# Sequential ALT

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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

• 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 ) = E1 {var( y p (1))}
= E1 {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

 E1 (var( y p ( x1 )))                      
                                             
                                             
Λ
E1 (var( y p ( xH 1 ))) 
                                             
                               var( y p (0)) 
                                             

Min E1 (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
E1 (var( y p ( xi )))  
1  1             
1

cΣi cd 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
Ii                                                              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  8.6171105 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:
                     
E1 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

              
E1 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 E1 (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

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
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 ~ Uniform0.8,1.2

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 Ii                           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 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%)

Constant-                        Step-Stress

reasons:
• 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 E1 (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 
 E1 (var( y p ( x1 )))          0                     0          0         
                                                                            
         0               E1 (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 

 1 ~ Uniform  0.8,1.2
Low Humidity Level: 60%
High Humidity Level: 90%
 p3                                                                  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.

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