## 6 Plate Partitioning Optimization
### Results  
The two optimization approaches are compared to elaborate how much improvement can be achieved by using a mathematical optimization (SimPP) over a heuristic optimization (SucPP). We created BTBs for lengths between 7000 mm and 69.500 mm with an increment of 10 mm. For a fair comparison of the two approaches, we restrict SimPP to determine the longest plates possible (i.e., not taking into account any segment thicknesses). The main results are shown in Figure 9.  
In the top of Figure 9, the resulting number of plates for the side of the middle section are shown for SucPP (black dashed lines) and SimPP (gray solid lines). As one can observe, SimPP slightly outperforms SucPP for longer booms (length \(\geq 39.000\,\mathrm{mm}\)). For the number of top and bottom plates, the differences in the number of plates are fewer. The respective plots are in the Supplementary Materials. While in terms of the number of plates the performance of the two approaches is rather similar, SimPP outperforms SucPP in terms of finding longer plate lengths, as shown in the histograms at the bottom of Figure 9. For each of the generated BTBs, the ratio of plate lengths among the first to second last plate lying within 1% of _MaxLength_ is determined (the last plate is disregarded since its length is merely set to reach the total length of the middle section _TotalLength_). Each histogram shows the frequencies of these ratios over all generated booms for the side plates. The respective histograms for the bottom and the top plates are in the Supplementary Materials. Additionally, in Table 2, some examples for the obtained lengths of the side plates for both approaches are shown. From these results, the advantages of using an optimization algorithm become apparent: Instead of setting the length successively as in SucPP (first top, then bottom, and finally side plate lengths), SimPP simultaneously searches for appropriate lengths, leading to a globally optimal solution. For example, in some cases, it is advantageous to shorten a bottom plate (while keeping the same number of plates) in order to obtain longer side plates, which lies outside the solution space of SucCP. When comparing SucPP and SimPP using the cost function of Equation (4), we can see in Figure 10 that the costs measured by this function are rather similar for the BTBs inferred by SucPP and SimPP. While this cost-saving with SimPP is marginal, there are other places where costs can be saved, as mentioned in Section 6.1. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Length of Section** & **SucPP—Lengths of Side Plates** & **SimPP—Lengths of Side Plates** \\ \hline
26,948 & 9683 **9948** 7317 (0.5 ) & 9679 **9952** 7317 (0.5) \\
43,036 & 9685 **9950** 9535 9532 4334 (0.25) & 9684 **9952** **9952** **9952** **9952** **3496** (0.75) \\
48,916 & 9685 **9951** 9536 9530 **9912** 302 (0.4) & 9497 **9952** **9952** **9952** **9563** (0.75) \\
52,606 & 9685 **9951** 9538 9527 **9952** **9952** **3953** (0.4) & 9497 **9952** **9952** **9950** **9952** **3302** (0.8) \\
61,915 & 9686 **9951** 9538 9526 **9952** **9952** **9952** **3310** (0.5) & 9497 **9952** **9952** **9950** **9952** **9952** **2660** (0.83) \\
69,405 & 9686 **9951** 9540** 9524 **9952** **9952** **9952** **9952** **9952** **9952** **9952** **9952** **9952** **9952** **9447** **703** (0.71) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Examples of resulting lengths of side plates, with _MaxLength_\(=9952\,\mathrm{mm}\). Bold values indicate plates whose length is within 1% of _MaxLength_. Figure 9: Comparison of the two approaches for calculating the side plate partitioning. Regarding the time requirements of SucPP and SimPP, the run times are compared by measuring wall-clock time. The results are presented in Figure 11 as a box-plot, showing the time requirement in seconds in dependence of the boom length. For both algorithms time requirements increase, when additional plates are introduced.