1
Accelerated Degradation Tests Planning With Competing
Failure Modes
Xiujie Zhao
a*†
, Jianyu Xu
b,c
and Bin Liu
a
a
Department of Systems Engineering and Engineering Management
City University of Hong Kong, Hong Kong
b
City University of Hong Kong Shenzhen Research Institute, Shenzhen, China
c
University of Chinese Academy of Sciences, Beijing, China
Abstract
Accelerated degradation tests (ADT) have been widely used to assess the reliability of prod-
ucts with long lifetime. For many products, environmental stress not only accelerates their
degradation rate but also elevates the probability of traumatic shocks. When random traumatic
shocks occur during an ADT, it is possible that the degradation measurements cannot be taken
afterward, which brings challenges to reliability assessment. In this paper, we propose an ADT
optimization approach for products suffering from both degradation failures and random shock
failures. The degradation path is modeled by a Wiener process. Under various stress levels, the
arrival process of random shocks is assumed to follow a non-homogeneous Poisson process.
Parameters of acceleration models for both failure modes need to be estimated from the ADT.
Three common optimality criteria based on the Fisher information are considered and com-
pared to optimize the ADT plan under a given number of test units and a pre-determined test
duration. Optimal two- and three-level optimal ADT plans are obtained by numerical methods.
We use the general equivalence theorems to verify the global optimality of ADT plans. A nu-
merical example is presented to illustrate the proposed methods. The result shows that the op-
timal ADT plans in the presence of random shocks differ significantly from the traditional ADT
plans. Sensitivity analysis is carried out to study the robustness of optimal ADT plans with
respect to the changes in planning input.
Keywords: accelerated degradation tests, competing failure modes, degradation modeling,
optimal design, Fisher information, reliability assessment
*
Corresponding author: xiujizhao2-c@my.cityu.edu.hk
†
The work described in this paper was partially supported by a theme-based project grant (T32-101/15-R) of University
Grants Council of Hong Kong, and a Key Project (71532008) supported by National Natural Science Foundation of China.
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Acronyms and Abbreviations
ADT Accelerated degradation test
GET General equivalence theorems
MLE Maximum likelihood estimation
PF0 Proportion of failure under use condition
PF1 Proportion of failure under the maximum stress
RE Relative efficiency
Notation
Degradation path of the test product
Lifetime of the test product
Degradation threshold
Standard Brownian motion
Inverse Gaussian distribution with mean
and scale
True parameters
MLE of
-th lifetime percentile
Inspection interval in the ADT
Drift parameter under standardized stress
Number of stress levels
Number of test units
Level of the
-th stress
Proportion of test units allocated to the
-th stress
Number of test units allocated to the
-th stress
Fisher information matrix
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CDF and PDF of the standard normal distribution
Trace of matrix
Determinant of
Expectation of random variable
Asymptotic variance of
Two-level test plan and optimal plan
The optimal compromise plan
1. Introduction
Accelerated reliability tests are commonly used to assess the reliability of new products,
especially those with extremely long lifespan under field use. In such tests, the products are
exposed to elevated stress conditions, such as higher temperature, pressure, humidity, or a com-
bination of them. Data analysis and optimal planning of accelerated tests have drawn
considerable attention from reliability researchers and engineers, who desire to predict the re-
liability as precise as possible through a more economical approach. Test information from a
well-planned accelerated reliability test can provide useful guidance for maintenance
scheduling and warranty prediction [1], [2]. For an overview, see Elsayed [3].
Inferences of lifetime distribution from accelerated life tests (ALT) is becoming very chal-
lenging because many highly reliable products have none or very few failures even under
elevated stresses in a reasonable test duration. In such situations, we can resort to accelerated
degradation tests (ADT) if the product has one or more measurable quality characteristics (QC)
that can be modeled as degradation processes [4]. Instead of observing the failure times as in
ALT, degradation levels of test units are measured periodically in ADT. The planning of ADT
or other types of degradation test, such as step-stress ADT (SSADT) [5] and accelerated
destructive degradation tests (ADDT) [6] have proved to be efficient in enhancing the accuracy
of reliability assessment and saving experimental resources. The optimal ADT plans based on
Wiener processes [7], [8], gamma processes [9]–[11] , and inverse Gaussian processes [12]
have been intensely studied in the literature.
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Most existing works on optimal ALT/ADT planning assumed that there was only one failure
mode. However, many products have more than one failure modes. Neglect of any failure mode
may significantly influence the optimality of reliability test plans and therefore the prediction
accuracy of lifetime, thus it is necessary to consider multiple failure modes when planning
accelerated tests. Bai and Chun [13] discussed the optimal simple step-stress ALT (SSALT)
plans with independent competing causes. Afterward, Pascual [14], [15] studied the ALT plan-
ning by considering independent Weibull or lognormal competing risks. For repairable sys-
tems, Liu and Tang [16] used a Bayesian D-optimality criterion to optimize ALT plans with
independent risks, and an extension to SSALT can be found in Liu and Qiu [17]. Similar ideas
have also been discussed for ALT with multiple stresses [18] and dependent failure modes
[19].
Although there are numerous studies on ALT planning with more than one failure modes,
fewer studies have addressed the ADT modeling and planning with multiple failure modes. Ye
et al. [20] developed a burn-in planning method by differentiating normal and mortality failure
modes. The optimal two-variable ADT planning method for gamma processes was discussed
in Tsai et al. [10]. Li and Jiang [21] proposed a SSADT planning method with independent
stochastic degradation processes. Furthermore, SSADT planning problem with two dependent
gamma processes was studied by Pan and Sun [22]. Haghighi and Bae modeled and analyzed
linear degradation data and traumatic failures with competing risks in an SSADT experiment
with the cumulative exposure model [23]. Nevertheless, to our knowledge, none of the studies
in literature has considered both degradation and shock failures in the ADT planning problem,
although the joint modeling of degradation and random shocks as well as related
maintenance/warranty problems have been very popular in recent literature [24]–[26]. It is
common that either performance degradation or random traumatic shocks could lead to fail-
ures. Generally, performance degradation is due to the natural aging and usage of a product,
and if the performance degrades to an unsatisfactory level, the product is deemed failed alt-
hough it may still work. For example, an LED lamp are commonly regarded failed if its light
intensity drops to a certain level. On the other hand, traumatic shocks are more likely to be
caused by external events that influence the whole system, such as the sudden change in cur-
rency and voltage for electronic devices. The shocks lead to immediate product failures. During
an ADT, the degradation measurability can be influenced by random shocks. Regarding the
LED lamp example, if a lamp suddenly goes out during an ADT, its brightness cannot be
tracked after the failure. In this situation, test planners need to consider the possibility that
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increasing number of random shocks under elevated stress conditions significantly decreases
the test information due to the partial missing of degradation measurements. Figure 1 shows
an illustration of such cases, where each 10 test units are allocated to low- and high-level
stresses, respectively. The measurements from high-level stress provide more information on
the acceleration relationship of degradation, but the stress also leads to more shock failures
during the test. As is shown there are only three test units that survive to the end of the test and
provide full degradation information during the test. In contrast, there is no shock failure for
the units under low-level stress, and the degradation measurements are complete, yet the deg-
radation increase is not significant so that the inference of the degradation rate can be greatly
influenced by random noises. In previous ADT studies, test planners usually took full ad-
vantage of the highest possible stress to obtain measurements with high degradation rate as
long as it is believed that the degradation mechanism remains the same for the highest stress
[27]. However, if random shocks are taken into account, higher stress may lead to much more
shock failures in the test process and the number of degradation measurements become con-
siderably less, which will cause loss in data to estimate unknown parameters and predict life-
time under use condition.
Figure 1 Stress-related shocks in ADT experiment
Considering the shock issue in ADTs, we propose an optimal ADT planning approach with
competing failure modes in this paper. The product to be tested is assumed to suffer from both
degradation and shock failures. Both failure modes are accelerated by a common experimental
![Table 10 Optimal ADT plans under [C3] (ᵃ� -optimaility) under different shock failure assumptions.](/figures/table-10-optimal-adt-plans-under-c3-optimaility-under-1fp7knhf.webp)
![Table 9 Optimal ADT plans under [C2] (ᵃ�-optimaility) under different shock failure assumptions.](/figures/table-9-optimal-adt-plans-under-c2-optimaility-under-jhmq9c9o.webp)
![Figure 4 Derivative function against lower stress level ᵅ� under [C1], [C2], [C3]](/figures/figure-4-derivative-function-against-lower-stress-level-n3jumr3w.webp)
