Determination of the Optimal Replacement Age for a Preventive Maintenance Problem

involving a Weibull Failure Probability Distribution Function

Ernesto Gutierrez-MiraveteRensselaer 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 PolicyComponent 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

MTBR(t0) = ∫0

t0 t f(t) dt

• If 0 < Pr(T>t0) < 1 then

MTBR(t0) = ∫0

t0 R(t) dt = ∫

0

t0 (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) =

= (c + k F(t0))/∫0

t0 (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

F = 1 – exp( - (t/a)b )a = location parameterb = shape parameter

Cost Rate Equation for F = 1 – exp( - (t/10)3 )

C = c/MTBR(t0) =

= (c + k F(t0))/∫0

t0 (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, Pseudo-

Random 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

1 2 3 4 5 6 7 8 9 10 110

0.5

1

1.5

2

2.5

3

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.