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Problems

A-59

subject to

( p r o t e i n , m g ) ( i r o n , m g ) ( c a r b o h y d r a t e s , m g ) x 1 , x 2 Ú 0 3 x 1 + 2 x 2 Ú 1 2 1 2 x 1 + 3 x 2 Ú 3 0 4 x 1 + 5 x 2 Ú 2 0

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

0.03

0.02

0

0

0

Quantity

x1

x2

s1

s2

s3

3.6

0

1

0

0.20

- 0.80

1.6

1

0

0

- 0.13

0.20

4.4

0

0

1

0.47

- 3.2

0.12

0.03

0.02

0

0

- 0.01

0

  • 0

    0

0

  • -

    0.01

c

j

Basic Variables

0.02

x2

0.03 0

x s

1 1

z j

z j - c j

  • a.

    Formulate the dual for this problem.

  • b.

    What do the dual variables equal, and what do they mean?

  • c.

    Determine the optimal ranges for c1 and c2.

  • d.

    Determine the ranges for q1, q2, and q3 (milligrams of protein, iron, and carbohydrates, respectively).

  • e.

    What would it be worth for the school dietitian to be able to reduce the requirement for carbohydrates, and what is the smallest number of milligrams of carbohydrates that would be required at that value?

50. The Overnight Food Processing Company prepares sandwiches (among other processed food items) for vending machines, markets, and business canteens around the city. The sandwiches are made at night and delivered early the following morning. Any sandwiches not purchased during the previous day are thrown away. Three kinds of sandwiches are made each night, a basic cheese sandwich (x1), a ham salad sandwich (x2), and a pimento cheese sandwich (x3). The profits are $1.25, $2.00, and $1.75, respectively. It takes 0.5 minutes to make a cheese sandwich, 1.2 minutes to make a ham salad sandwich, and 0.8 minutes to make a pimento cheese sandwich. The company has 20 hours of labor available to produce the sandwiches each night. The demand for ham salad sandwiches is at least as great as the demand for the two types of cheese sandwiches combined. However, the company has only enough ham salad to produce 500 sandwiches per night. The com- pany has formulated the following linear programming model in order to determine how many of each type of sandwich to make to maximize profit:

maximize Z = $1.25x1 + 2.00x2 + 1.75x3 subject to

( p r o d u c t i o n t i m e , m i n . ) ( d e m a n d f o r h a m s a l a d s a n d w i c h e s ) ( h a m s a l a d s a n d w i c h l i m i t ) x 1 , x 2 , x 3 Ú 0 x 2 5 0 0 x 1 + x 3 x 2 0 . 5 x 1 + 1 . 2 x 2 + 0 . 8 x 3 1 , 2 0 0

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