Taguchi's Orthogonal Arrays

Taguchi's orthogonal arrays are highly fractional orthogonal designs proposed by Dr. Genichi Taguchi, a Japanese industrialist. These designs can be used to estimate main effects using only a few experimental runs. These designs are not only applicable to two level factorial experiments, but also can investigate main effects when factors have more than two levels. Designs are also available to investigate main effects for certain mixed level experiments where the factors included do not have the same number of levels. [Note] As in the case of Plackett-Burman designs, these designs require the experimenter to assume that interaction effects are unimportant and can be ignored. A few of Taguchi's orthogonal arrays available in DOE++ are included in Appendix C.

Some of Taguchi's arrays, with runs that are a power of 2, are similar to the corresponding 2 designs. For example, consider the L4 array shown in Figure 8.3 (a). The L4 array is denoted as L4(2^3) in DOE++. L4 means the array requires 4 runs. 2^3 indicates that the design estimates up to three main effects at 2 levels each. The L4 array can be used to estimate three main effects using four runs provided that the two factor and three factor interactions can be ignored. Figure 8.3 (b) shows the 2 design (defining relation ) which also requires four runs and can be used to estimate three main effects, assuming that all two factor and three factor interactions are unimportant. A comparison between the two designs shows that the columns in the two designs are the same except for the arrangement of the columns. In Figure 8.3 (c), columns of the L4 array are marked with the name of the effect from the corresponding column of the 2 design.

 Figure 8.3: Taguchi's L4 orthogonal array - Figure (a) shows the design, (b) shows the 2 design with the defining relation and (c) marks the columns of the L4 array with the corresponding columns of the design in (b).

Similarly, consider the L8(2^7) array shown in Figure 8.4 (a). This design can be used to estimate seven main effects using eight runs. This array is again similar to the 2 design shown in Figure 8.4 (b), except that the aliasing between the columns of the two designs differs in sign for some of the columns (see Figure 8.4 (c)).

 Figure 8.4: Taguchi's L8 orthogonal array - Figure (a) shows the design, (b) shows the 2 design with the defining relation and (c) marks the columns of the L8 array with the corresponding columns of the design in (b).

The L8 array can also be used as a full factorial three factor experiment design in the same way as a 2 design. However, the orthogonal arrays should be used carefully in such cases, taking into consideration the alias relationships between the columns of the array. For the L8 array, Figure 8.4 (c) shows that the third column of the array is the product of the first two columns. If the L8 array is used as a two level full factorial design in the place of a 2 design, and if the main effects are assigned to the first three columns, the main effect assigned to the third column will be aliased with the two factor interaction of the first two main effects. The proper assignment of the main effects to the columns of the L8 array requires the experimenter to assign the three main effects to the first, second and fourth columns. These columns are sometimes referred to as the preferred columns for the L8 array. To know the preferred columns for any of the orthogonal arrays, the alias relationships between the array columns must be known. The alias relations between the main effects and two factor interactions of the columns for the L8 array are shown in Table 8.2. The cell value in any () cell of the table gives the column number of the two factor interaction for the th and th columns. For example, to know which column is confounded with the interaction of the first and second columns, look at the value in the () cell. The value of 3 indicates that the third column is the same as the product of the first and second columns. The alias relations for some of Taguchi's orthogonal arrays are available in Appendix D.

 Table 8.2: Alias relations for the L8 array.

Example 8.1

Recall the experiment to investigate factors affecting the surface finish of automobile brake drums discussed in Chapter 7, Two Level Factorial Experiments. The three factors investigated in the experiment were honing pressure (factor ), number of strokes (factor ) and cycle time (factor ). Assume that you used Taguchi's L8 orthogonal array to investigate the three factors instead of the 2 design that was used in Chapter 7. Based on the discussion in the previous section, the preferred columns for the L8 array are the first, second and fourth columns. Therefore, the three factors should be assigned to these columns. The three factors are assigned to these columns based on Figure 8.4 (c), so that you can easily compare results obtained from the L8 array to the ones included in Chapter 7. Based on this assignment, the L8 array for the two replicates, along with the respective response values, should be as shown in Table 8.3. Note that to run the experiment using the L8 array, you would use only the first, the second and the fourth column to set the three factors.

 Table 8.3: Using Taguchi's L8 array to investigate factors affecting the surface finish of automobile brake drums.

The experiment design for this example can be set using the properties shown in Figure 8.5. Note that for this design, the factor properties are set up as shown in the design folio using the Factor Properties icon in the Control Panel. The factor properties to be used are shown in Figure 8.6. The resulting design along with the response values is shown in Figure 8.7 and the results from DOE++ for the design are shown in Figure 8.8. The results identify honing pressure, number of strokes, and the interaction between honing pressure and cycle time to be significant effects. This is identical to the conclusion obtained from the 2 design used in Chapter 7, Two Level Factorial Experiments.

 Figure 8.5: Design properties for the experiment in Example 8.1.

 Figure 8.6: Factor properties for the experiment in Example 8.1.

 Figure 8.7: Experiment design for Example 8.1.

 Figure 8.8: Results for the experiment in Example 8.1.