Preliminaries

Orthogonal experimental design for parameter optimisation provides a systematic and efficient approach to determine near optimal parameter settings. The objective is to select the best combination of control factors (parameters) so that the product or process is most robust with respect to noise factors. The orthogonal experimental design applies orthogonal arrays

from experimental design theory to study a large number of variables with a small number of trials, significantly reducing the number of experimental configurations. Moreover, in case that parameter-interaction space is relatively smooth, the conclusions drawn from such small- scale experiments are valid over the entire experimental region spanned by the control factors and their settings.

6.4.1.1 Orthogonal Array

Orthogonal arrays are a special set of Latin squares (Dénes and Keedwell, 1974), constructed by Taguchi to lay out the experimental design. In this array, the columns are said to be mutually orthogonal or balanced. That is, for any pair of columns, all combinations of factor levels occur, and occur an equal number of times. By using the table, the required experimental situations are defined. Consider a 3-level and 4-factor orthogonal array shown in Table 6.1 below:

Table 6.1: Orthogonal array L9(3 4

)

The array is designated by the symbol L9(3 4

), involving four factors A, B, C, and D, each at three levels one (1), two (2), and three (3). The array has a size of nine rows and four columns. The numbers (1/2/3) in the row indicate the factor levels and each row represents

A B C D 1 1 1 1 1 2 1 2 2 2 3 1 3 3 3 4 2 1 2 3 5 2 2 3 1 6 2 3 1 2 7 3 1 3 2 8 3 2 1 3 9 3 3 2 1

specific test characteristics of each experiment. The vertical columns represent the experimental factors to be studied using that array. Each of the columns contains three assignments at each levels (1, 2, or 3) for the corresponding factors. These conditions can combine in nine possible ways (i.e. (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)) for two factors, with 34 possible combinations of levels for all the four factors.

The orthogonal array facilitates the experimental design process by assigning factors to the appropriate columns. In this case, referring to table 1, factors A, B, C, and D are arbitrarily assigned to columns 1, 2, 3, and 4 respectively. From the table, nine trials of experiments are needed, with the level of each factor for each trial-run as indicated in the array. The experimental descriptions are reflected through the condition level. The experimenter may use different designators for the columns, but the nine trial-runs will cover all combinations, independent of column definition. In this way, the orthogonal array assures consistency of the design carried out by different experimenters. The orthogonal array also ensures that factors influencing the quality of solutions are properly investigated and controlled during the initial design stage.

6.4.1.2 Comparison to the traditional method of factorial design

The traditional method of factorial design is to investigate all possible combinations and conditions in an experiment that involves multiple factors. Let A be the number of levels for each factor, and B be the number of factors involved, then the number of possible designs N (number of trials) by this method is

B

A

N = . (6.19) If the factoria l design is implemented for the four 3-level factors in Table 6.1, the total number of trials needed would be a full combination of 81 (34) trials, rather than 9 trials by the orthogonal array L9(3

4

The reason why by using orthogonal array superior parameter configurations could be found by only a small number of experiments is because of its mutual balance. For example, in Table 6.1, each column contains three ones, three twos, and three threes; and any pair of columns contain all combinations of levels (i.e. (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)) exactly once. It is the characteristic of mutual balance that guarantees the choice of combinations producing elite solutions.

Furthermore, in the factorial design process, the means of levels combination laid out is not specified, which may lead to different results on the same experimental subject each time a trial is conducted. However, Taguchi’s orthogonal array is able to simplify and standardize the factorial design in a manner that will produce consistent and similar results, even though the trials are implemented by different experimenters. Hence, two different investigators will have similar results and a standard design methodology.

The concept of consistent results and standard design methodology through orthogonal array analysis is important, because it allows the experimenter to produce two outcomes of the same quality standards, using the same materials, but with differences in the experimental process. This is possible since, through orthogonal array experimental analysis, the factors influencing the quality of results can be identified, controlled, and subsequently compensated during the early design stage. Thus, the quality of the outcome itself is able to adapt to the experimental process, rather than depends on the experimental process.

In summary, in case that parameter-interaction space is relatively smooth, compared with the traditional factorial design method, Taguchi’s orthogonal array is considered to be superior since:

• It can produce similar and consistent results, even though the experiments may be carried out by different experimenters;

• It enables determination of the contribution of each quality-influencing factor.

The limitation of orthogonal arrays is that they can only be applied at the initial design stage. There are some situations where orthogonal array techniques are not available, such as a process involving control factors that vary in time and cannot be quantified exactly.