# GA-Based Optimization Method for Mobile Crane Repositioning Route Planning
## 4 Method Verification
### Optimization  
Table 1 in Appendix A shows the results obtained using Equations (6) and (11) to identify the feasible supply points (\(E^{i,k}\)) and the feasible points for the mobile crane (\(F^{i,k}\)) for the demand points (\(S_{i}\)) in this test case. Therefore, based on Equation (12), the size of the optimal solution search range in this test case was 3.09878 \(\times\) 10\({}^{21}\) (= \((3+1+2+3+3+3)\times(13+5+4+1)\times(10+1+11+8)\times(2+12+13+4+2)\times(5+13+15+10) \times(3+14+2+11)\times(3+3+3+2+2)\times(10+6+9)\times(3+3+2+1+2+1+3)\times(5+4+12+4 )\times(1+4+6+6+10+5)\times(1+1+15+1+16)\times(5+15+16+13+2)\times(11+15+8+6) \times(14+11+1)\)).  
The GA parameters were initialized with [\(PS\) = 400, \(CR\) = 0.4, \(MR\) = 0.05, StallGenLimit = 200 times, TimeLimit = Inf, StallTimeLimit = Inf] to execute the GA optimization operation. The GA operation stopped as the allowable cumulative change of the objective function reached 1 \(\times\) 10\({}^{-6}\) after undergoing many generations.  
RPOS found [(3, 1), (3, 4), (3, 9), (3, 4), (2, 1), (2, 8), (3, 3), (1, 5), (7, 1), (3, 4), (5, 3), (3, 2), (2, 4), (2, 6), (1, 7)] as the reposition route that minimized the operating time of the mobile crane (Table 1). These results were expressed using the chromosomes designed for the GA operation. Converting these results into the location points of the mobile crane yields [124, 124, 124, 124, 124, 231, 231, 231, 231, 231, 176, 52, 52, 176, 52, 52]. In other words, the mobile crane operates by moving in a sequence of 124\(\rightarrow\)231\(\rightarrow\)176\(\rightarrow\)52. It is shown that three relocations were required for the task. Furthermore, transforming the optimization operation results of the GA into pairs of the supply points for the demand points yields [82, 82, 82, 161, 161, 247, 247, 247, 254, 254, 218, 53, 53, 49]. If these points are interpreted using the set of the column installation points (demand points), for which the installation sequence has been determined, and the placement sequence of the crane, the following results can be obtained. First, the mobile crane is positioned at point 124 to lift the columns to the demand points 148, 108, 106, 146, and 186. Here, the columns for demand points 148, 108, and 106 are fetched from the supply point 82, whereas the columns for demand points 146 and 186 are fetched from the supply point 161. Then, the crane moves to point 231 and lifts the columns to the demand points 188, 150, 190, 152, and 192. The columns for demand points 188, 150, and 190 are fetched from the loading point 247, whereas the columns for the demand points 152 and 192 are fetched from the supply point 254. Then, the crane moves to points 176 and 52 to continue loading (Figure 5).  
The total required operating time of the mobile crane according to the GA operation result was 4326 s. However, this result does not consider the time required for hanging a column on the hook of the crane, the waiting time after lifting a column to the demand point, and the time before installing/fixing it after lifting. They are not included in the result because that time does not affect the optimal solution search. In the test case, as the column lifting work is performed 15 times, the total work time can be derived if the waiting time for installing/fixing is estimated and multiplied by 15 and then added to the required operating time of 4326 s. The decision-maker can check the result (Table 1, Figure 5) obtained by using our method to effectively and easily determine where the mobile crane should be located, where it should travel, and the location and payload quantity that should be loaded.