Question

A farmer is planning his spring planting. He has 20 acres on which he can plant a combination of Corn, Pumpkins and Beans. He wants to maximize his profit but there is a limited demand for each crop. Each crop also requires fertilizer and irrigation water which are in short supply. There are only 50 acre ft of irrigation available and only 8,000 pounds/acre of fertilizer available. The following table summarizes the data for the problem.

Crop	Profit per Acre ($)	Yield per Acre (lb)	Maximum Demand (lb)	Irrigation (acre ft)	Fertilizer (pounds/acre)
Corn	2,100	21,000	200,000	2	500
Pumpkin	900	10,000	180,000	3	400
Beans	1,050	3,500	80,000	1	300

Formulate the LP for this problem.

Answer 1

Let corn be planted in x acres, pumpkin in y acres and beans in z acres

Objective Function is to maximize profit

Max Z = 2100x + 900y + 1050z

Constraints -

x + y + z <= 20 .... acres available

21000x <= 200000 .... demand for corn

10000y <= 180000 .... demand for pumpkin

3500z <= 80000 .... demand for beans

2x + 3y + z <= 50 ... irrigation available

500x + 400y + 300z <= 8000 ... fertilizer available

x, y, z >= 0

Solving this using excel solver -

Initially assuming x = y = z = 1 (in green cell). this will be optimized to give the feasible solution to maximize profit

run the solver to get -

Hence, maximum profit of $30450 is achieved if corn is planted in 9 acres and beans is planed in 11 acres, no pumpkin is planted.