Question Details

MAT 540
WEEK 8
MAT540 Homework
Week 8
Page 1 of 4
MAT540
Week 8 Homework
Chapter 4
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
D
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- 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- 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:
MAT540 Homework
Week 8
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ï‚· None of the programs can receive more than 40% of the total allocation.
ï‚· The amount allocated to parks cannot exceed the total allocated to both the sanitation project and the mobile library
ï‚· 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:
ï‚· Between 1,500 and 2,000 calories
ï‚· At least 5 mg of iron
ï‚· At least 20 but no more than 60 g of fat
ï‚· At least 30 g of protein
ï‚· At least 40 g of carbohydrates
ï‚· 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 (g/lb.)
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.
MAT540 Homework
Week 8
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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
MAT540 Homework
Week 8
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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- 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?
Instructions for Homework
1)	Homework problems are listed in the pdf file “mat540 hw wk8.pdf”.
There are 6 problems.  To reduce your workload, 
a.	Ignore 3.c, 5.c and 6.c.
b.	For problem #4, ignore the constraints on Ash and Sulfur content.
2)	You are required to use the Excel Solver to solve the LP problems.  Answers without embedded formulas for solver will not be given credit.
3)	Download the homework template Excel file “Hw8_2015_Winter_Template”.   You are required to use the template for the homework.
4)	Rename the downloaded file “Hw8_2015_Winter_Template” to “Hw8_2015_Winter_ your last name”.  Do not insert or remove columns or rows.
5)	Do the homework and enter your answers in the cells highlighted in yellow.  I will grade only answers in the cells highlighted in yellow.  You must show how you obtain the answers using embedded Excel formula.
6)	Click on the link “Week 8 Homework Submission”.  Submit the completed homework as attachment.
Note
These homework problems are quite challenging.  To help you, I have left some of the “usage” and “objective” formulas in the template.  I have also given you the LP modeling equations. Most of the solver setup are quite similar to Week 6 and week 7 homework.  It just has more variables and constraints.  You may refer to the Week 6 Homework Solution Key I have posted.  I have also uploaded a file “Chapter 4 Textbook Examples”.  It is the Excel solver solution of the examples given in Chapter 4 of your textbook.  Study them with the equations given in the textbook.  You will get a better idea how to use solver.  I will also go over some of these homework in Tuesday- Live Help Session.  To see the embedded formula, hold the Ctrl key and click ~ key or double-click the highlighted cells, it will show you which cells are being multiplied and added.  Study the pattern and how it relates to the LP equations.  To go back to normal display mode, hold the Ctrl key and click ~ key.  It will toggle between the two modes.

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