Multiple responses: The desirability approach
The
desirability
approach
is a
popular
method
that
assigns a
"score" to
a set of
responses
and
chooses
factor
settings
that
maximize
that score
The desirability function approach is one of the most widely used methods
in industry for the optimization of multiple response processes. It is based
on the idea that the "quality" of a product or process that has multiple
quality characteristics, with one of them outside of some "desired" limits,
is completely unacceptable. The method finds operating conditions x that
provide the "most desirable" response values.
Desirabilit
y functions
of
Derringer
and Suich
Depending on whether a particular response Yi is to be maximized,
minimized, or assigned a target value, different desirability functions di(Yi)
can be used. A useful class of desirability functions was proposed by
Derringer and Suich (1980). Let Li, Ui and Ti be the lower, upper, and
target values, respectively, that are desired for response Yi, with Li Ti
Ui.
Desirabilit
y function
for "target
is best"
If a response is of the "target is best" kind, then its individual desirability
function is
For each response Yi(x), a desirability function di(Yi) assigns numbers
between 0 and 1 to the possible values of Yi, with di(Yi) = 0 representing a
completely undesirable value of Yi and di(Yi) = 1 representing a
completely desirable or ideal response value. The individual desirabilities
are then combined using the geometric mean, which gives the overall
desirability D:
with k denoting the number of responses. Notice that if any response Yi is
completely undesirable (di(Yi) = 0), then the overall desirability is zero. In
practice, fitted response values i are used in place of the Yi.
with the exponents s and t determining how important it is to hit the target
value. For s = t = 1, the desirability function increases linearly towards Ti;
for s < 1, t < 1, the function is convex, and for s > 1, t > 1, the function is
concave (see the example below for an illustration).
Desirabilit
If a response is to be maximized instead, the individual desirability is
y function
for
maximizing
a response
defined as
with Ti in this case interpreted as a large enough value for the response.
Desirabilit
y function
for
minimizing
a response
Finally, if we want to minimize a response, we could use
with Ti denoting a small enough value for the response.
Desirabilit
y approach
steps
The desirability approach consists of the following steps:
1. Conduct experiments and fit response models for all k responses;
2. Define individual desirability functions for each response;
3. Maximize the overall desirability D with respect to the controllable
factors.
Example:
An example
using the
desirability
approach
Factor and
response
variables
Derringer and Suich (1980) present the following multiple response
experiment arising in the development of a tire tread compound. The
controllable factors are: x1, hydrated silica level, x2, silane coupling agent
level, and x3, sulfur level. The four responses to be optimized and their
desired ranges are:
Source
Desired range
PICO Abrasion index, Y1
120 < Y1
200% modulus, Y2
1000 < Y2
Elongation at break, Y3
400 < Y3 < 600
Hardness, Y4
60 < Y4 < 75
The first two responses are to be maximized, and the value s=1 was
chosen for their desirability functions. The last two responses are "target is
best" with T3 = 500 and T4 = 67.5. The values s=t=1 were chosen in both
cases.
Experiment
al runs
from a
central
composite
design
The following experiments were conducted using a central composite
design.
Run
Number
x1
x2
x3 Y1
Y2 Y3
Y4
Fitted
response
Using ordinary least squares and standard diagnostics, the fitted responses
are:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
-1.00
+1.00
-1.00
+1.00
-1.00
+1.00
-1.00
+1.00
-1.63
+1.63
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
-1.00
-1.00
+1.00
+1.00
-1.00
-1.00
+1.00
+1.00
0.00
0.00
-1.63
+1.63
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
-1.00
-1.00
-1.00
-1.00
+1.00
+1.00
+1.00
+1.00
0.00
0.00
0.00
0.00
-1.63
+1.63
0.00
0.00
0.00
0.00
0.00
0.00
102
120
117
198
103
132
132
139
102
154
96
163
116
153
133
133
140
142
145
142
(R2 = 0.8369 and adjusted R2 = 0.6903);
(R2 = 0.7137 and adjusted R2 = 0.4562);
(R2 = 0.682 and adjusted R2 = 0.6224);
900
860
800
2294
490
1289
1270
1090
770
1690
700
1540
2184
1784
1300
1300
1145
1090
1260
1344
470
410
570
240
640
270
410
380
590
260
520
380
520
290
380
380
430
430
390
390
67.5
65.0
77.5
74.5
62.5
67.0
78.0
70.0
76.0
70.0
63.0
75.0
65.0
71.0
70.0
68.5
68.0
68.0
69.0
70.0
(R2 = 0.8667 and adjusted R2 = 0.7466).
Note that no interactions were significant for response 3 and that the fit for
response 2 is quite poor.
Optimizati
on
performed
by DesignExpert
software
Optimization of D with respect to x was carried out using the DesignExpert software. Figure 5.7 shows the individual desirability functions
di( i) for each of the four responses. The functions are linear since the
values of s and t were set equal to one. A dot indicates the best solution
found by the Design-Expert solver.
Diagram of
desirability
functions
and
optimal
solutions
FIGURE 5.7 Desirability Functions and Optimal Solution for
Example Problem
Best
Solution
The best solution is (x*)' = (-0.10, 0.15, -1.0) and results in:
d1( 1) = 0.34 ( 1(x*) = 136.4)
*)
d2(
2)
= 1.0 (
2(x
d3(
3)
= 0.49 (
3(x
d4(
4)
= 0.76 (
4(x
= 157.1)
*)
= 450.56)
*)
= 69.26)
The overall desirability for this solution is 0.596. All responses are
predicted to be within the desired limits.
3D plot of
the overall
desirability
function
Figure 5.8 shows a 3D plot of the overall desirability function D(x) for the
(x2, x3) plane when x1 is fixed at -0.10. The function D(x) is quite "flat" in
the vicinity of the optimal solution, indicating that small variations around
x* are predicted to not change the overall desirability drastically. However,
the importance of performing confirmatory runs at the estimated optimal
operating conditions should be emphasized. This is particularly true in this
example given the poor fit of the response models (e.g., 2).