# GA-Based Optimization Method for Mobile Crane Repositioning Route Planning
## 3 Mobile Crane Repositioning Route Optimization Method
### Near-Global Optimal Mobile Crane Reposition Route
\[f_{OT}=\frac{1}{\sum_{i=1}^{l}ht(k_{i},f_{i,k})+\sum_{i=2}^{l}tt(f_{i-1,k},f_{ i,k})+\sum_{i=2}^{l}rt\times a_{i}}, \tag{13}\]  
where \(ht(k_{i},f_{i,k})\) is the time required when the mobile crane is positioned at a point \(f_{i,k}\)-th feasible point for the \(S^{i}-E^{i,k}\) pair and lifts the load of the supply point \(E^{i,k}\) to the demand point \(S^{i};tt(f_{i-1,k},f_{i,k})\) is the time required when the crane moves from \(i-1\)th crane placement point to \(i\)-th placement point; \(rt\) is the time required to install/dismantle the mobile crane; \(a_{i}\) is a binary number that determines whether to install or dismantle the mobile crane. \(ht(k_{i},f_{i,k})\) is calculated using the lifting time estimation function of the crane defined using an ANN by [PERSON] et al. [26] (Equation (14)). The input variables of the lifting time  
Figure 3: Encoding a chromosome. estimation function (\(f_{\text{{{hosit}}}}\)) are the weight of the load (\(W_{M}\)), rotation angle of the crane (\(A_{C}\)), distance between the supply point and demand point (\(D_{L}\)), and height (\(H_{I}\)) of the demand point. The rotation angle of the mobile crane (\(A_{C}\)\({}^{i}\)) and distance between the supply point (\(E^{i,k}\)) and demand point (\(S^{i}\)) are calculated using Equations (15) and (16), respectively. The height (\(H_{I}\)\({}^{i}\)) of the supply point (\(S^{i}\)) is called from the IfcProduct coordinate system. The weight of the load (\(W_{M}\)) is defined by the user. \[ht(k_{i},f_{i,k})=f_{\text{{hosit}}}\left(W_{M},A_{C}\)\({}^{i},D_{L}\)\({}^{i},H_{I}\)\({}^{i}\right), \tag{14}\]  
\[AC^{i}=atan\left(\frac{E^{i,k}(2)-f_{i,k}(2)}{E^{i,k}(1)-f_{i,k}(1)}\right)-atan \left(\frac{S^{i}(2)-f_{i,k}(2)}{S^{i}(1)-f_{i,k}(1)}\right), \tag{15}\]  
\[D_{L}\)\({}^{i}=\sqrt{\left(E^{i,k}(1)-S^{i}(1)\right)^{2}+\left(E^{i,k}(2)-S^{i}(2) \right)^{2}}, \tag{16}\]  
RPOS uses [PERSON]'s algorithm (DA) to calculate the traveling time \(tt(f_{i-1,k},f_{i,k})\) between the points of the mobile crane. DA is an algorithm used for finding the shortest paths from a starting point to all other points. The DA operation requires a two-dimensional array called a weighted adjacency matrix. Weights can be interpreted in various ways, including cost, distance, and time. In this study, the weight is the traveling time of the mobile crane. The weighted adjacency matrix (\(G\)) was constructed by setting the traveling time value by considering the link length, traveling performance of the crane, and geotechnical attributes (i.e., rolling resistance, grade resistance, etc.) if a link existed between two nodes or by setting an infinite value if there was no link. After constructing the weighted adjacency matrix (G), the traveling time from \(i\)-1 th crane placement point to \(i\)-th placement point was calculated using the distances function (Equation (17)). However, using DA to calculate the point-to-point traveling time is valid assuming that the crane operates on the shortest distance when traveling between two points. \[tt(f_{i-1,k},f_{i,k})=distances(G,f_{i-1,k},f_{i,k}). \tag{17}\]  
The time \(rt\) required for installing/dismantling the mobile crane is defined by the user. The mobile crane is installed in a series of processes, including pedestal installation, outrigger deployment, and horizontal balance inspection. On the other hand, the crane is dismantled by the processes of outrigger retraction, pedestal removal, etc. The user considers the mobile crane type, installation/dismantling procedures, and proficiency of the operator to determine an appropriate \(rt\) value and input it into the system. \(a^{i}\) is a binary number that determines whether to install/dismantle the crane; and if the \(i\)-th crane placement point (\(F^{i,k}\)) is the same as the previous (\(i-1\)) point (\(F^{i-1,k,f}\)), then 1 is assigned; if not, 0 is assigned (Equation (18)). \[a_{i}=\left\{ \begin{array}{rl}1,&if\ F^{i,k,f}==F^{i-1,k,f}\\ 0,&otherwise\end{array}