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

2. 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

3. 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 !

4. 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

6. Multiple Time Scales
Kh. Kordonsky,

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

8. 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 ?

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

10. 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)

11. 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

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

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

14. 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

15. 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

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

17. 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.

18. $ 0.47/hour