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Optimization: Minimizing Cost

  1. Apr 15, 2008 #1
    1. The problem statement, all variables and given/known data
    A hardware store sells approximately 10 000 light bulbs a year. The owner wishes to determine how large an inventory of x (thousand) bulbs should be kept to minimize the cost for inventory. The carrying cost for the bulbs is $40/1000 while the paperwork for ordering is $12 per order. The variable reorder costs are $24/1000

    a) If x (thousand) bulbs are ordered each time, find the yearly cost for ordering the 10 000 bulbs into inventory.
    b) On the average his inventory is x/2 (thousand) bulbs. Find the total cost for inventory.
    c) Find the size of each order to minimize the cost for inventory.


    2. Relevant equations
    none


    3. The attempt at a solution

    This is what I did:
    $40/1000 = x/10 000
    x= $400

    Then, the formula which I developed was [tex]400 +[/tex][tex]\frac{12(10000)}{x}[/tex]


    Please help me out and let me know if I am on the right track. Thanks.
     
  2. jcsd
  3. Apr 15, 2008 #2

    HallsofIvy

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    That's the formula for what? It makes no sense at all to write a formula with out saying what it is equal to!

    (a) asks for "ordering cost". That has nothing to do with "carrying cost" (the cost of storing the light bulbs after they arrive).

    (a) If you need 10000 light bulbs and you order them 1000x at a time, how many times do you have to order? If each order cost $12, how much will those orders cost?

    (b) Inventory (carrying cost) is $40 per 1000 light bulb stored. If you store, on average, (x/2)(1000) light bulbs, how much will that cost?


    (c) Find x that minimizes the total of (a) and (b).
     
  4. Apr 15, 2008 #3
    Thank you for your response HallsofIvy. Actually, that was my mistake for not reading the question properly. Your extended details have allowed me to grasp the question better.

    For part a, we have to order 10 000 bulbs. To find out how many times we have to order, we put: 1000x = 10 000. Then, we get x = 10.

    To find out the cost for ordering 10 times, we multiply this by $12. So the cost ordering 10 000 bulbs into the inventory is $120.

    For part b, I understand that we have to find the total cost for inventory. We know that there are (x/2)1000 = 500x bulbs in the inventory. The cost for storing is $40/1000. So I think that we multiply $40 by 500x and divide by thousand to get the cost.

    40/1000 = x/500x

    But at this point I get confused. Please help. Thanks.
     
  5. Apr 15, 2008 #4
    I really need this since it is due very soon. Please help I need this. Thanks.
     
  6. Apr 15, 2008 #5

    D H

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    x is a variable, so you don't know x=10. Your job here is to solve for x. Hint: The answer to part (a) should have x in it.

    You are ignoring the cost of the light bulbs. The cost to order 1000 light bulbs is $24/1000*1000 + $12 = $36. For 2000 light bulbs, the cost is $24/1000*2000 + $12 = $60. For 3000 bulbs, it is ...

    Again, you do not know x.
    That's because you assigned a value to x up front when you shouldn't have done that.
     
  7. Apr 16, 2008 #6

    HallsofIvy

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