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Math Help - Linear Programming / Graphical Models help!

  1. #1
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    Unhappy Linear Programming / Graphical Models help!

    Hi!

    I'm studying for a midterm coming up in a few days for my quantitative methods class.
    Could anyone help me? I was sick for a few lectures and am not sure how to go about this.

    These are two questions in particular that I am having issues with.

    1) A farmer prepares feed for livestock by combining two grains: Grain I and Grain II. Each kilogram of Grain I costs $0.60 and contains 2 units of protein, 5 units of iron, and 5 units of carbohydrates. Each kilogram of Grain II costs $0.80 and contains 4 units of protein, 1 unit of iron, and 6 units of carbohydrates. Each animal must receive at least 20 units of protein, 16 units of iron, and 46 units of carbohydrates per day. Formulatea linear programming model to help the farmer determine the amount of Grain I and Grain II to purchase so as to meet the nutritional requirements of the animals at the least possible cost. Do Not Solve. (<-- I'm not sure why it says 'do not solve' at the end of the question

    2) Use the Graphical Method to solve both of the two linear programming problems below, as follows:
    (i) Shade-in the feasible region;
    (ii) Label all cornerpoints and determine their coordinates; and
    (iii) Write the optimal values of the objective function and the decision variables clearly.



    A)
    Maximize Z= 2 x+ 4 y

    Subject to: 2x+ y≤ 40

    x+ y≤ 25

    3x - 2y0

    x 0, y
    0


    B) Minimize C= 6x + 2 y
    Subject to: 2 x+ y≥ 40

    2 x + 5 y ≥ 160

    2x ≤ 3y

    x ≥0, y≥ 0

    If anyone could help me or point into the right direction it would be greatly appreciated!!





    Last edited by jxx09; October 2nd 2012 at 08:00 PM.
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  2. #2
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    Re: Linear Programming / Graphical Models help!

    Quote Originally Posted by jxx09 View Post
    Hi!

    I'm studying for a midterm coming up in a few days for my quantitative methods class.
    Could anyone help me? I was sick for a few lectures and am not sure how to go about this.

    These are two questions in particular that I am having issues with.





    1. A farmer prepares feed for livestock by combining two grains: Grain I and Grain II. Each kilogram of Grain I costs $0.60 and contains 2 units of protein, 5 units of iron, and 5 units of carbohydrates. Each kilogram of Grain II costs $0.80 and contains 4 units of protein, 1 unit of iron, and 6 units of carbohydrates. Each animal must receive at least 20 units of protein, 16 units of iron, and 46 units of carbohydrates per day. Formulatea linear programming model to help the farmer determine the amount of Grain I and Grain II to purchase so as to meet the nutritional requirements of the animals at the least possible cost. Do Not Solve. (<-- I'm not sure why it says 'do not solve' at the end of the question)




    1. Use the Graphical Method to solve both of the two linear programming problems below, as follows:

    (i) Shade-in the feasible region;
    (ii) Label all cornerpoints and determine their coordinates; and
    (iii) Writethe optimal values of the objective functionand the decisionvariables clearly.


    A)
    Maximize Z= 2 x+ 4 y

    Subjectto: 2x+ y≤ 40


    x+ y≤ 25



    3x - 2y0


    x 0, y 0




    B) Minimize C= 6x+ 2 y

    Subjectto: 2 x+ y≥ 40

    2 x+ 5 y≥ 160

    2x≤ 3y

    x ≥0, y≥ 0

    If anyone could help me or point into the right direction it would be greatly appreciated!!





    Are questions 1, A and B all different questions?
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    Re: Linear Programming / Graphical Models help!

    Sorry! I realize I didn't sort the questions properly, I just edited the original post and numbered the questions/colour coordinated them so hopefully they are more clear!
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  4. #4
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    Re: Linear Programming / Graphical Models help!

    Sorry! I realize I didn't sort the questions properly, I just edited the original post and numbered the questions/colour coordinated them so hopefully they are more clear!
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  5. #5
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    Re: Linear Programming / Graphical Models help!

    Quote Originally Posted by jxx09 View Post
    Hi!

    I'm studying for a midterm coming up in a few days for my quantitative methods class.
    Could anyone help me? I was sick for a few lectures and am not sure how to go about this.

    These are two questions in particular that I am having issues with.

    1) A farmer prepares feed for livestock by combining two grains: Grain I and Grain II. Each kilogram of Grain I costs $0.60 and contains 2 units of protein, 5 units of iron, and 5 units of carbohydrates. Each kilogram of Grain II costs $0.80 and contains 4 units of protein, 1 unit of iron, and 6 units of carbohydrates. Each animal must receive at least 20 units of protein, 16 units of iron, and 46 units of carbohydrates per day. Formulatea linear programming model to help the farmer determine the amount of Grain I and Grain II to purchase so as to meet the nutritional requirements of the animals at the least possible cost. Do Not Solve. (<-- I'm not sure why it says 'do not solve' at the end of the question

    2) Use the Graphical Method to solve both of the two linear programming problems below, as follows:
    (i) Shade-in the feasible region;
    (ii) Label all cornerpoints and determine their coordinates; and
    (iii) Write the optimal values of the objective function and the decision variables clearly.



    A)
    Maximize Z= 2 x+ 4 y

    Subject to: 2x+ y≤ 40

    x+ y≤ 25

    3x - 2y0

    x 0, y
    0


    B) Minimize C= 6x + 2 y
    Subject to: 2 x+ y≥ 40

    2 x + 5 y ≥ 160

    2x ≤ 3y

    x ≥0, y≥ 0

    If anyone could help me or point into the right direction it would be greatly appreciated!!





    Well, with linear programming problems, you always have an objective of finding the proper quantities of two commodities that will maximise or minimise something, such as profit (maximising) or cost (minimising). So the steps you have to follow:

    1. The two commodities need to be written as variables (since you don't know the right values which will give the maximum or minimum).

    2. Determine the objective function in terms of the variables.

    3. Write down the constraints, which will be inequalities in terms of the variables.

    4. Graph the inequalities to determine the feasible region.

    5. It's important to note that linear functions are always maximised or minimised at their endpoints. The linear constraints are at their endpoints at the corner points of the feasible region. This means that the maximum and minimum has to each lie at one of the corner points. So substitute each of the corner points into your objective and see which gives the maximum or minimum.


    See how you go...
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