Design of Experiments (DoE)

Full-factorial designs combine all defined levels of all design variables. For instance, a full-factorial design investigating one two-level continuous variable, one three-level continuous variable and one four-level category variable will include 2x3x4=24 experiments.

Among other properties, full-factorial designs are perfectly balanced, i.e. each level of every design variable is studied an equal number of times in combination with every level of the other design variables.

Full-factorial designs include enough experiments to allow use of a model with all interactions. Thus, they are a logical choice if one intends to study interactions in addition to main effects.

In the specific case where there are only two-level variables (continuous with lower and upper levels, and/or binary variables), one can define fractions of full factorial designs that enable the investigation of as many design variables as the chosen full-factorial designs with fewer experiments. These “economic” designs are called fractional factorial designs.

Given that one already has defined a full-factorial design suitable for the investigation, the most natural way to build a fractional design is to use only half the experimental runs of the original design. For instance, one might try to study the effects of three design variables with only 4 (2(3-1)) instead of 8 (23) experiments. Larger factorial designs admit fractional designs with a higher degree of fractionality, i.e. even more economical designs, such as investigating nine design variables with only 16 (2(9-5) ) experiments instead of 512 (29). Such a design can be referred to as a fractional design; its degree of fractionality is 5. This means that one investigates nine variables at the usual cost of four (thus saving the cost of five).

If the experimental objective is to study the main effects only, and there are many design variables to investigate (i.e. > 10), Plackett-Burman designs may be the solution. They are very economical, since they require only one to four more experiments than the number of design variables.

Plackett–Burman designs [Plackett and Burman, 1946] are experimental designs developed while the authors were working in the British Ministry of Supply. Their goal was to find experimental designs for investigating the dependence of some measured responses on a number of independent variables (factors), each taking L levels. The designs were developed in such a way as to minimize the variance of the estimates of these dependencies using a limited number of experiments. Interactions between the factors were considered negligible. The solution to this problem is to find an experimental design in which each combination of levels for any pair of factors appears the same number of times. A complete factorial design would satisfy this criterion, but the idea was to find smaller designs.

Central composite designs (CCD) are extensions of two-level full factorial designs. A CCD enables a quadratic model to be fitted by including new levels, in addition to the specified lower and upper levels to fit such a model.

A CCD consists of three types of experiments:

• Cube samples are experiments which cross lower and upper levels of the design variables; they are the 'factorial' part of the design
• Center samples are the replicates of the experiment which cross the mid-levels of all design variables; they are usually used to determine the experimental error of the design
• Star (or axial) samples are used to make the design region spherical and are located at the midpoints of the “faces” of the factorial part of the design. These samples are specific to CCD designs

Box-Behnken designs are not built on a factorial basis, but they are nevertheless good optimization designs with simple properties.

In a Box-Behnken design, all design variables have exactly three levels: low cube, center, and high cube. Each experiment crosses the extreme levels of two or three design variables with the mid-levels of the others. In addition, the design includes a number of center samples.

The properties of Box-Behnken designs are the following:

• The actual range of each design variable is low cube to high cube, which makes it easy to handle
• All non-center samples are located on a sphere, thus achieving the property of rotatability

Mixture design deals with factors that are dependent in a way that their sum is constant. Depending on the goal of your experimentation you can choose between axial design (screning) simplex-lattice design (screning and optimization) and simplex-centroid designs (optimization).