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:

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 _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

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

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

with T_i 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:

(R² = 0.8369 and adjusted R² = 0.6903);

(R² = 0.7137 and adjusted R² = 0.4562);

(R² = 0.682 and adjusted R² = 0.6224);

(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. 

Optimization performed by Design-Expert software

Optimization of D with respect to x was carried out using the Design-Expert software. Figure 5.7 shows the individual desirability functions d_i(_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^*) = 1571.05)

d₃(₃) = 0.49    (₃(x^*) = 450.56)

d₄(₄) = 0.76    (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., ₂).

FIGURE 5.8  Overall Desirability Function for Example Problem