5.
Process Improvement
5.5. Advanced topics 5.5.3. How do you optimize a process?


Optimizing of a single response usually starts with line searches in the direction of maximum improvement 
The experimental optimization of a single response is usually
conducted in two phases or steps, following the advice of
Box and Wilson. The first
phase consists of a sequence of line searches in
the direction of maximum improvement. Each search in the sequence is
continued until there is evidence that the direction chosen does not
result in further improvements. The sequence of line searches is
performed as long as there is no evidence of lack of fit for a simple
firstorder model of the form
\( \hat{Y} = \beta_{0} + \beta_{1} X_{1} + \beta_{2} X_{2} + \cdots + \beta_{k} X_{k} \) 

If there is lack of fit for linear models, quadratic models are tried next 
The second phase is performed when there is lack of linear fit in Phase
I, and instead, a secondorder or quadratic polynomial regression model
of the general form
\( \begin{array}{lcl} \hat{Y} & = & \beta_{0} + \beta_{1}X_{1} + \beta_{2}X_{2} + \cdots + \beta_{k}X_{k} + \\ & & \beta_{11} X_{1}^{2} + \beta_{22} X_{2}^{2} + \cdots + \beta_{kk} X_{k}^{2} + \\ & & \beta_{12}X_{1}X_{2} + \beta_{13}X_{1}X_{3} + \cdots + \beta_{1k}X_{1}X_{k} + \\ & & \beta_{23}X_{2}X_{3} + \cdots + \beta_{2k}X_{2}X_{k} + \cdots + \beta_{k1,k}X_{k1}X{k} \end{array} \) is fit. Not all responses will require quadratic fit, and in such cases Phase I is stopped when the response of interest cannot be improved any further. Each phase is explained and illustrated in the next few sections. 

"Flowchart" for two phases of experimental optimization 
The following is a flow chart showing the two phases of experimental
optimization.
