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

## 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 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:

$$D = \left( d_{1}(Y_{1}) d_{2}(Y_{2}) \cdots d_{k}(Y_{k})\right) ^{1/k}$$
with k denoting the number of responses. Notice that if any response Yi is completely undesirable (di(Yi) = 0), then the overall desirability is zero. In practice, fitted response values $$\hat{Y}_{i}$$ are used in place of the Yi.
Desirability 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 LiTiUi.
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 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).
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 Ti 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 Ti denoting a small enough value for the response.
Desirability 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 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:
Factor and response variables
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.
Experimental 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

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:
$$\begin{array}{lcl} \hat{Y}_{1} & = & 139.12 + 16.49 x_{1} + 17.88 x_{2} + 2.21 x_{3} \\ & & -4.01 x_{1}^{2} - 3.45 x_{2}^{2} - 1.57 x_{3}^{2} \\ & & + 5.12 x_{1} x_{2} - 7.88 x_{1} x_{3} - 7.13 x_{2} x_{3} \end{array}$$
(R2 = 0.8369 and adjusted R2 = 0.6903);
$$\begin{array}{lcl} \hat{Y}_{2} & = & 1261.13 + 268.15 x_{1} + 246.5 x_{2} - 102.6 x_{3} \\ & & - 83.57 x_{1}^{2} - 124.92 x_{2}^{2} + 199.2 x_{3}^{2} \\ & & + 69.37 x_{1} x_{2} - 104.38 x_{1} x_{3} - 94.13 x_{2} x_{3} \end{array}$$
(R2 = 0.7137 and adjusted R2 = 0.4562);
$$\hat{Y}_{3} = 417.5 - 99.67 x_{1} - 31.4 x_{2} - 27.42 x_{3}$$
(R2 = 0.682 and adjusted R2 = 0.6224);

$$\begin{array}{lcl} \hat{Y}_{4} & = & 68.91 - 1.41 x_{1} + 4.32 x_{2} + 0.21 x_{3} \\ & & + 1.56 x_{1}^{2} + 0.058 x_{2}^{2} - 0.32 x_{3}^{2} \\ & & - 1.62 x_{1} x_{2} + 0.25 x_{1} x_{3} - 0.12 x_{2} x_{3} \end{array}$$

(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.

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 (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., $$\hat{Y}_{2}$$).
FIGURE 5.8: Overall Desirability Function for Example Problem