how many of each type of house to the company manufacture per month to obtain the best returns?

Question

A woodworking company handcrafts two types of houses one is large and the other is small. Both types of houses go through two technicians; first a shaper and then a painter. The large house requires two hours of the shaper’s time and one hour on the painter’s time. The small house requires one hour of the shaper’s time and two hours on the painter’s time. The shaper has 104 hours and the painter has 76 hours of time available this month. Profit on the large house is $6 and on the small house is $11. Assuming all of the houses that are produced can be sold. how many of each type of house to the company manufacture per month to obtain the best returns?

Answer 1

This is a linear programming problem, and the maximum (or minimum) occurs at one of the vertices of a convex polygonal region formed by the constraints.

 

Step 1. Identify variables and the function to be optimized

Let x be the number of large houses to be produced and

    y be the number of small houses to be produced.

 

Then the profit that the company can generate this month is

P(x, y) = 6x + 11y {Profit on the large house is $6 and on the small house is $11}

 

Step 2. Identify all constraint inequalities

x ≥ 0; y ≥ 0

2x + y ≤ 104 {The shaper has 104 hours available this month, 2 hrs/large house, 1 hr/small house}

x + 2y ≤  76 {The painter has 76 hours available this month, 1 hr/large house, 2 hrs/small house}

 

Step 3. Find the coordinates of the vertices.

Vertices are (0,0), (52, 0), (0, 38) and (44, 16). 

The last one is the solution of

2x + y = 104

x + 2y = 76

 

Step 4. Calculate the values of the function at the vertices

Evaluate P(x, y) = 6x + 11y for each of the vertices (0,0), (52, 0), (0, 38) and (44, 16), and you will find the (x, y) that maximizes the profit.