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    Probability Linear Programming and Demand Forecast

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    1. Roger's Regalia is a manufacturer of caps, gowns, and assorted party outfits, primarily for the commencement season. (80 percent of their yearly sales occur over a six-week period.) One of their popular products is a Parrot Head, sometimes worn by graduating college students as a sign of festive occasions. The Parrot Head is produced China, so Roger's Regalia must make a single order well in advance of the upcoming season. Roger, the owner, has prepared the following demand forecast.

    (dozens) Relative
    5,000 0.0183
    10,000 0.0733
    15,000 0.1465
    20,000 0.1954
    25,000 0.1954
    30,000 0.1563
    35,000 0.1042
    40,000 0.0595
    45,000 0.0298
    50,000 0.0132
    55,000 0.0053
    60,000 0.0019
    65,000 0.0006
    70,000 0.0002
    75,000 0.0001

    Roger's Regalia sell the Parrot Head for a wholesale price of $12. Their production cost, including the transportation, duty, and shipping, is $6 per Head. Leftover inventory is sold to discounters for $2.50. Due to production batch considerations, orders must be in multiples of 5000 units.

    a. Suppose Roger's Regalia orders 40,000 Parrot Heads. What is the chance they have to liquidate 10,000 or more Heads with a discounter?

    b. What order quantity maximizes Roger's Regalia's expected profit?
    c. If Roger's Regalia wants to have a 90 percent in-stock probability, then how many Parrot Heads should be ordered?

    d. If Roger's Regalia orders the quantity chosen in part c, what is Roger's Regalia's actual in-stock percentage?

    2. A professor built an LP model to maximize profits for his client (a maker of fan club souvenirs), solved the model, and generated some output from the solver. Unfortunately, in his absent-mindedness, he deleted the Excel workbook and all other information from his computer. All that remains is a one-page printout with the following information:

    Adjustable Cells
    Cell Name Final Value Reduced
    Cost Objective Coefficient Allowable Increase Allowable Decrease
    $C$5 Parrot Heads 31.25 0 30 32.66666667 0.666666667
    $D$5 Salt Shakers 125 0 22 0.5 14.5
    $E$5 Fins 100 0 25 1E+30 24.5

    Cell Name Final Value Shadow Price Constraint R.H. Side Allowable Increase Allowable Decrease
    $F$12 Molding LHS 900 0.125 900 100 166.6666667
    $F$13 Cutting LHS 500 7.25 500 250 150
    $F$14 Painting LHS 162.5 0 200 1E+30 37.5
    $F$15 Sales Max LHS 100 24.5 100 41.66666667 75

    Using the information in these tables, answer the following questions or state that "The answer can not be determined with the information given."

    a. What is the profit (the optimal value) of the optimal solution?

    b. Which constraints are binding?

    c. If the available amount of painting capacity decreased by 25 hours, what would be the change in the optimal values of the decision variables and the optimal value of the objective function? Would anything else change?

    d. If the available amount of molding capacity increased by 200 hours to 1100 hours, how would that affect the optimal value of the objective function?

    e. If the available amount of cutting capacity increased by 20 hours to 520 hours, how would that affect the optimal value of the objective function?

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