1. Introduction

Two popularly used Response Surface Methodology (RSM) designs in modeling second order effects are the Box-Benhken and Box-Wilson designs. Each of them has less number of experimental runs when compared with the three-level full factorial designs. The Box-Benhken design abbreviated BBD was introduced by Box and Benhken (1960) and constitutes an alternative to the Box-Wilson design (otherwise called Central Composite Design (CCD)) introduced by Box and Wilson (1951). Although some categories of the Box-Wilson designs are not rotatable, Box-Benhken designs are a class of rotatable or near-rotatable second-order designs based on three-level incomplete factorial designs. They are formed by combining two-level factorial designs with incomplete block design in a particular fashion. They are designs introduced for second-order models in order to limit the growing sample size as the number of model parameters increases. When compared with the CCDs, the BBDs have the advantage of reduced design points. In general, the number of design points of the BBDs is where is the number of factors and is the number of center points. On the other hand, the number of design points of the CCDs is where remains the number of factors and remains the number of center points.

The CCDs can be studied for however, the BBDs do not exist for and can be studied for In the simplest case of the number of design points of the Box-Benhken design is 12 plus center points and the number of design points of the central composite design is 14 plus center points. For the number of design points of the Box-Benhken design is 24 plus center points and the number of design points of the central composite design is 24 plus center points. For the number of design points of the Box-Benhken design is 40 plus center points and the number of design points of the central composite design is 42 plus center points. For the number of design points of the Box-Benhken design is 60 plus center points and the number of design points of the central composite design is 76 plus center points. The advantage in reduced design size becomes more pronounced as increases beyond seven. For instance, when , the number of design points of the Box-Benhken design is 180 plus center points and the number of design points of the central composite design is 1,044 plus center points.

Myres et.al (2009) observed that in many scientific studies that require Response Surface Methodology, researchers are inclined to require three evenly spaced factor levels of which Box-Benhken design is an efficient option and an important alternative to the central composite design Box-Benhken designs like some central composite designs are spherical and do not deviate substantially from being rotatable. Many standard ways of comparing designs, ranging from the use of numeric values to graphs, exist in Response Surface Methodolgy. Lucas (1976) compared Response Surface designs by their D- and G-efficiency values. A Box-Benhken design has been compared with a Uniform shell design of Doehlert (1970) in Crosier (1993). It was seen that the Box-Benhken design exhibited a very high G-efficiency value of 98.9% as against 59.7% of the Uniform shell design. Also, Box-Benhken design gave a better prediction at the perimeter of the design region while the Uniform shell design gave a better prediction near the center of the design region. Although the Uniform shell design generally requires fewer runs then CCDs and BBDs, they are not widely used in fitting second-order model. Crosier (1993) further compared the performance of three-factor central composite design having center points with the three-factor Box-Benhken design having center points. The Box-Benhken design performed better near the design center while the central composite design performed better near the perimeter.

Zolgharnein et.al (2013) carried out a comparative study of Box-Benhken, central composite and Doehlert matrix for multivariate optimization. Practical applications of Box-Benhken and Central Composite designs are numerous (see e.g Tekindal et.al (2012), Igder et.al (2012), Zolgharnein et.al (2013) and Sabela et.al (2014)). In this work, the effects of addition of center points on the optimality of Box-Benhken and Box-Wilson second-order designs shall be the focus. The aim is to see how increasing the number of center runs added to Box-Benhken and Box-Wilson designs affects A-, D-, E-, G- and T-optimality values for the designs.

2. Methodology

For a p parameter polynomial in k factors, each k-factor, N-point Box-Benhken and Box-Wilson second-order response surface design, written as an Nxk matrix, shall be studied for changes in A-, D-, E-, G- and T-optimality values when the center points, are increased. To estimate the parameters (coefficients) of the second-order polynomial, the matrix of the design shall be expanded into an Nxp model matrix having one column for each parameter of the polynomial model. The moment matrix or information matrix of the design shall be obtained. In order to compare designs of varying sizes, the information matrix shall be normalized to remove the effect of the varying design sizes. Although Myres et.al (2009) recommends the use of three to five center runs, this study shall consider Three categories of the Box-Wilson design shall be investigated alongside the Box-Benhken design. The categories of Box-Wilson design are the central composite designs with respective axial distance and where The designs shall be assessed with respect to statistical criteria, namely, A-, D-, E-, G- and T-optimality criteria, which are related to the variance-covariance matrix of the model parameter estimator. The chosen optimality criteria are commonly encountered in literatures on optimal design of experiment.

Rady et al. (2009) presented a survey on these optimality criteria as well as the relationships among them. As in literature, by A-optimality, a design in which the sum of the variances of the model coefficients is minimized is sought. It is defined as

where Min implies that minimization is over all designs and tr represents trace.

By D-optimality, a design in which the determinant of the moment matrix

is maximized over all designs is sought. Where represents the model matrix associate with the D-optimal design and represents its transpose. The criterion of D-optimality equivalently minimizes the determinant of

By E-optimality criterion, a design which maximizes the minimum eigen value of M or equivalently minimizes the maximum eigen value of is sought. By E-optimality, a design which minimizes the maximum variance of all possible normalized linear combination of parameter estimates is sought. E-optimality criterion is defined by

where and represent minimum eigen value and maximum eigen value, respectively. By G-optimality, a design which minimizes the maximum scaled prediction variance in the region of the design is sought. It is defined by

By T-optimality, a design which maximizes the trace of the information matrix is sought. It is defined as

where Max implies that maximization is over all designs and tr represents trace.

The second-order model to consider is the complete model having main effects, interaction effects and quadratic effects and is given as

To effectively compare the designs, the information matrix of any given design shall be normalized. By normalization, the effect of changing sample size shall be removed. In studying the effects of addition of center points on the optimality of the BBDs and CCDs, we shall consider , whose design sizes are moderately small and relatively close.

3. Results

The results of the investigation regarding the effects of center point on optimality of Box-Behnken and Box-Wilson designs are contained in Table 1. The first column entries on the table show the design type. The second column entries on the table show the number of controllable factors of experimentation. The third column entries on the table show the number of model parameters associated with the k variable model. The columns under Optimality Constant represent values obtained using A-, D-, E-, G- and T-optimality criteria. The design matrices are built from the designs and the models. For an illustration, assuming Box-Behnken design in k=3 variables, the associated 10-parameter model is

Table 1. Optimality Constants for the BBD and CCD

The 16-point Box-Behnken design is

With the model and the 16-point design, the associated model matrix is

The corresponding normalized information matrix is

and its inverse matrix is

The determinant value of information matrix is 3.0518e-005.

The determinant value of its inverse matrix is 32768.

The trace of the information matrix is 4.7500.

The trace of its inverse matrix is 34.

The eigenvalues of the information matrix are

0.1340

0.2500

0.2500

0.2500

0.2500

0.2500

0.5000

0.5000

0.5000

1.8660

The eigenvalues of the inverse matrix are

0.5359

2.0000

2.0000

2.0000

4.0000

4.0000

4.0000

4.0000

4.0000

7.4641

Each point of the design, apart from the center point, has variance of predicted value of 12.0. The variance of predicted value at the center point is 4.0. Each design follows these unique presentations for defined k, p and N. MATLAB R2007b was used in all computations and the results are as presented in Table 1. The MATLAB outputs for BBD, k= 5, p=21, N=43, CCD, k = 3, α =1.682, N=15 and k = 3, p = 10, α = 1.7321, N = 15 have been included in the Appendix. Presentation for only three categories is purely for space management.

4. Discussion of Results

The results on the effects of addition of center points on the optimality of Box-Benhken and Box-Wilson second-order designs show some interesting relationships. For the Box-Benhken designs considered, the trace of the variance-covariance matrix generally decreases as increases, thus showing a negative linear relationship. This is also true for the central composite designs defined at and For the central composite design defined at α = 1.0, the trace of the variance-covariance matrix increases as increases, thus showing a positive linear relationship. Trace of the variance-covariance matrix for the three-variable Box-Benhken design is minimized when For the four-variable Box-Benhken design, trace of the variance-covariance matrix is minimized when and for the five-variable Box-Benhken design, trace of the variance-covariance matrix is minimized when For the central composite design defined at the trace of the variance-covariance matrix is minimized when However, for the central composite design defined at and the trace of the variance-covariance matrix is minimized when

On D-optimality criterion, the determinant of the information matrix of the Box-Benhken design generally decreases as increases. Equivalently, the determinant of the variance-covariance matrix of the Box-Benhken design generally increases as increases. These are also true for all categories of the central composite design. For all cases considered, the determinant of information matrix is maximized at except for the three-variable Box-Benhken design where the determinant of information matrix is maximized at The minimum eigenvalue of the Box-Benhken design and the central composite design defined at and is maximized when and maximized when for the central composite design defined at α = 1.0. The maximum scaled predictive variance associated with the Box-Benhken designs and the Box-Wilson designs generally increases as increases and minimized when for CCD defined at when for CCD defined at and and the Box-Benhken design. For the three-variable 10-parameter Box-Benhken design, the minimum variance equals 10.5 which is approximately close to the number of model parameters. For the four-variable 15-parameter Box-Benhken design, the minimum variance equals 15.1667 which is approximately close to the number of model parameters. For the five-variable 21-parameter Box-Benhken design, the minimum variance equals 21.0 which is exactly the number of model parameters.

The closeness of the minimum variance of prediction to the number of model parameter associated with the CCDs defined at was not obvious, however for CCDs defined at and and for k=3 and k=4, near approximations exist. As in the literature, maximum value of the predictive variance equal tp or approximately equal to the number of model parameters show that the corresponding design is rotatable or near-rotatable. The trace of information matrix associated with all considered design types consistently decreases as increases. Generally, the relationships between addition of center points and the optimality properties are very strong between the Box-Benhken designs and the central composite design defined at and This is most likely due to the sphericity and rotatabilty properties of the three design types.

5. Conclusions

The effects of addition of center points on the optimality of Box-Benhken and Box-Wilson second-order designs have been investigated. Relationships are seen to exist between optimal design properties and changing size of the designs by the addition of center points. The relationships are very strong between the Box-Benhken designs and the central composite design defined at and Variations seem to exist with central composite designs defined at

Appendix