planning 2-stage accelerated life tests department of industrial and systems engineering national...

Planning 2-Stage Accelerated Life Tests

Department of Industrial and Systems Engineering

National University of Singapore

LC Tang (董润楨 ), Ph.D

Overview

• Planning a sequential Accelerated Life Test (ALT)

• Motivation of using an Auxiliary Stress (AS)

• An integrated planning framework for sequential ALT with an AS

• Numerical illustrations

A Constant-Stress ALT

Low Stress

Mid Stress

High Stress

Time

Stress LevelUse Stress

Maximum Test Duration

Probability distributions

Life-stress relationship

A Scale-Accelerated Weibull Lifetime Model

• Standardization of stress

• Weibull lifetime distribution at any stress

• A scale-accelerated failure time model

1

0

0 : use stress : the highest stressk k

k

x x s s s s s

s s

SEV

log -T

0 1 1, 0

is a constant independent of stress

x

Motivations of Sequential ALT Planning

• ALT planning based on the Maximum Likelihood theory

• Locally optimal for specified model parameters

• Problems:– There often exists a high margin of specification error

– Developed plans are usually sensitive to the specified value

Step 1:

Specify ALT model parameter values

Step 1:

Specify ALT model parameter values

Step 2:

Minimize the asymptotic variance of ML estimator

Step 2:

Minimize the asymptotic variance of ML estimator

Step 3:

Evaluate the plan using simulations

Step 3:

Evaluate the plan using simulations

0 1, ,

A Framework of Sequential ALT Planning

• Tang, L.C. and Liu, X. (2010) “Planning for Sequential Accelerated Life Tests”, Journal of Quality Technology, 42, 103-118.

• Liu, X. and Tang, L.C. (2009) “A Sequential Constant-Stress Accelerated Life Testing Scheme and Its Bayesian Inference”, Quality and Reliability Engineering International, 25, 91-109.

Information on the slope parameter

Preliminary information on

Plan the tests at lower stress levels

Planning information

e.g. test duration, specified parameter values, etc.

Planning information

e.g. test duration, number of stress levels, sample sizes, etc.

0( , )

Information Planning Procedure

Plan & Perform the test at the highest stress to quickly obtain failures

1

Part IPlanning Sequential Constant-Stress Accelerated Life Tests

Sample Size at the Highest Stress Level

0

Specify

the values of (or ) and

the censoring time

the expected number of failures

H

H

H

c

R

1/1 exp / exp( )H

H HH

Rp p c

n

Sample Size:

Inference at the Highest Stress Level

Stress

Time in log-scale

Low UseHigh0 1

; ,H H H H Hl θ θ D θ

Inference at the Highest Stress Level

1

2 2ˆ

ˆ ˆ ˆ | ~ ( , )

where

ˆ arg max ( )

ˆ ˆ

ˆ [ ( ; ) / ] H H

H H H H

H H

H H

H H H H

N

lθ θ

θ y θ Σ

θ θ

Σ I

I θ D θ

Generalized MLE

Covariance matrix

Observed information

1

Information on

the value of

,H

,k

1

is a constanti k ix

, for 0,1i ix

Construction of Prior Distributions

Construction of Priors at Low Stresses

1

1

1 1 1

11 1

2~ ( , )

3/ 2 1/ 2

1/ 21 1 1

( )( ) ( , ) ( )

ˆerf ( ) erf ( ) ( )exp

ˆ ˆ2 ( var( )) ( ) 2 var( )

where

ˆ ˆ ˆ ˆ ˆ ˆ, , , cov( , ) / var (

i ii i i

i

Ui i i H

H i i H

i H i i H i i H i H H

xx dF

x x x

θ

1/ 2

1/ 2 1/ 2 1/ 2

2 1/ 2

1/ 2 1/ 2 1/ 2

2 1/ 2

ˆ) var ( )

ˆ ˆ ˆ ˆvar ( ) var ( ) ( ) var ( )ˆ ˆ(2 var( ) var( )(1 ))

ˆ ˆ ˆ ˆvar ( ) var ( ) ( ) var ( )ˆ ˆ(2 var( ) var( )(1 ))

H H

i H i H i H Hi

H H

i H i H i H Hi

H H

for any 1,..., 1, there exists a one-one transformation ( )

with non-vanishing Jacobian / ,such thati H

H i

i H θ θ

θ θ

Illustration of the Sequential ALT

Stress

Time in log-scale

Low UseHigh0 1

Plan & Run the test at the highest stress

Deduction of Prior Distributions

Pre-Posterior Analysis & Optimization

The Bayesian Optimization Criterion

Given the information obtained under the highest stress, the optimum sample allocation and stress combinations for tests under lower stresses are chosen to minimize the pre-posterior expectation of the posterior variance of certain life percentile under use stress over the specified range of β1

1

1 0

Min ( ) = {var( (1))}

= { var( ) } [1, log( log(1 ))]

p

T

C E y

E p

ξ

c θ c c

Problem Formulation

1 1

1 1 1T

H Hx x xX

1

1 1

11 2 1

1 2

Min (var( (1))) ( )

s.t. {( , ,..., ) : 0 1} and 0

{( , ,..., ) : 1 and 0 1}

T Tp

HH i H

HH i ii

E y

x x x x x

1 X Λ X 1

x

π

Design Matrix

1

1

1

1

(var( ( )))

(var( ( )))

var( (0))

p

p H

p

E y x

E y x

y

Λ

Pre-Posterior Analysis

1

2 2

2 2

2

2

where

log log;

log

i

i

i

i

i

i i

i i

i

i

l lE y f y dy

θ

θ

Σ I I

θ θI

θ θ

θI

θ

Information contained in the prior density

Information expected to obtain at stress level i

1

11

11 1

1 (var( ( )))p i iE y x dcΣ c

Adhesive Bond Test (Meeker and Escobar 1998)

5

1 10 1 0 1 0

log( ) ~ ,

Activation energy, 1log (Arrhenius)

Boltzmann constant, 8.6171 10

, log , ( )

is a constant

SEV

a

B

a B H a B H

T

EA

k T

x A E k s E k s s

• Total Sample Size: 300 • Total Testing Duration: 6 months =183days• Standardized Testing Region: • Assumptions:

0 1H Ux x x

Planning at the Highest Temp

Planning

information:

4.72

0.6

15

60

H

H

H

R

c

50 samples are needed50 samples are needed H HR n p

Posterior DensitySimulated Failure times:

33.3, 48.4, 39.3, 58.8, 47.4, 60.0, 33.6, 19.4, 38.0, 28.6, 60.0, 53.2, 17.7, 25.4, 44.5, 34.6, 16.9, 60.0, 31.7, 60.0 ,49.2, 60.0, 10.953, 60.0, 18.8, 3.3, 1.4, 17.3, 46.8, 40.9, 60.0, 28.4, 60.0, 4.2, 21.9, 49.6, 20.6, 60.0, 46.6, 6.4, 25.2, 60.0, 13.6, 29.5, 60.0, 60.0, 31.3, 29.4, 54.3, 34.0

Normal Approximation

1 2 2ˆ

ˆ ˆ ˆ| data ~ ( , )

ˆ ˆ ˆ where and [ ( ) / ] H H

H H H

H H H H H

N

lθ θ

θ θ Σ

Σ I I θ θ

Planning of an ALT with 2 Stress Levels

1

1

Sample size

300 250

Test duration

183 123

Posterior density at

ˆ ˆ | ~ ,

Specified range of

3.84,5.12

( . . 0.6,0.8 )

L H

L H

H

H H H H

a

n n

c c

x

N

i e E

θ y θ Σ

0. 01

0. 1

1

10

100

0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1

1

var 1 in log-scalepE yPlanning Information:

LxHigh Low

0. 01

0. 1

1

10

100

0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1

Effects of the pre-specified slope parameter

Lx

Suppose we raise the expectation of the product reliability

Effect:

Run the test under a higher stress to produce more failures

Effect:

Run the test under a higher stress to produce more failures

1

var 1 in log-scalepE y

Beta1 ranges from 3.84 to 5.76

Beta1 ranges from 3.84 to 5.12

1 10.6,0.8 0.6,0.9 i.e. 3.84,5.12 3.84,5.76Ea Ea

High Low

Plan an ALT with 3 stress levels

1

250, 123

, , 3.84,5.12L L

H

n c

250 (1 )

250 for 0 1

2Minimum number of failure

and

L

M

LM

L M

n

n

xx

R R

Planning Information:

Additional constraints:

1 0.1Min (var( (1)); , )

. .

(1 ) ( )

( )

0 2 1

0 1

L

L M L L

L M M M

H L M

E y x

s t

n p x R

n p x R

x x x

0

0

where

( ) (1 exp( exp( ))) ( , )

( ) (1 exp( exp( ))) ( , )

L L L L L L

M M M M M M

p x d d

p x d d

The feasible region

Interior Penalty Function Method

Inference from Test Results

Stress

Level

Sample

Size

Test

Duration

Expected

Failures

Simulated

Failure Times

Observed

Failures

High50 60 15

33.3, 48.4, 39.3, 58.8, 47.4, 60.0, 33.6, 19.4, 38.0, 28.6, 60.0, 53.2, 17.7, 25.4, 44.5, 34.6, 16.9, 60.0, 31.7, 60.0 ,49.2, 60.0, 10.953, 60.0, 18.8, 3.3, 1.4, 17.3, 46.8, 40.9, 60.0, 28.4, 60.0, 4.2, 21.9, 49.6, 20.6, 60.0, 46.6, 6.4, 25.2, 60.0, 13.6, 29.5, 60.0, 60.0, 31.3, 29.4, 54.3, 34.0

38

Mid20 123 5

46.1 62.5 86.2 98.9 101.7 123 (×224) 5

Low230 123 5

22.8 44.8 59.1 84.4 87.7105.2 123 (×224) 60.78Lx

0Hx

0 1(assume 4, 4, 4) Simulated failure times

0.39Mx

0.0377 0.0060

~ 7.24,0.664 ,0.0060 0.0042

,L

N

0.0112 0.0003

~ 3.87,0.65 ,0.0003 0.0086

,H

N

• Results obtained under the high stress

• Results obtained under the mid and low stress

0.0156 0.0016

~ 5.28,0.594 ,0.0016 0.0080

,M

N

Decreasing

Increasing

Inference

Planning information:

Total Sample Size: 300 Total Test Duration: 183 Pre-specified ALT model parameters: 9 scenarios are considered

*For sequential plans:We set the expected number of failures at the high stress level at 15 within 60 days

*For each simulation scenario:a. both sequential and non-sequential plans are generated;b. failure data are generate according to the optimum plans;c. 10th percentile are use stress are estimated;d. repeat b and c for 100 times, and move to another scenario

Simulation Study

Simulation Design Table

Scenarios Pre-specified Pre-specified Pre-specified

(non-sequential)

Pre-specified

(sequential)

1 ( 0 ) ( 0 ) ( 0 ) - 20 % ~ + 20 %

2 - 25 % - 25 % - 20 % - 20 % ~ + 20 %

3 - 25 % - 25 % + 20 % - 20 % ~ + 20 %

4 - 25 % + 25 % - 20 % - 20 % ~ + 20 %

5 - 25 % + 25 % + 20 % - 20 % ~ + 20 %

6 + 25 % - 25 % - 20 % - 20 % ~ + 20 %

7 + 25 % - 25 % + 20 % - 20 % ~ + 20 %

8 + 25 % + 25 % - 20 % - 20 % ~ + 20 %

9 + 25 % + 25 % + 20 % - 20 % ~ + 20 %

110

- k %: the specified value is k% lower than the true value+k %: the specified value is k% higher than the true value(0): the specified value is the true value

Simulation Results

Precision1. Sequential plans yields more precise estimation

2. Sequential plans gives a conservative sense of statistical precision: Sample variance > Asymptotic variance

00. 1

0. 20. 30. 4

0. 50. 6

0. 70. 8

0 1 2 3 4 5 6 7 8 9 10Si mul ati on scenari os

Vari

ance

Sample variance (non-sequential plan)

Asymptotic variance (non-sequential plan)

Asymptotic variance (sequential plan)

Sample variance (sequential plan)

For sequential plan:

Since

1. Model parameters and are estimated at stage one;

2. An interval value of is used

Hence, the plan robustness to the mis-specification of model parameters has been enhanced

Effect of Parameter Mis-specification on Precision

Effect on the

expected variance

Effect on the

observed variance

0.270 0.1945

0.016 -0.2075

0.044 0.043

-0.007 -0.1180

0.083 0.0655

0.035 0.0185

-0.031 -0.009

0

1

0 1*

0 *

1 *

0 1* *

Effect on the expected variance

Effect on the observed variance

-0.053 -0.038

(0, 0.0001) (- 0.0001,0)

(- 0.0001,0) (- 0.0001,0)

0

0 *

Non-sequential Plans Sequential Plans

For non-sequential plan:

Results are sensitive to the specified model parameters and .

0

0

1

1

RobustnessDefine the Relative Error (RE) as:

3. Sequential plans is more robust to mis-specification of model parameters

sample variance - asymptotic variance asymptotic variance

0

0. 5

1

1. 5

2

2. 5

0 1 2 3 4 5 6 7 8 9 10

Si mul at i on scenar i os

RE

RE

(non-sequential plan)

RE

(sequential plan)


Since




Effect of Parameter Mis-specification on the Relative Error (RE)

Effect-0.7684

-0.7187

-0.2367

0.4905

0.1334

0.1201

-0.0532

0

1

0 1*

0 * 1 *

0 1* *

Effect0.0011

(0, 0.0001)

(0, 0.0001)

0

0 *



RE is sensitive to the pre-specified model parameters and .

0

0

1

1

0

20

40

60

80

100

120

0 1 2 3 4 5 6 7 8 9 10

Temp

erat

ure

Si mul ati on scenari os

Simulation Results4. Sequential plans reduce the degree of extropolation;

5. Sequential plans are especially robust to mis-specification of the intercept parameters (scenarios 6-9) due to the preliminary test under the high stress

Optimum low stress(non-sequential plan)

Optimum low stress(sequential plan)

Use stress


Since




Effect of Parameter Mis-specification on the Optimum Low Stress level

Effect12.25

-13.25

3.25

-1.25

3.25

1.75

0.75

0

1

0 1*

0 * 1 *

0 1* *

Effect-5

(- 0.0001,0)

(- 0.0001,0)

0

0 *



RE is sensitive to the pre-specified model parameters and .

0

0

1

1

Comparison with 4:2:1 Plan

|Ase Ase when model parameters are correctly specified| ASR

Ase when model parameters are correctly specified

: Test duration at the highest stress levelc

Extension from 2-Stage Planning to a Full Sequential Planning

2-Stage Planning

• Prior distributions for all low stresses are constructed simultaneously (all-at-one)

• Tests at all low stresses are planned simultaneously

Full Sequential Planning

• Only the prior distribution for one low stress is constructed

• Only the test at one low stresses are planned

• More tests at low stresses are planned iteratively

The basic framework still works !

Part IIPlanning Sequential Constant-Stress Accelerated

Life Tests with

Stepwise Loaded Auxiliary Stress

Liu X and Tang LC (2010), “Planning sequential constant-stress accelerated life tests with stepwise loaded auxiliary acceleration factor”, Journal of Statistical Planning and Inference, 140, 1968-1985.

Motivations of an Auxiliary Stress

• Testing more units near the use condition is intuitively appealing, because more testing is being done closer to the use condition

• Failures are elusive at low stress levels for highly reliable testing items – the lowest stress level is forced to be elevated, resulting in high,

sometimes intolerable, degree of extrapolation in estimating product reliability at use stress

Illustration

Candidate low stress 1

Candidate low stress 2

High Stress

Time

Stress LevelUse Stress

Maximum Test Duration

Low degree of extrapolation with zero failure

high degree of extrapolation with more failures

Auxiliary Stress

• An Auxiliary Stress (AS), with roughly known effect on product life, is introduced to further amplify the failure probability at low stress levels

• Examples of possible AS:– In the reliability test of micro relays operating at difference levels of

silicone vapor (ppm), the usage rate (Hz) might be used as an auxiliary factor (Yang 2005).

– In the temperature-accelerated life test, the humidity level controlled in the testing chamber might be used as an AS (Livingston 2000).

– Dimension of testing samples (Bai and Yun 1996)

• Joseph and Wu (2004) and Jeng et al. (2008) proposed a method known as the Failure Amplification Method (FAMe) for the Design of Experiments. – FAMe was developed for system optimization while ALT is used for

reliability estimation at user condition through extrapolation.

Model Extension

1use max use

use max

( ) ( )

: use stress : the highest stress

h v v v v

v v

0 1 2

is a constant independent of stress

x h

2[0,1]

An Integrated Framework of Sequential ALT Planning with an Auxiliary Stress

Planning Information

e.g. Sample size; Test duration; Specified model parameters

Step 1: Plan and perform the life test at the highest stress level

Step 2: Compute the number of failures at low stresses

Is an AS needed?

Step 3a: Plan the tests at low stresses without an AS

i.e. optimize sample allocation, and stress combinations

Step 3b: Plan the tests at low stresses with an AS

i.e. optimize sample allocation, stress combinations, and the loading profile of AS

Is an AS available?

yes

No

yes

No

Step 1Planning & Inference

at the Highest Temperature Level

To demontrate the 10% life quantile at use condition exceeds 2 years

Temperature (other factors, such as humidity, voltage, etc are set to use level)

Experiment Target:

Stress Factor:

Planning in

0

0

1). 120 sample units and 75 days are available.

2). The use temperature is 45 C 318

The highest temperature allowed in the test is 85 C 358

3). Failure t

formation and Assumptions:

K

K

ime follows Weibull distribution

log t

4). is a constant, independent of temperature; follows Arrhenius stress-life

relationship

T

F t

0 1

0 1

Activation energy, 1 log

Boltzmann constant,

where log 11605 /

i iB i

i i

EaA s

k T

A Ea s T

ALT for Electronic Controller

1/

0

target number of failures:

censoring time: exp exp

parameter values: ( ),

confidence level:

k

kk

k

kkR c

r

c

kn

Planning Inputs:

Planning Output:

Testing Output: 0ˆˆ ˆ( ) o r H

1

0

1 1

(Binomial Bogey test, Yang 2007)

k

k

k

ri n ii

n k ki

C R R

Risk of see less failures than expected

Test Planning at the Highest Stress

Planning

information:

7.5

0.677

6

720hr

0.9

k

k

k

r

c

44 samples are needed

Results

Weibull Probability Plot for Observed Failure Data

Data Obtained at the Highest Stress

Time-to-failure (hours)

79.559 210.47 590.03 400.56 491.41 138.94 673.98 109.4 149.95 204.7 425.32 643.31 117.15 328.99 351.87 720×29

Note: This is just a particular run

1

; ;

1exp log exp 1 exp

where ( , ); 0 if the data is censored, otherwise 1

k

k k

nj k j k k k

j jj

k k

y l

y y c

θ θ y

θ

Posterior distribution derived from a constant prior :

Normal Approximation to the Posterior distribution (Berger 1985)

1

2

2ˆ

1

ˆ ˆˆ ˆ~ ( , ) ( ,[ ] )

( ; ) ˆˆwhere = (observed Fisher information at )

0.1142 0.0529ˆ ˆ ˆ [7.35,0.90]symmetric 0.0489

k k

k k k k k

kk k

k

k k k

N N

l

θ θ

θ y θ Σ θ I

θ yI θ

θ

θ Σ I

Statistical Inference at the Highest Stress

• The quality of the approximation needs to be checked e.g. Kolmogorov-Smirnov (K-S) test (Martz et.al 1988, Technometrics). • The posterior normality needs to be checked e.g. Kass and Slate 1994 Ann. Statist. ).

Illustration

Step 2Computation of the Expected

Number of Failures at Low Stress Levels

1

Information on

the value of

,H

,k

1

is a constanti k ix

, for 0,1i ix

Construction of Prior Distributions

1

1

11 1 1

2

3/ 2 1/ 2

1

( ), ( , ) ( )

ˆ1 ( ) exp erf ( ) erf ( ) 1,..., 1

ˆ ˆ2 ( var( )) ( ) 2 var( )

where ( ) is a uniform distribution defined on an interval

i ii i i

i

i ii i

xx d

i k

2

1 1

1/ 2

0

1

1/ 2 1/1 1

[ , ]

erf is the error function given by the definite integral erf ( ) 2

ˆ

ˆ ˆ ˆ ˆ ˆ , , cov( , ) / var ( ) var

z t

i k i

i k i i k i k k

z e dt

x

x x

2

1/ 2 1/ 2 1/ 2

2 1/ 2

1/ 2 1/ 2 1/ 2

2 1/ 2

ˆ( )

ˆ ˆ ˆ ˆvar ( ) var ( ) ( ) var ( )

ˆ ˆ(2 var( ) var( )(1 ))

ˆ ˆ ˆ ˆvar ( ) var ( ) ( ) var ( )

ˆ ˆ(2 var( ) var( )(1 ))

i i ki

k

i i ki

k

Density Function of the Constructed Prior

Uncertainty over becomes larger for lower testing temperatureUncertainty over becomes larger for lower testing temperature

1Let 0.8,1.2 , i.e ~ Uniform 0.8,1.2Ea

Illustration of the Constructed Priors at 65⁰C and 45⁰C

In order to see 5 failures, the temperature is almost on the middle of the testing region !!

Expected Number of Failures at Low Stress

Another Point of View:Prior Information v.s Information To Be Obtained

2 2

2 2

det ( ) log ( ) where = and =

det i

i i ii i

i i

lE

I θ θ θI θ I θ

I θ θ

Information to be obtained by performing a test at stress level i

“Information” contained in the prior knowledge

Little Information obtained from low temp

Step 3Planning at the Lower

Temperature LevelWith Auxiliary Stress

•The choice of AS•The loading of AS•The integration of AS in test planning

1). The effect of AS is well understood

2). The failure mode does NOT change after an AS is introduced

Assumpotions:

Auxiliary Stres Humidity

Hallberg-Peck Model (Livingston,

s:

2000

0

00 1

: use humidity level, 60%

: humidity level in test ( 90%

l

)

og

):

js p

R

RH

H

RH

RH

The Choice of AS

Constant-Stress Loading

Step-Stress Loading

A 2-step step-stress loading profile is preferred due to the following reasons:

• The initial loading will not be too harsh• The stress can be dynamically monitored given a target time

compression factor (only amplify the failure as needed)• The verification of the effect of the selected AS is possible

A 2-step step-stress loading profile is preferred due to the following reasons:

• The initial loading will not be too harsh• The stress can be dynamically monitored given a target time

compression factor (only amplify the failure as needed)• The verification of the effect of the selected AS is possible

The Choice of Loading Profile for AS

( )equivalent test duration,

actual test duration, Time Compression Factor:

ei

ii

c

c

Setting a Target Acceleration Factor

LCEM Cumulative Exposure Model(Yeo and Tang 1999, Tang 2003)

A Bayesian Planning Problem1

1 1

11 2 1

111 2 1 1

11 2 1

11 2 1

Min (var( (1))) ( )

s.t. target time compression , for 1,..., 1

( , ,..., ) [0,1]

( , ,..., ) [0,1] : 1

( , ,..., ) [0,1]

( , ,..., ) [0, ]

whe

T Tp

i

kk

kkk i ki

kk

kk

E y

i k

x x x

h h h

c

1 X Λ X 1

1

1

1 2

1

2

re

1 1 1

(var( ( ))) 0 0 0

0 (var( ( ))) 0 0

0 0 0 var( ( ))

T

k

p

p

p k

x x x

E y x

E y x

y x

X

Λ

Stress levels

Sample allocation

Initial level of AS

Stress changing time for AS

1 2

1

12 2 2

1

Sample size

120 76

Test duration

1800 720 1080

Posterior density at

ˆ ˆ ~ ,

~ Uniform 0.8,1.2

3

Maximum RH = 90%

Use RH = 60%

3

H

n n

c

x

N

p

θ θ I θ

Planning Information:

Humidity Loading Profile at Low Temperature

Testing Condition

Temp(C)

RH(%)

Testing Duration

Sample Size

Use 45 60

Low 53 See Profile

1080hrs 76

High 85 60 720hrs 44

Low Humidity Level: 60%High Humidity Level: 90%Holding Time: 170.5 hrs Expected Failures: Interval [0, 170.5] : No failure Interval [170.5,1080]: 5 failures Interval [1080, ): 71 censored

Planning Results

Temperature

Relative Humidity

85 53

60%

Point A: (85, 60%)Failure Probability = 0.32Point A: (85, 60%)Failure Probability = 0.32

Point B: (53, 60%)Failure Probability < 0.01Point B: (53, 60%)Failure Probability < 0.01

Illustration: ALT without AS

Temperature

85 53

60%

90%

Point A: (85, 60%)Failure Probability = 0.32Point A: (85, 60%)Failure Probability = 0.32

Point C: (53, 60%)Failure Probability < 0.01Point C: (53, 60%)Failure Probability < 0.01

Point D: (53, 90%)Failure Probability = 0.08Point D: (53, 90%)Failure Probability = 0.08

Illustration: ALT with ASRelative Humidity

RHT:Relative change of low humidity holding time

RT/RHRelative change of low humidity/low temperature

Sensitivity of the Optimal Plan to p

RSD:Relative change of Asymptotic SD

Sensitivity of the Plan to the Activation Energy

RHT:Relative change of low humidity holding time

RT/RHRelative change of low humidity/low temperature

RSD:Relative change of Asymptotic SD

Evaluation of the Developed ALT Plan

References of Part II

• Liu X and Tang LC (2010), “Planning sequential constant-stress accelerated life tests with stepwise loaded auxiliary acceleration factor”, Journal of Statistical Planning and Inference, 140, 1968-1985.

planning 2-stage accelerated life tests department of industrial and systems engineering national...

Documents

stress combinations

auxiliary stress

information slide

sequential alt stress

lower stress levels

number of stress levels

d slide

integrated planning