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

costs $8,000.

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

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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

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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.

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

risk.

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

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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

week.

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

month possible.

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

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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.

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

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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

problem.

c. Solve the linear program to determine the minimum cost shipping schedule

for the problem.