# A-58

# Module A

# The Simplex Solution Method

a.

Formulate the dual for this problem.

b.

What do the dual variables equal, and what does this dual solution mean?

c.

Determine the optimal ranges for c

_{1 }and c_{2}.d.

Determine the range for q

_{1 }(process 1, labor hr.).e.

Due to a problem with a supplier, only 100 pounds of material will be available for production instead of 120 pounds. Will this affect the optimal solution mix?

48.

A manufacturer produces products 1, 2, and 3 daily. The three products are processed through three production operations that have time constraints, and the finished products are then stored. The following linear programming model has been formulated to determine the number of prod- uct 1 (x

_{1}), product 2 (x_{2}), and product 3 (x_{3}) to produce:

maximize Z = 40x_{1 }+ 35x_{2 }+ 45x_{3 }(profit, $) subject to

( o p e r a t i o n 1 , h r . ) ( o p e r a t i o n 2 , h r . ) ( o p e r a t i o n 3 , h r . ) x 1 , x 2 , x 3 Ú 0 x 1 + x 2 + x 3 … 4 0 ( s t o r a g e , f t . 2 ) 3 x 1 + 2 x 2 + 4 x 3 … 1 0 0 4 x 1 + 3 x 2 + x 3 … 1 6 0 2 x 1 + 3 x 2 + 2 x 3 … 1 2 0

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

40

35

45

0

0

0

0

Quantity

x_{1 }

x_{2 }

x_{3 }

s_{1 }

s_{2 }

s_{3 }

s_{4 }

10

-1>2

0

0

1

0

1>2

-4

60

2

0

0

0

1

1

-5

10

1/2

0

1

0

0

1/2

-1

30

1/2

1

0

0

0

-1>2

2

1,500

40

35

45

0

0

5

25

0

0

0

0

0

-5

- 25

z j

c j - z j

c j

Basic Variables

0 0 45 35

s s x x 1 2 3 2

a.

Formulate the dual for this problem.

b.

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

c.

How does the fact that this problem has multiple optimum solutions affect the

interpretation of the dual solution values? d . D e t e r m i n e t h e o p t i m a l r a n g e f o r e . 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 f o r ( s q u a r e f e e t o f s t o r a g e s p a c e ) . q 4 c f. What is the maximum price the manufacturer would be willing to pay to lease additional 2 .

storage space, and how many additional square feet could be leased at that price?

49. A school dietitian is attempting to plan a lunch menu that will minimize cost and meet certain minimum dietary requirements. The two staples in the meal are meat and potatoes, which provide protein, iron, and carbohydrates. The following linear programming model has been formulated to determine how many ounces of meat (x_{1}) and ounces of potatoes (x_{2}) to put in a lunch:

minimize Z = 0.03x_{1 }+ 0.02x_{2 }(cost, $)