**3.6 Identifying possible interactions**

The definition of interaction can be as follows: If the effect of a factor on the response variable depends on the value of the other factor, it is said that there is an interaction between two factors as seen in **Figure 2** [30]. The interactions can have a significant impact on performance characteristics. Taguchi thinks that interaction is not that important. The reason of this; the view is that in order to detect the interaction, the experimenter has to control the two main effects, and the interaction does not contribute anything when one or more of the main factors are under control [33]. Taguchi and Wu [35] suggest that one of the following techniques should be applied to reduce the interaction effects.


However, the experimenter must have the necessary attention and knowledge. It is difficult to add all interaction factors to the experiment due to the high cost and time required. On the other hand, including interaction factors believed to be important in the experiment will increase success. The existence of interaction between two factors can be determined by graphical procedure.

#### **Figure 2.**

*Graphical representation of interaction between two factors. (a) No interaction, (b) Weak interaction, (c) Strong interaction.*

## **3.7 Choosing appropriate orthogonal arrays**

Orthogonal Arrays (OA) take us all the way to Euler's Greco-Latin squares. But in Euler's time they were not known as OA. At that time they were known as mathematical games, like 36 office workers' problems. OA is a matrix of numbers arranged in rows and columns. Orthogonal arrays have a balanced property which entails that every factor setting occurs the same number of times for every setting of all other factors considered in the experiment. In an OA, each row represents the levels of the selected factors in a given experiment, and each column represents a specific factor whose effects on the process performance or product quality characteristic can be studied.

The idea of using OA in DOE independently of each other is originated in the USA and Japan after World War II [36]. The first use of OA was in the 1930s by Fisher in England. Taguchi added three OAs in 1956. And in the following years, three OAs were added by the American NIST [31]. Taguchi makes use of OA in performing multivariate experiments with a small number of trials. Using OA significantly reduces the size of the experiment to be studied [37]. The use of OA is not exclusive to Taguchi. However, Taguchi simplified their usage. Taguchi developed tabulated standard OA and corresponding linear graphs. A typical OA table is shown in **Table 2**.

In this array the columns are bilateral orthogonal. In each column there are all combinations of factor levels with an equal number. There are 4 factors (A, B, C, D) and three levels of each. This design is called the L9 design. The letter L indicates the orthogonal array, and 9 the row number, in other words the number of trials [4].

One point we should pay attention to that how much the OA reduces the number of trials. Due to the full factorial design (2<sup>k</sup> or 3<sup>k</sup> ), OA significantly reduces the number of attempts to be made in large numbers. For our example, 3<sup>4</sup> = 81 trials are required, but only 9 trials will be done to achieve the same results. It is obvious that it will provide more convenience in larger series. **Table 3** highlights the convenience that OA provides in terms of the number of trials [37].

OA allows working economically and simultaneously with many variables that are effective in product mean and variance. Two different OAs can be selected for CV and UCV. Using statistical DOE techniques, suitable subsets for CV and CIA can be demonstrated. Taguchi suggests using OA in planning DOE optimization. The multiplicity of CV and the emergence of interaction require very careful attention in the selection of OA and assignment of CV to columns. Target in establishing CV

matrix; It should be to setup a design where the most information can be obtained

**OA # of Factors and levels Full factorial design trial number**

L4 3 factors 2 levels 8 L8 7 factors 2 levels 128 L9 4 factors 3 levels 81 L16 15 factors 2 levels 32,768 L27 13 factors 3 levels 1,594,323 L64 21 factors 4 levels 4.4\*1012

**OA # of Row # of Maximum factor # of Maximum column**

L4 4 3 3 ——— L8 8 7 7 ——— L9 9 4 — 4 — — L12 12 11 11 ——— L16 16 15 15 ——— L16' 16 5 — — 5 — L18 18 8 1 7 — — L25 25 6 ——— 6 L27 27 13 — 13 — — L32 32 31 31 ——— L32' 32 10 1 — 9 — L36 36 23 11 12 — — L36' 36 16 3 13 — — .... .... ... .... .... ..... ....

**2 Levels 3 Levels 4 Levels 5 Levels**

*Frequently used OAs and full factorial design comparison.*

*Taguchi Method as a Robust Design Tool DOI: http://dx.doi.org/10.5772/intechopen.94908*

Depending on the levels of CV, an appropriate OA is chosen or some changes are made on the selected OA. The assignment of the CV and interaction variables to the columns is achieved by using standard linear graphs suitable for the selected OA. To determine a suitable OA for the experiment, the following procedure should be

with the least effort. **Table 4** presents a brief knowledge about the OAs.

1.Determination of the number of factors and their levels

2.Determining the degree of freedom

4.Consideration of interaction

followed.

**119**

**Table 4.**

*OA information table.*

**Table 3.**

3. Selection of OA


**Table 2.** *L9 orthogonal Array.*
