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 100 pounds of fertilizer in the whole growing season. Every acre planted with cantaloupes requires 75 gallons of water per day and must be prepared for planting with 80 pounds of fertilizer in the whole growing season. 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 $3 each, and cantaloupes will sell for $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 $0.5 per pound. Finally, the farmer can hire laborers to harvest the fields at a rate of $15 per hour. If the farmer sells all the watermelons and cantaloupes he produces, how many acres of each crop should the farmer plant to maximize profits?
a. Formulate the LP model and generate its sensitivity report.
b. If the unit price of watermelon increases from $3 to $4, is our current solution still optimal? Use your calculation to explain.
c. If the labor cost per hour increases from $15 to $20, (unit price of water melon is still $3), is our current solution still optimal? Use your calculation to explain. (Hint: you may need to apply the 100% rule we introduced in class.)
d. Assume the unit price of watermelon and labor costs remain unchanged. The farmer is now considering planting muskmelons on the farm. If one acre planted with muskmelons requires 40 gallons of water per day and must be prepared for planting with 70 pounds of fertilizer. Each acre can yield 150 salable units at $2/unit and it takes 3 hours of labor to harvest. Would it be profitable to plant muskmelons? Use your calculation to explain. (Only use the information from your existing sensitivity report. Do not formulate a new model in the spreadsheet.)