City Guilds Model Questions Vol1 2024 Engineering Question 23

A machine operates 8 hours per day, 250 days a year (total \(H = 2000\) h). It has two critical components, A and B, each subject to a constant failure rate (exponential distribution).\n\nComponent A: failure rate \(\lambda_{A}=0.001\) h\(^{-1}\), preventive‑maintenance (PM) cost \(C_{PM,A}=₦5\,000\) per PM, failure cost \(C_{F}=₦20\,000\) per unplanned failure.\nComponent B: failure rate \(\lambda_{B}=0.0005\) h\(^{-1}\), PM cost \(C_{PM,B}=₦3\,000\).\n\nPM is performed at a fixed interval \(\tau\) (hours) for each component, resetting its age to zero. The expected number of failures per year for a component with interval \(\tau\) is \[ N_{F}=\frac{H}{\tau}\bigl(1-e^{-\lambda\tau}\bigr). \]\nThe total annual cost for a component is \[ C(\tau)=\frac{H}{\tau}C_{PM}+N_{F}C_{F}. \]\nDetermine the optimal PM interval (in whole hours) for each component that minimises its annual cost, assuming the interval cannot exceed the annual operating time (2000 h). Also give the minimum annual cost for each component (rounded to the nearest naira).