Orthogonal experiment test case design and tool recommendation

In scientific research and production practice, people often do many experiments to carry out a certain research. Experimental conditions generally include many factors. When the values ​​of factors are different, the experimental results are also different. If you want to experiment with each value of each factor, the total number of experiments is equal to the product of the number of values ​​of each factor, and this number is often too large to exceed the acceptable cost.
For example, suppose an experiment consists of four factors A, B, C, and D, and each factor has 10 different values. If we want to take each factor into account, we need to do 10 10 10* 10 = 10000 experiments.
In order to reduce the number of experiments, we must select those most representative examples. Therefore, the Orthogonal Array Testing Strategy is used.

Basic Mathematical Properties of Orthogonal Tables

Let the strength of the orthogonal table be S, then the orthogonal table has the following mathematical properties:

• 1. Orthogonality: Orthogonality has two meanings
  • ⑴ In the last S column, a level value of each factor in each column collides once and only once with each level value in other columns. In other words, the subtable formed by the last S columns is full. Therefore, the number of records in the orthogonal table = the product of the number of levels in the last S column.
  • (2) For any S column at the same position, there is no repeated element in the set of S-dimensional ordered number pairs formed by it. In particular, when the levels of the factors are equal, for each set N composed of S columns, this set N traverses each point in the S-dimensional space (that is, the Cartesian product of the S column) once and Only traverse once, figuratively speaking, it is "neither heavy nor leaky". But when the factors are not equal, the set of the last S column must be full. That is the situation mentioned in (1).

From this we can see the role of strength, strength is like a sieve, screening out all the records in the solution space that conform to the principle of S-dimensional orthogonality, when S=1, only traverse all the values ​​of the last variable. When S=number of factors, the whole solution space is obtained.

• 2. Uniformity: The number of occurrences of the level value of each factor in the table is uniform. For each column in the last S column (they must be full), the number of occurrences of each level value is equal.

Orthogonal experimental design test case

Orthogonal tables are a method of screening experimental cases. Before introducing its specific content, we first introduce a few basic concepts

• (1) The number of factors, Factors, will be replaced by F in this article, and the factors correspond to a column in the orthogonal table
• ⑵Levels, which will be abbreviated as L in the future. Its meaning is the number of possible values ​​for each factor. Note that here we don't care about the number of each specific value, but the number.
  The specific value of the variable is called the level value. If it is not confused with the level number, it is called the level for short, and is represented by the variable name + number. For example, a factor A may have three levels, which can be recorded as A1, A2, A3.
• (3) Strength, hereafter abbreviated as S: Strength is one of the most important indicators for constructing an orthogonal table. The specific meaning will be explained in detail later. Here we simply say that the core nature of the orthogonal table is the last S factors. Each level value should touch each other once and only once.
• ⑷Runs: the number of record rows in the last generated orthogonal table, one row of records is an experiment.
• (5) The symbolic representation of the orthogonal table: start with the letter L first, and the subscript r indicates the number of records,

In parentheses is the multiplication of terms with the number of factors with the same level number.

To give a few specific examples:

• ⑴ There are 3 factors A, B, C, and when the level number of each factor is 3, the generated orthogonal table is L27(33) (take the case where the strength is equal to 3), and the number of records is 27=3 3 3
• (2) There are 5 factors, and when the levels of each factor are 2, 2, 2, 3, 3 respectively, the generated orthogonal tables of different strengths S are respectively:
  • When s=2, the result is that
    
    the number of records is the product of the level numbers of the last two variables 3*3=9
  • When s=3, the result is that
    the number of records is the product of the level numbers of the last three variables 2 3 3=18

Orthogonal table constructed by hand

Let’s look at a specific example: There are 4 variables A, B, C, D, the level number of the first three variables is 3, and the level of the last variable is 4, then, according to different strengths, different orthogonal tables can be obtained . When the strength s=2, first get the basic orthogonal table

It can be seen that each horizontal value of C collides with each horizontal value of D once and only once. The number of occurrences of each level of A and B is also very uniform. And any pair of ordered numbers consisting of two columns at the same position has no duplicate values.
In order to maintain the uniformity of the value, fill the items with 0 with the horizontal value of the factor, and get the final orthogonal table with
red numbers, which is the result of our circular filling with the horizontal value. Similarly, we can get an orthogonal table with a strength of 3 and

the number of records is 4 3 3 = 36.

It can be seen that it is not an easy task to manually construct an orthogonal table, so we provide a design test case based on the orthogonal version small tools.

Orthogonal experiment design test case tool download address: Click to download