5.
Process Improvement
5.5. Advanced topics 5.5.3. How do you optimize a process? 5.5.3.1. Single response case


Starting at the current operating conditions, fit a linear model  If experimentation is initially performed in a new, poorly understood production process, chances are that the initial operating conditions X_{1}, X_{2}, ...,X_{k} are located far from the region where the factors achieve a maximum or minimum for the response of interest, Y. A firstorder model will serve as a good local approximation in a small region close to the initial operating conditions and far from where the process exhibits curvature. Therefore, it makes sense to fit a simple firstorder (or linear polynomial) model of the form: Experimental strategies for fitting this type of model were discussed earlier. Usually, a 2^{kp} fractional factorial experiment is conducted with repeated runs at the current operating conditions (which serve as the origin of coordinates in orthogonally coded factors).  
Determine the directions of steepest ascent and continue experimenting until no further improvement occurs  then iterate the process 
The idea behind "Phase I" is to keep experimenting along the direction
of steepest ascent (or descent, as required) until there is no further
improvement in the response. At that point, a new fractional factorial
experiment with center runs is conducted to determine a new search
direction. This process is repeated until at some point significant
curvature in \( \hat{Y} \)
is detected. This implies that the operating conditions
X_{1}, X_{2}, ...,X_{k} are
close to where the maximum (or minimum, as required) of Y
occurs. When significant curvature, or lack of fit, is detected, the
experimenter should proceed with "Phase II". Figure 5.2 illustrates a
sequence of line searches when seeking a region where curvature exists
in a problem with 2 factors (i.e., k=2).


Two main decisions: search direction and length of step 
There are two main decisions an engineer must make in Phase I:


Flow chart of iterative search process 


Procedure for Finding the Direction of Maximum Improvement  
The direction of steepest ascent is determined by the gradient of the fitted model 
Suppose a firstorder model (like above) has
been fit and provides a useful approximation. As long as lack of fit
(due to pure quadratic curvature and interactions) is very small
compared to the main effects, steepest ascent can be attempted. To
determine the direction of maximum improvement we use


The direction of steepest ascent depends on the scaling convention  equal variance scaling is recommended 
The direction of the gradient, g, is given by the values
of the parameter estimates, that is,
g' = (b_{1}, b_{2}, ...,
b_{k}).
Since the parameter estimates b_{1},
b_{2}, ..., b_{k}
depend on the scaling convention for the factors, the steepest ascent
(descent) direction is also scale dependent. That is, two
experimenters using different scaling conventions will follow different
paths for process improvement. This does not diminish the general
validity of the method since the region of the search, as given by
the signs of the parameter estimates, does not change with scale. An
equal variance scaling convention, however, is recommended. The coded
factors x_{i}, in terms of the factors in the
original units of measurement, X_{i},
are obtained from the relation
\( \mbox{subject to:} \hspace{.2in} \sum_{i=1}^{k}{x_{i}^{2}} \le \rho^{2} \) 

Solution is a simple equation  This problem can be solved with the aid of an optimization solver (e.g., like the solver option of a spreadsheet). However, in this case this is not really needed, as the solution is a simple equation that yields the coordinates  
Equation can be computed for increasing values of 
An engineer can compute this equation for different increasing values
of \( \rho \)
and obtain different factor settings, all on the steepest ascent
direction.
To see the details that explain this equation, see Technical Appendix 5A. 

Example: Optimization of a Chemical Process  
Optimization by search example 
It has been concluded (perhaps after a factor screening experiment)
that the yield (Y, in %) of a chemical process is mainly
affected by the temperature (X_{1}, in C)
and by the reaction time (X_{2}, in minutes).
Due to safety reasons, the region of operation is limited to
150 ≤ X_{2} ≤ 500 

Factor levels 
The process is currently run at a temperature of 200
\( ^{\circ} \mbox{C} \)
and a reaction time of 200 minutes. A process engineer decides to run a
2^{2} full factorial experiment with factor levels at


Orthogonally coded factors 
Five repeated runs at the center levels are conducted to assess lack
of fit. The orthogonally coded factors are


Experimental results 
The experimental results were:


ANOVA table 
The corresponding ANOVA table for a firstorder polynomial model is
SUM OF MEAN F SOURCE SQUARES DF SQUARE VALUE PROB>F MODEL 503.3035 2 251.6517 4.7972 0.0687 CURVATURE 8.2733 1 8.2733 0.1577 0.7077 RESIDUAL 262.2893 5 52.4579 LACK OF FIT 37.6382 1 37.6382 0.6702 0.4590 PURE ERROR 224.6511 4 56.1628 COR TOTAL 773.8660 8 

Resulting model 
It can be seen from the ANOVA table that there is no significant lack
of linear fit due to an interaction term and there is no evidence of
curvature. Furthermore, there is evidence that the firstorder model
is significant. The resulting model (in the coded variables) is


Diagnostic checks  The usual diagnostic checks show conformance to the regression assumptions, although the R^{2} value is not very high: R^{2} = 0.6504.  
Determine level of factors for next run using direction of steepest ascent 
To maximize \( \hat{Y} \),
we use the direction of steepest ascent. The engineer selects
\( \rho \) = 1 since a point on the steepest ascent direction one unit
(in the coded units) from the origin is desired. Then from the equation
above for the predicted Y response, the coordinates of the factor
levels for the next run are given by:




Technical Appendix 5A: finding the factor settings on the steepest ascent direction a specified distance from the origin  
Details of how to determine the path of steepest ascent 
The problem of finding the factor settings on the steepest
ascent/descent direction that are located a distance \( \rho \)
from the origin is given by the optimization problem,
\( \mbox{subject to:} \hspace{.2in} \sum_{i=1}^{k}{x_{i}^{2}} \le \rho^{2} \) 

Solve using a Lagrange multiplier approach 
To solve it, use a Lagrange multiplier approach. First, add a penalty
\( \lambda \)
for solutions not satisfying the constraint (since we want a direction
of steepest ascent, we maximize, and therefore the penalty is
negative). For steepest descent we minimize and the penalty term is
added instead.
\( \frac{\partial L}{\partial \lambda} = (x'x  \rho^{2}) = 0 \) 

Solve two equations in two unknowns 
These two equations have two unknowns (the vector x and
the scalar \( \lambda \) )
and thus can be solved yielding the desired solution:


Multiples of the direction of the gradient  From this equation we can see that any multiple \( \rho \) of the direction of the gradient (given by \( \frac{b}{\parallel b \parallel} \) ) will lead to points on the steepest ascent direction. For steepest descent, use instead b_{i} in the numerator of the equation above. 