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DeLTA was developed at the Institute
for Reliability and Risk Analysis, and forms part of a suite of
programs designed to work in the Microsoft Windows environment.
OVERVIEW
The objective of DeLTA is to make predictions about the
"strength" of an item when it is to be subjected to high levels
of stress, based on the data obtained when similar items have been
subjected to low levels of stress. Hence the name, decelerated life
testing, as opposed to accelerated life testing. This method is
particularly useful to engineering planning when it is impractical to test
items at high levels of stress. DeLTA employs a Bayesian approach, to
incorporate the expert opinion of experts in the construction of the model.
The model created then assumes the lognormal distribution for the
"strength" of the items tested. This current version of the model
allows up to ten experts to give their opinion on the parameters of the
lognormal distribution, along with some other parameters. An impartial
analyst is then tasked to note the possible correlation between the
opinions of these experts and the model further incorporates this perceived
relationship in its calculations.
DeLTA has four main modules whereby various opinions are keyed into the
system. These cater to the experts’ opinion on "strength",
on the parameters of the other distributions, on the correlation mentioned
previously and also the observed "strength" under low levels of
stress. DeLTA then estimates the predictive mean and standard deviation of
the item’s strength after further tests. It also allows the user to
select the size of the "basket" for the grouping of data. For
details on the theory, see Singpurwalla, Wilson and Fuller (1996).
Description of the Program
DeLTA consists of several modules, all of which can be accessed from a main
module. The program itself comes as a shell. The successful execution of
one cycle of calculations requires the filling up of data in the modules
designated as Choice 1 to Choice 4.
CHOICE 1 : Entering the expert
opinion of strength
The opinion of up to ten experts can be keyed into the system through this
option. The numbers that have to be keyed in reflects the experts’
opinion on the surviving fraction of the strength of the material tested
after ten and twenty cycles of stress-loading respectively. As it is only
to be expected that stress degrades the material, the program only accepts
a reduction in surviving strength. Whether the numbers input will
work is reflected by the status blue column to the right of the entries.
Once the input is complete, it is imperative that the user clicks on the
UPDATE PARAMETERS button or the MAIN MENU button. By returning to the main
menu without pressing either button will cause the program not to recognize
the new numbers. The updated parameters are reflected in the Lambda and
Delta values directly below the input table. The default values for the
parameters are 21 and 32 respectively. See Figure 1.
Figure 1
CHOICE 2 : Entering the observed
strength of material.
A key feature of the decelerated life testing approach is its ability to
incorporate expert opinion and observed data. The observed data in this
case is the strength of the material after it had been subjected to certain
levels of stress loading. DeLTA allows the observed strength of the
material after up to ten cycles of testing to be submitted. The user can
key in the observed data in two ways: either as the observed strength
itself (raw data), or as the mean as well as the standard deviation of the
observed strength per level of stress (statistics). See Figure 2 below. If
the former choice is made, DeLTA accepts up to thirty observations per
stress level, calculates the mean and standard deviation and updates the
relevant parameters once the DONE button is pressed. If the latter is
chosen, the user will then be directed to another table where only the mean
and standard deviation of the strengths need be filled in. See Figure 3.
DeLTA does not require test data from consecutive levels of stress; for
example, it is able to accept data from the first, second, third, sixth and
ninth level of stress loading.
Figure 2
Figure 3
CHOICE 3: Entering the forecast
level.
This is the option that most engineers would be interested in. What is the
predicted strength of the material if it were to be subjected to high
levels of stress? The user, upon selecting this option, is directed to
decide on the stress level for which the strength is to be predicted.
Although DeLTA accepts data for the first ten levels of stress, it does not
forbid the user from entering any integer between 1 and 10. In fact, should
such an integer be selected, it would be interesting to see how the test
data correspond with the expert opinion. The default is set at ten.
CHOICE 4 : Entering expert opinion
and correlation.
The selection of this choice allows the user to key in the expert opinion
on the rest of the parameters, as well as the perceived correlation between
the opinions of some of these experts. Not all experts’ opinion are
independent; one might possibly find that some of the experts might be
former collaborators in some related field, which would therefore tilt
their opinion in a certain direction. Furthermore, even experts may have
tendencies to be overly cautious, or even exaggerative. Bias (multiply),
bias (add) and precision takes care of this. A Bias (multiply) of more than
1 is used to compensate for an expert who tends to exaggerate by a certain
multiple. Similarly, a positive Bias (add) is used to compensate for an
expert with a known history of exaggeration by a certain value. A precision
of more than one is also used to counter an expert who normally tends to
offer a larger standard deviation for his opinion.
Viewing the Posterior versus Prior
graphs.
Figure 4 shows the Posterior versus Prior graph. This graph allows the user
to compare the graphs of a key parameter of the decelerated life-testing
approach, alpha. Alpha controls the rate of degradation of the material. If
alpha is close to zero, the material will degrade slowly and reach its
"equilibrium" strength very quickly. If alpha is closer to one,
the rate of degradation decreases slowly, which means that the material
will reach its "equilibrium" strength much later. Figure 4 shows
the posterior and prior graph for the default settings; note how the
incorporation of the test data has shifted the graph to the left. At the
same time, the standard deviation has also shrunk, indicating that the
posterior information is more "certain" than the prior.
Figure 4
Viewing the "ungrouped
Data" graph.
This graph shows the predicted values generated by DeLTA using the range of
parameters defined plotted against the probability of them occurring. The
graph is plotted as it is. In other words, there is no sorting and no
grouping of the data points. As DeLTA generates the predicted values by
running through a list of possible values for the parameters, it becomes
rather obvious from this graph the effects of increasing these parameters.
The graph that DeLTA presents may be very small; however, the user may edit
the size of it by right clicking on the graph, and stretching the graph in
the appropriate direction by placing the mouse over one of the eight boxes
and dragging it with the left mouse button.
Figure 5
Viewing the "Grouped Data"
graph.
To generate the predicted values, DeLTA uses a range of values for the main
parameters. These predicted values come with their own probability of
occurring. Therefore, should the user require to know the probability of
the predicted strength being, say 250, then the obvious problem is how
close to 250 should we consider. Do we consider values like 249.58,
or 251.27? The bracket size that comes with this choice allows the user to
decide the interval length to which the data is grouped together. With a
"bracket size" of 1, DeLTA groups the values to the nearest integer,
combines the probability for all the predicted values that falls within the
interval and plots the point. The default size of the "bracket"
is five, which means that DeLTA groups all points in packets of 5 before
plotting the graph. Figure 6 shows the graph for the default settings.
Figure 6
Rerunning the whole program.
The selection of this choice commands the program to rerun itself. All
calculations will be re-performed. This step is absolutely essential should
any of the first 4 options be changed. Furthermore, as the original program
comes without any graphs or calculations performed, the selection of any of
the graph options will yield a blank screen if the this option is not
utilised. Once all the calculations are complete, DeLTA provides the
predicted mean and standard deviation in a small table below this option.
Guideline for Analysis
Examining the Posterior versus Prior graph in Figure 5 , it is obvious that
the prior graph is to the right of the posterior graph. The reason for this
is that the actual value of alpha (which is still unknown) is actually
smaller than that predicted by the experts. What is the significance of
having an alpha which is smaller than what it is thought to be? alpha is
the rate of degradation. A value of a closer to 1 means that the rate of
degradation remains fairly constant and only diminshes slowly. A smaller
value of alpha would, on the other hand, would mean that the rate of
degradation diminishes rapidly. This implies that the strength of the item
will stabilize much faster, and at a higher level, than for another
material with a higher value of alpha. Therefore, as the examination of the
default posterior versus prior graph shows, the value of alpha has
decreased with the inclusion of data. This should mean that the actual
observed strength should be higher than the predicted strength, since the
actual strength is expected to stabilise at a higher level than the
predicted strength.
Now, supposed we set the prediction level to a level that we have
experimental data on. In our case, we set our prediction level to i
= 4. The experimental data showed that the mean strength of the test item
is 285 whereas the predicted mean strength and standard deviation are
248.06 and 44.11 respectively The reason why the predicted strength is
below that of the actual strength could be that the experts prefer to err
on the side of caution. The earlier discussion on the difference in prior
and posterior mean of alpha serves to reinforce this point. Therefore, the
two observations serve to reinforce each other.
Hardware and Software Requirements.
The current PC-based version of DeLTA is programmed in Visual Basic, but
executes most of its calculations through Microsoft Excel. DeLTA is a
memory-hungry program which will perform close to a million iterations per
cycle of calculations. As such, it is advised, for the sake of the
user’s sanity, to run the program on at least a Pentium processor,
clock speed 166 MHz minimum, with 64 Mbytes of RAM on board. On a Pentium
II processor, clock speed 333 MHz with 128 Mbytes of RAM, the basic
calculations require about two minutes.
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