An introdution to alt planining


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Accelerated life tests (ALTs) are employed to generate failure time data at higher-than-normal-use stress levels. ALT planning is critical for achieving statistical efficiency and reducing experimental cost through design of experiments (DOE). In this talk, I will describe a real world example of ALT planning and its impact on decision making. I will present models for regression with failure time data, including exponential and Weibull regression. Censoring, which is present in many life testing experiments, and its effect on regression models is discussed. Graphical methods for data analysis of life testing experiments are discussed, as well as the software for ALT planning and data analysis.

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An introdution to alt planining

  1. 1. 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 Calendar/Webinars ‐_Chinese/Webinars_‐_Chinese.html
  2. 2. ASQ Reliability Division  ASQ Reliability Division Chinese Webinar Series Chinese Webinar Series One of the monthly webinars  One of the monthly webinars on topics of interest to  reliability engineers. To view recorded webinar (available to ASQ Reliability  Division members only) visit ) / To sign up for the free and available to anyone live  webinars visit and select English  Webinars to find links to register for upcoming events Calendar/Webinars ‐_Chinese/Webinars_‐_Chinese.html
  3. 3. An Introduction to ALT Planning g Rong Pan, Ph.D.Schools of Computing, Informatics, Decision Systems Engineering Arizona State University
  4. 4. 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
  5. 5. 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 • Conclusions3 1/21/2012 IEE 573: Reliability Engineering
  6. 6. 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 44 1/21/2012 IEE 573: Reliability Engineering
  7. 7. 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 55 1/21/2012 IEE 573: Reliability Engineering
  8. 8. An Example • Nelson (1980) data, obtained from a SSALT of cryogenic cable insulation - consists of four different test plans (groups) 66 1/21/2012 IEE 573: Reliability Engineering
  9. 9. 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 77 1/21/2012 IEE 573: Reliability Engineering
  10. 10. 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 + .... 88 1/21/2012 IEE 573: Reliability Engineering
  11. 11. 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 + .... 99 1/21/2012 IEE 573: Reliability Engineering
  12. 12. 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. 1010 1/21/2012 IEE 573: Reliability Engineering
  13. 13. 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) 1111 1/21/2012 IEE 573: Reliability Engineering
  14. 14. 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 1212 1/21/2012 IEE 573: Reliability Engineering
  15. 15. 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 1313 1/21/2012 IEE 573: Reliability Engineering
  16. 16. 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 1414 1/21/2012 IEE 573: Reliability Engineering
  17. 17. The Example - Analysis • Parameter Estimation 1515 1/21/2012 IEE 573: Reliability Engineering
  18. 18. 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 1616 1/21/2012 IEE 573: Reliability Engineering
  19. 19. 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 Conclusions17 1/21/2012 IEE 573: Reliability Engineering
  20. 20. 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 1818 1/21/2012 IEE 573: Reliability Engineering
  21. 21. 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, 2008.19 1/21/2012 IEE 573: Reliability Engineering
  22. 22. 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 cycle20 1/21/2012 IEE 573: Reliability Engineering
  23. 23. Test Instrument Temperature Cycle Chamber21 1/21/2012 IEE 573: Reliability Engineering
  24. 24. 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 function22 1/21/2012 IEE 573: Reliability Engineering
  25. 25. 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, 2005.23 1/21/2012 IEE 573: Reliability Engineering
  26. 26. 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)24 1/21/2012 IEE 573: Reliability Engineering
  27. 27. 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 minutes25 1/21/2012 IEE 573: Reliability Engineering
  28. 28. 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 12526 1/21/2012 IEE 573: Reliability Engineering
  29. 29. Designs Considered Standards based Orthogonal (23-1) Recommended1.90%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 red27 1/21/2012 IEE 573: Reliability Engineering
  30. 30. 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 cycles28 1/21/2012 IEE 573: Reliability Engineering
  31. 31. Results: How to read the graph True parameter value Sample sizes Data censoring Experimental D i E i t l Design M t i Matrix29 1/21/2012 IEE 573: Reliability Engineering
  32. 32. 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 p30 1/21/2012 IEE 573: Reliability Engineering
  33. 33. 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 specific31 1/21/2012 IEE 573: Reliability Engineering
  34. 34. 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, Minitab32 1/21/2012 IEE 573: Reliability Engineering
  35. 35. 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: 1/21/2012 IEE 573: Reliability Engineering
  36. 36. 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 experiments34 1/21/2012 IEE 573: Reliability Engineering
  37. 37. 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.35 1/21/2012 IEE 573: Reliability Engineering
  38. 38. 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…36 1/21/2012 IEE 573: Reliability Engineering
  39. 39. 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 DOE37 1/21/2012 IEE 573: Reliability Engineering
  40. 40. Thank you y