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.