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The Simplex Solution Method

maximize Z = 4x1 + 10x2 + 8x3 (profit, \$)

subject to

1 5x + 4x

2

1 2x + 5x

2

x1 + x

2

2x1 + 4x

2

3 + 4x

200 (pine, lb.)

3 + 2x

160 (spruce, lb.)

3 + 2x

50 (cutting, hr.)

3 + 2x

80 (pressing, hr.)

3 2 x1, x , x Ú

0

The optimal simplex tableau is as follows:

Basic Variables

Quantity

4 x1

10 x2

8 x3

0 s1

0 s2

0 s3

0 s4

s s x x

1 2 3 2

80 70 20 10

7>3 -1>3 1>3 1>3

0 0 0 1

0 0 1 0

1 0 0 0

0 1 0 0

-4>3 1>3 2>3 -1>3

-2>3 -4>3 -1>6 1/3

z

j

260

6

10

8

0

0

2

2

c -z

j

j

-2

0

0

0

0

-2

-2

c j

0 0 8 10

• a.

What is the marginal value of an additional pound of spruce? Over what range is this value valid?

• b.

What is the marginal value of an additional hour of cutting? Over what range is this value valid?

• c.

Given a choice between securing more cutting hours or more pressing hours, which should management select? Why?

• d.

If the amount of spruce available to the firm was decreased from 160 to 100 pounds, would this reduction affect the solution?

• e.

What unit profit would have to be made from Western paneling before management would consider producing it?

• f.

Management is considering changing the profit of Colonial paneling from \$8 to \$13. Would this change affect the solution?

54. A manufacturing firm produces four products. Each product requires material and machine pro- cessing. The linear programming model formulated to determine the number of product 1 (x1), product 2 (x2), product 3 (x3), and product 4 (x4) to produce is as follows:

maximize Z = 2x1 + 8x2 + 10x3 + 6x4 (profit, \$)

subject to

( m a t e r i a l , l b . ) ( m a c h i n e p r o c e s s i n g , h r . ) x 1 , x 2 , x 3 , x 4 Ú 0 x 1 + 2 x 2 + 2 x 3 + x 4 1 6 0 2 x 1 + x 2 + 4 x 3 + 2 x 4 2 0 0

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