Accelerated Life Test Plans

Template:ALTABOOK SUB Poor accelerated test plans waste time, effort and money and may not even yield the desired information. Before starting an accelerated test (which is sometimes an expensive and difficult endeavor), it is advisable to have a plan that helps in accurately estimating reliability at operating conditions while minimizing test time and costs. A test plan should be used to decide on the appropriate stress levels that should be used (for each stress type) and the amount of the test units that need to be allocated to the different stress levels (for each combination of the different stress types' levels). This section presents some common test plans for one-stress and two-stress accelerated tests.

General Assumptions

Most accelerated life testing plans use the following model and testing assumptions that correspond to many practical quantitative accelerated life testing problems.

1. The log-time-to-failure for each unit follows a location-scale distribution such that:

[math]\displaystyle{ \underset{}{\overset{}{\mathop{\Pr }}}\,(Y\le y)=\Phi \left( \frac{y-\mu }{\sigma } \right)\,\! }[/math]

where [math]\displaystyle{ \mu \,\! }[/math] and [math]\displaystyle{ \sigma \,\! }[/math] are the location and scale parameters respectively and [math]\displaystyle{ \Phi \,\! }[/math] ( [math]\displaystyle{ \cdot \,\! }[/math] ) is the standard form of the location-scale distribution.

2. Failure times for all test units, at all stress levels, are statistically independent.

3. The location parameter [math]\displaystyle{ \mu \,\! }[/math] is a linear function of stress. Specifically, it is assumed that:

[math]\displaystyle{ \begin{align} \mu =\mu ({{z}_{1}})={{\gamma }_{0}}+{{\gamma }_{1}}x \end{align}\,\! }[/math]

4. The scale parameter, [math]\displaystyle{ \sigma ,\,\! }[/math] does not depend on the stress levels. All units are tested until a pre-specified test time.

5. Two of the most common models used in quantitative accelerated life testing are the linear Weibull and lognormal models. The Weibull model is given by:

[math]\displaystyle{ Y\sim SEV\left[ \mu (z)={{\gamma }_{0}}+{{\gamma }_{1}}x,\sigma \right]\,\! }[/math]

where [math]\displaystyle{ SEV\,\! }[/math] denotes the smallest extreme value distribution. The lognormal model is given by:

[math]\displaystyle{ Y\sim Normal\left[ \mu (z)={{\gamma }_{0}}+{{\gamma }_{1}}z,\sigma \right]\,\! }[/math]

That is, log life [math]\displaystyle{ Y\,\! }[/math] is assumed to have either an [math]\displaystyle{ SEV\,\! }[/math] or a normal distribution with location parameter [math]\displaystyle{ \mu (z)\,\! }[/math], expressed as a linear function of [math]\displaystyle{ z\,\! }[/math] and constant scale parameter [math]\displaystyle{ \sigma \,\! }[/math].

Planning Criteria and Problem Formulation

Without loss of generality, a stress can be standardized as follows:

[math]\displaystyle{ \xi =\frac{x-{{x}_{D}}}{{{x}_{H}}-{{x}_{D}}}\,\! }[/math]

where:

• [math]\displaystyle{ {{x}_{D}}\,\! }[/math] is the use stress or design stress at which product life is of primary interest.

• [math]\displaystyle{ {{x}_{H}}\,\! }[/math] is the highest test stress level.

The values of [math]\displaystyle{ x\,\! }[/math], [math]\displaystyle{ {{x}_{D}}\,\! }[/math] and [math]\displaystyle{ {{x}_{H}}\,\! }[/math] refer to the actual values of stress or to the transformed values in case a transformation (e.g., the reciprocal transformation to obtain the Arrhenius relationship or the log transformation to obtain the power relationship) is used.

Typically, there will be a limit on the highest level of stress for testing because the distribution and life-stress relationship assumptions hold only for a limited range of the stress. The highest test level of stress, [math]\displaystyle{ {{x}_{H}},\,\! }[/math] can be determined based on engineering knowledge, preliminary tests or experience with similar products. Higher stresses will help end the test faster, but might violate your distribution and life-stress relationship assumptions.

Therefore, [math]\displaystyle{ \xi =0\,\! }[/math] at the design stress and [math]\displaystyle{ \xi =1\,\! }[/math] at the highest test stress.

A common purpose of an accelerated life test experiment is to estimate a particular percentile (unreliability value of [math]\displaystyle{ p\,\! }[/math]), [math]\displaystyle{ {{T}_{p}}\,\! }[/math], in the lower tail of the failure distribution at use stress. Thus a natural criterion is to minimize [math]\displaystyle{ Var({{\hat{T}}_{p}})\,\! }[/math] or [math]\displaystyle{ Var({{\hat{Y}}_{p}})\,\! }[/math] such that [math]\displaystyle{ {{Y}_{p}}=\ln ({{T}_{p}})\,\! }[/math]. [math]\displaystyle{ Var({{\hat{Y}}_{p}})\,\! }[/math] measures the precision of the [math]\displaystyle{ p\,\! }[/math] quantile estimator; smaller values mean less variation in the value of [math]\displaystyle{ {{\hat{Y}}_{p}}\,\! }[/math] in repeated samplings. Hence a good test plan should yield a relatively small, if not the minimum, [math]\displaystyle{ Var({{\hat{Y}}_{p}})\,\! }[/math] value. For the minimization problem, the decision variables are [math]\displaystyle{ {{\xi }_{i}}\,\! }[/math] (the standardized stress level used in the test) and [math]\displaystyle{ {{\pi }_{i}}\,\! }[/math] (the percentage of the total test units allocated at that level). The optimization problem can be formulized as follows.

Minimize:

[math]\displaystyle{ Var({{\hat{Y}}_{p}})=f({{\xi }_{i}},{{\pi }_{i}})\,\! }[/math]

Subject to:

[math]\displaystyle{ 0\le {{\pi }_{i}}\le 1,\text{ }i=1,2,...n\,\! }[/math]

where:

[math]\displaystyle{ \underset{i=1}{\overset{n}{\mathop{\sum }}}\,{{\pi }_{i}}=1\,\! }[/math]

An optimum accelerated test plan requires algorithms to minimize [math]\displaystyle{ Var({{\hat{Y}}_{p}})\,\! }[/math].

Planning tests may involve compromise between efficiency and extrapolation. More failures correspond to better estimation efficiency, requiring higher stress levels but more extrapolation to the use condition. Choosing the best plan to consider must take into account the trade-offs between efficiency and extrapolation. Test plans with more stress levels are more robust than plans with fewer stress levels because they rely less on the validity of the life-stress relationship assumption. However, test plans with fewer stress levels can be more convenient.