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x1 = 0 x2 = 6 x2 = 6 x1+ x2 = 8 x1+ x2 = 8 x1 = 5 x1 = 5 x2 = 0 x2 = 0 x1 = 0

x1 x2

(x1, x2) = (0, 6) (x1, x2) = (6, 2) (x1, x2) = (5, 3) (x1, x2) = (5, 0) (x1, x2) = (0, 0)

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x2 = 6 x1+ x2= 8 x1 = 5 x2 = 0 x1 = 0

P= 4500x1+ 4500x2

x1 1

x2 1

5000x1+ 4000x2 6000

400x1+ 500x2 600

x1, x2 0

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(P= 4500x1+ 4500x2)

(0, 1.5) (0, 1.2)

(0, 1)

(0.25, 1) (0.4, 1)

(1, 1) (0.67, 0.67)

(1, 0.4) (1, 0.25)

(1, 0) (1.2, 0)

(1.5, 0) (0, 0)

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M = (0, 0) C = (1, 0) J = (0, 1)

x1+ 2x2

x1+ 3x2+ x3 = 8

x1+ x2+ x4 = 4

x1, x2, x3, x4 0

(0, 0) (0, 0, 8, 4) x1, x2 x3, x4

0, 8

3

0, 8

3, 0, 4

3

x1, x3 x2, x4

(2, 2) (2, 2, 0, 0) x3, x4 x1, x2 (4, 0) (4, 0, 4, 0) x2, x4 x3, x4

x1 = x2 = 0

x3 = 8

x4 = 4

x1 = x3 = 0

3x2 = 8

x2 = 8

3

x2+ x4 = 4

x4 = 4

3

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x3 = x4 = 0

x1+ 3x2 = 8

x1+ x2 = 4

x1 = x2 = 2 x2 = x4 = 0

x1+ x3 = 8

x3 = 4

x1 = 4

(0, 4) (0, 4,4, 0) x1, x4 x2, x3 (8, 0) (8, 0, 0,4) x2, x3 x1, x4

x1 = x4 = 0

3x2+ x3 = 8

x3 = 4

x2 = 4

x2 = x3 = 0

x1 = 8

x1+ x4 = 4

x4 = 4

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Z = x1+ 2x2

x1+ 3x2+ x3 = 8

x1+ x2+ x4 = 4

x1, x2, x3, x4 0

x1 = x2= 0 = x3 = 8 x4 = 4 z= x1 + 2x2 = 0 x2

x1 = 0 x2

x3 = 8 3x2 0

x2 8

3

x4 = 4 x2 0

x2 4

8

3 x2 x3 = 0

Z = 1

3x1

2

3x3+

16

31

3x1+ x2+

1

3x3 =

8

32

3x1

1

3x3+ x4 =

4

3x1, x2, x3, x4 0

0, 8

3, 0, 4

3

x1

1

3>0 x1

x4

Z = 1

2x3

1

2x4+ 6

x2+1

2x3+

1

2x4 = 2

x1 1

2x3+

3

2x4 = 2

x1, x2, x3, x4 0

x3 x4 x =

(2, 2, 0, 0) Z = 6

x = (2, 2, 0, 0) Z = 6

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Z = 2x1+ 4x2+ 3x3

x1+ 3x2+ 2x3 30

x1+ x2+ x3 24

3x1+ 5x2+ 3x3 60

x1, x2, x3 0

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x1 >0 x2 = 0 x3 > 0

x2 = 0

Z = 2x1+ 3x3

x1+ 2x3 30

x1+ x3 24

3x1+ 3x3 60

x1, x3 0

Z = 2x1+ 3x3

x1+ 2x3 30

x1+ x3 20

x1, x3 0

x1 >0 x3 >0

(x1, x

2, x

3) = (10, 0, 10) Z = 50

x1 x3 x4 x5

z

x4

x5

x1 x3 x4 x5

z

x3

x5

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x1 x3 x4 x5

z

x3

x1

(x1, x

2) = (2, 2) Z = 6

x1 x3 x4 x5

z

x3

x4

x1 x3 x4 x5

z

x2

x4

x1 x3 x4 x5

z

x2

x1

Z = 5x1+ 9x2+ 7x3

x1+ 3x2+ 2x3 10

3x1+ 4x2+ 2x3 12

2x1+ x2+ 2x3 8

x1, x2, x3 0

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x1 = x2= x3 = 0

Z 5x1 9x2 7x3 = 0

x1+ 3x2+ 2x3+ x4 = 10 = x4 = 10

3x1+ 4x2+ 2x3+ x5 = 12 = x5 = 12

2x1+ x2+ 2x3+ x6 = 8 = x6 = 8

(0, 0, 0, 10, 12, 8) x2

x1+ 3x2+ 2x3+ x4 = 10

x4 = 10 3x2

x2 10

3

3x1+ 4x2+ 2x3+ x5 = 12

x5 = 12 4x2

x2 3....mn.

2x1+ x2+ 2x3+ x6 = 8

x6 = 8 x2

x2 8

x5 x1 = x5 = x3 = 0

Z+ 1.75x1 2.5x3+ 2.25x5 = 27

1.25x1+ 0.5x3+ x4 0.75x5 = 1 = x4 = 1

0.75x1+ x2+ 0.5x3+ 0.25x5 = 3 = x2 = 3

1.25x1+ 1.5x3 0.25x5+ x6 = 5 = x6 = 5

x3 (0, 3, 0, 1, 0, 5) x3

1.25x1+ 0.5x3+ x4 0.75x5 = 1 = x4= 1 0.5x3 2...mn.

0.75x1+ x2+ 0.5x3+ 0.25x5 = 3 = x2= 3 0.5x3 6

1.25x1+ 1.5x3 0.25x5+ x6 = 5 = x6= 5 1.5x3 10

3

x4 x1 = x5 = x4 = 0

Z 4.5x1+ 5x4 1.5x5 = 32

2.5x1+ x3+ 2x4 1.5x5 = 2 = x3 = 2

2x1+ x2 x4+ x5 = 2 = x2 = 2

5x1 3x4+ 2x5+ x6 = 2 = x6 = 2

x1 (0, 2, 2, 0, 0, 2) x1

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2.5x1+ x3+ 2x4 1.5x5 = 2 = x3 = 2 + 2.5x1

2x1+ x2 x4+ x5 = 2 = x2 = 2 2x1 = x1 1

5x1 3x4+ 2x5+ x6 = 2 = x6 = 2 5x1 = x1 0.4...mn.

x6 x6 = x5 = x4 = 0

Z+ 2.3x4+ 0.3x5+ 0.9x6 = 33.8

x3+ 0.5x4 0.5x5+ 0.5x6 = 3 = x3 = 3

x2+ 0.2x4+ 0.2x5 0.4x5 = 1.2 = x2 = 1.2

x1 0.6x4+ 0.4x5+ 0.2x6 = 0.4 = x1 = 0.4

(0.4, 1.2, 3, 0, 0, 0)

(x1.x

2, x

3) = (0.4, 1.2, 3) Z = 33.8

x1 x2 x3 x4 x5 x6

z

x4

x5

x6

x1 x2 x3 x4 x5 x6

z

x4

x2

x6

x1 x2 x3 x3 x4 x5

z

x4

x2

x6

x1 x2 x3 x3 x4 x5

z

x3

x2

x1

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