How to use it
- 1. Build the systemSeries kills you, parallel saves you. Add redundancy and watch system reliability jump.
- 2. Find the replacement ageReplace too early and you waste good life; too late and failures cost you. There is a minimum.
- 3. Stock the sparesPoisson demand over the lead time sets how many spares hit your target fill rate.
- 4. Verify by simulationThe age-replacement cost curve is checked against 20,000 simulated lifetimes.
Reliability, replacement, spares
The component
Replacement economics
System & spares
Reliability block diagram. Every unit in the chain must work; each stack is parallel redundancy. One weak stage sets the system - reliability multiplies, so series is unforgiving.
Age-replacement cost rate vs replacement age. The dot is the optimum; dashed = run-to-failure. Crosses are simulation checks.
Poisson demand over the lead time. Bars past the red line are the stock-outs your fill rate tolerates.
Try these experiments
- Set β = 1.0: the failure rate goes flat (memoryless), and the cost curve loses its minimum - preventive replacement becomes pointless. You cannot prevent random.
- Raise β to 4.5: sharp wear-out, and the optimal age tightens well below η. Prevention only makes sense when failures are ageing, not random.
- Go from 4 units in series at redundancy 1 to redundancy 2: system reliability leaps. Adding a spare in parallel beats making each unit slightly better.
About this tool
The concept
Three linked maintenance questions: how reliable is the system, when should we replace, and how many spares do we hold? All three fall out of one lifetime distribution.
How it works
Weibull R(t)=e−(t/η)β. Age replacement cost rate = [cpR(T) + cf(1−R(T))] / ∫0TR(t)dt. Spares: smallest s with Poisson CDF(s; λL) ≥ fill rate.
Winning strategy
Check β before you buy a preventive-maintenance programme. β≤1 means failures are random or infant-mortality - scheduled replacement then adds cost and achieves nothing.