## 6 Plate Partitioning Optimization
### Optimization Approaches
#### 6.2.2 Simultaneous Plate Partitioning  
For the task at hand, the simplex algorithm [17] is a suitable choice since the optimization problem can be re-written as a series of linear programs (LPs). The problem of optimizing the plate partitioning in the sense of achieving the longest plates possible (i.e., smallest number of plates possible) is encoded into a _series of LPs_. This is needed since the constraints in Equations (21) and (22) are not linear. We demonstrate with an example how this linearization and problem formulation is made using a series of LPs; the general formulation follows analogously. Assuming a length of the middle section of _TotalLength_ = 25,000 mm and a maximum plate length _MaxLength_ = 10,000 mm an LP is set up as follows. The Decision Variables **r**  
Informally speaking, to determine the number of bottom (\(n_{b}\)), side (\(n_{s}\)), and top (\(n_{t}\)) plates (and thus the number of variables), the total length of the middle section (25,000 mm) is divided by the maximum length of each plate (10,000 mm), and the ceiling of this number is taken as the result. If the simplex algorithm does not find a solution with this number of plates, it is successively increased first for the side, then for the bottom, and finally for the top plates. In our example, this means that there are three bottom (\(u_{i}\)), three side (\(v_{i}\)), and three top plates (\(w_{i}\)), i.e., nine variables in total. Note that the left and right side of the boom are identical. The Objective Function \(c_{1}r_{1}+\cdots+c_{n}r_{n}\)  
Since there are nine variables, the constant weights \(c_{1},\ldots,c_{9}\) have to be specified. The objective function is chosen such that the first to the second last plates on the bottom, side, and top, respectively, are made as long as possible (assigning equal weights \(c_{j}=1\)). The length of the last plate is merely used to reach the correct total length (setting the corresponding weights \(c_{i}=0\)). This means that the function \(u_{1}+u_{2}+v_{1}+v_{2}+w_{1}+w_{2}\) is to be _maximized_. The Constraints  
The requirements of obtaining the appropriate length for the boom, obeying manufacturing restrictions (minimal and maximal lengths of the plates), and taking the welding seams into account are formalized as follows. If not stated otherwise, the indices \(i,j\) and \(k\) range from 1 to 3. * Considering minimum and maximum lengths of plates as well as the sum of the single plates, the constraints are identical to the ones in Equations (18)-(20). These constraints have to be formulated for the bottom, side, and top plates. * Considering minimum gap between welding seams between plates and bulkheads (corresponding to Equation (21)), as mentioned in Section 4, the bulkheads are positioned at equal distances, starting from the pivot section. To be able to formulate the optimization problem as a linear one, in each LP, only the position of one bulkhead for the first to second last bottom plate can be considered at a time. Thus, a series of LPs is considered, differing in the selection of the bulkheads, among which the best solution is chosen. As an example, assume that, for the current formulation, the third and seventh bulkheads are selected. The constraints are set up such that the first and second bottom plate end between the third and fourth, and seventh and eighth bulkheads, respectively: \[u_{1} \geq 3\cdot\text{DistBulkheads}+\text{MinGap}\] \[u_{1} \leq(3+1)\cdot\text{DistBulkheads}-\text{MinGap}\] \[u_{1}+u_{2} \geq 7\cdot\text{DistBulkheads}+\text{MinGap}\] \[u_{1}+u_{2} \leq(7+1)\cdot\text{DistBulkheads}-\text{MinGap}\] The formulation for the welding seams between the side plates and the bulkheads is similar.