Final Exam Preparation Problems (IOM)
A Linear Programming Problem with two variables
1. Minimize g  12 x  8 y subject to
x  2 y  10
2 x  y  11
x y9
x  0, y  0
2. Maximize f  2 x  6 y subject to
x y 7
2 x  y  12
x  3 y  15
x  0, y  0
3. Minimize g  22 x  7 y subject to
8 x  5 y  100
12 x  25 y  360
x  0, y  0
4. Maximize f  x  2 y subject to
x y  4
2x  y  8
y4
5. At one of its factories, a jeans manufacturer makes two styles:#891 and #917. Each pair of
style-891 takes 10 minutes to cut out and 20 minutes to assemble and finish. Each pair of
style-917 takes 10 minutes to cut out and 30 minutes to assemble and finish. The plant has
enough workers to provide at most 7500 minutes per day for cutting and at most 19500
minutes per day for assembly and finishing. The profit on each pair of style-891 is $6.00 and
the profit on each pair of style-917 is $7.50. How many pairs of each style should be
produced per day to obtain maximum profit? Find the maximum profit.
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6. In a laboratory experiment, two separate foods are given to experimental animals. Each
food contains essential ingredients, A and B, for which the animals have a minimum
requirement, and each food also has an ingredient C, which can be harmful to the animals.
The table below summarizes this information.
Food 1
Food 2
Required
Ingredient A
10 units / g
3 units / g
49 units
Ingredient B
6 units / g
12 units / g
60 units
Ingredient C
3 units / g
1 unit / g
How many grams of foods 1 and 2 should be given to the animals in order to satisfy the
requirements for A and B while minimizing the amount of ingedient C ingested?
Solve the following LP problems using Simplex or Two-Phase Simplex Method
1.
Maximize f  2 x1  6 x 2
subject to
 x1  x 2  1
2 x1  x 2  2
x1 , x 2  0
2.
Maximize f  6 x1  7 x 2
subject to
7 x1  6 x 2  42
5 x1  9 x 2  45
x1  x 2  4
x1 , x 2  0
3.
Minimize f  x1  3x 2
subject to
 4 x1  3 x 2  12
x1  x 2  7
x1  4 x 2  2
x1 , x 2  0
2
4.
Maximize the function F  x1  3 x 2  2 x3
subject to
x1  3 x3  3
2 x1  x3  2
x1 , x 2 , x3  0
5.
Minimize the function f  2 x1  4 x 2  3 x3
subject to
3 x1  x 2  2 x3  15
x1  3 x 2  5 x3  20
x1 , x 2 , x3  0
6.
Maximize f  240 x1  104 x 2  60 x3  19 x 4
subject to
20 x1  9 x 2  6 x3  x 4  20
10 x1  4 x 2  2 x3  x 4  10
x1 , x 2 , x3 , x 4  0
7.
Minimize f  3 x1  4 x 2
subject to
3x1  2 x 2  10
4 x1  x 2  8
x1 , x 2  0
8.
Minimize f   x1  3x 2  5 x3
subject to
x1  x 2  3
x1  x3  4
x1 , x 2 , x3  0
9.
Minimize f  5 x1  x 2
subject to
5 x1  x 2  3
 2 x1  3 x 2  2
x1 , x 2  0
3
10.
Minimize f  3 x1  2 x 2  5 x3
subject to
2 x1  x 2  3 x3  30
x1  x 2  3 x3  20
x1 , x 2 , x3  0
11. A grocery store wants to buy five different types of vegetables from four farms in a
month. The prices of the vegetables at different farms, the capacities of the farms and the
minimum requirements of the grocery store are indicated in the following table
Determine the buying scheme that corresponds to a minimum cost.
Try to solve also the problems 5 and 6 from A Linear Programming Problem with two
variables.
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