Complex conjugate history of reliability


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Are you interested in significant-other reliability developments (SOD) that have not been adopted? Combined with adopted developments, they constitute real reliability, just like the product of a complex number and its complex conjugate yields a real number. SOD includes nonparametric estimates of age-specific field reliability and failure rate functions (actuarial rates), without life data. These estimates deal with renewal processes, repairable processes, and missing data. SOD also quantify uncertainty, not just sample uncertainty. Privacy protection is afforded by not tracking products or people by serial number or name to obtain ages at failures and survivors’ ages. SOD may help employ reliability people and induce, governments, companies, and consumers to make decisions and compare products based on real reliability and risk.

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Complex conjugate history of reliability

  1. 1. Complex Conjugate  History of Reliability Larry George ©2011 ASQ & Presentation Larry George Presented live on Jan 13th, 2011‐_English/Webinars_‐_English.html
  2. 2. ASQ Reliability Division  English Webinar Series One of the monthly webinars  on topics of interest to  reliability engineers. To view recorded webinar (available to ASQ Reliability  Division members only) visit To sign up for the free and available to anyone live  webinars visit and select English  Webinars to find links to register for upcoming events‐_English/Webinars_‐_English.html
  3. 3. Complex Conjugate Historyof Reliability•  SORD*SOTA = Real Reliability –  SORD = Significant Other Reliability Developments –  SOTA = State Of The (reliability) Art•  Why? –  Profit, save our jobs, and protect privacy –  Do something about reliability, risk, and uncertainty!•  What s in the future? What s needed?1/10/2011 Problem Solving Tools 1
  4. 4. What SORDs? Risk is present when future events occur with measurable probability. Uncertainty ispresent when the likelihood of future events is indefinite or incalculable. Frank Knight •  Nonparametric reliability and failure rate functions for: –  Grouped, left-and-right-censored, and truncated data –  Renewal and repairable processes •  Without life data •  Uncertainty: brooms, jackknives and bootstraps, extrapolations, scenarios,… 1/10/2011 Problem Solving Tools 2
  5. 5. Examples•  Component D (Weibull vs. nonparametric)•  M88A1 drivetrain parts (Renewal process)•  LED L70 reliability (Black-Scholes)•  Pleasanton O-D matrix and travel times (multivariate, network tomography)1/10/2011 Problem Solving Tools 3
  6. 6. ANCIENT HISTORY•  Discrete failure rate functions, aka actuarial rates ~220 AD –  Domitius Ulpianus: Roman Legion pension planning, life table –  John Graunt 1600s life tables –  Edmond Halley ca 1693 annuities•  Insurance –  James Dodson, Equitable Life, casualty (1762) –  Gompertz Curve (1825) death rate is •  a(t) = e t+ from a double exponential cdf (Weibull)1/10/2011 Problem Solving Tools 4
  7. 7. Gambling and Physics•  Gambling: Pascal, Laplace, Bernoullis, John Kelly, Ed Thorp, Dr. Z•  Utility, game, risk, credibility: Neumann, Morgenstern, Nash, Harsanyi, Hilary Seal, Bühlmann…•  Financial analysis, hedging, scenarios: Black- Merton-Scholes, Shannon, Thorp, Ziemba•  Physics: Schrödinger wave function !: |! (x;t)|2 is probability density: Myron Tribus statistical thermodynamics, entropy, and1/10/2011 Problem Solving Tools 5 reliability
  8. 8. Modern Times (outline)•  Modern histories•  Significant other reliability developments –  RAND and the US AFLC –  Barlow, Proschan, Marshall, Saunders, Block, et al. –  Lajos Takacs, Stephen Vajda –  Kaplan-Meier –  Sir David Cox –  Network tomography1/10/2011 Problem Solving Tools 6
  9. 9. Modern Histories•  Barlow and Proschan reviewed reliability in their first book (1965)•  Nowlan and Heap s RCM appendix D-1 contains more (1978)•  Recent publications about adopted developments [McLinn, Saleh and Marais]•  Psychologists hijack the meaning of reliability1/10/2011 Problem Solving Tools 7
  10. 10. RAND and US AFLC•  RAND adapted actuarial methods for managing expensive, repairable equipment such as aircraft engines ~1960 –  AFI 21-104 is current version –  Actuarial forecast = n(t)a(t); demand ~Poisson•  MOD-METRIC used to buy $4B of F100PW100 engines and spares ~1973•  USPO 5287267, Robin Roundy et al. patented negative binomial demand distribution ~19911/10/2011 Problem Solving Tools 8
  11. 11. Barlow, Proschan, et al.•  What if failure rate isn t constant? –  Tests and bounds: IFR, IFRA, DMRL… –  Renewal theory, replacement, availability, maintenance –  FTA, Bayes, system vs. parts •  Coherence, redundancy, multivariate,•  Russians too: Kolmgorov, Gnedenko, Belyayev, Gertsbakh,… –  Inspection, opportunistic maintenance1/10/2011 Problem Solving Tools 9
  12. 12. Hungarians Too•  Asymptotic alternating renewal process (up- down-up-down-) statistics are normally distributed, regardless (Takacs) –  Even with dependence (1960s) –  Improve production throughput and reduce variance,•  Gozintos N next-assembly matrix (Vajda) –  Products Vector*(I-N)-1 = Parts Vector1/10/2011 Problem Solving Tools 10
  13. 13. Kaplan-Meier npmle•  Nonparametric max. likelihood reliability function (npmle) estimate from right-censored ages at failures –  JASA made Ed Kaplan combine his vacuum tube reliability paper with Paul Meiers biostatistics paper (1957) –  For dead-forever systems, not repairable•  Odd Aalen did the same for the failure rate function (Nelson-Aalen estimator)1/10/2011 Problem Solving Tools 11
  14. 14. Sir David Cox PH Model•  Proportional hazards (aka relative risk) model is a semiparametric failure rate function of concomitant factors z (1971) –  az(t) = ao(t)e- z: is regression coeff. vector –  Easier than multivariate statistics: e.g., calendar time and miles, operating hours•  Biostatisticians adopt PH model for testing hypotheses about z –  Clinical trials1/10/2011 Problem Solving Tools 12
  15. 15. Finance and Reliability•  Risk and hedging –  Black-Scholes stochastic pde for stock price S dS = dt+ SdW: W is Brownian motion •  Nobel prize to Merton and Scholes (1997) for option price model •  Hedging, LTCM, SIVs, CDOs, CDSs, mortgage defaults, credit crises, deflation, deleveraging, inflation, unemployment??? –  LED deterioration resembles geometric Brownian motion –  Scenarios include some black swans1/10/2011 Problem Solving Tools 13
  16. 16. SORD Reliability (outline)•  Credible Reliability Prediction –  Not just MTBF (ASQ RD monograph advert)•  Parametric vs. nonparametric –  Component D•  LEDs L70•  Help! No life data•  Unforeseen consequences•  Renewal and repair1/10/2011 Problem Solving Tools 14
  17. 17. Parametric vs. Nonparametric Rule 1. Original data should be presented in a way that will preserve therelevant information derived from evidence in the original data for allpredictions assessed to be useful. Walter A Shewhart •  Parametric distribution if justified –  Normal variation or asymptotic, weakest link, exponential-Poisson-beta-binomial-Gamma-chi-square, lognormal (rate changes), inverse Gauss,… •  Nonparametric distribution –  Preserves all information in data –  Avoids opinions and mathematical convenience •  AIC balances overfitting and likelihood •  Entropy quantifies assumed information 1/10/2011 Problem Solving Tools 15
  18. 18. Component D Weibull vs.nonparametric•  AIC = 2k!2lnL: k = # estimated parameters and L is likelihood function•  Entropy ! p(t)ln(p(t)) is uncertainty in a random variable s pdf; less is better Weibull Npmle AIC 16.683 16.685 Entropy 0.0127 0.01351/10/2011 Problem Solving Tools 16
  19. 19. Black-Scholes and LEDs Scatter Plot of Data Set 1 Normalized1.021.01 10.990.980.970.96 0 730.5 1461 2191.5 2922 3652.5 4383 5113.5 5844 6574.5 Each Label is One Month in Hours1/10/2011 Problem Solving Tools 17
  20. 20. L70: P[Age at 70% initial lumens > t]? •  Lumens at age t ~N[ t, t], independent •  Deterioration fits Black-Scholes dSt = dt+ StdWt where St is 1-(% of initial lumens) –  Estimate and from geometric Brownian motion –  L70 ~inverse Gauss with parameters as functions of 70%, and 1/10/2011 Problem Solving Tools 18
  21. 21. L70 Weibull vs. InverseGauss LED L70 Inverse-Gaussian Mixture and Weibull Reliability Functions 1 0.9 0.8 0.7 0.6 Reliability IG Mixture 0.5 Weibull 0.4 0.3 0.2 0.1 0 0 2 4 6 8 10 12 14 16 18 20 Age, Years1/10/2011 Problem Solving Tools 19
  22. 22. Help! No Life Data? People s intuition about random sampling appears to satisfy the law ofsmall numbers, which asserts that the law of large numbers applies tosmall numbers as well. Tversky and Kahneman •  You need ages at failures and survivors ages •  It s too hard to estimate reliability from ships and returns counts –  Ships are counts of production, sales, installations, or other installed base –  Returns are counts of complaints, failures, repairs, or even spares sales •  Follow a sample by S/N? Ships and returns are population data, required by GAAP! 1/10/2011 Problem Solving Tools 20
  23. 23. •Cases •DeathsM/G/! and npmle •n1 •R1•  Npmle of service distribution •n2 •R2 from M/G/! queue input and output times (1975 NLRQ) Time•  Richard Barlow and I overlooked potential for reliability•  Works for Mt/G/! queues under mild conditions on the nonstationary Poisson Mt•  Extended to renewal processes (recycling)1/10/2011 Problem Solving Tools 21
  24. 24. Nplse: Actuarial Forecasts•  Orjan Hallberg (Ericsson ret.) researches medical problems•  Carl Harris and Ed Rattner used nplse to forecasts AIDS deaths from HIV+!AIDS conversions and death counts –  Carl died early of heart attack, and Ed claims he s fully retired.•  Dick Mensing: SSE = [Expected-Observed]2 –  Expected = actuarial forecast (hindcast)1/10/2011 Problem Solving Tools 22
  25. 25. Apple: UnforeseenConsequences•  Boss thinks ships and returns counts are sufficient. Lit. search =>1975 NRLQ article•  Estimate all service parts reliability, forecast failures and recommend stock levels•  Dealers scream! Apple had required dealers to buy obsolescent spares•  Apple bought back $36M of obsolescent spares, for $18M, and crushed them. Made me limit returns to ~$6M per quarter.1/10/2011 Problem Solving Tools 23
  26. 26. Repairable Reliability (outline)School Clip Art / TOASTER12/19/01 •  Triad Systems Corp. •  Brie Engineering M88A1 •  Larry Ellison, Oracle 1/10/2011 Problem Solving Tools 24
  27. 27. Triad Systems Corp.•  New Products manager proposes auto parts demand forecast = n(t)a(t): n(t) = cars by year –  Fails due to autocorrelation, no pun intended –  Auto parts sales might be the second, third, or ??? Stores don t know –  Derived the nplse failure rate estimates for renewal processes ~1994. Got job. Forecasts are better. –  Extended to generalized repairable processes (first TTF differs) and npmle ~1999•  Triad US Patent 5765143 actuarial forecast1/10/2011 Problem Solving Tools 25
  28. 28. M88A1•  In 2000, Brie engineer shares M88A1 drivetrain rebuilds counts for 1990s, $186k then. Laid off –  Estimate: ~25% fail in first year. Either problem wasn t fixed or faulty rebuild. TACOM uninterested. –  2005 AVDS 1790 engine backorders. RAND publishes Velocity Management. RAND uninterested in actuarial forecasts –  ASQ Quality Progress 2010 publishes article on greening the engine overhaul process1/10/2011 Problem Solving Tools 26
  29. 29. M88A1 Drivetrain Component ReliabilityM88A1 1 0.9 Engine 0.8 Trans RelayAsm 0.7 TransPTO 0.6 GenEngAC Generator 0.5 DrvAssy RtFdAsm 0.4 FuelPump EngPTO 0.3 Starter 0.2 TurboC TranCooler 0.1 0 0 5 10 15 20 25 Age at replacement, years1/10/2011 Problem Solving Tools 27
  30. 30. Oracle and Breast Cancer•  Oracle CMM dbs record ages at system failures and the parts that failed –  They don t identify parts by serial number, location: TOAD, AIMS?, Other? –  What if there were duplicate parts?•  Breast cancer recurrences: same side second time or other side???1/10/2011 Problem Solving Tools 28
  31. 31. EM and Hidden Renewals•  EM algorithm, (Estimation-Maximization), gives part reliability npmle – [Dempster, Laird, and Rubin]•  Nplse failure rate estimates and forecasts for renewal processes with missing data (2008) –  Provisional patent pending application is in procrastination1/10/2011 Problem Solving Tools 29
  32. 32. Two-Part System•  Least Sqs is for both parts, EM is for one Alternative Reliability Estimates 1 0.8 0.6 Least Sqs R(t) EM R(t) 0.4 0.2 0 0 4 8 12 Age, Quarters1/10/2011 Problem Solving Tools 30
  33. 33. You re Being Followed•  It s human nature to doubt statistically significant conclusions based ona sample that is a small fraction of the population Tversky and Kahneman –  Pleasanton residents complain about traffic cutting thru. City adjust signal timing to back cars onto freeway. Crash –  City cars follow intruders. Citizens arise (2000) –  Pleasanton gives traffic count data –  Nplse of O-D matrix and travel time distributions –  Traffic manager doesn t understand O-D, probability distributions, and their use –  City stations cheap labor at major intersections to record license numbers (2009) 1/10/2011 Problem Solving Tools 31
  34. 34. Pleasanton1/10/2011 Problem Solving Tools 32
  35. 35. Network Tomography Southbound: Foothill, Hopyard-Hacienda- Owens, Santa RitaEastbound: LasPositas, Westbound: Source-SinkStoneridge, Stanley BlvdFoothill Northbound: Sunol Blvd. 1/10/2011 Problem Solving Tools 33
  36. 36. Pleasanton PM OD matrix•  AKA network tomography Pmatrix Pton Thru origin Pton O from From 0 From N From S From E From W Lambda go g1 D to-> 0 To 0 0.0000 0.8640 0.0000 0.0000 0.7801 6.5128 0.9924 0.8541 To N 0.2136 0.0000 0.0135 0.0000 0.0721 5.9121 0.0001 0.0802 To S 0.1801 0.0285 0.0000 1.0000 0.0000 0.0000 0.0075 0.0656 To E 0.1755 0.0177 0.2679 0.0000 0.1479 0.0000 0.0000 0.0000 To W 0.4308 0.0899 0.7186 0.0000 0.00001/10/2011 Problem Solving Tools 34
  37. 37. Dealing with Uncertainty The analyst should provide a measure of the uncertainty that results from theassumptions underpinning the set of models applied in the analysis and thedeliberate and unconscious simplifications made. Terje Aven •  Randomness (aleatory uncertainty) –  Reliability function, bounds, and stochastic dominance •  Sample uncertainty vs. population –  Why sample if you can get population statistics? •  Epistemic, Knightian, unknown unknowns… –  PRA and Uncertainty in the URC –  Jackknife, bootstrap, broom charts… –  Nonparametric extrapolations –  Scenarios 1/10/2011 Problem Solving Tools 35
  38. 38. Component D•  Given first year of monthly failure counts, how many will fail in remainder of 3-year warranty? –  Data are left and right censored. All failure counts were collected on one calendar date. Monthly ships too –  Some failures are 12 months old, some 11 months….•  I do not think that a nonparametric approach would work. –  It works: facilitates extrapolation, uncertainty –  Weibull reliability under-forecasts failures1/10/2011 Problem Solving Tools 36
  39. 39. Alternative ReliabilityEstimates•  ! 12 months of ships and failures 1 0.9995 npmle Weibull mle nplse 0.999 Naïve mle Weibull 0.9985 lse Weibull 0.998 0 3 6 9 12 Age, Months1/10/2011 Problem Solving Tools 37
  40. 40. Failure Rate Extrapolation ! Uncertainty0.00050.0004 npmle0.0003 nplse mle Weibull0.0002 lse Weibull0.0001 0 0 3 6 9 12 15 18 21 24 27 30 33 36 Age, Months 1/10/2011 Problem Solving Tools 38
  41. 41. Actuarial Forecasts Method E[Failures] Npmle 2687 Nplse 2704 Mle Weibull 2066 Lse Weibull 2495 Meeker et al. (Weibull) 20321/10/2011 Problem Solving Tools 39
  42. 42. Extrapolation Scenarios•  Nonparametric linear extrapolations –  Jackknife; leave out one month s data –  Broom; all 12 months, first 11, first 10…•  W. Weibull recommends power functions for simplicity•  Sensitivity and delta method: –  derivatives of actuarial forecasts wrt linear extrapolation coeffs are n(t) and tn(t)•  Future uncertainty???1/10/2011 Problem Solving Tools 40
  43. 43. Possible Reliability Futures•  MTBF no longer a specification?•  Less Weibull? More inverse Gauss?•  Consumer bills of rights? WikiReliability? –  Do not track by serial number or name (privacy), unless reduced sample uncertainty is worth the costs•  More uncertainty and risk analysis? –  Risk equity, FMERD… –  Dempster-Shaefer Theory of Evidence, belief –  Statisticians work on causal inference and vv•  What do you think? What s needed?1/10/2011 Problem Solving Tools 41
  44. 44. REFERENCES•  AFI 21-104, Selective Management of Selected Gas Turbine Engines, Air Force Instruction 21-104, Air Force Material Command, June 1994,•  McLinn, James, A Short History of Reliability, ASQ Reliability Review, Vol. 30, No. 1, pp. 11-18, March 2010•  Barlow, Richard E. and Frank Proschan, Historical Background of the Mathematical Theory of Reliability, in chapter 1 of Mathematical Theory of Reliability, John Wiley, SIAM, New York, 1965•  Geisler, Murray and H. W. Karr, The design of military supply tables for spare parts, Operations Research, Vol. 4, No. 4, pp. 431-442, 1956•  Kamins, Milton and J. J. McCall, Rules for Planned Replacement of Aircraft and Missile Parts, RAND RM-2810-PR, Nov. 1961•  Saleh, J. H. and K. Marais, Highlights from the early (and pre-) history of reliability engineering, Reliability Engineering and System Safety, Vol. 91, No. 2, pp. 249-256, Feb. 2006•  ISO 26000, Guidance on Social Responsibility, Draft International Standard, 2009•  Lee, Miky, Craig Hillman, and Duksoo Kim, How to predict failure mechanisms in LED and laser diodes, Aug. 2005, Problem Solving Tools 42
  45. 45. References by George•  Estimation of a Hidden Service Distribution of an M/G/! Service System, Naval Research Logistics Quarterly, pp. 549-555, September 1973, Vol. 20, No. 3. co-author A. Agrawal•  A Note on Estimation of a Hidden Service Distribution of an M/G/! Service System, Random Samples, ASQC Santa Clara Valley June 1994•  Origin-Destination Proportions and Travel-Time Distributions Without Surveys, INFORMS Salt Lake City, May 2000, http:/•  Biomedical Survival Analysis vs. Reliability: Comparison, Crossover, and Advances, The J. of the RIAC, pp. 1-5. Q4-2003, id=85&Scope=reg•  Failure Modes and Effects Risk Diagnostics,•  Nonparametric Forecasts from Left-Censored Failures,, Dec. 2010•  LED Reliability Analysis, ASQ Reliability Review, Vol. 30. No. 4, pp.4-11,, Dec. 2010•  Credible Reliability Prediction, ASQ Reliability Division Monograph,, 2003•  Nonparametric Forecasts From Left-Censored Data,, Dec. 20101/10/2011 Problem Solving Tools 43