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Math Help - Big M method

  1. #1
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    South Africa
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    Big M method

    I cannot understand only a small bit of how this is solved. Please see below. I have written about what I cannot understand in red.

    "A company produces two brands of trail mix, regular and deluxe, by mixing dried fruit, nuts and cereal. The recipes for the mixes are given in the table. The company has 1,200 pounds of dried fruits, 750 pounds of nuts and 1,500 pounds of cereal to be used in producing the mixes. The company makes a profit of $0.40 on each pound of regular mix and $0.60 on each pound of deluxe mix. How many pounds of each ingredient should be used in each mix in order to maximize the company's profit?

    Type of Mix | Ingredients
    Regular | At least 20% nuts
    | At most 40% cereal
    Deluxe | At least 30% nuts
    | At most 25% cereal

    Note: State the mathematical model (objective function and all constraints); no solution is required.

    Then I identified the variables
    Variables:
    X1 = Nuts in Regular mix
    X2 = Nuts in Deluxe Mix
    X3 = Cereal in Regular Mix
    X4 = Cereal in Deluxe Mix
    X5 = Fruit in Regular Mix
    X6 = Fruit in Deluxe Mix

    Constraints:

    First we need to limit total supplies:





    Than we limit content:

    This one just means that at least 20% of total regular mix is the nuts in the regular. We can rewrite this one as:
    How does 0.2(X1+X3+X5) become -4X1+X3+X5<=0?

    Than we follow the same idea to guarantee the nut content of the deluxe and limit the cereal content of both:


    How are those three also rewritten into what they are right now?


    Finally, the objective function is to maximize:
    "
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  2. #2
    Super Member ILikeSerena's Avatar
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    Re: Big M method

    Quote Originally Posted by Yoodle15 View Post
    I cannot understand only a small bit of how this is solved. Please see below. I have written about what I cannot understand in red.

    "A company produces two brands of trail mix, regular and deluxe, by mixing dried fruit, nuts and cereal. The recipes for the mixes are given in the table. The company has 1,200 pounds of dried fruits, 750 pounds of nuts and 1,500 pounds of cereal to be used in producing the mixes. The company makes a profit of $0.40 on each pound of regular mix and $0.60 on each pound of deluxe mix. How many pounds of each ingredient should be used in each mix in order to maximize the company's profit?

    Type of Mix | Ingredients
    Regular | At least 20% nuts
    | At most 40% cereal
    Deluxe | At least 30% nuts
    | At most 25% cereal

    Note: State the mathematical model (objective function and all constraints); no solution is required.

    Then I identified the variables
    Variables:
    X1 = Nuts in Regular mix
    X2 = Nuts in Deluxe Mix
    X3 = Cereal in Regular Mix
    X4 = Cereal in Deluxe Mix
    X5 = Fruit in Regular Mix
    X6 = Fruit in Deluxe Mix

    Constraints:

    First we need to limit total supplies:





    Than we limit content:

    This one just means that at least 20% of total regular mix is the nuts in the regular. We can rewrite this one as:
    How does 0.2(X1+X3+X5) become -4X1+X3+X5<=0?

    Than we follow the same idea to guarantee the nut content of the deluxe and limit the cereal content of both:


    How are those three also rewritten into what they are right now?


    Finally, the objective function is to maximize:
    "
    Hi Yoodle15!

    Let's take a closer look at the 20% nuts in the regular mix.

    It means that the amount of nuts (x1) is at least 20% of the total (x1+x3+x5).
    In a formula: x_1 \ge 0.2(x_1+x_3+x_5).

    Let's simplify that:

    x_1 \ge 0.2(x_1+x_3+x_5)

    5 x_1 \ge x_1+x_3+x_5

    0 \ge -4 x_1+x_3+x_5

    -4 x_1+x_3+x_5 \le 0

    Does that make sense to you?
    How would the others look then?
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  3. #3
    Newbie
    Joined
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    South Africa
    Posts
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    Re: Big M method

    Thank you! It does!
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