```Kelson Sporting Equipment, Inc. makes two different types of
baseball gloves: a regular model and a catcher’s model. The firm
has 900 hours of production time available in its cutting and
sewing department, 300 hours available in its finishing department,
and 100 hours available in its packaging and shipping department.
The production time requirements and profit contribution per glove
are given in the following table.

Production Time (hours)
Model Cutting and Sewing Finishing Packaging and Shipping
Profit/Globe
Regular model 1 1/2 1/8 \$5
Catcher’s model 3/2 1/3 1/4 \$8

Assuming that the company is interested in maximizing the total

a) What is the linear programming model for this problem?
b) Find the optimal solution using the graphical solution procedure
AND Excel. How many gloves of each model should Kelson manufacture?
c) What is the total profit contribution Kelson can earn with the
given production quantities?
d) How many hours of production time will be scheduled in each
department?
e) What is the slack time in each department?

CHAPTER 2, CHAPTER 3

Applied Technology, Inc. (ATI) produces bicycle frames using two
fiberglass materials that improve the strength-to-weight ratio of
the frames. The cost of the standard grade material is \$7.50 per
yard and the cost of the professional grade material is \$9.00 per
yard. The standard and professional grade materials contain
different amounts of fiberglass, carbon fiber, and Kevlar as shown
in the following table.

Fiberglass 84% 58%
Carbon fiber 10% 30%
Kevlar 6% 12%

ATI signed a contract with a bicycle manufacturer to produce a new
frame with a carbon fiber content of at least 20% and a Kevlar
content of not greater than 10%. To meet the required weight
specification, a total of 30 yards of material must be used for
each frame.

a) Formulate a linear program to determine the number of yards of
each grade of fiberglass material that ATI should use in each frame
in order to minimize total cost. Define the decision variables and
indicate the purpose of each constraint.
b) Use the graphical solution procedure to determine the feasible
region. What are the coordinates of the extreme points?
c) Compute the total cost at each extreme point. What is the
optimal solution?

Use the graphical method AND Excel.

CHAPTER 3

Refer to the Kelson Sporting Equipment (problem 1)
USING EXCEL SOLVER

a) Determine and interpret the objective coefficient ranges
b) Determine and interpret the right-hand side ranges
c) How much will the value of the optimal solution improve if 20
extra hours of packaging and shipping time are made available?

CHAPTER 4

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. In 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 of 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 should be purchased?
c) Which departments are limiting Benson’s manufacturing
quantities? Use the shadow price to determine the value of an extra
hour in each of these departments.

HW#1

CHAPTER 2

Problem 1 (2.5 points)

Kelson Sporting Equipment, Inc. makes two different types of
baseball gloves: a regular model and a catcher’s model. The firm
has 900 hours of production time available in its cutting and
sewing department, 300 hours available in its finishing department,
and 100 hours available in its packaging and shipping department.
The production time requirements and profit contribution per glove
are given in the following table.

Production Time (hours)
Model   Cutting and Sewing      Finishing       Packaging and
Shipping  Profit/Globe
Regular model   1       1/2     1/8     \$5
Catcher’s model 3/2     1/3     1/4     \$8

Assuming that the company is interested in maximizing the total

a)     What is the linear programming model for this problem?
b)      Find the optimal solution using the graphical solution
procedure AND Excel. How many gloves of each model should Kelson
manufacture?
c)      What is the total profit contribution Kelson can earn with
the given production quantities?
d)      How many hours of production time will be scheduled in each
department?
e)      What is the slack time in each department?

CHAPTER 2, CHAPTER 3

Problem 2 (2.5 points)

Applied Technology, Inc. (ATI) produces bicycle frames using two
fiberglass materials that improve the strength-to-weight ratio of
the frames. The cost of the standard grade material is \$7.50 per
yard and the cost of the professional grade material is \$9.00 per
yard. The standard and professional grade materials contain
different amounts of fiberglass, carbon fiber, and Kevlar as shown
in the following table.

Fiberglass      84% 58%
Carbon fiber    10% 30%
Kevlar  6%  12%

ATI signed a contract with a bicycle manufacturer to produce a new
frame with a carbon fiber content of at least 20% and a Kevlar
content of not greater than 10%. To meet the required weight
specification, a total of 30 yards of material must be used for
each frame.

a)     Formulate a linear program to determine the number of yards
of each grade of fiberglass material that ATI should use in each
frame in order to minimize total cost. Define the decision
variables and indicate the purpose of each constraint.
b)      Use the graphical solution procedure to determine the
feasible region. What are the coordinates of the extreme points?
c)      Compute the total cost at each extreme point. What is the
optimal solution?

Use the graphical method AND Excel.

CHAPTER 3

Problem 3  (2.5 points)

Refer to the Kelson Sporting Equipment  (problem 1)
USING EXCEL SOLVER

a) Determine and interpret the objective coefficient ranges
b) Determine and interpret the right-hand side ranges
c) How much will the value of the optimal solution improve if 20
extra hours of packaging and shipping time are made available?

CHAPTER 4

Problem 4 (2.5 points)

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. In 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 of 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 should be
purchased?
c)      Which departments are limiting Benson’s manufacturing
quantities? Use the shadow price to determine the value of an extra
hour in each of these departments.
```

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