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Linear Program - Margaret Black's family

Margaret Black's family owns five parcels of farmland broken into a southeast sector, north sector, northwest sector, west sector, and southwest sector. Margaret is involved primarily in growing wheat, alfalfa, and barley crops and in currently preparing her production plan for next year. The Pennsylvania Water Authority has just announced its yearly water allotment, with the Black 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 2000 3200
North 2300 3400
Northwest 600 800
West 1100 500
Southwest 500 600

Each of Margarets 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

Margaret'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 $50 per ton. 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.
A. Formulate Margaret's production plan
B. What should the crop plan be, and what profit will it yield?
C. The Water Authority informs Margaret that for a special fee of $6,000 this year, her farm will qualify for an additional allotment of 600 acre-feet of water. How should she respond?
Create an Excel spreadsheet for this problem

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Solution Preview

A. Formulate Margaret's production plan

Please see the attached excel file - blue cells are given data, red cells are decision variables, constraints and objective function is ...