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This chapter is from the book

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

Z1

=

(Relative Humidity - 62.5)/7.5

Z2

=

(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 B2 - 0.051 C2 - 0.084 BC - 0.056 Z1 + 0.064 Z2 - 0.068 AZ1 + 0.084 B Z1 - 0.123 B Z2 - 0.036 C Z1 + 0.072 C Z2

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 σ2Z1 and σ2Z2, respectively. Estimates of the variances σ2Z1 and σ2Z2 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 B2 - 0.051 C2 - 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 s2 = (0.0373)2 = 0.00139.

Estimated Variance (predicted thickness)

=

(- 0.056 - 0.068 A 1 0.084 B 2 0.036 C)2σ2Z1

+ (0.064 - 0.123 B + 0.072 C)2 σ2Z2 + 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 σZ1 ≈ 0.5 and σZ2 ≈ 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)

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