Planning 2-Stage Accelerated Life Tests

1. Planning 2-Stage Accelerated Life Tests LC Tang (董润楨), Ph.D Department of Industrial and Systems Engineering National University of Singapore

2. 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

3. A Constant-Stress ALT Time ? Probability distributions Maximum Test Duration Life-stress relationship Use Stress Low Stress Mid Stress High Stress Stress Level

4. A Scale-Accelerated Weibull Lifetime Model • Standardization of stress • Weibull lifetime distribution at any stress • A scale-accelerated failure time model

5. 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 2: Minimize the asymptotic variance of ML estimator Step 3: Evaluate the plan using simulations

6. A Framework of Sequential ALT Planning Information Planning Procedure • 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. Planning information e.g. test duration, specified parameter values, etc. Plan & Perform the test at the highest stress to quickly obtain failures Preliminary information on Information on the slope parameter Planning information e.g. test duration, number of stress levels, sample sizes, etc. Plan the tests at lower stress levels

7. Part IPlanning Sequential Constant-Stress Accelerated Life Tests

8. Sample Size at the Highest Stress Level Sample Size:

9. Inference at the Highest Stress Level Time in log-scale Stress 0 1 Use High Low

10. Inference at the Highest Stress Level Generalized MLE Covariance matrix Observed information

11. Construction of Prior Distributions

12. Construction of Priors at Low Stresses

13. Illustration of the Sequential ALT Plan & Run the test at the highest stress Time in log-scale Deduction of Prior Distributions Pre-Posterior Analysis & Optimization Stress 0 1 Use High Low

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

15. Problem Formulation Design Matrix ?

16. Pre-Posterior Analysis Information contained in the prior density Information expected to obtain at stress level i

17. Adhesive Bond Test (Meeker and Escobar 1998) • Total Sample Size: 300 • Total Testing Duration: 6 months =183days • Standardized Testing Region: • Assumptions:

18. Planning at the Highest Temp 50 samples are needed

19. 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

20. Normal Approximation

21. Planning of an ALT with 2 Stress Levels Planning Information: High Low

22. Effects of the pre-specified slope parameter Suppose we raise the expectation of the product reliability Effect: Run the test under a higher stress to produce more failures Beta1 ranges from 3.84 to 5.76 Beta1 ranges from 3.84 to 5.12 High Low

23. Plan an ALT with 3 stress levels Planning Information: Additional constraints:

24. The feasible region

25. Interior Penalty Function Method

27. Inference from Test Results Simulated failure times

28. Inference • Results obtained under the high stress Increasing • Results obtained under the mid and low stress Decreasing

29. 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

30. 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

31. Simulation Results

32. Precision • Sequential plans yields more precise estimation • Sequential plans gives a conservative sense of statistical precision: Sample variance > Asymptotic variance Asymptotic variance (non-sequential plan) Sample variance (non-sequential plan) Asymptotic variance (sequential plan) Sample variance (sequential plan)

33. Effect of Parameter Mis-specification on Precision Sequential Plans Non-sequential Plans • For sequential plan: • Since • Model parameters and are estimated at stage one; • An interval value of is used • Hence, the plan robustness to the mis-specification of model parameters has been enhanced For non-sequential plan: Results are sensitive to the specified model parameters and .

34. Robustness Define the Relative Error (RE) as: 3. Sequential plans is more robust to mis-specification of model parameters RE (non-sequential plan) RE (sequential plan)

35. Effect of Parameter Mis-specification on the Relative Error (RE) Non-sequential Plans Sequential Plans • For sequential plan: • Since • Model parameters and are estimated at stage one; • An interval value of is used • Hence, the plan robustness to the mis-specification of model parameters has been enhanced For non-sequential plan: RE is sensitive to the pre-specified model parameters and .

36. Simulation Results • Sequential plans reduce the degree of extropolation; • 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

37. Effect of Parameter Mis-specification on the Optimum Low Stress level Non-sequential Plans Sequential Plans • For sequential plan: • Since • Model parameters and are estimated at stage one; • An interval value of is used • Hence, the plan robustness to the mis-specification of model parameters has been enhanced For non-sequential plan: RE is sensitive to the pre-specified model parameters and .

38. Comparison with 4:2:1 Plan

39. 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 !

40. 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.

41. 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

42. Illustration Low degree of extrapolation with zero failure Time high degree of extrapolation with more failures Maximum Test Duration Use Stress Candidate low stress 1 Candidate low stress 2 High Stress Stress Level

43. 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.

44. Model Extension • Standardization of the level of AS • The extended testing region: • A scale-accelerated failure time model Examples: Hallberg-Peck model Higher usage rate model (Yang 2005)

45. 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? No yes No Is an AS available? Step 3a: Plan the tests at low stresses without an AS i.e. optimize sample allocation, and stress combinations yes Step 3b: Plan the tests at low stresses with an AS i.e. optimize sample allocation, stress combinations, and the loading profile of AS

46. Step 1 Planning & Inference at the Highest Temperature Level

47. ALT for Electronic Controller

48. Test Planning at the Highest Stress Planning Inputs: Planning Output: Risk of see less failures than expected Testing Output:

49. Results 44 samples are needed

50. Data Obtained at the Highest Stress Weibull Probability Plot for Observed Failure Data Note: This is just a particular run