Kh. Kordonsky’s analysis of fatigue failure data (1965-1975) IL:-14 aircraft,36 passengers/cargo, 1950, fatigue wing failure

Notion central to reliability theory- LIFETIME Time to occurrence of certain event (failure) – i.e. the time span from the beginning of operation to the failure appearance. lifetime Time failure Time in air

Intuitive analysis does not work here ! Incomplete Fatigue Data – Interval-type and Censored. - Incomplete Fatique Data FIELD DATA: For i-th Aircraft : presence or absence of a crack on the interval [0,T] i i T Intuitive analysis does not work here !

Interval [t,t+d] will be failure-free ? Question: if during [0,t] there was no failure, what is the probability that the Interval [t,t+d] will be failure-free ? Time t t+d

Multiple Time Scales Kh. Kordonsky, 1989-1999

[1]. Kordonsky, Kh. and I. Gertsbakh. 1993. Choice of the best time scale for the reliability analysis. European Journal of Operational Research, 65, 235-246. [2]-[3]. Kordonsky, Kh. and I. Gertsbakh. 1995. System state monitoring and lifetime scales -I, II. Reliability Engineering and System safety, 47, 1-14, 49, 149-154. [4]. Kordonsky, Kh. and I. Gertsbakh. 1997. Multiple time scale and the coefficient of variation: engineering applications. Lifetime Data Analysis, 3, 139-156. [5]. Kordonsky, Kh. and I. Gertsbakh.1994. Best time scale for age replacement. Internat. Journal of Reliability, Quality and Safety Engineering, 219-229. [6]. Kordonsky, Kh. and I. Gertsbakh.1996. Fatigue crack monitoring in parallel time scales, ESREL Proceedings, 14851490 [7]. Kordonsky, Kh. and I. Gertsbakh.1997. Choice of the best time scale for preventive maintenance in heterogeneous environments. EJOR, 98, 64-74 [8]. Kordonsky, Kh. and I. Gertsbakh.1998. Parallel time scales and two-dimensional manufacturer and individual customer warranties, IIE Transactions, 30, 1041-1054

For a particular failure, there are usually two or more “parallel” relevant time scales. Calendar time Aircraft Time in air Number of cycles Which scale is the best ?

Mileage CARS Time in use Operation time Aircraft Engine Number of cycles Mileage CARS Time in use

The “BEST” time scale has the SMALLEST coefficient of variation for lifetime Why the C.V. ? 1.Only two moments are needed 2. C.V. is SCALE-INVARIANT 3.Smaller C.V. provides longer “safe” life 4. Smaller C.V. guarantees higher efficiency of preventive maintenance 5. New approach to warranty nomination (individual warranties)

Interval [0, t(0.1)] is the “safe” time W( beta=3) C.V.=0.36 Weibull d.f. t(0.1)/mean=0.53 t(0.1) W(beta=1) C.V.=1 Weibull d.f.-> exponential t(0.1)/mean=0.11 t(0.1) Interval [0, t(0.1)] is the “safe” time

“IDEAL” Lifetime distribution: This distribution has the C.V. near zero mean Time

Geometry of Two-dimensional Time: operational trajectories M, mileage in 10,000 failure censored intensive use Low use T, time in use, months

Geometry of V=(1-a)T + a M V-scale v V-time= Const. on a line orthogonal to (1-a,a) (t,m) a T 1-a

Individual warranty regions Iscandar & Blischke, “Reliability and Warranty Analysis of a Motorcycle”, Case Studies…, 2003 M V-lifetime 30,000 0.1-prob. Warranty line m (t,m)-failure 10,000 Individual Warranty Regions Individual warranty regions 1 year 2.5 years t T

Kordonsky, Kh.B., I.B. Gertsbakh (1998),Parallel time scales and two-dimensional manufacturer and individual customer warranties, IIE Transactions, vol 30, 1181-1189

K=H+L, K=6,000 ; C.V.=0 C(p)=$1,000; No failures allowed. Replacement: L, cycles 1/2 5,000 K 1/4 1/4 1/4 3,000 1/2 1/2 1,000 3,000 5,000 H-distribution L-distribution 1,000 1/4 10 H, hours 1,000 3,000 5,000 K=H+L, K=6,000 ; C.V.=0 Replacement: C(p)=$1,000; No failures allowed.

$ 0.47/hour