5.
Process Improvement
5.5.
Advanced topics
5.5.3.
How do you optimize a process?
5.5.3.1.
Single response case
5.5.3.1.3.

Single response: Choosing the step length


A procedure for choosing how far along the direction of steepest
ascent to go for the next trial run

Once the search direction is determined, the second decision needed
in Phase I relates to how far in that direction the process should be
"moved". The most common procedure for selecting a step length is
based on choosing a step size in one factor and then computing step
lengths in the other factors proportional to their parameter estimates.
This provides a point on the direction of maximum improvement. The
procedure is given below. A similar approach is obtained by choosing
increasing values of \( \rho \)
in
\( x_{i}^{*} = \rho \frac{b_{i}}{\sqrt{\sum_{j=1}^{k}{b_{j}^{2}}}}
\hspace{.3in} i = 1, 2, \ldots , k \)
However, the procedure below considers the original units of
measurement which are easier to deal with than the coded "distance"
\( \rho \).


Procedure: selection of step length

Procedure for selecting the step length

The following is the procedure for selecting the step length.
 Choose a step length \( \Delta X_{j} \)
(in natural units of measurement) for some factor j.
Usually, factor j is chosen to be the one engineers feel
more comfortable varying, or the one with the largest
b_{j}. The value of \( \Delta X_{j} \)
can be based on the width of the confidence cone around the
steepest ascent/descent direction. Very wide cones indicate that
the estimated steepest ascent/descent direction is not reliable,
and thus \( \Delta X_{j} \)
should be small. This usually occurs when the R^{2}
value is low. In such a case, additional experiments can be
conducted in the current experimental region to obtain a better
model fit and a better search direction.
 Transform to coded units:
\( \Delta x_{j} = \frac{\Delta X_{j}}{s_{j}} \)
with s_{j} denoting the scale
factor used for factor j (e.g.,
s_{j} = range_{j}/2).
 Set \( \Delta x_{i} = \frac{b_{i}}{b_{j}} \Delta x_{j} \)
for all other factors i.
 Transform all the \( \Delta x_{i} \)'s
to natural units:
\( \Delta X_{i} = (\Delta x_{i})(s_{i}) \).


Example: Step Length Selection.

An example of step length selection

The following is an example of the step length selection procedure.
 For the chemical process experiment described previously,
the process engineer selected
\( \Delta X_{2} = 50 \)
minutes. This was based on process engineering considerations.
It was also felt that \( \Delta X_{2} = 50 \)
does not move the process too far away from the current region
of experimentation. This was desired since the
R^{2} value of 0.6580 for the fitted model is quite
low, providing a not very reliable steepest ascent direction (and a
wide confidence cone, see
Technical
Appendix 5B).
 \( \Delta x_{2} = \frac{50}{50} = 1.0 \)
 \( \Delta x_{1} = \frac{1.2925}{11.14} = 0.1160 \)
 \( \Delta X_{2} = (0.1160)(30) = 3.48^{\circ}C \)
Thus the step size is \( \Delta X \)'
= (3.48^{o}C, 50 minutes).


Procedure: Conducting Experiments Along the Direction of Maximum
Improvement

Procedure for conducting experiments along the direction of
maximum improvement

The following is the procedure for conducting experiments along the
direction of maximum improvement.
 Given current operating conditions
\( X_{0}^{'} = (X_{1}, X_{2}, \dots , X_{k}) \)
and a step size
\( \Delta X' = (\Delta X_{1}, \Delta X_{2}, \dots ,
\Delta X_{k}) \),
perform experiments at factor levels
\( X_{0} + \Delta X, X_{0} + 2 \Delta X, X_{0} + 3 \Delta X,
\dots \)
as long as improvement in the response Y
(decrease or increase, as desired) is observed.
 Once a point has been reached where there is no further
improvement, a new firstorder experiment (e.g., a
2^{kp} fractional factorial)
should be performed with repeated center runs to assess lack
of fit. If there is no significant evidence of lack of fit,
the new firstorder model will provide a new search direction,
and another iteration is performed as indicated in Figure 5.3.
Otherwise (there is evidence of lack of fit), the experimental
design is augmented and a secondorder model should be fitted.
That is, the experimenter should proceed to "Phase II".


Example: Experimenting Along the Direction of Maximum
Improvement

Step 1: increase factor levels by \( \delta \)

Step 1:
Given X_{0} = (200^{o}C, 200 minutes) and
\( \Delta X \)
= (3.48^{o}C, 50 minutes), the next experiments were
performed as follows (the step size in temperature was rounded to
3.5^{o}C for practical reasons):

X_{1}

X_{2}

x_{1}

x_{2}

Y (= yield)

X_{0}

200

200

0

0


X_{0} + \( \Delta X \)

196.5

250

0.1160

1

56.2

X_{0} + \( 2 \Delta X \)

193.0

300

0.2320

2

71.49

X_{0} + \( 3 \Delta X \)

189.5

350

0.3480

3

75.63

X_{0} + \( 4 \Delta X \)

186.0

400

0.4640

4

72.31

X_{0} + \( 5 \Delta X \)

182.5

450

0.5800

5

72.10

Since the goal is to maximize Y, the point of maximum
observed response is
X_{1} = 189.5^{o}C,
X_{2} = 350 minutes. Notice that the search was
stopped after 2 consecutive drops in response, to assure that
we have passed by the "peak" of the "hill".

Step 2: new factorial experiment

Step 2:
A new 2^{2} factorial experiment is performed with
X' = (189.5, 350) as the origin. Using the same scaling
factors as before, the new scaled controllable factors are:
\( x_{1} = \frac{X_{1}  189.5} {30} \hspace{.5in} \)
and
\( \hspace{.5in} x_{2} = \frac{X_{2}  350} {50} \)
Five center runs (at X_{1} = 189.5,
X_{2} = 350)
were repeated to assess lack of fit. The experimental results were:
x_{1}

x_{2}

X_{1}

X_{2}

Y (= yield)

1

1

159.5

300

64.33

+1

1

219.5

300

51.78

1

+1

159.5

400

77.30

+1

+1

219.5

400

45.37

0

0

189.5

350

62.08

0

0

189.5

350

79.36

0

0

189.5

350

75.29

0

0

189.5

350

73.81

0

0

189.5

350

69.45

The corresponding ANOVA table for a linear model is
SUM OF MEAN F
SOURCE SQUARES DF SQUARE VALUE PROB > F
MODEL 505.376 2 252.688 4.731 0.0703
CURVATURE 336.364 1 336.364 6.297 0.0539
RESIDUAL 267.075 5 53.415
LACK OF FIT 93.896 1 93.896 2.168 0.2149
PURE ERROR 173.179 4 43.295
COR TOTAL 1108.815 8
From the table, the linear effects (model) are significant and there is
no evidence of lack of fit. However, there is a significant curvature
effect (at the 5.4 % significance level), which implies that the
optimization should proceed with Phase II; that is, the fit and
optimization of a secondorder model.
