Question 14.3: Equation 14.9 gives the empirical model of the response surf...
Equation 14.9 gives the empirical model of the response surface for the data in Table 14.4 when the factors are in coded form. Convert the equation to its uncoded form.
R=15.0+2.0A^{\star}+5.0B^{\star}+0.5A^{\star}B^{\star} (14.9)
Table 14.4 Example of Uncoded and Coded Factor Levels and Responses for a 2² Factorial Design
| Run | A | B | A* | B* | Response |
| 1 | 15 | 30 | +1 | +1 | 22.5 |
| 2 | 15 | 10 | +1 | –1 | 11.5 |
| 3 | 5 | 30 | –1 | +1 | 17.5 |
| 4 | 5 | 10 | –1 | –1 | 8.5 |
To convert the equation to its uncoded form, it is necessary to solve equation 14.3
x_{\mathrm{f}}=c_{\mathrm{f}}+d_{\mathrm{f}}x_{\mathrm{f}}^{\star} (14.3)
for each factor. Values for c_{\mathrm{f}} and d_{\mathrm{f}} are determined from the high and low levels for each factor; thus
\begin{array}{cl} c_{\mathrm{A}}=\frac{H_{\mathrm{A}}+L_{\mathrm{A}}}{2}=\frac{15+5}{2}=10 & d_{\mathrm{A}}=H_{\mathrm{A}}-c_{\mathrm{A}}=15-10=5 \\ c_{\mathrm{B}}=\frac{H_{\mathrm{B}}+L_{\mathrm{B}}}{2}=\frac{30+10}{2}=20 & d_{\mathrm{B}}=H_{\mathrm{B}}-c_{\mathrm{B}}=30-20=10\end{array}
Substituting known values into equation 14.3,
x_f=c_f+d_fx^*_f \quad 14.3\\A=10+5 A^{*} B=20+10 B^{\star}
and rearranging, gives
A^{\star}=0.2 A-2 \quad B^{\star}=0.1 B-2
Substituting these equations into equation 14.9 gives, after simplifying, the uncoded equation for the response surface.
\begin{aligned} R & =15.0+2(0.2 A-2)+5(0.1 B-2)+0.5(0.2 A-2)(0.1 B-2) \\ & =15.0+0.4 A-4+0.5 B-10+0.01 A B-0.2 A-0.1 B+2 \\ & =3.0+0.2 A+0.4 B+0.01 A B \end{aligned}
We can verify this equation by substituting values for A and B from Table 14.4 and solving for the response. Using values for the first run, for example, gives
R=3.0+(0.2)(15)+(0.4)(30)+(0.01)(15)(30)=22.5
which agrees with the expected value.