Introductory Chapter: Response Surface Methodology in Engineering Science

1. Introduction

RSM (response surface methodology) is a factorial/experimental design for optimising a strong bond between one or more target variables. This pragmatic method has been recommended by Box and Wilson [1]. Moreover, it is the key component for gaining the finest results through an array of exclusive experiments. To achieve it, normally, the second-degree polynomial model is used. Even though RSM is a hypothetical model, the scientists and researchers practice it for estimations.

RSM helps to examine the connection between the input variables and the responses (output) of any process or system. Its main intention is to advance the response time or to reach the scope of betterment in the work.

2. Response surface methodology - A view

The response surface methodology (RSM) is used to analyse the rapport when the input factors are quantitative. Additionally, the variables (x₁) and (x₂) maximise the yield of a process (Y). To put it briefly, the variables influence the process yield, as mentioned below:

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

Indeed, RSM is expedient in creating and analysing the system, as the ultimate aim is to optimise the impact response by various factors.

As RSM is very important in the formulation, creation, and implementation of new engineering research and products, it is more predominant in the field of industrial, manufacturing, clinical sciences, material development, social science, and physical, biological, food engineering, and engineering sciences. It is a known fact that it has many applications; therefore, the researchers explore its origin. In addition, many researchers have used RSM for manufacturing and engineering applications. For example, surface roughness and cutting force components are modelled analytically by using the response surface methodology (RSM). Its outcome determines the workpiece’s feed rate and its hardness impacts the cutting force components. Nevertheless, the surface roughness is affected by both the feed rate and the hardness of the workpiece [2].

RMS’s efficiency depends on the accuracy of y at various points throughout the response surface; therefore, the researcher determines its optimal or improved system’s reaction. From the very beginning of the research, the researcher studies the components or variables of the response surface. Correspondingly, non-significant independent variables are isolated from the critical ones with a view to conducting a good experiment. Also, it is necessary to focus on the essential components before the examination study. In addition, Hill and Hunter divide the response surface analysis into four phases [3]. The specifics are as follows:

1. The experiments can be planned as a statistically valid experiment.

2. Compute the coefficients for the response surface equation.

3. Test the equation by using the lack-of-fit test to see if this is correct.

4. Test the response surface regions target by examination.

RSM manipulates the precise design of experiments (DoE), which has recently acquired favour for formulation. Along with it, its statistical approach is used to assess the interaction effect between the process factors than the traditional approach.

RSM (response surface methodology) is a computational and scientific technique for modelling and analysing the situations that consider various factors that impact the desired response and attempt to maximise the result [4]. The connection between the output and the independent variables is unknown in most of the RSM issues [5]. In a similar fashion, RSM estimates the outcome variable ‘y’ and the set of independent variables ‘x’ first. In several portions of the response variable, a low-order polynomial is commonly used. The functional approximation in the first-order model is well represented by the independent parameters and it is given below [4]:

$$y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+\dots +{\beta }_k{x}_k+\epsilon$$E2

When the structure is having curvature in the second-order form, a higher-grade polynomial is used [4].

$$y={\beta }_0+{\displaystyle {\displaystyle \sum _{i=1}^k}}{\beta }_i{x}_i+{\displaystyle {\displaystyle \sum _{i=1}^k}}{\beta }_{ii}{x}_i^2+\sum _{i<j}\sum {\beta }_{ij}{x}_i{x}_j+\epsilon$$E3

Approximation polynomials’ parameters are estimated by using the least-square approach. So, a surface response analysis is performed on the linked surface. Besides that, the analysis of the installed surface is compared with the analysis of the actual system, when the installed surface is a reasonable approximation to the genuine response function [6].

Consecutively, the model variables envisage the most suitable experimental designs successfully for data collection. Eventually, the surface design is considered as the suitable responsive surface. Undoubtedly, RSM is a method for identifying the system’s optimal process parameters or a factor space area that successively meets [7]. Likewise, simultaneous analysis of multiple responses commences with appropriate response surface models for each result. Later, it pursues to optimise a set of operating conditions and keeps all responses within the necessary range at the lower limit [8].

3. Conclusions

From the literature and other details as on date, RSM is extensively used in engineering science and it has found numerous applications. By using it properly, the process output can be improved to many folds and RSM can be utilised as an important technique in engineering and science applications.