Economic Order Quantity (EOQ)

[ohhnicolette]

a) if it costs 2$ per unit to store an item for one year and 40$ setup cost every time you produce a lot, and you use 1000 units per year, how many lots of what size should be manufactured each year? b) how would your answer change if the setup cost can be reduced to 10$?

The formula that I have deciphered from the book goes:

Ct= i (Q/2) + S (R/Q)

Q= back of units
S= setup cost dollars
CT= total annual cost of that inventory item

I am having trouble taking the information from the problem set and putting it into the formula and making sense of this. Any help would be appreciated!
Thanks

 

[tkhunny]

You'll have to define 'R" and "i".

The real crux of the matter is the storage time. I explain.

If lot size is 1 unit, there is no storage time.
If lot size is 2 units, the second item must be stored while the first is in use.
If lot size is 3 units, the second and third must wait while the first is used. The third must wait again while the second is used.
and so on.

Since we use 1000 per year, it takes 365/1000 days to use a single unit. Let's call this value d = 365/1000.

Total waiting time for a lot of size S, is (S-1)*d + (S-2)*d + ... + (S-(S-1))1*d = ½*d*S*(S-1) days

Total Charge for Storage for one lot, is ($2 / 365 days)*[½*d*S*(S-1) days] = ($1 / 365)*[d*S*(S-1)]
Since d = 365/1000, we have ($1 / 365)*[(365/1000)*S*(S-1)] = ($1)*[(1/1000)*S*(S-1)]

If we decide there are 'n' lots in a year, we have simply S = 1000/n, giving us the lot size in terms of the number of lots.

Total Charge for Storage for one lot, is ($1)*[(1/1000)*(1000/n)*((1000/n)-1)] = ($1)*(1/n)*[(1000/n)-1] = (1000-n)/(n^2)

One more piece for this one. There are 'n' lots per year, so the TOTAL STORAGE COST for the WHOLE YEAR is n*(1000-n)/(n^2) = (1000-n)/n

One more piece for the WHOLE THING. The additional cost of each lot is $40. This gives, finally, 40*n + (1000-n)/n as the total annual cost of n lots in a single year.

Warning: If we cannot find an INTEGER minimum of this thing, we may have to rethink a little. If the algebraic minimum is 3.87058, for example, the answer is likely to be n = 3 or n = 4. We'll cross that bridge when we come to it.

Well, that's enough of that. Let's see what your definitions are and this will tell us how all this algebrea stacks up to the approximation formula.

And, yes, by the way. Dropping the Setup Cost to $10, doubles the number of lots for minimum cost for the year.

That was a delightful exploration.

 

[JeffM]

a) if it costs 2$ per unit to store an item for one year and 40$ setup cost every time you produce a lot, and you use 1000 units per year, how many lots of what size should be manufactured each year? b) how would your answer change if the setup cost can be reduced to 10$?

The formula that I have deciphered from the book goes:

Ct= i (Q/2) + S (R/Q)

Q= back of units
S= setup cost dollars
CT= total annual cost of that inventory item

I am having trouble taking the information from the problem set and putting it into the formula and making sense of this. Any help would be appreciated!
Thanks

You will need to provide more information.

Usually, the way these problems are framed is: (1) by setting up a cost function based on lot size and time in storage, (which may include interest) and then taking the derivative to find a minimum, (2) by setting up a profit function based on cost and revenue and then taking the derivative to find a maximum, or (3) optimizing expected profits where sales are a random variable. The problem may or may not include dealing with backlogs.

As you state the problem given, it APPEARS that you are being asked a cost minimization problem without backlogs. If so, there is a relatively easy answer, but it is not entirely clear to me whether the storage cost is a a fixed cost imposed discretely each year or a linear flow of cost. That is, is the storage cost $.50 for three months, $1.00 for six months, and $2.00 for twelve months, or is it $2.00 for a year or any fraction thereof.

It is also not clear whether Ct is a multiplication or a typo for the function C(t), and whatever it is, whether or not CT = Ct. By the way, in addition to R, Q, i, and S, definitions of C and t would be helpful.

 

[Denis]

Ct= i (Q/2) + S (R/Q)

Assuming Ct is a single variable, then make it simply C:
C = i (Q/2) + S (R/Q)

IF you need to solve that for Q, then:

Q = [C +- sqrt(C^2 - 2iSR)] / i : do you follow that?

Just want to see where you're at at solving equations...