# Increase of _Trichoderma harzianum_ Production Using Mixed-Level Fractional Factorial Design
## 2 Methodology  
The experimental strategy for the maximization of _Trichoderma harzianum_ spore production consists of five steps. _Step 1. Design structure._  
The proposed experiment is a \(2^{1}3^{2}4^{2}\) mixed-level factorial design, which contains 288 runs. The experiment includes three numerical factors (% humidity, temperature, and pH), two categorical factors (air and substrate), and the response variable spore production. The size of the full factorial makes this design economically infeasible. Therefore, running a fraction is a more suitable approach. _Step 2. Application of NOBA method._  
Many alternatives to fractionate mixed-level factorial designs have been proposed; however, these require complex programming techniques and a lot of computational time [11; 12; 13; 14; 15; 16; 17; 18]. The NOBA method offers the use of four simple steps to obtain the fraction using basic mathematics, allowing the experimenter to reduce the number of runs significantly. This method is flexible, and the experimenter can decide the size of the fraction to be constructed. The authors recommend choosing a size fraction with a high number of runs because this fraction will have better levels of balance and orthogonality, but of course, financial restrictions will play an important role in this decision [19]. _Procedure to apply the NOBA method:_  
(i) Evaluate the divisor factors and the sizes of the fractions that they produce from the full factorial. The divisor factor is a column of a factor that forms segments into its column and is used to divide the design into \(s\) segments and so to create a size \(s\) fraction. Therefore, \(s\) = number of runs in the full factorial \(n\)/size of each segment. Then, for the design (\(2^{1}3^{2}4^{2}\)) if we compute and analyze the divisor factor B, \(s=288/2=144\), for factor C, \(s=288/6=48\), and for the factor D, \(s=288/18=16\), and the factors A and E are discarded as possible divisor factors. To obtain as much information as possible while maintaining a budget, we select C as the divisor factor to produce a fraction with 48 runs. Note that it is unrecommended to use the first and last columns as divisor factors because they produce a fraction that is too big or too small [19]. (ii) Determine if the fraction will be balanced. To know this, it is necessary to calculate the least common multiple (_LCM_) for the factor's levels. The least common multiple for level 2,3,3,4,4 is 12, and since \(s\geq LCM\) (48 \(\geq\) 12), the fraction will be balanced. (iii) Assign positions. A position is a number assigned to each run within a segment, given that, in this case, each segment contains six runs. The positions take values from 1 to 6 within each segment (see Table 3). In this way, every run in the factorial design is labeled as segment \(i\), position \(j\), the number of positions in a segment can be denoted as \(p\) = \(n\)/\(s\), and the number of times that each position appears in the fraction is \(m\) = \(s\)/\(p\). Therefore, \(n=288\), \(s=48\) thus \(p=n\)/\(s=288/48=6\) and \(m=s\)/\(p=48/6=8\). (iv) Use of permuted vectors. Permuted vectors are used to assign the positions in the segments. According to the NOBA method for \(s=48\), \(p=6\), and \(m=8\), we need 8 vectors of size 6 to fill the 48 positions see Table 4. \begin{table}
\begin{tabular}{c c c c} \hline
**Run** & **Divisor Factor C** & **Segment** & **Position** \\ \hline
2 & 1 & & 2 \\
3 & 1 & & 3 \\
4 & 1 & & 4 \\
5 & 1 & & 5 \\
6 & 1 & & 6 \\ \hline
7 & 2 & & 1 \\
8 & 2 & & 2 \\
9 & 2 & & 3 \\
10 & 2 & & 4 \\
11 & 2 & & 5 \\
12 & 2 & & 6 \\ \hline \end{tabular}
\end{table} 
Table 3: Segments and positions for (288, \(2^{1}3^{2}4^{2}\)) using factor C as divisor factor. Next, we build the fraction by selecting one run for each segment, making sure that each position appears \(m\) times see Table 5.