Question

A farmer in the Midwest has 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 of labor, 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 preparation and has contracted with a union for 6,000 worker-days of labor. A midwestern 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) To 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 the loan amount so that additional loans will not have to be secured. (3) The farmer desires a profit of at 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. a. Formulate a goal programming model to determine the number of acres of each crop the farmer should plant to satisfy the goals in the best possible way b. Solve this model by using the computer.

Answer 1

Set Objective: SFS11 Value Of: Corn Wheat abeans constraints Preperation cost per acre Labour hours Profit per acre 100 200 By Changing Variable Cells: SCS8:SFS8 Subject to the Constraints: 10 wheat 500 30 Total 1000 wheat Add 200 1000 80000 loan SFS10SHS10 SFS11 SH$11 rep cost labour hours Profit per acre 080000 6000 labour hours fulfilled 10500 Profit fulfilled 10 1400 5000 6400| >= 6000 20000 0i 260001 Beset All 13 15 16 17 18 19 20 Make Unconstrained Variables Non-Negative Sclect a Soving Method: GRG Nonlinear Solving Method Select the GRG Nonlinear engine for Sover Problems that are smooth nonlinear. Select the L Simplex engine tor linear Solver Problems, and select the Evolutionary engine tor Solver problems that are non-smooth.