## 6 Results and Discussion
For a value of 20 MPa, the only a priori valid cross-section is 50 \(\times\) 25 mm\({}^{2}\), although it may be interesting to carry out a more exhaustive search through a new optimisation process focused on other cross-sections whose \(\sigma_{\text{m,min}}\) slightly exceeds this value, such as 50 \(\times\) 30 mm\({}^{2}\) or 55 \(\times\) 25 mm\({}^{2}\), in which some individual that meets this condition could perhaps be found. In this work, the maximum value for initial bending stress was set at 33 MPa. The staggered dashed line in Table 2 separates the cross-sections that meet this condition. The DM can choose from any of the cross-sections in the group or look at some of the nearby cross-sections that are left out, if necessary. In the case presented here, the 75 \(\times\) 30 mm\({}^{2}\) cross-section was chosen as a result of the minimum width imposed by the selected bolt \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Depth (mm)**} & \multicolumn{8}{c}{**Width (mm)**} \\ \cline{3-10}  & & **50** & **55** & **60** & **65** & **70** & **75** & **80** \\ \hline
25 & \(\sigma_{\text{m,min}}\) (MPa) & 19.96 & 21.54 & 23.19 & 27.85 & 30.69 & 31.92 & 31.70 \\  & \(\sigma_{\text{m,max}}\) (MPa) & 29.77 & 48.82 & 43.07 & 51.27 & 44.87 & 43.52 & 36.98 \\  & AI \({}^{1}\) (MPa) & 9.81 & 27.28 & 19.88 & 23.42 & 14.18 & 11.6 & 5.28 \\  & RI \({}^{2}\) & 0.33 & 0.56 & 0.46 & 0.46 & 0.32 & 0.27 & 0.14 \\  & n & 364 & 75 & 15 & 4 & 7 & 8 & 6 \\
30 & \(\sigma_{\text{m,min}}\) (MPa) & 21.00 & 22.53 & 24.30 & 29.24 & 31.37 & 32.88 & 31.49 \\  & \(\sigma_{\text{m,max}}\) (MPa) & 44.38 & 51.86 & 44.94 & 56.17 & 59.74 & 47.60 & 60.00 \\  & AI \({}^{1}\) (MPa) & 23.38 & 29.33 & 20.64 & 26.93 & 28.37 & 14.72 & 28.51 \\  & RI \({}^{2}\) & 0.53 & 0.57 & 0.46 & 0.48 & 0.47 & 0.31 & 0.48 \\  & n & 184 & 86 & 22 & 12 & 10 & 6 & 11 \\
35 & \(\sigma_{\text{m,min}}\) (MPa) & 22.39 & 23.92 & 25.41 & 29.80 & 29.08 \({}^{1}\) & 31.23 \({}^{1}\) & 33.90 \\  & \(\sigma_{\text{m,max}}\) (MPa) & 44.97 & 60.51 & 64.53 & 51.30 & 60.27 & 64.54 & 54.67 \\  & AI \({}^{1}\) (MPa) & 22.58 & 36.59 & 39.12 & 21.5 & 30.29 & 33.31 & 20.77 \\  & RI \({}^{2}\) & 0.50 & 0.60 & 0.61 & 0.42 & 0.50 & 0.52 & 0.38 \\  & n & 16 & 34 & 14 & 18 & 13 & 10 & 2 \\
40 & \(\sigma_{\text{m,min}}\) (MPa) & 25.31 & 26.56 & 27.80 & 29.40 & 31.02 & 32.20 & 55.92 \\  & \(\sigma_{\text{m,max}}\) (MPa) & 63.66 & 59.00 & 55.25 & 74.41 & 57.35 & 68.15 & 70.20 \\  & AI \({}^{1}\) (MPa) & 38.35 & 32.44 & 27.45 & 45.01 & 26.33 & 35.95 & 14.28 \\  & RI \({}^{2}\) & 0.60 & 0.55 & 0.50 & 0.60 & 0.46 & 0.53 & 0.20 \\  & n & 7 & 23 & 10 & 10 & 6 & 19 & 2 \\
45 & \(\sigma_{\text{m,min}}\) (MPa) & 30.11 & 29.18 & 31.58 & 36.14 & 35.24 & 34.95 & 38.07 \\  & \(\sigma_{\text{m,max}}\) (MPa) & 53.43 & 70.54 & 72.22 & 81.91 & 77.71 & 66.66 & 78.66 \\  & AI \({}^{1}\) (MPa) & 23.32 & 41.36 & 40.64 & 1 & 45.77 & 42.47 & 31.71 & 40.59 \\  & RI\({}^{2}\) & 0.44 & 0.59 & 0.56 & 0.56 & 0.55 & 0.48 & 0.52 \\  & n & 6 & 19 & 13 & 1 & 6 & 12 & 7 & 11 \\
50 & \(\sigma_{\text{m,min}}\) (MPa) & 33.52 & 32.24 & \(\cdot\) & 52.27 & 61.59 & 49.61 & 47.31 & 41.69 \\  & \(\sigma_{\text{m,max}}\) (MPa) & 44.51 & 75.87 & \(\cdot\) & 52.27 & 88.78 & 49.61 & 81.84 & 41.69 \\  & AI \({}^{1}\) (MPa) & 10.99 & 43.63 & \(\cdot\) & 0 & 27.19 & 0 & 34.53 & 0 \\  & RI\({}^{2}\) & 0.25 & 0.58 & \(\cdot\) & 0 & 0.31 & 0 & 0.42 & 0 \\  & n & 5 & 7 & 1 & 3 & 1 & 5 & 1 \\ \hline \hline \end{tabular}  
* Absolute improvement, AI = \(\sigma_{\text{m,max}}-\sigma_{\text{m,min}}\); \({}^{2}\) Relative improvement, RI = \(1-\sigma_{\text{m,max}}/\sigma_{\text{m,min}}\).  
\end{table} 
Table 2: Step 1 optimisation outcome. diameter (12 mm) and the maximum timber utilisation that could be achieved from the cross-sections available in the sawmill. As can be seen in Table 2, the optimisation algorithm in Step 1 only provided six individuals for the chosen cross-section, of which only 1 had a stress below 33 MPa (Figure 20), so a broader search was necessary.