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Question 481375: The Highlands Organic Farm in Cameron Highlands has a 50-acre farm on which to plant strawberries and tomatoes. The farm has 300 hours of labour per week and 800 tons of fertilizer available, and has contracted for shipping space for a maximum of 26 acres worth of strawberries and 37 acres worth of tomatoes. An acre of strawberries requires 10 hours of labour and 8 tons of fertilizer, whereas an acre of tomatoes requires 3 hours of labour and 20 tons of fertilizer. The profit from an acre of strawberries is RM400, and the profit from an acre of tomatoes is RM300. The farm management wants to know the number of acres of strawberries and tomatoes to plant to maximize profit.
a) Formulate the above as a linear programming problem
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! let x = number of acres of strawberries
let y = number of acres of tomatoes
your limitation equations are:
land:
x + y <= 50
labor:
10x + 3y <= 300
fertilizer
8x + 20y <= 800
shipping space
x <= 26
y <= 37
solve all these equations for y so you can graph them.
you get:
land:
y <= 50-x
labor:
3y <= 300-10x becomes:
y <= 100 - 10x/3
fertilizer
20y <= 800-8x becomes:
y <= 40-8x/10 which becomes:
y <= 40 - 4x/5
shipping space equations stay the same at:
x <= 26
y <= 37
there are 2 other restrictions that are there, but not shown explicitly.
those are:
x >= 0
y >= 0
your graph would look like this:
a more distant view would look like this:
linear programming theory states that the maximum / minimum points would be at the intersection points and never in between.
since all of your constraints have to be less than any line on the graph other than the x-axis and the y-axis, then your region of compatibility will be from x = 0 and y = 0 to any of the 7 intersection points.
those intersection points are:
x y profit
0 0 0
0 37 11,100
7.5 37 14,100
16.67 33.33 16,666.67
21.43 28.57 17,142.86
26 0 10,400
27 13.33 14,400
the profit equation is P = 400x + 300y.
you solve the profit equation at each of the intersection points and then choose the intersection point that provides you with the maximum profit.
the maximum profit point is when x = 21.43 and y = 28.57
that would be 21.43 acres of strawberries and 28.57 acres of tomatoes.
here's a reference that explains the process.
http://www.purplemath.com/modules/linprog.htm
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