Determination of the Optimal Replacement
Age for a Preventive Maintenance Problem
involving a Weibull Failure Probability
Distribution Function
Ernesto Gutierrez-Miravete
Rensselaer at Hartford
ICRA6-Barcelona
Reliability, Wear and Maintenance
of Complex Engineering Systems
• Modern engineering systems exhibit high
reliability.
• However, wear and deterioration are
inevitable.
• And maintenance is required to ensure proper
operation.
• Maintenance theory of reliability can be used
to help determine optimal age replacement
preventive maintenance policies.
Hard Alpha Defect in a Jet Engine Disk
Wear in a Journal Bearing Raceway
Age Replacement Policy
Component is replaced once it reaches
predetermined replacement age t0 OR as soon as it
fails, at time T, if T < t0.
• F(t) = failure time distribution function
• R(t) = 1 – F(t) = Reliability function
• c = Cost of preventive replacement
• k = r c = Cost of unplanned replacement (r >1)
• c+k = c(1+r) = Cost of replacing a failed
component
Mean Time Between Replacements
at replacement age t0, (MTBR(t0))
• If Pr(T>t0) = 1 then
MTBR(t0) = t0
• If Pr(T>t0) = 0 then
t0
MTBR(t0) =
• If 0 < Pr(T>t0) < 1 then
MTBR(t0) =
∫
∫0
t f(t) dt
t0
0
R(t) dt =
∫
t0
0
(1-F(t))dt
Cost per Replacement Period c and
Cost Rate with replacement age t0 ,C
c = c + k Pr(T <t0) = c + k F(t0)
C = c/MTBR(t0) =
t0
= (c + k F(t0))/∫0 (1-F(t))dt
• If the failure time distribution is Weibull, the integral in the
denominator can be readily obtained in closed form in terms
of the Whittaker M function using the symbolic manipulation
software Maple.
Weibull Failure Time Distribution
Function
b
F = 1 – exp( - (t/a) )
a = location parameter
b = shape parameter
Cost Rate Equation for
3
F = 1 – exp( - (t/10) )
C = c/MTBR(t0) =
t0
= (c + k F(t0))/∫0 (1-F(t))dt =
4 √t0 (k exp(-t/10)3 - c - k)) (exp( ½(t0/10)3 )2
= --------------------------------------------------t0 [3 √10 WM(1/6,2/3,(t0/10)3) exp( ½(t0/10)3 + 4 √t0 ]
Cost Rate Function
(for c=5, in terms of t0 and r)
C via Monte Carlo Simulation
• Generate a collection of independent, PseudoRandom numbers R uniformly distributed between 0
and 1
• Compute a collection of Weibull distributed Failure
Times T using the Inverse Transform Formula
T = a [ - ln (1 – R) ]1/b
• For given c and r, compute values of C for various
values of t0
• Determine the value of t0 that yields the lowest C
C via MC for c=r=5
3
2.5
2
1.5
1
0.5
0
0
2
4
6
8
10
12
Conclusions
• For items with Weibull distributed failure times
closed form expressions in terms of the Whittaker M
function can be obtained for the cost rate of age
replacement policies.
• Optimal replacement ages can then be determined.
• Alternatively, Monte Carlo simulation can be used to
determine optimal replacement ages if simulated
failure times can be computed from an inverse
transform formula.