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.
