8.
Assessing Product Reliability
8.3.
Reliability Data Collection
8.3.1.
How do you plan a reliability assessment test?
8.3.1.4.

Accelerated life tests


Accelerated
testing is needed when testing even large sample sizes at use stress would
yield few or no failures within a reasonable time 
Accelerated life tests are
component life tests with components operated at high stresses and failure
data observed. While high stress testing can be performed for the sole
purpose of seeing where and how failures occur and using that information
to improve component designs or make better component selections, we will
focus in this section on accelerated life testing for the following two
purposes:

To study how failure is accelerated by stress and fit an acceleration model
to data from multiple stress cells

To obtain enough failure data at high stress to accurately project (extrapolate)
what the CDF at use will be.
If we already know the acceleration model (or the acceleration factor to
typical use conditions from high stress test conditions), then the methods
described two pages ago can be used. We assume,
therefore, that the acceleration model is not known in advance. 
Test planning means picking stress levels
and sample sizes and test times to produce enough data to fit models and
make projections 
Test planning and operation for a (multiple)
stress cell life test experiment consists of the following:

Pick several combinations of the relevant stresses (the stresses that accelerate
the
failure mechanism under investigation). Each combination is a "stress
cell". Note that you are planning for only one mechanism of failure at
a time. Failures on test due to any other mechanism will be considered
censored
run times.

Make sure stress levels used are not too high  to the point where new
failure mechanisms that would never occur at use stress are introduced.
Picking a maximum allowable stress level requires experience and/or good
engineering judgment.

Put random samples of components in each stress cell and run the components
in each cell for fixed (but possibly different) lengths of time.

Gather the failure data from each cell and use the data to fit an acceleration
model and a life distribution model and use these models to project reliability
at use stress conditions.
Test planning would be similar to topics already covered in the chapters
that discussed modeling and experimental design except for one important
point. When you test components in a stress cell for a fixed length test,
it is typical that some (or possibly many) of the components end the test
without failing. This is the censoring problem, and it greatly complicates
experimental design to the point at which it becomes almost as much of
an art (based on engineering judgment) as a statistical science. 

An example will help illustrate the design issues.
Assume a metal migration failure mode is believed to follow the
2stress temperature voltage model given by
$$ t_f = A \, V^\beta \, \mbox{exp } \left( \frac{\Delta H}{kT} \right) \, . $$
Normal use conditions are 4 volts and 25 degrees Celsius, and the high
stress levels under consideration are 6, 8,12 volts and 85^{o},
105^{o} and 125^{o}. It probably would be a waste of resources
to test at (6v, 85^{o}), or even possibly (8v, 85^{o})
or (6v,105^{o}) since these cells are not likely to have enough
stress acceleration to yield a reasonable number of failures within typical
test times.
If you write all the 9 possible stress cell combinations in a 3x3 matrix
with voltage increasing by rows and temperature increasing by columns,
the result would look like the matrix below:
Matrix Leading to "Backward
L Design"
6v, 85^{o}

6v, 105^{o}

6v, 125^{o}

8v, 85^{o}

8v,105^{o}

8v,125^{o}

12v,85^{o}

12v,105^{o}

12v,125^{o}


"Backwards L" designs are common in accelerated
life testing. Put more experimental units in lower stress cells. 
The combinations in bold are the most likely
design choices covering the full range of both stresses, but still hopefully
having enough acceleration to produce failures. This is the socalled
"backwards
L" design commonly used for acceleration modeling experiments.
Note: It is good design practice to put more of your test units
in the lower stress cells, to make up for the fact that these cells will
have a smaller proportion of units failing. 
Sometimes simulation is the best way to learn
whether a test plan has a chance of working 
Design by Simulation:
A lengthy, but better way to choose a test matrix is the following:

Pick an acceleration model and a life
distribution model (as usual).

Guess at the shape parameter value of the life distribution model based
on literature studies or earlier experiments.
The shape parameter should remain the same for all stress cells. Choose
a scale parameter value at use so that the use stress CDF exactly meets
requirements (i.e., for the lognormal, pick a use \(T_{50}\)
that gives the desired use reliability  for a Weibull model choice, do
the same for the characteristic life parameter).

Guess at the acceleration model parameters values (\(\Delta H\) and \(\beta\),
for the 2stress model shown above). Again, use whatever is in the literature
for similar failure mechanisms or data from earlier experiments).

Calculate acceleration factors from any proposed test cells to use
stress and divide the use scale parameter by these acceleration factors
to obtain "trial" cell scale parameters.

Simulate cell data for each proposed stress cell using the derived cell
scale parameters and the guessed shape parameter.

Check that every proposed cell has sufficient failures to give good estimates.

Adjust the choice of stress cells and the sample size allocations until
you are satisfied that, if everything goes as expected, the experiment
will yield enough data to provide good estimates of the model parameters.

After you make advance estimates, it is sometimes
possible to construct an optimal experimental design  but software for
this is scarce 
Optimal Designs:
Recent work on designing accelerated life tests has shown it is possible,
for a given choice of models and assumed values of the unknown parameters,
to construct an optimal design (one which will have the best chance of
providing good sample estimates of the model parameters). These optimal
designs typically select stress levels as far apart as possible and heavily
weight the allocation of sample units to the lower stress cells. However,
unless the experimenter can find software that incorporates these optimal
methods for his or her particular choice of models, the methods described
above are the most practical way of designing acceleration experiments. 