Design
of Experiments (Experimental Design,
DOE) 

Design of Experiments (DOE) involves conducting
a systematic series of tests to discover
the relationship between the factors effecting
a process (X) and the critical outputs
(responses). The inputs are varied in a
systematic fashion and the effects on the
response(s) are observed.
The Design of Experiments (Experimental
Design) is used when a process is affected
by many separate factors. It is far more
efficient than the 'One Factor at a Time'
(OFAT) method in which each factor is varied
in turn whilst the others remain constant.
The Design of Experiments will give more
information for less testing. It will also
allow 'interactions' between factors to
be evaluated, and the effects of interactions
are usually important.
The two main approaches to experimental
design are the 'classical' and 'Taguchi' methods.
Experts are divided on the merits of the
Taguchi method, but the emphasis
on variation and the methods it uses to
address variation are important. The types
of design commonly used include Full
Factorial Designs, Fractional
Factorial Designs and
PlackettBurman Designs.
A more powerful approach is to use Response
Surface Methods; this group includes the
Box Behnken and Central Composite Designs
(CCD).
Designs that involve mixtures require
a different method of analysis, see the
topic on Mixture Designs.
Factorial designs are the most common
type of Experimental Design. In a factorial
design several factors are controlled at
two or more levels, and the effect on the
response is investigated.
There are two main types: Full Factorial
Designs and Fractional Factorial Designs.
In Fractional Factorial designs the amount
of testing is reduced, but the downside
is that some interactions and factors are
aliased. One of the limitations of Fractional
Factorial designs is that the number of
runs is 2^{n}, if there are more
than a few factors this leads to large
gaps in the available options (4, 8, 16,
32, 64, 128, etc.). PlackettBurman designs
overcome this, but at the expense of confounding.
Factorial designs involve
testing several factors at two levels,
high and low. In a Full Factorial experiment
every possible combination of factors and
permutations is tested. If there are n
factors this is 2^{n} levels.
The design matrix shows the combinations
for a 2^{2} design
Run

A

B

AB

1

1

1

+1

2

+1

1

1

3

1

+1

1

4

+1

+1

+1

Fractional
Factorial Designs 

A Fractional Factorial experiment
uses only a half (2^{n1}), a quarter
(2^{n2}), or some other division
by a power of two of the number of runs
that would be required for a Full Factorial
Experiment.
The example shows a 2^{31} design:
Run

A

B

C + AB

1

1

1

+1

2

+1

1

1

3

1

+1

1

4

+1

+1

+1

C is aliased with AB, thus it is a Resolution
III design.
These are Resolution III (screening designs).
The number of runs in a PlackettBurman
designs is a multiple of 4, thus avoiding
the limitations of factorial and fractional
factorial designs where the number of runs
is 2^{k}.
Taguchi designs are a type of Experimental
Design pioneered by Genichi Taguchi. They
are distinguished by the use of:
 the Signal to Noise Ratio
 inner and outer arrays
 Taguchi arrays
Taguchi designs are criticized by many
experts. Taguchi does not give the same
attention to interactions (Taguchi Designs
are often highly confounded screening designs)
as the classical Design of Experiments
approach. It is also easy to demonstrate
that the
Signal to Noise ratio method does not always
give the best results, particularly when
the aim is to achieve a target response
rather than maximize or minimize the response.
One of the original advantages of the Taguchi
approach was that it favored graphical
analysis and avoided complex analysis.
With modern software, such as Minitab,
this advantage is no longer as strong.
Response surface methods are types of
Experimental Design that investigate curvature
of the response surface. They achieve this
by using a quadratic regression equation
rather than the linear form of the regression
equation used in factorial designs.
When the true response surface is approximated
by a linear equation the maximum and minimum
values at a corner point. Response surface
designs can have the maximum or minimum
in the interior of the surface. This allows
the response to be optimized using hill
climbing methods.
There are various types of response surface
design including the Central Composite
(CCD) and Box Behnken designs.
Mixture designs are a type
of Experimental Design that is
used to find the best composition when
there is a mixture of ingredients. Mixtures
are different from other types of Experimental
Design because the proportions must add
up to 100%. Thus increasing the level of
one constituent must reduce the level of
the others.
The analysis is complex and
so a software package, such as Minitab,
would invariably be used to analyze the
results.
