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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, 2011



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Bayesian Reliability Demonstration
                 Test in a
       Design for Reliability Process


                 Mingxiao Jiang (Medtronic Inc.)


                                      2011


Mingxiao Jiang        MEDTRONIC CONFIDENTIAL
                                                   1
Outline

             - Design for Reliability (DFR) process

             - Challenges of Reliability Demonstration
             Test (RDT) in DFR Validation phase

             - Bayesian RDT (BRDT) with DFR

             - Concluding remarks




Mingxiao Jiang           MEDTRONIC CONFIDENTIAL
                                                         2
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) -> DFR

Mingxiao Jiang            MEDTRONIC CONFIDENTIAL
                                                                 3
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
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
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       manufacturing
Mingxiao Jiang    MEDTRONIC CONFIDENTIAL
                                                           6
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
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
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
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
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                                      1

Mingxiao Jiang           MEDTRONIC CONFIDENTIAL
                                                                         11
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
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 ) dR
Mingxiao Jiang             MEDTRONIC CONFIDENTIAL
                                                                  13
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
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 distribution
Mingxiao Jiang        MEDTRONIC CONFIDENTIAL
                                                               15
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
                                                         R
Mingxiao Jiang                  MEDTRONIC CONFIDENTIAL
                                                                                    16
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
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
                                                                  Reliability

Mingxiao Jiang                                 MEDTRONIC CONFIDENTIAL
                                                                                                        18
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
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
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 VRP
Mingxiao Jiang     MEDTRONIC CONFIDENTIAL
                                                         21
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 RP
Mingxiao Jiang     MEDTRONIC CONFIDENTIAL
                                                     22
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               L
G(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
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    263
Mingxiao Jiang            MEDTRONIC CONFIDENTIAL
                                                                             24
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
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
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
Q&A




                        Thank you!

            Mingxiao.Jiang@medtronic.com



Mingxiao Jiang     MEDTRONIC CONFIDENTIAL
                                            28

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Bayesian reliability demonstration test in a design for reliability process

  • 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, 2011 http://reliabilitycalendar.org/The_Reli ability_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 events http://reliabilitycalendar.org/The_Reli ability_Calendar/Webinars_‐ _Chinese/Webinars_‐_Chinese.html
  • 3. Bayesian Reliability Demonstration Test in a Design for Reliability Process Mingxiao Jiang (Medtronic Inc.) 2011 Mingxiao 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 remarks Mingxiao 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) -> DFR Mingxiao 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 manufacturing Mingxiao 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 1 Mingxiao 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 ) dR Mingxiao 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 distribution Mingxiao 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 R Mingxiao 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 Reliability Mingxiao 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 VRP Mingxiao 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 RP Mingxiao 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  L G(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 81 132 263 Mingxiao 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.com Mingxiao Jiang MEDTRONIC CONFIDENTIAL 28