Reliability Education Opportunity: Reliability Analysis of Field Data

Transcription

1 Reliability Education Opportunity: Reliability Analysis of Field Data Vasiliy Krivtsov, PhD Sr. Staff Technical Specialist Reliability & Risk Analysis Ford Motor Company 25th Anniversary of Reliability University of Maryland

2 Discussion Outline Introduction Practical Importance of Reliability Analysis of Field Data Modelling Peculiarities in Reliability Analysis of Field Data Staggered Production/Sales Bivariate Models (Time & Usage) Seasonality Data Maturation Issues Illustrative Case Studies Proposed Course Structure Conclusions 2

3 Practical Importance of Reliability Analysis of Field Data Root cause analysis and future failure avoidance through statistical engineering inferences on the failure rate trends and factors (covariates) affecting them Lab test calibration by equating percentiles of the failure time distributions in the field and in the lab Cost avoidance through early detection of field reliability problems Cash flow optimization through the prediction of the required warranty reserve and/or the expected maintenance costs 3

4 Staggered Production/Sales

5 Nonparametric Estimation Formalized Data Structure: Number of vehicles Time in service intervals Failure time intervals j = 1,, k i = 1,, k k v 1 1 r 11 v 2 2 r 21 r 22 v 3 3 r 31 r 32 r 33 v 4 4 r 41 r 42 r 43 r 44 v 5 5 r 51 r 52 r 53 r 54 r 55 v 6 6 r 61 r 62 r 63 r 64 r 65 r 66 v 7 7 r 71 r 72 r 73 r 74 r 75 r 76 r 77 v 8 8 r 81 r 82 r 83 r 84 r 85 r 86 r 87 r 88 v 9 9 r 91 r 92 r 93 r 94 r 95 r 96 r 97 r 98 r 99 v k k r k1 r k2 r k3 r k4 r k5 r k6 r k7 r k8 r k9 r kk Hazard function at the j-th failure time unit interval: ĥ j d n j j Number of failures at time unit interval j, with r 0 = 0: Risk set exposed at time unit interval j : k d j r pj n p j k j1 j ( vp rpq ) p j q1 5

6 Month in Service Sales Month Numerical Example Jan'02 Feb'02 Mar'02 Apr'02 May'02 Jun'02 Jul'02 Aug'02 Sep'02 Oct'02 Nov '02 Volume Repair Month Jan'02 10, Feb'02 10, Mar'02 10, Apr'02 10, May'02 10, Jun'02 10, Jul'02 10, Aug'02 10, Sep'02 10,000 Mechanical Transfuser Example: 1 3 Oct'02 10,000 24MIS/Unlm usage warranty plan 0 Nov '02 10,000 Time t Risk Set n(t) Risk Set (corr) n'(t) Repairs d(t) Hazard h(t)=d(t)/n'(t) Cum Hazard H(t)=Sh(t) Reliability R(t)=e{-H(t)} CDF F(t)=1-R(t) 0 110, , , , , , , , , , ,

7 C D F : F ( t ) Mechanical Transfuser: Nonparametric Inferences x Concavity is an indication of an IFR. Note: F(t) H(t), for small F(t). x x 88 x 79 ~1.4% 9 MIS x 90 x 72 x 46 x 25 x E Time: t 7

8 Nonparametric Estimation under a Bivariate Warranty Plan Risk set exposed at time unit interval j : n j ( k p j ( v p j1 q1 r pq )) j Probability of mileage not exceeding the warranty mileage limit at failure time unit interval j : j Mileage j 60,000 36,000 12, t, MIS 8

9 C D F : F ( t ) Weibull Probability Plot: Mechanical Transfuser Data ReliaSoft Weibull x 8 x 25 x 46 Time: t x 64 x 88 x 79 x x x 90 x 72 Probability-W eibull CB@ 95% 2-Sided [T] All Data W eibull-2p RRX SRM MED FM F=627/S= Data Points Probability Line Top CB-I Bottom CB-I Mechanical Transfuser Warranty Forecast Summary: Failure 24MIS: Population Size: 110,000 Total Expected Repairs: 15,004 Cost per repair: $30 Total Expected Warranty Cost: $450,120 Year-to-date Cost: $18,810 Required Warranty Reserve: $431,310 Vasiliy Krivtsov VVK 9/22/2007 4:51:35 PM 9

10 Calendarized Forecasting C D F : F ( t ) ReliaSoft Weibull Mechanical Transfuser Warranty Forecast Summary: Failure 24MIS: Population Size: 110,000 Total Expected Repairs: 15,004 Cost per repair: $30 Total Expected Warranty Cost: $450,120 Year-to-date Cost: $18,810 Required Warranty Reserve: $431,310 x 25 x 46 x 64 x 88 x 79 x x x 90 x 72 Probability-W eibull CB@ 95% 2-Sided [T] All Data W eibull-2p RRX SRM MED FM F=627/S= Data Points Probability Line Top CB-I Bottom CB-I How will this total number of repairs be distributed along the calendar time, i.e. how many repairs to expect next month, the following month, etc.? x Time: t Vasiliy Krivtsov VVK 9/22/2007 4:51:35 PM 10

11 Time in Service Calendarized Forecast (generic example) Calendar Time Population Exposed Predicted Number of Repairs Time Parametric thru in in PDF Oct'02 Nov'02 Dec'02 in in thru in in Sep'04 Oct'04 Oct'02 Nov'02 Dec'02 in in Sep'04 Oct' k d j f ( t 0 0i ) nij ( t 0 i1 t 0 i ) i 0j total -> ,004 11

12 Time vs. Usage

13 Note: DFR in mileage domain Note: DFR in mileage domain Time or usage? mileage mileage time time Note: DFR in time domain Note: IFR in time domain Depending on variability in mileage accumulation rates of individual vehicles, the same data may result in a contradicting inference in time and mileage domains. 13

14 Time or usage? (Hu, Lawless & Suzuki, 1998) H(t) ~1.1K/mo ~1K/mo ~0.9K/mo ~0.8K/mo H(t) ~0.9K/mo ~1K/mo ~1.1K/mo ~0.6K/mo ~0.6K/mo ~0.8K/mo Time (MIS) Mileage Note: cum haz functions in time domain appear to be dependant on mileage accumulation, which suggests that time may be NOT the appropriate domain for this failure mode. Note: cum haz functions in mileage domain appear to be independent of mileage accumulation, which suggests that mileage may be the appropriate domain for this failure mode. 14

15 Time or usage? (Kordonsky & Gertsbakh, 1997) f(t) f(t) Time (MIS) Mileage Choose the scale that provides a lower coefficient of variation of the respective failure distribution. 15

16 Data Maturity

17 Data Maturity: Lot Rot Data Maturity Problem: CDF estimates for a nominally homogeneous population at a fixed failure time change as a function of the observation time. F(t) Jan 06 May 06 Mar 06 Various observation times Possible cause: Lot Rot, i.e., vehicle reliability degrades from sitting on the lot prior to be sold. F(t) t 0 Units with 0-10 days on lot t Solution: May 06 Stratify vehicle population by the time spent on lot (the difference between sale date and production date). t 0 Jan 06 Mar 06 t 17

18 Data Maturity: Reporting Delays Data Maturity Problem: CDF estimates for a nominally homogeneous population at a fixed failure time change as a function of the observation time. Possible cause: The number of claims processed at each observation time is underreported due to the lag between repair date and warranty system entry date. Solution: Adjust* the risk set by the probability of the lag time, W : j k j1 n j ( ( v r )) W F(t) F(t) t 0 Jan 06 May 06 Mar 06 At each observation time, risk sets adjusted to account for the underreported claims Mar 06 Various observation times t May 06 p pq j Jan 06 p j q1 t 0 t * J. Kalbfleisch, J. Lawless and J. Robinson, "Method for the Analysis and Prediction of Warranty Claims", Technometrics, Vol. 33, # 1, 1991, pp

19 Data Maturity: Warranty Expiration Rush Data Maturity Problem: F(t) May 06 CDF estimates for a nominally homogeneous population disproportionably increases as a function of the observation time and proximity to the warranty expiration time. Possible cause: Soft (non-critical) failures tend to not get reported until the customer realizes the proximity of warranty expiration date. F(t) Mar 06 Jan 06 t 0 May 04 t w t Solution: Use historical data on similar components to empirically* adjust for the warranty-expiration rush phenomenon. Mar 04 t 0 t w *B. Rai, N. Singh Modeling and analysis of automobile warranty data in presence of bias due to customer-rush near warranty expiration limit, Reliability Engineering & System Safety, Vol. 86, Issue 1, pp A basis for adjustment t 19