Orthogonal Array (OA)
Orthogonal Array (OA)
An Orthogonal Array is a table designed to allocate combinations of levels in experiments, ensuring that combinations of levels for any two factors appear an equal number of times.
Orthogonal arrays come in different types, such as 2-level and 3-level systems, and mixed arrays like L12, L18, and L36. Mixed arrays are often used in parameter design.
L18
| Ｎｏ． | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| 3 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| 4 | 1 | 2 | 1 | 1 | 2 | 2 | 3 | 3 |
| 5 | 1 | 2 | 2 | 2 | 3 | 3 | 1 | 1 |
| 6 | 1 | 2 | 3 | 3 | 1 | 1 | 2 | 2 |
| 7 | 1 | 3 | 1 | 2 | 1 | 3 | 2 | 3 |
| 8 | 1 | 3 | 2 | 3 | 2 | 1 | 3 | 1 |
| 9 | 1 | 3 | 3 | 1 | 3 | 2 | 1 | 2 |
| 10 | 2 | 1 | 1 | 3 | 3 | 2 | 2 | 1 |
| 11 | 2 | 1 | 2 | 1 | 1 | 3 | 3 | 2 |
| 12 | 2 | 1 | 3 | 2 | 2 | 1 | 1 | 3 |
| 13 | 2 | 2 | 1 | 2 | 3 | 1 | 3 | 2 |
| 14 | 2 | 2 | 2 | 3 | 1 | 2 | 1 | 3 |
| 15 | 2 | 2 | 3 | 1 | 2 | 3 | 2 | 1 |
| 16 | 2 | 3 | 1 | 3 | 2 | 3 | 1 | 2 |
| 17 | 2 | 3 | 2 | 1 | 3 | 1 | 2 | 3 |
| 18 | 2 | 3 | 3 | 2 | 1 | 2 | 3 | 1 |
Using this L18 orthogonal array, 1 factor with 2 levels and 7 factors with 3 levels can be investigated. While a full-factorial experiment would require 4,374 runs, this can be accomplished with just 18 runs using the orthogonal array. However, reducing the number of experiments is not the main purpose of orthogonal array experiments.
Purpose of Orthogonal Array Experiments
Orthogonal array experiments are conducted to verify whether the gains from laboratory experiments can be reproduced under downstream conditions, such as at the production site or in the market.
〇Reliability Improvement: In an L18 experiment, the effect of each factor level is evaluated using 18 samples, which is comparable to 144 experiments in one-factor-at-a-time methods.
〇Interaction Assessment: While smaller interactions are preferable, determining their magnitude is not particularly useful since factor levels are ultimately fixed. The presence or absence of interactions is confirmed through validation experiments.
〇Stability Evaluation: The key advantage is that when investigating one factor, the levels of other factors are varied. The factor effect diagrams obtained from orthogonal arrays reflect average effects that are robust even when other factor levels change slightly. As a result, stable and favorable conditions are more likely to be selected.
〇Interaction with Noise: For further stability evaluation, external noise factors are deliberately introduced. This evaluates the interaction between control factors and noise factors to select conditions that are robust to noise. Improvement is only possible when there is interaction between control factors and noise, which is efficiently evaluated using the Signal-to-Noise (S/N) ratio.
Orthogonal Arrays and Degrees of Freedom
The total degrees of freedom that can be investigated depend on the size of the orthogonal array. The maximum degrees of freedom are equal to (Number of rows – 1).
- L8: Can investigate up to 7 factors with 2 levels each (2-1) × 7 = 7.
- L9: Can investigate up to 4 factors with 3 levels each (3-1) × 4 = 8.
- L25: Can investigate up to 6 factors with 5 levels each (5-1) × 6 = 24.
Follow me!
