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Homework Help: Total Cost Minimization

  1. Jan 18, 2013 #1
    1. The problem statement, all variables and given/known data
    A system is to be operated 5000 hours per year, where
    Total Cost (for 5 years) = Acquisition cost + Spares cost + Downtime cost
    System acquisition cost is related to MTBF, θ, as follows:

    CA(θ) = 685.2917e0.003779θ
    The average cost of a spare item is $1,000 and the average cost of downtime associated with failure is $5,000.

    2. Relevant equations
    Find MTBF or θ required for a system that will result in approximate minimum total cost over a five-year period if the

    3. The attempt at a solution
    This is my attempt but not sure
    Find θ = MTBF
    1/MTBF = R/T where R= rate of failure and T= total time = 5000hrs
    => MTBF =T/R = 5000/R
    Total Cost = Acquisition cost + Spares cost + Downtime cost
    = 685.2917e^0.003779(5000/R) + 5000R + 1000R
    = 685e^20/R + 6000R
    I need to find R such that it will minimize Total Cost and I am stuck. Please check if my approach is correct. Thanks.
  2. jcsd
  3. Jan 19, 2013 #2

    I like Serena

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    Homework Helper

    Welcome to PF, ptlnguyen! :smile:

    Overall your formula seem fine, but I'm missing references to the 5 year the system is supposed to operate.
    So I think you should have a couple of factors 5 in there somewhere...

    To minimize total cost, you're supposed to take the derivative with respect to R.
    Set it equal to zero, and solve R.
    Do you know how to take a derivative?
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