Applications of Response Surface Methodology (RSM) in Product Design, Development, and Process Optimization

Abstract

In this review chapter, the authors presented a systematic exposition to the concept of Response Surface Methodology (RSM) for applications by Scientists, Engineers, Technologists and Industries. (RSM) is an empirical model which employs the use of mathematical and statistical techniques in relating input variables otherwise known as factors to the response. RSM became very useful due to the fact that other methods available such as the theoretical model could be very cumbersome to use, time-consuming, inefficient, error prone and unreliable. In order to draw meaningful conclusions and findings, an experiment is required. In an effort to obtain an objective conclusion (between the factors and the response), an experimenter needs to plan and design the experiments, and analyze the results. An approximation of the response in relation to the variables is otherwise known as RSM. This chapter reviews RSM concept for easy understanding and adoption by researchers. In section 2.0, the various terminologies used in RSM were defined. In section 3.0, RSM design types were highlighted and RSM research phases exposed in section 4.0. Section 8.0 gave some scenario applications of RSM in various fields and section 9.0 defined the RSM research cycle process. General applications and conclusions stated.

1. Introduction

Experimentation, Data collection, Data processing, and Analysis of data are very basic and essential to Scientists, Engineers, Technologists, and Manufacturing Industries to design, develop, improve and validate their products, processes, and operations. Response surface methodology (RSM) which is available in MINITAB and other proprietary software is a collection of both statistical and mathematical techniques useful for developing, improving, and optimizing processes [1]. RSM is known to play a pivotal role in new product design and development as well as in improving existing ones. With response surface methodology we can determine the optimum factor needed to produce the best result. RSM is a critical and very robust tool for data manipulation and analysis of research data to obtain a quality result or an improvement [1]. RSM could be applied by an industry that desires to manufacture a component (from Al-Si Alloy material) with minimum surface roughness by combining three controllable variables (cutting speed, feed rate, and, depth of cut). Because of this, the Design of Experiments (DOE) could be used to carry out the study of the effect of the three machining variables (cutting speed, feed rate, and depth of cut) on the surface roughness (Ra) of Al-Si alloy [2]. With the use of response surface methodology (RSM), a mathematical prediction model of the surface roughness would be developed in terms of cutting speed, feed rate, and depth of cut. The effects of the three process parameters on both Ra can then be investigated by using the response surface methodology (RSM). The above approach can be adopted by any industry, scientist, or researcher in getting better results (response) from several variables otherwise known as factors. RSM helps to reduce the noise in an experiment, thereby ensuring optimization. Many researchers have conducted researches on the application of RSM or other DOE concept in which the results of their findings have been used to develop a predictive model in several fields such as; tool life modeling, surface roughness prediction, for monitoring and functionality or health condition of electronic devices also for the surface roughness of Inconel using full factorial design of experiment among other areas of applications [2, 3, 4]. The RSM looks into an adequate approximation relationship between input and output variables and determines the best operating circumstances for a system under study or a portion of the factor field that complies with the operating requirements or conditions [3, 5, 6].

Response surface methodology can be better referred to as a collection of statistical and mathematical techniques employed for product design and improvement, process development and improvement as well as process optimization. It has major applications in the design, development, and, formulation of new products as well as in improving existing product design. RSM is a robust tool for the design of experiments, analysis of experimental data, and process optimization. In RSM, the response is determined by the variables and the aim is to optimize the response [1, 7, 8]. There are two primary experimental designs used in response surface methodology: Box-Behnken designs (BBD) and central composite designs (CCD) [8, 9]. Recently, optimization studies have also used central composite rotatable design (CCRD) and face central composite design (FCCD) [8, 10, 11, 12, 13, 14].

Wong [15] employed RSM concept to carry out reliability analysis of soil slopes. Tandjiria et al. [16] used response surface method for reliability analysis of laterally loaded piles. Sivakumar Babu and Amit Srivastava [17] presented a study on the analysis of allowable bearing pressures on shallow foundation using response surface method and showed that a comparative study of the results of the analysis from conventional solution and numerical analysis in terms of reliability indices enables rational choice of allowable loads. For better understanding of the RSM concept in our daily life experiences as described in Figure 1. Take for example we have two variables (humidity and temperature) and we want to see the effects of these variables on human comfort. We can name these independent variables temperature and humidity, X1 and X2 and the response which is human comfort can be named Y. Response Surface Methodology is useful in this case for the modeling and optimization of the situation above in which the response of interest (human comfort) is influenced by the variables (humidity and temperature). In this model example, our objective is to optimize this response The visual representation of the above is otherwise known as Response Surface Methodology (RSM) or response surface modeling. To find the levels of temperature (X1) and pressure (X2) for maximum human comfort (y) in the above process.

Figure 1.

Response surface for humidity and Temperature on human comfort.

$$y=\left[\left(x_1,x_2\right)+ϵ\right]$$E1

ϵ is referred to as the error term inherent in the system

1.1 The concept of RSM

The concept of Response Surface Methodology can be used to establish an approximate explicit functional relationship between input random variables and output response through regression analysis and probabilistic analysis can be performed [15]. RSM involves a combination of metamodeling (i.e., regression) and sequential procedures (iterative optimization). Response Surface Methodology (RSM) is a collection of mathematical and statistical techniques useful for the modeling and analysis of problems. By careful design of experiments, the objective is to optimize a response (output variable) that is influenced by several independent variables (input variables). A collection of mathematical and statistical methods called Response Surface Methodology (RSM) can be used to simulate and analyze issues. The goal of meticulous experiment design is to maximize a response (output variable) that is affected by a number of independent variables (input variables). The motivation behind this work is the applicability of the concept of RSM to many areas of scientific research, engineering and manufacturing industries.

1.2 Objective of this present study

The applications of RSM is for product and process development are discussed through some general and scenario applications. The chapter review presented, is shown that with RSM we can;

1. identify the sensitive parameter that provides the greatest influence on the response.

2. easily take decision that will impact positively the product design and process optimization.

3. ensure reliability, acceptability and profitability of the product developed and/or optimized condition.

2. Some useful terminologies

• Factors are input variables/parameters that potentially affect the response. It can be controllable or uncontrollable, and quantitative and qualitative.

• Response is a dependent variable. It is the desired results obtained from combining the interaction of independent variables.

• Experiment is a series of tests, called runs, in which changes are made in the input variables to identify the reasons for changes in the output response.

• Experimenter is a person experimenting for research purposes.

• Treatment is a combination of one or more factors.

• Levels are the values a factor can take on

• Effect simply means how much a main factor or interaction between two or more factors influences the response.

• Design Points: simply means the assigned values of the individual factors for which the experiment was performed.

  • One design point = one treatment

  • Points are typically coded to more practical values.

  • example. 1 factor with 2 levels – levels coded as (−1) for low level and (+1) for high level

• Design Space is the range of values over which factors are to be varied or adjusted [18, 19].

• Response Surface is the unknown or experimental purpose. It is the mean response at any given level of the factors in the design space.

RSM Design Types; The summary of the various types of design available in response surface methodology is presented in Figure 2 according to [18].

Figure 2.

Available Designs in RSM source MINITAB 20.

(ii) Central Composite Design (CCD) (2 to 10 continuous factors)

(ii) Box-Behnken Design (3,4,5,6,7,9 or 10 continuous factors)

To do a visual analysis of the response surface design, the designer can use the following visualization tool to visualize the response in RSM.

• Residual plots

• Effect plots

3. RSM research phases

RSM involves four broad phases as highlighted below.

1. Use a simulation model e.g., Minitab to fit a linear regression model to the data points in the workspace and, find a better solution from the linear regression model.

2. Repeat the above process until the slope of the linear response surface obtained from the linear regression model is approximately zero.

3. Fit a nonlinear quadratic regression.

4. Lastly find the optimum of this equation.

4. Getting access to response surface methodology (RSM)?

MINITAB, STATISTICA, DESIGN EXPERT, etc. are software tools that can be used for experimental design and analyze data. RSM is one of the techniques that have been programmed in this software. Among all, MINITAB is highly rated when it comes to the design of experiments using response surface methodology. Minitab is a proprietary software tool, a computer program applied in statistical studies, developed in 1972. Its interface is similar to Microsoft Excel or Calc of OpenOffice, used in universities and companies, it has specific functions focused on process management and analysis of the Six Sigma suite. Minitab offers Quality Control tools, Experiment Planning (DOE) e.g., RSM, Reliability Analysis, and General Statistics [18, 20]. Figure 3 shows the navigation process in Minitab 18 to access response surface methodology interface (Table 1).

Figure 3.

Diagram showing the navigation of RSM with MINITAB software.

S/N	Factors and types	Responses
1.	mean interactive time (uncontrollable, quantitative)	Mean daily production rate
2.	Mean service time (controllable, uncontrollable, quantitative)	Meantime in the system for patient
3.	Number of servers (controllable, quantitative)	Mean inventory level
4.	Reorder point (controllable, quantitative)	Mean surface roughness
5.	Queuing discipline, qualitative)	Number of students who wait till the end of the lecture
6.	Mean winter demand time (uncontrollable, quantitative)	Mean material removal rate

Table 1.

Some examples of Factors and Response in RSM.

5. Advantages of RSM

The application of response surface methodology in research and industry comes with the following advantages

1. Seamless statistical analysis

2. Optimization of manufacturing system, process, or product.

3. Experimental layout and design.

4. Prediction

5. Interaction of variables is easily presented with clear curves and other visual aids.

6. Good visualization of responses or results with the use of surface plots, graphs, etc.

7. Associated empirical mathematical models.

6. Some scenarios of RSM Applications

6.1 A scenario of RSM in a manufacturing process

In a quest to manufacture a component during CNC turning operation. Proper selection of process parameters or variables (cutting speed, feed rate, and depth of cut) for optimal surface quality (Response) must be achieved. This requires a more methodical approach by using experimental methods and mathematical and statistical models. The design of experiments will play a pivotal role in this regard. This will require considerable knowledge and experience of the designer to design experiments and analyze data. Note that the traditional design-of-experiment (DOE) technique requires a large number of samples to be produced. To increase machining process efficiency, strategies for optimizing machining parameters using experimental methodologies as well as mathematical and statistical models have developed significantly over time. A full factorial approach may be required to look into all potential combinations to build an approximation model that can describe interactions between design variables in this CNC turning operation. An experimental approach known as a factorial experiment involves varying design variables simultaneously rather than one at a time. It is necessary to define the lower and upper bounds for each of the n design variables in the optimization problem. Then, at various levels, the permitted range is discounted. If just the lower and upper bounds (two levels) of each variable are defined. The experimental design is referred to as 2n full factorial if each variable is defined at just the upper and lower boundaries (two levels). Second-order models can be fitted using factorial designs. When a first-order model exhibits a lack of fit as a result of the interaction between variables and surface curvature, a second-order model can considerably enhance the optimization procedure. The goal of a meticulous experiment design is to optimize the response. (Surface quality of the machined part) which is influenced by several independent input variables (cutting speed, feed rate, and depth of cut).

6.2 A scenario of RSM in the energy industry

Due to the limited availability of high-grade coal for energy production, low-grade coal can be employed. High ash levels and high moisture content are characteristics of low-grade coal. With the use of the response surface methodology, the operational parameters were optimized to generate clean coal as effectively as possible. The impact of three independent variables, including hydrofluoric acid (HF) concentration (10–20 percent by volume), temperature (60–100^oC), and time (90–180 min), for ash reduction from the low-grade coal, was explored to attain this coal optimization target. By utilizing the central composite design (CCD) method, a quadratic model was presented to correlate the independent variables for maximal ash reduction at the ideal process condition. In comparison to time and temperature, the study finds that HF concentration was the most efficient parameter for ash reduction [16].

6.3 A scenario of RSM in extraction optimization

In order to maximize the extraction process of oil from leaves, fruits etc., it is important to optimize the extraction parameters so as to get the best yield. RSM concept has been used more often in recent years to optimize various oil extractions from plant sources [17, 21].

6.4 A scenario of RSM in drinking water treatment process

Both trihalomethanes (THMs) and Natural Organic Matter (NOM) has been characterized with cancer risk in drinking water According to [22]. The concept of RSM was used for the development of water treatment technologies and optimization of process variables in order to reduce THMs and NOM level of concentration in drinking water. A model was developed to control the process. The developed models can be effectively used to remove both THMs and NOM from drinking water.

6.5 A scenario of RSM in construction industry

The construction industry is a very germane industry in the technological advancement of any nation. The level of research-based construction has been improved lately. A study on the analysis of allowable bearing pressures on shallow foundation using response surface method was conducted and showed that a comparative study of the results of the analysis from conventional solution and numerical analysis in terms of reliability indices enables rational choice of allowable loads [15].

6.6 A scenario of RSM in product development

The effect of oven parameters such as air velocity, time, temperature etc. on formulations (sugar, water, fats, flavors, etc.,) of the quality of baked food product can the analyzed with the application of response surface methodology [23]. RSM model is a powerful tool to optimize the product quality (volume of baked product, crust and crumb color, bake loss among others). The data collected through RSM can further be used to obtain the variability of the response(s) with tested parameters [23]. In this scenario, the results of the optimization obtained is otherwise referred to as quality product.

RSM cycle processes is shown in Figure 4.

Figure 4.

RSM Research Cycle Process.

1. Experimental Plan/Data collection. This is the initial stage. The planning session precedes any other session involved in RSM modeling. In this session, all decisions involved in the project or experiment are clearly stated and defined. Some of the decisions under this heading include the research objective, methodology of the research, and variables that could influence the results are highlighted. This process takes care of all necessary information regarding the experimental strategy [20, 24, 25, 26]. To clarify the objective of the experiment, the objective must determine;

   1. What data is to be collected?

   2. How to measure it?

   3. How does the data relate to processing performances and experimental objectives?

2. Experimental Design. Experimental design can either be a conventional method or a statistical method. The conventional method has the following features;

   1. It is time-consuming.

   2. Can handle one factor over time (OVAT) or one factor at a time (OFAT).

   3. Interaction between two or more variables cannot be interpreted.

   Features of statistical method experimental design

   1. It is otherwise known as Design of Experiment (DOE)

   2. Apply the factorial concept

   3. It makes use of modeling to predict the behaviors of process variables e.g., RSM

   4. The process variables could be explained through interaction plots and graphs

   5. Saves time and improves efficiency.

   An experiment is designed based on the decisions during the designing or data collection stage. The experimental design clearly states the number of experiments and how the experiment will be carried out [20, 26, 27].

3. Conducting Experiment. The next step after the experimental design is to experiment with the exact research parameters and, in the order, defined by the layout for easy statistical validity. The person experimenting is called an experimenter.

4. Analyzing the Results. The primary focus in this analysis stage is to obtain useful information from the experiment conducted and ascertain the level of quality or improvement recorded. In this stage, the results obtained from the experiment conducted are analyzed. The analysis of the results is targeted toward specific conclusions. Since we have several samples tested in each experimental run, different analysis techniques can be selected.

5. Graphical Analysis. One of the powerful features of RM is the ability to present results or responses using visual aids for easy interpretation or understanding.

6. Confirmation of results/Ask questions relating to the research objective. A test can be carried out to ascertain if the actual performance of the product in-service condition matches the improvement stated in the results. The test here helps to determine the research gap (Figure 5).

Figure 5.

RSM Flow chart.

7. General applications of RSM

1. RSM found its major application in the industry to model and design a product. Also, to optimize the manufacturing process.

2. RSM is capable of Data analysis, prediction, product design, and optimization [28, 29].

3. RSM can predict the relationship or interaction that exists between the values of some measurable response variable(s) and those of a set of experimental factors presumed to affect the response(s).

4. RSM is capable to predict the response value at various process conditions.

5. Application of RSM can be used to screen independent variables in order to determine most significant main effect of factors among several potential ones.

6. With RSM a non-identified interaction effect could be determined.

7. With RSM application for optimization, one can easily identify the best factor(s), process interaction effect, that produce the response that brings the optimized condition. This is the real deal in parameter optimization.

8. RSM enhances product quality, product life span and increase productivity.

9. Application of response surface methodology exposes the best type of design: CCD or BBD needed to achieve optimization.

10. RSM can predict the relationship or interaction that exists between the values of some measurable response variables and those of a set of experimental factors presumed to affect the response.

11. RSM is capable to predict the response value at various process conditions.

12. Power visual aids such as plots graphs in 2D or 3D for easy understanding of the results.

8. Visualizing results in RSM

RSM uses a variety of surface visualization techniques according to Figures 6–11 to visually assess how factors affect the response. When a regression model is fitted as a result of interactions between two or more predictors, visualization better communicates the experimental results or responses. Effects plots, contour plots, residual plots, surface plots, etc. are a few examples of graphical visualization tools also known as response surface plots. These plots aid in determining the process conditions and desired response Values [18, 19, 27, 29, 30, 31, 32, 33, 34, 35].

Figure 6.

Figure showing standardized effect.

Figure 7.

Central composite design.

Figure 8.

Main effect plot in RSM.

Figure 9.

Interaction plot explored from RSM software.

Figure 10.

Residual plot for optimized point.

Figure 11.

Some response surface plots for visualization of RSM results.

9. Conclusions

In this chapter, the authors provided a detailed approach for the understanding and implementing Response Surface Methodology (RSM) for the various professionals or researchers who may be involved in the application of Response Surface Methodology. In an attempt to design a product or to optimize an existing process there are several methods that can be adopted. RSM has many advantages when compared to classical methods. It requires fewer runs of experiments to understand the effects of all the factors and the optimum combination of all factor input. RSM requires less time, removes trial by error and ensure high quality results. Having presented in this chapter the huge applications of RSM in various fields of research, it can be concluded that RSM is a great research tool for product design, development and process optimization. The chapter coverage is detailed enough to give the basic insight of RSM even to a novice hearing about RSM for the very first time. However, the chapter does not cover all the information required for mastery of the RSM concept.

Acknowledgments

The authors sincerely appreciates IntechOpen with the opportunity provided to publish this chapter review in the main book; Response Surface Methodology-Research Advances and Applications.

We also give kudos to the critical peer review process.

Notes/thanks/other declarations

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