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    Simplex

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    MATH 141 Homework Due Dec 2 Name :

    Solve the following linear programming problem using a graphical method

    A company makes two puddings, vanilla and chocolate. Each serving of vanilla pudding requires 2 teaspoons of sugar and 25 fluid ounces of water, and each serving of chocolate requires 3 teaspoons of sugar and 15 fluid ounces of water. The company has available each day 3,600 teaspoons of sugar and 22,500 fluid ounces of water. The company makes no more than 600 servings of vanilla pudding because it is all that it can sell each day. If the company makes a profit of 10 cents on each serving of vanilla pudding and 7 cents on each serving of chocolate pudding, how many servings of each pudding should it make to maximize its profit ?

    Step I : Set up your objective function and constraint inequalities

    Maximize: P = 10x + 7y
    Subject to 2x + 3y ≤ 3600
    5x + 3y ≤ 4500
    x ≤ 600
    x_1≥0 y_1≥0

    Step II : Shade the feasible region, Find corner points and evaluate profit at each corner point.

    Please complete the table below.

    Point

    The maximum Profit is ___________.
    It occurs when the number of vanilla puddings is ___________
    and the number of chocolate puddings is ___________

    Now solve the same linear programming problem in question #1 above using the SIMPLEX
    method. Clearly state the row operations used. For each tableau used, complete the leftmost column by providing the appropriate basic variables.

    Step I : Set up your objective function and constraint inequalities
    Maximize: P = +
    Subject to: = ≤
    = ≤
    X_1
    X_2

    Step II : Introduce slack variables
    Step III : Set up the initial Simplex tableau.

    1 0 1 0 0 0 600
    2 3 0 1 0 0 3600
    5 3 0 0 1 0 4500
    -10 -7 0 0 0 1 0
    R_2-2R_1 →R2
    R_3-5R_1 →R3
    R_4+10R_1 →R4
    Now carry out successive pivoting operations to obtain an optimum solution to the linear programming problem. In each case clearly state the row operations used. For each tableau used, complete the leftmost column by providing the appropriate basic variables. You must show all work leading to the final tableau. Use the tables in the next page. At the end of your work please answer the questions below.

    The maximum Profit is __$100__.
    It occurs when the number of vanilla puddings is 300 servings and the number of chocolate puddings is 1000 servings
    Does this result agree with your solution from question #1?

    Show your work in the following tables. You may need less than the six tables provided for you.

    1 0 1 0 0 0 600
    0 3 -2 1 0 0 2400
    0 3 -5 0 1 0 1500
    0 -7 10 0 0 1 6000
    R_2-R_3 →R2
    3R_4+7R_3→R4

    S1 1 0 1 0 0 0 600
    S2 0 0 3 1 -1 0 900
    S3 0 3 -5 0 1 0 1500
    P 0 0 -5 0 7 3 28500

    S1 3 0 0 -1 1 0 900
    S2 0 0 3 1 -1 0 900
    S3 0 9 0 5 -2 0 9000
    P 0 0 0 5 16 9 90000
    3R_1-R_2 →R1
    3R_3+5R_2 →R3
    3R_4+5R_2→R4

    3.
    Verik Engineering Company manufactures three different types of
    Calculators and classifies them as Scientific, Business and Graphing
    according to their computing capabilities. The production requirements
    are as follows
    Each Scientific calculator requires five circuit components, one
    assembly hour and one case.
    Each Business calculator requires seven circuit components, three
    assembly hours and one case.
    Each Graphing calculator requires ten circuit components, four
    assembly hours and one case.
    The company has a monthly limit of 90,000 circuit components,
    30,000 labor hours and 9,000 cases. The unit profits on Scientific,
    Business, and Graphing calculators are $6, $13 and $20 respectively.
    How many of each should be produced to yield the maximum profit?
    What is the maximum profit?

    a. Complete the production table below

    Scientific Business Graphing
    Circuit components
    Assembly Time (hours)
    Cases

    b. Construct the mathematical model for the linear programming problem.
    Include the objective equation and the constraints in standard form.
    Let
    = the number of Scientific calculators
    = the number of Business calculators
    = the number of Graphing calculators
    P = Profit

    c. Introduce slack variables to obtain the initial system.

    In order to answer questions d and e below you must first complete
    questions f, g and h below,

    d. How many of each calculator should Columbia Engineering Company
    produce each month to maximize profit.

    Scientific = ______ Business = ______ Graphing = ______

    e. What is the maximum profit?
    Maximum Profit = ______

    f. Are there any unused circuit components, hours, or cases?
    If there are, what are they and how many.

    g. Set up the initial simplex tableau. Circle the pivot element.

    h.
    Now carry out successive pivoting operations to obtain an optimum solution to
    the linear programming problem. In each case clearly state the row operations
    used. For each tableau used, complete the leftmost column by providing
    the appropriate basic variables.
    You must show all work leading to the final tableau.
    Show your work in the following tables. You may need less than the seven tables provided for you.

    The initial simplex tableau for this problem is :

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    https://brainmass.com/math/calculus-and-analysis/simplex-614839

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