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# The Simplex Solution Method

45. The Stratford House Furniture Company makes two kinds of tables—end tables (x1) and coffee tables (x2). The manufacturer is restricted by material and labor constraints, as shown in the fol- lowing linear programming formulation:

maximize Z = 200x1 + 300x2 (profit, \$) subject to

( l a b o r , h r . ) ( w o o d , b d . f t . ) x 1 , x 2 Ú 0 3 x 1 + 3 x 2 1 3 5 2 x 1 + 5 x 2 1 8 0

The final optimal simplex tableau for this problem is as follows:

c j

Basic Variables

### Quantity

x

2

x

1

300

30

200

15

c j - z j z j

12,000

200 x1

300 x2

0 1

1 0

200 0

300 0

1>3

-2>9

-1>3

5>9

0

0

s1

s2

100>3

400>9

-100>3

-400>9

• a.

Formulate the dual for this problem.

• b.

Define the dual variables and indicate their values.

• c.

What profit for coffee tables will result in no production of end tables, and what will the

new optimal solution values be? d. What will be the effect on the optimal solution if the available wood is increased from 135

to 165 board feet? e . D e t e r m i n e t h e o p t i m a l r a n g e s f o r a n d f . D e t e r m i n e t h e f e a s i b l e r a n g e s f o r ( l a b o r h o u r s ) a n d ( b o a r d f e e t o f w o o d ) . q 2 q 1 c 2 . g. What is the maximum price the Stratford House Furniture Company would be willing to pay c 1

for additional wood, and how many board feet of wood could be purchased at that price? h. If the furniture company wanted to secure additional units of only one of the resources,

labor or wood, which should it be?

46. A manufacturing firm produces electric motors for washing machines and vacuum cleaners. The firm has resource constraints for production time, steel, and wire. The linear programming model for determining the number of washing machine motors (x1) and vacuum cleaner motors (x2) to produce has been formulated as follows:

maximize Z = 70x1 + 80x2 (profit, \$)

subject to

( p r o d u c t i o n , h r . ) ( s t e e l , l b . ) ( w i r e , f t . ) x 1 , x 2 Ú 0 x 1 + 2 x 2 2 0 x 1 + x 2 1 4 2 x 1 + x 2 1 9

## The final optimal simplex tableau for this model is as follows:

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