Answer to Question #242472 in Operations Research for opr

Question

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

Answer 1

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 $Z=30X+40Y$ at these $4$ points:

$At \ (0,0): \\ Z=0$

$At \ (0,50): Z=0+2000=2000\\ At \ (37.5,0): Z=1125+0=1125\\ At \ (25,25): Z=750+1000=1750$

So, profit will be maximum when X=0, Y= 2000