Bayesian reliability demonstration test in a design for reliability process

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Bayesian reliability demonstration test in a design for reliability process

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This presentation starts with an introduction of a DFR process. Then the challenges of reliability demonstration test (RDT) in the Validation phase will be illustrated by applying a classical RDT......

This presentation starts with an introduction of a DFR process. Then the challenges of reliability demonstration test (RDT) in the Validation phase will be illustrated by applying a classical RDT (CRDT) approach, which may require a large sample size to demonstrate the required high reliability at acceptable confidence levels. This is true in demonstration of the required reliability at subsystem or component level, after product reliability requirement allocation activity in the DFR process.
Bayesian reliability demonstration test (BRDT) approach can be adopted to significantly reduce sample size or testing duration. In the present work, we will enhance BRDT in several aspects:
We will show how BRDT can be an integrated part of the whole DFR process, by linking to FMEA, PoF, and reliability requirement flow down or allocation.
Successful application of a Bayesian approach depends on the prior experience or life data (testing or field) from previous generations of the product under design. However, there is a case when the product or design is totally new and there is no prior product life data from testing or field. It can be shown in our present work that BRDT can still be used successfully for a totally new product design and development, with the DFR process.
Bayesian reliability approaches involve challenging mathematical operations for engineers, like integrations needing numerical methods. We simplify the BRDT approach based on the prior distribution characteristics of reliability in a DFR process. The approach given in the present paper can be used very easily by engineers with any standard spreadsheet application calculation.

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  • 1. Bayesian Reliability  Demonstration Test in a  Design for Reliability Process  (可靠性设计过程 – 贝叶斯可 靠性验证试验) Dr. Mingxiao Jiang  (蒋鸣晓博士) ©2011 ASQ & Presentation Jiang Presented live on Jul 13th, 2011http://reliabilitycalendar.org/The_Reliability_Calendar/Webinars_‐_Chinese/Webinars_‐_Chinese.html
  • 2. ASQ Reliability Division  Chinese 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 eventshttp://reliabilitycalendar.org/The_Reliability_Calendar/Webinars_‐_Chinese/Webinars_‐_Chinese.html
  • 3. Bayesian Reliability Demonstration Test in a Design for Reliability Process Mingxiao Jiang (Medtronic Inc.) 2011Mingxiao Jiang MEDTRONIC CONFIDENTIAL 1
  • 4. Outline - Design for Reliability (DFR) process - Challenges of Reliability Demonstration Test (RDT) in DFR Validation phase - Bayesian RDT (BRDT) with DFR - Concluding remarksMingxiao Jiang MEDTRONIC CONFIDENTIAL 2
  • 5. Why DFSS and DFR - Increasing competition - Increasing product complexity - Increasing customer expectations of product performance, quality and reliability - Decreasing development time - … - Higher product quality (“out-of-box” product performance often quantified by Defective Parts Per Million) -> DFSS - Higher product reliability (often as measured by failure rate, survival function, etc) -> DFRMingxiao Jiang MEDTRONIC CONFIDENTIAL 3
  • 6. DFSS vs. DFR DFSS ANOVA Environmental & Usage Conditions DFR Regression VOC Life Data Analysis Flowdown Physics of Failure Hypothesis Testing QFD FMEA Accelerated Life Testing General Linear Model Control Plans Reliability Growth MSA Sensitivity Analysis Parametric Data Analysis Modeling DOE Warranty Predictions Tolerancing FA recognition etc. etc. DFR utilizes unique tools to improve reliability.Mingxiao Jiang MEDTRONIC CONFIDENTIAL 4
  • 7. DFR Process Development Timeline Concept, Requirements, Prototype Design & Prioritization Design Optimization Validation Production Environment Warranty Analysis & Usage Stressors Reliability Risk Prioritization DFM & Manufacturing Control Strategy Requirements & allocation Prior Products Pareto Physics of Failure Stress Testing FMEA Parametric Data Analysis Reliability Demonstration Test Failure Analysis Corrective Action & Preventative Action DFR activities are paced with development.Mingxiao Jiang MEDTRONIC CONFIDENTIAL 5
  • 8. For Example: Parametric Data Analysis Few failures Iceberg Full distribution Look at all the parts, not just the few failures! • Degradation metrics: • Up-stream metrics: Performance Performance measured measured during from supplier and during reliability test manufacturingMingxiao Jiang MEDTRONIC CONFIDENTIAL 6
  • 9. Classical Reliability Demonstration Test (CRDT) [1] r n   1  R L k R L n  k  1  C    k 0  k  Or 1 RL  r 1 1 FC ;2r  2;2(n  r ) nr “Success Run” test (r = 0): RL  (1  C )1 / n where, n is the test sample size, r is the given allowable number of failures, C is the confidence level, F( ) is the F distribution function, and RL is the testing reliability goal.Mingxiao Jiang MEDTRONIC CONFIDENTIAL 7
  • 10. RDT Challenges in DFR Sample size, n, needed in RDT: r=0 C r=2 C RL 90% 95% 99% RL 90% 95% 99% 90% 22 29 44 90% 52 61 81 95% 45 59 90 95% 105 124 165 99% 230 299 459 99% 531 628 837 r=4 C r=6 C RL 90% 95% 99% RL 90% 95% 99% 90% 78 89 113 90% 103 116 142 95% 158 181 229 95% 209 234 287 99% 798 913 1157 99% 1051 1182 1452 After reliability allocation in DFR, it is very challenging to conduct RDT.Mingxiao Jiang MEDTRONIC CONFIDENTIAL 8
  • 11. RDT: Classical vs. Bayesian Prior distribution of RDT planning E.g. Bayesian RDT w/ uniform prior distribution of  tradeoff: Reliability R  one less sample needed than classical RDT F C, RL , n, r   0 for zero failure test. 0 0 1 0 Reliability, R 1 • Classical RDT: no prior knowledge of R. • Bayesian RDT (Ref. 1-5): prior knowledge of R; challenging math for engineers. • Bayesian RDT w/ DFR (Ref. 6): prior knowledge of R weighted more to the right side; math simplified by spreadsheet calculations.Mingxiao Jiang MEDTRONIC CONFIDENTIAL 9
  • 12. Bayesian Approach – Discrete Case [1] Posterior P(Hi is true | data) Prior P(Hi is true) Conditiona l P(data | H i )  n  Prior P(Hi is true) Conditiona l P(data | H i ) i 1 Hi (i = 1, …, n) represent a mutually exclusive exhaustive collection of hypothesis. Suppose that an event S exists and the conditional probabilities P(S|Hi) are known. P(Hi) is termed as the prior probability that Hi is true, and P(Hi|S) is the posterior probability that Hi is true upon observing S.Mingxiao Jiang MEDTRONIC CONFIDENTIAL 10
  • 13. Bayesian Approach – Discrete Case, cont’ Example: A large number of identical units are received from two vendors, A and B. Vendor A supplies with nine times the number of units that vendor B supplies. Based on records, defective rate from A is 2% and defective rate from B is 6%. Incoming inspection randomly selects one unit and finds it to be defective. Q: which vendor produced it? Prior Conditional (Prior P) x Posterior Vendor probability probability (Conditional P) Probability A 0.9 0.02 0.018 0.75 B 0.1 0.06 0.006 0.25 1 1Mingxiao Jiang MEDTRONIC CONFIDENTIAL 11
  • 14. Bayesian Approach – Continuous Case f ( ) h(S |  ) Prob( | S)    f ( ) h(S |  ) d where, S represents a group of observed events, θ is a random scalar or vector to describe the parameters or statistics of the underline event distribution, Prob(θ|S) is the posterior probability density function of θ, f(θ) is the prior probability density function of θ, and h(S|θ) is the conditional distribution of S.Mingxiao Jiang MEDTRONIC CONFIDENTIAL 12
  • 15. Bayesian Reliability Demonstration Test (BRDT) If θ is the reliability R, and S is RDT result, then f (R ) h (S | R ) Prob(R | S)  1 0 f (R ) h (S | R ) dR The confidence level C for the true reliability within interval [RL, 1] can be obtained as: 1 R L f (R ) h (S | R )dR C(R L  R  1)  1 0 f (R ) h (S | R ) dRMingxiao Jiang MEDTRONIC CONFIDENTIAL 13
  • 16. h(S|R) For a certain product with a true reliability R, with S denoting the outcome of testing the whole population of sample size n, we have the conditional probability density function of S given R:  n  nr r h( S | R )    R r  (1  R)  Mingxiao Jiang MEDTRONIC CONFIDENTIAL 14
  • 17. Prior Distribution of Reliability - 1 a R 1  Rb Beta distribution: f ( R)  Bea, b  a  1  b  1 Where, Bea, b  a  b  2 Properties of Beta distribution: - Richness: being able to represent many states of prior information; - Conjugation: Beta prior distribution generates Beta posterior distributionMingxiao Jiang MEDTRONIC CONFIDENTIAL 15
  • 18. Prior Distribution of Reliability - 2 a=0, b=0 a=5, b=5 a=10, b=1 a=5, b=0 6 5 4 f(R) 3 2 1 0 0 0.2 0.4 0.6 0.8 1 RMingxiao Jiang MEDTRONIC CONFIDENTIAL 16
  • 19. Trade-off: (C, RL, r, n) 1 R n  a  r 1-Rb  r dR R L C ( RL  R  1)  Be(n  a  r , b  r ) For Success Run test, r = 0: 1 R n  a 1-Rb dR R L C ( RL  R  1)  Be(n  a, b)Mingxiao Jiang MEDTRONIC CONFIDENTIAL 17
  • 20. Reliability Prior Distribution in DFR Process - 1 If a product development adopts a DFR process, the prior distribution of reliability for the components or subsystems to be validated can be reasonably assumed to be of Beta distribution being heavily weighted to the right end of (0, 1), with a > b. 20 16 a = 10, b = -1 a = 10, b = 0 a = 10, b = 1 12 Density a = 10, b = 2 a = 20, b = -1 8 a = 20, b = 0 a = 20, b = 1 a = 20, b = 2 4 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ReliabilityMingxiao Jiang MEDTRONIC CONFIDENTIAL 18
  • 21. Reliability Prior Distribution in DFR Process - 2 • In the DFR risk prioritization phase, the reliability allocated to a specific component or subsystem could be very high. For example, a product under development may have an overall reliability requirement of 90% (for example, first year). Through FMEA and prior product Pareto assessment, about 10 critical components and subsystems are identified. For the sake of argument, assuming equal allocation of reliability requirement to each critical component or subsystem (a much better allocation approach can be done based on consideration of cost, risk level, etc) we have approximately 99% reliability as the requirement at one of these individual components or subsystems. • Throughout the DFR process with stress testing and PoF driven corrective actions, the reliability growth is tracked. Of course, this is subject to RDT to validate.Mingxiao Jiang MEDTRONIC CONFIDENTIAL 19
  • 22. Bayesian RDT in DFR Monte Fit prior R Statistics Carlo by Beta of prior R simulation distribution Ref: http://www.barringer1.com/w dbase.htm; Construct Telcordia; Simplified Mil-HDBK-217; Prior R NSWC (Naval Surface algorithm [6] Warfare Center) HDBK of Reliability Prediction Procedure for Mechanical Key parameters Equipment (Software Trade-off identified by MechRel); study, using CALCE; DFR (FMEA, Firm developed; spreadsheet PoF …) etc (RL, C, n, r)Mingxiao Jiang MEDTRONIC CONFIDENTIAL 20
  • 23. Simplified Algorithm for BRDT in DFR Step 1: Construct a prior reliability: R P  F( x1, x 2 ,...) where, RP is the prior reliability, and xk is the key input variable (could be random) identified : in DFR. Step 2: Obtain the prior distribution of RP: Monte Carlo simulation results with mean of prior reliability mRP and variance of prior reliability VRPMingxiao Jiang MEDTRONIC CONFIDENTIAL 21
  • 24. Simplified Algorithm for BRDT in DFR Step 3: Fit the Beta distribution as the prior distribution of reliability [1]: m RP  1  m RP 2 V RP  m RP  2  b V RP : m RP  b  2  1 a 1  m RPMingxiao Jiang MEDTRONIC CONFIDENTIAL 22
  • 25. Simplified Algorithm for BRDT in DFR (Cont’) Step 4: Conduct the trade-off study among RL, C, r and n (Ref 6): 100 C   G (k , n, r ) k 0 Where, k   inta   n  r   1  R  1   k b r 1  k  LG(k , n, r )    k  b  r  1Beinta   n  r, b  r  Simple Excel spread sheet calculation; no programming is needed.Mingxiao Jiang MEDTRONIC CONFIDENTIAL 23
  • 26. Example Allocated Reliability goal > 99% @ 5-year Accelerated RDT w/ usage stress and PoF: AF = 50  TimeRDT = 0.1yr Wearout  Weibull shape:  ~ U1, 4 PoF  Weibull scale:  ~ U0.7, 1.4 yr Zero failure test Confidence sample size 0.9 0.95 0.99 Classical RDT 230 299 459 Bayesian RDT 81 132 263Mingxiao Jiang MEDTRONIC CONFIDENTIAL 24
  • 27. Remarks - 1 • Successful application of a Bayesian approach depends on the prior experience or life data (testing or field) from previous generations of the product under design. BRDT can still be used successfully for a totally new product design and development, based on the prior distribution characteristics of reliability in a DFR process. • DFR activities aid estimation of prior reliability. BRDT can be integrated into the whole DFR process by linking it to FMEA, PoF, and reliability requirement flow down or allocation.Mingxiao Jiang MEDTRONIC CONFIDENTIAL 25
  • 28. Remarks - 2 • Estimating prior reliability quantifies the interim effectiveness of the DFR process: the more effective upstream DFR effort, the more efficient and often earlier RDT. This can feed into reliability growth analysis useful for the BRDT design. • Bayesian reliability approaches involve challenging mathematical operations for engineers. The illustrated numerical approach can be used easily by engineers with any standard spreadsheet calculation methodology, for success run test or test with failures. • Bayesian RDT is more efficient and cost effective than Classical RDT.Mingxiao Jiang MEDTRONIC CONFIDENTIAL 26
  • 29. References [1] Kececioglu D, Reliability & Life Testing Handbook, Vol.2, PTR Prentice Hall, 1994. [2] Kleyner A et al., Bayesian Techniques to Reduce the Sample Size in Automotive Electronics Attribute Testing, Microelectronics Reliability, Vol. 37, No. 6, 879-883, 1997. [3] Krolo A et al., Application of Bayes Statistics to Reduce Sample-size Considering a Lifetime-Ratio, Proceedings of Annual Reliability and Maintainability Symposium, 577-583, 2002. [4] Lu M-W and Rudy R, Reliability Demonstration Test for a Finite Population, Quality and Reliability Engineering International, Vol. 17, 33-38, 2001. [5] Martz H and Waller R, Bayesian Reliability Analysis, Krieger Publishing Company, 1982. [6] Jiang M and Dummer D, Bayesian Reliability Demonstration Test in a Design for Reliability Process, PROCEEDINGS Annual Reliability and Maintainability Symposium, 2009.Mingxiao Jiang MEDTRONIC CONFIDENTIAL 27
  • 30. Q&A Thank you! Mingxiao.Jiang@medtronic.comMingxiao Jiang MEDTRONIC CONFIDENTIAL 28