1. Grafton Metalworks Company produces metal alloys from six different ores it mines. The company has an order from a customer to produce an alloy that contains four metals according to the following specifications: at least 21% of metal A, no more than 12% of metal B, no more than 7% of metal C and between 30% and 65% of metal D. The proportion of the four metals in each of the six ores and the level of impurities in each ore are provided in the following table:
|
Ore
|
Metal (%)
|
Impurities
(%)
|
Cost/Ton
|
|
A
|
B
|
C
|
0
|
|
1
|
19
|
15
|
12
|
14
|
40
|
27
|
|
2
|
43
|
10
|
25
|
7
|
15
|
25
|
|
3
|
17
|
0
|
0
|
53
|
30
|
32
|
|
4
|
20
|
12
|
0
|
18
|
50
|
22
|
|
5
|
0
|
24
|
10
|
31
|
35
|
20
|
|
6
|
12
|
18
|
16
|
25
|
29
|
24
|
When the metals are processed and refined, the impurities are removed. The company wants to know the amount of each ore to use per ton of the alloy that will minimize the cost per ton of the alloy.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.
2. As a result of a recently passed bill, a congressman's district has been allocated $4 million for programs and projects. It is up to the congressman to decide how to distribute the money. The congressman has decided to allocate the money to four ongoing programs because of their importance to his district - a job training program, a parks project, a sanitation project, and a mobile library. However, the congressman wants to distribute the money in a manner that will please the most voters, or, in other words, gain him the most votes in the upcoming election. His staff's estimates of the number of votes gained per dollar spent for the various programs are as follows.
|
Program
|
Votes/ Dollar
|
|
Job training
|
0.02
|
|
Parks
|
0.09
|
|
Sanitation
|
0.06
|
|
Mobile library
|
0.04
|
In order also to satisfy several local influential citizens who financed his election, he is obligated to observe the following guidelines:
None of the programs can receive more than 40% of the total allocation.
The amount allocated to parks cannot exceed the total allocated to both the sanitation project and the mobile library
The amount allocated to job training must at least equal the amount spent on the sanitation project.
Any money not spent in the district will be returned to the government; therefore, the congressman wants to spend it all. The congressman wants to know the amount to allocate to each program to maximize his votes.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.
3. Anna Broderick is the dietician for the State University football team, and she is attempting to determine a nutritious lunch menu for the team. She has set the following nutritional guidelines for each lunch serving:
Between 1,500 and 2,000 calories
At least 5 mg of iron
At least 20 but no more than 60 g of fat
At least 30 g of protein
At least 40 g of carbohydrates
No more than 30 mg of cholesterol
She selects the menu from seven basic food items, as follows, with the nutritional contributions per pound and the cost as given:
|
|
Calories
(per lb.)
|
Iron (mg/lb.)
|
Protein (g/lb.)
|
Carbo- hydrates (g/lb.)
|
Fat (Ob.)
|
Chol- esterol (mg/lb.)
|
Cost $/lb.
|
|
Chicken
|
520
|
4.4
|
17
|
0
|
30
|
180
|
0.80
|
|
Fish
|
500
|
3.3
|
85
|
0
|
5
|
90
|
3.70
|
|
Ground beef
|
860
|
0.3
|
82
|
0
|
75
|
350
|
2.30
|
|
Dried beans
|
600
|
3.4
|
10
|
30
|
3
|
0
|
0.90
|
|
Lettuce
|
50
|
0.5
|
6
|
0
|
0
|
0
|
0.75
|
|
Potatoes
|
460
|
2.2
|
10
|
70
|
0
|
0
|
0.40
|
|
Milk (2%)
|
240
|
0.2
|
16
|
22
|
10
|
20
|
0.83
|
The dietician wants to select a menu to meet the nutritional guidelines while minimizing the total cost per serving.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer
c. If a serving of each of the food items (other than milk) was limited to no more than a half pound, what effect would this have on the solution?
4. The Cabin Creek Coal (CCC) Company operates three mines in Kentucky and West Virginia, and it supplies coal to four utility power plants along the East Coast. The cost of shipping coal from each mine to each plant, the capacity at each of the three mines and the demand at each plant are shown in the following table:
|
|
Plant
|
|
|
Mine
|
1
|
2
|
3
|
4
|
Mine Capacity
(tons)
|
|
1
|
$ 7
|
$ 9
|
$10
|
$12
|
220
|
|
2
|
9
|
7
|
8
|
12
|
170
|
|
3
|
11
|
14
|
5
|
7
|
280
|
|
Demand (tons)
|
110
|
160
|
90
|
180
|
|
The cost of mining and processing coal is $62 per ton at mine 1, $67 per ton at mine 2, and $75 per ton at mine 3. The percentage of ash and sulfur content per ton of coal at each mine is as follows:
|
Mine
|
% Ash
|
% Sulfur
|
|
1
|
9
|
6
|
|
2
|
5
|
4
|
|
3
|
4
|
3
|
Each plant has different cleaning equipment. Plant 1 requires that the coal it receives have no more than 6% ash and 5% sulfur; plant 2 coal can have no more than 5% ash and sulfur combined; plant 3 can have no more than 5% ash and 7% sulfur; and plant 4 can have no more than 6% ash and sulfur combined. CCC wabts to determine the amount of coal to produce at each mine and ship to its customers that will minimize its total cost.
a. Formulate a linear programming model for this problem.
b. Solve this model by using the computer.
5. Joe Henderson runs a small metal parts shop. The shop contains three machines - a drill press, a lathe, and a grinder. Joe has three operators, each certified to work on all three machines. However, each operator performs better on some machines than on others. The shop has contracted to do a big job that requires all three machines. The times required by the various operators to perform the required operations on each machine are summarized as follows:
|
Operator
|
Drill Press (min)
|
Lathe
(min)
|
Grinder (min)
|
|
1
|
23
|
18
|
35
|
|
2
|
41
|
30
|
28
|
|
3
|
25
|
36
|
18
|
Joe Henderson wants to assign one operator to each machine so that the topal operating time for all three operators is minimized.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer
c. Joe's brother, Fred, has asked him to hire his wife, Kelly, who is a machine operator. Kelly can perform each of the three required machine operations in 20 minutes. Should Joe hire his sister-in-law?
6. The Cash and Carry Building Supply Company has received the following order for boards in three lengths:
|
Length
|
Order (quantity)
|
|
7 ft.
|
700
|
|
9 ft.
|
1,200
|
|
10 ft.
|
300
|
The company has 25-foot standard-length boards in stock. Therefore, the standard-length boards must be cut into the lengths necessary to meet order requirements. Naturally, the company wishes to minimize the number of standard-length boards used.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer
c. When a board is cut in a specific pattern, the amount of board left over is referred to as "trim-loss." Reformulate the linear programming model for this problem, assuming that the objective is to minimize trim loss rather than to minimize the total number of boards used, and solve the model. How does this affect the solution?