Optimizing Total Cost: Finding MTBF for Maximum Savings in 5 Years

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ptlnguyen
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Homework Statement


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.

Homework Equations


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

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.
 
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ptlnguyen said:

Homework Statement


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.

Homework Equations


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

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.

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?