# GA-Based Optimization Method for Mobile Crane Repositioning Route Planning
## 3 Mobile Crane Repositioning Route Optimization Method
### Feasible Location Identification for the Mobile Crane
For example, at demand point I1, as shown in Figure 2, loads can be lifted from two supply points (L1 and L2). RPOS identifies the available supply points (\(E^{i,k}\)) where loads can be lifted for each demand point (\(S^{i}\)) based on \(R_{W}\) (Equation (9)). \[E^{i,k}=P^{j};\,\sqrt{\left(S^{i}(1)-P^{j}(1)\right)^{2}+\left(S^{i}(2)-P^{j }(2)\right)^{2}}\leq\frac{2\times R_{\rm C}}{g^{s}},\,\,\forall j, \tag{9}\]  
where \(i\) is the index of the demand point (\(S^{i}\)), \(S^{i}(1)\) and \(S^{i}(2)\) are the x and y coordinates of the \(i\)-th demand point (\(S^{i}\)), respectively, \(j\) is the index of the supply point (\(P^{j}\)), \(P^{j}(1)\) and \(P^{j}(2)\) are the x and y coordinates of the \(j\)-th supply point (\(P^{j}\)), respectively. Figure 2: Working range of the mobile crane and the available supply point for the demand point. Next, RPOS creates a matrix \(M_{\text{C}}\)\({}^{i,k}\) to store the feasible points of the mobile crane for each pair of demand point (\(S^{i}\)) and supply point (\(E^{i,k}\)). \(M_{\text{C}}\)\({}^{i,k}\) is a matrix of \(M\times N\) size and has 0 as an attribute value: \(M_{\text{C}}\)\({}^{i,k}\) = zeros (\(M,N\)). RPOS identifies the coincident points (\(EF^{i,k,q}\)) for the \(S^{i}-E^{i,k}\) pair by intersecting the sets of points (\(SF^{i,p}\)) located within the range of \(R_{W}\) centering on the \(S^{i}\) and points (\(RF^{i,k,q}\)) located within the range of \(R_{W}\) centering on the \(E^{i,k}\) (Equation (10)). RPOS assigns a value of 2 to the component corresponding to the (x, y) coordinate of \(RF^{i,k,t}\) among the components of \(M_{\text{C}}\)\({}^{i,k}\) (Equation (11)). \[\left\{RF^{i,k,t}\right\}=\left\{SF^{i,p}\right\}\cap\left\{EF^{i,k,q}\right\},\ \forall p,\ \forall q\]  
\[ \begin{array}{l}SF^{i,p}=n;\ \sqrt{\left(S^{i}(1)-n(1)\right)^{2}+\left(S^{i}(2 )-n(2)\right)^{2}}\leq\frac{R_{W}}{8s},\ \forall n\\ EF^{i,k,q}=n;\ \sqrt{\left(E^{i,k}(1)-n(1)\right)^{2}+\left(E^{i,k}(2)-n(2)\right)^{2}} \leq\frac{R_{W}}{8s},\ \forall n\end{array}  \tag{10}\]  
\[M_{\text{C}}\)\({}^{i,k}\)\(\left(RF^{i,k,t}(1),RF^{i,k,t}(2)\right)=2,\ \forall t,\) (11)  
where \(n\) is the index of the on-site grid coordinate point, \(p\) is the index of the point (\(SF^{i,p}\)) located within the range of \(R_{W}\) centering on the demand point \(S^{i}\), and \(q\) is the index of the point (\(EF^{i,k,q}\)) located within the range of \(R_{W}\) centering on the supply point \(E^{i,k}\). RPOS performs the addition of \(M_{L}\) and \(M_{\text{C}}\)\({}^{i,k}\) to derive \(M_{i}\)\({}^{i,k}\) (i.e., \(M_{i}\)\({}^{i,k}\) = \(M_{L}\)\({}^{i,k}\)). Additionally, RPOS determines the set of points where the attribute value of \(M_{i}\)\({}^{i,k}\) is 2 (\(F_{L}\)\({}^{i,k}\) = \([F^{i,k,1},F^{i,k,2},...,F^{i,k,q}]\)) as feasible points (\(F^{i,k,f}\)) for the \(S^{i}-E^{i,k}\) pair. The points where the attribute value of \(M_{i}\)\({}^{i,k}\) is 1 or 3 indicate the points already occupied by buildings, etc., (\(B_{b}\)), points where the bearing capacity of soil is not secured (\(R^{b}\)), or points where the safe work distance is not secured (\(S_{B}\)\({}^{i,g}\)). \(F^{i,k,f}\) for all demand points (\(\{S^{1},S^{2},...,S^{i}\}\)) is stored in the system memory.