Question

A craftsman builds two kinds of birdhouses, one for wrens and one for bluebirds. Each wren birdhouse takes 3 hours of labor and 4 units of lumber. Each bluebird house requires 2 hours of labor and 10 units of lumber. The craftsman has available 80 hours of labor and 100 units of lumber, and he wants to build at least 6 wren houses. Wren houses profit $8 each and bluebird houses profit $16 each. How many of each kind of birdhouses should be built in order to maximize total profit? Formulate this as a linear programming problem (i.e., DO NOT solve it.)

Answer 1

Obj Function, Max P=8w+16b

Subject to the Constraints

3w+2l≤80

4w+10b≤100

w≥6

b>0

Explanation:

Let the number of Wren's birdhouse built=w

Let the number of bluebirds birdhouse built=b

Constraint:Labour

Each wren birdhouse takes 3 hours of labor.

Each bluebird house requires 2 hours of labor.

The craftsman has available 80 hours of labor

Formulated as an inequality:

3w+2l≤80

Constraint: Lumber

Each wren birdhouse requires 4 units of lumber.

Each bluebird birdhouse 10 units of lumber.

The craftsman has available 100 units of lumber

Formulated as an inequality:

4w+10b≤100

The craftsman wants to build at least 6 wren houses.

w≥6

Since he builds the two kinds of house, b>0 and w>0.

Objective Function

Wren houses profit $8 each and bluebird houses profit $16 each.

The Objective of the function is to make profit, so we maximize.

Obj Function, Max P=8w+16b

The linear programming problem is therefore stated below:

Obj Function, Max P=8w+16b

Subject to the Constraints

3w+2l≤80

4w+10b≤100

w≥6

b>0