Applied Statistics and Probability for Engineers

Table S14-9 summarizes the analysis of variance for this model. The response surface is

shown graphically in Fig. S14-12. Notice that viscosity increases as both time and tempera-

ture increase.

As in most response surface problems, the experimenter in this example had conflicting

objectives regarding the two responses. The objective was to maximize yield, but the accept-

able range for viscosity was 38y 2 42. When there are only a few independent variables,

an easy way to solve this problem is to overlay the response surfaces to find the optimum.

Figure S14-13 shows the overlay plot of both responses, with the contours y 1 69% conver-

sion, y 2 38, and y 2 42 highlighted. The shaded areas on this plot identify unfeasible

combinations of time and temperature. This graph shows that several combinations of time

and temperature will be satisfactory.

Example S14-4 illustrates the use of a central composite design(CCD) for fitting a

second-order response surface model. These designs are widely used in practice because they

are relatively efficient with respect to the number of runs required. In general, a CCD in k

14-17

Figure S14-10

Central composite

design for Example

S14-4.

(0, 0)

+2

• 2 +2
  
  
  • 2
  
  
  
  

(0, –1.414)

(–1.414, 0) (1.414, 0)

(0, 1.414)

(–1, –1)

(–1, 1)

x 2

x 1

(1, –1)

(1, 1)

Table S14-8 Analysis of Variance for the Quadratic Model, Yield Response

Source of Sum of Degrees of Mean

Variation Squares Freedom Square f 0 P-Value

Model 45.89 5 9.178 14.93 0.0013

Residual 4.30 7 0.615

Total 50.19 12

Independent Coefficient Standard tfor H 0

Variable Estimate Error Coefficient  0 P-Value

Intercept 69.100 0.351 197.1

x 1 1.633 0.277 5.891 0.0006

x 2 1.083 0.277 3.907 0.0058

x^21 0.969 0.297 3.259 0.0139

x^22 1.219 0.297 4.100 0.0046

x 1 x 2 0.225 0.392 0.5740 0.5839

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