Reliability can be Interesting, Especially Repairable Systems TTFFTBF1TBF2TTFFEtc. Kelly AFB engine...
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Transcript of Reliability can be Interesting, Especially Repairable Systems TTFFTBF1TBF2TTFFEtc. Kelly AFB engine...
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Reliability can be Interesting, Especially Repairable Systems
TTFF TBF1 TBF2 TTFFEtc.
Kelly AFB engine shop manager said, “Build that sucker so it doesn’t come back
for at least 600 hours!” [P&W F100 F15 engine]
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Don’t Believe Easy Solutions!• Failure rate is constant: theorem from when hell freezes over [Barlow and
Proschan]• Reliability is Weibull
– US Patent 7149659 Dec, 12, 2006 “System and Method for Performing Reliability Analysis” Alan Lesmerises, Standard Aerospace
• TBFs are independent and identically distributed• Demand is normally distributed
– US Patent 5287627 Feb. 15, 1994, “Method for Parts Procurement Quantity where Demand is Uncertain for the Product in which the Parts are Used” Robin Roundy et al.
• Examples:– Aircraft recips and gas turbine engines– Emergency diesel generators– Tesco supermarket kitchen appliances– Solyndra workstations
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Airline Engines’ pdf first 750 hours
• TTFF TBF1 TBF2 TBF3
• Real
• Weibull
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Airline engine correlation estimates
TTFF TBF1 TBF2
TBF1 0.073376
TBF2 0.298017 -0.33782
TBF3 0.896035 -0.52976 0.891329
TBF1 0.073376
TBF2 0.501917 0.039882
TBF3 0.820457 -0.42322 0.077551
• First set of correlations omits suspensions; second assumes suspensions are TBFs. Both sets are computed from same data. Real correlation is probably somewhere in between.
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F-18 Engines’ FOD CCDF
• TTFF TBF1 TBF2
• Real
• Weibull
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F-18 Engines FOD pdf(TTFF, TBF1)
01
23
4-0.02
0
0.02
0.04
0.06
0.08
0.1
0
1
2
3
4
5
pdf of TTFF and TBF1
TBF1TTFF
• Units are usage relative to average?
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F-18 Engine correlations
TTFF TBF1TBF1 -0.36962TBF2 0.104095 -0.04504Conditional TBF1 -0.2Conditional TBF2 Not enough data Not enough data
• Conditional TBF1 – corr(TTFF, TBF1|TTFF < 5 and TBF1 < 5)
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Emergency Diesel GeneratorsJAN FEB MARAPR MAY JUN JUL AUG SSP OCT NOV DEC
0% 6% 6% 6% 13% 13% 6% 13% 6% 0% 13% 25%
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EDG Sample Distribution Parameters
Parameter Value, Days Exponential
sum 5571mean 348.1875stdev 375.4963Exponential should have ~ same stdev and meanskew 1.7321Exponential should have skew ~ 2kurt 3.5300Exponential should have excess kurt ~ 6
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Tesco Bread Plant cdfs
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Tesco Bread Plant Bivariate CDF of TTFF and TBF1
• TTFF is 1-16 months and TBF1 is 1-8 months• Left is nonparametric; right is Gaussian copula
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Solyndra Workstations
• TBFs and TTRs were correlated within and between• Increase correlation of workstation (TBF(i),
TBF(i+1)), (TBF(i), TTR(i)), and correlation of workstation processing times at successive workstations– Increases throughput– Reduces variance of throughput– Reduces time from start to steady state
• ARS Symposium, Reno, 2010
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Recommendations• Take advantage of dependence to improve throughput,
http://pstlarry.home.comcast.net/genie.htm• Optimize opportunistic maintenance depending on objective (Max P[Life
> 600 hours]?)– If TBF(i) is small and corr(TBF(i),TBF(i+1|fix) > 0, fix a lot; If TBF(i) is small and
corr(TBF(i),TBF(i+1|fix) < 0, fix a little?– What else could you do while fixing something?
• Duane-Crow-AMSAA “Reliability [MTBF] Growth” estimation for dependence, joint distributions, and multiple systems starting successively, in perhaps different configurations– Use MCF (mean cumulative function) with caution
• Send field reliability data to [email protected] for nonparametric estimates of joint pdf and correlations, installed base and failure or repair counts are statistically sufficient
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References• G. R. Weckman et al., "Modeling the Reliability of Repairable Systems in the
Aviation Industry,“ Computers and Industrial Engineering, 40 2001, pp. 51-63 [Weibull]
• Jens-Uwe Kluegel, “Investigation of time-dependent trends in failure data of active components,” NPP Goesgen, CH-4658 Daeniken, Switzerland [Weibull]
• Moreira, Ana and Luis Meira-Machado, “survivalBIV: Estimation of the Bivariate Distribution Function for Sequentially Ordered Events Under Univariate Censoring,“ J. Statistical Software, Vol. 46, No. 13, March 2012
• Lin, D. Y., W. Sun, and Zhialiang Yin, “Nonparametric estimation fo the gap time distributions for serial events with censored data,“ Biometrika, Vol. 86, no. 1, pp. 59-70, 1999
• Millar, R. C. And David Olwell, “Parametric Models for Aircraft Engine Removals Resulting from Foreign Object Damage,“ Naval Engineers Journal, 2011 #3