In many cases, the experimenter has to optimize a number of responses at the same time. For the example in Chapter 9, Method of Steepest Ascent, assume that the experimenter has to also consider two other responses - cost of the product (which should be minimized) and the pH of the product (which should be close to 7 so that the product is neither acidic nor basic). The data is presented in Figure 9.23. The problem in dealing with multiple responses is that now there might be conflicting objectives because of the different requirements of each of the responses. The experimenter needs to come up with a solution that satisfies each of the requirements as much as possible without compromising too much on any of the requirements. The approach used in DOE++ to deal with optimization of multiple responses involves the use of desirability functions that are discussed next (for details see [4]).
|
Figure 9.23: Data for the additional responses of cost and pH for the example to investigate the yield of a chemical process. |
Under this approach, each 
th response is assigned a desirability
function, 
, where the value of 
varies between 0 and 1. The function,
is defined differently based on the
objective of the response. If the response is to be maximized, as in the
case of the previous example where the yield had to be maximized, then
is defined as follows:
(14)where 
represents the target value of the
th response, 
, 
represents the acceptable lower limit
value for this response and 
represents the weight. When 
the function 
is linear. If 
then more importance is placed on
achieving the target for the response, 
. When 
, less weight is assigned to achieving
the target for the response, 
. A graphical representation is shown
in Figure 9.24 (a).
If the response is to be minimized, as in the case when the response
is cost, 
is defined as follows:
(15)
Here 
represents the acceptable upper limit
for the response (see Figure 9.24 (b)).
|
Figure 9.24: Desirability function plots for different response optimizations - (a) the goal is to maximize the response, (b) the goal is to minimize the response and (c) the goal is to get the response to a target value. |
There may be times when the experimenter wants the response to be neither
maximized nor minimized, but instead stay as close to a specified target
as possible. For example, in the case where the experimenter wants the
product to be neither acidic nor basic, there is a requirement to keep
the pH close to the neutral value of 7. In such cases, the desirability
function is defined as follows (see Figure 9.24 (c)):
(16)
Once a desirability function is defined for each of the responses, assuming
that there are 
responses, an overall desirability
function is obtained as follows:
(17)
where the 
s represent the importance of each
response. The greater the value of 
, the more important the response with
respect to the other responses. The objective is to now find the settings
that return the maximum value of 
.
To illustrate the use of desirability functions, consider the previous
example with the three responses of yield, cost and pH as shown in Figure
9.23. The response surfaces for the two additional
responses of cost and pH are shown in Figures 9.25 (a)
and (b), respectively. In terms of actual variables, the models obtained
for all three responses are as shown next:
|
Figure 9.25: Response surfaces for (a) cost and (b) pH. |
Assume that the experimenter wants to have a target yield value of 95,
although any value of yield greater than 94 is acceptable. Then the desirability
function for yield is:
(18)
For the cost, assume that the experimenter wants to lower the cost to
400, although any cost value below 415 is acceptable. Then the desirability
function for cost is:
(19)
For the pH, a target of 7 is desired but values between 6.9 and 7.1
are also acceptable. Thus, the desirability function here is:
(20)
Notice that in the previous equations all weights used (
s) are 1. Thus, all three desirability
functions are linear. The overall desirability function, assuming equal
importance (
) for all the responses, is:
(21)
The objective of the experimenter is to find the settings of 
and 
such that the overall desirability,
, is maximum. In DOE++, the settings
for the desirability functions for each of the three responses can be
entered as shown in Figure 9.26 using the Optimization
icon in the Control Panel. Based on these settings, DOE++ solves this
optimization problem to obtain the following solution (see Figure 9.27):
|
Figure 9.26: Optimization settings for the three responses of yield, cost and pH. |
|
Figure 9.27: Optimum solution from DOE++ for the three responses of yield, cost and pH. |
Based on the response values, the individual desirability functions
are:
Then the overall desirability is:
This is the same as the Global Desirability displayed by DOE++ in Figure 9.27. At times, a number of solutions may be obtained from DOE++ and it is up to the experimenter to choose the most feasible one.
See Also:
Analysis of the Second Order Model
Taguchi's Robust Parameter Design Method
