Taguchi Orthogonal Array Designs
Taguchi Orthogonal Array (OA) design is a type of general fractional factorial
design. It is a highly fractional orthogonal design that is based on a design
matrix proposed by Dr. Genichi Taguchi and allows you to consider a selected subset
of combinations of multiple factors at multiple levels. Taguchi Orthogonal
arrays are balanced to ensure that all levels of all factors are considered
equally. For this reason, the factors can be evaluated independently of each
other despite the fractionality of the design. This article will show how to
perform a Taguchi OA design
using DOE++.
In the Taguchi OA design, only the main effects and twofactor interactions are
considered, and higherorder interactions are assumed to be nonexistent. In
addition, designers are asked to identify (based on their knowledge of the subject
matter) which interactions might be significant before conducting the design.
Consider an example where a truck front fender’s injectionmolded polyurethane
bumpers suffer from too much porosity [1]. A team of engineers
conducted a Taguchi OA design to study the effects of several factors related to
the porosity.
Table 1: Factors
The team also decided to consider interactions AB and BC. They took two
measurements of porosity (for porosity values, smaller is better). They also
decided to use the L8 orthogonal array for this design. Table 2 shows the alias
relation for L8 arrays.
Table 2: L8 (2^7) Arrays
In Table 2, column 3 is the alias of the interaction of column 1 and column 2
(in red), while column 6 is the alias of the interaction of column 2 and column 4
(in red). Since interactions AB and BC are considered significant in this example,
the main factors A, B, C, D and E are assigned to columns 1, 2, 4, 5
and 7. The interaction factors AB and BC are assigned to columns 3 and 6.
Table 3 shows the L8 orthogonal array and the porosity measurements the team
took.
Table 3: L8 Orthogonal Array
To use DOE++ to perform the Taguchi OA design, add a folio to a project
by choosing Project > Add Standard Folio. In Step 1 of the Design
Wizard, select the Factorial Design option because Taguchi OA design
actually is a factorial design.
In Step 2 of the Design Wizard, choose the Taguchi OA Factorial Design
option.
In Step 3 of the Design Wizard, select the 2 Level Design option with a
L8 (2^7) array and specify that there are five factors with two replicates and
one response.
Click Factor Properties and assign the factors as shown next:
After creating the design folio and entering the data from Table 3,
the DOE++ folio will look like this:
The interactions are not currently included in the reduced L8 array. In order
to take them into account, they need to be included as a 2Way Interaction using
the Effects window (Data > Select Effects).
Click the Calculate icon to analyze the
experimental data. On the Analysis tab, the ANOVA
table and the Regression Information table should look like this. The factors in
red means they are significant.
From the ANOVA table above, it is interesting to see that the interaction BC is
significant while both main factors B and C are not significant. According to the
Effect Heredity Principle [2]: "In order for an interaction
to be significant, at least one of its parent factors should be
significant." There is something hidden here. Look back to Table 2 again.
Column 6 is not
only the alias of column 2 and column 4 (in red, which is the interaction BC in
this example); it is also the alias of column 1 and column 7 (in green, which is
the interaction AE in this example). Factors B and C are not significant,
thus their interaction BC should not be significant either. There is a great
chance that it is the interaction AE which is significant, not BC. This is a
Resolution III design, where the interaction BC and AE are confounded with
each other. So the next step is to replace interaction BC by interaction AE in the
model and run the analysis again.
The ANOVA table for the new analysis is shown below. It is identical to the
previous result, which confirms that the significant interaction is AE not BC.
The Main Effects plot is shown next. The plot shows that factors A and E are
significant while the other three factors are not, which is the same as the results
shown in the ANOVA table.
The ANOVA table shows that factors B, C and D are not significant. However,
interactions AB and AE are significant. Thus in the reduced model, factor B cannot
be removed; the factors that can be removed are C and D. In the Effects window,
remove factor C and D and do the analysis again.
The new ANOVA table and Regression Information table are shown below.
To identify which factor settings can provide the best design, open the
Diagnostics Window by choosing Data > Diagnostics. The
"Fitted Value (YF)" column shows the expected porosity for the factor
settings under different runs.
The table shows that run order 4 (standard order 14) and run order 9 (standard
order 6) give the best result. The predicted porosity is 1.125. Note that because
the experiment used two replicates, run order 9 and run order 4 used the same
settings: mold temperature high, chemical temperature low, throughput high and cure
time low.
Conclusions
This article described the Taguchi orthogonal array design and provided an
example using DOE++ that showed how to use the
design to create an experiment and analyze the data.
References
[1] Kai Yang and Basem ElHaik. "Taguchi's Orthogonal
Array Experiment," in Design for Six Sigma: A Roadmap for Product
Development, McGrawHill, 2008, pp. 469497.
[2] C. F. Jeff Wu and Michael Hamad, "Full Factorial
Experiments at Two Levels," in Experiments: Planning, Analysis, and
Parameter Design Optimization, John Wiley & Sons, Inc. 2000, p. 112.
