A farmer in Georgia has a 100-acre farm on which to plant watermelons and cantaloupes. Every acre planted with watermelons requires 50 gallons of water per day and must be prepared for planting with 20 pounds of fertilizer. Every acre planted with cantaloupes requires 75 gallons of water per day and must be prepared for planting with 15 pounds of fertilizer. The farmer estimates that it will take 2 hours of labor to harvest each acre planted with watermelons and 2.5 hours to harvest each acre planted with cantaloupes. He believes that watermelons will sell for about $3 each, and cantaloupes will sell for about $1 each. Every acre planted with watermelons is expected to yield 90 salable units. Every acre planted with cantaloupes is expected to yield 300 salable units. The farmer can pump about 6,000 gallons of water per day for irrigation purposes from a shallow well. He can buy as much fertilizer as he needs at a cost of $10 per 50- pound bag. Finally, the farmer can hire laborers to harvest the fields at a rate of $5 per hour. The farmer sells all the watermelons and cantaloupes he produces and would like to determine how many acres of each crop to plant in order to maximize profits.
Formulate an LP model for this problem.
Implement the model in an Excel spreadsheet. What is the optimal solution?
Consider the following prices for the watermelons: (i) $2.75/unit, (ii) $2.50/unit, (iii) $2.25/unit, (iv) $2.00/unit. Which price points (if any) would result in the farmer maintain his planting strategy? Which price points (if any) should the farmer respond by changing his planting strategy? Please explain your reasoning.
Suppose the farmer can lease up to 20 acres of land from a neighboring farm to plant additional crops. How many acres should the farmer lease and what is the maximum amount he should pay to lease each acre?