A manufacturer of hospital supplies has a uniform annual demand for 320,000 boxes of bandages. It costs ​$10 to store one box of bandages for one year and $ 160 to set up the plant for production. How many times a year should the company produce boxes of bandages in order to minimize the total storage and setup​ costs?

Answer 1

Assume "store one year" means 365 box-days and not "a year or part thereof".

k=320000 boxes/year
= 320000/365 boxes/day
= 876.7 boxes/day

If supplies are produced every n days, then it must produce nk boxes at each production, and storage requirements are
P=(0+1+2+3+...n-1)k box-days for each production
=> P=(n/2)*(0+n-1)k=n(n-1)k/2 box-days/production

Over the whole year, there are 365/n productions, so total cost is
C(n)=160(365/n)+10(P/365)*(365/n)
=160*(365/n)+10(n-1)k/2

To minimize cost, C'(n)=0 =>
C'(n)=-58400/n²+5k=0 => n=3.65,
so to minimize cost the manufacturer must produce bandages
N=365/n=365/3.65=100 times a year, or every 3.65 days.
Check:
C(3)=28233.7899543379
C(3.65)=27616.43835616438
C(4)=27750.68493150684

So if production is every 4 days, it will cost $134 more than the optimum at 3.65 days which could be awkward to manage (overtime, shifts, etc.)