Establish the design-to MTBF required for a system that will result in an approx

Question

Establish the design-to MTBF required for a system that will result in an approximate minimum total cost over an eight year period if the system is to be operated 4000 hours per year, where Total Cost(for 8 years)=Acquisition cost + Spares cost + Downtime cost System acquisition cost is related to MTBF, theta, as follows: C_A(theta)=theta e^0.01 theta The average cost of a spare item is $1,000 and the average cost of downtime associated with failure is $10,000.

Explanation / Answer

Let = M for convienence sake Failure Rate, R = T/M Where T = Total time available = 4000*8 = 32000 hours Hence R = 32000/M ==> M = 32000/R Acquisition cost would be C(M) = M * e 0.01M C(M) = (32000/R)*e0.01(32000/R) C(M) = (32000/R)*e320/R Spares Cost = R * 1000 = 1000 * 32000/M Downtime Cost = R * 10000 = 10000 * 32000/M Hence Total equation of cost becomes TC = M*e0.01M + 1000*32000/M + 10000*32000/M TC = M*e0.01M + 110000*32000/M To find the minimum total cost, we have to differentiate this equation with R and equate it to 0. d(TC)/dM = 0.01*M3*e0.01M+0.01*M2*e0.01M-11000*32000 =0 we can use goal seek function in excel to solve this easily. which would result in M = 529.53 Hours

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