Question

Use EXCEL to format this and solve using solve and explain. Biggest problem is once have variables (which i have half done) is setting up in Excel.

A farmer in the Midwst haas 1,000 acres of land on which she intends to plant corn, wheat, and soybeans. Each acre of corn costs $100 for preparation, requires 7 worker-days of labor, and yields a profit of $30. An acre of wheat costs $120 to prepare, requires 10 worker-days, and yields $40 profit. An acre of soybeans costs $70 to prepare, requires 8 worker-days, and yields $20 profit. The farmer has taken out a loan of $80,000 for crop preparations and ahs contracted with a union for 6,000 worker-days o labor. A midwesters granary has agreed to purchase 200 acres of corn, 500 acres of wheat, and 300 acres of soybeans. The farmer has established the following goals, in order of their importance:

(1) Maintain good relations with the union, the labor contract must be honored; that is, the full 6,000 worker-days of labor contracted for must be used.

(2) Preparation costs should not exceed teh loan amount so that additional loans will not have to be secured.

(3) The farmer desires a profit of a least $105,000 to remain in good financial condition.

(4) Contracting for excess labor should be avoided.

(5) The farmer would like to use as much of the available acreage as possible.

(6) The farmer would like to meet the sales agreement with the granary. However, the goal should be weighted according to the profit returned by each crop.

Solve:

a) Formulate a goal programming model to determine the number of acres of each crop the famer shoud plat to satisfy the goals in the best possible way.

b) Solve this model using the computer (Excel and Solver)

Answer 1

Formulation

Let C, W, and S are the acres of corn, wheat, and soybeans to be cultivated.
Also, let Ui and Oi be the underachievement and overachievement for the i-th goal.

Objective function: Minimize Z = 6U1 + 5O2 + 4U3 + 3O1 + 2U5 + (3/9)U6+(4/9)U7+(2/9)U8

Subject to,
Goal constraints

Goal 1: 7C + 10W + 8S + U1 - O1 = 6000
Goal 2: 100C + 120W + 70S + U2 - O2 = 80000
Goal 3: 30C + 40W + 20S + U3 - O3 = 105000
Goal 4: 7C + 10W + 8S + U2 - O2 = 6000 (same as goal 1)
Goal 5: C + W + S + U5 - O5 = 1000
Goal 6: C + U6 - O6 = 200; W + U7 - O7 = 500; S + U8 - O8 = 300

Non-goal constraints

C+W+S <= 1000 i.e. O5=0

C, W, S >= 0

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Implementation