Answer to Question #202020 in Microeconomics for Athi

Question

 

A farmer has 50 hectares of land and 120 hours of labour. He can produce either maize or soyabeans. It takes an acre of land and 3 hours of labour to produce an acre of maize. A hectare of maize costs R154,00 to produce. It yields 7,5 tonnes per hectare, and as a consequence it generates a net profit of R56,00 per hectare. On the other hand, if a farmer produces soyabeans, it takes an acre of land and 2 hours of labour to produce an acre of soyabeans. It costs R133,00 to produce an acre of soyabeans. An acre of soyabeans yields 30 tonnes and generates a profit of R47,00 per acre. Using LP, determine the combination of activities which will maximize profits for the farmer.                                  

Formulate a linear programming model for this problem

Answer 1

Solution:

The linear programming model for this problem will be as follows:

Variety                     Costs per hectare  profit    hectares      labor        Yields (Tonnes)

Maize                       154                        56            1                3               7.5

Soyabeans                133                        47            1                2              30

 

Let the total cost for growing maize = X (in hectares)

Let the total cost for growing soyabeans = Y (in hectares)

X and Y are decision variables

Let the objective function be Z

Max Z = 56X + 47Y

Writing the constraints:

Total costs = 120*50 = 6,000

154X + 133Y ≤ 6,000

3X + 2Y ≤ 120

X + Y ≤ 50

 

The non-negativity constrain:

X ≥ 0, Y ≥ 0