(87a) On Mapping the Operating Window of Processes

When you have a new process on
your hands, be it a lab scale or a large scale manufacturing process, how do
you map your process operating window?

Theoretical modeling can only
take you so far, and the answer it provides can be too generic for your rather
specific case. A good deal of empiricism seems to be generally applied when it
comes to mapping the operating window.

Practitioners typically espouse educated
trial-and-error in order to search for upper and lower levels of the process
variables which can render a fail-safe outcome: the so called operating range
of these variables. More often than not, a restricted range is established for the
lack of a systematic search procedure. The amount of time necessary to conduct
a full search of the operating range of each variable may be easily prohibitive
under trial-and-error. When operating ranges are restricted, so are the
possibilities of process optimization.

In the theory of design of
experiments, Screening designs are the very first line of attack to process improvement.
In order to screen variables for their relative importance, Screening designs
place experimental runs at the (high/low) edges of the operating ranges. It is sine
qua non that these experimental runs be fail-safe so that model parameters
can be properly estimated with a few runs. Hence, prevention of fail runs is the
motivation for restricting the operating window, i.e. making it narrower than
it could be, so as to be able to implement Screening designs smoothly.

As such, the establishment of the
operating window is a pre-condition for the optimization of new processes. If
you do not know the high/low levels, you cannot screen variables. And if the
operating window is established in a restricted form, it reduces the level of optimization
that can be attained if some sweet spot within the true (and larger) operating
window is left out.  

The work presented here offers a
systematic method for mapping the operating window with minimal restriction. This
method makes use of Sequential Design of Experiments. It starts out with an
ordinary Screening design and, as fail runs surface, it makes use of Logistic
regression to establish the initial location of the boundary of the operating window
within the multivariate space of process input variables. Iteratively, new runs
are placed at the boundary, new fail runs surface, and the boundary is relocated
after re-estimating the Logistic model with new information.  A heuristic is
used to induce convergence to the true boundary after a few iterations.

With this method, experimental
runs are employed with the sole objective of optimally learning the location of
the operating window, which is in average a more cost-effective manner of
mapping vs. trial-and error. This, in turn, opens up the possibility of mapping
the operating window with minor restrictions within a manageable timeline. 

An additional advantage of this
method is that the fail-safe runs it generates form, in some cases with the
need for augmentation, an experimental design which can be used for estimating
Screening models altogether.