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Transcript of 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
Page 8
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:
Page 9
Inference at the Highest Stress Level
Stress
Time in log-scale
Low UseHigh0 1
; ,H H H H Hl θ θ D θ
Page 10
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
Page 12
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
Page 13
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
Page 14
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
Page 15
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
Page 16
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
Page 17
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
Page 18
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
Page 19
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
Page 20
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
Page 21
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
Page 22
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
Page 23
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
Page 24
The feasible region
Page 25
Interior Penalty Function Method
Page 26
Page 27
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
Page 29
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
Page 30
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
Page 31
Simulation Results
Page 32
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)
Page 33
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
Page 34
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)
Page 35
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 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 *
Non-sequential Plans Sequential Plans
For non-sequential plan:
RE is sensitive to the pre-specified model parameters and .
0
0
1
1
Page 36
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
Page 37
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 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 *
Non-sequential Plans Sequential Plans
For non-sequential plan:
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.