Question

Find to the nearest whole number, the optimum number of batches of an item that should be produced annually (in order to minimize cost) if 80,000 units are to be made, it cost $3 to store a unit for one year, and it costs $380 to set up the factory to produce each batch.

A.15 batches
b. 13 Batches
C 18 batches
D 20 batches

how do i solve this?

Answer 1

If you are making 80,000 units a year, you might ship 80,000/12 = 6667 per month. You could run one batch of 80,000 for $380 then store 6667 units for 1 month, 6667 units for 2 months, 6667 units for 3 months..., up to 6667 units for 12 months. If you ran two batches a year, you could produce 40,000 units in each batch and never have to store any units more than six months.

Answer 2

i got the same answer but im confused as to why our answers are stil not on the ones listed
the closest one i would say is 20 batches

Answer 3

Let the total cost = x and the number of batches = n then:
\[x = $380n+\frac{$3}{12}\frac{80000}{12}*\frac{12}{n}*({\frac{12}{n}+1})*\frac{1}{2}\]

Answer 4

Thanks

Answer 5

Simplify and we have:
\[x = $380n+\frac{$2500}{3}({\frac{144}{n^2}+\frac{12}{n}}) = $380n+$2500({\frac{48}{n^2}+\frac{4}{n}})=$380n+{\frac{$120,000}{n^2}+\frac{$10,000}{n}}\]
\[x' = $380-$240,000n^{-3}-$10,000n^{-2}\]
Setting x' = 0 I get n between 9 and 10.

Answer 6

The part I don't like is not being sure if you should count any storage costs if you produce and ship units in the same month. Is the max amount of storage months 11 or 12? It shouldn't actually make a difference for the calculation of the number of ideal batches, but it does make a difference to our total cost calculation.