104 American Academic Scientific Research Journal for Engineering, Technology, and Sciences ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 http://asrjetsjournal.org/ Construction of Alternative Axial Points Using Standard Axial Points of Central Composite Design Thomas Ugbe a *, Edet Bassey b , Richmond Ofonodo c , Uduakobong Umondak d a,b,c,d Department of Statistics, University of Calabar, Calabar, Nigeria a Email: ugbethomas1@gmail.com , b Email: eddybassey93@yahoo.com c Email: richmondofonodo@yahoo.com Abstract There has been an over-flogged attention given to propositions on how one can make good choice of the existing axial points rather than procedural techniques for constructing axial points about the existing axial points. In order to curb this oversight, this work has constructed axial points about the standard axial points. The construction has given rise to 𝛼* = 0.99k (where k is the number of factors) in comparison to the standard axial points 𝛼 = 𝑓 1 4 (where f is the number of factorial points). Both axial points have been implemented on a central composite design used for maximizing a four-factor process. The constructed axial points produced yields of about 87.211%, better than the yield of 87.187% produced by the standard axial points. Furthermore, the central composite design resulting from the constructed axial points satisfied the D-, A- and E-optimality criteria in comparison to that obtained from the standard or existing axial points. Keywords: Axial points; Optimality Criteria; Factorial points; Response Surface; Central composite design. 1. Introduction Response Surface Methodology (RSM) is one of the frequently used statistical techniques for achieving process optimization. It was developed by [2]. RSM is a collection of mathematical and statistical techniques useful in modeling and analyzing a problem, where a set of controllable factors influence a response , the aim is to optimize the response,( [11,12]) .Response Surface Methodology bases it methods on a supposed set of data containing observations, a response variable y and the independent variables ([10;9]). Response surface designs are designs used to model response surface. These designs can be classified as first-order or second-order design ([3;1]). ------------------------------------------------------------------------ * Corresponding author. about:blank American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 105 First-order design is used to model a response surface when the current operating condition of the process is remote from its optimum [5]. Several first-order designs such as 2 k factorial, Plackett-Burman and simplex designs have been developed overtime; nevertheless, the most commonly used first-order design is the 2 k factorial [7]. In the course of implementing response surface methodology, the process engineer is eventually guided to the optimal region of his process via the method of steepest ascent (or descent) and least squares. In such circumstance, the need for a second-order model becomes imperative to account for system curvature; but fitting a second-order model requires the use of second-order design. Overtime, a variety of second re–order designs such as 3 k factorial design , central composite designs and Box-Behnken designs have been developed; the most commonly used remains the central composite design. Central composite design consists of 2 k factorial points, 2k axial points and n0 center points. Overtime, many researchers such as [6,13], studied the central composite design in response surface analysis. These researchers opined that the selection of axial points is dependent on how many factorial points there are in the design; precisely, 𝛼 = 𝑓 1 2 is known to produce central composite designs that are effective with f denoting factorial points. In this research, we shall construct an alternative axial points about the standard axial points formulated by [2]. 2. Material and Methods 2.1 Standard Central Composite Design A Box-Wilson central composite design, commonly called central composite design contains an imbedded factorial or fractional design with center points that is augmented with a group of axial points that allow estimation of curvature. If the distance from the center of the design space to a factorial point is ± unit for each factor, the distance from the center of the design space to a star point is |𝛼|>1. The precise value of 𝛼 depends on certain properties desired for the design and on the number of factors involved. Furthermore, the axial points are at some distance 𝛼 from the center based on the properties desired for the design and the number of factors in the design. The axial points establish new extremes for the low and high settings for all factors. These designs have circular, spherical or hyper-spherical symmetry and also require 5 levels for each factor. Augment an existing factorial or resolution (v) fractional design with axial points can produce this design. Figure 1 shows a diagram of the standard central composite design, for those situations in which the limits specified for factor settings are truly limits, the inscribed centered central composite design uses the factor settings as the axial points and creates a factorial or fractional design within those limits; in other words, an inscribed central composite design is a scaled down standard central composite design with the level of central composite design divided by 𝛼. This design also requires 5 levels of each factor. Figure 2 shows an inscribed central composite design. 2.2 Face-centered central composite design In the face-centered central composite design, the axial points are at the center of each face of the factorial space, so that 1 . This variant of the standard central composite design requires 3 levels of each factor. However, augmenting an existing factorial or a resolution fractional v factorial design with appropriate axial points can also produce this design. Figure 3 depicts the face-centered central composite design. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 106 Figures 1, 2: Standard central composite design (circumscribed central composite design) and face-centred cube. Figure 3: Inscribed central composite design (ICCD). 2.2 Rotatable design An experimental design is said to be rotatable if the variance of the predicted response �̂� is equal at all points equidistance from the design center. A design categorizing with the attribute leaving v(�̂�) constant shows that the design is rotatable about the center. Central composite design is rotatable if 𝛼 = 𝑓 1 4 with 𝜆𝛼 times observed by individual star point. To this, the design is rotatable if 𝛼 = ( 𝑓 𝜆𝛼 ) 1 4 2.3 Optimality Criteria for Testing Design Efficiency An optimality Criterion is a single number that summarizes how good a design is, and it is maximized or minimized by an optimal design. The D-optimality criterion maximizes the determinant of the Information matrix 𝑋𝑇𝑋 or minimizes the determinant of the dispersion matrix (𝑋𝑇𝑋)−1 . Symbolically, a design is D- optimal if it gives max{det(𝑋𝑇𝑋)} or min{det(𝑋𝑇𝑋)−1}. The G-optimality criterion minimizes the maximum American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 107 variance of the estimated response function min{max d(𝑋, ) } over the design region. The experimenter optimizing a design according to the G-optimality criterion intends to get a good estimate of all the observed responses. The A-optimality criterion minimizes the trace of the dispersion matrix (𝑋𝑇𝑋)−1. Symbolically, A- optimal design is a design that gives min{trace(𝑋𝑇𝑋)−1}. The E-optimality criterion minimizes the maximum eigenvalue of the dispersion matrix (𝑋𝑇𝑋)−1. Symbolically, A-optimal design is a design that gives min{trace((𝑋𝑇𝑋)−1}. The E-optimality criterion minimizes the maximum eigenvalue of the dispersion matrix (𝑋𝑇𝑋)−1. Symbolically, a design is E-optimal if it gives min𝜆−1, where 𝛌 is the largest eigenvalue of the information matrix 𝑋𝑇𝑋. 2.4 Procedure for construction and testing axial points The following steps are used for the construction and testing of axial points: (i) Draw lines from the center points of the design to each factorial point. (ii) Construct perpendiculars to straight lines drawn to the factorial points from the center point. (iii) Mark out the points of intersection on the star or axial axes. (iv) join the constructed axial points to form a crystal (see figure 4) (v) The construction gives rise to 𝛼∗ = 1.988 for a two-factor case-study. By proportionality, a one-factor case study gives 𝛼∗ = 𝛼∗ 2 = 1.988 2 = 0.99 But for a three-factor case study 𝛼∗ = o.99 x3 = 2.97. As a generalization, we therefore have 𝛼∗ = 0.99k, for k factors. (vi) Run central composite design on research case study using the standard axial points and the constructed axial points. (vii) Compare results and established properties. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 108 Figure 4: Diagram showing the constructed axial points. 3. Results and Discussion 3.1 Numerical illustration Case study (Success Foods International, Calabar Municipal Council) The data used was from food success International (Calabar, Municipal Council). The chemical engineer was interested in determining the operating conditions that improves the yield of his process. Four controllable factors influenced process yield. The factors are: temperature, pressure, concentration, and stirring rate. A factorial experiment was carried out in the pilot plant to study how these factors influenced the percentage yield of the product. He was operating the process at an operating condition around a reaction temperature of 24 0 Fahrenheit, reaction pressure of 35 atmosphere, percentage concentration of 155 percent and stirring rate of 75 percent which resulted in yields around 72 percent. Since it was unlikely that this region contained the optimum, a first order model was fitted and the method of steepest ascent applied. He decided that the region of exploration for fitting the first order model should be (19, 29) degrees Fahrenheit, (30, 40) atmosphere of pressure, (150, 160) percent of concentration and (70, 80) percent of stirring rate. To simplify the calculations, we coded the independent variables to a (-1, 1) interval. Thus, if 1  denotes the natural variable temperature, 2  denotes the natural variable pressure, 3  denotes the natural variable concentration and 4  denotes the natural variable stirring rate then the coded variables are: American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 109 5 75 , 5 155 , 5 35 5 24 4 4 3 3 2 2 1 1          xxxx . The data is displayed in Table1 below Table 1: Process data for fitting the first first-order model. Natural variables Coded variables Responses 1  2  3  4  1 x 2 x 3 x 4 x y 19 30 150 70 -1 -1 -1 -1 72.3 19 30 150 80 -1 -1 -1 1 72.8 19 30 160 80 -1 -1 1 1 71.2 19 40 160 80 -1 1 1 1 72.9 29 40 160 80 1 1 1 1 73.1 29 40 160 70 1 1 1 -1 71.9 29 40 150 70 1 1 -1 -1 70.6 29 30 150 70 1 -1 -1 -1 69.9 29 30 160 70 1 -1 1 -1 70.9 19 40 150 80 -1 1 -1 1 67.9 29 30 150 80 1 -1 -1 1 69.9 19 40 160 70 -1 1 1 -1 71.9 29 40 150 80 1 1 -1 1 72.9 29 30 160 80 1 -1 1 1 73.9 19 30 150 70 -1 -1 1 -1 72.9 19 40 150 70 -1 1 -1 -1 68.9 24 35 155 75 0 0 0 0 72.9 24 35 155 75 0 0 0 0 72.9 24 35 155 75 0 0 0 0 72.6 24 35 155 75 0 0 0 0 72.7 24 35 155 75 0 0 0 0 72.8 24 35 155 75 0 0 0 0 72.7 24 35 155 75 0 0 0 0 72.8 24 35 155 75 0 0 0 0 72.9 24 35 155 75 0 0 0 0 72.8 24 35 155 75 0 0 0 0 72.6 24 35 155 75 0 0 0 0 72.8 24 35 155 75 0 0 0 0 72.9 The design used to collect the data is a k2 factorial augmented by 12 center points. Repeat observations at the center were used to estimate the experimental error. The design is centered about the current operating conditions for the process. Using MINITAB, a first order model was fitted to this data by least squares as displayed below. Since overall lack of fit indicates model adequacy of the first-order model. We continue the procedure along the path of steepest ascent. In order to shift from the center of the design on the path of steepest ascent a movement of 0.176 units is made in the direction of 𝑋1, for all 199.0 unit in the direction of 𝑋2, 812.0 unit in the direction of 𝑋3 and 363.0 unit in the direction of 𝑋4. Hence, the path of steepest passed the center of the design with slope 𝑋2 𝑋1 = −0.199 0.176 𝑋3 𝑋1 = 0.812 0.176 , 𝑋4 𝑋1 = 0.363 0.176 A basic step size of five minutes of temperature was employed. With knowledge of the relationship between 1  and 𝑋1 it was observed that a reaction time of five degrees Fahrenheit is the same as one step in the coded American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 110 variable. This implies that the steps along the path of steepest ascent are:  𝑋1 = 1.0000,  𝑋2 = ( −0.199 0.176 ) 𝑋1 = -1.1307,  𝑋3 = ( 0.812 0.176 ) 𝑋1 = 4.6136,  𝑋4 = ( 0.363 0.176 ) 𝑋1 = 2.0625. We computed points on the resulting path steepest ascent, observing responses per point until no obvious response increase was observed. Table 2 below shows our observations for response increased up to step five. But step six produced a response decrease. Table 2: Steepest ascent procedure using the first first-order model. Coded Variables Natural Variables Response X1 X2 X3 X4 1  2 3 4 Y Origin 0.000 0.000 0.000 0.000 24 35 155 75 72.0 ∆ 1.000 -1.131 4.164 2.063 Origin∆ 1.000 -1.131 4.164 2.063 29 29.347 175.818 83.313 73.5 Origin+2∆ 2.000 -2.261 8.327 4.125 34 23.693 196.636 95.625 78.5 Origin+3∆ 3.000 -3.392 12.491 6.188 39 18.040 217.454 105.938 79.6 Origin+4∆ 4.000 -4.523 16.654 8.250 44 12.386 238.272 116.250 81.2 Origin+5∆ 5.000 -5.654 20.818 10.313 49 6.733 259.090 126.563 87.3 Origin+6∆ 6.000 -6.784 24.982 12.375 54 1.079 279.908 136.875 79.4 Therefore, another first-order model had to be fitted within the region about the point ( ξ1 = 49, ξ2 = 6.7325, ξ3 = 259.090, ξ1 = 126.5625). Exploration region about ξ1 was (44, 54), (2,12) about ξ2, (254, 264) about ξ3, and (122, 132) about ξ4. Thus the coded variables were: 𝑋1 = 𝜉1−49 5 , 𝑋2 = 𝜉2−7 5 , 𝑋3 = 𝜉3−259 5 , 𝑋4 = 𝜉4−127 5 , The same design and augmentation was employed (see table 3 below) . In collecting our data we employed a American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 111 42 factorial design technique using 12 center points to augment. In order to estimate the error of experiment, observations were repeated at the design center. Table 3: Process data for fitting the second first-order model. Natural variables Coded variables Responses 1  2  3  4  1 x 2 x 3 x 4 x y 44 2 254 122 -1 -1 -1 -1 87.6 44 2 254 132 -1 -1 -1 1 87.5 44 2 264 132 -1 -1 1 1 87.4 44 12 264 132 -1 1 1 1 87.0 54 12 264 132 1 1 1 1 87.0 54 12 264 122 1 1 1 -1 86.6 54 12 254 122 1 1 -1 -1 87.1 54 2 254 122 1 -1 -1 -1 87.9 54 2 264 122 1 -1 1 -1 87.8 44 12 254 132 -1 1 -1 1 87.0 54 2 254 132 1 -1 -1 1 87.1 44 12 264 122 -1 1 1 -1 87.0 54 12 254 132 1 1 -1 1 87.0 54 2 264 132 1 -1 1 1 87.7 44 2 264 122 -1 -1 1 -1 87.5 44 12 254 122 -1 1 -1 -1 87.2 49 7 259 127 0 0 0 0 87.3 49 7 259 127 0 0 0 0 87.1 49 7 259 127 0 0 0 0 87.2 49 7 259 127 0 0 0 0 87.3 49 7 259 127 0 0 0 0 87.4 49 7 259 127 0 0 0 0 87.2 49 7 259 127 0 0 0 0 87.3 49 7 259 127 0 0 0 0 87.3 49 7 259 127 0 0 0 0 87.2 49 7 259 127 0 0 0 0 87.3 49 7 259 127 0 0 0 0 87.1 49 7 259 127 0 0 0 0 87.2 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 112 The lack of fit test (see Appendix B) indicates that the model does not fit the data at an overall level of significance of P =0.01. This curvature in the true surface indicates that we are near the optimum. At this point, additional analysis had to be done to locate the optimum more precisely. He cannot fit a second-order model in 𝑋1, 𝑋2 , 𝑋3 and 𝑋4 variables with the data in Table 3. So we decided to augment this data with more points to fit a second-order model. To get this done, we got four observation : ( 𝑋1 ± 2.000, 𝑋2 = 0, 𝑋3 = 0, 𝑋4 = 0); ( 𝑋1 = 0, 𝑋2 ± 2.000, 𝑋3 = 0, 𝑋4 = 0); ( 𝑋1 = 0, 𝑋2 = 0, 𝑋3 ± 2.000, 𝑋4 = 0) ( 𝑋1 = 0, 𝑋2 = 0, 𝑋3 = 0, 𝑋4 ± 2.000 ). The complete data set is displayed in Table 4 below. Table 4: Process data for fitting the second-order model using the standard axial point. Natural variables Coded variables Responses 1  2  3  4  1 x 2 x 3 x 4 x y 44 2 254 122 -1 -1 -1 -1 87.6 44 2 254 132 -1 -1 -1 1 87.5 44 2 264 132 -1 -1 1 1 87.4 44 12 264 132 -1 1 1 1 87.0 54 12 264 132 1 1 1 1 87.0 54 12 264 122 1 1 1 -1 86.6 54 12 254 122 1 1 -1 -1 87.1 54 2 254 122 1 -1 -1 -1 87.9 54 2 264 122 1 -1 1 -1 87.8 44 12 254 132 -1 1 -1 1 87.0 54 2 254 132 1 -1 -1 1 87.1 44 12 264 122 -1 1 1 -1 87.0 54 12 254 132 1 1 -1 1 87.0 54 2 264 132 1 -1 1 1 87.7 44 2 264 122 -1 -1 1 -1 87.5 44 12 254 122 -1 1 -1 -1 87.2 49 7 259 127 0 0 0 0 87.3 49 7 259 127 0 0 0 0 87.1 49 7 259 127 0 0 0 0 87.2 49 7 259 127 0 0 0 0 87.3 49 7 259 127 0 0 0 0 87.4 49 7 259 127 0 0 0 0 87.2 49 7 259 127 0 0 0 0 87.3 49 7 259 127 0 0 0 0 87.3 49 7 259 127 0 0 0 0 87.2 49 7 259 127 0 0 0 0 87.3 49 7 259 127 0 0 0 0 87.1 49 7 259 127 0 0 0 0 87.2 59 7 259 127 2.00 0 0 0 87.4 39 7 259 127 -2.00 0 0 0 87.1 49 17 259 127 0 2.00 0 0 87.4 49 -3 259 127 0 -2.00 0 0 87.3 49 7 269 127 0 0 2.00 0 87.5 49 7 249 127 0 0 -2.00 0 87.4 49 7 259 137 0 0 0 2.00 87.4 49 7 259 117 0 0 0 -2.00 87.3 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 113 The Appendix C shows the analysis done using Table 4, and which clearly indicates model adequacy of the second order model. In Table 5 (APPENDIX A) a display of the coded variables accompanied by the constructed axial points is presented. Appendix D shows the analysis done based on Table 5. Clearly the second-order model developed via Table 5 is also adequate for explaining the curvature in the system. The construction produced an approximate axial point of 1.980 for a two-factor process compared to that of the standard axial point which gives an axial point of 1.414. Using proportionality, we have obtained the general relation k99.0*  for producing other axial points for values of k (where k is number of factors) compared to the existing relation 4 F (where F is number of factorial point) which produces axial points. Table 5: Process data for fitting the second-order model using the constructed axial point. Natural variables Coded variables Responses 1  2  3  4  1 x 2 x 3 x 4 x y 44 2 254 122 -1 -1 -1 -1 87.6 44 2 254 132 -1 -1 -1 1 87.5 44 2 264 132 -1 -1 1 1 87.4 44 12 264 132 -1 1 1 1 87.0 54 12 264 132 1 1 1 1 87.0 54 12 264 122 1 1 1 -1 86.6 54 12 254 122 1 1 -1 -1 87.1 54 2 254 122 1 -1 -1 -1 87.9 54 2 264 122 1 -1 1 -1 87.8 44 12 254 132 -1 1 -1 1 87.0 54 2 254 132 1 -1 -1 1 87.1 44 12 264 122 -1 1 1 -1 87.0 54 12 254 132 1 1 -1 1 87.0 54 2 264 132 1 -1 1 1 87.7 44 2 264 122 -1 -1 1 -1 87.5 44 12 254 122 -1 1 -1 -1 87.2 49 7 259 127 0 0 0 0 87.3 49 7 259 127 0 0 0 0 87.1 49 7 259 127 0 0 0 0 87.2 49 7 259 127 0 0 0 0 87.3 49 7 259 127 0 0 0 0 87.4 49 7 259 127 0 0 0 0 87.2 49 7 259 127 0 0 0 0 87.3 49 7 259 127 0 0 0 0 87.3 49 7 259 127 0 0 0 0 87.2 49 7 259 127 0 0 0 0 87.3 49 7 259 127 0 0 0 0 87.1 49 7 259 127 0 0 0 0 87.2 69 7 259 127 3.96 0 0 0 87.1 29 7 259 127 -3.96 0 0 0 87.0 49 27 259 127 0 3.96 0 0 87.2 49 13 259 127 0 -3.96 0 0 87.0 49 7 279 127 0 0 3.96 0 87.3 49 7 239 127 0 0 -3.96 0 87.1 49 7 259 147 0 0 0 3.96 87.2 49 7 259 107 0 0 0 -3.96 87.3 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 114 3.2 Testing optimality criteria 3.2.1 Testing D-optimality criteria Let A and * A denote the design matrices from Table 4 and Table 5 respectively. By the D-optimality criterion A is D-optimal if 3737 ** AAAA TT  * A is D-optimal, otherwise. Now, we have the information matrices from A and * A respectively as follows:                    675676.002703.0027027.0027027.0 02703.0675676.002703.002703.0 027027.002703.0675676.0027027.0 027027.002703.0027027.0675676.0 AA T 0.206531 37  AA T                    307114.102703.0027027.0027027.0 02703.0307114.102703.002703.0 027027.002703.0307114.1027027.0 027027.002703.0027027.0307114.1 ** AA T 911845.2 37 **  AA T Since 3737 ** AAAA TT  , we conclude that 37 ** AA T is maximized. Therefore, the matrix * A is D- optimal in comparison with the matrix A . American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 115 3.2.2 Testing A-optimality criteria Recall that a design is A-optimal if          N trace 1 min AA T . But,                    675676.002703.0027027.0027027.0 02703.0675676.002703.002703.0 027027.002703.0675676.0027027.0 027027.002703.0027027.0675676.0 AA T                    018262.000073.000073.000073.0 00073.0018262.000073.000073.0 00073.000073.0018262.000073.0 00073.000073.000073.0018262.0 37 AA T                          00445.550374427.203718.203718.2 037427.200449.55037427.2037427.2 03718.2037427.200445.5503718.2 03718.2037427.203718.200445.55 37 1 AA T   01784.220 1   AA Ttrace Similarly,                    307114.102703.0027027.0027027.0 02703.0307114.102703.002703.0 027027.002703.0307114.1027027.0 027027.002703.0027027.0307114.1 ** AA T                    035327.000073.000073.000073.0 00073.0035327.000073.000073.0 00073.000073.0035327.000073.0 00073.000073.000073.0035327.0 37 ** AA T American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 116                         34155.28562805.056274.056274.0 562805.034155.28562805.0562805.0 56274.0562805.034155.2856274.0 56274.0562805.056274.034155.28 37 1 ** AA T   3662.113 1**   AA Ttrace Now,   3662.113 37 min 1         AA T trace Therefore, the matrix * A is A-optimal in comparison to the matrix A . 3.2.3 Testing E-optimality criteria Recall that a design matrix A is E-optimal if  1maxmin  . But,                    018262.000073.000073.000073.0 00073.0018262.000073.000073.0 00073.000073.0018262.000073.0 00073.000073.000073.0018262.0 37 AA T 8950.48020452.0 1 11   0386.57017532.0 1 22   0386.57017532.0 1 33   0386.57017532.0 1 44   , 0386.57max 1  Also, American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 117                    035327.000073.000073.000073.0 00073.0035327.000073.000073.0 00073.000073.0035327.000073.0 00073.000073.000073.0035327.0 37 ** AA T 6546.26037517.0 1* 1 * 1   9042.28034597.0 1* 2 * 2   , 9042.28034597.0 1* 3 * 3   9042.28034597.0 1* 4 * 4   , 9042.28max 1  Since,   9042.28maxmin 1  , we conclude that the matrix * A is E-optimal in comparison to the matrix A . 3.3 Testing optimum yields 3.3.1 Testing optimum yield using the standard axial points Now, we have that bXy T 00 2 1ˆˆ 0  β But using the second order model for the standard axial points, we have that                    0143.00889.00736.00139.0 0889.00393.00611.00264.0 0736.00611.00143.00639.0 0139.00264.00639.00107.0 B                     031041.0284432.8177056.239847.7 284432.8241278.014882.0277998.9 177056.214882.08174.26301.13 739847277998.96301.134433.20 1 B American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 118                   0343.0 0074.0 184.0 0241.0 b ,                   320626.0 01748.0 05817.0 10019.1 2 1 bBX 1 0  320626.001748.005817.010019.1 T 0 X 01347.0 2 1  bX T 0 . Therefore, 1865.8701347.02.87 2 1ˆˆ 0  bXy T 00 β 3.3.2 Testing optimum yield using constructed axial points Now, we have that *** 0 * 0 2 1ˆˆ bXy T 0  β , but using the second - order model for the constructed axial points, we have that;                    0004.00897.00728.00147.0 0897.00036.00603.00272.0 0728.00603.00099.00647.0 0147.00272.00647.00131.0 * B                     647137.3905127.7256409.376207.3 905127.7739504.299003.1646095.6 256409.399003.1449458.00059.10 76207.3646095.60059.1089613.8 * 1 B American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 119                  0302.0 0090.0 0811.0 0076.0 * b ,                 165842.0 001089.0 114375.0 45865.0 2 1 *** 0 bBX 1  165842.0001089.0114375.045865.0* T 0 X 00888.0 2 1 **  bX T 0 . Therefore , 2112.870888.03.87 2 1ˆˆ *** 0 * 0  bXy T 0 β . On making comparisons of the results, we observe that the optimum yield via the constructed axial point is better than that obtained via the standard axial point. Table 6: Comparative analysis of the standard and constructed axial points. Test Standard Constructed Optimality criteria D-optimality 0.206531 (Not D-optimal) 2.911845 (D-optimal) A-optimality 220.01784 (Not A-optimal) 113.3662 (A-optimal) E-optimality 57.0386 (Not E-optimal) 28.9042 (E-optimal) Optimum yield 87.1865 87.2112 Alpha value 4 F k99.0*  Optimum yield df= 0.0247 3.4 discussion of results In section 3.2 both the standard and the constructed axial points were used in testing the D-, E-, and A- optimality criteria. In this case, the results showed that whereas the constructed axial points made the central composite design D-, E-, and A-optimal (with respective values of 2.911845, 113.3662, and 57.0386), the standard axial points gave a central composite design that was not D-, E-, and A-optimal (with respective values of 0.206531, 220.0178, and 28.9042. Similarly, in section 3.3 both the standard and the constructed axial points were again used in testing and comparing the optimum yields. Here, the result showed that the optimum yield (of about 87.2112 percent) obtained using the constructed axial point was better than the obtained (87.18653 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 120 percent) via the standard axial points. This is summarized in table 6. 3.5 conclusion This research has constructed axial points about the standard axial points of central composite designs. The construction produced an approximate axial point of 1.980 for a two-factor process compared to that of the standard axial point which gives an axial point of 1.414. The research obtained a general relation k99.0*  for producing other axial points for values of k (number of factors) compared to the existing relation 4 F which produces axial points. The design matrix obtained from the constructed axial point was found to be D-, A-, and E- optimal compared to the design matrix obtained from the standard axial points. The yield obtained from the constructed axial points (87.2112 percent) was observed to be better than that of the standard axial points (87.18653). Both the constructed and standard axial points produced rotatable central composite designs. References [1] A. Andrew. “Statistical details: Design selection”. Journal of Scientific Findings, vol. 14(3), pp.1-30, 2015. [2] G. E. P. Box & K. P. Wilson. “Response surface methodology”. Journal Storage, vol. 3(5), pp. 256-263, 1951. [3] M. Cavazzuh. “Optimization methods: From theory to design”. Springer-Verlag Berlin Heidelberg Journal, vol.2(3), pp. 13-43, 2013. [4] A. I. Khuri & J. A. Cornell. Response Surfaces, design and analysis. 2nd edition, Marcel Dekker Inc, Newyork, 1996 [5] I. A. Khuri & S. Mukhopahyay. “Response surface methodology. Wiley Interdisciplinary Reviews”. Computational Statistics, vol.2(2), pp.128-149, 2010. [6] D. C. Montgomery. Response surface methodology: Design and Analysis of Experiments. USA: Wiley & Sons, 1995, pp. 246-298. [7] D. C. Montgomery. Response surface methodology: Design and Analysis of Experiments. USA: Wiley & Sons, 2013, pp. 478-553. [8] R. H. Myers, D. C. Montgomery and C. M. Anderson-Cook. Response Surface Methodology: Process and Product optimization using designed experiments. USA: 3rd edition Wiley, NY, 2009. [9] T. A. Ugbe, S. S. Akpan & J. E. Usen. “Reduction of Syrup Loss Owing to Frothing in Soft Drinks American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 121 using Response Surface Methodology.” Global Journal of Mathematics, vol. 9(1), pp.663-672, 2017. [10] T. A. Ugbe, S. S. Akpan, U. J. Umondak, I. J. Udoeka & A. O. Ofem. “Response Surface Methodology and its Improvement in the Yield of Pineapple Fruit Drinks.”International Journal of Scientific & Engineering Research, vol.7(1), pp. 541-552, 2016. [11] J. E. Usen, S. S. Akpan, T. A. Ugbe, I. N. Ikpang, J. O. Uket and B. O. Obeten.“Multivariate- Based Technique for Solving Multi-Response Surface Optimization (MRSO) Problems: The Case of a Maximization Problem”. Asian Journal of Probability and Statistics, vol. 11(4), pp. 60-85, 2021. [12] J. E. Usen, E. J. Okoi, E. M. Egomo, E. N. Henshaw & B. E. Hogan. “A Critique on the Foundational Response Surface Methodology for Exploring Optimal Regions.” Asian Journal of Probability and Statistics, vol.8(2), pp.1-16, 2020. [13] C. F.J. Wu & D. Yuan. “Construction of response surface designs for qualitative and quantitative factors.” Journal of Statistical Planning and Inference, vol. 71(2), pp. 331-348, 1998. Appendix a Regression analysis for Table 1 via MINITAB The regression equation is y = 72.1 + 0.176 x1 - 0.199 x2 + 0.812 x3 + 0.363 x4 Predictor Coef StDev T P VIF Constant 72.0646 0.2420 297.75 0.000 x1 0.1755 0.3167 0.55 0.585 1.0 x2 -0.1995 0.3167 -0.63 0.535 1.0 x3 0.8120 0.3167 2.56 0.017 1.0 x4 0.3630 0.3167 1.15 0.263 1.0 S = 1.299 R-Sq = 25.7% R-Sq(adj) = 13.3% American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 122 Analysis of Variance Source DF SS MS F P Regression 4 14.003 3.501 2.08 0.116 Error 24 40.489 1.687 Total 28 54.492 Source DF Seq SS x1 1 0.326 x2 1 0.846 x3 1 10.615 x4 1 2.216 Unusual Observations Obs x1 y Fit StDev Fit Residual St Resid 10 -1.00 67.900 71.241 0.669 -3.341 -3.00R R denotes an observation with a large standardized residual Durbin-Watson statistic = 0.88 Lack of fit test Possible curvature in variable x1 (P = 0.084) Possible interactions with variable x1 (P = 0.094) American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 123 Possible lack of fit at outer X-values (P = 0.000) Overall lack of fit test is significant at P = 0.000 Pure error test - F = 87.66 P = 0.0000 DF(pure error) = 12 15 rows with no replicates Appendix B Regression analysis for Table 3 via MINITAB The regression equation is y = 87.3 + 0.0080 x1 - 0.280 x2 - 0.0330 x3 - 0.0545 x4 Predictor Coef StDev T P VIF Constant 87.2653 0.0317 2756.43 0.000 x1 0.00799 0.04143 0.19 0.849 1.0 x2 -0.27951 0.04143 -6.75 0.000 1.0 x3 -0.03299 0.04143 -0.80 0.434 1.0 x4 -0.05451 0.04143 -1.32 0.201 1.0 S = 0.1699 R-Sq = 66.9% R-Sq(adj) = 61.4% Analysis of Variance Source DF SS MS F P Regression 4 1.39895 0.34974 12.12 0.000 Error 24 0.69278 0.02887 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 124 Total 28 2.09172 Source DF Seq SS x1 1 0.00142 x2 1 1.33217 x3 1 0.01538 x4 1 0.04998 Unusual Observations Obs x1 y Fit StDev Fit Residual St Resid 6 1.00 86.6000 87.0153 0.0907 -0.4153 -2.89R 11 1.00 87.1000 87.5312 0.0876 -0.4312 -2.96R R denotes an observation with a large standardized residual Durbin-Watson statistic = 1.94 Possible lack of fit at outer X-values (P = 0.011) Overall lack of fit test is significant at P = 0.011 Pure error test - F = 6.36 P = 0.0016 DF(pure error) = 12 15 rows with no replicates Appendix C Regression analysis for Table 4 via MINITAB using the standard axial points The regression equation is American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 125 y = 87.2 + 0.0241 x1 - 0.184 x2 - 0.0074 x3 - 0.0343 x4 - 0.0639 x1x2 + 0.0264 x1x3 - 0.0139 x1x4 - 0.0611 x2x3 + 0.0736 x2x4 + 0.0889 x3x4 - 0.0611 x1x2x3 + 0.0736 x1x2x4 + 0.0639 x1x3x4 + 0.0014 x2x3x4 - 0.0107 x1^x1 + 0.0143 x2^x2 + 0.0393 x3^x3 + 0.0143 x4^x4 Predictor Coef StDev T P VIF Constant 87.2417 0.0583 1495.46 0.000 x1 0.02405 0.04078 0.59 0.563 1.0 x2 -0.18428 0.04078 -4.52 0.000 1.0 x3 -0.00739 0.04078 -0.18 0.858 1.0 x4 -0.03428 0.04078 -0.84 0.412 1.0 x1x2 -0.06392 0.04965 -1.29 0.214 1.0 x1x3 0.02642 0.04965 0.53 0.601 1.0 x1x4 -0.01392 0.04965 -0.28 0.782 1.0 x2x3 -0.06108 0.04965 -1.23 0.234 1.0 x2x4 0.07358 0.04965 1.48 0.156 1.0 x3x4 0.08892 0.04965 1.79 0.090 1.0 x1x2x3 -0.06108 0.04965 -1.23 0.234 1.0 x1x2x4 0.07358 0.04965 1.48 0.156 1.0 x1x3x4 0.06392 0.04965 1.29 0.214 1.0 x2x3x4 0.00142 0.04965 0.03 0.977 1.0 x1^x1 -0.01065 0.03569 -0.30 0.769 1.0 x2^x2 0.01435 0.03569 0.40 0.692 1.0 x3^x3 0.03935 0.03569 1.10 0.285 1.0 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 126 x4^x4 0.01435 0.03569 0.40 0.692 1.0 S = 0.2021 R-Sq = 67.3% R-Sq(adj) = 34.6% Analysis of Variance Source DF SS MS F P Regression 18 1.51299 0.08406 2.06 0.068 Error 18 0.73511 0.04084 Total 36 2.24811 Source DF Seq SS x1 1 0.00721 x2 1 0.84573 x3 1 0.00130 x4 1 0.02747 x1x2 1 0.05912 x1x3 1 0.00721 x1x4 1 0.00103 x2x3 1 0.06902 x2x4 1 0.09032 x3x4 1 0.12153 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 127 x1x2x3 1 0.06422 x1x2x4 1 0.08582 x1x3x4 1 0.06646 x2x3x4 1 0.00001 x1^x1 1 0.00355 x2^x2 1 0.00669 x3^x3 1 0.04970 x4^x4 1 0.00660 Unusual Observations Obs x1 y Fit StDev Fit Residual St Resid 5 1.00 87.0000 87.2250 0.1845 -0.2250 -2.73R 32 0.00 87.4000 86.9305 0.1532 0.4695 3.56R 33 0.00 87.3000 87.6676 0.1542 -0.3676 -2.81R R denotes an observation with a large standardized residual Durbin-Watson statistic = 2.53 * Not enough data for lack of fit test Pure error test - F = 13.61 P = 0.0001 DF(pure error) = 12 23 rows with no replicates Appendix D Regression analysis for Table 5 via MINITAB using the constructed axial points American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 128 The regression equation is y = 87.3 + 0.0076 x1 - 0.0811 x2 + 0.0090 x3 - 0.0302 x4 - 0.0647 x1x2 + 0.0272 x1x3 - 0.0147 x1x4 - 0.0603 x2x3 + 0.0728 x2x4 + 0.0897 x3x4 - 0.0603 x1x2x3 + 0.0728 x1x2x4 + 0.0647 x1x3x4 + 0.0022 x2x3x4 - 0.0131 x1x1 - 0.0099 x2x2 - 0.0036 x3x3 - 0.0004 x4x4 Predictor Coef StDev T P VIF Constant 87.2706 0.0581 1503.29 0.000 x1 0.00763 0.03760 0.20 0.841 1.0 x2 -0.08113 0.03760 -2.16 0.045 1.0 x3 0.00900 0.03760 0.24 0.813 1.0 x4 -0.03020 0.03760 -0.80 0.432 1.0 x1x2 -0.06465 0.06390 -1.01 0.325 1.0 x1x3 0.02715 0.06390 0.42 0.676 1.0 x1x4 -0.01465 0.06390 -0.23 0.821 1.0 x2x3 -0.06035 0.06390 -0.94 0.357 1.0 x2x4 0.07285 0.06390 1.14 0.269 1.0 x3x4 0.08965 0.06390 1.40 0.178 1.0 x1x2x3 -0.06035 0.06390 -0.94 0.357 1.0 x1x2x4 0.07285 0.06390 1.14 0.269 1.0 x1x3x4 0.06465 0.06390 1.01 0.325 1.0 x2x3x4 0.00215 0.06390 0.03 0.973 1.0 x1x1 -0.01312 0.01259 -1.04 0.311 1.0 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 129 x2x2 -0.00993 0.01259 -0.79 0.441 1.0 x3x3 -0.00356 0.01259 -0.28 0.781 1.0 x4x4 -0.00037 0.01259 -0.03 0.977 1.0 S = 0.2603 R-Sq = 46.0% R-Sq(adj) = 0.0% Analysis of Variance Source DF SS MS F P Regression 18 1.04119 0.05784 0.85 0.630 Error 18 1.21989 0.06777 Total 36 2.26108 Source DF Seq SS x1 1 0.00144 x2 1 0.32223 x3 1 0.00423 x4 1 0.04338 x1x2 1 0.06201 x1x3 1 0.00809 x1x4 1 0.00135 x2x3 1 0.06658 x2x4 1 0.08723 x3x4 1 0.12498 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 86, No 1, pp104-130 130 x1x2x3 1 0.06190 x1x2x4 1 0.08294 x1x3x4 1 0.06896 x2x3x4 1 0.00006 x1x1 1 0.06081 x2x2 1 0.03960 x3x3 1 0.00535 x4x4 1 0.00006 Unusual Observations Obs x1 y Fit StDev Fit Residual St Resid 1 -1.00 87.6000 87.3091 0.2226 0.2909 2.16R 5 1.00 87.0000 87.2783 0.2235 -0.2783 -2.08R 8 1.00 87.9000 87.5829 0.2229 0.3171 2.36R 15 -1.00 87.5000 87.2271 0.2233 0.2729 2.04R 32 0.00 87.2000 86.7936 0.2352 0.4064 3.64R 33 0.00 87.0000 87.4361 0.2357 -0.4361 -3.95R R denotes an observation with a large standardized residual Durbin-Watson statistic = 2.05 * Not enough data for lack of fit test, Pure error test = 23.91, P = 0.0000 DF(pure error) = 12, 23 rows with no replicates