math

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?

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asked by akim
  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.)

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    posted by MathMate

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