This document discusses fractional factorial designs of experiments. It explains that full factorial designs require testing all possible combinations of factors and levels, which can become infeasible with more than a few factors. Fractional factorial designs exploit this redundancy by selecting a subset of the full factorial design. This allows studying many factors using fewer experimental runs. The document provides examples of selecting fractional factorial designs and discusses orthogonal arrays, which provide balanced and efficient experimental designs according to Taguchi methods.

1. Fractional Factorial Designs of Experiments
P M V Subbarao
Professor
Mechanical Engineering Department
Selection of Few Significant Parameters
for Experimentation…..

2. Full Factorial Design : Design Matrix

3. Full Factorial Design : Result Matrix

4. Most Significant Factor

5. Sample 23 Experiment
7
9
9
9
8
3
8
3
9 + 9 + 3 + 3
6
7 + 9 + 8 + 8
8
Main Effect of X1 6 – 8 = -2

6. Why Fractional Factorials?
Full Factorials
No. of combinations

For
two-levels
In engineering, this is the sample
size -- no. of prototypes to be built.
Continuous development of
Knowledge is introducing
more and more factors

7. Why so many Treatments?
“There tends to be a redundancy in full factorial designs”
– redundancy in terms of an excess number of interactions that can be
estimated …
Fractional factorial designs exploit this redundancy …” 
philosophy

8. How to select a subset of 4 runs from a 23=8 -run design?
Many possible “fractional” designs

9. First Choice

10. Second Choice

11. Third Choice
Wow!
Balanced design
All factors occur and low and high levels
same number of times; Same for interactions
Columns are orthogonal. Projections …

12. How to select a subset full factorial design
• We note that the product of any two columns is zero.
• Also the column sums are zero.
• Hence the three columns may be considered as vectors that form an
orthogonal set.
• In fact while calculating the sample variance earlier these properties
were used without being spelt out.

13. Want to study 5 factors (1,2,3,4,5) using a 2^4 = 16-run design
i.e., construct half-fraction of a 2^5 design
= 2^{5-1} design

14. DOE - Taguchi Method
• Dr. Taguchi of Nippon Telephones and Telegraph
Company, Japan has developed a method based on "
ORTHOGONAL ARRAY " experiments.
• This gives much reduced " variance " for the
experiment with " optimum settings " of control
parameters.
• "Orthogonal Arrays" (OA) provide a set of well
balanced (minimum) experiments serve as objective
functions for optimization.

15. Taguchi Method : When to Select a ‘larger’ OA
to perform “Factorial Experiments”
• We always ‘think’ about ‘reducing’ the number of
experiments (to minimize the ‘resources’ – equipment,
materials, manpower and time)
• However, doing ALL / Factorial experiments is a good
idea if
– Conducting experiments is ‘cheap/quick’ but
measurements are ‘expensive/take too long’
– The experimental facility will NOT be available later to
conduct the ‘verification’ experiment
– We do NOT wish to conduct separate experiments for
studying interactions between Factors

16. Taguchi Method Design of Experiments
• The general steps involved in the Taguchi Method are as follows:
• 1. Define the process objective, or more specifically, a target value
for a performance measure of the process.
• 2. Determine the design parameters affecting the process.
• The number of levels that the parameters should be varied at must
be specified.
• 3. Create orthogonal arrays for the parameter design indicating the
number of and conditions for each experiment.
• The selection of orthogonal arrays is based on the number of
parameters and the levels of variation for each parameter, and will
be expounded below.
• 4. Conduct the experiments indicated in the completed array to
collect data on the effect on the performance measure.
• 5. Complete data analysis to determine the effect of the different
parameters on the performance measure.

18. Determining Parameter Design Orthogonal Array
• The effect of many different factors on the performance characteristic
in a condensed set of experiments can be examined by using the
orthogonal array experimental design proposed by Taguchi.
• The main factors affecting a process that can be controlled (control
Factors) should be determined.
• The levels at which these parameters should be varied must be
determined.
• Determining what levels of a variable to test requires an in-depth
understanding of the process, including the minimum, maximum, and
current value of the parameter.
• If the difference between the minimum and maximum value of a
parameter is large, the values being tested can be further apart or more
values can be tested.
• If the range of a parameter is small, then less values can be tested or
the values tested can be closer together.
• Typically, the number of levels for all parameters in the experimental
design is chosen to be the same to aid in the selection of the proper
orthogonal array.

20. Orthogonal Array Selector
Number of Factors
Number
of
Levels

21. Taguchi Method :
How to Select a ‘larger’ OA
to perform “Factorial Experiments”
Control Factors
and Levels
Factorial
Combinations
Suitable
OA
2 CF / 2-levels 4 L4
3 CF / 2-levels 8 L8
4 CF / 2-levels 16 L16
5 CF / 2-levels 32 L32

22. L27 Array

23. L 50 Array