SOLUTION: "Extremista" produces two products, A and B, which have profits of $9 and $7, respectively. Each unit of product must be processed on two assembly lines, where the required product

Question 1205183: "Extremista" produces two products, A and B, which have profits of $9 and $7, respectively. Each unit of product must be processed on two assembly lines, where the required production times are as follows:
Product Hours/Unit
Line 1 Line 2
A 12 4
B 4 8
Total hours 60 40

a. Formulate a Linear Programing model and determine the optimal mix of products that will
maximize profits. (9 marks)
b. Sketch the graph of the model you have developed. (5 marks)
c. What would be the effect on the optimal solution if the profit for Product B was moved from
$7 to $15. (6 marks)
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
your objective function is 9x + 7y which you want to maximize.

your constraint inequalities are:
12x + 4y <= 60
4x + 8y <= 40

using the desmos.com calculator, you would:
graph the opposite of the inequalities.
the area on the graph not shaded is the region of feasibility.
at least one of the corner points of this region contain he maximum profit.
you find this by evaluating the objective function at each corner point.

this is what the graph looks like.

maximum profit is at the coordinate point of (4,3).
the profit is 4 * 9 + 3 * 7 = 57
all the constraints are satisfied, as they need to be.
12x + 4y = 12 * 4 + 4 * 3 = 60 <= 60
4x + 8y = 4 * 4 + 8 * 3 = 40 <= 40

if the profit for product B was 15 instead of 7, .....
the constraints would remain the same.
the profit for each option:
at (0,5) would become 0 * 9 + 5 * 15 = 75 instead of 0 * 9 + 5 * 7 = 35.
at (4,3) would become 4 * 9 + 3 * 15 = 81 instead of 4 * 9 + 3 * 7 = 57.
at (5,0) would become 5 * 9 + 0 * 15 = 45 same as 5 * 9 + 0 * 7 = 45.
the maximum profit would remain at (4,3).
that means 4 units of product A plus 3 units of product B.
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