Problems:
Benson Electronics manufactures a number of components and products for a variety of commercial applications. Each product places different demands on the various departments within the company. In one instance, Benson manufactures three components used to produce cellular telephones and other communication devices. In a given production period, demand for the three components may exceed Benson's overall 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
|
$3.50
|
$6.00
|
$3.75
|
|
Purchase
|
$5.50
|
$9.80
|
$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 and Packaging
|
1.5
|
2
|
5
|
For example, 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. Component demands that must be satisfied are 6000 units for component 1, 4000 units for component 2, and 3500 units for component 3.
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.
b. Determine the optimal plan that minimizes the total manufacturing and purchasing costs, including the number units of each component to be manufactured and the number units of each component to be purchased.
c. Determine which departments, if any, are limiting Benson's manufacturing quantities. Use the dual prices to determine the value of an extra hour in each of these departments. Determine the level of unused capacity, if any, resulting from the optimal schedule for each of the associated departments.
d. Suppose that Benson needs to obtain one additional unit of component 1. Interpret what the dual price for the component 2 constraint illustrates with respect to the cost to obtain the additional unit.