## 5 Boom Configuration Optimization  
### Problem Definition  
The optimization problem was introduced by [PERSON] et al. [3] as a constrained single objective optimization problem. Following their description, the objective is to find a boom configuration that minimizes material and welding costs for the plates and stiffeners while satisfying the structural and some other constraints (see below). Collecting the variables of a boom configuration in \(\mathbf{x}=(x_{1,b},x_{1,s},x_{1,t},\ldots,x_{n,b},x_{n,s},x_{nt,t})\), \(\mathbf{y}=(y_{1,b},y_{1,s},\ldots,y_{n,b},y_{n,s})\) and \(\mathbf{z}=(z_{b},z_{s})\), the objective function is defined as  
\[f(\mathbf{x},\mathbf{y},\mathbf{z})=wc_{boom}(\mathbf{x})+\sum_{i=1}^{n}\left( \sum_{j\in\{b\text{s},t\}}mc_{met}(x_{i,j})+\sum_{k\in\{b\text{s}\}}y_{i,k}\ (mc_{\text{siffener}}(z_{k})+wc_{\text{siffener}}(z_{k}))\right) \tag{4}\]  
with \begin{tabular}{l l} \(mc_{met}\) & material costs of the metal depending on the thickness \(x_{i,j}\); \\ \(mc_{\text{siffener}}\) & material costs of the stiffener depending on the stiffener type \(z_{k}\); \\ \(wc_{\text{siffener}}\) & welding costs of the stiffener depending on the stiffener type \(z_{k}\); and \\ \(wc_{boom}\) & welding costs of the plates depending on all thicknesses \(x_{i,j}\). \\ \end{tabular}   
While the material and welding costs of the stiffeners can be calculated independently for each segment, the welding costs of the boom depends on all thicknesses, since the metal sheets of the segments can be combined to larger plates. This may also affect the material costs of the metal sheets, in the case segments with different thicknesses would be combined to a single plate (typically then using the thicker one). Since, at this stage, the exact costs are not yet available, a cost approximation is used (and verified in Section 5.3 against the true cost function). To approximate the welding costs of the plates, a welding seam is calculated if  
Figure 4: Steps of the Box-Type Boom optimization. two consecutive segments have different metal thicknesses; or two consecutive segments have the same metal thicknesses and the previous three segments were already combined (i.e., were also of the same thickness). This approximation assumes that up to three segments can be combined into a long plate (based on metal plate size restrictions) and that the welding seams are exactly placed at the end of a segment. On the one hand, the placement of the welding seams in this way is in practice not allowed due to manufacturability constraints (no two welding seams are closer by than a certain offset), but, on the other hand, allows us to calculate the material costs of the metal sheet independently for each segment. Besides defining the cost function, the optimization problem also contains a set of constraints, which ensure that the designed boom configuration yields a functioning crane. These constraints are formally defined within the optimization problem below and elaborated thereafter (following [PERSON] et al. [3]). \[\min_{x_{i,j},y_{i,k}\geq k}f(\mathbf{x},\mathbf{y},\mathbf{z})\]  
subject to  
\[\sigma_{i,j}<1 (5) x_{i,b}\in\{\rho_{0},\ldots,\rho_{p}\}\subset\mathbb{N} (8) y_{i,b}\in\{0,1,2,3\} \tag{12}\] \[\phi_{i,j}<1 (6) x_{i,s}\in\{\tau_{0},\ldots,\tau_{q}\}\subset\mathbb{N} (9) y_{i,s}\in\{0,1,2\}\] (13) \[\beta_{i,k}<1 (7) x_{i,t}\in\{\omega_{0},\ldots,\omega_{r}\}\subset\mathbb{N} (10) y_{i,k}\geq y_{i+1,k}\] (14) \[x_{i,j}\geq x_{i+1,j} z_{k}\in\{1,2,3,4\} \tag{15}\]  
for all \(i\in\{1,\ldots,n\},j\in\{\mathsf{b},\mathsf{s},\mathsf{t}\},k\in\{\mathsf{b}, \mathsf{s}\}\). The first set of constraints (Equations (5)-(7)) ensures that the utility constraints with respect to stress (denoted by \(\sigma\)) and fatigue (\(\phi\)) on the bottom, side, and top as well as with respect to buckling (\(\beta\)) on the bottom and side are fulfilled for each segment. Any value larger than 1 is a violation, and, the larger the value, the higher is the violation.