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Problems

A-61

a . F i n d t h e o p t i m a l r a n g e s f o r a l l v a l u e s . b . F i n d t h e f e a s i b l e r a n g e s f o r a l l v a l u e s . q i c j

1 2 20x + 10x + 16x

3

1 2 10x + 20x + 16x

3

x1 + 2x2 + 2x

3

1 2 x +x +x

3

1 2 2x + x + x

3

1 x , x2, x

3

52. The Sunshine Food Processing Company produces three canned fruit products—mixed fruit (x1), fruit cocktail (x2), and fruit delight (x3). The main ingredients in each product are pears and peaches. Each product is produced in lots and must undergo three processes—mixing, canning, and packaging. The resource requirements for each product and each process are shown in the fol- lowing linear programming formulation:

• 320 (pears, lb.)

• 400 (peaches, lb.)

• 43 (mixing, hr.)

• 60 (canning, hr.)

• 40 (packaging, hr.)

Ú0

maximize Z = 10x1 subject to

• +

6x2 + 8x3 (profit, \$)

The optimal simplex tableau is as follows:

Basic Variables

Quantity

10 x1

6 x2

8 x3

0 s1

0 s2

0 s3

0 s4

0 s5

x x s s s

1 2 3 4 5

8 16 3 36 8

1 0 0 0 0

0 1 0 0 0

8/15 8/15 2/5 - 1>15 - 3>5

1/15 -1>30 0 -1>30 -1

1/30 1/15 - 1>10 - 1>30 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

z

j

176

10

6

128/15

7/15

1/15

0

0

0

c -z

j

j

0

0

- 8>15

- 7>15

- 1>15

0

0

0

c j

10 6 0 0 0

• a.

What is the maximum price the company would be willing to pay for additional pears? How much could be purchased at that price?

• b.

What is the marginal value of peaches? Over what range is this price valid?

• c.

The company can purchase a new mixing machine that would increase the available mixing time from 43 to 60 hours. Would this affect the optimal solution?

• d.

The company can also purchase a new packaging machine that would increase the available packaging time from 40 to 50 hours. Would this affect the optimal solution?

• e.

If the manager was to attempt to secure additional units of only one of the resources, which should it be?

53. The Evergreen Products Firm produces three types of pressed paneling from pine and spruce. The three types of paneling are Western (x1), Old English (x2), and Colonial (x3). Each sheet must be cut and pressed. The resource requirements are given in the following linear programming formulation:

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