Методы оптимальных решений. Практикум
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Сырье |
Расход на 1 т |
Суточный запас |
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|
материал А |
материал В |
|
I |
12 |
8 |
120 |
II |
9 |
20 |
180 |
III |
0 |
8 |
64 |
1.29. Для изготовления деталей A и B фабрика расходует в качестве сырья сталь и цветные металлы, имеющиеся в ограниченном количестве. Указанные изделия производят с помощью токарных и фрезерных станков. Исходные данные приведены в таблице. Определить план выпуска продукции, при котором будет достигнута максимальная прибыль.
Вид ресурса |
Нормы расхода на одно изделие |
Объем |
|
|
A |
B |
|
сталь, кг |
20 |
60 |
580 |
цветные металлы, кг |
20 |
40 |
400 |
токарные станки, ст. ч |
100 |
100 |
1500 |
фрезерные станки, ст. ч |
80 |
40 |
1120 |
Прибыль, у.е. |
3 |
8 |
|
1.30. Завод выпускает резину по двум технологиям из ресурсов трёх видов: А, В, С. Известны запасы ресурсов, затраты каждого ресурса на 1 час работы по определенной технологии и прибыль завода от реализации продукции, изготовленной за 1 час работы с использованием той или иной технологии. Общее время работы завода по обеим технологиям – 400 часов. Найти, сколько времени по каждой технологии должен работать завод, чтобы обеспечить максимум прибыли от реализации выпускаемой продукции.
Вид ресурса |
Затраты на 1 час работы |
Запасы, кг |
|
|
по технологии |
|
|
|
№ 1 |
№ 2 |
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|
|
|
|
A |
1 |
1 |
350 |
B |
3 |
5 |
750 |
C |
1 |
3 |
900 |
|
|
|
|
Прибыль, руб. |
300 |
300 |
|
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Задание 2. Графическое решение задачи ЛП
Решить задачу линейного программирования графически.
2.1. F = x1 + 2x2 → max(min) 2x1 − 3x2 ≥ −9, 5x1 + 3x2 ≤ 30, 3x1 + 4x2 ≥ 12, x2 ≥ 1, x1, x2 ≥ 0.
2.2. F = x1 + 3x2 → max(min) x1 + 2x2 ≥ 2,
2x1 − 3x2 ≥ −12, −2x1 + 3x2 ≥ 0, x1 ≤ 3, x1, x2 ≥ 0.
2.3. F = x1 + 4x2 → max(min) −4x1 + 3x2 ≤ 12, x1 − x2 ≥ −5, −x1 + 2x2 ≥ 2,
3x1 + 4x2 ≥ 12, x1, x2 ≥ 0.
2.4. F = −x1 + 2x2 → max(min) 2x1 − x2 ≥ −2, −x1 + 2x2 ≤ 7, x1 + 3x2 ≤ 18, 4x1 − 3x2 ≤ 12,
x1, x2 ≥ 0.
2.5. F = 4x1 − 3x2 → max(min)
−x1 + x2 ≤ 5,
5x1 − 2x2 ≤ 20, 8x1 − 3x2 ≥ 0, 5x1 − 6x2 ≤ 0, x1, x2 ≥ 0.
2.6. F = 2x1 + x2 → max(min) x1 + x2 ≤ 12,
2x1 − x2 ≤ 12,
2x1 − x2 ≥ 0,
2x1 + x2 ≥ 4, x1, x2 ≥ 0.
2.7. F = 3x1 + 7x2 → max(min) 5x1 − x2 ≥ 0, x1 + x2 ≥ 5, x2 ≥ 3,
2x1 − 3x2 ≤ 0, x1, x2 ≥ 0.
2.8. F = 3x1 + 2x2 → max(min) x1 − x2 ≥ −2,
3x1 − 2x2 ≤ 6, 2x1 + x2 ≥ 2, x2 ≤ 3, x1, x2 ≥ 0.
2.9. F = x1 + x2 → max(min) 3x1 + 5x2 ≤ 15,
−x1 + x2 ≤ 2,
2x1 + x2 ≤ 7, x1 + x2 ≥ 1, x1, x2 ≥ 0.
2.10. F = 2x1 − 3x2 → max(min) −x1 + 3x2 ≤ 12, 5x1 + 6x2 ≤ 30, 3x1 + 2x2 ≥ 6, x1 − 2x2 ≤ 2, x1, x2 ≥ 0.
32
2.11. F = 3x1 + 4x2 → max(min) |
2.16. F = 3x1 − 2x2 → max(min) |
−3x1 + 4x2 ≤ 0, |
2x1 − x2 ≥ −1, |
x1 + x2 ≤ 7, |
x1 − 3x2 ≥ −13, |
3x1 + 8x2 ≥ 24, |
4x1 + x2 ≤ 26, |
x1 ≤ 6, |
x1 − 3x2 ≤ 0, |
x1, x2 ≥ 0. |
x1, x2 ≥ 0. |
2.12. F = 4x1 + x2 → max(min) |
2.17. F = 4x1 + x2 → max(min) |
3x1 − 3x2 ≤ 3, |
x1 − 4x2 ≤ 4, |
x1 + x2 ≤ 7, |
3x1 − x2 ≥ 0, |
8x1 + 3x2 ≥ 24, |
x1 + x2 ≥ 4, |
2x1 − x2 ≥ −1, |
2x1 + 4x2 ≤ 17, |
x1, x2 ≥ 0. |
x1, x2 ≥ 0. |
2.13. F = 4x1 + x2 → max(min) |
2.18. F = −3x1 + 2x2 → max(min) |
−3x1 + 5x2 ≤ 15, |
x1 + 2x2 ≤ 12, |
3x1 − x2 ≤ 9, |
2x1 − 3x2 ≤ 6, |
2x1 + x2 ≥ 4, |
x1 − x2 ≤ 4, |
2x1 − x2 ≥ 0, |
2x1 − x2 ≥ −1, |
x1, x2 ≥ 0. |
x1, x2 ≥ 0. |
2.14. F = x1 − 2x2 → max(min) |
2.19. F = 3x1 + 2x2 → max(min) |
−x1 + x2 ≤ 3, |
3x1 − 2x2 ≤ 12, |
2x1 + x2 ≤ 6, |
−x1 + 2x2 ≤ 8, |
2x1 + 5x2 ≥ 10, |
2x1 + 3x2 ≥ 6, |
−x1 + 2x2 ≥ 2, |
x1 + x2 ≤ 9, |
x1, x2 ≥ 0. |
x1, x2 ≥ 0. |
2.15. F = −2x1 − 3x2 → max(min) |
2.20. F = x1 + 5x2 → max(min) |
5x1 − 2x2 ≥ 7, |
−x1 + 2x2 ≤ 2, |
−x1 + 2x2 ≤ 5, |
x1 − 3x2 ≤ 1, |
4x1 + 2x2 ≥ 12, |
x1 + 2x2 ≥ 6, |
x1 + 2x2 ≥ 4, |
2x1 + x2 ≥ 8, |
x1, x2 ≥ 0. |
x1, x2 ≥ 0. |
33
2.21. F = −2x1 + x2 → max(min) 3x1 − 2x2 ≤ 12,
−x1 + 2x2 ≤ 8,
2x1 + x2 ≥ 12, x2 ≤ 7, x1, x2 ≥ 0.
2.22. F = x1 + 3x2 → max(min) 2x1 − x2 ≤ 6,
−2x1 + 5x2 ≤ 10, x1 + 2x2 ≥ 8, x1 ≥ 3, x1, x2 ≥ 0.
2.23. F = 2x1 + x2 → max(min) x1 + 2x2 ≤ 14, −5x1 + 3x2 ≤ 15, 2x1 + 3x2 ≥ 12, x1 − x2 ≤ 4, x1, x2 ≥ 0.
2.24. F = 2x1 + 3x2 → max(min) 2x1 + x2 ≤ 10, −2x1 + 3x2 ≤ 6, 3x1 + 4x2 ≥ 12, x1 − x2 ≤ 2, x1, x2 ≥ 0.
2.25. F = −2x1 + 3x2 → max(min)
−2x1 |
+ x2 |
≤ 3, |
x1 − 2x2 |
≤ 4, |
|
x1 |
+ x2 |
≤ 6, |
3x1 |
+ x2 |
≥ 3, |
|
x1, x2 ≥ 0. |
|
2.26. F = 2x1 + x2 → max(min) −2x1 + x2 ≤ 2,
−x1 + 2x2 ≤ 8, x1 + x2 ≥ 5. x1 + 3x2 ≥ 6, x1, x3 ≥ 0.
2.27. F = 2x1 − x2 → max(min) x1 + x2 ≥ 4, 2x1 − x2 ≥ 2,
2x1 + 3x2 ≤ 18, x1 − 3x2 ≤ 6, x1, x2 ≥ 0.
2.28. F = 2x1 + x2 → max(min) x1 − 4x2 ≤ 4, 3x1 − x2 ≥ 0, x1 + x2 ≥ 4,
−x1 + 2x2 ≤ 10, x1, x2 ≥ 0.
2.29. F = 2x1 − x2 → max(min) x1 + x2 ≥ 4, −x1 + 2x2 ≤ 2, x1 + 2x2 ≤ 10, x1 − x2 ≤ 5, x1, x2 ≥ 0.
2.30. F = 2x1 − 6x2 → max(min) x1 + x2 ≥ 2, −x1 + 2x2 ≤ 4, x1 + 2x2 ≤ 8,
2x1 − 3x2 ≤ 6, x1, x2 ≥ 0.
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Задание 3. Симплекс-метод
Решить задачу линейного программирования симплекс-методом.
3.1. F = 2x1 + 3x2 + 3x3 → max |
3.6. |
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x1 + 2x2 + 3x3 ≤ 6, |
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3x1 + 2x2 + x3 ≤ 6, |
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2x1 + 3x2 + x3 ≤ 7, |
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2x1 − 3x2 + 4x3 ≤ 8, |
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x1, x2, x3 ≥ 0. |
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3.2. F = 2x1 + 3x2 + 3x3 → max |
3.7. |
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x1 + 2x2 + 3x3 ≤ 8, |
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3x1 − 2x2 + x3 ≤ 8, |
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2x1 + 3x2 + 2x3 ≤ 10, |
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x1 + x2 − 3x3 ≤ 12, |
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x1, x2, x3 ≥ 0. |
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3.3. |
F = 9x1 + 10x2 + 10x3 → max |
3.8. |
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4x1 + 2x2 + 3x3 ≤ 35, |
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3x1 − x2 − 2x3 ≤ 10, |
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x1 + 3x2 + 2x3 ≤ 22, |
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3x1 − x2 + 2x3 ≤ 23, |
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x1, x2, x3 ≥ 0. |
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3.4. |
F = 11x1 + 10x2 + 10x3 → max 3.9. |
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x1 − x3 ≤ 2, |
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3x1 − x2 + x3 ≤ 6, |
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4x1 − 3x2 + 3x3 ≤ 3, |
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x1 + 3x2 + 2x3 ≤ 22, |
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x1, x2, x3 ≥ 0. |
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3.5. |
F = 8x1 − x2 + 6x3 → max |
3.10. |
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3x1 − x2 + x3 ≤ 7, |
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4x1 − 3x2 + 3x3 ≤ 6, x1 + 3x2 + 2x3 ≤ 24,
2x1 − x3 ≤ 6, x1, x2, x3 ≥ 0.
F = 5x1 + 9x2 − 3x3 → max 6x1 + 2x2 + 3x3 ≤ 20, x1 + 3x2 − 2x3 ≤ 6,
2x1 + 5x2 − 2x3 ≤ 13, 2x1 + x2 + x3 ≤ 7, x1, x2, x3 ≥ 0.
F = x1 + x2 + x3 → max 2x1 + 5x3 ≤ 14, 2x1 + 4x2 + 3x3 ≤ 14, 5x1 + 6x2 − 3x3 ≤ 10,
−x1 + 2x2 + 3x3 ≤ 10, x1, x2, x3 ≥ 0.
F = x1 + x2 + 2x3 → max 2x2 + 3x3 ≤ 5, 3x1 − 2x2 + 3x3 ≤ 4,
−x1 + 3x2 − x3 ≤ 3, x1 + 4x2 + x3 ≤ 8, x1, x2, x3 ≥ 0.
F = 8x1 + 5x2 + 2x3 → max 6x1 + 2x2 + x3 ≤ 10, 3x1 + 2x2 + 3x3 ≤ 8, x1 + 2x2 − x3 ≤ 3,
2x1 + x2 + x3 ≤ 8, x1, x2, x3 ≥ 0.
F = 3x1 + 4x2 + 6x3 → max 4x1 − 3x2 + x3 ≤ 10, 3x1 + 2x2 + x3 ≤ 20, 3x1 − x3 ≤ 10, x1 + 3x2 + 4x3 ≤ 21, x1, x2, x3 ≥ 0.
35
3.11. F = 9x1 + 10x2 + 7x3 → max |
3.16. F = 7x1 + 10x2 + 9x3 → max |
x1 + 2x2 + 3x3 ≤ 20, |
x1 + 2x2 + 3x3 ≤ 16, |
3x1 + 2x2 + x3 ≤ 16, |
2x1 + 3x2 + 2x3 ≤ 15, |
2x1 + 3x2 + 2x3 ≤ 15, |
3x1 + 2x2 + x3 ≤ 20, |
x1 + 4x2 + x3 ≤ 21, |
x1 + 4x2 + x3 ≤ 21, |
x1, x2, x3 ≥ 0. |
x1, x2, x3 ≥ 0. |
3.12. F = x1 + 2x2 + 3x3 → max |
3.17. F = 7x1 + 5x2 + 2x3 → max |
2x1 + 3x3 ≤ 4, |
x1 − x3 ≤ 2, |
3x1 − 2x2 + 3x3 ≤ 3, |
6x1 + 2x2 + 3x3 ≤ 10, |
−x1 + 3x2 − x3 ≤ 3, |
3x1 + 2x2 + 3x3 ≤ 8, |
x1 + 2x2 + x3 ≤ 8, |
x1 + 2x2 − x3 ≤ 3, |
x1, x2, x3 ≥ 0. |
x1, x2, x3 ≥ 0. |
3.13. F = 3x1 + 2x2 + 3x3 → max |
3.18. F = 4x1 + 4x2 + 3x3 → max |
4x1 + 2x2 − 3x3 ≤ 8, |
4x1 − 3x2 + 2x3 ≤ 8, |
x1 + 3x2 + 2x3 ≤ 6, |
x1 + 2x2 + 3x3 ≤ 6, |
x1 + 2x2 + 3x3 ≤ 6, |
2x1 + 3x2 + 2x3 ≤ 7, |
3x1 + x2 + 2x3 ≤ 7, |
3x1 + 2x2 + x3 ≤ 6, |
x1, x2, x3 ≥ 0. |
x1, x2, x3 ≥ 0. |
3.14. F = x1 + x2 + x3 → max |
3.19. F = 3x1 + 2x2 + 3x3 → max |
−x1 + 3x2 + 2x3 ≤ 10, |
x1 + x2 − 3x3 ≤ 14, |
2x1 + 5x2 ≤ 14, |
2x1 + x2 + 3x3 ≤ 8, |
5x1 − 3x2 + 6x3 ≤ 10, |
3x1 + 2x2 + 2x3 ≤ 10, |
2x1 + 3x2 + 4x3 ≤ 14, |
−2x1 + 3x2 + x3 ≤ 8, |
x1, x2, x3 ≥ 0. |
x1, x2, x3 ≥ 0. |
3.15. F = x1 + 2x2 + x3 → max |
3.20. F = 4x1 + 4x2 + 3x3 → max |
3x1 + 3x2 − 2x3 ≤ 4, |
4x1 − 3x2 + 2x3 ≤ 8, |
x1 + x2 + 4x3 ≤ 8, |
x1 + 2x2 + 3x3 ≤ 6, |
3x2 + 2x3 ≤ 5, |
2x1 + 3x2 + 2x3 ≤ 7, |
−x1 − x2 + 3x3 ≤ 3, |
3x1 + 2x2 + x3 ≤ 6, |
x1, x2, x3 ≥ 0. |
x1, x2, x3 ≥ 0. |
36
3.21. F = 10x1 − 10x2 + 3x3 → max 3.26. |
F = 3x1 + 4x2 + 4x3 → max |
x1 + x3 ≤ 8, |
x1 + x2 + 2x3 ≤ 6, |
2x1 − 3x2 + 3x3 ≤ 12, |
2x1 − 3x2 + 4x3 ≤ 8, |
2x1 − 2x2 − x3 ≤ 4, |
3x1 + 2x2 + x3 ≤ 6, |
x1 − 2x2 + 2x3 ≤ 4, |
2x1 + 3x2 + 2x3 ≤ 7, |
x1, x2, x3 ≥ 0. |
x1, x2, x3 ≥ 0. |
3.22. F = 9x1 − 3x2 + 5x3 → max 5x1 − 2x2 + 2x3 ≤ 13, 3x1 − 2x2 + x3 ≤ 6, x1 + x2 + 2x3 ≤ 7,
2x1 + 3x2 + 6x3 ≤ 20, x1, x2, x3 ≥ 0.
3.23. F = x1 + x2 + 2x3 → max 3x1 − x2 − x3 ≤ 3,
4x1 + x2 + x3 ≤ 8, 2x1 + 3x3 ≤ 5, −2x1 + 3x2 + 3x3 ≤ 4, x1, x2, x3 ≥ 0.
3.24. F = 4x1 + 3x2 + 4x3 → max x1 + 3x2 − 2x3 ≤ 8,
−3x1 + x2 + x3 ≤ 10, x1 + 3x2 − 2x3 ≤ 8,
2x1 + 2x2 + 3x3 ≤ 10, x1, x2, x3 ≥ 0.
3.25.F = 10x1 + 11x2 + 11x3 → max
−x2 + x3 ≤ 2,
−x1 + 3x2 + 4x3 ≤ 3, 3x1 + 2x2 + x3 ≤ 22,
−x1 + x2 + 3x3 ≤ 6, x1, x2, x3 ≥ 0.
3.27. F = 7x1 + 4x2 + 5x3 → max x1 + 4x2 − 3x3 ≤ 10, −x1 + 3x2 ≤ 10, 4x1 + x2 + 3x3 ≤ 21, x1 + 3x2 + 2x3 ≤ 20, x1, x2, x3 ≥ 0.
3.28. F = 2x1 + 8x2 + 5x3 → max 3x1 + 3x2 + 2x3 ≤ 8, x1 + 2x2 + x3 ≤ 8, −x1 + x2 + 2x3 ≤ 3, x1 + 6x2 + 2x3 ≤ 10, x1, x2, x3 ≥ 0.
3.29. F = 3x1 + x2 + 2x3 → max
−x1 − x2 + 3x3 ≤ 3, x1 + x2 + 2x3 ≤ 8,
3x1 + 2x2 ≤ 4, 3x1 + 3x2 − 2x3 ≤ 3, x1, x2, x3 ≥ 0.
3.30. F = 2x1 + 3x2 + 2x3 → max 2x1 + 2x2 + 3x3 ≤ 10, x1 − 3x2 + x3 ≤ 14,
3x1 + x2 − 2x3 ≤ 8, x1 + 3x2 + 2x3 ≤ 8, x1, x2, x3 ≥ 0.
37
Задание 4. Двухэтапный метод
Решить каноническую задачу линейного программирования.
4.1. F = x1 + 4x2 + x3 → max −x1 + 2x2 + x3 = 4, 3x1 + x2 + 2x3 + x4 = 9, 2x1 + 3x2 + x3 − x5 = 6, xj ≥ 0, j = 1, . . . , 5.
4.2. F = −2x1 − x2 + x3 → max 2x1 + x2 − x3 − x4 = 5, x1 + 2x2 + x3 + x5 = 7, x1 − x2 + 2x3 = 1, xj ≥ 0, j = 1, . . . , 5.
4.3. F = x1 − x2 + x3 → max 4x1 + 2x2 + x3 − x4 = 6,
−x1 + x2 + x3 = 1, x1 + x2 + 4x3 + x5 = 24, xj ≥ 0, j = 1, . . . , 5.
4.4. F = 5x1 + 2x2 + x3 → max x1 + x2 + x3 − x4 = 3, x1 + 2x2 + 2x3 = 4,
3x1 + 4x2 + 2x3 + x5 = 12, xj ≥ 0, j = 1, . . . , 5.
4.5. F = x1 − 8x2 − 3x3 → max 3x1 + x2 + 2x3 − x4 = 6, x1 + x2 + x3 = 4, −x1 + 3x2 − x3 − x5 = 4, xj ≥ 0, j = 1, . . . , 5.
4.6. F = −x1 − 3x2 − x3 → max 3x1 + x2 + x3 − x4 = 6, x1 + 3x2 + x3 = 10, −x1 + 3x2 − x3 + x5 = 12, xj ≥ 0, j = 1, . . . , 5.
4.7. F = x1 + 4x2 + 3x3 → max x1 − 3x2 + 2x3 = 3, 2x1 + 4x2 + x3 + x4 = 18,
−x1 + x2 + 3x3 − x5 = 10, xj ≥ 0, j = 1, . . . , 5.
4.8. F = 2x1 + 2x2 + 2x3 → max x1 + x2 + 2x3 + x4 = 4, x1 − x2 + x3 = 2,
3x1 + x2 + 2x3 − x5 = 6, xj ≥ 0, j = 1, . . . , 5.
4.9. F = 3x1 + 2x2 + 2x3 → max x1 + x2 + x3 − x4 = 3, x1 + x3 + x5 = 2, −x1 + x2 + x3 = 1, xj ≥ 0, j = 1, . . . , 5.
4.10. F = 2x1 − 8x2 − 3x3 → max 3x1 + x2 + 2x3 − x4 = 12, x1 + x2 + x3 = 8,
−2x1 + 3x2 − x3 + x5 = 8, xj ≥ 0, j = 1, . . . , 5.
38
4.11. F = −6x1 − 7x2 − 9x3 → max x1 + 2x2 + 2x3 − x4 = 5,
−x1 + x2 + x3 = 2, x1 + x2 − x3 + x5 = 2, xj ≥ 0, j = 1, . . . , 5.
4.12. F = 5x1 + 2x2 + x3 → max x1 + x2 + x3 = 3,
2x1 + x2 + x3 − x4 = 4, 3x1 + 2x2 − 2x3 + x5 = 12, xj ≥ 0, j = 1, . . . , 5.
4.13. F = −6x1 − x2 − 3x3 → max 2x1 + 2x2 + x3 − x4 = 6,
−x1 + x2 + x3 = 1, x1 − x2 − x3 + x5 = 7, xj ≥ 0, j = 1, . . . , 5.
4.14. F = 2x1 + 2x2 − 3x3 → max 3x1 + x2 + 3x3 − x4 = 6, x1 + x2 + x3 = 4,
−3x1 + 3x2 − x3 + x5 = 4, xj ≥ 0, j = 1, . . . , 5.
4.15. F = −8x1 + 2x2 − 3x3 → max x1 + 3x2 + 2x3 − x4 = 12, x1 + x2 + x3 = 8,
3x1 − 2x2 − x3 + x5 = 8, xj ≥ 0, j = 1, . . . , 5.
4.16. F = −4x1 − 3x2 − 2x3 → max 4x1 + x2 + 2x3 − x4 = 8, 2x1 + x2 − x3 = 6, −x1 + 3x2 + x3 + x5 = 4, xj ≥ 0, j = 1, . . . , 5.
4.17. F = −4x1 − x2 − 3x3 → max 4x1 − x2 − 2x3 = 3, x1 + 3x2 + x3 − x4 = 4,
3x1 − x2 + x3 + x5 = 12, xj ≥ 0, j = 1, . . . , 5.
4.18. F = x1 − 3x2 − 2x3 → max 3x1 + x2 − 2x3 − x4 = 13, x1 − 3x2 + x3 = 1, x1 + 2x2 + 3x3 + x5 = 11, xj ≥ 0, j = 1, . . . , 5.
4.19. F = −3x1 − 2x2 − 2x3 → max 3x1 + 2x2 − 2x3 − x4 = 11, −x1 + 6x2 + 3x3 = 23, x1 − x2 + 2x3 + x5 = 2, xj ≥ 0, j = 1, . . . , 5.
4.20. F = 3x1 + 2x2 + 3x3 → max x1 + x2 − 2x3 − x4 = 2,
−x1 + 2x2 + x3 + x5 = 4, x1 + 2x3 = 2, xj ≥ 0, j = 1, . . . , 5.
39
4.21. F = x1 + 2x2 + x3 → max x1 + x2 − x3 − x4 = 1, x1 + x2 + x3 = 10, x1 + x3 + x5 = 1, xj ≥ 0, j = 1, . . . , 5.
4.22. F = 2x1 + x2 + 2x3 → max x1 + 2x2 − x3 − x4 = 2,
2x1 + x2 + 2x3 = 2, 2x1 + x2 − 2x3 + x5 = 6, xj ≥ 0, j = 1, . . . , 5.
4.23. F = x1 + 3x2 + x3 → max 3x1 + x2 + x3 − x4 = 6, x1 + 3x2 + x3 = 10,
−x1 + 3x2 − x3 + x5 = 6, xj ≥ 0, j = 1, . . . , 5.
4.24. F = x1 + 2x2 + 2x3 → max 2x1 + x2 − x3 − x4 = 2, x1 + 2x2 + 2x3 = 2, x1 + 2x2 − 2x3 + x5 = 6, xj ≥ 0, j = 1, . . . , 5.
4.25. F = −2x1 − 2x2 + 5x3 → max 2x1 − 2x2 + 3x3 − x4 = 12,
−x1 + x2 − x3 + x5 = 2,
2x1 − x2 + 2x3 = 24, xj ≥ 0, j = 1, . . . , 5.
4.26. F = −x1 − 2x2 − 2x3 → max x1 + x2 − 4x3 − x4 = 1, x1 − 2x2 + 2x3 = 2, x1 + 2x2 − 2x3 + x5 = 6, xj ≥ 0, j = 1, . . . , 5.
4.27. F = 5x1 + 7x2 + 9x3 → max x1 − x2 + x3 = 3,
−x1 + 2x2 + x3 + x4 = 4, −2x1 + x2 + 2x3 + x5 = 8, xj ≥ 0, j = 1, . . . , 5.
4.28. F = 5x1 − x2 + 4x3 → max x1 + 3x2 − 2x3 − x4 = 3,
−x1 + 2x2 + x3 + x5 = 5, x1 − x2 − 3x3 = 7, xj ≥ 0, j = 1, . . . , 5.
4.29. F = −3x1 − 2x2 + 3x3 → max
−x1 + 2x2 + x3 − x4 = 6,
2x1 − x2 + x3 + x5 = 12, 2x1 − x2 − 2x3 = 3, xj ≥ 0, j = 1, . . . , 5.
4.30. F = −3x1 + x2 + 2x3 → max −x1 + x2 + 2x3 = 2, x1 + x3 + x4 = 4, x1 + x2 + x3 − x5 = 6, xj ≥ 0, j = 1, . . . , 5.
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