## 5 Results and Discussions
### Impact of the Number of Vessels at the Current Voyage Speed under FAFU  
Figure 4 illustrates the results of case C2, in which the optimization model determines the number of each type of vessel at the current travel speed. #LV in Figure 4 represents the number of large vessels, and #SV is that of small vessels. Figure 4a shows the system throughput, which is equivalent to the transported volume per year. A few cases could not meet the target throughput (3248 KT). Either three large vessels or small vessels could transport the target product volume. As one fewer large vessel is used, one more small vessel is needed to meet the throughput target.  
In addition, we analyze the cost breakdown structure of case C2 as shown in Figure 4b only for the feasible cases that meet the target throughput. The left \(y\)-axis is the total cost, and the right \(y\)-axis is the ratio of the total cost of each case of large and small vessels to the highest total cost when (#LV, #SV) = (3, 4).  
In most cases, the port dues ratios are similar and vary slightly from 0.13 to 0.16. Port dues decrease slightly as the number of large vessels increases. Total dues decrease because a large vessel needs fewer trips than a small vessel to transport the same amount of products. Similarly, the voyage cost decreases slightly as the number of large vessels increases because a large vessel has a lower unit voyage cost than a small vessel.  
However, the rental cost of a large vessel is larger than that of a small vessel. Thus, as the number of large vessels increases, the rental cost increases. There is a trade-off between the operating cost (voyage cost and port dues) and rental cost according to the number of large and small vessels. The smallest total cost occurs when only three large vessels  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{**No**} & \multicolumn{2}{c}{**Voyage Speed Factor**} & \multicolumn{2}{c}{**Product Volume (KT)**} & \multicolumn{3}{c}{**Costs (USD/Year)**} \\ \cline{3-10}  & & & **FS** & **BS** & **Produced** & **Transported** & **Total** & **Rental** & **Port** & **Voyage** \\  & & & & & & **Cost** & **Cost** & **Dues** & **Cost** \\ \hline C0 & - & 1 & 1 & 3274.83 & 3259.32 & 10,020,600 & 4,258,330 & 1,889,720 & 3,872,540 \\ \hline \multirow{6}{*}{C1} & 1 & 1 & 0.9 & 3274.83 & 3260.30 & 9,836,700 & 4,258,330 & 1,890,240 & 3,688,120 \\ \cline{2-10}  & 2 & 0.9 & 1 & 3274.83 & 3258.92 & 9,839,440 & 4,258,330 & 1,891,120 & 3,689,990 \\ \cline{2-10}  & 3 & 1 & 0.8 & 3274.83 & 3258.12 & 9,639,210 & 4,258,330 & 1,886,680 & 3,494,200 \\ \cline{2-10}  & 4 & 0.8 & 1 & 3274.83 & 3259.67 & 9,643,280 & 4,258,330 & 1,887,840 & 3,497,110 \\ \cline{2-10}  & 5 & 1 & 0.7 & 3274.83 & 3260.58 & 9,450,080 & 4,258,330 & 1,885,480 & 3,306,270 \\ \cline{2-10}  & 6 & 0.7 & 1 & 3274.83 & 3260.70 & 9,445,440 & 4,258,330 & 1,883,960 & 3,303,150 \\ \cline{2-10}  & 7 & 0.9 & 0.9 & 3274.83 & 3257.89 & 9,648,830 & 4,258,330 & 1,889,720 & 3,500,780 \\ \cline{2-10}  & 8 & 0.8 & 0.8 & 3274.83 & 3257.95 & 9,261,100 & 4,258,330 & 1,884,220 & 3,118,550 \\ \cline{2-10}  & 9 & 0.7 & 0.7 & 3274.83 & 3258.18 & 8,894,980 & 4,258,330 & 1,886,080 & 2,750,570 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Impact of lowering voyage speed with the current number of vessels.  
and no small vessels are used, that is, (#LV, #SV) = (3, 0). From the result, we infer a large vessel's economic benefit from saving voyage cost and port dues may be better than the disadvantage from the increasing rental cost under the considered cost structure.