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# The Simplex Solution Method

maximize Z = 6x1 + 10x2 (profit, \$) subject to

( b r a s s , l b . ) ( l a b o r , h r . ) x 1 , x 2 Ú 0 2 x 1 + 2 x 2 6 0 x 1 + 4 x 2 9 0

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

x

2

x

1

c j

10 6

Basic Variables

c j - z j z j

6

10

0

0

x1

x2

s1

s2

0

1

1>3

-1>6

1

0

-1>3

2>3

6

10

4>3

7>3

0

0

-4>3

-7>3

## Quantity

20 10

260

a. Formulate the dual of this model. b. Define the dual variables and indicate their value. 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 o u n d s o f b r a s s ) a n d ( l a b o r h o u r s ) . q 2 q 1 c 2 . e. What is the maximum price the company would be willing to pay for additional labor c 1

hours, and how many hours could be purchased at that price?

43. The Southwest Foods Company produces two brands of chili—Razorback and Longhorn—from several ingredients, including chili beans and ground beef. The number of 100-gallon batches of Razorback chili (x1) and Longhorn chili (x2) that can be produced daily is constrained by the avail- ability of chili beans and ground beef, as shown in the following linear programming model:

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

( c h i l i b e a n s , l b . ) ( g r o u n d b e e f , l b . ) x 1 , x 2 Ú 0 3 4 x 1 + 2 0 x 2 8 0 0 1 0 x 1 + 5 0 x 2 5 0 0

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

x

2

x

1

c j

300 200

Basic Variables

c j - z j z j

6

0

20

1

Quantity

200 x1

5,800

200 0

1

17/750

-1>150

0

-1>75

1/30

300

0

0

x2

s1

s2

300

310/75

70/15

0

-310>75

-70>15

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