MAT540 Complete Homework Assignment
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Thursday, 30 July 2015

MAT540 Complete Homework Assignment


MAT 540 MAT540 Complete Homework Assignment

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MAT540

Week 1 Homework

Chapter 1

1.      The Retread Tire Company recaps tires. The fixed annual cost of the recapping operation is $55,000.The variable cost of recapping a tire is $8.The company charges $21 to recap a tire. 


a.  For an annual volume of 10,000 tires, determine the total cost, total revenue, and profit.

b.  Determine the annual break-even volume for the Retread Tire Company operation. 


2.      Evergreen Fertilizer Company produces fertilizer. The company’s fixed monthly cost is $30,000, and its variable cost per pound of fertilizer is $0.16. Evergreen sells the fertilizer for $0.40 per pound. Determine the monthly break-even volume for the company.

3.      If Evergreen Fertilizer Company in Problem 2 changes the price of its fertilizer from $0.40 per pound to $0.60 per pound, what effect will the change have on the break-even volume?

4.      If Evergreen Fertilizer Company increases its advertising expenditures by $14,000 per year, what effect will the increase have on the break-even volume computed in Problem 2?

5.      Annie McCoy, a student at Tech, plans to open a hot dog stand inside Tech’s football stadium during home games. There are seven home games scheduled for the upcoming season. She must pay the Tech athletic department a vendor’s fee of $2,500 for the season. Her stand and other equipment will cost her $3,100 for the season. She estimates that each hot dog she sells will cost her $0.35. She has talked to friends at other universities who sell hot dogs at games. Based on their information and the athletic department’s forecast that each game will sell out, she anticipates that she will sell approximately 2,000 hot dogs during each game.

a.  What price should she charge for a hot dog in order to break even?

b.  What factors might occur during the season that would alter the volume sold and thus the break-even price Annie might charge?

6.  The College of Business at Kerouac University is planning to begin an online MBA program. The initial start-up cost for computing equipment, facilities, course development, and staff recruitment and development is $360,000.The college plans to charge tuition of $17,000 per student per year. However, the university administration will charge the college $12,000 per student for the first 100 students enrolled each year for administrative costs and its share of the tuition payments.

a.  How many students does the college need to enroll in the first year to break even?

b.  If the college can enroll 75 students the first year, how much profit will it make?

c. The college believes it can increase tuition to $22,000, but doing so would reduce enrollment to


MAT540 Homework

Week 1
Page 2 of 3

35. Should the college consider doing this?

Chapter 11

7.      The following probabilities for grades in management science have been determined based on past records:

Grade          Probability
A                                      0.15

B                                     0..25

C                                     0..38
D                                     0..12

F                                       0.10

1.00

The grades are assigned on a 4.0 scale, where an A is a 4.0, a B a 3.0, and so on. Determine the expected grade and variance for the course.

8.      An investment firm is considering two alternative investments, A and B, under two possible future sets of economic conditions, good and poor. There is a .60 probability of good economic conditions occurring and a .40 probability of poor economic conditions occurring. The expected gains and losses under each economic type of conditions are shown in the following table:


Economic Conditions

Investment
Good
Poor

A
$350,000
-$350,000

B
120,000
70,000


Using the expected value of each investment alternative, determine which should be selected.

9.      The weight of bags of fertilizer is normally distributed, with a mean of 50 pounds and a standard deviation of 7 pounds. What is the probability that a bag of fertilizer will weigh between 45 and 55 pounds?

10.  The Polo Development Firm is building a shopping center. It has informed renters that their rental spaces will be ready for occupancy in 19 months. If the expected time until the shopping center is completed is estimated to be 16 months, with a standard deviation of 4 months, what is the probability that the renters will not be able to occupy in 19 months?

11.  The manager of the local National Video Store sells videocassette recorders at discount prices. If the store does not have a video recorder in stock when a customer wants to buy one, it will lose the sale because the customer will purchase a recorder from one of the many local competitors. The problem is that the cost of renting warehouse space to keep enough recorders in inventory to meet all demand is excessively high. The manager has determined that if 90% of customer demand for


MAT540 Homework

Week 1
Page 3 of 3

recorders can be met, then the combined cost of lost sales and inventory will be minimized. The manager has estimated that monthly demand for recorders is normally distributed, with a mean of 180 recorders and a standard deviation of 60. Determine the number of recorders the manager should order each month to meet 90% of customer demand.


MAT540 Homework

Week 2
Page 1 of 3

MAT540

Week 2 Homework

Chapter 12

1.      A local real estate investor in Orlando is considering three alternative investments: a motel, a restaurant, or a theater. Profits from the motel or restaurant will be affected by the availability of gasoline and the number of tourists; profits from the theater will be relatively stable under any conditions. The following payoff table shows the profit or loss that could result from each investment:



Gasoline Availability


Investment
Shortage
Stable Supply
Surplus

Motel
$-8,000
$15,000
$20,000

Restaurant
2,000
8,000
6,000

Theater
6,000
6,000
5,000


Determine the best investment, using the following decision criteria.

a.       Maximax

b.      Maximin

c.       Minimax regret

d.      Hurwicz (α = 0.4)

e.       Equal likelihood

B       A concessions manager at the Tech versus A&M football game must decide whether to have the vendors sell sun visors or umbrellas. There is a 30% chance of rain, a 15% chance of overcast skies, and a 55% chance of sunshine, according to the weather forecast in College Junction, where the game is to be held. The manager estimates that the following profits will result from each decision, given each set of weather conditions:



Weather Conditions


Decision
Rain
Overcast
Sunshine


.30
.15
.55

Sun visors
$-500
$-200
$1,500

Umbrellas
2,000
0
-900


a.       Compute the expected value for each decision and select the best one.

b.      Develop the opportunity loss table and compute the expected opportunity loss for each decision.

3.  Place-Plus, a real estate development firm, is considering several alternative development projects. These include building and leasing an office park, purchasing a parcel of land and building an office building to rent, buying and leasing a warehouse, building a strip mall, and building and selling


MAT540 Homework

Week 2
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condominiums. The financial success of these projects depends on interest rate movement in the next 5 years. The various development projects and their 5-year financial return (in $1,000,000s) given that interest rates will decline, remain stable, or increase, are shown in the following payoff table. Place-Plus real estate development firm has hired an economist to assign a probability to each direction interest rates may take over the next 5 years. The economist has determined that there is a .50 probability that interest rates will decline, a .40 probability that rates will remain stable, and a .10 probability that rates will increase.

a.       Using expected value, determine the best project.

b.      Determine the expected value of perfect information.





Interest Rate


Project
Decline
Stable
Increase

Office park
$0.5
$1.7
$4.5

Office building
1.5
1.9
2.5

Warehouse
1.7
1.4
1.0

Mall
0.7
2.4
3.6

Condominiums
3.2
1.5
0.6


4.      The director of career advising at Orange Community College wants to use decision analysis to provide information to help students decide which 2-year degree program they should pursue. The director has set up the following payoff table for six of the most popular and successful degree programs at OCC that shows the estimated 5-year gross income ($) from each degree for four future economic conditions:



Economic Conditions



Degree Program
Recession
Average
Good
Robust


Graphic design
145,000
175,000
220,000
260,000


Nursing
150,000
180,000
205,000
215,000


Real estate
115,000
165,000
220,000
320,000


Medical technology
130,000
180,000
210,000
280,000


Culinary technology
115,000
145,000
235,000
305,000


Computer information
125,000
150,000
190,000
250,000


technology









Determine the best degree program in terms of projected income, using the following decision criteria:

a.       Maximax

b.      Maximin

c.       Equal likelihood


MAT540 Homework

Week 2
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d.  Hurwicz (α = 0.50)

5.      Construct a decision tree for the following decision situation and indicate the best decision.

Fenton and Farrah Friendly, husband-and-wife car dealers, are soon going to open a new dealership. They have three offers: from a foreign compact car company, from a U.S. producer of full-sized cars, and from a truck company. The success of each type of dealership will depend on how much gasoline is going to be available during the next few years. The profit from each type of dealership, given the availability of gas, is shown in the following payoff table:
Gasoline Availability

Dealership
Shortage
Surplus


.6
.4

Compact cars
$ 300,000
$150,000

Full-sized cars
-100,000
600,000

Trucks
120,000
170,000


Decision Tree diagram to complete:
















MAT540 Homework

Week 3
Page 1 of 2

MAT540

Week 3 Homework

Chapter 14

2.      The Hoylake Rescue Squad receives an emergency call every 1, 2, 3, 4, 5, or 6 hours, according to the following probability distribution. The squad is on duty 24 hours per day, 7 days per week:

Time Between


Emergency Calls (hr.)
Probability

1
0.05

2
0.10

3
0.30

4
0.30

5
0.20

6
0.05


1.00


f.       Simulate the emergency calls for 3 days (note that this will require a “running”, or cumulative, hourly clock), using the random number table.

g.      Compute the average time between calls and compare this value with the expected value of the time between calls from the probability distribution. Why are the results different?

C       The time between arrivals of cars at the Petroco Service Station is defined by the following probability distribution:

Time Between


Arrivals (min.)
Probability

1
0.25

2
0.30

3
0.35

4
0.10


1.00


c.       Simulate the arrival of cars at the service station for 20 arrivals and compute the average time between arrivals.

d.      Simulate the arrival of cars at the service station for 1 hour, using a different stream of random numbers from those used in (a) and compute the average time between arrivals.

e.       Compare the results obtained in (a) and (b).

4.      The Dynaco Manufacturing Company produces a product in a process consisting of operations of five machines. The probability distribution of the number of machines that will break down in a week follows:


MAT540 Homework

Week 3
Page 2 of 2

Machine Breakdowns


per Week
Probability

0
0.05

1
0.15

2
0.20

3
0.30

4
0.25

5
0.05


1.00


            Simulate the machine breakdowns per week for 20 weeks.

            Compute the average number of machines that will break down per week.

4.      Simulate the following decision situation for 20 weeks, and recommend the best decision.

A concessions manager at the Tech versus A&M football game must decide whether to have the vendors sell sun visors or umbrellas. There is a 30% chance of rain, a 15% chance of overcast skies, and a 55% chance of sunshine, according to the weather forecast in College Junction, where the game is to be held. The manager estimates that the following profits will result from each decision, given each set of weather conditions:



Weather Conditions

Decision
Rain
Overcast
Sunshine


.30
.15
.55

Sun visors
$-500
$-200
$1,500

Umbrellas
2,000
0
-900


5.      Every time a machine breaks down at the Dynaco Manufacturing Company (Problem 3), either 1, 2, or 3 hours are required to fix it, according to the following probability distribution:

Repair Time (hr.)
Probability

1
0.25

2
0.55

3
0.20


1.00


a.  Simulate the repair time for 20 weeks and then compute the average weekly repair time.


MAT540 Homework

Week 4
Page 1 of 4

MAT540

Week 4 Homework

Chapter 15

3.      The manager of the Carpet City outlet needs to make an accurate forecast of the demand for Soft Shag carpet (its biggest seller). If the manager does not order enough carpet from the carpet mill, customers will buy their carpet from one of Carpet City’s many competitors. The manager has collected the following demand data for the past 8 months:


Demand for Soft Shag
Month
Carpet (1,000 yd.)
1
9
2
8
3
7
4
8
5
10
6
11
7
13
8
12


h.      Compute a 3-month moving average forecast for months 4 through 9.

i.        Compute a weighted 3-month moving average forecast for months 4 through 9. Assign weights of 0.50, 0.30, and 0.20 to the months in sequence, starting with the most recent month.

j.        Compare the two forecasts by using MAD. Which forecast appears to be more accurate?

D      The manager of the Petroco Service Station wants to forecast the demand for unleaded gasoline next month so that the proper number of gallons can be ordered from the distributor. The owner has accumulated the following data on demand for unleaded gasoline from sales during the past 10 months:

Month
Gasoline Demanded (gal.)

October
800

November
725

December
600

January
500

February
625

March
690

April
810

May
935

June
1,200

July
1,100


MAT540 Homework

Week 4
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f.       Compute an exponentially smoothed forecast, using an α value of 0.30.

g.      Compute the MAPD.

5.      Emily Andrews has invested in a science and technology mutual fund. Now she is considering liquidating and investing in another fund. She would like to forecast the price of the science and technology fund for the next month before making a decision. She has collected the following data on the average price of the fund during the past 20 months:

Month
Fund Price
1
$57 3/4
2
54 1/4
3
55 1/8
4
58 1/8
5
53 3/8
6
51 1/8
7
56 1/4
8
59 5/8
9
62 1/4
10
59 1/4
11
62 3/8
12
57 1/1
13
58 1/8
14
62 3/4
15
64 3/4
16
66 1/8
17
68 3/4
18
65 1/2
19
69 7/8
20
70 1/4
            Using a 3-month average, forecast the fund price for month 21.

            Using a 3-month weighted average with the most recent month weighted 0.60, the next most recent month weighted 0.30, and the third month weighted 0.10, forecast the fund price for month 21.

            Compute an exponentially smoothed forecast, using α=0 .40, and forecast the fund price for month 21.

            Compare the forecasts in (a), (b), and (c), using MAD, and indicate the most accurate.



5.      Carpet City wants to develop a means to forecast its carpet sales. The store manager believes that the store’s sales are directly related to the number of new housing starts in town. The manager has gathered data from county records on monthly house construction permits and from store records on monthly sales. These data are as follows:


MAT540 Homework

Week 4
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Monthly Carpet Sales
Monthly Construction

(1,000 yd.)
Permits

9
17

14
25

10
8

12
7

15
14

9
7

24
45

21
19

20
28

29
28




            Develop a linear regression model for these data and forecast carpet sales if 30 construction permits for new homes are filed.

            Determine the strength of the causal relationship between monthly sales and new home construction by using correlation.

5.      The manager of Gilley’s Ice Cream Parlor needs an accurate forecast of the demand for ice cream.

The store orders ice cream from a distributor a week ahead; if the store orders too little, it loses business, and if it orders too much, the extra must be thrown away. The manager believes that a major determinant of ice cream sales is temperature (i.e., the hotter the weather, the more ice cream people buy). Using an almanac, the manager has determined the average daytime temperature for 14 weeks, selected at random, and from store records he has determined the ice cream consumption for the same 14 weeks. These data are summarized as follows:


Average Temperature
Ice Cream Sold
Week
(degrees)
(gal.)
1
68
80
2
70
115
3
73
91
4
79
87
5
77
110
6
82
128
7
85
164
8
90
178
9
85
144
10
92
179
11
90
144
12
95
197
13
80
144
14
75
123


MAT540 Homework

Week 4
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a.       Develop a linear regression model for these data and forecast the ice cream consumption if the average weekly daytime temperature is expected to be 85 degrees.

b.      Determine the strength of the linear relationship between temperature and ice cream consumption by using correlation.

6.  Report the coefficient of determination for the data in Problem 5 and explain its meaning.


MAT540 Homework

Week 6
Page 1 of 2

MAT540

Week 6 Homework

Chapter 2


1.      A company produces two products that are processed on two assembly lines. Assembly line 1 has

vvvv.                  available hours, and assembly line 2 has 42 available hours. Product 1 requires 10 hours of processing time on line 1 and product 2 requires 14 hours of processing on line 1. , On line 2, product 1 requires 7 hours and product 2 requires 3 hours. The profit for product 1 is $6 per unit, and the profit for product 2 is $4 per unit.

            Formulate a linear programming model for this problem.

            Solve the model by using graphical analysis.


2.      The Pinewood Furniture Company produces chairs and tables from two resources – labor and wood. The company has 80 hours of labor and 36 board-ft. of wood available each day. Demand for chairs is limited to 6 per day. Each chair requires 8 hours of labor and 2 board-ft. of wood, whereas a table requires 10 hours of labor and 6 board-ft. of wood. The profit derived from each chair is $400 and from each table, $100. The company wants to determine the number of chairs and tables to produce each day in order to maximize profit. Formulate a linear programming model for this problem.

            Formulate a linear programming model for this problem.

            Solve the model by using graphical analysis.


3.      In Problem 2, how much labor and wood will be unused if the optimal numbers of chairs and tables are produced?

4.      The Elixer Drug Company produces a drug from two ingredients. Each ingredient contains the same three antibiotics, in different proportions. One gram of ingredient 1 contributes 3 units and one gram of ingredient 2 contributes 1 unit of antibiotic 1; the drug requires 6 units. At least 4 units of antibiotic 2 are required and the ingredients contribute 1 unit each per gram. At least 12 units of antibiotic 3 are required; a gram of ingredient 1 contributes 2 units, and a gram of ingredient 2 contributes 6 units. The cost for a gram of ingredient 1 is $80, and the cost for a gram of ingredient

2  is $50. The company wants to formulate a linear programming model to determine the number of grams of each ingredient that must go into the drug in order to meet the antibiotic requirements at the minimum cost.

a.       Formulate a linear programming model for this problem.

b.      Solve the model by using graphical analysis.


                          A clothier makes coats and slacks. The two resources required are wool cloth and labor. The


MAT540 Homework

Week 6
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clothier has 150 square yards of wool and 200 hours of labor available. Each coat requires 3 square yards of wool and 10 hours of labor, whereas each pair of slacks requires 5 square yards of wool and 4 hours of labor. The profit for a coat is $50, and the profit for slacks is $40. The clothier wants to determine the number of coats and pairs of slacks to make so that profit will be maximized.

a         Formulate a linear programming model for this problem.

b        Solve the model by using graphical analysis.



f.       Solve the following linear programming model graphically: Maximize Z = 5x1 + 8x2

Subject to 4x1 + 5x2 ≤ 50 2x1 + 4x2 ≤ 40 x1 ≤ 8

x2 ≤ 8 x1, x2 ≥ 0


MAT540 Homework

Week 7
Page 1 of 4

MAT540

Week 7 Homework

Chapter 3

  1. Southern Sporting Good Company makes basketballs and footballs. Each product is produced from two resources rubber and leather. The resource requirements for each product and the total resources available are as follows:


Resource Requirements per Unit
Product
Rubber (lb.)
Leather (ft2)
Basketball
3
4
Football
2
5
Total resources
500 lb.
800 ft2
available



k.      State the optimal solution.

l.        What would be the effect on the optimal solution if the profit for a basketball changed from $12 to $13? What would be the effect if the profit for a football changed from $16 to $15?

m.    What would be the effect on the optimal solution if 500 additional pounds of rubber could be obtained? What would be the effect if 500 additional square feet of leather could be obtained?



E       A company produces two products, A and B, which have profits of $9 and $7, respectively. Each unit of product must be processed on two assembly lines, where the required production times are as follows:


Hours/ Unit
Product
Line 1
Line2
A
12
4
B
4
8
Total Hours
60
40



            Formulate a linear programming model to determine the optimal product mix that will maximize profit.

            Transform this model into standard form.


  1. Solve problem 2 using the computer.

            State the optimal solution.

            What would be the effect on the optimal solution if the production time on line 1 was reduced to 40 hours?


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c.       What would be the effect on the optimal solution if the profit for product B was increased from $7 to $15 to $20?

d.      For the linear programming model formulated in Problem 2 and solved in Problem 3.

            What are the sensitivity ranges for the objective function coefficients?

            Determine the shadow prices for additional hours of production time on line 1 and line 2 and indicate whether the company would prefer additional line 1 or line 2 hours.

e.       Formulate and solve the model for the following problem:

Irwin Textile Mills produces two types of cotton cloth – denim and corduroy. Corduroy is a heavier grade of cotton cloth and, as such, requires 7.5 pounds of raw cotton per yard, whereas denim requires 5 pounds of raw cotton per yard. A yard of corduroy requires 3.2 hours of processing time; a yard of denim requires 3.0 hours. Although the demand for denim is practically unlimited, the maximum demand for corduroy is 510 yards per month. The manufacturer has 6,500 pounds of cotton and 3,000 hours of processing time available each month. The manufacturer makes a profit of $2.25 per yard of denim and $3.10 per yard of corduroy. The manufacturer wants to know how many yards of each type of cloth to produce to maximize profit. Formulate the model and put it into standard form. Solve it.

.

a.       How much extra cotton and processing time are left over at the optimal solution? Is the demand for corduroy met?

b.      What is the effect on the optimal solution if the profit per yard of denim is increased from $2.25 to $3.00? What is the effect if the profit per yard of corduroy is increased from $3.10 to $4.00?

c.       What would be the effect on the optimal solution if Irwin Mils could obtain only 6,000 pounds of cotton per month?

  1. Continuing the model from Problem 5.

    1. If Irwin Mills can obtain additional cotton or processing time, but not both, which should it select? How much? Explain your answer.

    1. Identify the sensitivity ranges for the objective function coefficients and for the constraint quantity values. Then explain the sensitivity range for the demand for corduroy.


MAT540 Homework

Week 7
Page 3 of 4

7.      United Aluminum Company of Cincinnati produces three grades (high, medium, and low) of aluminum at two mills. Each mill has a different production capacity (in tons per day) for each grade as follows:


Mill

Aluminum Grade
1

2
High
6

2
Medium
2

2
Low
4

10

The company has contracted with a manufacturing firm to supply at least 12 tons of high-grade aluminum, and 5 tons of low-grade aluminum. It costs United $6,000 per day to operate mill 1 and $7,000 per day to operate mill 2. The company wants to know the number of days to operate each mill in order to meet the contract at minimum cost.

a.  Formulate a linear programming model for this problem.



8.   Solve the linear programming model formulated in Problem 16 for Unite Aluminum Company by using the computer.

a. Identify and explain the shadow prices for each of the aluminum grade contract requirements. b. Identify the sensitivity ranges for the objective function coefficients and the constraint quantity

values.

c.   Would the solution values change if the contract requirements for high-grade alumimum were increased from 12 tons to 20 tons? If yes, what would the new solution values be?

9.  Solve the linear programming model developed in Problem 22 for the Burger Doodle restaurant by

using the computer.






a.
Identify and explain the shadow prices for each of the resource constraints
b.
Which of the resources constrains profit the most?



c.
Identify the sensitivity ranges for the profit of a sausage biscuit and the amount of sausage

available. Explain these sensitivity ranges.




Reference Problem 22. The manager of a Burger Doodle franchise wants to determine how

many sausage biscuits and ham biscuits to prepare each morning for breakfast customers. The

two types of biscuits require the following resources:














Biscuit
Labor (hr.)
Sausage (lb.)

Ham (lb.)
Flour (lb.)



Sausage
0.010
0.10

---
0.04



Ham
0.024
---

0.15
0.04


The franchise has 6 hours of labor available each morning. The manager has a contract with a local grocer for 30 pounds of sausage and 30 pounds of ham each morning. The manager also


MAT540 Homework

Week 7
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purchases 16 pounds of flour. The profit for a sausage biscuit is $0.60; the profit for a ham biscuit is $0.50. The manager wants to know the number of each type of biscuit to prepare each morning in order to maximize profit. Formulate a linear programming model for this problem.


MAT540 Homework

Week 8
Page 1 of 4

MAT540

Week 8 Homework

Chapter 4

6.      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:



Metal (%)

Impurities

Ore
A
B
C
D
(%)
Cost/Ton
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.

n.        Formulate a linear programming model for this problem.
o.      Solve the model by using the computer.


F        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:


MAT540 Homework

Week 8
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2        None of the programs can receive more than 40% of the total allocation.

3        The amount allocated to parks cannot exceed the total allocated to both the sanitation project and the mobile library

4        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.



c.       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
Iron
Protein
Carbo-
Fat
Chol-
Cost

(per lb.)
(mg/lb.)
(g/lb.)
hydrates
(g/lb.)
esterol





(g/lb.)

(mg/lb.)
$/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 Capacity
Mine
1
2

3
4
(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
Lathe
Grinder

(min)
(min)
(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?


MAT540 Homework

Week 9
Page 1 of 2

MAT540

Week 9 Homework

Chapter 5

  1. The Livewright Medical Supplies Company has a total of 12 salespeople it wants to assign to three regions – the South, the East, and the Midwest. A salesperson in the South earns $600 in profit per month of the company, a salesperson in the East earns $540, and a salesperson in the Midwest earns $375. The southern region can have a maximum assignment of 5 salespeople. The company has a total of $750 per day available for expenses for all 12 salespeople. A salesperson in the South has average expenses of $80 per day, a salesperson in the East has average expenses of $70 per day, and a salesperson in the Midwest has average daily expenses of $50. The company wants to determine the number of salespeople to assign to each region to maximize profit.

    1. Formulate an integer programming model for this problem

    1. Solve this model by using the computer.

  1. Solve the following mixed integer linear programming model by using the computer:
Maximize Z = 5 x1 + 6 x2 + 4 x3

Subject to
5 x1 + 3 x2 + 6 x3 ≤ 20 x1 + 3 x2 ≤ 12
x1, x3 ≥ 0
x2  ≥ 0 and integer

  1. The Texas Consolidated Electronics Company is contemplating a research and development program encompassing eight research projects. The company is constrained from embarking on all projects by the number of available management scientists (40) and the budget available for R&D projects ($300,000). Further, if project 2 is selected, project 5 must also be selected (but not vice versa). Following are the resource requirements and the estimated profit for each project.

Project
Expense
Management
Estimated Profit

($1,000s)
Scientists required
(1,000,000s)
1
$ 60
7
$0.36
2
110
9
0.82
3
53
8
0.29
4
47
4
0.16
5
92
7
0.56
6
85
6
0.61
7
73
8
0.48
8
65
5
0.41


MAT540 Homework

Week 9
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Formulate the integer programming model for this problem and solve it using the computer.

D      During the war with Iraq in 1991, the Terraco Motor Company produced a lightweight, all-terrain vehicle code-named “J99-Terra” for the military. The company is now planning to sell the Terra to the public. It has five plants that manufacture the vehicle and four regional distribution centers. The company is unsure of public demand for the Terra, so it is considering reducing its fixed operating costs by closing one or more plants, even though it would incur an increase in transportation costs. The relevant costs for the problem are provided in the following table. The transportation costs are per thousand vehicles shipped; for example, the cost of shipping 1,000 vehicles from plant 1 to warehouse C is $32,000.



Transportation Costs ($1000s)

Annual Fixed
From

to Warehouse

Annual Production
Operating
Plant
A
B
C
D
Capacity
Costs
1
$56
$21
$32
$65
12,000
$2,100,000
2
18
46
7
35
18,000
850,000
3
12
71
41
52
14,000
1,800,000
4
30
24
61
28
10,000
1,100,000
5
45
50
26
31
16,000
900,000
Annual
6,000
14,000
8,000
10,000


Demand








Formulate and solve an integer programming model for this problem to assist the company in determining which plants should remain open and which should be closed and the number of vehicles that should be shipped from each plan to each warehouse to minimize total cost.


MAT540 Homework

Week 10
Page 1 of 2

MAT540

Week 10 Homework

Chapter 61. Consider the following transportation problem:



To (cost)


From
1
2
3
Supply
A
$ 5
$ 4
$3
130
B
2
3
5
70
C
4
8
7
100
Demand
80
110
60



Formulate this problem as a linear programming model and solve it by using the computer.

2.  Consider the following transportation problem:



To (cost)


From
1
2
3
Supply
A
$ 5
$ 12
$ 2
130
B
2
9
5
70
C
4
24
7
100
Demand
80
110
60



Solve it by using the computer.

  1. World Foods, Inc., imports food products such as meats, cheeses, and pastries to the United States from warehouses at ports in Hamburg, Marseilles and Liverpool. Ships from these ports deliver the products to Norfolk, New York and Savannah, where they are stored in company warehouses before being shipped to distribution centers in Dallas, St. Louis and Chicago. The products are then distributed to specialty foods stores and sold through catalogs. The shipping costs ($/1,000 lb.) from the European ports to the U.S. cities and the available supplies (1000 lb.) at the European ports are provided in the following table:



To (cost)


From
4. Norfolk
5. New York
6. Savannah
Supply
1. Hamburg
$420
$390
$610
55
2. Marseilles
510
590
470
78
3. Liverpool
450
360
480
37

The transportation costs ($/1,000 lb.) from each U.S. city of the three distribution centers and the demands (1,000 lb.) at the distribution centers are as follows:

Warehouse
Distribution Center


MAT540 Homework

Week 10
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7. Dallas
8. St. Louis
9. Chicago
4.
Norfolk
$ 75
$ 63
$ 81
5.
New York
125
110
95
6.
Savannah
68
82
95
Demand
60
45
50

Determine the optimal shipments between the European ports and the warehouses and the distribution centers to minimize total transportation costs.


4        The Omega pharmaceutical firm has five salespersons, whom the firm wants to assign to five sales regions. Given their various previous contacts, the salespersons are able to cover the regions in different amounts of time. The amount of time (days) required by each salesperson to cover each city is shown in the following table:



Region (days)


Sales-
A
B
C

D
E
person






1
17
10
15

16
20
2
12
9
16

9
14
3
11
16
14

15
12
4
14
10
10

18
17
5
13
12
9

15
11

Which salesperson should be assigned to each region to minimize total time? Identify the optimal assignments and compute total minimum time.

MAT 540 MAT540 Complete Homework Assignment

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