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Another slide set for an ASQ Reliability Division webinar. This one on Field data analysis and statistical warranty forecasting.

Another slide set for an ASQ Reliability Division webinar. This one on Field data analysis and statistical warranty forecasting.

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  • 1. Field  Data  Analysis  &  Sta0s0cal  Warranty  Forecas0ng  Vasiliy  V.  Krivtsov,  Ph.D.  ©2013  ASQ  &  Presenta0on  Krivtsov  hCp://reliabilitycalendar.org/webinars/  
  • 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  asq.org/reliability    To  sign  up  for  the  free  and  available  to  anyone  live  webinars  visit  reliabilitycalendar.org  and  select  English  Webinars  to  find  links  to  register  for  upcoming  events  hCp://reliabilitycalendar.org/webinars/  
  • 3. Field Data Analysis &Statistical Warranty ForecastingVasiliy V. Krivtsov, Ph.D.Ford Motor CompanyASQ Reliability Division Seminar13 June 2013
  • 4. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 2 Introduction Sources of field data Statistical engineering inferences from field data analysis Importance of managing warranty reserve Probabilistic models Non-repairable items/systems, Repairable systems Bivariate models (time & usage) Statistical estimation Non-repairable items/systems, Repairable systems Bivariate models Special Topics Calendarized warranty forecasting Field data maturity issues Survival regression in root-cause analysis of field failures Key ReferencesDiscussion Outline
  • 5. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 3General Purposes of Field Data Analysis failure avoidance through statistical engineering inferences on thefailure rate trends and factors (covariates) affecting them, lab test calibration by equating percentiles of the failure timedistributions in the field and in the lab cost avoidance through early detection of field reliability problems,and cash flow optimization through the prediction of the requiredwarranty reserve and/or the expected maintenance costs.
  • 6. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 4Non-repairable vs. Repairable SystemsUnit 11Non-repairable RepairableF(t)t1L(t)tExpected number offailures per unit canNOT exceed a unity!Expected number offailures per unit canexceed a unity!Unit 2Unit 3TimeNote: same component on different units
  • 7. Probabilistic Models forNon-Repairable Systems
  • 8. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 6Consider a histogram of failure times, t, for alarge population of nominally identical parts.tf(t)A mathematical model that approximates thebehavior of this histogram is know as thedensity function and is denoted by f(t).Probabilistic models for non-repairable systemst=t0tF(t)1F(t0)t=t0The distribution function, F(t), represents a proportion ofparts failing before a given time t.F(t0)The reliability function, R(t), represents a proportionsurviving beyond a given time t.R(t)R(t0)R(t0)
  • 9. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 7Hazard Function (Instantaneous Failure Rate)The Hazard function, h(t), describes the propensity to failin the next small interval from t0 to t0+Dt, given thesurvival up to t0 , with an approximate probability off(t0)Dt.Mathematically, the hazard function is the ratio of thedensity function to the reliability function: h(t)= f(t)/R(t).In simpler terms, the estimate of the hazard function:number of part failures at a point in timenumber of parts at risk at that timef(t)tR(t0)t=t0th(t)Dtf(t0)t=t0h(t0)The Hazard (failure rate) function can take variousforms: increasing (IFR), decreasing (DFR), constant(CFR) and non-monotonous (e.g., first decreasing, thenconstant, then increasing). Shown on the left is anincreasing failure rate (IFR) case.
  • 10. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 8Popular Univariate ModelsNotation: t – independent variable; l, b, Q, m, s, b1, b2, k, p – parameters; F(.) – thecumulative standard normal distribution.
  • 11. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 9Bivariate ModelsJoint failure density:f(m,t)Marginal in mileage failure density: dttmfmf ),()(Marginal in time failure density: dmtmftf ),()(tm
  • 12. Probabilistic Models forRepairable Systems
  • 13. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co.Probabilistic Models for Repairable SystemsA system is a collection of two or more parts designed to perform one ormore functions. A repairable system is a system, which, after a failure toperform at least one of the functions satisfactorily, can be restored to fullysatisfactory performance by any method, other than replacement of theentire system.Comp 1Comp 2Comp 3TimeSystemSystem
  • 14. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 12A connection between non-repairable & repairable systemstN(t)E[N(t)]012345repairf(t) = dF(t)/dt – underlying (TTFF)distribution of non-repairable componentsX1 X2 X3 X5X4])([)()()]([0 tNdEtFtFtNE Fundamental renewal equation:CumulativeFailures
  • 15. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 13Repair Assumptions &Respective Point ProcessesSystem’s Age (Time)0 failuretime1) Good-as-New (ORP)2) Same-as-Old (NHPP)3) Better-than-Old-butWorse than-New (GRP)4) Worse-than-Old (GRP)Legend:ORP – Ordinary Renewal ProcessNHPP – Non-Homogeneous Poisson ProcessGRP – Generalized Renewal Process
  • 16. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 14Ordinary Renewal Process Repair Assumption: “good-as-new” Expected number of failures in [0, t]:F(t) = CDF of the time to first failure (TTFF) distribution Iterative Model (Smith, Leadbetter - 1963)TTFF distribution: Weibull Recursive Model (White - 1964)TTFF distribution: Weibull Numerical Integration Approach (Baxter, Scheuer - 1982)TTFF distributions: Weibull, Gamma, lognormal, truncated Gaussian Pade Approximation Model (Garg, Kalagnanam - 1998)TTFF distribution: truncated Gaussian LLtdtFtFt0)()()()( 
  • 17. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 15Nonhomogeneous Poisson Process (NHPP) Repair Assumption: “same-as-old” Expected number of failures in [0, t]:l() = the rate of occurrence of failures (ROCOF) Common Models for l() Loglinear Model (Cox, Lewis - 1966) Power Law Model (Crow - 1974) Monte Carlo SimulationX1 = time to first failure distributed according to F(t),X2 = time to second failure distributed according to F(t2| X1), whereFailure Interarrival Timet0d)()t( lL)()(1)|(11212XRXtRXttFLni inn XStSnEt 1)],|[max()(  111)](1[1  iii SStFFX 
  • 18. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 16Generalized Renewal Process (GRP) Repair Assumptions: "good-as-new", "same-as-old", "worse-than-old","better-than-old-but-worse-than-new" General Repair Model (Kijima & Sumita, 1986)Virtual Age :Sn , An = age of the system before the failure and after the repair,respectively; q = repair effectiveness factorq = 0 -> “good-as-new”, i.e. ORPq = 1 -> “same-as-old”, i.e., NHPPq > 1 -> “worse-than-old”0 < q < 1 -> “better-than-old-but-worse-than-new”nn SqA 
  • 19. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 17Fundamental G-Renewal Equation Expected number of failures in [0, t]:where Closed solution is impossible and numerical solutions via Laplace transform or power seriesexpansion also fail (Kijima, 1988) Numerical integration approximations are difficult to apply, because of a recurrent infinitesystem issue (Filkenstein, 1997) A Monte Carlo Solution is, however, possible (Kaminskiy & Krivtsov, 1998)Failure Interarrival Time,)|()()0|()(0 0ddxxxgxhgtt  L0,,)(1)()(  xtqXFqXtfxtgLni inn XStSnEt 1)],|[max()(1-i1-i1-i1-i Sq,)SqF(t-1)SqF(t-1-1FX 
  • 20. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 18Generalized Renewal Process: Monte Carlo Simulation024681012140 20 40 60 80 100timeExpectedNumberofFailures"worse-than-old" (q = 1.5)"same-as-old" (q = 1)"better-than-old-but-worse-than-new" (q = 0.5)"as-good-as-new" (q = 0)NHPPORPThe GRP Cumulative Intensity Functions for an IFR (Weibull, b = 1.5, Q = 20) TTFFDistribution Under Various Repair Assumptions.
  • 21. Statistical Estimation of Probabilistic Modelsfor Non-Repairable Systems
  • 22. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 20CensoringIn the context of warranty data analysis, right censoring is primarily associatedwith staggered entry times, while left censoring with “soft” failures.Typically, when test or field data are analyzed, some parts can be found unfailed, and theonly knowledge about their failure times is that these times are beyond (i.e., to the right onthe time axis of) the observation time. These data are said to be right censored.A failure time know only to be before (i.e., to the left on the time axis of) a given time issaid to be left censored. For example, a part is found to have already failed at the time ofits first examination.observation timeright censored observationleft censored observationstarting time
  • 23. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 21Typical Format for Warranty Data:Basics of Nonparametric EstimationJan02 Feb02 Mar02 Apr02 May02 Jun02 Jul02 Aug02 Sep02 Oct02 Nov 02VolumeJan02 10,000 1 3 6 9 15 17 20 22 41 64Feb02 10,000 0 2 5 10 12 18 19 24 45Mar02 10,000 1 4 5 10 14 18 20 23Apr02 10,000 1 2 7 11 16 17 20May02 10,000 0 1 6 12 17 18Jun02 10,000 1 3 4 9 16Jul02 10,000 2 3 7 11Aug02 10,000 1 4 6Sep02 10,000 1 3Oct02 10,000 0Nov 02 10,000TimetRisk Setn(t)Repairsd(t)0 110,000 01 100,000 82 90,000 253 80,000 464 70,000 725 60,000 906 50,000 887 40,000 798 30,000 699 20,000 8610 10,000 64295921952394376992159849497593967111000010000089992799670.013960.00956CDFF(t)=1-R(t)Cum HazardH(t)=Sh(t)0.999070.999640.99992010.007200.009510.013870.02053Repair Month0.998040.996540.994780.000080.000360.000930.001960.003460.005220.000360.000930.001960.003470.005240.007230.001990.002330.004410.006780.992800.990490.986130.979470.02075ReliabilityR(t)=e{-H(t)}MonthinService00.000080.000280.000580.001030.001500.001770SalesMonthHazardh(t)=d(t)/n(t)Risk Set (corr)n(t)0.00008Mechanical Transfuser Example:24MIS/Unlm usage warranty plan
  • 24. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 22Reliasoft’s Weibull++ Warranty Data Analysis ModuleSalesDataRepairData
  • 25. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 23Time: tCDF:F(t)1.000E-4 12.0002.400 4.800 7.200 9.6000.0000.0200.0040.0080.0120.016x 8x 25x 46x 72x 90x 88x 79x 69x 86Mechanical Transfuser: Nonparametric Inferences~1.4% failing@ 9 MISConcavity is anindication of an IFR.Note: F(t)≈H(t),for small F(t).
  • 26. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 24Number of failures at timeunit interval j, with r0 = 0: kjppjj rd kjp1j1qpqpj )rv(nRisk set exposed at time unitinterval j :Number ofvehiclesTime inserviceintervalsFailure time intervalsj = 1, …, ki = 1, …, k 1 2 3 4 5 6 7 8 9 kv1 1 r11v2 2 r21 r22v3 3 r31 r32 r33v4 4 r41 r42 r43 r44v5 5 r51 r52 r53 r54 r55v6 6 r61 r62 r63 r64 r65 r66v7 7 r71 r72 r73 r74 r75 r76 r77v8 8 r81 r82 r83 r84 r85 r86 r87 r88v9 9 r91 r92 r93 r94 r95 r96 r97 r98 r99vk k rk1 rk2 rk3 rk4 rk5 rk6 rk7 rk8 rk9 rkkSummary of Nonparametric Estimation:Univariate Warranty PlanFormalizedData Structure:jjjndhˆ Hazard function atthe j-th failure time unit interval:
  • 27. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 25 kjpjqjpqpj rvn11))((Risk set exposed at timeunit interval j :Probability of mileage not exceeding the warrantymileage limit at failure time unit interval j :Summary of Nonparametric Estimation:Bivariate (Automotive) Warranty Plan12 24 36 48 t, MIS12,00036,00060,000Mileagej
  • 28. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 26Weibull Probability Plot: Mechanical Transfuser DataReliaSoft Weibull++ 7 - www.ReliaSoft.comb,,Time: tCDF:F(t)0.100 100.0001.000 10.0000.0010.0050.0100.0500.1000.5001.0005.00010.00050.00090.00099.0000.001x 8x 25x 46x 72x 90x 88x 79x 69x 86x 640.50.60.70.80.91.01.21.41.62.03.04.06.0bProbability-W eibullCB@ 95% 2-Sided [T]All DataW eibull-2PRRX SRM MED FMF=627/S=99373Data PointsProbability LineTop CB-IBottom CB-IVasiliy KrivtsovVVK9/22/20074:51:35 PMMechanical Transfuser –Warranty Forecast Summary:Failure probability @ 24MIS: 0.1364Population Size: 110,000Total Expected Repairs: 15,004Cost per repair: $30Total Expected Warranty Cost: $450,120Year-to-date Cost: $18,810Required Warranty Reserve: $431,31013.6424
  • 29. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 27Estimation of Bivariate ModelsModel (Weibull):mmmttt))t/(m(1mm)/t(1ttemt)tm(et)t()t()tm()t,m(fbbbbbbAdvantages:• No assumption about usage accumulationmodel is required, and hence• No “penalty” from conditional failure cyclesand conditional usage accumulation not being thesame• Extrapolation is automatically available in timeand usage domains simultaneouslyLog-Likelihood:))t,m(F1ln(n)t,m(fln(r)Lln(jjwjj,ijiij  Challenges/Limitations:• Selection of the bivaritaeparametric model• Graphical goodness-of-fit• Convergence issues: flatnessof the likelihood functionunder (typically) heavycensoring
  • 30. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 28Bivariate Weibull Example04,0008,00012,00016,00020,00024,00028,00032,00036,000051015202530350.0000.0050.0100.0150.0200.0250.0300.0350.0400.045CDFMileageMIS0.00-0.01 0.01-0.01 0.01-0.02 0.02-0.02 0.02-0.03 0.03-0.03 0.03-0.04 0.04-0.04 0.04-0.05MIS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 351 30 16 9 12 3 1 12 9 21 15 21 11 3 5 3 2 4 1 13 1 9 16 20 15 12 14 5 13 6 1 2 3 1 3 14 2 4 5 9 4 11 9 10 5 13 4 5 11 3 6 1 2 15 2 2 2 3 7 14 9 9 3 14 10 8 7 5 5 2 4 3 2 1 1 16 1 1 2 1 5 5 6 8 6 10 8 9 9 7 6 3 8 6 5 1 1 27 2 4 4 2 4 4 8 10 8 5 8 16 11 9 11 7 7 2 2 2 2 1 1 2 1 18 1 1 3 4 2 2 2 3 3 9 5 10 14 4 12 8 10 6 7 4 5 2 2 1 1 19 4 3 6 3 3 5 5 3 15 15 11 11 9 10 8 5 1 4 3 1 1 110 2 1 1 2 1 6 5 8 4 10 7 12 11 12 7 1 1 1 1 1 111 1 4 1 4 2 1 3 7 5 8 8 10 9 4 9 5 5 3 2 2 1 1 112 1 1 1 1 2 2 2 5 3 5 8 9 7 4 8 7 7 8 4 2 4 1 113 1 1 2 1 1 3 4 2 5 7 11 5 7 6 7 6 4 1 3 1 2 114 1 2 1 3 2 1 3 2 3 7 5 4 3 4 5 5 2 3 2 1 1 115 1 1 1 3 1 1 2 2 2 3 1 1Mileage in 1000 miles
  • 31. Statistical Estimation of Probabilistic Modelsfor Repairable Systems
  • 32. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 30Jan02 Feb02 Mar02 Apr02 May02 Jun02 Jul02 Aug02 Sep02 Oct02 Nov 02VolumeJan02 100 12 14 17 20 26 28 31 33 52 60Feb02 100 11 13 16 21 23 29 30 35 56Mar02 100 12 15 16 21 25 29 31 34Apr02 100 12 13 18 22 27 28 31May02 100 11 12 17 23 28 29Jun02 100 12 14 15 20 27Jul02 100 13 14 18 22Aug02 100 12 15 17Sep02 100 12 14Oct02 100 11Nov 02 100TimetRisk Setn(t)Repairsd(t)0 1100 01 1000 1182 900 1243 800 1344 700 1495 600 1566 500 1437 400 1238 300 1029 200 10810 100 600.636130.89613Repair MonthCum IntensityL(t)=Sl(t)0.255780.423280.260000.28600 1.182130.30750 1.489632.969632.369631.829630.600000.340000.54000SalesMonthRepair Intensityl(t)=d(t)/n(t)0.118000MonthinService00.118000.137780.167500.2128600.511.522.533.50 5 10Time, tCumRepairs/System,L(t)Basics of Nonparametric Estimationfor a Repairable SystemCamrolla GT Example:24MIS/Unlm usage warranty plan
  • 33. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 31Number ofvehiclesTime inserviceintervalsFailure time intervalsj = 1, …, ki = 1, …, k 1 2 3 4 5 6 7 8 9 kv1 1 r11v2 2 r21 r22v3 3 r31 r32 r33v4 4 r41 r42 r43 r44v5 5 r51 r52 r53 r54 r55v6 6 r61 r62 r63 r64 r65 r66v7 7 r71 r72 r73 r74 r75 r76 r77v8 8 r81 r82 r83 r84 r85 r86 r87 r88v9 9 r91 r92 r93 r94 r95 r96 r97 r98 r99vk k rk1 rk2 rk3 rk4 rk5 rk6 rk7 rk8 rk9 rkkNumber of failures attime unit interval j, withr0 = 0:kjppjj rdkjppj vnRisk set exposed at time unitinterval j :Formalized Basics of Nonparametric Estimation for aRepairable SystemFormalizedData Structure:jjjndˆ lRepair intensity function at the j-thfailure time unit interval:Cumulative IntensityFunction (a.k.a., MeanCumulative Function) attime t :t1jjtˆˆ lL
  • 34. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 32024681012140 10 20 30Time, tCumRepairs/System,L(t)Nonpar EstimatePower Law NHPPPower Law NHPP: Camrolla GT Example72.1)42.5/t()t( L
  • 35. Special Topics
  • 36. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 34Calendarized ForecastingReliaSoft Weibull++ 7 - www.ReliaSoft.comb,,Time: tCDF:F(t)0.100 100.0001.000 10.0000.0010.0050.0100.0500.1000.5001.0005.00010.00050.00090.00099.0000.001x 8x 25x 46x 72x 90x 88x 79x 69x 86x 640.50.60.70.80.91.01.21.41.62.03.04.06.0 bProbability-W eibullCB@ 95% 2-Sided [T]All DataW eibull-2PRRX SRM MED FMF=627/S=99373Data PointsProbability LineTop CB-IBottom CB-IVasiliy KrivtsovVVK9/22/20074:51:35 PM13.6424Mechanical Transfuser –Warranty Forecast Summary:Failure probability @ 24MIS: 0.1364Population Size: 110,000Total Expected Repairs: 15,004Cost per repair: $30Total Expected Warranty Cost: $450,120Year-to-date Cost: $18,810Required Warranty Reserve: $431,310How will this total number of repairs bedistributed along the calendar time, i.e.how many repairs to expect next month,the following month, etc.?
  • 37. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 35TimeParametricPDFthruOct02inNov02inDec02…inSep04inOct04thruOct02inNov02inDec02…inSep04inOct040 0 110000 0 0 0 0 01 0.0001 100000 10000 0 0 0 6 1 0 0 02 0.0003 89992 10008 10000 0 0 27 3 3 0 03 0.0006 79967 10025 10008 0 0 49 6 6 0 04 0.0010 69921 10046 10025 0 0 69 10 10 0 05 0.0014 59849 10072 10046 0 0 84 14 14 0 06 0.0019 49759 10090 10072 0 0 92 19 19 0 07 0.0023 39671 10088 10090 0 0 93 24 24 0 08 0.0029 29592 10079 10088 0 0 84 29 29 0 09 0.0034 19523 10069 10079 0 0 66 34 34 0 010 0.0039 9437 10086 10069 0 0 37 40 40 0 011 0.0045 0 9437 10086 0 0 0 43 46 0 012 0.0051 0 0 9437 0 0 0 0 48 0 013 0.0057 0 0 0 0 0 0 0 0 0 014 0.0063 0 0 0 0 0 0 0 0 0 015 0.0069 0 0 0 0 0 0 0 0 0 016 0.0076 0 0 0 0 0 0 0 0 0 017 0.0082 0 0 0 0 0 0 0 0 0 018 0.0088 0 0 0 0 0 0 0 0 0 019 0.0094 0 0 0 0 0 0 0 0 0 020 0.0100 0 0 0 0 0 0 0 0 0 021 0.0106 0 0 0 0 0 0 0 0 0 022 0.0112 0 0 0 0 0 0 0 0 0 023 0.0118 0 0 0 10000 0 0 0 0 118 024 0.0124 0 0 0 10008 10000 0 0 0 124 124609 222 272 … 242 124Population Exposed Predicted Number of Repairstotal ->Calendarized Forecasting: Mechanical Transfuser15,004
  • 38. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 36Data Maturity: Lot RottF(t)Jan’06Mar’06May’06t0Data Maturity Problem:CDF estimates for a nominallyhomogeneous population at a fixedfailure time change as a function of theobservation time.Possible cause:“Lot Rot”, i.e., vehicle reliabilitydegrades from sitting on the lot prior tobe sold.VariousobservationtimesSolution:Stratify vehicle population by the timespent on lot (the difference betweensale date and production date).tF(t)Jan’06Mar’06May’06t0Units with 0-10 days on lot
  • 39. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 37Data Maturity: Reporting DelaystF(t)Jan’06Mar’06May’06t0Data Maturity Problem:CDF estimates for a nominallyhomogeneous population at a fixed failuretime change as a function of theobservation time.Possible cause:The number of claims processed at eachobservation time is under-reported due tothe lag between repair date and warrantysystem entry date.VariousobservationtimesSolution:Adjust* the risk set by the probability ofthe lag time, Wj :tF(t)Jan’06Mar’06May’06t0At each observation time, risk setsadjusted to account for the under-reportedclaims kjp1j1qjpqpj ))rv((n W* J. Kalbfleisch, J. Lawless and J. Robinson, "Method for the Analysis and Prediction ofWarranty Claims", Technometrics, Vol. 33, # 1, 1991, pp. 273-285.
  • 40. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 38Data Maturity: Warranty Expiration RushtF(t)Jan’06Mar’06May’06t0Data Maturity Problem:CDF estimates for a nominallyhomogeneous populationdisproportionably increases as a functionof the observation time and proximity tothe warranty expiration time.Possible cause:“Soft” (non-critical) failures tend to notget reported until the customer realizesthe proximity of warranty expiration date.Solution:Use historical data on similar componentsto empirically* adjust for the warranty-expiration rush phenomenon.*B. Rai, N. Singh “Modeling and analysis of automobile warranty data inpresence of bias due to customer-rush near warranty expiration limit”,Reliability Engineering & System Safety, Vol. 86, Issue 1, pp. 83-94.twtF(t)Mar’04May’04t0twA basis foradjustment
  • 41. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 39Survival Regression In Root-Cause Analysis of Field Failuresh(t,x) = h0(t) exp(bT x)where:t is the time to failure (survival) on rig test,h(t,x) is the hazard rate of tread separations,dependent on a vector of covariates:• Tire age  Wedge gauge• Inter-belt gauge  Belt 2 to Buttress• Peel force  Percent of Carbon Blackh0(t) is the baseline hazard ratebT is the transposed vector of coefficients
  • 42. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 40Survival Data & Estimation ResultsTireAgeWedgeGaugeInterbtGaugeEB2B PeelForce%CarbonBlackWedgeGaugex PeelForceSurv. Censoring(1-compl,0-cens)1.22 0.81 0.88 1.07 0.63 1.02 0.46 1.02 01.19 0.69 0.77 0.92 0.68 1.02 0.43 1.05 10.93 0.77 1.01 1.11 0.72 0.99 0.49 1.22 00.85 0.80 0.57 0.98 0.75 1.00 0.42 1.17 10.85 0.85 1.26 1.03 0.70 1.02 0.64 1.09 00.91 0.89 0.94 1.00 0.77 1.03 0.59 1.09 1… … … … … … … … …Covariate Beta St. Error t-value p-valueWedge gauge -9.313 4.069 -2.289 0.022Interbelt gauge -7.069 2.867 -2.466 0.014Peel force -27.411 10.578 -2.591 0.010Wedge A xPeel force18.105 7.057 2.566 0.010
  • 43. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 41Model Adequacy & Predictions1.005.0010.0050.00100.000 10.002.00 4.00 6.00 8.00Exponential Probability PlotCox-Snell ResidualsProbabilityExponentialData 1P=1, A=RRX-SF=11 | S=23CB/FM: 90.00%2 Sided-BC-Type 200.020.040.060.080.10.120.140.160 0.5 1 1.5SurvivalCumulativeHazardFunctionPoor TireGood Tire
  • 44. V. Krivtsov: Field Data Analysis & Statistical Warranty Forecasting | 2013 © Ford Motor Co. 42Key References W. Blischke and D. Murphy, Warranty Cost Analysis, 1994; Marcel Dekker, Inc., New York. J. Lawless, J. Hu and Cao, J., "Methods for Estimation of Failure Distributions and Rates fromAutomobile Warranty Data", Lifetime Data Analysis, Vol. 1, 1995, pp. 227-240. V.V. Krivtsov, Field Data Analysis & Statistical Warranty Forecasting – RAMS Alan O. Plait BestPaper, 2011, IEEE Catalog No CFP11RAM-CDR, ISBN: 978-1-4244-8855-1. J. Hu, J. Lawless and K. Suzuki, "Nonparametric Estimation of a Lifetime Distribution WhenCensoring Times Are Missing", Technometrics, Vol. 40, # 1, 1998, pp. 3-13. J. Kalbfleisch, J. Lawless and J. Robinson, "Method for the Analysis and Prediction of WarrantyClaims", Technometrics, Vol. 33, # 1, 1991, pp. 273-285. M.P. Kaminskiy and V.V. Krivtsov, "A Statistical Estimation of the Cost Impact from Introducing aMileage Limit in Automobile Warranty Policy", Institute of Mathematical Statistics Bulletin, Vol.28, # 2, 1999, p. 73-78. M.P. Kaminskiy and V.V. Krivtsov, "A Monte Carlo Approach to Warranty Repair Predictions" -SAE Technical Paper Series, 1997, # 972582. J.F. Lawless, and J.D. Kalbleisch, "Some issues in the collection and analysis of field reliabilitydata", Survival Analysis: State of the Art. Editors J.P. Klein and P.K. Goel, Kluwer AcademicPublishers, 1992, pp. 141-152. M.W. Lu, "Automotive reliability prediction based on early field failure warranty data", Qualityand Reliability Engineering International, 1998, Vol. 14, 2, pp 103-108. K. Suzuki, "Estimation of lifetime parameters from incomplete field data", Technometrics, 1985,27, pp.263-272. V.V. Krivtsov, D.E. Tananko and T.P. Davis, "A Regression Approach to Tire Reliability Analysis",Reliability Engineering & System Safety, 2002,vol. 78, pp. 267-273.

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