Fleet Maintenance Simulation With Insufficient Data

Transcription

1 Fleet Maintenance Simulation With Insufficient Data Zissimos P. Mourelatos Mechanical Engineering Department Oakland University Ground Robotics Reliability Center (GRRC) Seminar 17 March

2 Report Documentation Page Form Approved OMB No Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 17 MAR REPORT TYPE Briefing Charts 3. DATES COVERED to TITLE AND SUBTITLE Fleet Maintenance Simulation With Insufficient Data 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) Zissimos Mourelatos 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Mechanical Engineering Department,Oakland University,2200 N. Squirrel Road,Rochester,Mi, SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) U.S. Army TARDEC, 6501 East Eleven Mile Rd, Warren, Mi, PERFORMING ORGANIZATION REPORT NUMBER ; # SPONSOR/MONITOR S ACRONYM(S) TARDEC 11. SPONSOR/MONITOR S REPORT NUMBER(S) # DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution unlimited 13. SUPPLEMENTARY NOTES For Ground Robotics Reliability Center (GRRC) Seminar 17 March ABSTRACT briefing charts 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified ABSTRACT Public Release 18. NUMBER OF PAGES 54 19a. NAME OF RESPONSIBLE PERSON Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18

3 Goal Apply time-dependent reliability/durability concepts to address prognostic CBM using Available data (limited, censored) Expert opinion Computer simulations (physics-of-failure data) 2

4 What is Reliability? Reliability at time t is the probability that the system has not failed before time t. failure 0 T time t R ( t) P( T > t) = P( T t) = 1 Time-Dependent Reliability 3

5 Background Information SPL (t) (t) Time t Random Field Stress Time t Response(t) = f [ E(t), Degradation/Wear(t), Load(t) ] Random Process approach to reliability-based design is needed time-dependent reliability Limit States: g( X( t ),Y,t ) Random Process Random Variable 4

6 What can we Get from Time- Dependent Reliability? Define lifecycle cost and design for it. Use R(t) in CBM to determine time to maintenance. Design for: Lifecycle cost Quality Warranty Maintenance schedule System Reliability R(t) Quality Time for Maintenance Acceptable Reliability Time t 5

7 Definitions / Observations Reliability: Ability of a system to carry out a function in a time period [0, t L ] c ( t t ) F ( t ) p = P = Prob. of Time to Failure c f L T L F c T ( t ) = P( t [ 0, t ], such that g( X( t), t) 0) L L Cumulative Prob. of Failure F i T ( t ) P( g( X( t ), t ) 0) L = L L Instantaneous Prob. of Failure Time-Invariant Reliability 0 tl ( ) F i t T L time Time-Variant Reliability 0 tl time ( ) F c t T L 6

8 Design for Lifecycle Cost Lifecycle Cost = Production Cost + Inspection Cost + Expected Variable Cost Quality Time-Dependent System Reliability Accurate and efficient predictive tools are needed to estimate Timedependent System Reliability 7

9 C L Design for Lifecycle Cost ( ) E d, X, t, r = C ( d, X) + C ( d, X, t ) C (,, t, r) 0 + d X f P I V f Lifecycle Cost Production Cost Inspection Cost Expected Variable Cost F c T C E V ( t ) = P( t [ 0, t ], such that g( X( t), t) 0) L Final time t ( t r) rt c d, X, f, = cf ( t) e ft ( t)dt L f 0 Cost of failure at time t Interest rate PDF of time to failure time 8

10 How Can we Use it in Design? Specify a Desired System Reliability in Time s. t. min d, µ F F F d Q X, σ X C ( d, µ, σ, t r) L X X f, ( ) t d, X, t p ( t ) d 0 L d U µ σ c R c R X µ X µ X ( ) t d, X, t p ( t ) L X U σ X 1 ( ) t d, X, t p ( t ) σ L X U f f f f 1 0 f System Reliability t 1 Target System Reliability Time t f 9

11 How Can we Use it in Design? Determine Optimal Time to Maintenance in CBM s. t. max d, µ X, σ X C L t M ( ) t dµ X, σ, t, r C, X M L Acceptable Reliability R t ( t ) M F d d L d U µ σ c R X µ X µ X ( ) t d, X, t 1 R ( t ) L X U σ X M σ L X U M System Reliability t M Time for Maintenance Time t 10

12 System (Vehicle) Reliability PDF Comp. 1 MTBF TTF PDF... Vehicle Vehicle R( t) R( t) 1 R 1 Comp. n MTBF TTF R 2 R 3 0 t1 t2 t3 T t 11

13 We Need A Tool to Estimate the PDF of Time Between Failures (TBF) using limited, censored data Frequentist approach (Method 1) Bayesian updating approach (Method 2) Enhances data with expert opinion A Tool to Estimate System (Vehicle) Reliability Monte Carlo Simulation 12

14 Reliability Basics for Non-Repairable Systems 13

15 Reliability of Non-Repairable Systems failure 0 T time t R ( t) = P( T > t) = P( T t) R( t) = 1 F( t) 1 (1) λ ( t) Failure Rate = P ( t < T t + dt T > t) P( t < T t + dt) = dt dt P( T > t) ( t + dt) F( t) dt R( t) ( t) ( t) F f = λ ( t) = (2) R = 14

16 Reliability of Non-Repairable Systems 0 T time t R dr dt failure due to (2) dr dt ( t) = 1 F( t) = f ( t) = λ( t) R( t) dr R ( t) ( 0) R = λdt d( ln R) = λdt ln 1 R = t 0 λdt t R( t) = exp[ λdt] All we need is the 0 failure rate 15

17 Reliability of Non-Repairable Systems 0 T time Failure Target useful life 0 T time M λ( t) 0 T time t λ L ( t) N N t = # of safe systems at t s f # of failures in t t 16

18 Reliability Calculation All we need for calculating the reliability of a system (non-repairable or repairable) is the system PDF of time to failure (TTF) We use : Data to estimate the PDF of TTF for each component Monte Carlo simulation to estimate the PDF of TTF for the system 17

19 Estimation of the PDF (or CDF) of the TTF (TBF) using Limited, Censored Data Two approaches will be presented: Censored MLE approach (Method 1) Bayesian updating approach (Method 2) Enhances data with expert opinion 18

20 Group L1 Original data Updated data Vehicle# mileage Vehicle# mileage Limited Data 0 Time Between Failures (TBF) Censoring Mileage 19

21 Censored MLE Approach (Method 1) Using available limited data (TBFs and censoring mileage), estimate PDF of TBF using a censored MLE approach. Tail sample the PDF of previous step to enhance the original limited data. Using enhanced data from previous step, better estimate the PDF of TBF using an uncensored MLE approach. Using the PDF of previous step, a Bootstrap approach estimates statistics of TBF (e.g. distribution of MTBF, distribution of TBF standard deviation, etc.) 20

22 Bayesian Updating Approach (Method 2) Use a Bayesian approach to estimate statistics of TBF (e.g. distribution of MTBF, distribution of TBF standard deviation, etc.). The Bayesian approach: Refines estimate by progressively collecting data on a as needed basis. Allows fusion of available data with expert opinion. 21

23 Group L1 Original data Updated data Vehicle# mileage Vehicle# mileage Notation 0 Time Between Failures (TBF)

24 Observation / Assumption f ( A, B, p, q), ( A X B, and p > 0, 0) d M = X ~ β q > i i 1 p 1 q 1 p+ q 1 ( x, A, B, p, q) =β( p,q) ( x A) ( B x) ( B A), ( A x B, and p > 0, q > 0) i Beta Distribution 6.0E-05 PDF CDF 1.2 Probability density function (PDF) 5.0E E E E E Cumulative distribution function ( CDF ) A = 0 B = 45,000 miles p = 3, q = 5 0.0E Mileage 23

25 24 Observation / Assumption Beta distribution family is used to model TBF. ( ) ( ) ( ) ( ) ( ) ( ) 0 0, and,,,,,, > > + q p B x A A B x B A x p,q =β q p B A x f q p q p A=0, B = 30000

26 MLE Approach Determines parameters (A, B, p, q) of most likely Beta distribution using available data. It provides Likelihood function in Bayesian estimation. Censored MLE # of recorded failures Max A, B, p, q N i= 1 ( xi, A, B, p, q) [ 1 F( x j, A, B, p, q) ] F N s f Beta PDF j= 1 # of survivals Beta CDF Uncensored MLE Max A, B, p, q N i= 1 f ( x, A, B, p, q) i 25

27 Bayesian Updating Progressively updates estimated Beta parameters (A, B, p, q) using prior knowledge and available new data. It allows to fuse available data with expert opinion. Posterior ( θ) L( θ / DATA) Pr ior( θ) with θ = { A B p q} where: DATA = failures { } DATA F DATA S survivals and L N F N S ( θ / DATA) = L( θ / DATAF ) L( θ / DATAS ) = f ( xi, θ) [ 1 F( x j, θ) ] i= 1 j= 1 26

28 Censored MLE Approach (Method 1) 1. Enter recorded failure data 2. Data sorting 3. Histogram of recorded failure data 4. Maximum Likelihood Estimation (MLE) with censored data 5. Tail sampling to get inferred failure mileage 6. Histogram of both recorded and tailed failure data 7. MLE with uncensored data (considering tailed data) 8. Failure probability bounds are calculated by Bootstrap method 27

29 Censored MLE Approach (Method 1) 1. Enter recorded failure data Artificial data used: 15 vehicles, 4 tires each side, M threshold =30,000 miles Beta distribution: A=0, B=45,000, p = 3, and q = 5 ( A, B, p, q), ( A X B, and p > 0, 0) d M = X ~ β q > i i i 28

30 Censored MLE Approach (Method 1) 2. Data sorting Sort recorded failure data (white cells) Retrieve failure mileage data (164) and survival mileage data (120) 29

31 Censored MLE Approach (Method 1) 3. Histogram of recorded failure data Considers failure mileage data DOES NOT consider survival mileage data Histogram shape may change with different number of bins and ranges Histogram, PDF, and CDF of the failure data 30

32 Censored MLE Approach (Method 1) 31

33 Censored MLE Approach (Method 1) 4. Maximum Likelihood Estimation (MLE) with censored data Considers failure mileage data CONSIDERS survival mileage data as censored data The beta distributed CDF by MLE with censored data, shows that the CDF without survival mileage data is left-biased 32

34 Censored MLE Approach (Method 1) 5. Tail sampling to get inferred failure mileage Tailed failure mileage data represents inferred failure mileage data of the survived tires 33

35 Censored MLE Approach (Method 1) 6. Histogram of both recorded and tailed failure data Includes failure mileage data Includes also tailed failure mileage data The tailed samples may go beyond the threshold mileage of 30,000 MLE with censored data fits a beta distributed CDF to sample data with tailed mileage 34

36 Censored MLE Approach (Method 1) 7. MLE with uncensored data considering tailed failure data Includes both recorded failure data and tailed data Using MLE with uncensored data, a beta distributed CDF is fitted to the recorded and tailed data Failure probability is calculated 35

37 Censored MLE Approach (Method 1) 8. Failure probability bounds are calculated using Bootstrap Both recorded and tailed data are used samples (sets of sample points) are randomly generated from the recorded and tailed sample. Failure probability bounds with confident level of 0.9 are calculated. Statistics of other parameters are provided (mean of failure mileage, std dev of failure mileage, parameters p and q, and probability of failure). 36

38 Censored MLE Approach (Method 1) 37

39 Censored MLE Approach (Method 1) 38

40 Bayesian Updating Approach (Method 2) Specify PRIOR distribution Calculate LIKELIHOOD distribution Calculate POSTERIOR distribution 39

41 Bayesian Updating Approach (Method 2) 1. Specify PRIOR distribution PRIOR source Option 0: Uniform (non-informative) distribution PRIOR source Option 3: Normal distribution for each parameter -- Expert opinion 40

42 Bayesian Updating Approach (Method 2) 1. Specify PRIOR distribution (Cont d) Updated Parameter Distribution Table and 2-D Diagram PRIOR source option is automatically set to 1 41

43 Bayesian Updating Approach (Method 2) 2. Calculate LIKELIHOOD distribution Updated Parameter Distribution Table and 2-D Diagram 42

44 Bayesian Updating Approach (Method 2) 3. Calculate POSTERIOR distribution Updated Parameter Distribution Table and 2-D Diagram 43

45 Bayesian Updating Approach (Method 2) 3. Calculate POSTERIOR distribution (Cont d) Updated PRIOR source Best estimated (most probable) parameters (peak point of posterior distribution) Means and stand. deviations of parameters; Obtained by sampling posterior 5000 times Ranges (min, max) of parameters 44

46 Bayesian Updating Approach (Method 2) 3. Calculate POSTERIOR distribution (Cont d) PDF and CDF of Failure Probability and Its Bounds (sampling posterior 5000 times) 45

47 Summary Two methods have been presented to estimate statistics of Time Between Failures (TBF) using limited, censored data Censored MLE approach (Method 1) Bayesian updating approach (Method 2) Enhances data with expert opinion 46

48 Potential Developments in Durability, Reliability, Availability and Maintainability 47

49 System (Vehicle) Reliability PDF Comp. 1 MTBF TTF PDF... Vehicle Vehicle R( t) R( t) 1 R 1 Comp. n MTBF TTF R 2 Second Software Demonstration R 3 0 t1 t2 t3 T t 48

50 Vehicle TTF Histogram System (Vehicle) Reliability Monte Carlo Simulation t R( t) R 1 R 2 1 Reliability For Vehicle : MTBF = 0 ( ) R t dt R 3 0 t1 t2 t3 T t 49

51 Reliability Allocation Specify system (vehicle) reliability Optimization R( t) R 1 R 2 R t1 t2 t3 Relia bility T t Determine required reliability of EACH component This optimization problem DOES NOT have a unique solution 50

52 Reliability Allocation One way to get a unique solution is to trade-off reliability and associated cost min R comp Cost Target system reliability s. t. System Reliability = t R t By varying R, we get the so called Pareto Frontier. 51

53 Reliability vs Risk of Failure (Cost) We want to maximize Reliability and simultaneously minimize Risk of failure (cost) Reliability t R Utopia Pt Infeasible Domain Feasible Domain Pareto Front Cost 52

54 Uptime Putting it All Together!!! Minimum-Failure-Free-Operating Downtime Period MFFOP Failure 0 time p MFFOP : Probability of achieving MFFOP Determine component hazard rates to: Max Reliability Min Cost Max Availability Max MFFOP Availability Multi-Objective Optimization = E[ Uptime] [ ] + E[ Downtime] E Uptime 53

55 Q & A 54