# GA-Based Optimization Method for Mobile Crane Repositioning Route Planning
## 3 Mobile Crane Repositioning Route Optimization Method
### Feasible Location Identification for the Mobile Crane  
RPOS identifies the points occupied by building (\(B^{b}\)) and assigns a value of 1 to the positions corresponding to the (x, y) coordinates of \(B^{b}\) among the \(M_{L}\) components (Equation (2)). Mobile crane cannot be installed at the supply points (\(P^{i}\)). RPOS assigns a value of 1 to the position corresponding to the (x, y) coordinate of \(P^{i}\) among the \(M_{L}\) components (Equation (3)). \[M_{L}\Big{(}B^{b}(1),B^{b}(2)\Big{)}=1,\;\forall b, \tag{2}\]  
\[M_{L}\Big{(}P^{i}(1),P^{i}(2)\Big{)}=1,\;\forall j. \tag{3}\]  
The user selects an appropriate mobile crane by considering the site conditions and inputs the data into the system. RPOS queries the physical attribute data of the user-selected mobile crane from the equipment database and stores them in \(C_{I}\), as shown in Equation (4). \[C_{I}=(W_{C},R_{C},L_{B},H_{C},H_{R}), \tag{4}\]  
where \(W_{C}\) is the weight of the mobile crane, \(R_{C}\) is the minimum radius for safe lifting, \(L_{B}\) is the maximum boom length, \(H_{C}\) is the height of the mobile crane body, and \(H_{R}\) is the minimum rope length. Next, RPOS considers the weight of the selected mobile crane (\(W_{C}\)), geotechnical investigation report, and GIS data to determine whether the outriggers should be installed on the ground and identifies the points (\(U^{r}\)) where the mobile crane cannot be installed. RPOS assigns a value of 1 to the position corresponding to the (x, y) coordinate of \(U^{r}\) among the \(M_{L}\) components (Equation (5)). \[M_{L}(U^{r}(1),U^{r}(2))=1,\;\forall r. \tag{5}\]  
For safe lifting work, the mobile crane must maintain a certain distance from the facilities that hinder the lifting work, such as the outer walls of buildings. This is called the minimum radius of the mobile crane (\(R_{C}\)). RPOS extracts the points (\(E^{r}\)) corresponding to the outer walls of the buildings among the points (\(B^{b}\)) occupied by the building components(\(E_{B}=\{E^{1},E^{2},...,E^{c}\}\)). If it is determined that a point adjacent to the outer wall of a building is located within \(R_{\rm C}\), then it is excluded from the feasible locations of the mobile crane. RPOS identifies the points (\(S_{B}{}^{e,\mathcal{E}}\)) where \(R_{\rm C}\) is not secured from each outer wall point (\(E^{c}\)) of the building (Equation (6)) and assigns a value of \(1\) to the position corresponding to the (x, y) coordinate of \(S_{B}{}^{e,\mathcal{E}}\) among the \(M_{L}\) components (Equation (7)). \[S_{B}{}^{e,\mathcal{E}}=N_{D}{}^{n};\,\sqrt{\left(E^{e}(1)-n(1)\right)^{2}+ \left(E^{e}(2)-n(2)\right)^{2}}\leq\frac{R_{\rm C}}{g^{s}},\,\,\forall n, \tag{6}\]  
where \(e\) is the index of the outer wall point (\(E^{e}\)), \(E^{e}(1)\) and \(E^{e}(2)\) are the x and y coordinates of the \(e\)-th outer wall point, respectively, \(n\) is the index of the on-site grid coordinate point, and \(N_{D}{}^{n}\) is on-site grid coordinate point. \[M_{L}(S_{B}{}^{e,\mathcal{E}}(1),S_{B}{}^{e,\mathcal{E}}(2))=1,\,\,\forall e,\,\,\forall g. \tag{7}\]  
RPOS calculates the maximum radius (\(R_{W}\)) of the mobile crane. \(R_{W}\) is calculated using Equation (8) by considering not only the physical characteristics of the mobile crane but also the length of the payload (\(L_{M}\)) and height of demand point (\(H_{I}\)) simultaneously. \[R_{W}=\sqrt{L_{B}{}^{2}-\left(H_{I}-H_{B}+L_{M}+H_{R}\right)^{2}}, \tag{8}\]  
where \(L_{B}\), \(H_{B}\), and \(H_{R}\) call the corresponding values from \(C_{I}\) defined in Equation (4), and \(L_{M}\) is defined by the user. Furthermore, \(H_{I}\), which is called from the IfcProduct coordinate system, is the height information (z value) of a particular load demand point (\(S^{i}\)). Lifting is performed by pairing a demand point and a supply point. The demand and supply points must be located within the working range of the mobile crane (\(R_{W}\)); otherwise, the lifting work is impossible. However, the demand and supply points may be in the one-to-many relationships.