ROBUST DESIGN FOR SOFTWARE DEVELOPMENT - Software Design for Six-Sigma: A Roadmap for Excellence - page 480

18.7 ROBUSTNESS CONCEPT #5: ORTHOGONAL ARRAYS

This aspect of Taguchi robust design methods is the one most similar to traditional
design of experience (DOE) technique. Taguchi has developed a system of tabu-
lated designs (arrays) that allow for the maximum number of main effects to be
estimated in an unbiased (orthogonal) manner, with a minimum number of runs in
the experiment. Latin square designs, 2 k − p designs (Plackett-Burman designs, in
particular) and Box-Behnken designs also are aimed at accomplishing this goal. In
fact, many standard orthogonal arrays tabulated by Taguchi are identical to fractional
two-level factorials, Plackett-Burman designs, Box-Behnken designs, Latin square,
Greco-Latin squares, and so on.
Orthogonal arrays provide an approach to design efficiently experiments that will
improve the understanding of the relationship between software control factors and the
desired output performance (functional requirements and responses). This efficient
design of experiments is based on a fractional factorial experiment, which allows
an experiment to be conducted with only a fraction of all possible experimental
combinations of factorial values. Orthogonal arrays are used to aid in the design
of an experiment. The orthogonal array will specify the test cases to conduct the
experiment. Frequently, two orthogonal arrays are used: a control factor array; and
a noise factor array; the latter used to conduct the experiment in the presence of
difficult-to-control variation so as to develop robust software.
In Taguchi's experimental design system, all experimental layouts will be derived
from about 18 standard “orthogonal arrays.” Let us look at the simplest orthogonal
array, L 4 array, (Table 18.1).
The values inside the array, that is, 1 and 2, represent two different levels of a
factor. By simply use “ − 1” to substitute “1,” and “ + 1” to substitute “2,” we find that

this L 4 array becomes Table 18.2.
Clearly, this is a 2 3 − 1

fractional factorial design, with the defining relation, 9

I =

ABC Where “column 2” of L 4 is equivalent to the “A column” of the 2 3-1 − design,

“column 1” is equivalent to the “B column” of the 2 3 − 1 design, and “column 3” is

equivalent to “C column” of the 2 3 − 1

=−

AB.
In each of Taguchi's orthogonal array, there are “linear graph”(s) to go with it. A
linear graph is used to illustrate the interaction relationships in the orthogonal array,
for example, the L 4 array linear graph is given in Figure 18.8. The numbers “1” and
“2” represent column 1 and column 2 of the L 4 array, respectively, “3” is above the
line segment connecting “1” and “2,” which means that 'the interaction of column
1 and column 2 is confounded with column “3,” which is perfectly consistent with
C

design, with C

AB in the 2 3 − 1 fractional factorial design.
For larger orthogonal arrays, not only are there linear graphs but there are also
interaction tables to explain interaction relationships among columns. For example,
The L 8 array in Table 18.3 has the linear graph and table shown in Figure 18.9.
This approach to designing and conducting an experiment to determine the effect of
control factors (design parameters) and noise factors on a performance characteristic
is represented in Figure 18.10.

=−

9 The defining relation is covered in Chapter 12 of El-Haik and Roy (2005).