A farmer has 50 ha of land on which to plant maize and beans. He has a workforce of 150 laborers and it takes 4 laborers to work on 1 ha of maize and 2 laborers to work on 1 ha of beans . He has a capital of $4500 and 1 ha of maize requires $50 to cultivate while 1 ha of beans requires $100 to cultivate. Suppose that the farmer wishes to maximize profit and the profit per ha is $30 for maize and $40 for beans. Set up a linear programming problem and solve it graphically
The linear programming model for this problem will be as follows:
Variety Costs per hectare Profit Hectares Labour
Maize 50 30 1 4
Beans 100 40 1 2
Let the total cost for growing maize = X (in hectares)
Let the total cost for growing beans = Y (in hectares)
X and Y are decision variables
Let the objective function be Z
Max Z = 30X + 40Y
Writing the constraints:
Total costs = 150*50 = 7500
50X + 100Y ≤ 7500
4X + 2Y ≤ 150
X + Y ≤ 50
The non-negativity constrain:
X ≥ 0, Y ≥ 0
So, checking value of at these points:
So, profit will be maximum when X=0, Y= 2000
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