Hi, I need to know how to formulate the goal programming model for this problem.

Question

Hi, I need to know how to formulate the goal programming model for this problem.
I would also like to know how to input this information into Excel and/or QM for Windows and how to obtain these optimal solutions for this goal programming problem:
x2 = 666.67
d1+ = 666.667
d3? = 78,333.33
d4? = 333.333
d5? = 200.00
d6+ = 166.67
d7? = 300.00
Priority 1 and 2 are achieved

Thank you very much!

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

Explanation / Answer

The MS Excel Solver is not able to satisfy all the constrains, Particulary the constraint on Profit of $105000,The maximum profit avaible is only $ 24285.71

The step by step solution to the problem is given below.

The objective of the business is to maximise profit

Let X1,X2,X3 are Corn, Wheat, Soyabeans respectively.

The objective function is

Max Z= 30x1+40x2+20x3

Subject to

100x1+120x2+70x3<=80000

Preparation cost

7x1+10x2+8x3=6000

Labour Availability

x1>=200

Agreed to purchase 200 acre of Corn

x2>=500

Agreed to purchase 500 acre of Wheat

x3>=300

Agrees to puyrchase 300 acre of soyabean

30x1+40x2+20x3>=105000

The profit should be more than 105000

x1,x2,x3>=0

The solver solutions to the problem is given below

Max Z

24285.7143

Decisions Variable

x1

142.857143

x2

500

x3

0

Constraints

74285.71429

80000

6000

6000

142.8571429

200

500

500

0

300

24285.71429

105000

The objective of the business is to maximise profit

Let X1,X2,X3 are Corn, Wheat, Soyabeans respectively.

The objective function is

Max Z= 30x1+40x2+20x3

Subject to

100x1+120x2+70x3<=80000

Preparation cost

7x1+10x2+8x3=6000

Labour Availability

x1>=200

Agreed to purchase 200 acre of Corn

x2>=500

Agreed to purchase 500 acre of Wheat

x3>=300

Agrees to puyrchase 300 acre of soyabean

30x1+40x2+20x3>=105000

The profit should be more than 105000

x1,x2,x3>=0

The solver solutions to the problem is given below

Max Z

24285.7143

Decisions Variable

x1

142.857143

x2

500

x3

0

Constraints

74285.71429

80000

6000

6000

142.8571429

200

500

500

0

300

24285.71429

105000

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