4.3. Data Collection for Process Modeling
|Classical Designs Heavily Used in Industry||The most heavily used designs in industry are the "classical designs" (full factorial designs, fractional factorial designs, Latin square designs, Box-Behnken designs, etc.). They are so heavily used because they are optimal in their own right and have served superbly well in providing efficient insight into the underlying structure of industrial processes.|
|Reasons Classical Designs May Not Work||
Cases do arise, however, for which the tabulated classical
designs do not cover a particular practical situation.
That is, user constraints
preclude the use of tabulated classical designs
because such classical designs do not
accommodate user constraints. Such constraints include:
|What to Do If Classical Designs Do Not Exist?||
If user constraints are such that classical designs do not exist
to accommodate such constraints, then what is the user to do?
The previous section's list of design criteria (capability for the primary model, capability for the alternate model, minimum variation of estimated coefficients, etc.) is a good passive target to aim for in terms of desirable design properties, but provides little help in terms of an active formal construction methodology for generating a design.
|Common Optimality Criteria||
To satisfy this need, an "optimal design" methodology has
been developed to generate a design when user
constraints preclude the use of tabulated classical designs.
Optimal designs may be optimal in many different ways, and
what may be an optimal design according to one criterion may
be suboptimal for other criteria. Competing criteria have
led to a literal alphabet-soup collection of optimal
design methodologies. The four most popular ingredients in
that "soup" are:
|Need 1: a Model||
The motivation for optimal designs is the practical
constraints that the user has. The advantage of optimal
designs is that they do provide a reasonable
design-generating methodology when no other mechanism
exists. The disadvantage of optimal designs is that they
require a model from the user. The user may not have this model.
All optimal designs are model-dependent, and so the quality of the final engineering conclusions that result from the ensuing design, data, and analysis is dependent on the correctness of the analyst's assumed model. For example, if the responses from a particular process are actually being drawn from a cubic model and the analyst assumes a linear model and uses the corresponding optimal design to generate data and perform the data analysis, then the final engineering conclusions will be flawed and invalid. Hence one price for obtaining an in-hand generated design is the designation of a model. All optimal designs need a model; without a model, the optimal design-generation methodology cannot be used, and general design principles must be reverted to.
|Need 2: a Candidate Set of Points||The other price for using optimal design methodology is a user-specified set of candidate points. Optimal designs will not generate the best design points from some continuous region--that is too much to ask of the mathematics. Optimal designs will generate the best subset of points from a larger superset of candidate points. The user must specify this candidate set of points. Most commonly, the superset of candidate points is the full factorial design over a fine-enough grid of the factor space with which the analyst is comfortable. If the grid is too fine, and the resulting superset overly large, then the optimal design methodology may prove computationally challenging.|
|Optimal Designs are Computationally Intensive||
The optimal design-generation methodology is computationally
intensive. Some of the designs (e.g., D-optimal) are better
than other designs (such as A-optimal and G-optimal) in
regard to efficiency of the underlying search algorithm.
Like most mathematical optimization techniques, there is no
iron-clad guarantee that the result from the optimal design
methodology is in fact the true optimum. However, the
results are usually satisfactory from a practical point of
view, and are far superior than any ad hoc designs.
For further details about optimal designs, the analyst is referred to Montgomery (2001).