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Response Surface Methodology (RSM) involves the construction and analysis of mathematical models to depict the relationship between input variables and the response of a system or process. This method circumvents the need for exhaustive experimentation by strategically designing a limited set of experiments while maximizing the information gathered. Experimentation and optimization are integral processes across various scientific disciplines. The utilization of Response Surface Models (RSMs) has emerged as an indispensable tool in achieving optimal experimental outcomes. The foundational understanding of RSM involves its core components, emphasizing the relationship between independent variables and their impact on a response of interest by employing statistical techniques. RSM enables researchers to comprehend the intricate behavior of systems, identify critical factors influencing the response, and subsequently optimize the process. Response surface techniques facilitates not only the improvement of processes but also the minimization of costs, reduction of waste, enhancement of product quality, facilitating efficient exploration and analysis of complex systems. Response surface analysis could be explore in all fields to generate optimal condition for all the variables in an experiment.
Department of Agricultural and Environmental Engineering, School of Engineering and Engineering Technology, Federal University of Technology, Akure, Nigeria
*Address all correspondence to: ooolabinjo@futa.edu.ng, binjo_joko@yahoo.co.uk
1. Introduction
Optimization of experiments is of utmost importance in the field of research. It involves the identification of the optimal set of experimental conditions, which leads to the best possible result. Response Surface Methodology (RSM) is a statistical approach that is widely used for optimization of experiments in various fields, including the food industry, chemical engineering, and material science [1]. RSM is based on a mathematical model that describes the relationship between the response variable and the independent variables. This model helps to identify the optimal set of experimental conditions, which minimizes or maximizes the response variable of interest. RSM is a powerful tool for optimization of experiments, as it allows for the identification of the optimal set of experimental conditions with a minimal number of experiments. Jensen [2] highlighted the importance of RSM in product and process optimization using designed experiments. The study emphasized the efficiency of RSM in reducing the number of experiments required for optimization, which results in cost savings and faster results. Moreover, Schönbrodt et al. [3] demonstrated the use of RSM in testing similarity effects with dyadic response surface analysis. The study showed that RSM improved the efficiency of testing similarity effects. RSM has been widely used in the food industry to optimize various processes, such as extraction, drying, and fermentation [1]. For instance, Aydar [4] utilized RSM to optimize the extraction of plant materials. The study found that RSM improved the extraction efficiency, and the optimal conditions were obtained by adjusting the independent variables. Similarly, Mohammed et al. [5] used RSM to optimize the rubbercrete mixture. The study found that RSM improved the compressive strength and water absorption of the rubbercrete mixture. RSM has also been used in the optimization of chemical processes. Asfaram et al. [6] used RSM to optimize the removal of the basic dye Auramine-O from aqueous solutions. The study found that RSM improved the efficiency of the removal process, and the optimal conditions were obtained by adjusting the independent variables. Similarly, Sarabia et al. [7] discussed the use of RSM in chemical engineering, and highlighted its role in optimization of various processes. Therefore, RSM is an inevitable tool in optimization of experiments in research.
RSM is a statistical technique used to optimize the response of a process by exploring the relationship between the response and the input variables. The technique involves building a mathematical model that represents the relationship between the response and the input variables, which is then used to predict the response for any combination of input variables. RSM has been applied in various fields of engineering and science, including environmental engineering, manufacturing, and reliability analysis [8].
There are various types of RSM models, including central composite design (CCD), Box-Behnken design (BBD), and Taguchi design. The selection of an appropriate RSM model depends on the nature of the response variable and the number of input variables. CCD is commonly used when the response variable is continuous, and there are three or more input variables, while BBD is preferred for a smaller number of input variables. Taguchi design is used to optimize the process parameters for quality improvement [9].
3.1 Box-Behnken design
The Box-Behnken design is a type of response surface design that uses a set of three-level factorial designs with a central point. It was developed as an alternative to the more complex and time-consuming full-factorial designs. This design is particularly useful when the number of factors is high, which makes the full-factorial design impractical. The Box-Behnken design uses fewer experimental runs while maintaining a high level of accuracy in estimating the response surface. The design involves a series of levels that are evenly spaced between the high and low levels of each factor, along with a central point to estimate the curvature of the response surface.
The Box-Behnken design has several advantages over other designs. First, it has fewer experimental runs, which saves time and resources. Second, it provides a better estimation of the response surface curvature compared to other designs. Third, it allows for the assessment of the effects of each factor and their interactions, which is useful for understanding the behavior of the response variable. Finally, it is easy to implement and analyze using standard statistical software.
3.2 Central composite design
Central composite design (CCD) is a popular technique used in Response Surface Methodology (RSM) for optimization of experiments. CCD is a method for designing experiments in which the response of a system is modeled as a function of several independent variables. It is a widely used design method for fitting second-order response surface models in RSM experiments. CCD is a powerful statistical tool that is suitable for investigating non-linear relationships between experimental factors and response variables. The use of CCD in RSM has been shown to be effective in optimizing various processes, such as food processing, drug discovery, and engineering design [10, 11].
Olabinjo et al. [12] carried out the effects of oven temperature (46–74°C) and number of cycle (bath) (2–4), on the extraction yield (g/10 g) of citrus peels and antioxidant of the extract using ABTS (2,2-azinobis-(3-ethylbenzothiazoline-6-sulfonic acid)). This was investigated using statistical experimental design based on 22 Central Composite Design (CCD) with four axial points and five central points. The extraction employing pressurized liquid was decided to optimize only two process variables, temperature (T) and number of cycles (C). In the study, thirteen experimental treatments were assigned based on CCD with two independent variables at five levels of each variable (Table 1).
Levels
Independent variables
Axiaal (−α)
Low (−1)
Center (0)
High (1)
Axial (+α)
Temperature (°C)
46
50
60
70
74
Number of cycles (C)
1.6(2*)
2
3
4
4.4(4*)
Table 1.
Levels of independent variables (T and C) for the central composite design.
α = ± 1.41 * real time in the equipment. Olabinjo et al. [12].
One of the primary advantages of CCD is its ability to model both linear and non-linear relationships between experimental parameters and response variables. CCD is also able to capture the interactions between the experimental factors, which is essential in determining the optimal conditions for a process. Ferreira et al. [13] demonstrated the efficacy of CCD in optimizing the design of frequency selective surfaces for electromagnetic applications. The authors employed CCD to model the interactions between the geometry and dimensions of the surfaces and their electromagnetic response. The results showed that CCD was able to optimize the frequency selective surfaces more effectively than other design methods.
3.3 Taguchi design
The Taguchi design is a systematic approach to design experiments that ensure efficiency and effectiveness. It was developed by Dr. Genichi Taguchi, a Japanese engineer, in the late 1940s and early 1950s to improve the quality and reliability of products. The Taguchi design approach emphasizes the importance of choosing appropriate factors and levels to ensure that the optimization process is successful. The design approach involves conducting a series of experiments using a set of factors and levels, and then analyzing the results to determine the optimal settings for the factors. The Taguchi design approach has been used in many fields, including manufacturing, medicine, and engineering [14, 15].
The response surface model (RSM) is an effective tool for optimization of experiments using the Taguchi design approach. RSM is a mathematical model that can be used to predict the response of a system to certain inputs. The RSM can be used to optimize a system by identifying the input variables that affect the response and determining the optimal settings for these variables. The RSM can be used to develop a mathematical relationship between the input variables and the response variables. This relationship can then be used to optimize the system by predicting the optimum values for the input variables [16, 17].
The Taguchi design approach, when combined with the RSM, can lead to significant improvements in the optimization process. The Taguchi design approach can be used to identify the input variables that are most important in the optimization process, while the RSM can be used to develop a mathematical model that can predict the response of the system to these variables. The RSM can also be used to identify any interactions between the input variables. Interaction effects can significantly affect the optimization process, and it is essential to take them into account when optimizing a system [18, 19].
Response surface methodology (RSM) has been widely used in different fields to optimize experiments. In the field of engineering, RSM has been used to optimize the design of products and processes. For instance, Panwar et al. [20] used RSM to optimize surface roughness in turning of EN 36 alloy steel. The researchers used a combination of RSM and genetic algorithm to optimize the turning process. The results showed that the combination of RSM and genetic algorithm was an effective tool for optimizing the turning process and reducing surface roughness.
In analytical chemistry, Bezerra et al. [21] used RSM to optimize the analytical parameters for the determination of metals in soil samples. The researchers used a central composite design to determine the optimum conditions for the analysis. The results showed that RSM was an effective tool for the optimization of analytical methods and could be used to reduce the number of experimental runs required for optimization.
In information systems research, Sedera and Atapattu [22] used RSM to model and optimize the relationship between the performance of information systems and the factors that influence it. The researchers used RSM to develop a response surface model that characterized the relationship between system performance and the input variables. The results showed that RSM was an effective tool for modeling and optimizing complex systems.
4.1 Advantages and limitations of response surface model
One of the most significant advantages of response surface model is that it reduces the number of experiments needed to optimize a system. RSM can identify the optimal combination of input variables using only a fraction of the experiments that would be required using the traditional one-factor-at-a-time (OFAT) method [23]. This reduction in the number of experiments not only saves time and resources but also allows researchers to explore the system more efficiently. In addition, RSM can obtain a more comprehensive understanding of the relationship between input variables and the output response, which enhances the accuracy of the optimization.
Another advantage of RSM is that it can handle multiple input variables simultaneously. Unlike OFAT, which examines one input variable at a time while holding all other variables constant, RSM can analyze the effect of multiple input variables on the output response at the same time. This feature allows researchers to identify complex interactions between input variables that would be missed by OFAT. As a result, RSM provides a more realistic optimization model that considers all the input variables in a system [24].
Furthermore, response surface model can be used to model and predict the behavior of a system. Researchers can use RSM to develop a mathematical model that describes the relationship between input variables and output response. This model can be used to predict the behavior of the system under different input variable combinations. Thus, RSM can help researchers optimize a system without conducting numerous experiments, saving time and resources while ensuring accuracy [25].
Despite the numerous advantages of response surface model, there are some limitations to its use. One limitation is that RSM assumes that the relationship between input variables and output response is continuous and smooth. If the relationship is discontinuous or has non-smooth regions, RSM may not be suitable for optimizing the system. Additionally, RSM may not be appropriate for optimizing a system that has a large number of input variables. In such cases, RSM may require a large number of experiments to optimize the system adequately, making it less practical [26].
Another limitation of response surface model is that it assumes that the input variables are independent of each other. In reality, some input variables may be correlated, so an optimization model that assumes independence may not be accurate [27]. There is also a possibility of model overfitting when there is limited data available for modeling. In such cases, the model may be overly complex, leading to poor predictions on new datasets.
The data collected determines the accuracy of the response surface model and can be carried out using various methods, such as manual collection, automated data collection, and simulation-based data collection. The data collection process should be well-designed to obtain the best results from the experimental design. In many cases, a limited number of experiments is conducted to reduce the cost and time required to obtain the data [28]. The data collected should be representative of the entire process, and the input variables should be significant factors that affect the output response. Manual collection is usually used for simple experiments that require few observations, while automated data collection is used for complex experiments that require a significant amount of data [29]. Simulation-based data collection is used to generate data by simulating the system under different parameter combinations. This method is suitable for experiments that are dangerous, costly, or time-consuming to conduct.
RSM also enables the analysis of the effects of various factors on the response variable, which can help in identifying the critical factors that affect the response variable and in improving the understanding of the system under investigation [30]. One of the main advantages of RSM is its ability to find the optimal values of the independent variables that can maximize the response variable. Li et al. [31] applied RSM to a least squares support vector machine model for annual power load forecasting, and the results showed that RSM can improve the accuracy of the model by identifying the optimal values of the input variables. Similarly, Lyu et al. [32] used RSM to optimize the aerodynamic shape of the Common Research Model wing benchmark, and the results showed that RSM can reduce the number of experimental runs required to obtain accurate results while improving the accuracy of the optimization process.
Another advantage of RSM is its ability to analyze the effects of multiple factors on the response variable. Li et al. [33] applied the d-Level Nested Logit Model to assortment and price optimization problems, and the results showed that RSM can identify the effects of different factors on the response variable, such as the price of the products and the preferences of the consumers. This information can be used to optimize the experimental design by adjusting the factors that have the most significant effects on the response variable.
RSM is used to establish a relationship between the response variable and the input variables. The response variable is the output of the experiment, and the input variables are the factors that influence the response variable. The response surface is a graphical representation of the relationship between the input variables and the response variable as shown in Figure 1. The surface plot provides a visual representation of the relationship between the input variables and the response variable. The contour plot represents the same information as the surface plot but in a two-dimensional form. The contour plot is used to determine the optimum levels of the input variables that result in the highest response variable [34, 35].
Figure 1.
Response surface of essential oil extract yield (A) and antioxidants using ABTS (B) as a function of T and SC generated by quadratic mode. Source: Olabinjo et al. [12].
The pressurized liquid extractor was used to extract the essential oil of dried sweet orange peels of moisture content 6.5% (dry basis) using the central composite design (CCD) model of experiment. It has thirteen experiments of factorial, axial and central point as reported by Olabinjo et al. [12]; Olabinjo and Oliveira [36] as shown in Table 1. The PLE extractor yield result ranged from 21.1 to 49.3% with the mean of 27.7%. The positive axial temperature and negative axial static cycles had the highest yield of 49.3%. The lowest yield of 21.1% was recorded by the negative axial temperature and positive axial static cycles. Trolox was used as a reference standard for the antioxidant using ABTS, and the results were expressed as mg of trolox equivalent (mg TE) by grams of extract. In the analysis of the yield as function of process variables using response surface analysis (RSA), two optimized regions were achieved (Figure 1) as reported by Olabinjo et al. [12]. The result from response surface analysis gave 70°C and 60 min static extraction cycle time as the optimal condition for the maximum yield, while moderate yield was reported at 46°C and 45 min of static extraction cycles. The extraction yield and antioxidant property using ABTS radical scavenging ability under the above conditions were 49.3% and 22.1%; and 11.56 and 11.53 mg trolox g−1 [36]. Best yield occurred when high temperature and number of cycles (C) was used and region with low temperature for a moderate number of circles, high yield was also obtained. Similar observation also occurred with the extracted antioxidant. The fact of using low temperature (T) and fewer batches or cycles (C) and thus achieve good yields and good antioxidant extracts (Figure 1), is characterized as an economic condition in the extraction process, with lower energy consumption and solvent. Thus, the extraction with PLE can present a great advantage over conventional methods, in which sometimes must be used at high temperatures, more solvent and take more extraction time. The low temperature of moderate yield was preferred which may prevent loss of some thermo sensitive compounds in the essential oil less solvent and reduce extraction time, power consumption is less which is economical.
Several biochemical reactions in our body generate reactive oxygen species (ROS) which are capable of damaging crucial bio-materials, if they are not effectively scavenged by cellular constituents and can lead to disease conditions. The harmful action of free radicals can be blocked by antioxidant substances, which can scavenge the free radicals and detoxify the organism. Oboh [37] reported that food materials that exhibited high antioxidant properties have the highest radical scavenging ability.
The response surface model has been shown to be an inevitable tool in the optimization of experiments. The model allows researchers to identify the optimal conditions for the variables of interest, which helps to reduce the number of experiments conducted and saves time and resources. The model can be used in various fields, including biochemical networks, aircraft design, and control systems. Response surface analysis could be explore in all fields to generate optimal condition for all the variables in an experiment.
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Written By
Oyebola Odunayo Olabinjo
Submitted: 23 January 2024Reviewed: 24 January 2024Published: 10 July 2024
Open access peer-reviewed chapter
