MAT
540 MAT540 Complete Homework Assignment
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.
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 breakeven
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 breakeven volume?
4.
If Evergreen Fertilizer Company
increases its advertising expenditures by $14,000 per year, what effect will
the increase have on the breakeven 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 breakeven price Annie might charge?
6. The
College of Business at Kerouac University is planning to begin an online MBA
program. The initial startup 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
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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
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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. PlacePlus,
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
Page 2 of 3
condominiums.
The financial success of these projects depends on interest rate movement in
the next 5 years. The various development projects and their 5year financial
return (in $1,000,000s) given that interest rates will decline, remain stable,
or increase, are shown in the following payoff table. PlacePlus 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 2year 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 5year 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
Page 3 of 3
d. Hurwicz (α
= 0.50)
5. Construct
a decision tree for the following decision situation and indicate the best
decision.
Fenton
and Farrah Friendly, husbandandwife 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 fullsized 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


Fullsized 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 3month moving average forecast for months 4 through 9.
i.
Compute a weighted 3month 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
Page 2 of 4
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 3month average, forecast the
fund price for month 21.
Using a 3month 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
Page 3 of 4
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

Week 4
Page 4 of 4
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 boardft. of wood
available each day. Demand for chairs is limited to 6 per day. Each chair
requires 8 hours of labor and 2 boardft. of wood, whereas a table requires 10
hours of labor and 6 boardft. 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
Page 2 of 2
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 = 5x_{1} + 8x_{2}
Subject
to 4x_{1} + 5x_{2} ≤ 50 2x_{1} + 4x_{2} ≤ 40 x_{1}
≤ 8
x_{2}
≤ 8 x_{1}, x_{2} ≥ 0
MAT540 Homework
Week 7
Page 1 of 4
MAT540
Week 7 Homework
Chapter
3
 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 (ft^{2})

Basketball

3

4

Football

2

5

Total
resources

500 lb.

800 ft^{2}

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.
 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?
MAT540 Homework
Week 7
Page 2 of 4
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?
 Continuing
the model from Problem 5.
 If Irwin
Mills can obtain additional cotton or processing time, but not both,
which should it select? How much? Explain your answer.
 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
highgrade aluminum, and 5 tons of lowgrade 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 highgrade 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
Page 4 of 4
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
Page 2 of 4
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
Page 3 of 4
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
Page 4 of 4
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 sisterinlaw?
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 25foot standardlength boards in
stock. Therefore, the standardlength boards must be cut into the lengths
necessary to meet order requirements. Naturally, the company wishes to minimize
the number of standardlength 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 “trimloss.” 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
 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.
 Formulate
an integer programming model for this problem
 Solve
this model by using the computer.
 Solve
the following mixed integer linear programming model by using the
computer:
Maximize Z = 5 x_{1}
+ 6 x_{2} + 4 x_{3}
Subject to
5
x_{1} + 3 x_{2} + 6 x_{3} ≤ 20 x_{1} + 3 x_{2}
≤ 12
x_{1}, x_{3}
≥ 0
x_{2} ≥ 0 and integer
 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
Page 2 of 2
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,
allterrain vehicle codenamed “J99Terra” 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.
 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
Page 2 of 2
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