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Comparing Individual Reliability to Population Reliability for Aging Systems

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An ASQ Reliability Division webinar by Dr. Christine Anderson-Cook

An ASQ Reliability Division webinar by Dr. Christine Anderson-Cook

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  • 1. Comparing  Individual   Reliability  to   Popula6on  Reliabil6y   for  Aging  Systems   Dr.  Chris6ne  M.  Anderson-­‐Cook   ©2013  ASQ  &  Presenta6on  Anderson-­‐Cook   hHp://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   hGp://reliabilitycalendar.org/ webinars/  
  • 3. Comparing Individual Reliability to Population Reliability for Aging Systemsp y g g y Dr. Christine M. Anderson-Cook, LANL (candcook@lanl.gov) Dr Lu Lu University of South FloridaDr. Lu Lu, University of South Florida July 2013 https://sites google com/site/poprellu/home Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED https://sites.google.com/site/poprellu/home
  • 4. | Los Alamos National Laboratory | Outline 1. Individual and Population Reliability a. Definition b. When to use which c. Overview of methods for population reliability 2. Age Only examples (QE paper, 2011) a. Weibull (observations: time to failure) b. Probit (obs: Pass/Fail at specific age) 3. Age + Usage example (QREI, 2011) ( / f &a. Probit (obs: Pass/Fail at specific age & usage) 4 Conclusions Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 2 4. Conclusions
  • 5. | Los Alamos National Laboratory | Focus on ReliabilityFocus on Reliability  Definition of reliability:Definition of reliability: “the probability that a system will continue to perform its intended functions until a specified point in time under encountered use conditions.” Define boundaries of system (peripherals, human interface) Often exposure to environmental conditions Multiuse systems may have different thresholds for working (most severe, typical) environmental conditions may impact reliability Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 3 yp )
  • 6. | Los Alamos National Laboratory | Individual vs Population Reliability ility ability ystemReliabi pulationReli Age (months) Time into future (months) Sy Po  Individual System Summary (IndRel): For a given system with specified age, what is its reliability?  Population Aggregate Summary (PopRel): For a Population Aggregate Summary (PopRel): For a population of systems (each with possibly different ages), what is the probability that a randomly chosen system will work at the current or some future time? Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 4 system will work at the current or some future time?
  • 7. | Los Alamos National Laboratory | Two Different SummariesTwo Different Summaries  Relevance – IndRel: for managing individual units, perhaps to remo e them from the pop lation if the become tooremove them from the population if they become too unreliable, or to send them in for scheduled maintenance. – PopRel: for managing the population and require aPopRel: for managing the population and require a given performance level across the population at this or some future points in time.  Information needed – Summary of results from testing various systems (both) – An appropriate statistical model for the reliability given age (both) lplus – the age demographics of the population of interest at the current time (PopRel) Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 5
  • 8. | Los Alamos National Laboratory | Calculation of PopRel IndRelIndRel Age Demographics ReliabilitySystem For an individual system we Age (months) Age (months) For an individual system, we can predict its reliability now and into the future given its current age onReliability Time into future (months) Populatio Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 6 Time into future (months)
  • 9. | Los Alamos National Laboratory | Calculation of PopRel (cont’d) IndRelIndRel Age Demographics ReliabilitySystem Age (months) Age (months) abilitySystemRelia For each system in the population, we can determine its predicted li bilit b d it t Time into future (months) Time into future (months) Time into future (months) S Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 7 reliability based on its current age
  • 10. | Los Alamos National Laboratory | Calculation of PopRel (cont’d)Calculation of PopRel (cont d) iabilitySystemReli Now we use the estimates of all the individual predicted reliabilities to determine Time into future (months) Time into future (months) Time into future (months) PopRel the overall reliability of the population ReliabilityPopulation Time into future (months) Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 8 Note: this could be calculated for any sub-population Time into future (months)
  • 11. | Los Alamos National Laboratory | Reliability summary questionsReliability summary questions IndRel: For a given system with specified, what is its reliability? PopRel: For a population of systems (each with possibly different ages),p p p y ( p y g ) what is the probability that a randomly chosen system will work at a given point in time? Or what fraction of the parts will work at a given point in time? Questions: Which summary is more of interest, 1. If you own a single item? IndRely g 2. If you are own a collection of items used by your department? 3. If you work on maintaining the systems? 4 If id i h i t t l t h t IndRel PopRel IndRel or PopRel 4. If you are considering purchasing new systems to supplement what is currently available? PopRel Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 9
  • 12. | Los Alamos National Laboratory | Example 1: LCD j t l i W ib ll Di t ib tiLCD projector lamps using Weibull Distribution LCD Projection LCD Projection LCD Projection M d l H M d l H M d l H Observed failure time of 31 Model Hours Model Hours Model Hours 1 182 1 974 2 380 1 230 1 1755 2 418 1 244 2 50 2 584 lamps (3 different models) 182 h 230 h 1 387 2 81 2 1205 1 464 2 131 2 1407 1 473 2 158 2 1752 1 600 2 174 3 34 230 h 244 h … 1895 h 1 627 2 300 3 39 1 660 2 332 3 274 1 798 2 345 3 1895 1 954 1895 h 1 954 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 10
  • 13. | Los Alamos National Laboratory | Step 1: Estimate Individual Reliabilityy  Weibull model: 1 ( | ) exp( )f t t t        Bayesian analysis – specify priors ( | , ) exp( )f t t t       ~ (2.5, 2350) ~ (1 1) G am m a G am m a    Estimate the posterior (WinBUGS) ~ (1,1)G am m a ( , | ) ( | , ) ( , )f y f y f      NMCMC λ, β estimates are generated by WinBUGS to  i t th t i di t ib ti Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 11 approximate the posterior distribution
  • 14. | Los Alamos National Laboratory | Individual Reliability EstimatesIndividual Reliability Estimates ility ility Model 1 Model 2 ystemReliab ystemReliab Age (hours) Sy Age (hours) Sy   eliability Model 3   At age=t NMCMC estimates SystemRe Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 12 Age (hours)
  • 15. | Los Alamos National Laboratory | Step 2: Estimate Population Reliability  For a population of 51 Model 1 units Frequency A (h )Age (hours)   At each time,t, estimate reliability for eliability   NMCMC estimates estimate reliability for each unit, then combine to get PopRel estimate 1 ( ) ( )ip t p t  PopulationRe Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 13 ( ) ( )r i i U p t p t N   P Time into future (hours)
  • 16. | Los Alamos National Laboratory | Population Reliability ResultsPopulation Reliability Results ncy PopRel nReliability Frequen Population Age (hours) Time into future (hours) Reliability IndRel Could we have predicted this poor population reliability? SystemR Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 14 Age (hours)
  • 17. | Los Alamos National Laboratory | Answering Questions – IndRel or PopRel 1. For a unit that is 100 hours old, what is the probability that it will work? 2. What is the probability that a random unit will work ?now? 3 What is the probability of the unit I currently have3. What is the probability of the unit I currently have working when I turn it on? 4. My team has 5 units which we use regularly. What is the probability that a random unit from there will k? Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 15 work?
  • 18. | Los Alamos National Laboratory | PopRel for More Complex PopulationPopRel for More Complex Population Overall abilityPopulationRelia Time into future (hours) Model 1 Model 2 Model 3 Reliability nReliability nReliability Population Population Population Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 16 Time into future (hours) Time into future (hours) Time into future (hours)
  • 19. | Los Alamos National Laboratory | Frequentist Options for EstimationFrequentist Options for Estimation  Estimate λ,β (and their covariance matrix) using  maximum likelihood  IndRel: From this confidence intervals for reliability  are possible at all ages of the system  PopRel: a. Generate M draws from the bivariate normal  di t ib ti th d t bt i Mdistribution, use these draws to obtain M  estimates of λ,β and use to build an empirical  C.I. for PopRelp b. Sample with replacement from the original data,  use this to obtain M estimates of λ,β ….  Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 17
  • 20. | Los Alamos National Laboratory | Example 2: Missiles using Probit ModelExample 2: Missiles using Probit Model 227 missiles tested (destructive testing) Current Population Model: ~ ( )i iY Bernoulli p 1 Pass Y   Age = 40 months Age = 90 months 0 1( )i i age p         0 iY Fail    Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 18 s    
  • 21. | Los Alamos National Laboratory | Step 1: Estimating IndRel  Bayesian analysis to estimate parameters (WinBUGS) 0 1( )i i age p s         0 1 s  NMCMC estimatesestimates Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 19
  • 22. | Los Alamos National Laboratory | Step 2: Estimate PopRel  Current Population At each time,t, estimate reliability for each unit, then combine to get PopRel estimate 0 1 s  to get PopRel estimate 1 ( ) ( )r i i U p t p t N    NMCMC estimates abilityPopulationRelia PopRel IndRel P Time into future (months) PopRel Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 20
  • 23. | Los Alamos National Laboratory | Modeling Reliability as a Function of Age d Oth I f ti (U E )and Other Information (Usage or Exposure)  For Example 2, reliability was estimated as a function of the age of the systemfunction of the age of the system – If a system is x months old today, then in a month it will be (x+1) months old  But what if reliability is a function of age and usage? (eg. car reliability typically modeled with age and mileage)mileage) – If a car is 24 months old and has gone 30000 miles, what will these values be in 1 month? – Historical usage pattern can be helpful for prediction, but will introduce some additional variability Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 21 variability.
  • 24. | Los Alamos National Laboratory | Example3: Missiles Modeled with Probit for Ageg and Usage Model: ~ ( )Y Bernoulli pModel: ~ ( )i iY Bernoulli p 1 0 i Pass Y Fail     0 1 2( ) ( )i i i age usage p           0 Fail ip s     Estimation of IndRel – unchanged C ld B i i i f d l Here, #t f- Could use Bayesian estimation for model parameters - Could use maximum likelihood to get estimates usage = #transfers Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 22
  • 25. | Los Alamos National Laboratory | Estimating PopRel – need to predict f tfuture usage  By looking at the rate at hi h i ge which usage increases, we can predict what future usage values are Usag g likely, assuming the same pattern of usage P ibl t Age Usage range predicted from historical data  Possible sources to describe the pattern – Test data ge Current values Test data – Current population – User specified Usag Added variability Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 23 distribution Age
  • 26. | Los Alamos National Laboratory | Obtaining the Usage Rate Distribution  Historical (test data, current population, or both) Index Age Usage Usage Rate Use as population to draw from by sampling with 1 a1 u1 2 a2 u2 u1/a1 u2/a2 from by sampling with replacement Create a distribution  User specified distribution … … C eate a d st but o which adequately represents usage rate center and spread  User specified distribution – Allows flexibility to specify change in anticipated usage Create a distribution which adequately represents usage rate center and spread Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 24
  • 27. | Los Alamos National Laboratory | Estimating PopRel R t f tiEstimating PopRel Individual system reliability PopRel at Repeat for many times 0 1 2, , ,s   Usage rate r(1) Individual system reliability estimates at each time, t (1) (1) (1) (1) 0 1 2, , ,s   PopRel at each time, t r(2) … 0 1 2   (2) (2) (2) (2) 0 1 2, , ,s   … ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 1 2( ) ( * ) ( ) j j j j j j j i i iage usage rate age        NMCMC …… ( ) 0 1 2 ( ) ( ) ( ) ( )j i i i i j age usage ate age p t s          1 ( ) ( )t tMCMC estimates Obtain NMCMC values from characterizing distribution ( ) ( )r i i U p t p t N    Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 25 from characterizing distribution
  • 28. | Los Alamos National Laboratory | PopRel for Missile population Used historical usage information from both testedfrom both tested samples and current population onReliabilityPopulatio PopRel (assuming continued same pattern of usage for overall population) Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 26 Time into future (months) p p )
  • 29. | Los Alamos National Laboratory | Conclusions  Understanding which summary is appropriate to answer which question is key to good decision-making – IndRel answers “For a given system with specified age (and ) h i i li bili ?usage), what is its reliability? – PopRel answers “For a population of systems, what is the probability that a randomly chosen system will work at the current or some future time?current or some future time?  What is needed? – Summary of results from testing various systems [both] A statistical model for the reliability given age and usage [both]– A statistical model for the reliability given age and usage [both] plus – The age (and usage) demographics of the population at the current time [PopRel]  Predicting the age of systems into the future is straightforward, but additional assumptions about future usage of units in the population are needed to obtain a sensible PopRel estimate Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 27
  • 30. | Los Alamos National Laboratory | ReferencesReferences 1. Lu, L., Anderson-Cook, C.M. (2011) “Prediction of, , , ( ) Reliability of an Arbitrary System from a Finite Population” Quality Engineering 23 71-83. 2. Lu, L., Anderson-Cook, C.M. (2011) “Using Age and Usage for Prediction of Reliability of an ArbitraryUsage for Prediction of Reliability of an Arbitrary System from a Finite Population” Quality and Reliability Engineering International 27 179-190. Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA UNCLASSIFIED July 2013 | UNCLASSIFIED | 28