胡政宏 國立成功大學工業與資訊管理學系 Cheng-Hung Hu OPTIMUM STEP-STRESS ACCELERADTED DEGRADATION TEST FOR WIENER DEGRADATION PROCESS UNDER TIME CONSTRAINT 胡政宏 國立成功大學工業與資訊管理學系 Cheng-Hung Hu Department of Industrial and Information Management National Cheng Kung University

ABOUT ME （胡政宏）

Reliability (可靠度) Reliability is defined as the probability that a product, system, or service will perform its intended function without failure over a specified period of time. Reliability information is important for managers to Establish warranty period Schedule preventive maintenance Pricing extended warranties, etc. Random failure times and Probability distributions are used to assess the reliability.

Challenges from Modern Products Modern products are normally designed to operate without failure for relatively longer periods of time. Ex: Meeker and Hahn (1985) discussed test plans for the time to failure of an adhesive-bonded power element designed to survive for 10 years under 50 degree. Ex: LED light http://www.pantallasledsupergigantes.com/ Ex: Rear suspension aft lateral link Source: Step-Stress Accelerated Life Tests (Lu 2006)

Accelerated Degradation Test (ADT) Products are exposed and tested at higher than nominal levels of stress to incur failure early. e.g. pressure, temperature, humidity,… etc. Data under ADT can then be modeled and inferences are made from the observable data. Goal : Estimate MTTF, p-th percentile, or reliability function under normal use conditions. Pan & Crispinb (2010)

Stress Load Functions Several types of stress load strategies have been proposed and studied in literatures including constant stress, step-stress and ramp test Source: Elsayed et al. (2009)

Step-Stress LED Degradation Paths Under a SSADT, the degradation paths would be Tseng and Wen (2002)

Motivated Example Later, Liao and Tseng (2006) discussed the optimum SSADT Decision variables: N sample size measurement frequency li i=1,2,3,4,5 Objective Function: Min Var (Estimated tp) Constraints: Test cost constraint Integer constraint l1 l2 l3 l4 l5

Results Later, Liao and Tseng (2006) discussed the optimum SSADT

Liao and Tseng’s Results Later, Liao and Tseng (2006) discussed the optimum SSADT

Another Example Ge et al. (2011) proposed another optimum SSADT plan Decision variables: N sample size measurement frequency li i=1,2,3,4,5 Objective Function: Max Determinant (FIM) Constraints: Test cost constraint Integer constraint

Ge et al. (2011) Results

Observations Simple SSADT plan is better than SSADT plan with more steps This is true for more than one objective function l1 l2 l3 l4 l5

The Model 2.2 Model Assumption – (1) 2.2 Model Assumption – (2) Assumed that there exists an upper stress bound Assumed the failure mode has the same one under the normal use stress level Assumed the degradation path follows a stochastic Wiener process (denoted as ) with the drift and dispersion parameter and 2.2 Model Assumption – (2) The transformed degradation path at time under is where is a standard Brownian motion

Model Assumptions The increments of the process for each measurement is then The stress-parameter relationship satisfies

Inverse Gaussian Distribution Model Assumption Under model (1), the products failure time under the use condition can be defined as the first-passage time of the degradation process over a constant threshold a, follows an inverse Gaussian distribution, denoted by with location and scale parameters and

Fisher Information Matrix In models that meet standard regularity conditions, the large sample asymptotic variance-covariance matrix of the MLE is inverse of the Fisher matrix. The expected Fisher information matrix, is

Optimization Criteria Ng, Balakrishnan, Chan (2007) [C1] Maximize the determinant of the Fisher information matrix. [C2] Minimize the asymptotic variance of estimated MTTF under use condition [C3] Minimize the trace of the variance-covariance matrix of the MLE's of model parameters. [C4] Minimize the asymptotic variance of estimated p-th percentile of failure time distribution under normal use condition

Nonlinear Programming Model Decision Variables: Number of inspections or Proportion of inspections Objective Function: Any objective function from [C1] - [C4] Constraints: The Total Test Duration is Fixed. The Total Number of inspections (L) is fixed. Fixed measurement interval Non-negativity

Lemma 1 (Murthy and Sethi) Lemma1: Let X be a discrete random variable with finite outcomes . To maximize the variance of X, One should assign probabilities

Lemma 2 Lemma 2: Let X be a discrete random variable with finite outcomes . For any probability vector One can find another probability vector so that the expectations are the same but the second moment and variance of X is larger.

Proof of Lemma 2

Lemma 2 Lemma 2: Let X be a discrete random variable with finite outcomes . For any probability vector One can find another probability vector so that the expectations are the same but the second moment and variance of X is larger.

Objective Function [C1] Recall the Fisher information matrix is The determinant of this matrix is

Proof for Ge et al. (2011) By Lemma 1, to maximize We have

Objective Function [C2] and [C4] [C2] Minimize the asymptotic variance of estimated MTTF under use condition [C4] Minimize the asymptotic variance of estimated p-th percentile of failure time distribution under normal use condition

Objective Function [C2] and [C4] [C2] Minimize the variance of estimated MTTF under use condition By Delta-Method, the variance of estimated MTTF under x0 is [C4] Minimize the variance of estimated p-th percentile of failure time distribution under normal use condition

Proof for [C2] and [C4] By Lemma 2, for any , there exists a such that Moreover, if any of

Objective Function [C3] [C3] Minimize the trace of the variance-covariance matrix of the MLE's of model parameters. Trace=

Proof for [C3] By Lemma 2, for any , there exists a such that Similarly, if any of

Optimum SSADT Plan Proposition 2 4.2 Optimal Simple SSADT Plan For [C1], the optimum plan assigns inspections: For [C2] and [C4], the optimum plan assigns inspections: For [C3], the optimum plan assigns inspections:

A Numerical Example 22 LED lamps were tested under this SSADT plan and the light intensities are measured every 168 hours. Temperatures and the temperature change are as follows:

LED Example The estimated relationship between and the temperature stress is A simple linear regression fitting is

LED Example Table 1 presents comparisons between the above optimal SSADT plans and the original SSADT plan used by Tseng and Wen(2000) Optimum Simple SSADT plan could significantly improve the efficiency. The plan for [C3] provides relatively high efficiency for most criteria

LED Example Table 1 presents comparisons between the above optimal SSADT plans and the original SSADT plan used by Tseng and Wen(2000) Optimum Simple SSADT plan could significantly improve the efficiency. The plan for [C3] provides relatively high efficiency for most criteria

Simulation Study To investigate the performance of the optimum SSADT from large sample theorem when the sample size is small Optimum Simple SSADT still performs well even sample size is small.

Summary Our proposition suggests that the optimal SSADT would use only the two most extreme values of stress for several commonly used optimization criteria. Furthermore, we derive the optimal proportion of inspections at each stress level. Both theoretical and simulation results suggest that for many optimization criteria, the efficiencies could be improved by using the optimum simple SSADT plan.

Future Work To see if the results continue to hold for other stochastic process (e.g., Gamma Process). To see if the results continue to hold for other Life-Stress relationship such Arrhenius reaction rate model. Investigate the optimum SSADT for small sample size.

Thank you!