How this calculation works
Replacing a unit at failure or at a planned age creates repeating renewal cycles. Expected cost per cycle alone does not measure long-run cost: a policy with short cycles can incur that cost more often. This page divides expected replacement cost by expected cycle duration under an entered Weibull model. It evaluates a chosen age rather than claiming to optimize the replacement policy automatically.
Expected cycle = integral from 0 to T S(t) dt; expected cost = Cp S(T)+Cf[1−S(T)]; rate = cost/cycleA worked example
With exponential shape one, scale one hundred and planned age fifty, survival to the planned age is exp(−0.5). Expected cycle duration is one hundred times [1−exp(−0.5)], about 39.3469 time units. At planned cost ten and failure cost one hundred, expected cycle cost is about 45.4122. Dividing by cycle duration gives about 1.15415 currency units per time unit.
Good to know
The Weibull parameters are supplied, not fitted. Uncertain parameters, imperfect repair, differing replacement times and multiple failure modes can change the policy comparison substantially. No safety consequence is converted automatically into a cost, and the page does not recommend a maintenance interval. Numerical integration is an approximation and should be checked for sensitive decisions.
Reference: Durham University — renewal age replacement cost model (opens in a new tab) ↗
What this calculation answers
Replacing a unit at failure or at a planned age creates repeating renewal cycles. Expected cost per cycle alone does not measure long-run cost: a policy with short cycles can incur that cost more often.
This page divides expected replacement cost by expected cycle duration under an entered Weibull model. It evaluates a chosen age rather than claiming to optimize the replacement policy automatically.
The Weibull survival quantiles and conditional remaining life calculator supplies the survival function integrated across a renewal cycle. This policy also needs replacement cost assumptions.
Use the right input conventions
Lifetime scale and planned age share one time unit; both costs share one currency and price basis. Each replacement is assumed to restore the unit to an as-good-as-new condition.
Expected cycle duration is the area under survival up to the replacement age. A smoothing substitution handles small-shape curvature; 4096 and 8192 midpoint grids are compared before extrapolation. Age is limited to fifty scale units.
Interpret the result
The planned replacement occurs only if the unit survives to T. Otherwise the failure cost applies first. Weighting those outcomes by their probabilities avoids charging both costs during the same ideal cycle.
Cost rate is a renewal average, not a forecast of monthly cash spending. It also omits discounting and downtime, so its units and exclusions matter before comparing policies.
Work through the example
With exponential shape one, scale one hundred and planned age fifty, survival to the planned age is exp(−0.5). Expected cycle duration is one hundred times [1−exp(−0.5)], about 39.3469 time units.
At planned cost ten and failure cost one hundred, expected cycle cost is about 45.4122. Dividing by cycle duration gives about 1.15415 currency units per time unit.
Where the model stops
The Weibull parameters are supplied, not fitted. Uncertain parameters, imperfect repair, differing replacement times and multiple failure modes can change the policy comparison substantially.
No safety consequence is converted automatically into a cost, and the page does not recommend a maintenance interval. Numerical integration is an approximation and should be checked for sensitive decisions.
A few questions, answered
Why not divide expected cost by the planned age?
Failures end some cycles early, so expected cycle duration is the survival integral rather than always T.
What changes in the comparison?
Change lifetime aging behavior while keeping costs and planned age fixed.