Question 1171134: A small firm that assembles computers is about to start production of two new types of personal computers. Each type will require assembly time, inspection time, and storage space. The amount of each of these resources that can be devoted to the production of the computers is limited. The manager of the firm would like to determine the quantity of each computer to produce in order to maximize the profit generated by sales of these computers. In order to develop a suitable model of the problem, the manager has met with design and manufacturing personnel. As a result of those meetings, the manager has obtained the following information:
Type 1 Type 2
Profit per unit $60 $50
Assembly time per unit 4 hours 10 hours
Inspection time per unit 2 hours 1 hour
Storage space per unit 3 cubic feet 3 cubic feet
The manager also has acquired information on the availability of company resources. These daily amounts are:
resources available each day
Assembly time 100 working hours
Inspection time 22 working hours
Storage space 39 cubic feet
The manager met with the firm’s marketing manager and learned that demand for the computers was such that whatever combination of these two types of computers is produced, all of the output can be sold.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! x = number of type 1 computers
y = number of type 2 computers
your objective function is:
60x + 50y = profit.
your constraint inequalities are:
4x + 10y <= 100 for assembly time per unit.
2x + y <= 22 for inspection time per unit.
3x + 3y <= 39 for storage space per unit.
x >= 0 for number of type 1 computers can't be negative.
y >= 0 for number of type 2 computers can't be negative.
in the desmos.com graphing software, you would graph the opposite of the constraint inequalities.
specifically, you would graph the following inequalities.
4x + 10y >= 100 for assembly time per unit.
2x + y >= 22 for inspection time per unit.
3x + 3y >= 39 for storage space per unit.
x <= 0
y <= 0
the area on the graph that is not shaded is your region of feasibility.
the corner points of the region of feasibility are where your solution lies.
the graph looks like this.
you evaluate the objective function at each of these corner points to find the corner points that gives you the maximum profit.
i found that to be at (x,y) = (9,4), where the profit was 9 * 60 + 4 * 50 = 740
the profit at the other corner points was:
500 for (0,10), 700 for (5,8), 660 for (11,0).
all the constraints needed to be met.
at (9,4):
4x + 10y <= 100 became 36 + 40 <= 100 which became 76 <= 100 which is true.
2x + y <= 22 became 18 + 4 <= 22 which became 22 <= 22 which is true.
3x + 3y <= 39 became 27 + 12 <= 39 which became 39 <= 39 which is true.
all the constraints were met, so the solution is cofirmed to be good.
the solution is that the maximum profit can be achieved when the number of type 1 computers built and sold is 9 and the number of type 2 computers built and sold is 4.
the desmos.com graphing software can be found at https://www.desmos.com/calculator
|
|
|
