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 profit contribution, answer the following:
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.
Standard Grade Professional Grade
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.
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?
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.