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Basic Variables

Quantity

70 x1

80 x2

0 s1

0 s2

0 s3

x1

6

1

0

2>3

0

-1>3

s x

2 2

1 7

0 0

0 1

-1>3 -1>3

1 0

-1>3 2>3

z

j

980

70

80

20

0

30

c -z

j

j

0

0

- 20

0

- 30

c j

70

0 80

Problems

A-57

a. Formulate the dual for this problem. b. What do the dual variables equal, and what do they mean? c . 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 d . 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 ( p r o d u c t i o n h o u r s ) , ( p o u n d s o f s t e e l ) , a n d ( f e e t q 3 q 2 q 1 c 2 . c 1

of wire). e. Managers at the firm have determined that the firm can purchase a new production

machine that will increase available production time from 19 to 25 hours. Will this change affect the optimal solution?

47. A manufacturer produces products 1 and 2, for which profits are $9 and $12, respectively. Each product must undergo two production processes that have labor constraints. There are also mater- ial constraints and storage limitations. The linear programming model for determining the num- ber of product 1 to produce (x1) and the number of product 2 to product (x2) is given as follows:

maximize Z = 9x1 + 12x2 (profit, $)

subject to

( p r o c e s s 1 , l a b o r h r . ) ( p r o c e s s 2 , l a b o r h r . ) ( m a t e r i a l , l b . ) x 1 , x 2 Ú 0 x 2 7 ( s t o r a g e s p a c e , f t . 2 ) x 1 7 ( s t o r a g e s p a c e , f t . 2 ) 1 5 x 1 + 8 x 2 1 2 0 5 x 1 + 5 x 2 5 0 4 x 1 + 8 x 2 6 4

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

Basic Variables

Quantity

9 x1

12 x2

0 s1

0 s2

0 s3

0 s4

0 s5

x s s s x

1 5 3 4 2

4 1 12 3 6

1 0 0 0 0

0 0 0 0 1

-1>4 -1>4 7>4 1>4 1>4

2>5 1>5 -22>5 -2>5 -1>5

0 0 1 0 0

0 0 0 1 0

0 1 0 0 0

z

j

108

9

12

3>4

6>5

0

0

0

c -z

j

j

0

0

-3>4

-6>5

0

0

0

c j

9 0 0 0 12

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