Margaret Young's family owns five parcels of farmland broken into a southeast sector, north sector, northwest sector, west sector, and southwest sector. Young is involved primarily in growing wheat, alfalfa, and barley crops and is currently preparing her production plan for next year. The Pennsylvania Water Authority has just announced its yearly water allotment, with the Young farm receiving 7,400 acre-feet. Each parcel can only tolerate a specified amount of irrigation per growing season as specified in the following table:
Parcel
Area (Acres) Water Irrigation Limit (Acre-feet)
Southeast 2,000 3,200
North 2,300 3,400
Northwest 600 800
West 1,100 500
Southwest 500 600
Each of Young's crops needs a minimum amount of water per acre, and there is a projected limit on sales of each crop. Crop data follow.
Crop Maximum Sales Water Needed Per Acre (Acre-feet)
Wheat 110,000 bushels 1.6
Alfalfa 1,800 tons 2.9
Barley 2,200 tons 3.5
One acre of land yields an average of 1.5 tons of alfalfa and 2.2 tons of barley. The wheat yield is approximately 50 bushels per acre. Young's best estimate is that she can sell wheat at a net profit of $2 per bushel, alfalfa at $40 per ton, and barley at $40 per ton. The wheat yield is approximately 50 bushels per acre.
Write the objective function and constraints for the linear programming problem if we want to maximize the profit?