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.