Quantitative Analysis
BA 452 Homework 3 Questions
19.
Better
Products, Inc., manufactures three products on two machines. In a typical week,
40 hours are available on each machine. The profit contribution and production
time in hours per unit are as follows:
Category

Product 1

Product 2

Product 3

Profit/unit

$30

$50

$20

Machine 1 time/unit

0.5

2.0

0.75

Machine 2
time/unit

1.0

1.0

0.5

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Two operators are required for machine 1; thus, 2
hours of labor must ne scheduled for each hour of machine 1 time. Only one
operator is required for machine 2. A maximum of 100 laborhours is available
for assignment to the machines during the coming week.
Other production requirements are that product 1
cannot account for more than 50% of the units produced and that product 3 must
account for at least 20% of the units produced.
a. How many units of each product should be produced
to maximize the total profit contribution? What is the projected weekly profit
associated with your solution?
b. How many hours of production time will be
scheduled on each machine?
c. What is the value of an additional hour of labor?
d.
Assume that
labor capacity can be increased to 120 hours. Would you be interested in using
the additional 20 hours available for this resource? Develop the optimal
product mix assuming the extra hours are made available.
20.
Adirondack
Savings Bank (ASB) has $1 million in new funds that must be allocated to home
loans, personal loans, and automobile loans. The annual rates of return for the
three types of loans are 7% for home loans, 12% for personal loans, and 9% for
automobile loans. The bank’s planning committee has decided that at least 40%
of new funds must be allocated to home loans. In addition, the planning
committee has specified that the amount allocated to personal loans cannot
exceed 60% of the amount allocated to automobile loans.
a.
Formulate a
linear programming model that can be used to determine the amount of finds ASB
should allocate to each type of loan in order to maximize the total annual
return for the new funds.
b. How much should be allocated to each type of
loan? What is the total annual return? What is the annual percentage return?
c. If the interest rate on home loans increased to
9%, would the amount allocated to each type of loan change? Explain.
d. Suppose the total amount of new funds available
was increased by $10,000. What effect would this have on the total annual
return? Explain.
e.
Assume that
ASB has the original $1 million in new funds available and that the planning
committee has agreed to relax the requirement that at least 40% of the new
funds must be allocated to home loans by 1%. How much would the annual return
change? How much would the annual percentage return change?
Quantitative
Analysis BA 452 Homework 3 Questions
21.
Round Tree
Manor is a hotel that provides two types of rooms with three rental classes:
Super Saver, Deluxe, and Business. The profit per night for each type of room
and rental class is as follows:
Room

Super Saver

Deluxe

Business

Type I

$30

$35



Type II

$20

$30

$40





Type I rooms do not have Internet access and are not available for the
Business rental class.
Round Tree’s
management makes a forecast of the demand by rental class for each m=night in
the future. A linear programming model developed to maximize profit is used to
determine how many reservations to accept for each rental class. The demand
forecast for a particular night is 130 rentals in the Super Saver class, 60
rental sin the Deluxe class, and 50 rentals in the Business class. Round Tree
has 100 Type I rooms and 120 Type II rooms.
a.
Use linear
programming to determine how many reservations to accept in each rental class
and how the reservations should be allocated to room types. Is the demand by
any rental class not satisfied? Explain.
b. How many reservations can be accommodated in each
rental class?
c.
Management
is considering offering a free breakfast to anyone upgrading from a Super Saver
reservation to Deluxe class. If the cost of the breakfast to Round Tree is $5.
Should this incentive be offered?
d.
With a
little work, an unused office area could be converted to a rental room. If the
conversion cost is the same for both types of rooms, would you recommend
converting the office to a Type I or a Type II room? Why?
e.
Could the
linear programming model be modified to plan for the allocation of rental
demand for the next night? What information would be needed and how would the
model change?
Quantitative
Analysis BA 452 Homework 3 Questions
22.
Industrial Design has been awarded a contract
design label for a new wine produced by Lake View Winery. The company estimates
that 150 hours will be required to complete the project. The firm’s three
graphics designers available for assignment to this project are Lisa, a senior
designer and team leader; David, a senior designer; and Sarah, a junior
designer. Because Lisa has worked on several projects for Lake View Winery,
management specified that Lisa must be assigned at least 40% of the total
number of hours assigned to the two senior designers. To provide
labeldesigning experience for Sarah, Sarah must be assigned at least 15% of
the total project time. However, the total number of hours assigned to Sarah
must not exceed 25% of the total number of hours assigned to the two senior
designers. Due to other project commitments, Lisa has a maximum of 50 hours
available to work on this project. Hourly wage rates are $30 for Lisa, $25 for
David, and $18 for Sarah.
a. Formulate a linear program that can be used to
determine the number of hours each graphic designer should be assigned to the
project in order to minimize total cost.
b.
How many
hours should each graphic designer be assigned to the project? What is the
total cost?
c.
Suppose
Lisa could be assigned more than 50 hours. What effect would this have on the
optimal solution? Explain.
d. If Sarah were not required to work a minimum
number of hours on this project, would the optimal solution change? Explain.
23.
Vollmer
Manufacturing makes three components for sale to refrigeration companies. The
components are processed on two machines: a sharper and a grinder. The times
(in minutes) required on each machine are as follows:

Machine


Component

Sharper

Grinder

1

6

4

2

4

5

3

4

2




The sharper is available for 120 hours, and the grinder is available
for 110 hours. No more than 200 units of component 3 can be sold, but up to
1000 units of each of the other components can be sold. In fact, the company
already has orders for 600 units of component 1 that must be satisfied. The
profit contributions for components 1,2, and 3 are $8, $6, and $9,
respectively.
a. Formulate and solve for the recommended
production quantities.
b.
What are
the objective coefficient ranges for the three components? Interpret these
ranges for company management.
c. What are the righthandside ranges? Interpret
these ranges for company management.
d. If more time could be made available on the
grinder, how much would it be worth?
e. If more units of component 3 can be sold by
reducing the sales price by $4, should the company reduce the price?
Quantitative
Analysis BA 452 Homework 3 Questions
24.
National
Insurance Associates carries an investment portfolio of stocks, bonds, and
other investment alternatives. Currently $200,000 of funds are available and
must be considered fro new investment opportunities. The four stock options
National is considering and the relevant financial data are as follows:


Stock




A

B

C

D

Price per share

$100

$50

$80

$40

Annual rate of return

0.12

0.08

0.06

0.10

Risk measure per dollar

0.10

0.07

0.05

0.08

invested










The risk measure indicates the relative
uncertainty associated with the stock in terms of it realizing the projected
annual return; higher values indicate greater risk. The risk measures are
provided by the firm’s top financial advisor.
National’s top
management has stipulated the following investment guidelines: The annual rate
of return for the portfolio must be at least 9% and o one stock can account of
more than 50% of the total dollar investment.
a. Use linear programming to develop an investment
portfolio that minimizes risk.
b.
If the firm
ignores risk and uses a maximum returnoninvestment strategy, what is the
investment portfolio?
c.
What is the
dollar difference between the portfolios in parts (a) and (b)? Why might the
company prefer the solution developed in part (a)?
Quantitative
Analysis BA 452 Homework 3 Questions
25.
Georgia
Cabinets manufactures kitchen cabinets that are sold to local dealers
throughout the Southeast. Because of a large backlog of orders for oak and
cherry cabinets, the company decided to contract with three smaller
cabinetmakers to do the final finishing operation. For the three cabinetmakers,
the number of hours required to complete all the oak cabinets, the number of
hours required to complete all the cherry cabinets, the number of hours
available for the final finishing operation, and the cost per hour to perform
the work are shown here.

Cabinetmaker 1

Cabinetmaker 2

Cabinetmaker 3

Hours required to

50

42

30

complete all the




oak cabinets




Hours required to

60

48

35

complete all the




cherry cabinets




Hours available

40

30

35

Cost per hour

$36

$42

$55





For example, Cabinetmaker 1 estimates it will
take 50 hours to complete all the oak cabinets and 60 hours to complete all the
cherry cabinets. However, Cabinetmaker 1 only has 40 hours available for the
final finishing operations. Thus, Cabinetmaker 1 can only complete 40/50 =
0.80, or 80%, of the oak cabinets if it worked only on oak cabinets. Similarly,
Cabinetmaker 1 can only complete 40/60 = 0.67, or 67%, of the cherry cabinets
if it worked only on cherry cabinets.
a.
Formulate a
linear programming model that can be used to determine the percentage of the
oak cabinets and the percentage of the cherry cabinets that should be given to
each of the three cabinetmakers in order to minimize the total cost of
completing both projects.
b.
Solve the
model formulated in part (a). What percentage of the oak cabinets and what
percentage of the cherry cabinets should be assigned to each cabinetmaker? What
is the total cost of completing both projects?
c. If Cabinetmaker 1 has additional hours available,
would the optimal solution change? Explain.
d.
If
Cabinetmaker 2 has additional hours available, would the optimal solution
change? Explain.
e.
Suppose
Cabinetmaker 2 reduced its cost to $38 per hour. What effect would this change
have on the optimal solution? Explain.
Quantitative
Analysis BA 452 Homework 3 Questions
26.
Benson
Electronics manufactures three components used to produce cell telephones and
other communication devices. In a given production period, demand for the three
components may exceed Benson’s manufacturing capacity. Tin this case, the
company meets demand by purchasing the components from another manufacturer at an
increased cost per unit. Benson’s manufacturing cost per unit and purchasing
cost per unit for the three components are as follows:
Source

Component
1

Component
2

Component 3

Manufacture

$4.50

$5.00

$2.75

Purchase

$6.50

$8.80

$7.00





Manufacturing times in minutes per unit for Benson’s three departments
are as follows:
Department

Component 1

Component 2

Component
3

Production

2

3

4

Assembly

1

1.5

3

Testing
& Packaging

1.5

2

5





For instance, each unit for component 1 that Benson manufactures
requires 2 minutes of production time, 1 minute of assembly time, and 1.5
minutes of testing and packaging time. For the next production period, Benson
has capacities of 360 hours in the production department, 250 hours in the
assembly department, and 300 hours in the testing and packaging department.
a.
Formulate a
linear programming model that can be used to determine how many units of each
component to manufacture and how many units of each component to purchase.
Assume that component demands that must be satisfied are 6000 units for
component 1, 4000 units for component 2, and 3500 units for component 3. The
objective is to minimize the total manufacturing and purchasing costs.
b.
What is the
optimal solution? How many units of each component should be manufactured and
how many units of each component should be purchased?
c. Which departments are limiting Benson’s
manufacturing quantities? Use the dual value to determine the value of an extra
hour in each of these departments.
d.
Suppose
that Benson had to obtain one additional unit of component 2. Discuss what the
dual value for the component 2 constraints tells us about the cost to obtain
the additional unit.