(25)Residual plots can be used to check if the model obtained, based on
the MLE estimates, is a good fit to the data. DOE++ uses standardized residuals for R-DOE analyses.
If the data follows the lognormal distribution, then standardized residuals
are calculated using the following equation:(25)
For the probability plot, the standardized residuals are displayed on a normal probability plot. This is because under the assumed model for the lognormal distribution, the standardized residuals should follow a normal distribution with a mean of 0 and a standard deviation of 1.
For data that follows the Weibull distribution, the standardized residuals
are calculated as shown next:
(26)
The probability plot, in this case, is used to check if the residuals
follow the extreme-value distribution with a mean of 0. Note that in all
residual plots, when an observation, 
, is censored the corresponding residual
is also censored.
Example 11.3
|
Figure 11.5: The 2 |
|
Figure 11.6: Results of the R-DOE analysis for the experiment in Example 11.3. |
This example illustrates the use of R-DOE analysis to design reliability
into the products. An experiment was carried out to investigate the effect
of five factors (each at two levels) on the reliability of fluorescent
lights (Taguchi, 1987, p. 930). The factors, 
through 
, were studied using a 2
design (with the defining relations
and 
) under the assumption that all interaction
effects, except 
, can be assumed to be inactive. For
each treatment, two lights were tested (two replicates) with the readings
taken every two days. The experiment was run for 20 days and, if a light
had not failed by the 20th day, it was assumed to be a suspension. The
experimental design and the corresponding failure times are shown in Figure
11.5. The short duration of the experiment and
failure times were probably because the lights were tested under conditions
which resulted in stress higher than normal conditions. The failure of
the lights was assumed to follow the lognormal distribution.
The analysis results from DOE++ for this experiment are shown in Figure
11.6. The results are obtained by selecting the
main effects of the five factors and the interaction 
using the Select Effects icon in the
Control Panel. The results show that factors 
, 
and 
are active at a significance level
of 0.05. The MLE estimates of the effect coefficients corresponding to
these factors are 
, 
and 
, respectively. Based on these coefficients,
the best settings for these effects to improve the reliability of the
fluorescent lights (by maximizing the response, which in this case is
the failure time) are:
should be set at the lower level of
since its coefficient is negative
should be set at the higher level
of 
since its coefficient is positive
should be set at the lower level of
since its coefficient is negative
Note that, since actual factor levels are not disclosed (presumably for proprietary reasons), predictions beyond the test conditions cannot be carried out in this case.
Example 11.4
Consider a product whose reliability is thought to be affected by eight
potential factors - 
(temperature), 
(humidity), 
(load), 
(fan-speed), 
(voltage), 
(material), 
(vibration) and 
(current). Assuming that all interaction
effects are absent, a 2
design is used to investigate the
eight factors at two levels. The generators used to obtain the design
are 
, 
, 
and 
. The design and the corresponding
life data obtained are shown in Figure 11.7. Readings
for the experiment are taken every 20 time units and the test is terminated
at 200 time units. The life of the product is assumed to follow the Weibull
distribution.
The results from DOE++ for this experiment are shown in Figure 11.8. The results show that only factors 
and 
are active at a significance level
of 0.1. Assume that, in terms of the actual units, the 
level of factor 
corresponds to a temperature of 333
and the 
level corresponds to a temperature
of 383 
. Similarly, assume that the two levels
of factor 
are 1000 
and 2000 
respectively. From the MLE estimates
of the effect coefficients it can be noted that to improve reliability
(by maximizing the response) factors 
and 
should be set as follows:
should be set at the lower level of
333 
since its coefficient is negative
should be set at the higher level
of 2000 
since its coefficient is positive
|
Figure 11.7: The 2 |
|
Figure 11.8: Results for the experiment in Example 11.4. |
Now assume that the use conditions for the product for the significant
factors, 
and 
, are a temperature of 298 
and a fan-speed of 3000 
respectively. The analysis can be
taken a step further to obtain an estimate of the reliability of the product
at the use conditions using ReliaSoft's ALTA software. The data is entered into ALTA as shown
in Figure 11.9. ALTA allows for modeling of the
nature of relationship between life and stress. It is assumed that the
relation between life of the product and temperature follows the Arrhenius
relation [18] while the relation between life and fan-speed
follows the inverse power law relation [18]. Using these relations
ALTA fits the following model for the data in Figure 11.9:
(27)
|
Figure 11.9: Additional reliability analysis for Example 11.4, conducted using ReliaSoft's ALTA software. |
Based on this model the B10 life of the product at the use conditions
is obtained as shown next. [Note]
The Weibull reliability equation is: 
(28)
Substituting the value of 
from Eqn. (27)
and the value of 
as obtained from ALTA, the reliability
equation becomes:
Finally, substituting the use conditions (Temp 
, Fan-Speed 
) and the desired reliability value
of 90%, the B10 life is obtained:
Therefore, at the use conditions, the B10 life of the product is 225 time units. This result and other reliability metrics can be directly obtained from ALTA.
See Also:
experiment design for Example 11.3 to study
factors affecting the reliability of fluorescent lights.
design to investigate the reliability of
a product for Example 11.4.