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Math Help - Help with identifying certain constraints in a Standard maximization Problem

  1. #1
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    Exclamation Help with identifying certain constraints in a Standard maximization Problem

    i am asked to consider the following senario:
    making a trail mix with particular ingredients: Peanuts, Raisins, M&Ms, Mini-preztels (x1+x2+x3+x4) respectively
    Calories(kcal):855 ; 435 ; 1024 ; 162
    Protein(g): 34.57; 4.67; 9.01; 3.87
    Fat(g):72.5; .67; 43.95; 1.49
    Carbs(g): 31.4; 114.74; 148.12; 33.86
    ( table listed is for a serving size of 1 cup )
    make at most 10 cups, using all the ingredients. here's the catch.. so that the mix isn't dominated by any one ingredient, each ingredient will contribute at least (>= ) 10% of the total volume of the mix made

    more constraints, x1+x2+x3+x4<= 7000 calories

    Objective function: Max Carbs 31.4x1+ 114.74x2+ 184.12x3+ 33.86x4

    considering this... how would i write the constraints when forming a Simplex tableau??

    i am using excel for this problem.. but i think i am missing some constraints
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  2. #2
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    Hello, Apedroso!

    Making a trail mix with particular ingredients:

    \begin{array}{cccccccccc}&|& \text{Calories} &|& \text{Protein} &|& \text{Fat} &|& \text{Carb} &| \\ \hline\text{Peanuts }(x_1) &|& 85.5 &|& 34.57 &|& 72.5 &|& 31.4 &| \\\text{Raisins }(x_2) &|& 43.5 &|& 4.67 &|& 0.67 &|& 114.74 &| \\\text{M \& M }(x_3) &|& 102.4 &|& 9.01 &|& 43.95 &|& 148.12 &| \\\text{Pretzels }(x_4) &|& 162 &|& 3.87 &|& 1.49 &|&33.86 &|\end{array} . . . Serving size: 1 cup

    Make at most 10 cups, using all the ingredients.

    Constraints
    . . Each ingredient will be at least 10% of the total mix.
    . . Total calories will be at most 7000 calories.

    Objective function
    . . Max Carbs: 31.4x_1+ 114.74x_2+ 148.12x_3+ 33.86x_4

    Considering this... how would i write the constraints when forming a Simplex tableau??
    The total mixture is at most 10 cups: . {\color{blue}x_1 + x_2+x_3+x_4\:\leq\:10}


    Each ingredient must be at least 10% of the mixture.

    Consider the amount of peanuts: . x_1 \:\geq\:10\%\text{ of mixture}

    . . That is: . x_1 \:\geq \:\frac{1}{10}\left(x_1+x_2+x_3+x_4\right)\quad\Ri  ghtarrow\quad{\color{blue}9x_1 - x_2 - x_3 - x_4 \:\geq \:0}

    Similarly, we have: . \begin{array}{c}{\color{blue}9x_2-x_1-x_3-x_4 \:\geq\:0} \\ {\color{blue}9x_3-x_1-x_2-x_4\:\geq\:0} \\ {\color{blue}9x_4-x_1-x_2-x_3\:\geq\:0} \end{array}


    Calories: . {\color{blue}855x_2 + 435x_2 + 1024x_3 + 162x_2\:\leq\:7000}

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