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Math Help - Optimization

  1. #1
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    Optimization

    McDuff Preserves expects to bottle and sell 7000000 32-oz jars of jam. The company orders its containers from Consolidated Bottle Company. The cost of ordering a shipment of bottles is $700, and the cost of storing each empty bottle for a year is $.80. How many orders should McDuff place per year and how many bottles should be in each shipment if the ordering and storage costs are to be minimized? (Assume that each shipment of bottles is used up before the next shipment arrives.)
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  2. #2
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    Hello,Tim!

    McDuff Preserves expects to bottle and sell 7,000,000 32-oz jars of jam. The company orders
    its containers from Consolidated Bottle Company. The cost of ordering a shipment of bottles
    is $700 per shipment, and the cost of storing each empty bottle for a year is $0.80.
    How many orders should McDuff place per year and how many bottles should be in each
    shipment if the ordering and storage costs are to be minimized? (Assume that each
    shipment of bottles is used up before the next shipment arrives.)

    Let N = the number of shipments.
    The cost of shipping is: . 700N dollars.

    Then there are: . \frac{7,000,000}{N} bottles per shipment.
    The cost of storage is: . (0.08)\left(\frac{7,000,000}{N}\right) \:=\:\frac{560,000}{N} dollars.

    Hence, the cost is: . C \;=\;700N + 560,000N^{-1}


    Differentiate and equate to zero: . C\,' \;=\;700 - 560,000N^{-2} \;=\;0

    Multiply by N^2\!:\;\;\;700N^2 - 560,000 \:=\:0\quad\Rightarrow\quad N^2 \:=\:800

    Hence: . N \;=\;\sqrt{800} \;=\;20\sqrt{2}\;\approx\;28.2842...


    Therefore, McDuff should order 28 shipments
    . . . . . . . with 250,000 bottles per shipment.

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