Problema Solution
Lay manufacturing makes TV sets. It produces a bargain one that sells for $100 profit and a deluxe that has a $150 profit.
Assembly line. The bargain takes 3 hours each with deluxe taking 5 hours
Cabinet shop. The bargain takes 1 hour for each bargain and 3 hours each for deluxe
Testing takes 2 hours each for both models.
On this production run I have 3900 work hours for assembly.
2100 for cabinet
2200 for testing.
How many of each type should I produce to make the maximum profit?
Answer provided by our tutors
let
x = the number of bargain one, x>=0
y = the number of delux, y>=0
Assembly line. The bargain takes 3 hours each with deluxe taking 5 hours and I have 3900 work hours for assembly
3x + 5y <= 3900
Cabinet shop. The bargain takes 1 hour for each bargain and 3 hours each for deluxe and I have 2100 work hour for cabinet
x + 3y <= 2100
Testing. Testing takes 2 hours each for both models and I have 2200 work hours for testing
2x + 2y <= 2200
the profit is determined by the objective function
F(x , y) = 100x + 150y
lets draw the graph of the system of inequalities
x >= 0
y >= 0
3x + 5y <= 3900
x + 3y <= 2100
2x + 2y <= 2200
click here to see the graph
look at the graph and find the notice the corner points!
there are 5 corner points and we calculate F(x ,y) for each point to find the maximum value (the biggest value for F):
1. Corner point (300,600) gives F(300, 600) = $120,000
the solution of the system
3x + 5y = 3900
x + 3y = 2100
click here to see the step by step solution
gives the corner point (300,600) and
F(300, 600) = 100*300 + 150*600 = $120,000
2. Corner point (800, 300) gives F(800, 300) = $125,000
the solution of the system
3x + 5y = 3900
2x + 2y = 2200
click here to see the step by step solution
gives the corner point (800, 300) and
F(800, 300) = 100*800 + 150*300 = $125,000
3. Corner point (600, 500) gives F(600, 500) = $135,000
the solution of the system
x + 3y = 2100
2x + 2y = 2200
click here to see the step by step solution
gives the corner point (600, 500) and
F(600, 500) = 100*600 + 150*500 = $135,000
4. Corner point (0, 700) gives F(0, 700) = $105,000
the solution of the system
x + 3y = 2100
x = 0
gives the corner point (0, 700) and
F(0, 700) = 150*700 = 105,000
5. Corner point (1100, 0) gives F(1100, 0) = $110,000
the solution of the system
2x + 2y = 2200
y = 0
gives the corner point (1100, 0) and
F(1100, 0) = 100*1100 = 110,000
By comparing the values of the objective function for all 5 corner points we conclude that for
Corner point (600, 500) gives maximum F(600, 500) = $135,000 thus
To maximize the profit the company should produce 600 bargains and 500 delux TV sets.