A general accelerated life model for step-stress testing

IIE Transactions, Nov, 2005 by Wenbiao Zhao, Elsayed A. Elsayed

Notation

[beta]               = Weibull shape parameter;
[eta]                = Weibull scale parameter;
[[alpha].sub.AF]     = acceleration factor;
A                    = parameter of the Eyring relationship;
B                    = parameters of the Arrhenius and Eyring
                       relationship associated with the activation
                       energy;
C                    = parameter of the Arrhenius relationship;
K                    = parameter of the Inverse Power Law (IPL)
                       relationship;
n                    = parameter of the IPL relationship;
[phi], h             = parameters of the temperature-humidity
                       relationship;
U, V                 = applied stresses;
[U.sub.0], [V.sub.0] = normal operating stresses;
[U.sub.s], [V.sub.s] = accelerated stresses;
S                    = stress;
[[theta].sub.i]      = unknown parameter in the proposed model;
f()                  = probability density function;
R()                  = reliability function;
h()                  = hazard rate function.

1. Introduction

Accelerated Life Testing (ALT) using a time-varying stress application is often used to induce failures in relatively short times. The most basic and useful type of time-varying stress testing is step-stress testing. This is a special type of ALT in which the stress on each unit is increased step-by-step over time; it can substantially shorten the duration of the reliability test without affecting the accuracy of the reliability estimates at normal operating conditions.

This has lead to the development and planning of appropriate step-stress ALT models and test plans. Nelson (1980, 1990) and Miller and Nelson (1983) suggest that the life-stress model must take into account the cumulative effect of the applied stresses when dealing with data from accelerated tests with time-varying stresses. Based on this idea Nelson (1980, 1990) proposes a cumulative damage (exposure) model which has gained acceptance in the reliability engineering field. Bai et al. (1989) extend the results of Miller and Nelson (1983) to the case where a prescribed censoring time is involved. DeGroot and Goel (1979) propose a tampered random variable model and investigate the optimal design of a partially accelerated life test under the framework of Bayesian decision theory. Pan and Ayala (2003) discuss the relationship between Nelson's model and the Degroot-Goel model. Tang et al. (1996) use a linear cumulative model to analyze data from a step-stress ALT (SSALT) using a two-parameter Weibull distribution. Bai and Chun (1991) obtain an optimum simple SSALT with competing causes of failure and Chung and Bai (1996) consider optimal designs of SSALTs in which the life distributions of the test units at each stress level are lognormally-distributed. It is assumed that a log-linear relation exists between the lognormal location parameter and stress, and that the cumulative exposure model for the effect of changing stress holds. Bhattacharyya and Soejoeti (1989) develop a tampered failure-rate model. Bhattacharyya (1986) also derives an approach using a Gaussian stochastic process which was later modified and extended by Doksum and Hoyland (1991) and Lu and Storer (2001).

Khamis and Higgins (1996, 1998) propose a model for SSALT which is based on a time-transformation of the exponential model. Xiong (1998) presents the inferences of parameters in the simple step-stress model in accelerated life testing with Type II censoring. Xiong and Milliken (1999) study statistical models in SSALT when the stress change times are random and obtain the marginal life distribution for test units. Yin and Sheng (1987) investigate ALT by subjecting the units to progressive stresses.

Nonparametric approaches for step-stress testing are also proposed by Shaked and Singpurwalla (1983), and Schmoyer (1991). Schmoyer (1991) relates the ALT model with the proportional hazards model, suggested by Cox (1972) for reliability analysis, and discusses a nonparametric approach to overcome model uncertainty.

In this paper we present a general approach for modeling SSALT. We first develop a general SSALT model, formulate the maximum likelihood function of the failure time distribution and obtain estimates of the distribution parameters. We then discuss the relationship between the proposed model and other models in the literature and show that most of the widely used SSALT models such as the cumulative damage model of Nelson (1990), the model of Gouno (2001) and the model of Xiong and Milliken (2002) are special cases of the proposed model. Finally, we demonstrate the proposed SSALT model in reliability engineering field by using data from a laboratory experiment.

2. Model development

Accelerated life testing models are based on having an underlying failure time distribution and a life-stress relationship. Normally, a percentile of the failure distribution is chosen to be represented by the life-stress relationship. The objective is to obtain the parameters of the failure distribution and the life-stress relationship. There are several approaches to obtain these parameters such as the Maximum Likelihood Estimation (MLE) procedure. A general likelihood function can be formulated as: