Question 14.1: Find the optimum response for the response surface in Figure...

Letting a=0, b=0, s_{a}=1 and s_{b}=1 gives the vertices for the initial simplex as

Vertex 1: (a, b)=(0,0)

Vertex 2: \left(a+s_{\mathrm{A}}, b\right)=(1,0)

Vertex 3: \left(a+0.5 s_{\mathrm{A}}, b+0.87 s_{\mathrm{B}}\right)=(0.5,0.87)

The responses (calculated using equation 14.2)

R=5.5+1.5 A+0.6 B-0.15 A^2-0.0245 B^2-0.0857 A B      (14.2)

for the three vertices are shown in the following table

\begin{array}{cccc} \text{Vertex }& \text{Factor }A & \text{Factor }B & \text{Response }\\ \hline \mathrm{V}_{1} & 0 & 0 & 5.50 \\ \mathrm{~V}_{2}& 1.00 & 0 & 6.85 \\ \mathrm{~V}_{3} & 0.50 & 0.87 & 6.68 \end{array}

with V_{1} giving the worst response and V_{3} the best response (rule 1). We reject \mathrm{V}_{1} and replace it with a new vertex whose factor levels are calculated using rule 2 ; thus

New a=2 \times \left(\frac{a \text { for } \mathrm{V}_{2}+a \text { for } \mathrm{V}_{3}}{2}\right)-\left(a\right. for \left.\mathrm{V}_{1}\right)=2 \times \frac{1.00+0.50}{2}-0=1.50

\text { New } b=2 \times \left(\frac{b \text { for } \mathrm{V}_{2}+b \text { for } \mathrm{V}_{3}}{2}\right)-\left(b \text { for } \mathrm{V}_{1}\right)=2 \times \frac{0+0.87}{2}-0=0.87

The new simplex, therefore, is

\begin{array}{cccc} \text{Vertex }& \text{Factor }A & \text{Factor }\boldsymbol{B} & \text{Response }\\ \hline \mathrm{V}_{2} & 1.00 & 0 & 6.85 \\ \mathrm{~V}_{3} & 0.50 & 0.87 & 6.68 \\ \mathrm{~V}_{4} & 1.50 & 0.87 & 7.80 \end{array}

The worst response is for vertex 3 , which we replace with the following new vertex

a=2 \times \frac{1.00+1.50}{2}-0.5=2.00 \quad b=2 \times \frac{0+0.87}{2}-0.87=0

The resulting simplex now consists of the following vertices

\begin{array}{cccc} \text{Vertex }& \text{Factor }A & \text{Factor }B & \text{Response }\\ \hline V_{2} & 1.00 & 0 & 6.85 \\ V_{4} & 1.50 & 0.87 & 7.80 \\ V_{5} & 2.00 & 0 & 7.90 \end{array}

The calculation of the remaining vertices is left as an exercise. The progress of the completed optimization is shown in Table 14.3 and in Figure 14.10. The optimum response of (3,7) first appears in the twenty-fourth simplex, but a total of 29 steps is needed to verify that the optimum has been found.