- Quantifying Robust Design Performance
- The Taguchi Approach to Robust Design
- Robust Design Example
- Alternative Approaches to Robust Design
- Dealing with Variation
Alternative Approaches to Robust Design
Any design that includes replicate runs at each combination of settings of the controllable inputs can be used as a robust design, even if the noise variables creating the variation among the replicate runs are unknown. The analysis of results is similar to that just described for the data in Table 31-5. The mean response and the variation in response, evaluated using the logarithm of the standard deviation of the response values, are calculated for each combination of settings of the controllable inputs. The type of design used for the controllable inputs will determine the degree of detail provided to describe the individual and joint influences of the controllable inputs on the average response and variation in response.
Borror, Montgomery, and Myers (2002)^{3} proposed the use of response surface designs as robust designs in which both the controllable inputs and the noise variables are treated as factors. This approach requires the noise variables to be controlled at specified settings, which may be possible in a special environment such as a laboratory. The response surface approach is now illustrated using a modified version of the candy wrapper example employed in the preceding section of this chapter.
Robust Design with Response Surface Techniques
This illustration involves three controllable inputs over the ranges shown in Table 31-6 and two noise variables over the ranges shown in Table 31-7.
Table 31-6. Controllable Inputs for Response Surface Design
Controllable Input |
Range of Interest |
||
% Additive |
5% |
to |
25% |
Temperature |
160 °F |
to |
200 °F |
Belt Speed |
50 ft/min |
to |
70 ft/min |
Table 31-7. Noise Variables for Response Surface Design
Noise Variable |
Range of Interest |
||
Relative Humidity |
50% |
to |
70% |
Ambient Particulate Level |
1% |
to |
5% |
The performance variable is, again, film thickness. The objective is to achieve a film thickness of 1.00 with minimum variation.
As described in Chapter 30, Minitab offers two types of response surface designs, Central Composite Designs and Box-Behnken Designs. For five factors (three controllable inputs and two noise variables), the 32-run Central Composite Design shown in Table 31-8 is the more economical choice.
Table 31-8. 32-Run Central Composite Design
% Additive |
Temperature |
Belt Speed |
Relative Humidity |
Particulate Level |
10 |
170 |
55 |
55.0 |
4 |
20 |
170 |
55 |
55.0 |
2 |
10 |
190 |
55 |
55.0 |
2 |
20 |
190 |
55 |
55.0 |
4 |
10 |
170 |
65 |
55.0 |
2 |
20 |
170 |
65 |
55.0 |
4 |
10 |
190 |
65 |
55.0 |
4 |
20 |
190 |
65 |
55.0 |
2 |
10 |
170 |
55 |
70.0 |
2 |
20 |
170 |
55 |
70.0 |
4 |
10 |
190 |
55 |
70.0 |
4 |
20 |
190 |
55 |
70.0 |
2 |
10 |
170 |
65 |
70.0 |
4 |
20 |
170 |
65 |
70.0 |
2 |
10 |
190 |
65 |
70.0 |
2 |
20 |
190 |
65 |
70.0 |
4 |
5 |
180 |
60 |
62.5 |
3 |
25 |
180 |
60 |
62.5 |
3 |
15 |
160 |
60 |
62.5 |
3 |
15 |
200 |
60 |
62.5 |
3 |
15 |
180 |
50 |
62.5 |
3 |
15 |
180 |
70 |
62.5 |
3 |
15 |
180 |
60 |
47.5 |
3 |
15 |
180 |
60 |
77.5 |
3 |
15 |
180 |
60 |
62.5 |
1 |
15 |
180 |
60 |
62.5 |
5 |
15 |
180 |
60 |
62.5 |
3 |
15 |
180 |
60 |
62.5 |
3 |
15 |
180 |
60 |
62.5 |
3 |
15 |
180 |
60 |
62.5 |
3 |
15 |
180 |
60 |
62.5 |
3 |
15 |
180 |
60 |
62.5 |
3 |
Additional economy can be realized by omitting the axial points (shown in italics in Table 31-8) for the two noise variables. The effect of these deletions is to give up information about the quadratic effects of the noise variables, which is not a serious loss. The resulting design is shown in Table 31-9 with the measured film thickness for each run. The order of execution of the runs is randomized.
Table 31-9. Response Surface Design with Experimental Results
%Additive |
Temp |
Belt Speed |
Rel.Humidity |
Particulate |
Thickness |
10 |
170 |
55 |
55.0 |
4 |
1.03 |
20 |
170 |
55 |
55.0 |
2 |
0.87 |
10 |
190 |
55 |
55.0 |
2 |
1.26 |
20 |
190 |
55 |
55.0 |
4 |
1.08 |
10 |
170 |
65 |
55.0 |
2 |
0.83 |
20 |
170 |
65 |
55.0 |
4 |
1.47 |
10 |
190 |
65 |
55.0 |
4 |
0.94 |
20 |
190 |
65 |
55.0 |
2 |
1.03 |
10 |
170 |
55 |
70.0 |
2 |
0.75 |
20 |
170 |
55 |
70.0 |
4 |
0.78 |
10 |
190 |
55 |
70.0 |
4 |
1.24 |
20 |
190 |
55 |
70.0 |
2 |
1.31 |
10 |
170 |
65 |
70.0 |
4 |
1.11 |
20 |
170 |
65 |
70.0 |
2 |
0.44 |
10 |
190 |
65 |
70.0 |
2 |
1.06 |
20 |
190 |
65 |
70.0 |
4 |
0.93 |
5 |
180 |
60 |
62.5 |
3 |
1.15 |
25 |
180 |
60 |
62.5 |
3 |
1.17 |
15 |
160 |
60 |
62.5 |
3 |
0.45 |
15 |
200 |
60 |
62.5 |
3 |
0.95 |
15 |
180 |
50 |
62.5 |
3 |
0.99 |
15 |
180 |
70 |
62.5 |
3 |
0.91 |
15 |
180 |
60 |
62.5 |
3 |
1.15 |
15 |
180 |
60 |
62.5 |
3 |
1.16 |
15 |
180 |
60 |
62.5 |
3 |
1.14 |
15 |
180 |
60 |
62.5 |
3 |
1.21 |
15 |
180 |
60 |
62.5 |
3 |
1.18 |
15 |
180 |
60 |
62.5 |
3 |
1.11 |
The settings of the five factors are coded as follows:
A |
= |
(% Additive - 15)/5 |
B |
= |
(Temperature - 180)/10 |
C |
= |
(Belt Speed - 60)/5 |
Z_{1} |
= |
(Relative Humidity - 62.5)/7.5 |
Z_{2} |
= |
(Ambient Particulate Level - 3)/1 |
and a full quadratic equation (without quadratic terms in the two noise variables) is fitted to the coded data. After deleting nonsignificant terms, the fitted equation for measured film thickness, expressed in the coded units of the five factors, is:
Predicted Thickness |
= |
1.165 - 0.011 A + 0.107 B - 0.028 C - 0.113 B^{2} - 0.051 C^{2} - 0.084 BC - 0.056 Z_{1} + 0.064 Z_{2} - 0.068 AZ_{1} + 0.084 B Z_{1} - 0.123 B Z_{2} - 0.036 C Z_{1} + 0.072 C Z_{2} |
Minitab's output for this fitted model is shown in Figure 31-6.
Figure 31-6. Analysis of Data in Table 31-9
Response Surface Regression: thickness versus A, B, C, Z1, Z2 |
||||
The analysis was done using coded units. |
||||
Estimated Regression Coefficients for thickness |
||||
Term |
Coef |
SE Coef |
T |
P |
Constant |
1.16525 |
0.011794 |
98.796 |
0.000 |
A |
-0.01125 |
0.007613 |
-1.478 |
0.162 |
B |
0.10708 |
0.007613 |
14.065 |
0.000 |
C |
-0.02792 |
0.007613 |
-3.667 |
0.003 |
Z1 |
-0.05562 |
0.009324 |
-5.966 |
0.000 |
Z2 |
0.06437 |
0.009324 |
6.904 |
0.000 |
B*B |
-0.11306 |
0.007223 |
-15.654 |
0.000 |
C*C |
-0.05056 |
0.007223 |
-7.001 |
0.000 |
A*Z1 |
-0.06813 |
0.009324 |
-7.306 |
0.000 |
B*C |
-0.08438 |
0.009324 |
9.049 |
0.000 |
B*Z1 |
0.08437 |
0.009324 |
9.049 |
0.000 |
B*Z2 |
-0.12313 |
0.009324 |
-13.205 |
0.000 |
C*Z1 |
-0.03562 |
0.009324 |
-3.821 |
0.002 |
C*Z2 |
0.07187 |
0.009324 |
7.708 |
0.000 |
S = 0.03730 |
R-Sq = 98.7% |
R-Sq(adj) = 97.4% |
The corresponding fitted equation expressed in the original units of the five factors is:
Predicted Thickness |
= |
1.165 - 0.005 (% Add) 1 0.054 (Temp) - 0.014(Speed)- 0.028 (Temp)^{2} - 0.013 (Speed)^{2}- 0.021 (Temp)(Speed) - 0.056 (Humidity) + 0.064 (Particulate) - 0.034 (% Add)(Humidity) + 0.042 (Temp)(Humidity) - 0.062 (Temp)(Particulate) - 0.018 (Speed)(Humidity) + 0.036 (Speed)(Particulate) |
To use the fitted equation in the coded factors, A, B, C, Z1 and Z2 as a predictor, the controllable inputs are regarded as fixed factors, whereas the noise variables are regarded as random variables, each with mean 0 and variances σ^{2}_{Z1} and σ^{2}_{Z2}, respectively. Estimates of the variances σ^{2}_{Z1} and σ^{2}_{Z2} must be available from previous experience. In fact, in this example the coded values of -1 and 1 for each coded noise variable are assumed to be about two standard deviations away from the average value of that noise variable.
Under these assumptions, the predictive equation for the average thickness consists of terms from the fitted equation in the coded factors that involve only the coded controllable inputs A, B, and C. In coded units this is:
Predicted Thickness |
= |
1.165 - 0.011 A + 0.107 B - 0.028 C - 0.113 B^{2} - 0.051 C^{2} - 0.084 BC |
An equation describing how the variance of the film thickness depends on the settings of the controllable inputs can also be obtained from the fitted equation in the coded factors by evaluating the variance of each term and adding an estimate of the random error variance from the fitted equation, in this case s^{2} = (0.0373)^{2} = 0.00139.
Estimated Variance (predicted thickness)
= |
(- 0.056 - 0.068 A 1 0.084 B 2 0.036 C)^{2}σ^{2}_{Z1} + (0.064 - 0.123 B + 0.072 C)^{2} σ^{2}_{Z2} + 0.00139 |
If the coded values of -1 and 1 for each noise variable are assumed to span a range of approximately four standard deviations for that coded noise variable, then in this example σ_{Z1} ≈ 0.5 and σ_{Z2} ≈ 0.5. Using these estimates,
Estimated Variance of Predicted Thickness
= |
(- 0.056 - 0.068 A + 0.084 B - 0.036 C)^{2} (0.5)^{2} + (0.064 - 0.123 B + 0.072 C)^{2} (0.5)^{2} + 0.00139 |
The optimal solution is found by minimizing the estimated variance of the predicted thickness subject to the constraint that the predicted mean thickness = 1.00, the target value. A grid search over values of A, B, and C indicates an approximately optimal solution at the following settings:
A |
= |
-1.0 (% Additive = 10 %) |
B |
= |
-0.6 (Temperature = 174 °F) |
C |
= |
-1.0 (Belt Speed = 55 ft/min) |