## 4 Result and Discussion
When the time horizon considered is less than 30 months, the fixed-time horizon method always shows a better performance than the moving window method regarding the optimality of the solution (i.e., less value of \(Obj_{opt}\)), generating bang-bang solutions stably. As the time horizon exceeds 36 months, however, the fixed-time horizon method begins to fail in converging to a bang-bang solution, so thus the optimality of the solutions becomes worse to the levels compatible with the moving window method. In the extreme cases of the time horizon of 60 months, the moving window method even shows a better optimality. It should be noted that the convergence to a bang-bang solution is critical to the optimality of the solution because both methods compared here basically adopts the multi-period OCP framework where the rounding up scheme is employed when the convergence fails [33]. The failure in convergence to a bang-bang solution observed in the fixed-time horizon method can be attributed to the increased complexity of the problem by the increase of the time horizon to be considered. Our moving window decision-making \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{Fixed-Time Horizon} & \multicolumn{4}{c}{Moving Window} \\ \cline{2-9}
**Time** & \begin{tabular}{c} **Bang-Bang** \\ **Solution** \\ **Satisfied** \\ **(True/False)** \\ \end{tabular} & \begin{tabular}{c} **Objective** \\ **No. of** \\ **Cleaning** \\ **[k\(\ell\)]** \\ \end{tabular} & \begin{tabular}{c} **Optimiza** \\ **Function** \\ **Value** \\ **(min)** \\ \end{tabular} & \begin{tabular}{c} **Bang-Bang** \\ **Solution** \\ **Value** \\ **(True/False)** \\ \end{tabular} & \begin{tabular}{c} **Objective** \\ **Solution** \\ **Satisfied** \\ **[k\(\ell\)]** \\ \end{tabular} & \
\begin{tabular}{c} **Optimiza** \\ **Function** \\ **Value** \\ **(line)** \\ \end{tabular} &
\begin{tabular}{c} **Optimal** \\ **Moving** \\ **Window** \\ **Size** \\ \end{tabular} \\ \hline
12 & True & 2 & 385 & 0.46 & True & 2 & 385 & 0.17 & 4 \\
18 & True & 10 & 620 & 2.35 & True & 7 & 627 & 0.51 & 4 \\
24 & True & 16 & 852 & 6.39 & True & 18 & 863 & 0.49 & 5 \\
30 & True & 22 & 1085 & 9.43 & True & 21 & 1099 & 0.51 & 5 \\
36 & False & 25 & 1329 & 12.10 & True & 28 & 1332 & 0.67 & 5 \\
42 & False & 35 & 1556 & 21.49 & True & 33 & 1568 & 0.75 & 5 \\
48 & False & 44 & 1791 & 27.37 & True & 39 & 1807 & 0.91 & 5 \\
54 & False & 47 & 2022 & 43.04 & True & 46 & 2035 & 1.02 & 5 \\
60 & False & 55 & 2393 & 35.95 & True & 52 & 2271 & 1.13 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optimization performance of the fixed-time horizon and moving window methods for the problems with different time horizons. algorithm, however, can be free from this complexity issue because its complexity only depends on the moving window size, not on the whole-time horizon. Another important aspect to consider when solving an optimization problem is the computational time which is taken to obtain the solution. As can be seen in Table 2, the computational time of the presented moving window method always outperforms the fixed-time horizon method. As we used the same initial guess of \(y_{up}=1\ (\forall n,p)\) in all simulation studies, it can be reasonably concluded that the moving window method is more efficient in obtaining an optimized solution regarding the computational time. The details of the optimal solutions determined by two methods were analyzed more in depth for the case of \(t_{F}=18\) months, of which results are summarized in Figure 4. The total number of cleanings determined by the fixed-time horizon method is 10 while that obtained by our moving window algorithm is only 7. Moreover, the detailed cleaning schedules are clearly different from each other, showing only two common cleanings at the 7 th and 12 th month. Yet, the difference of the objective function values of two methods is just 7 k\(\varepsilon\) (see Table 2), which is very trivial indeed.