1. Chapter 11 Response Surface Methods and Other Approaches to Process Optimization

2. 11.1 Introduction to Response Surface Methodology
Response Surface Methodology (RSM) is useful for the modeling and analysis of programs in which a response of interest is influenced by several variables and the objective is to optimize this response.
For example: Find the levels of temperature (x1) and pressure (x2) to maximize the yield (y) of a process.

3. Response surface: (see Figure 11.1 & 11.2)
The function f is unknown
Approximate the true relationship between y and the independent variables by the lower-order polynomial model.
Response surface design

4. A sequential procedure
The objective is to lead the experimenter rapidly and efficiently along a path of improvement toward the general vicinity of the optimum.
First-order model => Second-order model
Climb a hill

5. 11.2 The Method of Steepest Ascent
Assume that the first-order model is an adequate approximation to the true surface in a small ragion of the x’s.
The method of steepest ascent: A procedure for moving sequentially along the path of steepest ascent.

6. Based on the first-order model,
The path of steepest ascent // the regression coefficients
The actual step size is determined by the experimenter based on process knowledge or other practical considerations

7. Example 11.1
Two factors, reaction time & reaction temperature
Use a full factorial design and center points (see Table 11.1):
Obtain an estimate of error
Check for interactions in the model
Check for quadratic effect
ANOVA table (see Table 11.2)
Table 11.3 & Figure 11.5
Table 11.4 & 11.5

9. The step size is 5 minutes of reaction time and 2 degrees F
What happens at the conclusion of steepest ascent?

10. Assume the first-order model
Choose a step size in one process variable, xj.
The step size in the other variable,
Convert the xj from coded variables to the natural variable

11. 11.3 Analysis of a Second-order Response Surface
When the experimenter is relative closed to the optimum, the second-order model is used to approximate the response.
Find the stationary point. Maximum response, Minimum response or saddle point.
Determine whether the stationary point is a point of maximum or minimum response or a saddle point.

12. The second-order model:

13. Characterizing the response surface:
Contour plot or Canonical analysis
Canonical form (see Figure 11.9)
Minimum response: i are all positive
Maximum response: i are all negative
Saddle point: i have different signs

14. Central composite design (CCD) (Table 11.6 & Figure 11.10) Table 11.7
Example 11.2
Continue Example 11.1
Central composite design (CCD) (Table 11.6 & Figure 11.10)
Table 11.7
ANOVA for Response Surface Quadratic Model Analysis of variance table [Partial sum of squares]
Sum of Mean F Source Squares DF Square Value Prob > F Model < A < B A < B < AB Residual Lack of Fit Pure Error Cor Total

15. The contour plot is given in the natural variables (see Figure 11.11)
The optimum is at about 87 minutes and degrees

16. The relationship between x and w:
M is an orthogonal matrix and the columns of M are the normalized eigenvectors of B.
Multiple response:
Typically, we want to simultaneously optimize all responses, or find a set of conditions where certain product properties are achieved
Overlay the contour plots (Figure 11.16)
Constrained optimization problem

17. 11.4 Experimental Designs for Fitting Response Surfaces
Designs for fitting the first-order model
The orthogonal first-order designs
X’X is a diagonal matrix
2k factorial and fractions of the 2k series in which main effects are not aliased with each others
Besides factorial designs, include several observations at the center.
Simplex design

18. Designs for fitting the second-order model
Central composite design (CCD)
nF runs on 2k axial or star points, and nC center runs
Sequential experimentation
Two parameters: nC and 
The variance of the predicted response at x:
Rotatable design: The variance of predicted response is constant on spheres
The purpose of RSM is optimization and the location of the optimum is unknown prior to running the experiment.

19.  = (nF)1/4 yields a rotatable central composite design
The spherical CCD: Set  = (k)1/2
Center runs in the CCD, nC: 3 to 5 center runs
The Box-Behnken design: three-level designs (see Table 11.8)
Cuboidal region:
face-centered central composite design (or face-centered cube)
 = 1
nC=2 or 3