( 
) represents the 
response values obtained at the 
th treatment combination of the inner
array (corresponding to the 
levels of the noise factors), then
the mean response at the 
th level is:
(2)Depending upon the objective of the robust parameter design experiment, Taguchi defined three different statistics called signal to noise ratios. These ratios were defined as the means to measure the variation of the response with respect to the noise factors. Taguchi's approach essentially consists of two models - a location model and a dispersion model.
This section is divided into the following subsections:
The location model is the regression model for the mean value of the
response at each treatment combination of the inner array. If ( ) represents the response values obtained at the th treatment combination of the inner
array (corresponding to the levels of the noise factors), then
the mean response at the th level is:(2)
The location model is obtained by fitting a regression model to all
values, by treating these values as
the response at each of the 
th treatments of the inner array. As
an example, the location model for an inner array with two factors can
be written as:
(3)where:
is the intercept.
is the coefficient for the first factor.
is the coefficient for the second
factor.
is the coefficient for the interaction.
and 
are respectively the variables for
the two factors.
The objective of using the location model is to bring the response to
its goal regardless of whether this is a target value, maximum value or
minimum value. This is done by identifying significant effects and then
using the least square estimates of the corresponding coefficients, 
s, to fit the location model. The fitted
model is used to decide the settings of the variables that bring the response
to the goal.
The dispersion model measures the variation of the response due to the
noise factors. The standard deviation of the response values at each treatment
combination, 
, is used. Usually, the standard deviation
is used as a log function of 
because 
are approximately normally distributed.
These values can be calculated as follows:
(4)
Thus, the dispersion model consists of using 
as the response and investigating
what treatment of the control factors results in the minimum variation
of the response. Clearly, the objective of using the dispersion model
is to minimize variation in the response. Instead of using standard deviations
directly, Taguchi defined three signal to noise ratios (abbreviated 
) based on the objective function for
the response. If the response is to be maximized, the 
ratio is defined as follows:
(5)
The previous ratio is referred to as the larger-the-better ratio and is defined to decrease variability when maximizing the response.
When the response is to be minimized, the 
ratio is defined as:
(6)
This ratio is referred to as the smaller-the-better ratio and is defined to decrease variability when minimizing the response.
If the objective for the response is to achieve a target or nominal
value, then the 
ratio is defined as follows:
(7)
This ratio is referred to as the nominal-the-best ratio and is defined to decrease variability around a target response value.
The dispersion model for any of the three signal to noise ratios can
be written as follows for an inner array with two factors:
where:
is the intercept.
is the coefficient for the first factor.
is the coefficient for the second
factor.
is the coefficient for the interaction.
and 
are respectively the variables for
the two factors.
The dispersion model is fit by identifying significant effects and then
using the least square estimates of the coefficients 
s. Once the fitted dispersion model
is known, settings for the control factors are found that result in the
maximum value of 
, thereby minimizing the response variation.
The location and dispersion regression models are usually obtained by using graphical techniques to identify significant effects. This is because the responses used in the two models are such that only one response value is obtained for each treatment of the inner array. Therefore, the experiment design in the case of the two models is an unreplicated design. [Note]
Once the location and dispersion models have been obtained by identification
of the significant effects, the following analysis strategy may be used
[27]:
ratio.
ratio.
Factors that do not show up as significant in both the models should be set at levels that result in the greatest economy. Generally, a follow-up experiment is usually carried out with the best settings to verify that the system functions as desired.
Example 10.2
This example illustrates the procedure to obtain the location and dispersion models for the experiment in Example 10.1.
The response values used in the location model can be calculated using
Eqn. (2). As an example, the response value for
the third treatment is:
Response values for the remaining seven treatments can be calculated
in a similar manner. These values are shown in Figure 10.7
under the column Y Mean. Once the response values for all the treatments
are known, the analysis to fit the location model can be carried out by
treating the experiment as a single replicate of the 2
design. The results obtained from
DOE++ are shown in
Figure 10.8. The results are displayed by selecting
Y Mean in the Response drop-down menu. The normal probability plot of
effects for this model shows that only the main effect of factor 
is significant for the location model
(see Figure 10.9). Using the corresponding coefficient
from Figure 10.9, the location model can be written
as:
(8)
where 
is the variable representing factor
.
|
Figure 10.7: Response values for the location and dispersion models in Example 10.2. |
|
Figure 10.8: Results for the location model in Example 10.2. |
|
Figure: 10.9: Normal probability plot of effects for the location model in Example 10.2. |
For the dispersion model, the applicable signal to noise ratio is given
by Eqn. (7) since this example is a case of nominal-the-best:
The response values for the dispersion model can now be calculated.
As an example, the response value for the third treatment is:
Other 
values can be obtained in a similar
manner. The values are shown in Figure 10.7 under
the column Signal Noise Ratio. As in the case of the location model, the
analysis to fit the dispersion model can be carried out by treating the
experiment as a single replicate of the 2
design. The results obtained from
DOE++ are shown in Figure 10.10. The results are
displayed by selecting Signal Noise Ratio in the Response drop-down menu.
The normal probability plot of effects for this model shows that the interaction
is the only significant effect for
this model (see Figure 10.11). Using the corresponding
coefficient from Figure 10.10, the dispersion model
can be written as:
(9)
where 
is the variable representing factor
.
|
Figure 10.10: Results for the dispersion model in Example 10.2. |
|
Figure 10.11: Normal probability plot of effects for the dispersion model in Example 10.2. |
Following the analysis strategy mentioned above in Analysis
Strategy, for the nominal-the-best case, the dispersion model should
be considered first. Eqn. (9) shows that to maximize
, either one of the following options
can be used:
and 
or
and 
Then, considering the location model of Eqn. (8),
to achieve a target response value as close to 7 as possible, the only
significant
effect for this model, 
, should be set at the level of 
. Therefore, option 1 should be selected
as the settings for the dispersion model. The final settings for the three
factors, as a result of the robust parameter design, are:
is set at the low level
is set at the high level and
, which is not significant in any of
the two models, can be set at the level that is most economical.
With these settings the predicted pH value for the product is:
The predicted signal to noise ratio is:
To make the signal to noise ratio model hierarchical, 
and 
have to be included in the model.
[Note]
Then, the predicted 
ratio is:
See Also:
Taguchi's Robust Parameter Design Method
Limitations of Taguchi's Approach
