Question

The Hogg company are a group of Oklahoma agricultural producers. The Hoggs want to derive a production process that will satisfy their goals. These farmers can produce corn, wheat, and/or hogs. The acreage they can cultivate for crops is 1100 acres, of which 600 acres is for corn production and 500 acres for wheat production. The Hogs have a maximum of 2000 hr of labor available for crop operations and 600 hr of labor for their hog operation. Corn production requires 1.6 hr of labor per acre, wheat requires 2.0 hr of labor per acre, and the hog operation requires 20 hr of labor per hog unit. This farm can produce 120 bushel of corn per acre and 32 bushel of wheat per acre. Corn and wheat can be sold on the market at $2.50 per bushel and $2.35 per bushel, respectively, and/or can be used as feed in the hog operation. Also, corn and wheat may be purchased from the city feed store at $2.70 per bushel and $2.55 pr bushel, respectively, for feeding the hogs. The hog unit requires 169.1 bushel of feed during a single production period. The corn and wheat used as hog feed can be transformed into corn equivalent units; that is, 1.1 bushel of wheat equals 1 bushel of corn.

The cost per acre to produce corn and wheat are $146.40 per acre and $48.60 per acre. The returns to the hog operation are $919.20 per hog unit. A hog unit is a sow and her litter, which are sold on the market. Returns to the hog operation are net values, for example, after maintenance and replacement stock cost.

The Hogg company is considering the following strategy:

As the goal with the highest priority, P1 they would like to see returns to the agriculture operation of at least $100,000. As their second priority, P2, at least three fourths of the land available for corn production and at least one-half of the land available for wheat production must be planted. The producers also desire that the under-planting of corn be weighed three times greater than that for wheat.

The third and last priority goal P3, is that the hog operation be no greater than 40 hog units.

Formulate the problem in a goal programming format and solve. Discuss the results of whether the goals were achieved in each strategy and the corn, wheat and hog production output

Answer 1

Goal Programming (GP) model :

Let

Cs and Ws be the production (in acres of land) of Corn and Wheat resp that is sold in the market

Cf and Wf be the production (in acres of land) of Corn and Wheat resp that is used as feed for hog operation.

Cp and Wp be the quantity (in bushels) of Corn and Wheat purchased from the city feed store for feeding the hogs.

H be the number of hog units produced.

pi, ni be the positive and negative deviation variables for respective goals.

Objective: Minimize 8n1 + 6n2a + 2n2b + 1p3

where n1 is the negative deviation variable for P1 goal, n2a and n2b are the negative deviation variables for P2 goal and p3 is the positive deviation variable for P3 goal.

s.t.

Cs+Cf <= 600

Ws+Wf <= 500

1.6Cs+1.6Cf+2Ws+2Wf <= 2000

20H <= 600

120Cf+32Wf+Cp+Wp-169.1H = 0

P1 goal :  (120*2.5-146.4)*Cs+(32*2.35-48.6)*Ws-2.7Cp-2.55Wp+919.2H-p1+n1 = 100000

P2a goal :  Cs+Cf-p2a+n2a = (3/4)*600

P2b goal :  Ws+Wf-p2b+n2b = (1/2)*500

P3 goal :  H-p3+n3 = 40

All variables >= 0

Solution of the GP model is following:

- O X Lingo 17.0 - Solution Report - Lingo1 File Edit Solver Window Help Lindo Model - Lingol DO Solution Report - Lingo1 Total solver iterations: Elapsed runtime seconds: Min 8n1 + 6n2a + 2n2b + 1p3 Model class : s.t. Cs+Cf <= 600 Total variables: Nonlinear variables: Integer variables: Ws+Wf <= 500 Total constraints: Nonlinear constraints: 1.6Cs+1.6Cf+2Ws+2Wf <= 2000 Total nonzeros: Nonlinear nonzeros: 20H <= 600 120Cf+32Wf+Cp+Wp-169.1H = 0 Variable 153.6Cs+26.6Ws-2.7Cp-2.55Wp+919.2H-p1+n1 = 100000 Cs+Cf-p2a+n2a = 450 N2A N2B P3 CS Ws+Wf-p2b+n2b = 250 H-p3+n3 = 40 Value 0.000000 0.000000 0.000000 0.000000 455.6701 0.000000 91.46875 158.5312 30.00000 0.000000 0.000000 0.000000 5.670125 0.000000 10.00000 Reduced Cost 8.000000 6.000000 2.000000 1.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 P2A P2B N3 Row IWNE Slack or Surplus 0.000000 144.3299 250.0000 770.9278 Dual Price -1.000000 0.000000 0.000000 0.000000 For Help, press F1 NUM Ln 1, Col1 9:52 am

Results:

P1 goal:

Total returns from agriculture operation (including hog) = 153.6Cs+26.6Ws-2.7Cp-2.55Wp+919.2H

= 153.6*455.67+26.6*91.47-2.7*0-2.5*0+919.2*30

= $ 100,000

P2 goal:

Total acres of corn planted = Cs+Cf = 455.67+0 = 455.67

Total acres of wheat planed = Ws+Wf = 91.47+158.53 = 250

P3 goal:
Hog units = 30

We see that all the goals were achieved.

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