1.1 Eastman publishing Company is considering publishing a paperback textbook on
spreadsheet applications for business. The fixed cost of manuscript preparation, textbook design, and production setup is estimated to be $80,000. Variable production
and material costs are estimated to be $3 per book. Demand over the life of the book
is estimated to be 4,000 copies. The publisher plans to sell the text to college and
university bookstores for $20 each.
a. What is the breakeven point?
b. What profit or loss can be anticipated with a demand of 4,000 copies?
c. With a demand of 4,000 copies, what is the minimum price per copy that
the publisher must charge to break even?
1.2 As part of a quality improvement initiative, Consolidated Electronics employees
complete a three-day training program on teaming and a two-day training program
on problem solving. The manager of quality improvement has requested that at
least 8 training programs on teaming and at least 10 training program on problem
solving be offered during the next six months. In addition, senior-level management
has specified that at least 25 training programs must be offered during this period.
Consolidated Electronics uses a consultant to teach the training programs. During
the next quarter, the consultant has 84 days of training time available. Each training
program on teaming costs $10,000 and each training program on problem solving
a. Formulate a linear programming model that can be used to determine the
number of training programs on teaming and the number of training programs on
problem solving that should be offered in order to minimize total cost.
b. Graph the feasible region.
c. Determine the coordinates of each extreme point.
d. Solve for the minimum cost solution.
1.3 Creative Sports Design (CSD) manufactures a standard-size racket and an oversize
racket. The firm’s rackets are extremely light due to the use of a magnesium-graphite
alloy. Each standard-size racket uses 0.125 kilograms of the alloy and each oversize
racket uses 0.4 kilograms; over the next two-week production period only 80 kilograms
of the alloy are available. Each standard-size racket uses 10 minutes of manufacturing
time and each oversize racket uses 12 minutes. Also, 40 hours of manufacturing time
are available each week. The profit contributions are $10 for each standard-size
racket and $15 for each oversize racket. How many rackets of each type should CSD
manufacture over the next two weeks to maximize the total profit contribution?
a. Define decision variables and formulate the problem.
b. Solve the problem using the graphical method.
1.4 Management of High Tech Services (HTS) would like to develop a model that
will help allocate their technician’s time between service calls to regular contract customers and new customers. A maximum of 80 hours of technician time is available
over the two-week planning period. To satisfy cash flow requirements, at least $800 in
revenue (per technician) must be generated during the two-week period. Technician
time for regular customers generates $25 per hour. However, technician time for new
customers only generates an average of $8 per hour. To ensure that new customer
contracts are being maintained, the technician time spent on new customer contracts
must be at least 60% of the time spent on regular customer contracts. Given these
revenue and policy requirements, HTS would like to determine how to allocate technician time between regular customers and new customers so that the total number
of customers contracted during the two-week period will be maximized. Technicians
require an average of 50 minutes for each regular customer contract and 1 hour for
each new customer contract.
a. Develop a linear programming model for the problem.
b. Find the optimal solution via Excel.
1.5 Industrial Designs has been awarded a contract to design a 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
label-designing experience for Sarah, Sarah must be assigned at least 15% of the total
project time. However, the 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
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.
1.6 National Insurance Associated carries an investment portfolio of stocks, bonds,
and other investment alternatives. Currently $200,000 of funds are available and must
be considered for new investment opportunities. The four stock options National is
considering and the relevant financial data are as in Table 1.
Table 1: Problem 1.6
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 invested 0.10 0.07 0.05 0.08
National’s top management has stipulated the following investment guidelines:
The annual rate of return for the portfolio must be at least 9% and no one stock can
account for more than 50% of the total dollar investment.
a. Use linear programming to develop an investment portfolio that minimizes
b. If the firm ignores risk and uses a maximum return-on-investment strategy,
what is the investment portfolio?
1.7 Hawkins Manufacturing Company produces connecting rods for 4- and 6-cylinder
automobile engines using the same production line. The cost required to set up
the production line to produce the 4-cylinder connecting rod is $2,000, and the cost
required to set up the production for the 6-cylinder connecting rod is $3,500. Manufacturing costs are $15 for each 4-cylinder connecting rod and $18 for each 6-cylinder
connecting rod. Hawkins makes a decision at the end of each week as to which product will be manufactured the following week. If a production changeover is necessary
from one week to the next, the weekend is used to reconfigure the production line.
Once the line has been set up, the weekly production capacities are 6,000 6-cylinder
connecting rods and 8,000 4-cylinder connecting rods. Let
x4 = the number of 4-cylinder connecting rods produced next week,
x6 = the number of 6-cylinder connecting rods produced next week,
s4 = 1 if the production line is set up to produce the 4-cylinder connecting
rods and =0 otherwise,
s6 = 1 if the production line is set up to produce the 6-cylinder connecting
rods and =0 otherwise.
a. Using the decision variables x4 and s4, write a constraint that limits next
week’s production of the 4-cylinder connecting rods to either 0 or 8,000 units.
b. Using the decision variables x6 and s6, write a constraint that limits next
week’s production of the 6-cylinder connecting rods to either 0 or 6,000 units.
c. Write three constraints that, taken together, limit the production of connecting rods for next week.
d. Write an objective function for minimizing the cost of production for next
1.8 EZ-Windows, Inc. manufacturers replacement windows for the home remodeling
business. In January, the company produces 15,000 windows and ended the month
with 9,000 windows in inventory. EZ-Windows’ management team would like to develop a production schedule for the next three moths. A smooth production schedule
is obviously desirable because it maintains the current workforce and provides a similar month-to-month operation. However, given the sales forecasts, the production
capacities, and the storage capabilities as shown in Table 2, the management team
does not think a smooth production schedule with the same production quantity each
The company’s cost accounting department estimates that increasing production
by one window from one month to the next will increase total costs by $1.00 for each
Table 2: Problem 1.8
February March April
Sales forecast 15,000 16,500 20,000
Production capacity 14,000 14,000 18,000
Storage capacity 6,000 6,000 6,000
unit increase in the production level. In addition, decreasing production by one unit
from one month to the next will increase total costs by $0.65 for each unit decrease in
the production level. Ignoring production and inventory carrying costs, formulate a
linear programming model that will minimize the cost of changing production levels
while still satisfying the monthly sales forecasts.
1.9 A local television station plans to drop three Friday evening programs at the end
of the season. Steve Botuchis, the station manager, developed a list of three potential
replacement programs. Estimates of the advertising revenue (in dollars) that can
be expected for each of the new programs in the three vacated time slots are as in
Table 3: Problem 1.9
5–6PM 6–7 PM 7–8 PM
Home Improvement 5000 3000 6000
World News 7500 8000 7000
Hollywood Briefings 7000 8000 3000
Mr. Botuchis asked you to find the assignment of programs to time slots that will
maximize total advertising revenue.
1.10 Adirondack Paper Mills, Inc. operates paper plants in Augusta, Maine, and
Tupper Lake, New York. Warehouse facilities are located in Albany, New York,
and Portsmouth, New Hampshire. Distributors are located in Boston, New York,
and Philadelphia. The Augusta plant has a capacity of 300 units, and the Tupper
Lake plant has a capacity of 100 units. Boston has a demand of 150 units, New
York has a demand of 100 units, and Philadelphia has a demand of 150 units. The
unit transportation costs (in dollars) for shipments from the two plants to the two
warehouses are presented in Table 4 and those from the two warehouses to the three
distributors are presented in Table 5.
Table 4: Problem 1.10a
Plant/Warehouse Albany Portsmouth
Augusta 7 5
Tupper Lake 3 4
Table 5: Problem 1.10b
Warehouse/Distributor Boston New York Philadelphia
Albany 8 5 7
Portsmouth 5 6 10
a. Draw the network representation of the Adirondack Paper Mills problem.
b. Formulate the Adirondack Paper Mills problem as a linear programming
c. Solve the linear program to determine the minimum cost shipping schedule
for the problem.