For the corresponding weights, Equation (8) is used by setting $p=0$ in Equation (9) matrix $B_d$ is obtained and for optimal variance, use Equation (10),

$B_d=UC{U}^{\prime }$.(11)

Factorial and axial weight corresponding to D-optimal are $D_{fw}=\frac{\sqrt{b_{11}}}{{{\displaystyle \sum }}_{j\le N}\sqrt{b_{jj}}}$ and $D_{aw}=\frac{\sqrt{b_{NN}}}{{{\displaystyle \sum }}_{j\le N}\sqrt{b_{jj}}}$ respectively with $b_{11},\cdots ,b_{NN}$ being diagonal entries of matrix $B_d$. Replicating the factorial and axial parts as per the weights gives the design matrix $X_d$. The D-optimal moment matrix $C_d=\frac{{X^{\prime }}_d{X}_d}{N_d}$ with $N_d$ being the number of runs in the design. The corresponding optimal variance for the design is given as $V\left({\varphi }_0\right)={\left(trace{C}_d\right)}^{\frac{1}{S}}$. S being the number of parameters in the model. Given two designs $X_1$ and $X_2$, their relative efficiency as per D-criterion is;

$D_e={\left[\frac{\left|{\left({X^{\prime }}_2{X}_2\right)}^{-1}\right|}{\left|{\left({X^{\prime }}_1{X}_1\right)}^{-1}\right|}\right]}^{\frac{1}{S}}$.(12)

S is the number of model parameters [1] . Relative efficiency of the general to D-optimal design is

${\varphi }_{Deff\left(\xi \right)}=\frac{{\left(\det C\right)}^{\frac{1}{S}}}{{\left(\det C_d\right)}^{\frac{1}{S}}}$. (13)

E-optimality aims at minimizing the largest eigen value of ${\left(X^{\prime }X\right)}^{-1}$ [18] . Let the minimum eigen value of the general design be ${\lambda }_{\min }\left(C\right)$ and the normalized eigen vector be Z. If ${\lambda }_{\min }\left(C\right)$ has a multiplicity of one, then matrix $E=\frac{z{z}^{\prime }}{z}$ such that trace(E) = 1 and ${x^{\prime }}_{i}E{x}_i\le {\lambda }_{\min }\left(C\right)$ for all $x_i\in \chi$ for ${\varphi }_{-\infty }$-optimal is used to determine the E-optimal weights for factorial and axial parts respectively [17] . The optimal variance is the minimum eigen value of the resulting moment matrix of the design i.e. ${\lambda }_{\min }\left(C_e\right)$ where $C_e=E^{\prime }E/N$ and N is the number of rows of matrix E. E-efficiency becomes

${\varphi }_{Eeff}=\frac{{\lambda }_{\min }\left(C\right)}{{\lambda }_{\min }\left(C_e\right)}$.(14)

Substituting $p=-1$ in Equation (9), matrix $B_a$ for computing the A-Optimal weights is

$B_a=U{U}^{\prime }$. (15)

The factorial ( $A_{fw}$) and axial ( $A_{aw}$) weights are $\frac{\sqrt{b_{11}}}{{{\displaystyle \sum }}_{j\le N}\sqrt{b_{jj}}}$ and $\frac{\sqrt{b_{NN}}}{{{\displaystyle \sum }}_{j\le N}\sqrt{b_{jj}}}$ respectively with $b_{11},\cdots ,b_{NN}$ being diagonal entries of $B_a$. A-optimal design is by replicating the factorial and axial parts as per the weights to give the design matrix $X_a$. The moment matrix $C_a=\frac{{X^{\prime }}_a{X}_a}{N_a}$ with $N_a$ runs in the design. Optimal variance for the design is $V\left({\varphi }_{-1}\right)={\left(\frac{1}{15}trace{C}_a^{-1}\right)}^{-1}={\left(\frac{1}{15}{\left({\displaystyle {\sum }_{j\le N}\sqrt{b_{jj}}}\right)}^2\right)}^{-1}$(16)

Relative efficiency of the general to the A-Optimal design is

${\phi }_{Aeff}=\frac{{\left(\frac{1}{15}trace{C}^{-1}\right)}^{-1}}{{\left(\frac{1}{15}trace{C}_a^{-1}\right)}^{-1}}$.(17)

By putting $p=1$ in Equation (9), matrix $B_t$ is given as

$B_t=U{C}^2{U}^{\prime }$.(18)

Factorial ( $T_{fw}$) and axial ( $T_{aw}$) weights are $\frac{\sqrt{b_{11}}}{{{\displaystyle \sum }}_{j\le N}\sqrt{b_{jj}}}$ and $\frac{\sqrt{b_{NN}}}{{{\displaystyle \sum }}_{j\le N}\sqrt{b_{jj}}}$ respectively with $b_{11},\cdots ,b_{NN}$ being diagonal elements of matrix $B_t$. T-optimal design is by replicating the factorial and axial parts as per the weights to give the design matrix $X_t$ which is employed to obtain the moment matrix $C_t=\frac{{X^{\prime }}_t{X}_t}{N_t}$ with $N_t$ being the number of runs in the design. Corresponding optimal variance is

$\begin{array}{c}V\left({\varphi }_1\right)={\left(\frac{1}{S}trace{C}_t^1\right)}^1\\ ={\left(\frac{1}{S}{\left({\displaystyle {\sum }_{j\le N}\sqrt{b_{jj}}}\right)}^2\right)}^{1}32\times 2+8\times 3\\ =88\end{array}$ (19)

The T-efficiency for T-optimal design is

$\frac{\varphi \left(C\right)}{\varphi \left(C_t\right)}=\frac{\left\{\frac{1}{S}traceC\right\}}{\left\{\frac{1}{S}trace{C}_t\right\}}$.(20)

Factorial and axial weights are $D_{fw}=0.02387036$ $D_{aw}=0.02951858$. Hence a D-optimal design has factorial part replicated twice i.e. ( $n_f=2$) and axial part thrice (i.e. $n_a=3$) with matrix X having runs. The corresponding D-optimal moment matrix $M_d=\frac{1}{88}{X}^{\prime }X$, as shown in Table 2 .

The D-optimal value is $V\left({\varphi }_0\right)={\left(\det M_d\right)}^{1/s}$ i.e. $V\left({\varphi }_0\right)=0.6965612$, the general design value is 0.6796529 according to [15] . Relative efficiency of the general design to the D-optimal design is

${\varphi }_{eff\left(\xi \right)}=\frac{0.6796529}{0.6965612}=97.71\%\cong 98\%$.(21)

The smallest eigen value of general design is ${\lambda }_{\min }\left(M_G\right)=0.002893284$ with a normalized eigenvector:

$Z^{\prime }=\left[0.895\begin{array}{cccccccccccccc}0.00& 0.00& 0.00& 0.00& -0.233& 0.00& 0.00& 0.00& -0.233& 0.00& 0.00& -0.233& 0.00& -0.233\end{array}\right]$

Matrix $E=\frac{z{z}^{\prime }}{z}$ of trace one and $x^{\prime }Ex\le {\lambda }_{\min }\left(M_G\right)$ for all $x\in \chi$ for ${\varphi }_{-\infty }$-optimal for all $x_i$ where $i=1,\cdots ,32$, such that $x^{\prime }Ex=0.0009351244$ and for all $x_i$ and $i=33,\cdots ,40$. $x^{\prime }Ex=0.01089277>{\lambda }_{\min }\left(M_G\right)=0.002893284$. E-optimal design allocates a weight of 1 to factorial and 0 to the axial part of the design. The ${\varphi }_{-\infty }$-optimal design moment matrix is $M_e=\frac{1}{32}{X}^{\prime }X$, see Table 3 . $M_e$ an optimal variance of 0.4182000. The E-efficiency is given by

${\varphi }_{Eeff}=\frac{{\lambda }_{\min }\left(C\right)}{{\lambda }_{\min }\left(C_e\right)}=\frac{0.002856958}{0.4182000}=0.0068315590626\cong 1\%$

The factorial and axial weights are 0.01704711 and 0.05681391 respectively. The optimal design is formed by setting $n_f=2$ and $n_a=6$ with $32\times 2+8\times 6=112$ being the total number of runs and A-optimal moment matrix is $M_A=\frac{1}{112}{X}^{\prime }X$.

The A-Optimal value is $V_{-1}={\left(\frac{1}{15}trace{M}_a^{-1}\right)}^{-1}=0.05798174$ while the optimal value for the general design is 0.04154701. Giving a relative efficiency of $0.71655334938\cong 71.7\%$.

Factorial, $T_{fw}=0.01691205$ and axial $T_{aw}=0.05735179$ weights after setting $p=1$ in Equations (9). Again the T-optimal design is formed by replicating factorial twice ( $n_f=2$) and axial six times ( $n_a=6$) for a total of 112 runs, with a moment matrix as shown in Table 4 . T-optimal value $V_1=\frac{1}{15}trace\left(M_G\right)=1.29828$, giving a relative efficiency of $\frac{1.136227}{1.29828}\cong 87.5\%$.