5.
Process Improvement
5.5.
Advanced topics
5.5.3.
How do you optimize a process?
5.5.3.2.
Multiple response case
5.5.3.2.2.

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.
For each response Y_{i}(x),
a desirability function
d_{i}(Y_{i}) assigns numbers
between 0 and 1 to the possible values of Y_{i},
with
d_{i}(Y_{i}) = 0 representing
a completely undesirable value of
Y_{i} and
d_{i}(Y_{i}) = 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:
\( D = \left( d_{1}(Y_{1}) d_{2}(Y_{2}) \cdots
d_{k}(Y_{k})\right) ^{1/k} \)
with k denoting the number of responses. Notice that if
any response Y_{i} is completely undesirable
(d_{i}(Y_{i}) = 0),
then the overall desirability is zero. In practice, fitted response
values \( \hat{Y}_{i} \)
are used in place of the Y_{i}.

Desirability functions of Derringer and Suich

Depending on whether a particular response
Y_{i} is to be maximized, minimized, or assigned
a target value, different desirability functions
d_{i}(Y_{i}) can be used. A
useful class of desirability functions was proposed by
Derringer and Suich (1980).
Let L_{i}, U_{i} and
T_{i} be the lower, upper, and target values,
respectively, that are desired for response
Y_{i}, with L_{i} ≤
T_{i} ≤ U_{i}.

Desirability function for "target is best"

If a response is of the "target is best"
kind, then its individual desirability function is
\(
d_{i}(\hat{Y}_{i}) =
\left\{ \begin{array}{ll}
0 & \mbox{if } \hspace{.1in} \hat{Y}_{i}(x) < L_{i} \\
\left( \frac{\hat{Y}_{i}(x)  L_{i}} {T_{i}  L_{i}}
\right) ^{s} &
\mbox{if } \hspace{.1in}
L_{i} \le \hat{Y}_{i}(x) \le T_{i} \\
\left( \frac{\hat{Y}_{i}(x)  U_{i}} {T_{i}  U_{i}}
\right) ^{t} &
\mbox{if } \hspace{.1in}
T_{i} \le \hat{Y}_{i}(x) \le U_{i} \\
0 & \mbox{if } \hspace{.1in} \hat{Y}_{i}(x) > U_{i}
\end{array}
\right.
\)
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
T_{i}; 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).

Desirability function for maximizing a response

If a response is to be maximized instead, the individual desirability
is defined as
\(
d_{i}(\hat{Y}_{i}) =
\left\{ \begin{array}{ll}
0 & \mbox{if } \hspace{.1in} \hat{Y}_{i}(x) < L_{i} \\
\left( \frac{\hat{Y}_{i}(x)  L_{i}} {T_{i}  L_{i}}
\right) ^{s} &
\mbox{if } \hspace{.1in}
L_{i} \le \hat{Y}_{i}(x) \le T_{i} \\
1.0 & \mbox{if } \hspace{.1in} \hat{Y}_{i}(x) > T_{i}
\end{array}
\right.
\)
with T_{i} in this case interpreted as a large
enough value for the response.

Desirability function for minimizing a response

Finally, if we want to minimize a response, we could use
\(
d_{i}(\hat{Y}_{i}) =
\left\{ \begin{array}{ll}
1.0 & \mbox{if } \hspace{.1in} \hat{Y}_{i}(x) < T_{i} \\
\left( \frac{\hat{Y}_{i}(x)  U_{i}} {T_{i}  U_{i}}
\right) ^{s} &
\mbox{if } \hspace{.1in}
T_{i} \le \hat{Y}_{i}(x) \le U_{i} \\
0 & \mbox{if } \hspace{.1in} \hat{Y}_{i}(x) > U_{i}
\end{array}
\right.
\)
with T_{i} denoting a small enough value for
the response.

Desirability approach steps

The desirability approach consists of the following steps:
 Conduct experiments and fit response models for all k
responses;
 Define individual desirability functions for each response;
 Maximize the overall desirability D with
respect to the controllable factors.


Example:

An example using the desirability approach

Derringer and Suich (1980)
present the following multiple response experiment arising in the
development of a tire tread compound. The controllable factors are:
x_{1}, hydrated silica level,
x_{2}, silane coupling agent level, and
x_{3}, sulfur level. The four responses to be
optimized and their desired ranges are:

Factor and response variables

Source

Desired range


PICO Abrasion index, Y_{1}

120 < Y_{1}

200% modulus, Y_{2}

1000 < Y_{2}

Elongation at break, Y_{3}

400 < Y_{3} < 600

Hardness, Y_{4}

60 < Y_{4} < 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
T_{3} = 500 and
T_{4} = 67.5. The values
s=t=1 were chosen in both cases.

Experimental runs from a central composite design

The following experiments were conducted using a central
composite design.
Run
Number

x_{1}

x_{2}

x_{3}

Y_{1}

Y_{2}

Y_{3}

Y_{4}


1

1.00

1.00

1.00

102

900

470

67.5

2

+1.00

1.00

1.00

120

860

410

65.0

3

1.00

+1.00

1.00

117

800

570

77.5

4

+1.00

+1.00

1.00

198

2294

240

74.5

5

1.00

1.00

+1.00

103

490

640

62.5

6

+1.00

1.00

+1.00

132

1289

270

67.0

7

1.00

+1.00

+1.00

132

1270

410

78.0

8

+1.00

+1.00

+1.00

139

1090

380

70.0

9

1.63

0.00

0.00

102

770

590

76.0

10

+1.63

0.00

0.00

154

1690

260

70.0

11

0.00

1.63

0.00

96

700

520

63.0

12

0.00

+1.63

0.00

163

1540

380

75.0

13

0.00

0.00

1.63

116

2184

520

65.0

14

0.00

0.00

+1.63

153

1784

290

71.0

15

0.00

0.00

0.00

133

1300

380

70.0

16

0.00

0.00

0.00

133

1300

380

68.5

17

0.00

0.00

0.00

140

1145

430

68.0

18

0.00

0.00

0.00

142

1090

430

68.0

19

0.00

0.00

0.00

145

1260

390

69.0

20

0.00

0.00

0.00

142

1344

390

70.0


Fitted response

Using ordinary least squares and standard diagnostics, the fitted
responses are:
\( \begin{array}{lcl}
\hat{Y}_{1} & = & 139.12 + 16.49 x_{1} + 17.88 x_{2} + 2.21 x_{3} \\
& & 4.01 x_{1}^{2}  3.45 x_{2}^{2} 
1.57 x_{3}^{2} \\
& & + 5.12 x_{1} x_{2}  7.88 x_{1} x_{3} 
7.13 x_{2} x_{3}
\end{array}
\)
(R^{2} = 0.8369 and adjusted R^{2} = 0.6903);
\( \begin{array}{lcl}
\hat{Y}_{2} & = & 1261.13 + 268.15 x_{1} + 246.5 x_{2} 
102.6 x_{3} \\
& &  83.57 x_{1}^{2}  124.92 x_{2}^{2} +
199.2 x_{3}^{2} \\
& & + 69.37 x_{1} x_{2}  104.38 x_{1} x_{3} 
94.13 x_{2} x_{3}
\end{array}
\)
(R^{2} = 0.7137 and adjusted R^{2} = 0.4562);
\( \hat{Y}_{3} = 417.5  99.67 x_{1}  31.4 x_{2}  27.42 x_{3} \)
(R^{2} = 0.682 and adjusted R^{2} = 0.6224);
\( \begin{array}{lcl}
\hat{Y}_{4} & = & 68.91  1.41 x_{1} + 4.32 x_{2} + 0.21 x_{3} \\
& & + 1.56 x_{1}^{2} + 0.058 x_{2}^{2} 
0.32 x_{3}^{2} \\
& &  1.62 x_{1} x_{2} + 0.25 x_{1} x_{3} 
0.12 x_{2} x_{3}
\end{array}
\)
(R^{2} = 0.8667 and adjusted R^{2} = 0.7466).
Note that no interactions were significant for response 3 and that
the fit for response 2 is quite poor.

Best Solution

The best solution is (x^{*})' =
(0.10, 0.15, 1.0) and results in:
\( d_{1}(\hat{Y}_{1}) = 0.34 \)

\( (\hat{Y}_{1}(x^{*}) = 136.4) \)

\( d_{2}(\hat{Y}_{2}) = 1.0 \)

\( (\hat{Y}_{2}(x^{*}) = 1571.05) \)

\( d_{3}(\hat{Y}_{3}) = 0.49 \)

\( (\hat{Y}_{3}(x^{*}) = 450.56) \)

\( d_{4}(\hat{Y}_{4}) = 0.76 \)

\( (\hat{Y}_{4}(x^{*}) = 69.26) \)

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
(x_{2}, x_{3}) plane when
x_{1} 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., \( \hat{Y}_{2} \)).
FIGURE 5.8: Overall Desirability Function for Example
Problem
