✱ tr♦♥❣ ✤â ai

✈➔

ai

❧➔ ❣✐ỵ✐ ❤↕♥ ❞÷ỵ✐ ✈➔ ❣✐ỵ✐ ❤↕♥ tr➯♥

❝❤♦ sè t✉②➳♥ ①❡ ❜✉þt ♠➔ tr✉♥❣ t➙♠ ✐ ✤÷đ❝ ♣❤➨♣ q✉↔♥ ❧þ✱ ✤✐➲✉ ❦✐➺♥ ✭✷✳✹✮

Số hóa bởi trung tâm học liệu

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✷✵
✈ỵ✐

bj ≡ 1

✈ỵ✐ ♠å✐ ❥✱ ❜↔♦ ✤↔♠ ♠é✐ t✉②➳♥ ①❡ ❜✉þt ✤➲✉ ✤÷đ❝ ♠ët tr✉♥❣

t➙♠ ♥➔♦ ✤â q✉↔♥ ❧þ✳
❈â t❤➸ t❤➜② ✭✷✳✷✮ ✲ ✭✷✳✻✮ ❧➔ ♠ët ❜➔✐ t♦→♥ q✉② ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ❞↕♥❣
✤➦❝ ❜✐➺t ✈ỵ✐ ♠ët sè ❜✐➳♥ ❜à ❝❤➦♥ tr➯♥ ✭❞♦ ✭✷✳✻✮✮✳ ❱➻ t❤➳✱ ❝→❝ ❦❤→✐ ♥✐➺♠
♣❤÷ì♥❣ →♥✱ ♣❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥✱ ♣❤÷ì♥❣ →♥ tè✐ ÷✉ ✭❤❛② ❧í✐ ❣✐↔✐✮ ✤÷đ❝
❤✐➸✉ t❤❡♦ ♥❣❤➽❛ q✉❡♥ t❤✉ë❝✳ ❘ã r➔♥❣ ♠✐➲♥ r➔♥❣ ❜✉ë❝ ❝õ❛ ❜➔✐ t♦→♥ ❦❤→❝
ré♥❣ ✭❞♦ ❝â ❣✐↔ t❤✐➳t ✭✷✳✶✮✮ ✈➔ ❜à ❝❤➦♥✿

xi0 ≤ ei

0 ≤ xij ≤ bj

✈ỵ✐ ♠å✐

i, j = 0, 0 ≤

✈ỵ✐ ♠å✐ ✐✳ ❱➻ t❤➳ ❜➔✐ t♦→♥ ❝❤➢❝ ❝❤➢♥ ❝â ❧í✐ ❣✐↔✐✳

❉♦ ❜➔✐ t♦→♥ ✭P✮ ❝â ❝➜✉ tró❝ ❦❤→ ✤➦❝ t❤ò ✭❣➛♥ ✈ỵ✐ ❝➜✉ tró❝ ❜➔✐ t♦→♥
✈➟♥ t↔✐ q✉❡♥ t❤✉ë❝✱ ❝❤➾ ❦❤→❝ ð ❝❤é ❝â t❤➯♠ ❝→❝ ❜✐➳♥ ❜à ❝❤➦♥ tr➯♥

xi0

✈ỵ✐

♠å✐ ✐✮ ♥➯♥ ♥➳✉ ❝❤➾ sû ❞ư♥❣ ✤ì♥ t❤✉➛♥ t❤✉➟t t♦→♥ ①û ❧þ ❜✐➳♥ ❜à ❝❤➠♥ tr➯♥
✤è✐ ✈ỵ✐ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ tê♥❣ q✉→t t❤➻ s➩ ❦➨♠ ❤✐➺✉ q✉↔✱ ❞♦ sè ❜✐➳♥
tr♦♥❣ ❜➔✐ t♦→♥ t➠♥❣ t❤❡♦ t➼❝❤

m × n✳

❱➻ t❤➳✱ ❦❤❛✐ t❤→❝ ❝➜✉ tró❝ ✤➦❝ ❜✐➺t

❝õ❛ ❜➔✐ t♦→♥ ✤➸ t➻♠ r❛ t❤✉➟t t♦→♥ ❣✐↔✐ ❤✐➺✉ q✉↔ ❧➔ r➜t ❝➛♥ t❤✐➳t ✈➔ ❝â þ
♥❣❤➽❛ ❝↔ ✈➲ ❧þ t❤✉②➳t ❧➝♥ ù♥❣ ❞ư♥❣ t❤ü❝ t✐➵♥✳
❚❛

❝ơ♥❣

❝â

t❤➸

✤➦t

❜➔✐

t♦→♥

t❤❡♦

♠ët

❝→❝❤

♥❤➻♥

❦❤→❝✿

✣➦t

T

y = (y1 , ..., ym ) ✱ H = {y ∈ Rm : ai ≤ yi ≤ ai , i = 1, 2, ..., m} ✈➔ S =
{y ∈ Rm : y1 + y2 + ... + ym = b}✳ ❈â t❤➸ ❞➵ ❞➔♥❣ ❦✐➸♠ tr❛ ❧↕✐ ❧➔ ✈ỵ✐
y ∈ H ∩ S t❤➻
m

n

f (y) = min

n

xij = yi , ∀i;

cij xij :
i=1 j=1

m

j=1

xij = bj , ∀j; xij ≥ 0, ∀i, j
i=1

❧➔ ♠ët ❤➔♠ ❧ç✐ t✉②➳♥ t➼♥❤ tø♥❣ ❦❤ó❝ ✭yi ❜✐➸✉ t❤à tê♥❣ ❧÷đ♥❣ ❤➔♥❣ ✈➟♥
❝❤✉②➸♥ tø tr↕♠ ♣❤→t ✐ tỵ✐ ♠å✐ tr↕♠ t❤✉✮✳ ❱➻ ✈➟②✱ ❜➔✐ t♦→♥ ✤➦t r❛ ð tr➯♥
❝ơ♥❣ ❝â t❤➸ ♠ỉ t↔ ♥❣➢♥ ❣å♥ ❞÷ỵ✐ ❞↕♥❣✿

min {f (y) |y ∈ H ∩ S }
✣➙② ❧➔ ♠ët ❜➔✐ t♦→♥ q✉② ❤♦↕❝❤ ❧ç✐ ✭❤➔♠ ♠ư❝ t✐➯✉ ❧ç✐✱ t✉②➳♥ t➼♥❤ tø♥❣
❦❤ó❝✮ tr➯♥ ❣✐❛♦ ❝õ❛ s✐➯✉ ❤ë♣ ❍ ✈ỵ✐ s✐➯✉ ♣❤➥♥❣ ❙✳ ❇➔✐ t♦→♥ ♥➔② ❝â ❝➜✉
tró❝ ❦❤→ ✤➦❝ ❜✐➺t ✈➔ ❝â t❤➸ ❣✐↔✐ t❤❡♦ ❝→❝ ❝→❝❤ r✐➯♥❣ ✭❝❤➥♥❣ ❤↕♥✱ ♣❤÷ì♥❣
♣❤→♣ ❝❤✐➳✉ ❣r❛✤✐➯♥✮✳ ❚✉② ♥❤✐➯♥✱ ✤✐➲✉ ❝➛♥ ❝❤ó þ ❧➔ ❤➔♠ ♠ư❝ t✐➯✉ ❢✭②✮

Số hóa bởi trung tâm học liệu

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✷✶
❦❤ỉ♥❣ ð ❞↕♥❣ ❤✐➸♥ ✈➔ ❧í✐ ❣✐↔✐ ❝õ❛ ❜➔✐ t♦→♥ ❝â t❤➸ ✤↕t t↕✐ ♠ët ✤✐➸♠ ❜✐➯♥
❤❛② ♠ët ✤✐➸♠ tr♦♥❣ ❝õ❛ t➟♣ r➔♥❣ ❜✉ë❝

H ∩ S✳

✷✳✷✳ ❇➔✐ t♦→♥ t÷ì♥❣ ✤÷ì♥❣ ✈➔ t➼♥❤ ❝❤➜t
❚r♦♥❣ ♠ư❝ ♥➔②✱ t❛ ①➨t tr÷í♥❣ ❤đ♣ ✤ì♥ ❣✐↔♥ ❤ì♥ ❦❤✐ ♠å✐

ai = +∞✱

tù❝ ❧➔ ♠ù❝ ❝✉♥❣ tè✐ ✤❛ ❝õ❛ ❝→❝ tr↕♠ ♣❤→t ❦❤ỉ♥❣ ❜à ❤↕♥ ❝❤➳✳ ❈ư t❤➸ ❧➔
❜➔✐ t♦→♥✿

m

n

f (x) ≡

(Q)

cij xij → min

✭✷✳✼✮

i=1 j=1
n

xij − xi0 = ai , i = 1, 2, ..., m,

✭✷✳✽✮

xij = bj , j = 1, 2, ..., n,

✭✷✳✾✮

j=1
m

i=1

xij ≥ 0, i = 1, ..., m, j = 1, ..., n,

✭✷✳✶✵✮

●✐↔ t❤✐➳t ✭✷✳✶✮ ❜➙② ❣✐í ❧➔

a1 + a2 + ... + am ≤ b1 + b2 + ... + bn .
❱➨❝tì ❤➺ sè ❝õ❛ ❜✐➳♥

✭✷✳✶✶✮

xij tr♦♥❣ ✭✷✳✽✮ ✲ ✭✷✳✾✮ ❧➔ Aij ∈ Rm+n ✈ỵ✐ ❝→❝ t❤➔♥❤

m + j ❜➡♥❣ 1✱ ♠å✐ t❤➔♥❤ ♣❤➛♥ ❦❤→❝ ❜➡♥❣ ✵ ✭✐ ❂ ✶✱ ✷✱ ✳✳✳ ✱ ♠✱ ❥
m+n
❂ ✶✱ ✷✱ ✳✳✳ ✱ ♥✮❀ Ai0 ∈ R
✈ỵ✐ t❤➔♥❤ ♣❤➛♥ ✐ ❜➡♥❣ ✲ ✶✱ ♠å✐ t❤➔♥❤ ♣❤➛♥
m+n

❦❤→❝ ❜➡♥❣ ✵ ❤❛② −Ai0 ❧➔ ✈➨❝tì ✤ì♥ ✈à t❤ù ✐ tr♦♥❣ R
✭✐ ❂ ✶✱ ✳✳✳ ✱ ♠✮✳
♣❤➛♥ ✐ ✈➔

✣➸ ❣✐↔✐ ❜➔✐ t♦→♥ ✭◗✮✱ t❛ ❝â t❤➸ →♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✤ì♥ ❤➻♥❤ ✈➟♥
t↔✐✳ ❱➲ ✤↕✐ t❤➸✱ t❤✉➟t t♦→♥ t❤ü❝ ❤✐➺♥ ❝→❝ t❤❛♦ t→❝ t÷ì♥❣ tü ♥❤÷ ❦❤✐ ❣✐↔✐
♠ët ❜➔✐ t♦→♥ ✈➟♥ t↔✐ t❤ỉ♥❣ t❤÷í♥❣✳ ✣✐➸♠ ✤→♥❣ ❝❤ó þ ❧➔ ð ❝❤é t❤✉➟t t♦→♥
✤➣ ❦❤❛✐ t❤→❝ ❝➜✉ tró❝ ✤➦❝ ❜✐➺t ❝õ❛ ❜➔✐ t♦→♥ ✤➸ ❝â ✤÷đ❝ ❝→❝ t➼♥❤ t♦→♥
✤ì♥ ❣✐↔♥✱ ✤➦❝ ❜✐➺t tr♦♥❣ ①➙② ❞ü♥❣ ♣❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ ❜❛♥ ✤➛✉✱ ❧➟♣ ✈➔
①û ❧þ ❝❤✉ tr➻♥❤✱ t➻♠ ♣❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ ♠ỵ✐✳
◆❤➡♠ ♠ư❝ ✤➼❝❤ ♥➯✉ tr➯♥✱ t❛ ❧➟♣ ♠ët ❜↔♥❣ ❚ ❣ç♠ ♠ ❤➔♥❣ ✐ ❂ ✶✱ ✷✱
✳✳✳ ✱ ♠ ✈➔ ✭♥ ✰ ✶✮ ❝ët ❥ ❂ ✵✱ ✶✱ ✳✳✳ ✱ ♥✳ ▼é✐ ❤➔♥❣ t÷ì♥❣ ù♥❣ ✈ỵ✐ ♠ët tr↕♠
♣❤→t✱ ♠é✐ ❝ët

j ≥1

t÷ì♥❣ ù♥❣ ✈ỵ✐ ♠ët tr↕♠ t❤✉✱ ❝ët ❥ ❂ ✵ ❣❤✐ ❣✐→ trà

Số hóa bởi trung tâm học liệu

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✷✷
❝→❝ ❜✐➳♥

xi0

✱ ❜✐➳♥ ♥➔② ❜✐➸✉ t❤à tê♥❣ ❧÷đ♥❣ ❤➔♥❣ ❝❤✉②➸♥ tø tr↕♠ ♣❤→t ✐

✭tỵ✐ ♠å✐ tr↕♠ t❤✉✮ ✈÷đt ❧÷đ♥❣ ❝✉♥❣ tè✐ t❤✐➸✉

ð ❤➔♥❣ ✐✱ ❝ët ❥ ❝õ❛ ❜↔♥❣✳ ❈÷ỵ❝ ♣❤➼

cij

ai ✳

❣❤✐ ð tr➯♥✱

❑þ ❤✐➺✉ ✭✐✱ ❥✮ ❧➔ ỉ ♥➡♠

xij

❣❤✐ ð ❞÷ỵ✐ ♠é✐ ỉ✳

❚r÷ỵ❝ ❤➳t✱ t❛ ♥➯✉ ♠ët sè t➼♥❤ ❝❤➜t ✤→♥❣ ❝❤ó þ ❝õ❛ ❜➔✐ t♦→♥ ✭◗✮ ✤➸
❧➔♠ ❝ì sð ❧þ ❧✉➟♥ ❝❤♦ t❤✉➟t t♦→♥ ❣✐↔✐✳

❚➼♥❤ ❝❤➜t ✶

✳

❍↕♥❣ ❝õ❛ ❤➺ r➔♥❣ ❜✉ë❝ ✭✷✳✽✮ ✲ ✭✷✳✾✮ ❝õ❛ ✭◗✮ ❜➡♥❣ m + n✳
✳ ◆❤➙♥ ❤➔♥❣ ✐ ð ✈➳ tr→✐ ❝õ❛ ✭✷✳✽✮ ✈ỵ✐ sè αi ✈➔ ❤➔♥❣ ❥ ð
✈➳ tr→✐ ❝õ❛ ✭✷✳✾✮ ✈ỵ✐ sè βj rç✐ ❝ë♥❣ t➜t ❝↔ ❧↕✐ ✈➔ ❝❤♦ ❜➡♥❣ ✈➨❝tì ❦❤ỉ♥❣

❈❤ù♥❣ ♠✐♥❤

✭❣ç♠ ♠ ✰ ♥ ✰ ✶ t❤➔♥❤ ♣❤➛♥ ❜➡♥❣ ✵✮✳ ❚ø ♠ t♦↕ ✤ë ❝õ❛ ✤➥♥❣ t❤ù❝ ✈➨❝tì
♥➔② t÷ì♥❣ ù♥❣ ✈ỵ✐ ❝→❝ ❜✐➳♥

xi0

✭✐ ❂ ✶✱ ✷✱ ✳✳✳ ✱ ♠✮ t❛ ♥❤➟♥ ✤÷đ❝

α1 = 0, α2 = 0, ..., αm = 0
✣è✐ ✈ỵ✐ ♥ t♦↕ ✤ë t÷ì♥❣ ù♥❣ ✈ỵ✐ ❝→❝ ❜✐➳♥

x1j

✭❥ ❂ ✶✱ ✷✱ ✳✳✳ ✱ ♥✮ t❛ ♥❤➟♥

✤÷đ❝

α1 + β1 = 0, α1 + β2 = 0, ..., α1 + βn = 0
β1 = β2 = ... = βn = 0✳

❚ø ✤â s✉② r❛

❱➟② ❤↕♥❣ ❝õ❛ ❤➺ r➔♥❣ ❜✉ë❝

✭✷✳✽✮ ✲ ✭✷✳✾✮ tr♦♥❣ ❜➔✐ t♦→♥ ✭◗✮ ❜➡♥❣ ♠ ✰ ♥✳

❚➼♥❤ ❝❤➜t ✷

❈❤♦ ● ❧➔ ♠ët t➟♣ ❤đ♣ ỉ ♥➔♦ ✤â ❝õ❛ ❜↔♥❣ ❚✳ ❍➺ ❝→❝
✈➨❝tì {Aij : (i, j) ∈ G} ❝õ❛ ❜➔✐ t♦→♥ ✭◗✮ ❧➔ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ❦❤✐ ✈➔ ❝❤➾
❦❤✐ t➟♣ ❤đ♣ ❝→❝ ỉ t❤✉ë❝ ● ❦❤ỉ♥❣ t↕♦ t❤➔♥❤ ❝❤✉ tr➻♥❤✳
✳

Số hóa bởi trung tâm học liệu

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✷✸

❈❤ù♥❣ ♠✐♥❤

✳ ●✐↔ sû ❤➺

{Aij : (i, j) ∈ G}

✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✳

◆➳✉ ● ❝❤ù❛ ♠ët ❝❤✉ tr➻♥❤ ❦❤ỉ♥❣ ✤✐ q✉❛ ❝ët ❥ ❂ ✵ ❣ç♠ ❝→❝ ỉ✿

(i1 , j1 ) , (i2 , j1 ) , (i2 , j2 ) , (i3 , j2 ) , ..., (ik , jk ) , (i1 , jk ) , k ≥ 2 ✈➔ ∀j1 , ..., jk >
0 ❤♦➦❝ ● ❝❤ù❛ ♠ët ❝❤✉ tr➻♥❤ ✤✐ q✉❛ ❝ët ❥ ❂ ✵ ❣ç♠ ❝→❝ ỉ✿
(i0 , 0) , (i1 , 0) , (i1 , j1 ) , (i2 , j1 ) , ..., (ik , jk ) , (i0 , jk ) , k ≥ 1 ✈➔ ∀j1 , ..., jk > 0
t❤➻ rã r➔♥❣ t❛ ❝â ✤➥♥❣ t❤ù❝ ✈➨❝tì

Ai1 j1 − Ai2 j1 + Ai2 j2 − ... + Aik jk − Ai1 jk = 0

✭tr÷í♥❣ ❤đ♣ ✤➛✉✮

❤♦➦❝

tù❝ ❧➔

−Ai0 0 + Ai1 0 + Ai1 j1 − ... + Aik jk − Ai0 jk = 0 ✭tr÷í♥❣ ❤đ♣ s❛✉✮✱
❤➺ {Aij : (i, j) ∈ G} ♣❤ư t❤✉ë❝ t✉②➳♥ t➼♥❤✱ tr→✐ ✈ỵ✐ ❣✐↔ t❤✐➳t✳

◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû ❝→❝ ỉ t❤✉ë❝ ● ❦❤ỉ♥❣ t↕♦ t❤➔♥❤ ❝❤✉ tr➻♥❤✳ ◆➳✉ ●
❦❤ỉ♥❣ ❝❤ù❛ ỉ t❤✉ë❝ ❝ët ❥ ❂ ✵ t❤➻ ♥❤÷ ✤➣ ❜✐➳t✱ ❤➺

{Aij : (i, j) ∈ G}

✤ë❝

❧➟♣ t✉②➳♥ t➼♥❤ ✭①❡♠ ❬✶❪✮✳ ❈á♥ ♥➳✉ ● ❝â ❝❤÷❛ ỉ t❤✉ë❝ ❝ët ❥ ❂ ✵ t❤➻ ❜➡♥❣
❧➟♣ ❧✉➟♥ ❣✐è♥❣ ♥❤÷ ✤➣ ❧➔♠ ✤è✐ ✈ỵ✐ ❜➔✐ t♦→♥ ✈➟♥ t↔✐ t❤ỉ♥❣ t❤÷í♥❣ t❛ ❝ơ♥❣
t❤➜② tø ✤➥♥❣ t❤ù❝ ✈➨❝tì

αij Aij = 0
(i,j)∈G
s➩ s✉② r❛

αij = 0

✈ỵ✐ ♠å✐

(i, j) ∈ G✱

tù❝ ❧➔ ❤➺

{Aij |(i, j) ∈ G}

✤ë❝ ❧➟♣

t✉②➳♥ t➼♥❤✳

❚➼♥❤ ❝❤➜t ✸

▼❛ tr➟♥ x = [xij ]m×(n+1) ❧➔ ♠ët ♣❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥
❝õ❛ ❜➔✐ t♦→♥ ✭◗✮ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ t➟♣ ❤đ♣ ❝→❝ ỉ ✭✐✱ ❥✮ ♠➔ xij > 0 ❦❤ỉ♥❣ t↕♦
t❤➔♥❤ ❝❤✉ tr➻♥❤✳
✳

❈❤ù♥❣ ♠✐♥❤✳

❇➔✐ t♦→♥ ✭◗✮ ❧➔ ♠ët ❜➔✐ t♦→♥ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤

❞↕♥❣ ❝❤➼♥❤ t➢❝✳ ❚ø ❧þ t❤✉②➳t q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ❝❤♦ t❤➜② ♠❛ tr➟♥

x = [xij ]m×(n+1) ❧➔ ♠ët ♣❤÷ì♥❣ →♥
✈➨❝tì Aij ❝õ❛ ❤➺ r➔♥❣ ❜✉ë❝ ✭✷✳✽✮ ✲

❝ü❝ ❜✐➯♥ ❝õ❛ ✭◗✮ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❝→❝
✭✷✳✾✮ t÷ì♥❣ ù♥❣ ✈ỵ✐

xij > 0

✤ë❝ ❧➟♣

t✉②➳♥ t➼♥❤✳ ❚❤❡♦ ❚➼♥❤ ❝❤➜t ✷✱ ✤✐➲✉ ♥➔② ①➞② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ t➟♣ ❤đ♣ ❝→❝
ỉ ✭✐✱ ❥✮ ❝õ❛ ❚ ✈ỵ✐

xij > 0

❦❤ỉ♥❣ t↕♦ t❤➔♥❤ ❝❤✉ tr➻♥❤✳

❈❤♦ ● ❧➔ ♠ët t➟♣ ❤đ♣ ỉ ❜➜t ❦ý ❝õ❛ ❜↔♥❣ ❚✳ ❚❛ ♥❤➢❝ ❧↕✐ ♠ët ỉ t❤✉ë❝
● ❣å✐ ❧➔

ỉ tr❡♦

♥➳✉ ♥â ❧➔ ỉ ❞✉② ♥❤➜t ❝õ❛ ● tr➯♥ ❤➔♥❣ ❤❛② tr➯♥ ❝ët ❝❤ù❛

ỉ ✤â✳ ●✐↔ sû ● ❧➔ ♠ët t➟♣ ❣ç♠ ♠ ✰ ♥ ỉ ❝õ❛ ❜↔♥❣ ❚✱ ❦❤ỉ♥❣ t↕♦ t❤➔♥❤

Số hóa bởi trung tâm học liệu

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✷✹
❝❤✉ tr➻♥❤✳ ❑❤✐ ✤â✱ ♠é✐ ỉ

(r, s) ∈
/ G

s➩ t↕♦ ✈ỵ✐ ❝→❝ ỉ t❤✉ë❝ ● ♠ët ❝❤✉

tr➻♥❤ ❞✉② ♥❤➜t✳ ❈❤✉ tr➻♥❤ ♥➔② ❝â t❤➸ t➻♠ ❜➡♥❣ ❝→❝❤ ❧♦↕✐ ❞➛♥ ❝→❝ ỉ tr❡♦
❝õ❛

G = G ∪ {(r, s)}✳

▼ët t➟♣ ❤đ♣ ỉ ● ❝õ❛ ❚ ❣å✐ ❧➔ ♠ët

t➟♣ ❤đ♣ ỉ ❝❤å♥

♥➳✉ ● ❝â ❝→❝ t➼♥❤

❝❤➜t✿
❛✮ ● ❣ç♠ ✤ó♥❣ ♠ ✰ ♥ ỉ ❦❤ỉ♥❣ t↕♦ t❤➔♥❤ ❝❤✉ tr➻♥❤✳
❜✮ ❇➔✐ t♦→♥ ✭◗✮ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜ê s✉♥❣

xij = 0

✤è✐ ✈ỵ✐

(i, j) ∈
/ G

❝â

♥❣❤✐➺♠✳

x = [xij ] ❧➔ ♠ët ♥❣❤✐➺♠ ♥❤÷ t❤➳✳ ❑❤✐ ✤â✱ t❤❡♦ ❚➼♥❤ ❝❤➜t ✸✱
x = [xij ]m×(n+1) ❧➔ ♠ët ♣❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ ❝õ❛ ✭◗✮ tr➯♥ ●✳ ❉➵

●✐↔ sû
♠❛ tr➟♥

❞➔♥❣ t❤➜② r➡♥❣ ✈ỵ✐ ♠é✐ t➟♣ ❤đ♣ ỉ ❝❤å♥ ● ❝â t❤➸ t➻♠ ✤÷đ❝ ✭❞✉② ♥❤➜t✮
❝❤♦ ♠é✐ ❤➔♥❣ ✭tr↕♠ ♣❤→t✮ ♠ët sè
t❤✉✮ ♠ët sè

ui = 0,
❈→❝ sè

vj

✭✐ ❂ ✶✱ ✳✳✳ ✱ ♠✮ ✈➔ ❝❤♦ ♠é✐ ❝ët ✭tr↕♠

✭❥ ❂ ✶✱ ✳✳✳ ✱ ♥✮ s❛♦ ❝❤♦

(i, 0) ∈ G ✈➔ ui + vj = cij , ∀ (i, j) ∈ G, j = 0

♥➳✉

ui , vj

ui

❣å✐ ❧➔ ❝→❝

t❤➳ ✈à

✭✷✳✶✷✮

❝õ❛ ❝→❝ ❤➔♥❣ ✈➔ ❝ët ✭✤è✐ ✈ỵ✐ ●✮✳ ❈❤ó♥❣

❝â t❤➸ t➻♠ ♥❤í ❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✷✮ ❝â ❞↕♥❣ t❛♠ ❣✐→❝ t❤❡♦ ❝→❝❤
s❛✉✿
✶✮ ✣➦t

ui = 0

(0, i) ∈ G❀
t❤➳ ✈à ✈➔ (i, j) ∈ G, j = 0❀

✈ỵ✐ ♠å✐

✷✮ ❑❤✐ ❤➔♥❣ ✐ ✤➣ ❝â

t❤➻ ✤➦t

vj = cij − ui

♥➳✉ ❝ët ❥ ❝❤÷❛ ❝â t❤➳ ✈à❀
✸✮ ❑❤✐ ❝ët ❥ ✤➣ ❝â t❤➳ ✈à ✈➔

(i, j) ∈ G

t❤➻ ✤➦t

ui = cij − vj

♥➳✉ ❤➔♥❣ ✐

❝❤÷❛ ❝â t❤➳ ✈à✳ ❉ø♥❣ ❦❤✐ ♠å✐ ❤➔♥❣ ✈➔ ❝ët ✤➣ ❝â t❤➳ ✈à✳
❚ø ❧þ t❤✉②➳t ✤è✐ ♥❣➝✉ tr♦♥❣ q✉✐ ❤♦↕❝❤ t✉②➳♥ t➼♥❤ s✉② r❛

❚➼♥❤ ❝❤➜t ✹✭❚✐➯✉ ❝❤✉➞♥ tè✐ ÷✉✮

P❤÷ì♥❣ →♥ ❝ü❝ ❜✐➯♥ ① tr➯♥ t➟♣ ❤đ♣
ỉ ❝❤å♥ ● ❧➔ ♠ët ♣❤÷ì♥❣ →♥ tè✐ ÷✉ ❝õ❛ ✭◗✮ ❦❤✐ ❝→❝ t❤➳ ✈à ui , vj ✭✤è✐ ✈ỵ✐
●✮ t❤♦↔ ♠➣♥
✳

ui ≥ 0, ∀i, ui + vj ≤ cij, ∀ (i, j) , j = 0

✭✷✳✶✸✮

❈❤ù♥❣ ♠✐♥❤

✳✣è✐ ♥❣➝✉ ❝õ❛ ✭◗✮ ❧➔ ❜➔✐ t♦→♥

m
✭❉✮

max

n

bj vj : ui + vj ≤ cij , ∀ (i, j) , j = 0; ui ≥ 0, ∀i

ai ui +
i=1

j=1

Số hóa bởi trung tâm học liệu

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