Introductory Chapter: Response Surface Methodology

1. Introduction

The development of empirical models and optimization is the focus of the mathematical and statistical methodologies, which is called as the response surface methodology (RSM). In RSM, the experiment optimizes a response (output variable) that is sensitive to many factors. As a result of it, study designs will accomplish these aims (input variables). Many trials are conducted using different variables, and the best variables are selected to accomplish the goals successfully.

RSM is the first model to incorporate observed responses [1]. Afterward, the numerical experimentation-based modeling technique are emerged. Also, several possible errors exist in the solution. Insufficient convergence of iterative methods, rounding mistakes, and a discrete representation of a continuous physical event are all potential sources of numerical noise in computer experiments. If the measurements are inaccurate in a scientific experiment, the conclusions will be affected at the end. If the measurements are inaccurate, then the findings of a physical experiment will be inaccurate [2, 3, 4, 5], at last. As a result of it, the RSM methodology adopts the position where errors occur at random. When used for design optimization, RSM reduces the need for high-priced analytical approaches like the computational fluid dynamics (CFD) analysis.

For the purpose of optimizing designs, many researchers have discussed the use of RSM. For example, the optimum surface roughness on machining is optimized by determining the optimum cutting speed (x₁) and feed (x₂). It is observed that cutting speed and feed affect the surface roughness:

$$\mathrm{y}=f\left({\mathrm{x}}_1,{\mathrm{x}}_2\right)+\epsilon$$E1

where ε denotes experimental error due to back ground noise, manual error, etc.

RSM is graphically displayed in either a two- or three-dimensional plot known as contour plot, and it is interpreted in two different ways. When all the factors are held at constant except for the x_i and x_j coordinates, the resulting curves are called contours. It specifies the highest reaction on each form’s surface. Figure 1 shows the steps in RSM, which clearly shows how the optimization is carried out by RSM.

Figure 1.

Response surface methodology.

2. Approximate model function

The relationship between the response and a nonresponse variable is unclear. First, RSM determines the best near model. Further the vast majority of models used in practice are polynomials of low order (first or second order). When a structure of the model has curvature, it includes the interaction effects and square effects. This model is observed as:

$$y={\beta }_0+{\displaystyle \sum }_{i=1}^k{\beta }_i{x}_i+{\displaystyle \sum }_{i=1}^k{\beta }_{ii}{x}_i^2+{\displaystyle \sum }_{i<j}\sum {\beta }_{ij}{x}_i{x}_j+\epsilon$$E2

Variety of work has been carried out in response surface modeling. This degree of usefulness is quantified by the goodness-of-fit metric. Sensitivity data is used to reduce the number of computer simulation analyses for model fitting, though it is not always readily available or affordable.

Some of the important methods used for RSM are as follows:

• Design of experiments: RSM relies mainly on design of experiments, abbreviated as DoE [1]. This method is mainly developed to fit the models for physical experiment data. Also, it is used for numerical experiment data.

• Full factorial design: It is necessary to do a full factorial analysis [6] for creating approximate model in establishing the relationship between the design variables.

• Central composite design (CCDs): CCDs are considered, as the first-order designs with an extra center and axial points in order to estimate the tuning parameters of second-order models,

• D-optimal designs: The experimental points provide the most reliable estimations of the response model coefficients using quadratic models considering the D-optimality criterion [7].

• Taguchi’s contribution to experimental design: Taguchi has used the orthogonal arrays for designing the experiment. Taguchi’s design allows for fewer testing than a full factorial design.

Other approaches like [8, 9, 10] are also considered by some researchers. It is clearly known that RSM is an important method in statistical design, and it is applied for different kind of experiments. Research advances are carried out in different fields of Engineering, Science, and Technology [11]. The application of RSM in the different fields is innumerable. Hence, RSM method finds its applications in textile industry, food industries, chemical industries, mechanical industries, pharmaceutical industries, etc. By using proper design of experiments, the modeling and optimization are carried out without cumbersome efforts.

3. Conclusion

RSM is very much useful for analyzing and optimizing the experimental and numerical responses. It has got close approximations of both experimental and numerical results by choosing the correct method of experimentation. The best plans for experiments have the experimental locations dispersed around the area of interest. Recently, many new plans are implemented for analyzing the data. In spite of computer-based many modeling and optimization techniques are developed, RSM is an indispensable method and is used for modeling and optimization in the field of Science, Engineering, and Technology.