*Design Selection Guideline Completely randomized designs ( Number of Factor =1) Randomized block designs ( Number of Factor =2-4) Fractional factorial designs Box-Behnken Design Response Surface Experiment*

**Design Selection Guideline**

**Completely randomized designs ( Number of Factor =1)**

Here we consider completely randomized designs that have one primary factor. The experiment compares the values of a response variable based on the different levels of that primary factor.

For completely randomized designs, the levels of the primary factor are randomly assigned to the experimental units. By randomization, we mean that the run sequence of the experimental units is determined randomly. For example, if there are 3 levels of the primary factor with each level to be run 2 times, then there are 6 factorial possible run sequences (or 6! ways to order the experimental trials). Because of the replication, the number of unique orderings is 90 (since 90 = 6!/(2!*2!*2!)). An example of an unrandomized design would be to always run 2 replications for the first level, then 2 for the second level, and finally 2 for the third level. To randomize the runs, one way would be to put 6 slips of paper in a box with 2 having level 1, 2 having level 2, and 2 having level 3. Before each run, one of the slips would be drawn blindly from the box and the level selected would be used for the next run of the experiment.

All completely randomized designs with one primary factor are defined by 3 numbers:

*k*= number of factors (= 1 for these designs)

*L*= number of levels

*n*= number of replications and the total sample size (number of runs) is *N = k x L x n*.

**Randomized block designs ( Number of Factor =2-4)**

For randomized block designs, there is one factor or variable that is of primary interest. However, there are also several other nuisance factors.

Nuisance factors are those that may affect the measured result, but are not of primary interest. For example, in applying a treatment, nuisance factors might be the specific operator who prepared the treatment, the time of day the experiment was run, and the room temperature. All experiments have nuisance factors. The experimenter will typically need to spend some time deciding which nuisance factors are important enough to keep track of or control, if possible, during the experiment.

The general rule is:*“Block what you can, randomize what you cannot.”*

Blocking is used to remove the effects of a few of the most important nuisance variables. Randomization is then used to reduce the contaminating effects of the remaining nuisance variables.

Name of Design | Number of Factors k | Number of Runs n |

2-factor RBD | 2 | L_{1} * L_{2} |

3-factor RBD | 3 | L_{1} * L_{2} * L_{3} |

4-factor RBD | 4 | L_{1} * L_{2} * L_{3} * L_{4} |

. | . | . |

k-factor RBD | k | L_{1} * L_{2} * … * L_{k} |

with L_{1}= number of levels (settings) of factor 1

L_{2}= number of levels (settings) of factor 2

L3= number of levels (settings) of factor 3

L_{4}= number of levels (settings) of factor 4

. . L*k*= number of levels (settings) of factor *k
*

A common experimental design is one with all input factors set at two levels each. These levels are called `high’ and `low’ or `+1′ and `-1′, respectively. A design with all possible high/low combinations of all the input factors is called a full factorial design in two levels.

*If there are k factors, each at 2 levels, a full factorial design has 2 ^{k} runs.*

**Fractional factorial designs**

*A factorial experiment in which only an adequately chosen fraction of the treatment combinations required for the complete factorial experiment is selected to be run.*” Even if the number of factors, *k*, in a design is small, the 2^{k}runs specified for a full factorial can quickly become very large. For example, 2^{6} = 64 runs is for a two-level, full factorial design with six factors. To this design we need to add a good number of center point runs and we can thus quickly run up a very large resource requirement for runs with only a modest number of factors.

Fractional 2 level factorial design

•The number of experiments in 2 level full factorial design is still a lot, it is 2k, for example, if k=7, that is, if we have 7 factors, the number of experiments will be 27= 128!

•To reduce the number of experiments, 2 level fractional factorial design is developed,

•It will only use a small fraction of experiments as that of a full factorial experiment.

•Fractional factorial experiments mainly only study main effects,and 2 factor interactions. They ignore higher order interactions.

Resolution III design: Only study main effects

Resolution IV design: can study main effects and some 2 factor interactions

Resolution V design: can study main effects and all 2 factor interaction

**Response Surface Experiment**

**Central Composite Design**

–2-Level full factorial design augmented with a center point and two “star”points for each factor

–Useful in studying higher order effects

–Larger number of runs, requires 2n+2n+1 pointsX2X1X3

**Box-Behnken Design**

The Box-Behnken design is an independent quadratic design in that it does not contain an embedded factorial or fractional factorial design. Inthis experimental design the treatment combinations are at the midpoints of edges of the design space and at the center. These designs are rotatable (or near rotatable) and require 3 levels of each factor.