# A-62

# Module A

# The Simplex Solution Method

maximize Z = 4x_{1 }+ 10x_{2 }+ 8x_{3 }(profit, $)

subject to

1 5x + 4x

2

1 2x + 5x

2

x_{1 }+ x

2

2x_{1 }+ 4x

2

3 + 4x …

200 (pine, lb.)

3 + 2x …

160 (spruce, lb.)

3 + 2x …

50 (cutting, hr.)

3 + 2x …

80 (pressing, hr.)

3 2 x_{1}, x , x Ú

0

## The optimal simplex tableau is as follows:

Basic Variables

Quantity

4 x_{1 }

10 x_{2 }

8 x_{3 }

0 s_{1 }

0 s_{2 }

0 s_{3 }

0 s_{4 }

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 (x_{1}), product 2 (x_{2}), product 3 (x_{3}), and product 4 (x_{4}) to produce is as follows:

maximize Z = 2x_{1 }+ 8x_{2 }+ 10x_{3 }+ 6x_{4 }(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