Introduction to Response Surface Methodology

15.11 Introduction to Response Surface Methodology

In Case Study 15.2, a regression model was fitted to a set of data with the specific goal of finding conditions on those design variables that optimize (maximize) the cleansing efficiency of coal. The model contained three linear main effects, three two-factor interaction terms, and one three-factor interaction term. The model re-

sponse was the cleansing efficiency, and the optimum conditions on x 1 ,x 2 , and x 3

640 Chapter 15 2 k Factorial Experiments and Fractions were found by using the signs and the magnitude of the model coefficients. In this

example, a two-level design was employed for process improvement or process op- timization. In many areas of science and engineering, the application is expanded to involve more complicated models and designs, and this collection of techniques is called response surface methodology (RSM). It encompasses both graph- ical and analytical approaches. The term response surface is derived from the appearance of the multidimensional surface of constant estimated response from a second-order model, i.e., a model with first- and second-order terms. An example will follow.

The Second-Order Response Surface Model In many industrial examples of process optimization, a second-order response sur-

face model is used. For the case of, say, k = 2 process variables, or design variables, and a single response y, the model is given by

y=β 0 +β 1 x 1 +β 2 x 2 +β 11 x 2 1 +β 22 x 2 2 +β 12 x 1 x 2 + ǫ. Here we have k = 2 first-order terms, two pure second-order, or quadratic, terms,

and one interaction term given by β 12 x 1 x 2 . The terms x 1 and x 2 are taken to be in the familiar ±1 coded form. The ǫ term designates the usual model error. In

general, for k design variables the model will contain 1 + k + k + k 2 model terms, and hence the experimental design must contain at least a like number of design points. In addition, the quadratic terms require that the design variables be fixed in the design with at least three levels. The resulting design is referred to as a second-order design. Illustrations will follow.

The following central composite design (CCD) and example is taken from Myers, Montgomery, and Anderson-Cook (2009). Perhaps the most popular class of second-order designs is the class of central composite designs. The example given

in Table 15.17 involves a chemical process in which reaction temperature, ξ 1 , and reactant concentration, ξ 2 , are shown at their natural levels. They also appear in coded form. There are five levels of each of the two factors. In addition, we have the order in which the observations on x 1 and x 2 were run. The column on the right gives values of the response y, the percent conversion of the process. The first four design points represent the familiar factorial points at levels ±1. The next four points are called axial points. They are followed by the center runs that were discussed and illustrated earlier in this chapter. Thus, the five levels of each of the two factors are −1, +1, −1.414, +1.414, and 0. A clear picture of the geometry of the central composite design for this k = 2 example appears in Figure 15.16. This figure illustrates the source of the term axial points.These four points are on the √ factor axes at an axial distance of α =

2 = 1.414 from the design center. In fact, for this particular CCD, the perimeter points, axial and factorial, are all at the √ distance

2 from the design center, and as a result we have eight equally spaced

points on a circle plus four replications at the design center.

Example 15.8: Response Surface Analysis: An analysis of the data in the two-variable example may involve the fitting of a second-order response function. The resulting response surface can be used analytically or graphically to determine the impact that x 1

15.11 Introduction to Response Surface Methodology 641

Table 15.17: Central Composite Design for Example 15.8 Temperature ( ◦

C) Concentration (%)

Observation Run ξ 1 ξ 2 x 1 x 2 y

ξ 1 , Temperature ( C)

Figure 15.16: Central composite design for Example 15.8.

and x 2 have on percent conversion of the process. The coefficients in the response function are determined by the method of least squares developed in Chapter 12 and illustrated throughout this chapter. The resulting second-order response model is given in the coded variables as

2 − 8.50x 1 − 5.25x 2 − 7.75x 1 x 2 , whereas in the natural variables it is given by ˆ

y = 79.75 + 10.18x ˆ

1 + 4.22x

2 − 0.0136ξ 1 − 0.2100ξ 2 − 0.0620ξ 1 ξ 2 . Since the current example contains only two design variables, the most illumi-

y = −1080.22 + 7.7671ξ 2

642 Chapter 15 2 k Factorial Experiments and Fractions nating approach to determining the nature of the response surface in the design

region is through two- or three-dimensional graphics. It is of interest to determine what levels of temperature x 1 and concentration x 2 produce a desirable estimated percent conversion, ˆ y. The estimated response function above was plotted in three dimensions, and the resulting response surface is shown in Figure 15.17. The height of the surface is ˆ y in percent. It is readily seen from this figure why the term re- sponse surface is employed. In cases where only two design variables are used, two-dimensional contour plotting can be useful. Thus, make note of Figure 15.18. Contours of constant estimated conversion are seen as slices from the response sur- face. Note that the viewer of either figure can readily observe which coordinates of temperature and concentration produce the largest estimated percent conver- sion. In the plots, the coordinates are given in both coded units and natural units. Notice that the largest estimated conversion is at approximately 240 ◦

C and 20% concentration. The maximum estimated (or predicted) response at that location is 82.47%.

mp 0 eratur 15.29

Figure 15.17: Plot for the response surface prediction conversion for Example 15.8.

Other Comments Concerning Response Surface Analysis

The book by Myers, Montgomery, and Anderson-Cook (2009) provides a great deal of information concerning both design and analysis of RSM. The graphical illustration we have used here can be augmented by analytical results that provide information about the nature of the response surface inside the design region.

15.12 Robust Parameter Design 643

x 2 0 ation, %)

ξ 1 (Temperature, °C)

Figure 15.18: Contour plot of predicted conversion for Example 15.8.

Other computations can be used to determine whether the location of the optimum conditions is, in fact, inside or remote from the experimental design region. There are many important considerations when one is required to determine appropriate conditions for future operation of a process.

Other material in Myers, Montgomery, and Anderson-Cook (2009) deals with further experimental design issues. For example, the CCD, while the most generally useful design, is not the only class of design used in RSM. Many others are discussed in the aforementioned text. Also, the CCD discussed here is a special case in which k = 2. The more general k > 2 case is discussed in Myers, Montgomery, and Anderson-Cook (2009).