An introdution to alt planining

An Introduction to ALT
Planning (加速寿命试验计划
Pl i (加速寿命试验计划
简介)

Dr. Rong Pan (潘荣博士)
©2011 ASQ & Presentation Pan
Presented live on Aug 10th, 2011

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An Introduction to ALT Planning
g

Rong Pan, Ph.D.
Schools of Computing, Informatics, Decision Systems Engineering
Arizona State University

Outline

• Topic 1: Statistical Inferences in ALT

• T i 2: Experimental Design in ALT
Topic 2 E i lD i i

• Topic 3: Software
p

• Q&A

• References:

• Wayne B. N l
W B Nelson (1990) Accelerated Testing: Statistical Models, Test
A l t d T ti St ti ti l M d l T t
Plans, and Data Analysis, John Wiley & Sons, Inc., Hoboken, NJ.

• William Q M k and L i A E
Willi Q. Meeker d Luis A. Escobar (1998) Statistical Methods for
b St ti ti l M th d f
Reliability Data, John Wiley & Sons, Inc., New York, NY.

2 1/21/2012 IEE 573: Reliability Engineering

Topic 1: Statistical Inferences in ALT

• Backgrounds of topics: ALT and SSALT

• E
Exponential and W ib ll regression
i l d Weibull i

• Statistical inference methods

• Parameter estimation

• Conclusions


Accelerated Life Testing (ALT)

• The need for highly reliable components and materials are widely
required for long-term performance
– unacceptable length of time and cost of product life testing experiments
under use condition

• Units are tested under more severe conditions (or stresses) than the
use condition
Stress

Stressed condition

Use-condition

Failure Time
4

Step-Stress Accelerated Life Testing (SSALT)

• SSALT is an advanced case of ALT
• Under SSALT, test units are run at different stress levels over time
(
(usually increased stress levels) while ALT is conducted at a constant
y )
stress level

Stress

x2

x1

Failure Time

5

An Example

• Nelson (1980) data, obtained from a SSALT of cryogenic cable
insulation
- consists of four different test plans (groups)

6

Things to Remember

• Failure time data are often censored
• Right censoring
• Interval censoring
• By test plan: type I censoring, type II censoring
type-I censoring type-II

• Failure time distribution cannot be normal distribution
• E ponential
Exponential
• Weibull
• L
Lognormal
l

• Data need to be extrapolated
• Use condition is outside of experimental region
• Extrapolation model is needed

7

Exponential Regression

• Failure time distribution is assumed to be exponential distribution
p
1
f (t ) = e −t / α , t>0
α
• Mean failure time (or mean time to failure, MTTF) and failure rate (or
hazard function)

MTTF = α = 1 / λ
• Relationship with covariates (stresses)

log MTTF = β 0 + β1 x1 + β 2 x2 + ....

8

Weibull Regression

• Weibull distribution generalizes exponential distribution
g p
γ γ −1 −( t / α )γ
f (t ) = γ t e , t>0
α
• Characteristic life α
• Relationship with covariates (stresses)

log α = β 0 + β1 x1 + β 2 x2 + ....

9

Acceleration Model

• Acceleration Factor (AF): the acceleration constant relating times
( ) g
to fail at the two stresses

(time to fail at lower stress) = AF x (time to fail at higher stress)

- through AF, we can project the results obtained from experiments
to the use condition

- even if information on AF is unknown, the results at higher stress
levels can b extrapolated t th use-condition b an appropriate
l l be t l t d to the diti by i t
physical acceleration model

e.g., Arrhenius, inverse power, Eyring models, etc.

10

Statistical Inference Methods in Reliability

• The statistical inference technique for ALT/SSALT
q
– Estimate model parameters
– Predict failure behavior at the use condition

• Two main statistical approaches:
– i) Classical approach (based on MLE)

– ii) Bayesian approach

• (-) the limitation due to its inherent model complexity and
computational intractability
• (+) the advent of Markov chain Monte Carlo (MCMC)

11

Dept. of Industrial Engineering

Classical Approach

• Find the contributions of each observation to the total likelihood
function

• Failure time – probability of failure (probability density function)

• Right censoring time – probability of survival (reliability function)

• Interval censoring – probability of failure in the interval (difference of two failure
functions)

• Loglikelihood

• Maximize loglikelihood

12

Example - likelihood
• Total likelihood over l stress levels:
l ki
L = ∏∏ f ( yij ) R ki : risk set at stress level i
cij 1− cij
( yij )
i =1 j =1
l ki
= ∏∏ λ exp(−λi yij ) cij: indicator variable
cij
i
i =1 j =1 for censoring

Using μij = λi yij and K = ∑ i =1 ki
l

l ki
L = ∏∏ μijij exp(− μij ) × yij
c − cij

i =1 j =1
K
= ∏ μ kck exp(− μ k ) × yk ck
−

k =1

ck ~ Poisson (μk) Offset: does not depend on λk 13

Bayesian Approach

• Assume some prior distribution for parameters
• Prior information is subjective

• Noninformative priors reduce the subjectivity of Bayesian analysis and minimize the
impact of priors on posterior distributions

• Combine likelihood function with prior distributions

posterior ∝ likelihood × prior
• Inferences are made based on posterior distributions

14

The Example - Analysis

• Parameter Estimation

15

Conclusions

• ALT/SSALT is an designed experiment for test-to-failure
• Pay attention to failure data type
• Select an appropriate regression model
• Data analysis is not difficult …
• If you know how to use a computer tool

16

Topic 2: Experimental Design in ALT

Problem Statement

Model & Model Parameters

Use Condition & Simulation

Design Region constraints & Feasibility Region

Parameter Estimation comparison between different experimental
designs

Conclusions


Things to Remember

• How to plan an ALT
• E i
Engineering concerns – material, equipment, h
i t i l i t human resources, b d t
budgetary and ti
d time
constraints
• Statistical concerns – sample size, q
p quality of inference
y

• Standard DOE
• Unsuitable to failure time data – non-normal distribution, non linear regression,
non normal non-linear
censoring
• Optimal test plans
• Difficult to obtain
• Could be sensitive to model assumption

18

A Real-World Example

Situation:
– ALT plan for evaluating solder joint reliability
Problem:
– Minimize uncertainty about the model parameter estimates
– Equipment, Materials, and Time to Market constraints
– Use of “industry standard” test conditions lead to sub-optimal model
parameter estimates
Scope:
– Eyring based acceleration model
– Weibull life distributions simulated based on “known” model parameters
– Interval and right censored data
Objective:
– Compare the design matrix influence to other design factors (n censoring)
(n,
– Identify designs that reduce parameter estimation variance (D-optimality
criteria)

See more, Monroe and Pan, Journal of Quality Technology, 2008.

Model Parameters

Eyring based model

TTF = A ⋅ (ΔT )− a (t −b exp ⎡− c⎛ ⎞⎤
dwell )
1
⎢ ⎜ ⎜
⎟⎥
⎢
⎣ ⎝ Tmax ⎟⎥
⎠⎦

t dwell

Tmax
T

ΔT = Temp
Amplitude
A lit d
Tmin

1 cycle


Test Instrument

Temperature Cycle Chamber


Log-Linear Transformation

Eyring based model

⎡ ⎛ 1 ⎞⎤
TTF = A ⋅ (ΔT ) (tdwell )
−a −b
exp ⎢ − c⎜
p ⎜ T ⎟⎥
⎟
⎣ ⎝ max ⎠⎦

⎛ 1 ⎞
ln (TTF ) = ln ( A) − a ⋅ ln (ΔT ) − b ⋅ ln (t dwell ) − c⎜
⎜T
⎟
⎟
⎝ max ⎠

Log-linear
Log linear function


Case Study Via Simulation
Assume Product lifetime ~ Weibull (αuse=10,000; β=2)

Assume Use environment
– t-dwell = 10 minutes; ΔT = 65°C; Tmax = 85°C

“Known” parameter estimates [1]
– a= 2.65;
; b= 0.136;
; c= 2185

Used to generate expected lifetime distributions for each test condition
– Characteristic life α = αuse /AF
life,

Compare various censoring conditions
– None (exact cycles to failure)
– Right censoring at characteristic life
– Interval censoring (every 250 cycles)

[1] = “An Acceleration Model for Sn-Ag-Cu Solder Joint Reliability under Various Thermal Cycle Conditions”. Hewlett Packard Company.
Surface Mount Technology Association International (SMTA). September 25, 2005.

Constraints

Materials:
– T-max ≤ 125°C
T max 125 C (test boards melt)
– tdwell ≥ 3 minutes (stress relaxation threshold)

Equipment: Temperature Cycle Chamber
– T min ≥ -55°C
T-min 55°C (condenser limit)
– T-max≤ 150°C (pressure vessel limit)
– tdwell ≤ 24 minutes (availability)

Time to Market
– ΔT ≥ +80°C
– AF ≥ 3.5x use condition (
(test time limit of 6 months)
)

Unique Test Conditions (N=4)


Design Region

View
– td ll=24 minute plane
dwell

Constraints
– Material constraint (blue)
– Equipment (green)
– Time to market ΔT (red)
– Time to market AF (orange)

Design region
– In plane: between 5 vertices
– Out of plane: between dwell
time of 3 and 24 minutes


Standards Based Testing

Originated prior to the implementation of “mechanism based” or “use
mechanism based use
condition” based testing strategies

Goal: simply meet the performance set by the previous product

These conditions were not selected in a Design of Experiment context

However, customers are very familiar with these benchmarks and often
request these tests from their suppliers

Example: Temp Cycle “B”
B
– Temperature range: 180°C [-55°C, +125°C]
– t-dwell ~10 minutes (total cycle time specified)
– Tmax 125


Designs Considered

Standards based Orthogonal (23-1) Recommended

1.90%
1 90% 24.87%
24 87% 70.71%
70 71%

= in plane (tdwell = 24 minutes)
= out of plane (tdwell=8 minutes)

D-efficiency scores are percentages in red

Censoring Options Considered

No censoring – exact cycles to failure over entire lifetime

Right censoring at characteristic life (63.2%)

Interval censoring – readouts taken every 250 cycles


Results: How to read the graph

True parameter value

Sample sizes

Data censoring

Experimental D i
E i t l Design M t i
Matrix


Results: Parameter b

D-optimal design converges to true estimate much faster
Is robustness to both right and interval censoring
Is efficient with minimal sample sizes required

Instability of estimates for Standard design with small sample size
y g p


Conclusions
D-Optimal based experimental designs:

Work well with constrained regions

Improved precision on parameter estimates
– Recommended design w/ n=25 outperformed Standards design w/ n=500.
– Slightly better results than orthogonal design (fractional factorial)

Test planning is an influential step
– O t i h b th sample size and censoring effects in terms of influence
Outweigh both l i d i ff t i t f i fl
– Yet they are not often considered in practice

Enable model form to be validated without masking of variables

Assume that the model form is known
A th t th d lf i k
– Orthogonal designs may be a preferred choice for robustness when
model form is unknown a priori
– Optimality solution in model specific


Topic 3: Software

Data
D t analysis
l i
– Most statistical software can handle it, e.g., SAS, MiniTab, S-plus, R
– Some reliability engineering software dedicated to failure time data
analysis: Weibull++, ALTA
ALT planning
– Not many tools available, so may need special codes for specific task
– A few of them: JMP, SPLIDA, Minitab


SAS & JMP

SAS Proc for failure time data analysis
P f f il ti d t l i
– Proc LIFEREG fits Weibull, lognormal, loglogistic regression models for
censored data
– Proc PHREG fit the proportional hazard regression model
– Bayesian data analysis can be requested in these two procedures
JMP
– JMP is a business unit of SAS Inc., specializing design of experiments
– JMP9 has enhanced ALT test planning
– Helpful tutorial website: www.jmp.com/applications/reliability


R & SPLIDA

R is f
i free
– Supported by the statistics community
– Survival package
– library(survival)
– Surv() defines a survival data objects
– survreg() fits ALT regression model

SPLIDA
– Developed by Dr. Meeker
p y
– Originally a free add-on program to S-Plus, recently converted it to R
– Provides the functions for single variable ALT data analysis, multiple
regression ALT data analysis, residual diagnosis, and ALT planning
analysis diagnosis
– Simulating and evaluating ALT experiments


Weibull++ & ALTA

Reliability
R li bilit engineering software f
i i ft from R li S ft
ReliaSoft
Developed for solving engineering problems
– Interactive user interface
– Spreadsheet format
– Graphical displays
Weibull++ can fit most lifetime distributions for censored data
ALTA is for ALT and ADT data analysis
– Physical acceleration model is explicitly defined
– Can handle more complicated tests, such as SSALT.


Minitab

Reliability functions are li it d b t enough f most engineering
R li bilit f ti limited, but h for t i i
applications
– Reliability data analysis Stat->Reliability/Survival->Accelerated life
testing… or Regression with life data…
– ALT test planning Stat->Reliability/Survival->Accelerated life test plans…


Summary

Topics discussed
T i di d
– Statistical inference concerns with how to estimate model parameters
– Design of experiments concerns with how to plan experiments
efficiently

ALT data analysis and test planning require advanced
statistical methods

Following techniques introduced: Weibull regression, MLE,
Bayesian inference D-optimal experimental design
inference,

Appreciation of DOE


An introdution to alt planining

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