## 2 Mathematical Models Used for Route Development
### Technical Specifications of the Ship  
To determine the environmental forces acting on the ship, it is also essential to specify the ship's technical parameters, such as dimensions, surface areas, and thruster parameters. Table 1 presents the data of the vessel used for the study.  
In the developed algorithm for determining the ship's passage route, it is assumed that the vessel is equipped with four thrusters: bow thruster, stern thruster and two propellers. Thrusters' data are shown in Table 2.  
The arrangement of individual thrusters on the ship is shown in Figure 3.  
The creation of the plot shown in Figure 1 begins with analyzing the environmental forces acting on the ship at an angle of 0 degrees. Initially, the components of environmental forces acting on each axis of the ship are determined for a wind speed of 50 knots. After these forces are identified, a suitable method for distributing thrust among the thrusters is applied (for example, [PERSON]'s allocation formulation [36]) to verify whether the ship can keep its position. If the position can be kept, the wind speed is incremented by a selected value, and the analysis is repeated. If the ship cannot keep its position at a given wind speed, the speed is reduced until it is capable of doing so. The final wind speed at which the ship can keep its position is recorded on the plot, after which the wind angle is increased by 10\({}^{\circ}\), and the analysis is repeated. Once all angles have been tested, the plot is drawn with a line connecting the individual points. A detailed step-by-step algorithm illustrating the process of finding the maximum wind speed at a specified angle of impact on the hull is shown in Figure 4.  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Value** \\ \hline Length overall (\(L_{\text{au}}\)) [m]: & 72.7 \\ Length between perpendiculars (\(L_{pp}\)) [m]: & 64 \\ Breadth [m]: & 11.6 \\ Drawbat [m]: & 3.4 \\ Displacement [T]: & 1886 \\ Distance between foremost and aftmost point of the hull below the surface at & 67.1 \\ design draft even keel [m]: & 639 \\ Water plane area [m\({}^{2}\)]: & 437 \\ Projected longitudinal area above water [m\({}^{2}\)]: & 0.1 \\ Surge position of geometric center of the projected longitudinal area above & 223 \\ Water with respect to \(L_{pp}/2\) [m]: & \(-\)2.9 \\ Projected longitudinal area below water [m\({}^{2}\)]: & \(-\)1.5 \\ Surge position of geometric center of the projected longitudinal area below & 36 \\ water with respect to \(L_{pp}/2\) [m]: & 246 \\ Surge position of water line center with respect to \(L_{pp}/2\) [m]: & 41 \\ Projected transverse area above water [m\({}^{2}\)]: & \(-\)1.5 \\ Projected transverse area above water [m\({}^{2}\)]: & 140 \\ Fuel consumption at 100\% main engine load [l/h] & 246 \\ Propulsion system efficiency [\%] & 41 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Technical specification of the ship.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Thruster** & **Thrust Max [kN]** & **Thrust Min [kN]** & **Power [kW]** \\ \hline Bow thruster & 94 & \(-\)94 & 447 \\ Stern thruster & 94 & \(-\)94 & 447 \\ Port propeller & 179 & \(-\)179 & 2000 \\ Stbd propeller & 179 & \(-\)179 & 2000 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Thrusters’ data.  
Figure 3: The arrangement of thrusters.