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Optimization: Minimizing Cost

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1. Homework Statement
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. Homework 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.
 

Answers and Replies

HallsofIvy
Science Advisor
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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).
 
89
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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.
 
89
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I really need this since it is due very soon. Please help I need this. Thanks.
 
D H
Staff Emeritus
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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.
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.

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.
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 ...

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.
Again, you do not know x.
40/1000 = x/500x

But at this point I get confused. Please help. Thanks.
That's because you assigned a value to x up front when you shouldn't have done that.
 
HallsofIvy
Science Advisor
Homework Helper
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898
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.
NO! you do not order 1000 bulbs x times, you order 1000x bulbs how many times?
What is 10000/(1000x)?

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.
Yes, it is 12 times the number of times you order: but that is NOT 10! And don't forget that the "variable reorder costs are $24/1000". In addition to the cost of the paper work, you must pay $24 for each 1000 bulbs and so how much for x thousand bulbs?


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
You said "multiply $40 by 500x and divide by thousand". Why didn't you do that?

But at this point I get confused. Please help. Thanks.
 

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