# Ferry Scheduling Optimization Considering Arrival Time Uncertainty and In-Place Time Differences
## 4 Numerical Example Analysis
### Research on Algorithm Parameters  
In Section 3, a multi-parameter configurable GA was designed to effectively and quickly solve the model. This section studies the range of values for population size, iteration times and sample size. Firstly, population size is an important indicator that affects the duration of a single iteration. In a single iteration, the larger the population size, the larger the search space of the algorithm. The more times the fitness of each individual is calculated, the lower the search efficiency. In contrast, the smaller the population size, the greater the number of iterations required. For the above case, assuming that the estimated arrival and departure time of the flight is strictly consistent with the actual time, the problem is the static scheduling problem of the ferry. By using precise algorithms to solve the final determination, without considering the uncertainty of flight arrival and departure time, the minimum number of ferries used is 10. In Table 2, the changes in relevant indicators during the population size change from 10 to 100 are shown. For obtaining the optimal solution from the indicator of the maximum number of iterations, when the population size is small, the number of iterations needed to obtain the optimal solution is relatively similar and even limited by the  
Figure 7: Deviation in flight departure and arrival time. set number of iterations. Similarly, there is a strong positive correlation between the solving time of a single experiment and the population size because individual fitness calculation is the main content of the iterative process. In Figure 8, the statistical results of 1000 repeated experiments with different population sizes are shown. The rightmost column shows the proportion of optimal solutions obtained through repeated experiments with different population sizes. The results showed that when the population size was 10, only 23.5% of the samples obtained the optimal solution. When the scale was 80-100, almost all experimental times obtained the optimal solution. It indicates that the population size is not suitable to be set too small, leading to premature convergence of the results. In addition, when the population size is between 80 and 100, the results in terms of obtaining the optimal solution are not significantly different. However, the single iteration time for a population of 100 increases by 0.22 s compared to 80. It indicates that it is not suitable to set the population size too large. Therefore, the population size should be set at 100 or slightly greater. In Table 3, when the number of iterations is set to [10; 40], the maximum number of iterations to obtain the optimal solution is 10, 20, 30 and 40. The proportion of samples to obtain the optimal solution cannot reach 100%. This phenomenon indicates that the number of iterations may limit the convergence of the algorithm. When the value is 40, there is still a small number of experiments that have not achieved the theoretically optimal solution. When the number of iterations is set to 50, the maximum number of iterations to obtain the optimal solution in each experiment is less than the set value. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Population Size**} & \multicolumn{2}{c}{**The Maximum Number of**} & \multirow{2}{*}{**Single Experiment**} & \multirow{2}{*}{**Proportion of Experiments**} \\  & **Iterations to Get the** & & **Run Time/s** & **Achieving** \\  & **Optimal Solution** & & **Optimal Solutions** \\ \hline
10 & 50 & 0.55 & 23.5 \\
20 & 48 & 0.80 & 55.4 \\
30 & 49 & 0.97 & 76.3 \\
40 & 47 & 1.15 & 88.4 \\
50 & 46 & 1.50 & 95.0 \\
60 & 46 & 1.67 & 96.4 \\
70 & 43 & 1.88 & 98.9 \\
80 & 44 & 1.97 & 99.3 \\
90 & 42 & 2.10 & 99.9 \\
100 & 42 & 2.48 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The experimental results of different population sizes, with \(iMax=50\). Figure 8: The influence of population size on convergence.