Question

Question 9 (AGRICULTURAL PRODUCTION PLANNING) Margaret Kowalchuk's family owns five parcels of farmland broken into a southeast sector, north sector, northwest sector, west sector, and southwest sector. Kowalchuk 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 Kowalchuk farm receiving 7400 acre-feet. Each parcel can only tolerate a certain amount of irrigation per growing season, as specified in the following table: WATER IRRIGATION LIMIT (ACRE-FEET) PARCEL AREA (ACRES) Southeast North Northwest West Southwest 2000 2300 600 1100 500 3200 3400 800 500 600 1 Each of Kowalchuk's crops needs a minimum amount of water per acre, and there is a projected limit on sales of each crop. Crop data follow: WATER NEEDED PER ACRE (ACRE-FEET) CROP MAXIMUM SALES Wheat Alfalfa Barley 110 000 bushels 1 800 tons 2 200 tons 1.6 2.9 3.5 Kowalchuk'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 Kowalchuk's production plan. (b) What should the crop plan be, and what profit will it yield? Use Solver to create the Answer and Sensitivity Reports for this LP problem. Now answer the following questions using these reports. Each question is independent of the others. (a) (b) (C) (d) Is this solution a unique optimal solution? Why or why not? If there are alternative solutions, use Solver to identify at least one other optimal solution Would it help Margaret's total profit if she could increase barley sales by 10%. If so, how? How would the availability of more water affect Margaret's total profit?

Answer 1

a.

Let Xij = acres of crop i planted on parcel j

Where

  i = 1 for wheat, 2 for alfalfa, 3 for barley

j = 1 to 5 for SE, N, NW, W, and SW parcels

Irrigation limits:

1.6X11 + 2.9X21 + 3.5X31   ≤ 3,200 acre-feet in SE

1.6X12 + 2.9X22 + 3.5X32   ≤ 3,400 acre-feet in N

1.6X13 + 2.9X23 + 3.5X33   ≤ 800 acre-feet in NW

1.6X14 + 2.9X24 + 3.5X34   ≤ 500 acre-feet in W

1.6X15 + 2.9X25 + 3.5X35   ≤ 600 acre-feet in SW

5 5 Σ1.6X1, +Σ2.9X -Σ3.5X, 47,400 , 400 + -1 -1

Water acre-feet total

Sales limits:

X11 + X12 + X13 + X14 + X15 ≤ 2,200 wheat in acres (= 110,000 bushels)

X21 + X22 + X23 + X24 + X25 ≤ 1,200 alfalfa in acres (= 1,800 tons)

X31 + X32 + X33 + X34 + X35 ≤ 1,000 barley in acres (= 2,200 tons)

Acreage availability:

  X11 + X21 + X31 ≤ 2,000 acres in SE parcel

  X12 + X22 + X32 ≤ 2,300 acres in N parcel

  X13 + X23 + X33 ≤ 600 acres in NW parcel

  X14 + X24 + X34 ≤ 1,100 acres in W parcel

  X15 + X25 + X35 ≤ 500 acres in SW parcel

Objective function:

maximize profit
5 5 5 = S2(50 bushels) X2} + $40(1.5 tons) X3; +(550) (2.2 tons) X3.;

b.   

The solution is to plant

   X12 = 1,250 acres of wheat in N parcel

   X13 = 500 acres of wheat in NW parcel

   X14 = 312 acres of wheat in W parcel

   X15 = 137 acres of wheat in SW parcel

   X25 = 131 acres of alfalfa in SW parcel

   X31 = 600 acres of barley in SE parcel

   X32 = 400 acres of barley in N parcel

Profit will be $337,862.10. Multiple optimal solutions exist.