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Planning 2-Stage Accelerated Life Tests
LC Tang (董润楨), Ph.D
Department of Industrial and Systems Engineering
National University of Singapore
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
Time
Probability
distributions
Maximum
Test
Duration
Life-stress
relationship
Use Low Mid High
Stress Level
Stress Stress Stress Stress
A Scale-Accelerated Weibull Lifetime Model
• Standardization of stress x x s s sk s0 sk 1
s0 : use stress sk : the highest stress
• Weibull lifetime distribution at any stress
log T -
SEV
• A scale-accelerated failure time model
0 1 x , 1 0
is a constant independent of stress
Motivations of Sequential ALT Planning
• ALT planning based on the Maximum Likelihood theory
Step 1: Step 2: Step 3:
Specify ALT model Minimize the asymptotic Evaluate the plan
parameter values variance of ML estimator using simulations
• Locally optimal for specified model parameters 0 , 1,
• Problems:
– There often exists a high margin of specification error
– Developed plans are usually sensitive to the specified value
A Framework of Sequential ALT Planning
Information Planning Procedure
Planning information
Plan & Perform the test at the
e.g. test duration, specified highest stress to quickly obtain
parameter values, etc. failures
Information on the slope Preliminary information on ( 0 , )
parameter 1
Planning information
e.g. test duration, number of Plan the tests at lower stress levels
stress levels, sample sizes,
etc.
• 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.
Part I
Planning Sequential Constant-Stress Accelerated
Life Tests
Sample Size at the Highest Stress Level
Specify
the values of 0 (or H ) and
the censoring time cH
the expected number of failures RH
Sample Size:
RH
nH
p
p 1 exp cH / exp( H )
1/
Page 8
Inference at the Highest Stress Level
Time in log-scale
θH l θH ; DH θH H ,
Stress
0 1
High Low Use
Inference at the Highest Stress Level
ˆ ˆ ˆ
θ H | y H ~ N (θ H , Σ H )
where
ˆ
θ H arg max (θ H ) Generalized MLE
ˆ ˆ
Σ I 1 Covariance matrix
H H
ˆ
I H [ 2l (θ H ; D H ) / θ H 2 ]θ
H
ˆ
θ H Observed information
Page 10
Construction of Prior Distributions
k ,
i , for xi 0,1
H ,
i k 1 xi
is a constant
Information on
the value of 1
Construction of Priors at Low Stresses
for any i 1,..., H 1, there exists a one-one transformation θi (θH )
with non-vanishing Jacobian θH / θi ,such that
1 ( i 1 xi )
(θi ) ( i 1 xi , ) dF ( 1 )
1 i
1 ~U ( 1 , 1 )
erf ( i ) erf ( i ) ( i H ) 2
ˆ
exp
23/ 2 ( var( H ))1/ 2 (i i )
ˆ 2 var( H )
ˆ
where
i H 1 xi , i H 1 xi , i H 1 xi , cov( H , H ) / var1/ 2 ( H ) var1/ 2 ( H )
ˆ ˆ ˆ ˆ ˆ ˆ ˆ
i var1/ 2 ( H ) i var1/ 2 ( H ) ( i H ) var1/ 2 ( H )
ˆ ˆ ˆ ˆ
(2 var( H ) var( H )(1 2 ))1/ 2
ˆ ˆ
i
i var1/ 2 ( H ) i var1/ 2 ( H ) ( i H ) var1/ 2 ( H )
ˆ ˆ ˆ ˆ
(2 var( H ) var( H )(1 2 ))1/ 2
ˆ ˆ
i
Page 12
Illustration of the Sequential ALT
Time in log-scale Plan & Run the
test at the highest
stress
Deduction of Prior
Distributions
Pre-Posterior
Analysis &
Stress Optimization
0 1
High Low Use
Page 13
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
Min C (ξ ) = E1 {var( y p (1))}
= E1 {c var(θ0 )cT } c [1, log( log(1 p))]
Page 14
Problem Formulation
T
Design Matrix X 1 1 1
x1 xH 1 xH
E1 (var( y p ( x1 )))
Λ
E1 (var( y p ( xH 1 )))
var( y p (0))
Min E1 (var( y p (1))) 1( XT Λ 1X) 11T
H 1
s.t. x {( x1 , x2 ,..., xH 1 ) : 0 xi 1} and xH 0
π {( 1 , 2 ,..., H ) H
: i i 1 and 0 i 1}
Page 15
Pre-Posterior Analysis
1 1
E1 (var( y p ( xi )))
1 1
1
cΣi cd 1
i 1 Information
Σ i I θi I
contained in
the prior
where density
2 log l θi 2 log l θi
I θi E ; y f y dy
θi 2
θi 2
Information
2 log θ i expected to
Ii obtain at
θi 2
stress level i
Page 16
Adhesive Bond Test (Meeker and Escobar 1998)
• Total Sample Size: 300
• Total Testing Duration: 6 months =183days
• Standardized Testing Region: 0 xH x xU 1
• Assumptions:
log(T ) ~ SEV ,
Activation energy, Ea 1
log A (Arrhenius)
Boltzmann constant, k B 8.6171105 T
0 1 x, 0 log A Ea k B 1 sH , 1 Ea k B 1 ( s0 sH )
is a constant
Page 17
Planning at the Highest Temp
Planning
information:
H 4.72
0.6
RH 15
cH 60
50 samples are needed RH nH p
Page 18
Posterior Density
Simulated 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 19
Normal Approximation
ˆ ˆ ˆ
θ H | data ~ N (θH , Σ H )
ˆ ˆ ˆ
where Σ H I 1 and I H [ 2l (θ H ) / θH 2 ]θ ˆ
H H θ H
Page 20
Planning of an ALT with 2 Stress Levels
Planning Information:
E1 var y p 1 in log-scale
Sample size
nL 300 nH 250 100
Test duration
cL 183 cH 123 10
Posterior density at xH
ˆ ˆ
θH | y H ~ N θH , ΣH 1
Specified range of 1
0.1
1 3.84,5.12
(i.e.Ea 0.6, 0.8) 0.01
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
High Low
xL Page 21
Effects of the pre-specified slope parameter
Suppose we raise the expectation of the product reliability
Ea 0.6,0.8 Ea 0.6,0.9 i.e. 1 3.84,5.12 1 3.84,5.76
E1 var y p 1 in log-scale
Effect:
100
Run the test under a
10 higher stress to
Beta1 ranges from produce more
3.84 to 5.76
1
failures
0.1 Beta1 ranges from
3.84 to 5.12
0.01
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
xL Page 22
High Low
Plan an ALT with 3 stress levels
Planning Information:
Min E1 (var( y0.1 (1)); xL , )
nL 250, cL 123
H , , 1 3.84,5.12 s.t.
nL M (1 ) p( xL ) RL
Additional constraints: nL M p ( xM ) RM
nL 250 (1 ) 0 x H x L 2 xM 1
nM 250 for 0 1
0 1
xL
xM
2
where
Minimum number of failure
p ( xL ) (1 exp( exp( L ))) ( L , L )d L d L
RL and RM 0
p( xM ) (1 exp( exp( M ))) ( M , M )d M d M
0
Page 23
The feasible region
Page 24
Interior Penalty Function Method
Page 25
Page 26
Inference from Test Results
Simulated failure times (assume 0 4, 1 4, 4)
Stress Sample Test Expected Simulated Observed
Level Size Duration Failures Failure Times Failures
33.3, 48.4, 39.3, 58.8, 47.4, 60.0, 33.6, 19.4, 38.0,
High 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,
50 60 15 17.3, 46.8, 40.9, 60.0, 28.4, 60.0, 4.2, 21.9, 49.6, 38
xH 0 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
Mid 46.1 62.5 86.2 98.9 101.7 123
20 123 5 (×224) 5
xM 0.39
Low 22.8 44.8 59.1 84.4 87.7
230 123 5 105.2 123 (×224) 6
xL 0.78
Page 27
Inference
• Results obtained under the high stress
H ,
0.0112 0.0003
~ N 3.87, 0.65 ,
0.0003 0.0086
Increasing
• Results obtained under the mid and low stress
Decreasing
M ,
0.0156 0.0016
~ N 5.28, 0.594 ,
0.0016 0.0080
L ,
0.0377 0.0060
~ N 7.24, 0.664 ,
0.0060 0.0042
Simulation Study
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
Page 29
Simulation Design Table
- 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
Scenarios Pre-specified Pre-specified Pre-specified 1 Pre-specified 1
0 (non-sequential) (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 %
Page 30
Simulation Results
Page 31
Precision
1. Sequential plans yields more precise estimation
2. Sequential plans gives a conservative sense of statistical
precision: Sample variance > Asymptotic variance
Asymptotic variance
0.8 (non-sequential plan)
0.7 Sample variance
(non-sequential plan)
0.6
Variance
0.5
Asymptotic variance
0.4
(sequential plan)
0.3
0.2 Sample variance
0.1 (sequential plan)
0
0 1 2 3 4 5 6 7 8 9 10
Simulation scenarios
Page 32
Effect of Parameter Mis-specification on Precision
Non-sequential Plans Sequential Plans
Effect on Effect on Effect on the Effect on the
the the expected observed
expected observed variance variance
variance variance
0 -0.053 -0.038
0 0.270 0.1945
(0, 0.0001) (- 0.0001,0)
1 0.016 -0.2075
0 * (- 0.0001,0) (- 0.0001,0)
0.044 0.043
0 * 1 -0.007 -0.1180 For sequential plan:
0 * 0.083 0.0655 Since
1 * 0.035 0.0185 1. Model parameters 0 and are
0 * 1 * -0.031 -0.009 estimated at stage one;
For non-sequential plan: 2. An interval value of 1 is used
Results are sensitive to the specified Hence, the plan robustness to the mis-
model parameters 0 and 1 . specification of model parameters
Page 33
has been enhanced
Robustness
Define the Relative Error (RE) as:
sample variance - asymptotic variance asymptotic variance
3. Sequential plans is more robust to mis-specification of model
parameters
2.5 RE
(non-sequential plan)
2
1.5
RE
RE
1
(sequential plan)
0.5
0
0 1 2 3 4 5 6 7 8 9 10
Simulation scenarios
Page 34
Effect of Parameter Mis-specification on the Relative Error (RE)
Non-sequential Plans Sequential Plans
Effect Effect
0 -0.7684 0 0.0011
1 -0.7187 (0, 0.0001)
-0.2367 0 * (0, 0.0001)
0 * 1 0.4905
0 * 0.1334
For sequential plan:
1 * 0.1201 Since
0 * 1 * -0.0532 1. Model parameters 0 and are
estimated at stage one;
For non-sequential plan: 2. An interval value of 1 is used
RE is sensitive to the pre-specified Hence, the plan robustness to the mis-
model parameters 0 and 1 . specification of model parameters
has been enhanced
Page 35
Simulation Results
4. 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
120 Optimum low stress
(non-sequential plan)
100
Temperature
80
60
40
Optimum low stress
20 Use stress
(sequential plan)
0
0 1 2 3 4 5 6 7 8 9 10
Simulation scenarios
Page 36
Effect of Parameter Mis-specification on the Optimum Low Stress level
Non-sequential Plans Sequential Plans
Effect Effect
0 12.25 0 -5
1 -13.25 (- 0.0001,0)
3.25 0 * (- 0.0001,0)
0 * 1 -1.25
0 * 3.25
For sequential plan:
1 * 1.75 Since
0 * 1 * 0.75 1. Model parameters 0 and are
estimated at stage one;
For non-sequential plan: 2. An interval value of 1 is used
RE is sensitive to the pre-specified Hence, the plan robustness to the mis-
model parameters 0 and 1 . specification of model parameters
has been enhanced
Page 37
Comparison with 4:2:1 Plan
|Ase Ase when model parameters are correctly specified|
ASR
Ase when model parameters are correctly specified
c: Test duration at the highest stress level
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 II
Planning 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
Low degree of extrapolation
with zero failure
Time
high degree of extrapolation
with more failures
Maximum
Test
Duration
Use Candidate low Candidate low High
Stress stress 1 stress 2 Stress
Stress Level
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
• Standardization of the level of AS
h (v vuse ) (vmax vuse ) 1
vuse : use stress vmax : the highest stress
• The extended testing region: [0,1]2
• A scale-accelerated failure time model
0 1 x 2 h
is a constant independent of stress
Examples: Hallberg-Peck model
Higher usage rate model (Yang 2005)
An Integrated Framework of Sequential ALT Planning
with an Auxiliary Stress
Planning Information Step 1: Plan and perform the life
test at the highest stress level
e.g. Sample size; Test
duration; Specified model
parameters Step 2: Compute the number of
failures at low stresses
No
Is an AS needed?
yes
No
Is an AS available?
yes Step 3a: Plan the tests at low
stresses without an AS
Step 3b: Plan the tests at low
stresses with an AS i.e. optimize sample allocation,
and stress combinations
i.e. optimize sample allocation,
stress combinations, and the
loading profile of AS
Step 1
Planning & Inference
at the Highest Temperature Level
ALT for Electronic Controller
Experiment Target:
To demontrate the 10% life quantile at use condition exceeds 2 years
Stress Factor:
Temperature (other factors, such as humidity, voltage, etc are set to use level)
Planning information and Assumptions:
1). 120 sample units and 75 days are available.
2). The use temperature is 450C 318K
The highest temperature allowed in the test is 850C 358 K
3). Failure time T follows Weibull distribution
F t log t
4). is a constant, independent of temperature; follows Arrhenius stress-life
relationship
Activation energy, Ea 1
i log A 0 1 si
Boltzmann constant, k B Ti
where 0 log A 1 Ea si 11605 / Ti
Test Planning at the Highest Stress
Planning Inputs:
target number of failures: rk
censoring time: ck 1/ k
Rk exp ck exp
parameter values: k ( 0 ),
confidence level:
Planning Output: nk
rk 1
Risk of see less
Cnk 1 Rk Rk nk i 1 failures than expected
i i
i 0
(Binomial Bogey test, Yang 2007)
ˆ
Testing Output: H ( 0 ) or
ˆ ˆ
Results
Planning
information:
k 7.5
0.677
rk 6
ck 720hr
0.9
44 samples are needed
Data Obtained at the Highest Stress
Time-to-failure (hours) Weibull Probability Plot for
Observed Failure Data
79.559 210.47 590.03
0
400.56 491.41 138.94 R² = 0.8995
673.98 109.4 149.95 -0.5
204.7 425.32 643.31 -1
117.15 328.99 351.87 -1.5
720×29 -2
-2.5
Note: This is just a
particular run 1.7 2.2 2.7 3.2
Statistical Inference at the Highest Stress
Posterior distribution derived from a constant prior :
θk ; y l θk ; y
nk 1 y j k
y j k ck k
exp j log
exp 1 j exp
j 1
where θk ( k , ); 0 if the data is censored, otherwise 1
Normal Approximation to the Posterior distribution (Berger 1985)
ˆ ˆ ˆ ˆ
θk y ~ N (θk , Σ k ) N (θk ,[I k ]1 )
ˆ 2l (θk ; y ) ˆ
where I k = (observed Fisher information at θk )
θk 2 θ ˆ
θ k
k
ˆ ˆ ˆ 0.1142
θk [7.35, 0.90] Σ k I k 1
0.0529
symmetric 0.0489
Illustration
• 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. ).
Step 2
Computation of the Expected Number of
Failures at Low Stress Levels
Construction of Prior Distributions
k ,
i , for xi 0,1
H ,
i k 1 xi
is a constant
Information on
the value of 1
Density Function of the Constructed Prior
1
( i 1 xi )
i , ( i 1 xi , ) ( 1 )d 1
1
i
( ) 2
ˆ
erf ( i ) erf ( i )
1
3/ 2 exp i 1,..., k 1
2 ( var( )) (i i )
ˆ 1/ 2
2 var( )
ˆ
where ( 1 ) is a uniform distribution defined on an interval [1 ,1 ]
z
erf is the error function given by the definite integral erf ( z ) 2
1/ 2 t 2
e dt
0
i k 1 xi
ˆ
i k 1 xi , i k 1 xi , cov( k , ) / var1/ 2 ( k ) var1/ 2 ( )
ˆ ˆ ˆ ˆ ˆ ˆ
i var1/ 2 ( ) i var1/ 2 ( ) ( ) var1/ 2 ( k )
ˆ ˆ ˆ ˆ
(2 var( k ) var( )(1 2 ))1/ 2
ˆ ˆ
i
i var1/ 2 ( ) i var1/ 2 ( ) ( ) var1/ 2 ( k )
ˆ ˆ ˆ ˆ
(2 var( k ) var( )(1 2 ))1/ 2
ˆ ˆ
i
Illustration of the Constructed Priors at
65⁰C and 45⁰C
Let Ea 0.8,1.2 , i.e 1 ~ Uniform0.8,1.2
Uncertainty over becomes larger for lower testing temperature
Expected Number of Failures at Low Stress
In order to see 5 failures, the
temperature is almost on the middle
of the testing region !!
Another Point of View:
Prior Information v.s Information To Be Obtained
det I θi 2l (θi ) 2 log (θi )
where I θi =E and I θi =
det Ii θi 2 θi 2
Information to be “Information”
obtained by performing contained in the
a test at stress level i prior knowledge
Little
Information
obtained from
low temp
Step 3
Planning at the Lower Temperature Level
With Auxiliary Stress
•The choice of AS
•The loading of AS
•The integration of AS in test planning
The Choice of AS
Assumpotions:
1). The effect of AS is well understood
2). The failure mode does NOT change after an AS is introduced
Auxiliary Stress: Humidity
Hallberg-Peck Model (Livingston, 2000):
RH j
0 1s p log
RH 0
RH 0 : use humidity level, 60%
RH : humidity level in test ( 90%)
The Choice of Loading Profile for AS
Constant- Step-Stress
Stress Loading 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
Setting a Target Acceleration Factor
equivalent test duration, ci( e)
Time Compression Factor: i
actual test duration, ci
LCEM Cumulative Exposure Model
(Yeo and Tang 1999, Tang 2003)
A Bayesian Planning Problem
Min E1 (var( y p (1))) 1( XT Λ 1X) 11T
s.t. target time compression i , for i 1,..., k 1
( x1 , x2 ,..., xk 1 ) [0,1]k 1 Stress levels
(1 , 2 ,..., k 1 ) [0,1]k 1 : i 1 i 1 k
k 1
Sample allocation
(h1 , h2 ,..., hk 1 ) [0,1]k 1 Initial level of AS
( 1 , 2 ,..., k 1 ) [0, c]k 1 Stress changing time for AS
where
T
1 1 1
X
x1 x2 xk
E1 (var( y p ( x1 ))) 0 0 0
0 E1 (var( y p ( x2 ))) 0 0
Λ
0 0 0 var( y p ( xk ))
Planning Results
Testing Temp RH Testing Sample
Planning Information: Condition (C) (%) Duration Size
Sample size Use 45 60
n1 120 n2 76 Low 53 See 1080hrs 76
Test duration Profile
c1 1800 720 1080 High 85 60 720hrs 44
Posterior density at xH
θ2 ~ N θ2 , I 1 θ2
ˆ ˆ Humidity Loading Profile at Low Temperature
1 ~ Uniform 0.8,1.2
Low Humidity Level: 60%
High Humidity Level: 90%
p3 Holding Time: 170.5 hrs
Maximum RH = 90% Expected Failures:
Use RH = 60% Interval [0, 170.5] : No
failure
3 Interval [170.5,1080]: 5
failures
Interval [1080, ): 71 censored
Illustration: ALT without AS
Relative
Humidity
Point B: (53, 60%) Point A: (85, 60%)
Failure Probability < 0.01 Failure Probability = 0.32
60%
Temperature
53 85
Illustration: ALT with AS
Relative
Humidity Point D: (53, 90%)
Failure Probability = 0.08
90%
Point C: (53, 60%) Point A: (85, 60%)
Failure Probability < 0.01 Failure Probability = 0.32
60%
Temperature
53 85
Sensitivity of the Optimal Plan to p
RHT: RT/RH RSD:
Relative change Relative change of Relative change of
of low humidity low humidity/low Asymptotic SD
holding time temperature
Sensitivity of the Plan to
the Activation Energy
RHT: RT/RH RSD:
Relative change Relative change of Relative change of
of low humidity low humidity/low Asymptotic SD
holding time temperature
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.
