(1)The experiment designs mentioned in Chapters 7 and 8 help the experimenter identify factors that affect the response. Once the important factors have been identified, the next step is to determine the settings for these factors that result in the optimum value of the response. The optimum value of the response may either be a maximum value or a minimum value, depending upon the product or process in question. For example, if the response in an experiment is the yield from a chemical process, then the objective might be to find the settings of the factors affecting the yield so that the yield is maximized. On the other hand, if the response in an experiment is the number of defects, then the goal would be to find the factor settings that minimize the number of defects. Methodologies that help the experimenter reach the goal of optimum response are referred to as Response Surface Methods. These methods are exclusively used to examine the "surface" or the relationship between the response and the factors affecting the response. Regression models are used for the analysis of the response, as the focus now is on the nature of the relationship between the response and the factors, rather than identification of the important factors.
Response surface methods usually involve the following steps:
The experimenter needs to move from the present operating conditions to the vicinity of the operating conditions where the response is optimum. This is done using the method of steepest ascent in the case of maximizing the response. The same method can be used to minimize the response and is then referred to as the method of steepest descent.
Once in the vicinity of the optimum response the experimenter needs to fit a more elaborate model between the response and the factors. Special experiment designs, referred to as RSM designs, are used to accomplish this. The fitted model is used to arrive at the best operating conditions that result in either a maximum or minimum response.
It is possible that a number of responses may have to be optimized at the same time. For example, an experimenter may want to maximize strength, while keeping the number of defects to a minimum. The optimum settings for each of the responses in such cases may lead to conflicting settings for the factors. A balanced setting has to be found that gives the most appropriate values for all the responses. Desirability functions are useful in these cases.
This section contains the following subsections:
The first step in obtaining the optimum response settings, after the
important factors have been identified, is to explore the region around
the current operating conditions to decide what direction needs to be
taken to move towards the optimum region. Usually, a first order regression
model (containing just the main effects and no interaction terms) is sufficient
at the current operating conditions because the operating conditions are
normally far from the optimum response settings. The experimenter needs
to move from the current operating conditions to the optimum region in
the most efficient way by using the minimum number of experiments. This
is done using the method of steepest ascent. In this method, the contour
plot of the first order model is used to decide the settings for the next
experiment, in order to move towards the optimum conditions. [Note]
Consider a process where the response has been found to be a function
of two factors. To explore the region around the current operating conditions,
the experimenter fits the following first order model between the response
and the two factors:(1)
The response surface plot for the model, along with the contours, is
shown in Figure 9.1. It can be seen in the figure,
that in order to maximize the response, the most efficient direction in
which to move the experiment is along the line perpendicular to the contours.
This line, also referred to as the path of steepest ascent, is
the line along which the rate of increase of the response is maximum.
The steps along this line to move towards the optimum region are proportional
to the regression coefficients, 
of the fitted first order model.
|
Figure 9.1: Path of steepest ascent for the model |
Experiments are conducted along each step of the path of steepest ascent until an increase in the response is not seen. Then, a new first order model is fit at the region of the maximum response. If the first order model shows a lack-of-fit, then this indicates that the experimenter has reached the vicinity of the optimum. RSM designs are then used explore the region thoroughly and obtain the point of the maximum response. If the first order model does not show a lack-of-fit, then a new path of steepest ascent is determined and the process is repeated.
The yield from a chemical process is found to be affected by two factors:
reaction temperature and reaction time. The current reaction temperature
is 230 
and the reaction time is 65 minutes.
The experimenter wants to determine the settings of the two factors such
that maximum yield can be obtained from the process. To explore the region
around the current operating conditions, the experimenter decides to use
a single replicate of the 2
design. The range of the factors for
this design are chosen to be (225, 235) 
for the reaction temperature and (55,
75) minutes for the reaction time. The unreplicated 2
design is also augmented with five
runs at the center point to estimate the error sum of squares, 
, and check for model adequacy. [Note]
The response values obtained for this design are shown in Figure 9.2.
In DOE++, this
design can be set up using the properties shown in Figure 9.3.
The resulting design is shown in Figure 9.4 and
the analysis results are shown in Figure 9.5. Note
that the results shown are in terms of the coded values of the factors
(taking -1 as the value of the lower settings for reaction temperature
and reaction time and +1 as the value for the higher settings for these
two factors). The results show that the factors, 
(temperature) and 
(time), affect the response significantly
but their interaction does not affect the response. Therefore the interaction
term can be dropped from the model for this experiment. The results also
show that Curvature is not a significant factor. This indicates that the
first order model is adequate for the experiment at the current operating
conditions. Using these two conclusions, the model for the current operating
conditions, in terms the coded variables is:
(2)
where 
represents the yield and 
and 
are the predictor variables for the
two factors, 
and 
, respectively. To further confirm
the adequacy of the model of Eqn. (2), the experiment
can be analyzed again after dropping the interaction term, 
(using the Select Effects icon in
the Control Panel). The results are shown in Figure 9.6.
The results show that the lack-of-fit for this model (because of the deficiency
created in the model by the absence of the interaction term) is not significant,
confirming that the model of Eqn. (2) is adequate.
[Note]
|
Figure 9.2: The 2 |
|
Figure 9.3: Design properties for the 2 |
|
Figure 9.4: The 2 |
|
Figure 9.5: Results for the 2 |
|
Figure 9.6: Results for the 2 |
The contour plot for the model of Eqn. (2) is
shown in Figure 9.7. The regression coefficients
for the model are 
and 
. To move towards the optimum, the
experimenter needs to move along the path of steepest ascent, which lies
perpendicular to the contours. This path is the line through the center
point of the current operating conditions (
, 
) with a slope of 
. Therefore, in terms of the coded
variables, the experiment should be moved 1.1625 units in the 
direction for every 0.4875 units in
the 
direction. To move along this path,
the experimenter decides to use a step-size of 10 minutes for the reaction
time, 
. The coded value for this step size
can be obtained as follows. Recall from Chapter
5 that the relationship between coded and actual values is:
|
Figure 9.7: Contour plot for the experiment in Figure 9.2, after the interaction term has been dropped from the model. |
(3)
(4)
Thus, for a step-size of 10 minutes, the equivalent step size in coded
value for 
is: 
In terms of the coded variables, the path of steepest ascent requires
a move of 1.1625 units in the 
direction for every 0.4875 units in
the 
direction. The step-size for 
, in terms of the coded value corresponding
to any step-size in 
, is:
Therefore, the step-size for the reaction temperature, 
, in terms of the coded variables is:
[Note
]
This corresponds to a step of approximately 12 
for temperature in terms of the actual
value as shown next:
Using a step of 12 
and 10 minutes, the experimenter conducts
experiments until no further increase is observed in the yield. The yield
values at each step are shown in Table 9.1. The yield starts decreasing
after the reaction temperature of 350 
and the reaction time of 165 minutes,
indicating that this point may lie close to the optimum region. To analyze
the vicinity of this point, a 2
design augmented by five center points
is selected. The range of exploration is chosen to be 345 to 355 
for reaction temperature and 155 to
175 minutes for reaction time. The experiment design is similar to the
design of Figure 9.4 and the response values recorded
are shown in Figure 9.8. The results for this design
are shown in Figure 9.9.
In the results in Figure 9.9, Curvature is displayed as a significant factor. This indicates that the first order model is not adequate for this region of the experiment and a higher order model is required. As a result, the methodology of steepest ascent can no longer be used. The presence of curvature indicates that the experiment region may be close to the optimum. Special designs that allow the use of second order models are needed at this point. [Note ]
|
Table 9.1: Response values at each step of the path of steepest ascent for the experiment to investigate the yield of a chemical process. Units for factor levels and the response have been omitted. |
|
Figure 9.8: The 2 |
|
Figure 9.9: Results for the 2 |
.
design augmented with five center points
to explore the region around current operating conditions for a chemical
process.
design in Example 9.1 to explore the current
operating conditions.
experiment design in Example 9.1 to explore
the current operating conditions.
experiment in Example 9.1 to explore the
current operating conditions.
experiment in Example 9.1 after the interaction
term is dropped from the model.
design augmented with five center points
to explore the region of maximum response obtained from the path of steepest
ascent in Table 9.1. Note that the center point of this design is the
new origin.
experiment in Example 9.1 to explore the
region of maximum response.