Introduction to Response Surface Methodology

## **Chapter 1**

## Introductory Chapter: Response Surface Methodology

*Palanikumar Kayarogannam*

## **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 [2–5], 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:

$$\mathbf{y} = f\left(\mathbf{x}\_1, \mathbf{x}\_2\right) + \varepsilon \tag{1}$$

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 xi and xj 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.

## **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:

$$\boldsymbol{y} = \boldsymbol{\beta}\_{0} + \sum\_{i=1}^{k} \boldsymbol{\beta}\_{i} \boldsymbol{\varkappa}\_{i} + \sum\_{i=1}^{k} \boldsymbol{\beta}\_{ii} \boldsymbol{\varkappa}\_{i}^{2} + \sum\_{i$$

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:


*Introductory Chapter: Response Surface Methodology DOI: http://dx.doi.org/10.5772/intechopen.110353*


Other approaches like [8–10] 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.

## **Author details**

Palanikumar Kayarogannam Department of Mechanical Engineering, Sri Sai Ram Institute of Technology, Chennai, India

\*Address all correspondence to: palanikumar@sairamit.edu.in

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

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[3] Audze P, Eglais V. New approach for planning out of experiments. Problems of Dynamics and Strengths. 1977;**35**:104-107

[4] Toropov V, Fred V, Markine V. Refinements in the multi-point approximation method to reduce the effects of noisy structural responses. In: 6th Symposium on Multidisciplinary Analysis and Optimization. Bellevue, WA, U.S.A. 1996

[5] Van Kampen A, Huiskes R. The threedimensional tracking pattern of the human patella. Journal of Orthopaedic Research. 1990;**8**(3):372-382

[6] Montgomery DC. Response surface methods and other approaches to process optimization. In: Montgomery DC, editor. Design and Analysis of Experiments. New York: John Wiley & Sons; 1997. pp. 427-510

[7] Myers RH, Montgomery DC. Response Surface Methodology: Process and Product Optimization Using Designed Experiment. Hoboken, New Jersey, U.S.: John Wiley & Sons. Inc; 1995

[8] McKay MD et al. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics. 1979;**21**(2):239-245

[9] Van Keulen F, Toropov VV. The multipoint approximation method in a parallel computing environment. Zeitschrift Fur Angewandte Mathematik Und Mechanik. 1999;**79**:S67-S70

[10] Venter JC, Smith HO, Hood L. A new strategy for genome sequencing. Nature. 1996;**381**(6581):364-366

[11] Palanikumar K. Introductory chapter: Response surface methodology in engineering science. In: Response Surface Methodology in Engineering Science. London, UK: IntechOpen; 2021. DOI: 10.5772/intechopen.100484

Section 2
