bayesian reliability demonstration test in a design for reliability process

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 JiangPresented live on Jul 13th, 2011

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Mingxiao Jiang MEDTRONIC CONFIDENTIAL1

Bayesian Reliability Demonstration

Test in a

Design for Reliability Process

Mingxiao Jiang (Medtronic Inc.)

2011


Outline

- Design for Reliability (DFR) process

- Challenges of Reliability Demonstration

Test (RDT) in DFR Validation phase

- Bayesian RDT (BRDT) with DFR

- Concluding remarks


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


DFSS vs. DFR

DFSS DFR

VOC

MSA

DOE

Control Plans

ANOVA

QFD

FMEA

RegressionFlowdown

Environmental &

Usage Conditions

Life Data Analysis

Physics of Failure

Accelerated Life Testing

Reliability Growth

Warranty Predictions

Parametric Data Analysis

General Linear Model

Tolerancing

Sensitivity AnalysisModeling

Hypothesis Testing

FA recognition

DFR utilizes unique tools to improve reliability.

etc.etc.


DFR Process

DFR activities are paced with development.

Corrective Action & Preventative Action

Reliability Demonstration Test

Warranty

Analysis

DFM & Manufacturing Control Strategy

Development Timeline

Parametric Data Analysis

Failure Analysis

Environment

& Usage Stressors

Stress Testing

Concept, Requirements,

& Prioritization

Prototype

Design

Design

Optimization Validation Production

Re

liab

ility

Ris

k P

rio

ritiz

atio

nP

rior

Pro

duct

s P

are

to

Req

uirem

ents

& a

lloca

tion

FM

EA

Physics of Failure


For Example: Parametric Data Analysis

Iceberg

Few failures

Full

distribution

Look at all the parts, not just the few failures!

• Up-stream metrics:

Performance measured

from supplier and during

manufacturing

• Degradation metrics:

Performance

measured during

reliability test


Classical Reliability Demonstration Test (CRDT) [1]

r

k

knL

kL CRR

k

n

0

1 1

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.

)(2;22; 1

1

1

rnrC

L

Frn

rR

Or

“Success Run” test (r = 0): n

L CR /1)1(


RDT Challenges in DFR

After reliability allocation in DFR, it is very

challenging to conduct RDT.

Sample size, n, needed in RDT:

r = 0 r = 2

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 r = 6

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

C

C C

C


RDT: Classical vs. Bayesian

• 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.

0

0 1

Reliability, RPrio

r d

istr

ibu

tio

n o

f

Relia

bili

ty

10

E.g. Bayesian RDT w/

uniform prior distribution of

R one less sample

needed than classical RDT

for zero failure test.

RDT planning

tradeoff:

0,,, rnRCF L


Bayesian Approach – Discrete Case [1]

n

1iii

ii

i

)H|P(data lConditiona true)is P(HPrior

)H|P(data lConditiona true)is P(HPrior

data)| trueis P(HPosterior

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.


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?

Vendor

Prior

probability

Conditional

probability

(Prior P) x

(Conditional P)

Posterior

Probability

A 0.9 0.02 0.018 0.75

B 0.1 0.06 0.006 0.25

1 1


Bayesian Approach – Continuous Case

d )|S(h )(f

)|S(h )(f)S|(Prob

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.


Bayesian Reliability Demonstration Test (BRDT)

If θ is the reliability R, and S is RDT result, then

10 dR )R|S(h )R(f

)R|S(h )R(f)S|R(Prob

10

1R

LdR )R|S(h )R(f

dR)R|S(h )R(f)1RR(C L

The confidence level C for the true reliability

within interval [RL, 1] can be obtained as:


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:

rrn RRr

nRSh )1( )|(


Prior Distribution of Reliability - 1

baBe

RRRf

ba

,

1)(

Beta distribution:

2

11,

ba

babaBeWhere,

Properties of Beta distribution:

- Richness: being able to represent many

states of prior information;

- Conjugation: Beta prior distribution generates

Beta posterior distribution


Prior Distribution of Reliability - 2

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1

R

f(R

)

a=0, b=0 a=5, b=5 a=10, b=1 a=5, b=0


Trade-off: (C, RL, r, n)

),(

1

)1(

1

rbranBe

dR-RR

RRC LRrbran

L

For Success Run test, r = 0:

),(

1

)1(

1

banBe

dR-RR

RRC LRban

L


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.

0

4

8

12

16

20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Reliability

De

ns

ity

a = 10, b = -1

a = 10, b = 0

a = 10, b = 1

a = 10, b = 2

a = 20, b = -1

a = 20, b = 0

a = 20, b = 1

a = 20, b = 2


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.


Bayesian RDT in DFR

Construct

Prior R

Key parameters

identified by

DFR (FMEA,

PoF …)

Monte

Carlo

simulation

Fit prior R

by Beta

distribution

Trade-off

study, using

spreadsheet

(RL, C, n, r)

Simplified

algorithm [6]

Statistics

of prior R

Ref:

http://www.barringer1.com/w

dbase.htm;

Telcordia;

Mil-HDBK-217;

NSWC (Naval Surface

Warfare Center) HDBK of

Reliability Prediction

Procedure for Mechanical

Equipment (Software

MechRel);

CALCE;

Firm developed;

etc


Simplified Algorithm for BRDT in DFR

,...)x,x(FR 21P

Step 1: Construct a prior reliability:

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


Simplified Algorithm for BRDT in DFR

:

Step 3: Fit the Beta distribution as the prior

distribution of reliability [1]:

RP

RPRPRPRP

V

mVmmb

21 2

RP

RP

m

bma

1

12


Simplified Algorithm for BRDT in DFR (Cont’)

Step 4: Conduct the trade-off study among RL,

C, r and n (Ref 6):

100

0

),,(k

rnkGC

Where,

rbrnaBerbk

Rk

rna

rnkG

rbkL

k

,int1

1int

1

),,(

1

Simple Excel spread sheet calculation; no

programming is needed.


Example

Allocated Reliability goal > 99% @ 5-year

Accelerated RDT w/ usage stress and PoF:

AF = 50 TimeRDT = 0.1yr

WearoutWeibull shape:

PoF Weibull scale:

4 ,1U~

yr 1.4 ,7.0U~

0.9 0.95 0.99

Classical RDT 230 299 459

Bayesian RDT 81 132 263

ConfidenceZero failure test

sample size


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.


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.


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.


Q & A

Thank you!

[email protected]

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