Question

A factory can produce two products, x and y, with a profit approximated by P=14x+22y-900. The production of y must exceed the production of x by at least 100 units. Moreover, production levels are limited by the formula x+2y<1400.

a) Identify the vertices of the feasible region.
b) What production levels yield the maximum profit, and what is the maximum profit?

Answer 1

a) The vertices of the feasible region are (0,100) (0,700) (400,500)

The minimum profit is at (0,100) and the maximum profit is at (400,500)

Explanation:

P=14x+22y-900 where p is profit

y > x +100 y must exceed the production of x by at least 100 units

x+2y<1400

x>0

y>0

We cannot produce negative quantities

Substitute y = x+100 into x+2y <1400

x+2(x+100) < 1400

x+2x+200 <1400

3x+200<1400

Subtract 200 from each side

3x<1200

Divide by 3

x<400

y = x+100

y = 400+100

y = 500

(400,500)

y > x +100

when x=0 y > 100

x+2y <1400

0+2y <1400

2y <1400

y <700

When x=0 y = 700

a) The vertices of the feasible region are (0,100) (0,700) (400,500)

b) Maximum and minimum profit occur at the vertices.

P=14x+22y-900

P(0,100) = 14(0) +22(100)-900 =2200-900=1300

P(0,700) = 14(0) +22(700)-900 =15400-900=14500

P(400,500) = 14(400) +22(500)-900 =5600+11000-900=15700

The minimum profit is at (0,100) and the maximum profit is at (400,500)