# A Multi-Agent Optimization Approach for Multimodal Collaboration in Marine Terminals
## 5 Case Study  
### Generation of Test Instances  
The developed model and solution methodologies were tested for a medium-sized terminal in Norway over one week, with time-units measured in minutes and aggregated into hourly time-periods. A medium-sized terminal typically processes approximately 500,000 to 1,000,000 TEUs per year, which corresponds to around 10,000 to 20,000 TEUs per week [36]. The instances in this study are characterized by three distinct factors.  
1. The number of vessels approaching the terminal during the examined working period.
2. The number of available berths for these ships.
3. The level of incoming truck traffic to the terminal.  
We report the values used for these three factors in all numerical experiments, together with other important parameters, in Table 5.  
For the generated vessels, the expected arrival time \(A_{v}\) is considered deterministic and known, while the length \(L_{v}\) follows a uniform distribution within their respective vessel categories. The cargo amounts \(I_{v}\) and \(E_{v}\) are correlated with the size of the vessel. They are also proportional to incoming traffic obtained from loop detector data for heavy vehicles that were gathered in a road axis adjacent to the examined terminal. These were transformed to preferred pickup \(T^{P}\) and delivery slots \(T^{D}\).  
Truck traffic is distributed among 50 logistics companies, each with a deviation cost factor \(\Lambda_{l}\) drawn from \(\mathcal{U}(0,1)\). Trucks are assigned randomly to vessels based with a constraint ensuring trucks are within 12 time-periods of the vessel's arrival. The assignment to \(I_{v}\) and \(E_{v}\) depends on whether the sampled window is before or after the vessel's arrival. Overall, traffic volumes varied from 3773 to 4449 trucks in low-traffic instances, 10,661 to 14,705 in medium traffic, and 17,275 to 21,748 in high-traffic scenarios.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Factors** & **Levels** \\ \hline Level of truck traffic & Low, Medium, High \\ Number of berths available & 2, 3, 4 \\ Number of quay cranes per berth & 1, 2, 3 \\ Number of logistics companies & 50 \\ Number of vessels approaching the terminal & 20, 25, 30, 35, 40, 45, 50 \\ Number of TEUs per time-period per mode (\(S^{B}\), \(S^{S}\)) & 50 (double), 31 (single) \\ Terminal Parameters: (\(K_{\max}\), \(T_{\max}\), \(L_{\max}\), \(V_{\max}\)) & 2000 TEUs, 250 trucks, 20 trucks, 50 vehicles \\ Vessel Categories (based on length) & \(<\)100 m, 100 m–200 m, \(>\)200 m \\ \hline \hline \end{tabular}
\end{table}
Table 5: Summary of test data.  
Regarding the terminal geometry, the terminal features between two and four berths. With two berths, one is dedicated to vessels smaller than 100 m, while the other accommodates vessels of all sizes. When three or four berths are available, additional berths serve vessels up to 200 m in length. The berth for vessels under 100 m is equipped with a single quay crane, whereas the larger berths each have two or three quay cranes. Travel time from any berth to the import area is five minutes, while travel time from the berth to the export area is four minutes. The travel time between the import and export areas is set at one minute.  
Regarding terminal productivity parameters, the crane throughput is set to 50 TEUs per period in double mode (\(S^{B}\)) and 31 TEUs per period in single mode (\(S^{S}\)). Loading and unloading times for internal trucks are each set to one minute. Gate terminal productivity is 50 trucks per lane per period, with a total of two lanes available, both of which can be used for pickup and deliveries. Finally, regarding terminal operational parameters, the maximum truck arrival rate per period (\(T_{\text{max}}\)) is set to 125 trucks per lane. The maximum storage capacity in both the import and export areas (\(K_{\text{max}}\)) is 2000 TEUs. The maximum allowable queue length at any time (\(L_{\text{max}}\)) is 20 trucks, and the maximum number of internal trucks in use at any time (\(V_{\text{max}}\)) is 50.