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

*Sheriff Lamidi, Nurudeen Olaleye, Yakub Bankole, Aishat Obalola, Emmanuella Aribike and Idris Adigun*

## **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.

**Keywords:** response surface methodology, RSM design, optimization, RSM applications, product design

## **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 [2–4]. 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, 10–14].

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 *Applications of Response Surface Methodology (RSM) in Product Design, Development,… DOI: http://dx.doi.org/10.5772/intechopen.106763*

**Figure 1.** *Response surface for humidity and Temperature on human comfort.*

**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.

$$\mathbf{y} = [(\mathbf{x}\_1, \; \mathbf{x}\_2) + \mathbf{c}] \tag{1}$$

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;


## **2. Some useful terminologies**

	- 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

*Applications of Response Surface Methodology (RSM) in Product Design, Development,… DOI: http://dx.doi.org/10.5772/intechopen.106763*


#### **Figure 2.**

*Available Designs in RSM source MINITAB 20.*

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

(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.


## **3. RSM research phases**

RSM involves four broad phases as highlighted below.


## **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.*


#### **Table 1.**

*Some examples of Factors and Response in RSM.*

*Applications of Response Surface Methodology (RSM) in Product Design, Development,… DOI: http://dx.doi.org/10.5772/intechopen.106763*

## **5. Advantages of RSM**

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


## **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 (60–100o C), 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**.

*Applications of Response Surface Methodology (RSM) in Product Design, Development,… DOI: http://dx.doi.org/10.5772/intechopen.106763*

**Figure 4.** *RSM Research Cycle Process.*

	- a. What data is to be collected?
	- b. How to measure it?
	- c. How does the data relate to processing performances and experimental objectives?
	- a. It is time-consuming.
	- b. Can handle one factor over time (OVAT) or one factor at a time (OFAT).
	- c. Interaction between two or more variables cannot be interpreted.

Features of statistical method experimental design


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].


## **7. General applications of RSM**


*Applications of Response Surface Methodology (RSM) in Product Design, Development,… DOI: http://dx.doi.org/10.5772/intechopen.106763*


## **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, 29–35].

*Applications of Response Surface Methodology (RSM) in Product Design, Development,… DOI: http://dx.doi.org/10.5772/intechopen.106763*

#### **Figure 9.** *Interaction plot explored from RSM software.*

**Figure 10.** *Residual plot for optimized point.*

*Applications of Response Surface Methodology (RSM) in Product Design, Development,… DOI: http://dx.doi.org/10.5772/intechopen.106763*

**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.

## **Acknowledgements**

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**

I hereby testify to the good work by Intechopen by way of contributing to research across all the fields from different part of the world. Indeed, there is no better way for the advancement of research than you are doing. Please, keep it up.

## **Author details**

Sheriff Lamidi\*, Nurudeen Olaleye, Yakub Bankole, Aishat Obalola, Emmanuella Aribike and Idris Adigun Lagos State University of Science and Technology (Formerly Lagos State Polytechnic), Lagos, Nigeria

\*Address all correspondence to: lamidi.s@mylaspotech.edu.ng

© 2022 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.

*Applications of Response Surface Methodology (RSM) in Product Design, Development,… DOI: http://dx.doi.org/10.5772/intechopen.106763*

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## **Chapter 4**

## Perspective Chapter: Cyclic Generation of Box-Behnken Designs and New Second-Order Designs

*Nam-Ky Nguyen, John J. Borkowski and Mai Phuong Vuong*

## **Abstract**

Box-Behnken designs (BBDs) are three-level second-order spherical designs with all points lying on a sphere, introduced by Box and Behnken, for fitting the second-order response surface models. They are available for 3–12 and 16 factors. Together with the central composite designs for the second-order model, BBDs are very popular response surface designs, especially for 3–7 factors. This chapter introduces an algorithm to produce cyclic generators for BBDs and similar designs, which we call cyclic BBDs (CBBDs). The new CBBDs offer more flexibility in choosing the designs for a specified number of factors. Comparisons between some BBDs and the new CBBDs indicate the superiority of the new CBBDs with respect to multiple design quality measures and graphical tools assessing prediction variance properties. A catalog of 24 new CBBDs, which includes orthogonally blocked CBBDs for 11, 13, and 14 factors, will be given.

**Keywords:** circulant matrices, foldover designs, interchange algorithm, response surface designs, spherical designs

## **1. Introduction**

Box-Behnken designs (BBDs) are three-level response surface designs (RSDs), introduced by Box and Behnken [1, 2], to fit a second-order response surface model

$$\mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\varepsilon} \tag{1}$$

For *<sup>m</sup>* factors in *<sup>n</sup>* runs. Here, **<sup>y</sup>***<sup>n</sup>*�<sup>1</sup> is a response vector; **<sup>X</sup>***<sup>n</sup>*�*<sup>p</sup>* the model matrix having an intercept term, *m* main effect (ME) terms, *m* quadratic effect (QE) terms, and *m* 2 2-factor interaction (2FI) terms; vector *<sup>β</sup><sup>p</sup>*�<sup>1</sup> of *<sup>p</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup>*<sup>m</sup>* <sup>þ</sup> *m* 2 parameters; and error vector *<sup>ε</sup><sup>n</sup>*�<sup>1</sup> with zero mean and covariance matrix **<sup>I</sup>***σ*2. BBDs are currently available for 3–12 and 16 factors [3]. Except for 11 factors, BBDs can be constructed by superimposing the two-level factorial design onto treatments in each block of a balanced incomplete block design (IBD) or partially balanced IBD. BBDs have the following properties:

i. Each factor has the same number of runs at high (+1) and low (�1) levels;


$$\left(\begin{array}{c|c}\mathbf{M}\_{11} & \mathbf{M}\_{12} \\ \hline \mathbf{M}\_{21} & \mathbf{M}\_{22} \end{array}\right),\tag{2}$$

where **M**<sup>11</sup> is a square matrix of order 1 þ *m*, and **M**<sup>22</sup> is a square matrix of order *<sup>m</sup>* <sup>þ</sup> *<sup>m</sup>* 2 � �. For a BBD, **<sup>M</sup>**<sup>21</sup> <sup>¼</sup> **<sup>0</sup>**, **<sup>M</sup>**<sup>12</sup> <sup>¼</sup> **<sup>0</sup>**<sup>0</sup> and **<sup>M</sup>**<sup>22</sup> <sup>¼</sup> **<sup>D</sup>**, where **<sup>D</sup>** is a diagonal matrix. Matrix **M** in (2) reduces to:

$$
\left(\begin{array}{c|c}\mathbf{M}\_{11} & \mathbf{0}'\\\hline \mathbf{0} & \mathbf{D} \end{array}\right).
\tag{3}
$$

As an example, we construct a 6-factor BBD. Consider an IBD of size ð Þ *v*, *k*, *r* = (6, 3, 4) for six varieties, arranged in blocks of size three, each with three replications per variety. Superimposing a 23 factorial onto the corresponding varieties of this IBD will result in the following 6-factor BBD without center points:


In each row, ð Þ �<sup>1</sup> � <sup>1</sup> � <sup>1</sup> represents the eight points of a 2<sup>3</sup> design and 0 is a column vector of eight 0's. Czyrski and Sznura [4] applied the 6-factor BBD in the optimization of HPLC separation of fluoroquinolones.

Next, we examine a foldover design in 48 runs (with no center points) generated by four cyclic generators: (�1, 0, 0, �1, 1, 0), (0, 1, 0, 0, 1, 1), (0, 0, 1, �1, 0, �1), and (0, 0, �1, �1, 0, 1). The first generator, for example, cyclically generates six design points:

$$
\begin{pmatrix}
0 & -1 & 0 & 0 & -1 & 1 \\
1 & 0 & -1 & 0 & 0 & -1 \\
0 & -1 & 1 & 0 & -1 & 0 \\
0 & 0 & -1 & 1 & 0 & -1
\end{pmatrix}.
$$

*Perspective Chapter: Cyclic Generation of Box-Behnken Designs and New Second-Order Designs DOI: http://dx.doi.org/10.5772/intechopen.107178*

The four cyclic generators produce 24 runs. The next 24 runs are obtained by folding over the first 24 runs (i.e., changing the signs of the factor levels). All points lie on a sphere of radius *<sup>ρ</sup>* <sup>¼</sup> ffiffiffi <sup>3</sup> <sup>p</sup> . It can be shown that these design points are also points in the 6-factor BBD. In this chapter, we call this type of design a cyclic BBD or CBBD.

Each factor of this BBD has half of its runs at the 0-level and the remaining at �1 levels. Now assume that the researchers are looking for an alternative spherical design with fewer 0-levels and more �1 levels for each factor. This allows the experimenter to increase the volume of the spherical design region by increasing the radius associated with CBBD points. This chapter introduces an algorithm that can generate CBBDs of varying radii. Designs with the same number of factors and runs but with different radii are compared with respect to D-criterion values (or *d*-values), variances of the parameter estimates, and the correlation among the main (ME), quadratic (QE), and interaction (2FI) effects. Concepts, such as rotatability, orthogonal blocking, and spherical designs, are well-described in Box and Behnken [2] and textbooks on response surface methodology, such as Myers et al. [5] or Box and Draper [6].

## **2. Calculating the elements of M of a CBBD**

The design matrix *D* of a CBBD has the form *C*<sup>0</sup> <sup>1</sup> … *C*<sup>0</sup> *<sup>r</sup>* <sup>0</sup><sup>0</sup> � �<sup>0</sup> where *C*1, … ,*Cr* are the circulant matrices of order *m* generated by *r* generating vectors *c*1, … ,*cr* and 0 is a matrix containing center points. For the information matrix **M** to have the form in (3), the elements *D* must satisfy the following conditions:

$$\sum\_{u=1}^{n} \mathfrak{x}\_{ui} = \mathbf{0} \left( \forall i \right) \tag{4}$$

$$\sum\_{u=1}^{n} \mathfrak{x}\_{ui} \mathfrak{x}\_{u\circ} = \mathbf{0} \ (i \neq j) \tag{5}$$

$$\sum\_{u=1}^{n} \mathfrak{x}\_{ui}^{2} \mathfrak{x}\_{uj} = \mathbf{0} \ (i \neq j) \tag{6}$$

$$\sum\_{u=1}^{n} \mathfrak{x}\_{ui}^{2} \mathfrak{x}\_{uj} \mathfrak{x}\_{uk} = \mathbf{0} \ (i \neq j \neq k) \tag{7}$$

$$\sum\_{u=1}^{n} \mathfrak{x}\_{ui} \mathfrak{x}\_{uj} \mathfrak{x}\_{uk} = \mathbf{0} \ (i \neq j \neq k) \tag{8}$$

$$\sum\_{u=1}^{n} \mathbf{x}\_{ui} \mathbf{x}\_{uj} \mathbf{x}\_{uk} \mathbf{x}\_{ul} = \mathbf{0} \ (i \neq j \neq k \neq l) \tag{9}$$

where *xui* is the level of the factor *i* for run *u* (Cf. Appendix A of [2]). The condition in (4) implies that *D* is a balanced design; that is, each column of *D* has the same number of þ1 and �1. To make *D* balanced, we just have to restrict the sum of the elements of the generating vectors *c*1, … ,*cr* to 0. As *D* is constructed from the circulant matrices, conditions (5)–(9) can be written as:

$$\sum\_{t=1}^{r} \sum\_{i=1}^{m-1} c\_{ti} c\_{t(i+j)\text{mod }m} = \mathbf{0} \ (\mathbf{1} \le j < m) \tag{10}$$

*Response Surface Methodology - Research Advances and Applications*

$$\sum\_{t=1}^{r} \sum\_{i=1}^{m-1} c\_{ti}^{2} c\_{t(i+j)\text{mod }m} = 0 \ (1 \le j < m) \tag{11}$$

$$\sum\_{t=1}^{r} \sum\_{i=1}^{m-1} c\_{\text{tr}}^2 c\_{t(i+j)\text{mod }m} c\_{t(i+k)\text{mod }m} = 0 \ (1 \le j < k < m) \tag{12}$$

$$\sum\_{t=1}^{r} \sum\_{i=1}^{m-1} c\_{ti} c\_{t(i+j)\text{mod }m} c\_{t(i+k)\text{mod }m} = 0 \ (1 \le j < k < m) \tag{13}$$

$$\sum\_{t=1}^{r} \sum\_{i=1}^{m-1} c\_{ti} c\_{t(i+j)\text{mod }m} c\_{t(i+k)\text{mod }m} c\_{t(i+l)\text{mod }m} = 0 \ (1 \le j < k < l < m) \tag{14}$$

where *cti* is the value of the factor *i* on the generating vector *t*. It can be seen that there are *<sup>m</sup>* � 1 summations in (10) and (11), *<sup>m</sup>*�<sup>1</sup> 2 � � in (12) and (13), and *<sup>m</sup>*�<sup>1</sup> 3 � � in (14). This explains why the lengths of the vectors *Jq* and *J* in Section 3 are <sup>2</sup>ð Þþ *<sup>m</sup>* � <sup>1</sup> <sup>2</sup> *<sup>m</sup>*�<sup>1</sup> 2 � � <sup>þ</sup> *<sup>m</sup>*�<sup>1</sup> 3 � �.

## **3. The CBBD algorithm**

Our CBBD algorithm is the generalization of the algorithm in Nguyen et al. [7] and Pham et al. [8]. Using the results in Section 2, we present the steps of the algorithm for generating a CBBD for *m* factors in *n* ¼ 2*rm* þ *nc* runs (where *nc* is the number of center points) with points on a sphere of radius *ρ*, and <sup>1</sup> <sup>3</sup> *<sup>m</sup>* <sup>≤</sup>*ρ*<sup>2</sup> <sup>&</sup>lt; *<sup>m</sup>* � �.


#### **Remarks**

1.These two steps make up one trial. Among all trials with *f* ¼ 0, we select the CBBD with the highest *D*-criterion value, which is defined as:

$$d\text{-value} = \frac{1}{n} |\mathbf{M}|^{1/p} \tag{15}$$

for the information matrix **M** and the number of parameters *p* for the secondorder model.

2.There are situations, where there is no CBBD with *f* ¼ 0 for particular values of *m*, *ρ*<sup>2</sup> and *r*. In this case, we compute two values *f* <sup>1</sup> and *f* <sup>2</sup>, set *f* <sup>1</sup> equal to the sum of squares of the first 2ð Þþ *<sup>m</sup>* � <sup>1</sup> <sup>2</sup> *<sup>m</sup>*�<sup>1</sup> 2 � � elements of *<sup>J</sup>* (or the first 2ð Þþ *<sup>m</sup>* � <sup>1</sup>

*Perspective Chapter: Cyclic Generation of Box-Behnken Designs and New Second-Order Designs DOI: http://dx.doi.org/10.5772/intechopen.107178*

*m*�1 2 elements of *<sup>J</sup>*) and *<sup>f</sup>* <sup>2</sup> the sum of squares of the remaining elements. A design is selected if *f* <sup>1</sup> ¼ 0, *f* <sup>2</sup> is minimum and the *d*-value in (15) is maximum.

3. If *D* is a foldover design, the sums in Eqs. (11) and (13) will be 0, and the length of the vector *Jq* and *<sup>J</sup>* is shortened to ð Þþ *<sup>m</sup>* � <sup>1</sup> *<sup>m</sup>*�<sup>1</sup> 2 <sup>þ</sup> *<sup>m</sup>*�<sup>1</sup> 3 .

## **4. BBDs and new CBBDs**

**Table 1** displays the quality measures of BBDs whose run sizes (excluding the two center runs) are multiples of the number of factors *m* and 24 CBBDs. **Table 1** does not include two BBDs for *<sup>m</sup>*, *<sup>ρ</sup>*<sup>2</sup> ð Þ = (9, 3) and (16,4) due to their over-abundance of 0-factor levels. This table includes *m* (the number of factors), *ρ*<sup>2</sup> (the square of the radius), *n* (the run size of each BBD which includes two center points), and the quality measures of the designs. These measures are the *d*-value in (15), *v*<sup>Q</sup> , *v*M, and *v*<sup>I</sup> (the maximum scaled variances of the QEs, MEs, and 2FIs, respectively), *r*QQ , *r*QI, *r*MI, and *r*II (the maximum of the absolute values of the correlations between two QEs, between a QE and a 2-FI, between a ME and a 2FI, and between 2FIs, respectively). Note that *r*QM (the correlation between a QE and a ME) and *r*MM (the correlation between two MEs) for all designs in **Table 1** are always zero.

Out of 24 CBBDs in **Table 1**, there are 15 CBBDs with *f* ¼ 0 using the foldover technique with the first half-fraction being balanced with factors having the same number of �1's. The first half-fraction of the CBBDs for 3–7 and for 8–14 factors in this table require four and eight cyclic generators, respectively. Like BBDs, these CBBDs have *r*QI ¼ *r*MI ¼ *r*II ¼ 0. Also, like BBDs, they can be orthogonally blocked, with each half-fraction forming a block. The four CBBDs that are identical to BBDs in terms of quality measures are the ones for 5, 6, 7, and 12 factors. Note that for 3 and 4 factors, the CBBDs have more runs than the corresponding BBDs, and, hence, provide more error degrees of freedom. Also, the 8-factor BBD requires many more runs (nearly 200) than the CBBD. The BBD for 11 factors cannot be orthogonally blocked, and BBDs for 13 and 14 factors are not available. It is necessary to mention that the designs in Nguyen and Borkowski [9] are not the foldover CBBDs in **Table 1**, and as such cannot be blocked in the same way.

There are nine CBBDs for 3–8 factors that are constructed without applying the foldover technique to the first half-fraction. We denote these CBBDs as CBBD\*s. The CBBD\* for three factors requires four cyclic generators, while all others require eight. CBBD\*s for 5–8 factors have *f* <sup>1</sup> ¼ 0 (see Remark 2 of Section 3). These designs cannot be blocked in the same way as the CBBDs in **Table 1**. They can, however, be nearly orthogonally blocked using suitable software (see [10]).

These CBBDs and CBBD\*s offer additional design choices to an experimenter. Comparisons of CBBDs and CCBD\*s to the BBDs for the same number of factors and runs indicate that they, in general, have higher *d*-values, smaller variances of the estimates, and smaller *r*QQ (the correlation between two different quadratic effects). **Figure 1** displays the color cell plots (CCPs) of BBDs for 5–8 factors, that is, 5a, 6a, 7a, and 8a, and the corresponding CBBD\*s with *<sup>ρ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>m</sup>* � 1, that is, 5c, 6b, 7d, and 8f. CCPs, proposed by Jones and Nachtsheim [11], display the magnitude of the correlation between the columns of the model matrix **X** (in terms of the absolute values). The color of each cell ranges from white (no or near-zero correlation) to dark (one or nearone correlation). It can be seen from these CCPs that the information matrices **M** of


*† Each design run size n includes two center runs. All BBDs can be orthogonally blocked except BBDs for m* ¼ 3, 11 *factors (3a and 11a). CBBDs require r* <sup>¼</sup> ð Þ *<sup>n</sup>* � <sup>2</sup> *<sup>=</sup>*2*m cyclic generators. CBBD\*s require r* <sup>¼</sup> ð Þ *<sup>n</sup>* � <sup>2</sup> *<sup>=</sup>m cyclic generators. ‡ The two BBDs for m* ¼ 8, 9 *(8a and 9a) appear in Box and Behnken [1].*

#### **Table 1.**

*Quality measures of BBDs, CBBDs, and CBBD\*s,†.*

the mentioned CBBD\*s do not have the form in (3), but all QEs are orthogonal to all MEs and 2FIs. Note that the BBD for 8 factors has 194 runs, while the corresponding CBBD\* has only 66 runs.

Appendices A and B display the cyclic generators of the CBBDs and CBBD\*s respectively, in **Table 1**.

*Perspective Chapter: Cyclic Generation of Box-Behnken Designs and New Second-Order Designs DOI: http://dx.doi.org/10.5772/intechopen.107178*

**Figure 1.** *CCPs for BBDs and CBBD\*s with <sup>ρ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>m</sup>* � <sup>1</sup> *(m* <sup>¼</sup> 5,6,7,8*).*

## **5. FDS plot and VDG comparisons**

When assessing the prediction properties of an RSD, we want a design that will produce predicted values *<sup>Y</sup>*^ð Þ *<sup>x</sup>*1, … , *xm* with low variance for points ð Þ *<sup>x</sup>*1, … , *xm* in the design space. The prediction variance at ð Þ *<sup>x</sup>*1, … , *xm* is var *<sup>Y</sup>*^ð Þ *<sup>x</sup>*1, … , *xm* � � <sup>¼</sup> *<sup>σ</sup>*<sup>2</sup>**x X**<sup>0</sup> ð Þ **<sup>X</sup>** �<sup>1</sup> **x**0 , where *<sup>σ</sup>*<sup>2</sup> is the error variance and **<sup>x</sup>** is ð Þ *<sup>x</sup>*1, … , *xm* expanded to contain the *m*<sup>2</sup> second-order model terms. Re-scaling by *n=σ*<sup>2</sup> yields the scaled prediction variance *V x*ð Þ¼ 1, … , *xm <sup>n</sup>***x X**<sup>0</sup> ð Þ **<sup>X</sup>** �<sup>1</sup> **x**0 .

Although a design efficiency measure (such as the *d*-value) may provide useful information regarding the overall quality of prediction, it does not provide information regarding the distribution of the prediction variance throughout the design region. This is addressed by studying a design's spherical prediction variance (SPV) properties.

*V<sup>ρ</sup>* is defined to be the *average of the scaled prediction variance* function taken over *Sρ*, the sphere of radius *ρ*. (See [12]) Thus,

$$V\_{\rho} = \frac{1}{\alpha\_{\rho}} \int\_{\mathbb{S}\_{\rho}} V(\mathbf{x}\_1, \dots, \mathbf{x}\_m) \, d\mathbf{x}\_1 \dots d\mathbf{x}\_m \tag{16}$$

where *ωρ* is the surface area of *Sρ*. Also of interest are the *minimum* and *maximum scaled prediction variances* defined as:

$$\text{VMIN}\_{\rho} = \min\_{(\mathbf{x}\_1, \dots, \mathbf{x}\_m) \in \mathbb{S}\_{\rho}} V(\mathbf{x}\_1, \dots, \mathbf{x}\_m) \tag{17}$$

$$\text{VMAX}\_{\rho} = \min\_{(\mathbf{x}\_1, \dots, \mathbf{x}\_m) \in \mathbb{S}\_{\rho}} V(\mathbf{x}\_1, \dots, \mathbf{x}\_m) \tag{18}$$

Fraction of design space (FDS) plots and variance dispersion graphs (VDGs) will be utilized to assess the prediction variance properties of designs in **Table 1**. Giovannitti-Jensen and Myers [13] introduced the VDG, which superimposes plots of *VMAXρ*, *VMIN<sup>ρ</sup>*, and *V<sup>ρ</sup>* against the radius *ρ* within a spherical design space. Modified VDGs that also include the SPV values of *V x*ð Þ 1, … , *xm* for a large set of random points in the spherical region [9] will be presented. Note that the proportion of the volume of the design region is small for values of *ρ* near-zero but increases rapidly with increase *ρ*. Thus, a large proportion of the design space is associated with a relatively small interval *ρ* near the design space boundary. To address this issue, Zahran et al. [14] introduced the FDS plot of the quantiles of *V x*ð Þ 1, … , *xm* against the fraction (or proportion) of the volume of the design region. Unlike single-valued design efficiency measures, both VDGs and FDS plots allow a more thorough assessment throughout the design region. For a summary of graphical methods for assessing the prediction variance properties of RSDs, see Borkowski [15].

Before a comparison of designs using these graphical tools can be made, a critical issue involving factor scaling needs to be addressed. A major difficulty in comparing a BBD to a CBBD or CBBD\* with the same design size *n* is that the design spaces are not the same. For example, consider the BBD with *<sup>m</sup>*, *<sup>ρ</sup>*<sup>2</sup> ð Þ¼ (5 , 2), that is, 5a. Calculation of *vQ* , *vM*, and *vI* is based on the assumption that the design region includes points within the 5-dimensional hypersphere of radius ffiffi 2 <sup>p</sup> . However, for the CBBD\* with *<sup>m</sup>*, *<sup>ρ</sup>*<sup>2</sup> ð Þ¼ (5 , 4), that is, 5c, the

### *Perspective Chapter: Cyclic Generation of Box-Behnken Designs and New Second-Order Designs DOI: http://dx.doi.org/10.5772/intechopen.107178*

calculation of *vQ* , *vM*, and *vI* are based on the assumption that the design region includes points within the 5-dimensional hypersphere of radius ffiffiffi 4 <sup>p</sup> .

Consider the following five-factor experiment presented in Myers et al. [5]. The response to be analyzed is rayon whiteness (RW), which is associated with fabric quality. The experimenters believed that RW can be affected by process variables, which include acid bath temperature in °C (temp1), percent acid concentration (conc1), water temperature in °C (temp2), sulfide concentration (conc2), and amount of chlorine bleach in lb./min (bleach). The experimental levels and the coded levels *x*1, *x*2, *x*3, *x*4, *x*<sup>5</sup> for the five variables are as follows:


**Table 2** shows the 42 design points for the BBD with *<sup>ρ</sup>*<sup>2</sup> <sup>¼</sup> 2, the CBBD with *<sup>ρ</sup>*<sup>2</sup> <sup>¼</sup> 3, and the CBBD\* with *<sup>ρ</sup>*<sup>2</sup> <sup>¼</sup> 4 (designs 5a, 5b, and 5c, respectively). If any 0-factor level is replaced with a value >0 or < 0 in any of these designs, then that point is outside that experiment's design space. There is an important implicit assumption that the fitted model will be appropriate when extrapolating outside the design space. This can be dangerous because it can not only result in predictions with increased bias but also result in larger prediction variances. Whether or not bias is introduced when extrapolating, increasing variances will occur and can be seen in the comparison of VDGs.

Therefore, to make comparisons between designs 5a, 5b, and 5c when choosing a design, it is reasonable to assume that the coded factor levels of �1, 0, 1 representing the same levels when uncoded. This will be true for all design comparisons made for *m* ¼ 3, … ,11 factors in **Table 1**.

We begin our comparisons between designs 5a, 5b, and 5c by generating FDS plots and VDGs over the maximum *<sup>ρ</sup>*2, which are seen in **Figure 2**. For *<sup>m</sup>* <sup>¼</sup> 5, that would be *<sup>ρ</sup>*<sup>2</sup> <sup>¼</sup> 4. In the VDGs, vertical reference lines are placed at *<sup>ρ</sup>* <sup>¼</sup> ffiffi 2 <sup>p</sup> and *<sup>ρ</sup>* <sup>¼</sup> ffiffiffi <sup>3</sup> <sup>p</sup> , which represent the maximum *ρ* for points in designs 5a and 5b, respectively. The FDS plots are based on the distribution of the SPV values for 10,000 randomly selected points in a sphere of radius ffiffiffi 4 <sup>p</sup> . The 10,000 (*<sup>m</sup>* <sup>¼</sup> 4, 5, 6, 7) or 20,000 (*<sup>m</sup>* <sup>¼</sup> 8) SPV values are also plotted in the VDGs (as suggested in [9]).

To compare the five-factor designs, the VDGs in **Figure 2** should be examined over three disjoint intervals for the radius: (i) 0, ffiffi <sup>2</sup> � � <sup>p</sup> , (ii) ffiffi 2 <sup>p</sup> , ffiffiffi <sup>3</sup> � � <sup>p</sup> , and (iii) ffiffiffi <sup>3</sup> <sup>p</sup> , ffiffiffi <sup>4</sup> � � <sup>p</sup> . For (i), the maximum and average SPV is best for the BBD followed by the CBBD\* 5c and CBBD 5b. This should not be surprising because every BBD design point is within ffiffi 2 <sup>p</sup> of the origin. However, for (ii) and (iii), it is clear that the CBBD\* is best for having smaller maximum, average, and minimum SPV values over *ρ*∈ ffiffi 2 <sup>p</sup> , ffiffiffi <sup>4</sup> � � <sup>p</sup> . These plots indicate that the BBD is best only if the experimenter does not plan to predict the mean response at points with *ρ* > ffiffi 2 <sup>p</sup> (such as at ð Þ �1, �1, �1,0,0 or ð Þ �1, �1, �1, �1, 0 ). This seems unrealistic. As stated earlier, if any 0-factor level is changed, then the negative consequences of extrapolation must be acknowledged.


*The first point (row) in each circulant block of five points generates the other four points cyclically. For the BBD and CBBD, the first 20 points are folded over to form the second 20 points. Each design has two center points to form these 42-point designs.*

#### **Table 2.**

*Design points for the five-factor BBD, CBBD, and CBBD\*.*

Predictions based on the BBD at such points are extrapolations leading to larger SPV values. This is reflected in *vQ* , *vM*, *vI* <sup>¼</sup> ð Þ *:*198, *:*063, *:*<sup>250</sup> for the BBD and ð Þ *:*031, *:*068, *:*050 for the CBBD\*. These values indicate that the estimated

*Perspective Chapter: Cyclic Generation of Box-Behnken Designs and New Second-Order Designs DOI: http://dx.doi.org/10.5772/intechopen.107178*

**Figure 2.**

*FDS plots and VDGs for designs with 5 factors (n* ¼ 42*). FDS lines: blue for BBD, green for CBBD, and red for CBBD\*. VDGs include solid black lines for the minimum, average, and maximum SPV. Vertical reference lines are plotted at* ffiffiffi <sup>2</sup> <sup>p</sup> *and* ffiffiffi 3 p *.*

parameter variances associated with the CBBD\* are smaller than those for the BBD for *ρ*∈ ffiffi 2 <sup>p</sup> , ffiffiffi <sup>4</sup> � � <sup>p</sup> . Thus, we would expect better predictions with the CBBD\*. This is supported by the VDGs and the CBBD\* having the largest *d*-value. The CBBD is the least desirable of the *m* ¼ 5 factor designs primarily due to the large *vQ* ¼ *:*208 value.

Using the comparison approach applied to the five-factor designs, we now summarize the comparison of equal-sized designs for *m* ¼ 4,6,7, and 8 factors.

For the four-factor designs with *n* ¼ 34, the FDS plot and VDG in column 1 of **Figure 3** for the CBBD with *<sup>ρ</sup>*<sup>2</sup> <sup>¼</sup> 3, that is, 4c, are superior to the BBD with *<sup>ρ</sup>*<sup>2</sup> <sup>¼</sup> 2, that is, 4b, especially over the interval ffiffi 2 <sup>p</sup> , ffiffiffi <sup>3</sup> � � <sup>p</sup> , where it has the smaller maximum, average, and minimum SPV values. This is expected because no extrapolation occurs over this interval for 4c, while it does for 4b. These plots indicate that design 4b is best only if the experimenter does not plan to predict the mean response at points with *ρ*> ffiffi 2 <sup>p</sup> . This is reflected in the larger *<sup>d</sup>*-value and smaller *vQ* , *vM*, *vI* � � for design 4b.

For the six-factor designs with *n* ¼ 50, the FDS plot and VDG in column 2 of **Figure 3** for the CBBD\* with *<sup>ρ</sup>*<sup>2</sup> <sup>¼</sup> 5 (design 6b) are superior to the BBD/CBBD with *<sup>ρ</sup>*<sup>2</sup> <sup>¼</sup> 3 (design 6a) for most of the design space. The only exception is for a small fraction of the design space, where *ρ*<sup>2</sup> is close to ffiffi 5 <sup>p</sup> and maximum SPV values are larger for design 6b. Despite this small subregion, design 6b has the smaller average and minimum SPV values over the interval ffiffiffi <sup>3</sup> <sup>p</sup> , ffiffi <sup>5</sup> � � <sup>p</sup> , which comprises most of the spherical volume. Design 6b also has a larger *d*-value and smaller *vQ* , *vM*, *vI* � �.

For seven factors, there are four designs with *n* ¼ 58. The FDS plots in **Figure 4** indicate that the CBBD\* with *ρ*<sup>2</sup> = 4, that is, 7b, is the best design over a spherical design space of radius ffiffiffi 6 <sup>p</sup> . The VDGs also indicate that this design has the smallest maximum, average, and minimum SPV values for *ρ*> ffiffiffi <sup>3</sup> <sup>p</sup> , and based on the concentration of SPV values near the maximum for any *ρ*, the distribution of SPV values is highly skewed-left. The experimenter, however, must realize that beyond *ρ*> ffiffiffi 4 <sup>p</sup> , extrapolation occurs for 7b and the experimenter is ignoring the possibility that increased bias may exist with predictions when using the fitted model that results from the experimental data. The VDG and FDS plot for 7c indicates that the geometry of the design points in the design space is poor despite having *<sup>ρ</sup>*<sup>2</sup> <sup>¼</sup> 5. This indicates that in certain cases, a design with a larger *ρ*<sup>2</sup> value does not necessarily guarantee a better design. It is important to note, however, that this case is a rare exception. The BBD/CBBD 7a is rotatable. Therefore, the minimum, maximum, and average SPVs are all equal for a given radius. This is reflected in the single curve in its VDG. The VDG for the *<sup>ρ</sup>*<sup>2</sup> <sup>¼</sup> 5 CBBD\* is truncated at SPV = 240 for scale clarity when making VDG comparisons.

For eight factors, there are three CBBB\* designs with *n* ¼ 66. The FDS plots in **Figure 5** suggest that the CBBD\* with *ρ*<sup>2</sup> = 4 (design 8e) is the best design over a spherical design space of radius ffiffiffi <sup>7</sup> <sup>p</sup> . The VDGs also indicate that this design has the smallest maximum and average SPV values for *ρ*> ffiffiffi <sup>3</sup> <sup>p</sup> . It is important to remind the experimenter that between *<sup>ρ</sup>* <sup>¼</sup> ffiffiffi 4 <sup>p</sup> and ffiffiffi <sup>7</sup> <sup>p</sup> , extrapolation is occurring for 8e. Thus, although 8e appears better than 8f, there may be increased bias with any prediction associated with using a fitted model for 8e in comparison with 8f over this interval. Note that although the minimum SPV curve for *<sup>ρ</sup>*<sup>2</sup> <sup>¼</sup> 7 CBBD\* (design 8f) is the lowest for *ρ*> ffiffiffi <sup>3</sup> <sup>p</sup> , it is associated with only a small fraction of the design space as evidenced by the sparsity of points near the minimum. The VDG for the *<sup>ρ</sup>*<sup>2</sup> <sup>¼</sup> 3 CBBD\* is truncated at SPV = 375 for scale clarity when making VDG comparisons.

Based on the comparisons for *m* ¼ 7and 8 factors, the design with the largest *d*-value is not necessarily the best design when using FDS plots and VDGs as criteria. A larger *d*-value does not ensure a good distribution of SPV values throughout the design space. It should be noted that the best design based on the FDS plots and VDGs always had the smallest *vQ* value. That is, those designs are associated with the smallest estimated variances for the quadratic effects (QEs).

What is clear in these comparisons is that there exists a CBBD or a CBBD\* that is superior to every BBD of the same size based on *d*-values, FDS plots, and VDGs. This is most likely due to the over-abundance of 0-factor levels in BBDs leading to poor prediction for larger radii.

*Perspective Chapter: Cyclic Generation of Box-Behnken Designs and New Second-Order Designs DOI: http://dx.doi.org/10.5772/intechopen.107178*

*FDS plots and VDGs for designs with 4 and 6 factors. FDS lines for m* <sup>¼</sup> <sup>4</sup>*: blue for <sup>ρ</sup>*<sup>2</sup> <sup>¼</sup> <sup>2</sup> *CBBD and red for <sup>ρ</sup>*<sup>2</sup> <sup>¼</sup> <sup>3</sup> *CBBD. FDS lines for m* <sup>¼</sup> <sup>6</sup>*: blue for <sup>ρ</sup>*<sup>2</sup> <sup>¼</sup> <sup>3</sup> *BBD and red for <sup>ρ</sup>*<sup>2</sup> <sup>¼</sup> <sup>5</sup> *CBBD\*. VDGs include solid black lines for the minimum, average, and maximum SPV. A vertical reference line is plotted at* ffiffiffi <sup>2</sup> <sup>p</sup> *for m* <sup>¼</sup> <sup>6</sup> *and at* ffiffiffi <sup>3</sup> <sup>p</sup> *for m* <sup>¼</sup> <sup>6</sup>*.*

#### **Figure 4.**

*FDS plots and VDGs for designs with 7 factors (n* ¼ 58*). FDS lines: blue for BBD/CBBD, green, magenta, and red for CBBD\*s with <sup>ρ</sup>*<sup>2</sup> <sup>¼</sup> 3,4,5,6*, respectively. VDGs include solid black lines for the minimum, average, and maximum SPV. Vertical reference lines are plotted at* ffiffiffi 3 p *,* ffiffiffi 4 p *and* ffiffiffi 5 p *.*

*Perspective Chapter: Cyclic Generation of Box-Behnken Designs and New Second-Order Designs DOI: http://dx.doi.org/10.5772/intechopen.107178*

**Figure 5.**

*FDS plots and VDGs for designs with 8 factors (n* ¼ 66*). FDS lines: blue, green, and red for CBBD\*s with <sup>ρ</sup>*<sup>2</sup> <sup>¼</sup> 3,4,7*, respectively. VDGs include solid black lines for the minimum, average, and maximum SPV. Vertical reference lines are plotted at* ffiffiffi 3 p *and* ffiffiffi 4 p *.*

## **6. Conclusions**

This chapter offers the cyclic-generating approach to create new designs (CBBDs and CBBD\*s) as alternatives to existing BBDs. Our new designs offer a compromise between the definitive screening designs [11] (where each factor has just three 0's) and BBDs (where the number of 0's for each factor is more than the number of �1's). In addition to quality measures, FDS plots and VDGs were generated to assess the prediction variance properties in ð Þ *m* � 1 -dimensional spherical regions. These were used to compare designs of equal size but with varying *ρ*2. The comparisons indicate that for each number of design factors *m*, there exists a CBBD or CBBD\* that is superior to a BBD based on these quality measures and graphical methods. Because of extrapolation concerns related to points extending beyond the maximum value of *ρ* associated with a design, it is stressed that comparisons of BBDs to CBBDs or CBBD\*s should take into account for the differences in the spherical design regions based on differing *ρ*<sup>2</sup> values. Once implemented, experimental data resulting from a CBBD or CBBD\* can be analyzed analogously to a data analysis for a BBD using currently available statistical software. A catalog of the RSDs in **Table 1**, which includes 15 CBBDs and nine CBBD\*s is given at the link https://designcomputing.net/cbbd/.

## **Appendix A. Cyclic generators for the first half-fractions of 15 CBBDs in Table 1**


*Perspective Chapter: Cyclic Generation of Box-Behnken Designs and New Second-Order Designs DOI: http://dx.doi.org/10.5772/intechopen.107178*


## **Appendix B. Cyclic generators for 9 CBBD\*s in Table 1**

## **Author details**

Nam-Ky Nguyen<sup>1</sup> \*, John J. Borkowski<sup>2</sup> and Mai Phuong Vuong<sup>3</sup>

1 Vietnam Institute for Advanced Study in Mathematics, Hanoi, Vietnam


\*Address all correspondence to: nknam@viasm.edu.vn

© 2022 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.

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## Section 3
