Applied Statistics and Probability for Engineers

where represents the noise or error observed in the response Y. If we denote the expected re-

sponse by E(Y)f(x 1 , x 2 ), then the surface represented by

is called a response surface.

We may represent the response surface graphically as shown in Fig. S14-5, where is

plotted versus the levels of x 1 and x 2. Notice that the response is represented as a surface plot

in a three-dimensional space. To help visualize the shape of a response surface, we often plot

the contours of the response surface as shown in Fig. S14-6. In the contour plot, lines of con-

stant response are drawn in the x 1 , x 2 plane. Each contour corresponds to a particular height of

the response surface. The contour plot is helpful in studying the levels of x 1 and x 2 that result

in changes in the shape or height of the response surface.

In most RSM problems, the form of the relationship between the response and the inde-

pendent variables is unknown. Thus, the first step in RSM is to find a suitable approximation

for the true relationship between Yand the independent variables. Usually, a low-order poly-

nomial in some region of the independent variables is employed. If the response is well

modeled by a linear function of the independent variables, the approximating function is the

first-order model

(S14-13)

If there is curvature in the system, then a polynomial of higher degree must be used, such as

the second-order model

(S14-14)

Many RSM problems use one or both of these approximating polynomials. Of course, it is

unlikely that a polynomial model will be a reasonable approximation of the true functional re-

lationship over the entire space of the independent variables, but for a relatively small region

they usually work quite well.

The method of least squares, discussed in Chapters 11 and 12, is used to estimate the

parameters in the approximating polynomials. The response surface analysis is then done in

terms of the fitted surface. If the fitted surface is an adequate approximation of the true

Y 0 a

k

i 1

ixia

k

i 1

iix

2

ib

ij

ijxixj

Y 0  1 x 1  2 x 2 pkxk

f 1 x 1 , x 22

14-12

Figure S14-5 A three-dimensional response surface showing the

expected yield as a function of temperature and feed concentration.

1.41.8

2.22.6

3.0

(^140160180)
(^100120)
34
44
54
64
74
84
Temperature, °C
Feed concentration, %
Yield
1.0 1.0 100 120 140 160 180
1.4
1.8
2.2
2.6
3.0 55
(^6065)
(^7075)
80
Current^85
operating
conditions Region
of the
optimum
Temperature, °C
Feed concentration, %
Figure S14-6 A contour plot of the yield re-
sponse surface in Figure S14-5.
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