## 2 Methods
### Fuel Consumption  
The consideration of resistance plays an important role in route planning. Resistance includes hull resistance due to the shape of the hull in calm water, and additional resistance is added by wind and waves during navigation. The ocean current is also an important factor affecting ship speed when the resistance in calm water is calculated [29]. Resistance affects the ship's fuel consumption, which is related to emissions and vessel operator costs. Considering that ships in the ocean will be affected by irregular wind waves and ocean currents, we will divide fuel consumption into three parts according to [PERSON] et al. when calculating the fuel consumption of a ship: fuel consumption in calm water conditions, and additional fuel consumption levels under wind and wave resistance conditions [30]. The formula for calculating the fuel consumption of ships under the influence of wind and waves \((F)\) is as follows:  
\[F=F_{water}+F_{wind}+F_{unve} \tag{11}\]  
\[F=P\cdot S_{SFOC} \tag{12}\]  
\[P=\frac{R\cdot V_{S}\cdot 10^{-6}}{\eta_{S}\cdot\eta_{D}} \tag{13}\]  
where \(P\) is the power consumed by the main engine of a ship in calm water conditions, \(V_{S}\) is the ship's speed through the water, \(P_{SFOC}\) is the fuel efficiency of the main engine, \(\eta_{S}\) is the transmission efficiency of shafting, and \(\eta_{D}\) is the propulsion efficiency. 1. Fuel consumption in calm water conditions \((F_{water})\): \[F_{water}=\frac{R_{f}\cdot V_{S}\cdot P_{SFOC}\cdot 10^{-6}}{\eta_{S}\cdot\eta_{D}}\] (14)  
\[R_{f}=\Big{(}C_{f}+\Delta C_{f}\Big{)}\frac{1}{2}\rho_{S}\cdot S_{s}\cdot{V_{ S}}^{2} \tag{15}\]  
where \(R_{f}\) is frictional resistance to a ship, \(C_{f}\) is the friction coefficient, \(\Delta C_{f}\) is the roughness allowance factor accounting for the roughness of the hull surface, \(\rho_{S}\) is the density of water, and \(S_{s}\) is the wetted surface area of the hull. \(C_{f}\) is usually calculated from the Reynolds number \(R_{e}\) using the following equation:  
\[C_{f}=\frac{0.4631}{(\text{lgRe})^{2.6}} \tag{16}\]  
The Reynolds number \(R_{e}\) is given by Equation (17). \[R_{e}=\frac{V_{S}L_{pp}}{v} \tag{17}\]  
where \(L_{pp}\) is the length of the ship perpendicular to the waterline, and \(v\) is the kinematic viscosity of water. 2. Additional fuel consumption under wind resistance conditions \((F_{wind})\) \[F_{wind}=\frac{R_{AA}\cdot V_{S}\cdot P_{SFOC}\cdot 10^{-6}}{\eta_{S}\cdot \eta_{D}}\] (18)  
\[R_{AA}=\frac{1}{2}\rho_{A}{V_{S}}^{2}A_{T}\cos\theta_{wi} \tag{19}\]where \(R_{AA}\) is the air resistance to a ship, calculated with reference to the International Towing Tank Conference (ITTC-1957) [31], \(\rho_{A}\) is the air density, \(A_{T}\) is the waterplane area, and \(\cos\theta_{wi}\) is the angle between the wind and the bow. 3. Additional fuel consumption under wave resistance conditions \((F_{wave})\)  
\[F_{wave}=\frac{R_{AW}\cdot V_{S}\cdot P_{SFOC}\cdot 10^{-6}}{\eta_{S}\cdot\eta_{D}} \tag{20}\]  
\[F=P\cdot S_{SFOC} \tag{12}\]0  
where \(R_{AW}\) is the mean resistance increase in short crested irregular waves, \(R_{wave}\) is the transfer function of the mean resistance increase in regular waves in accordance with the SNNM method adopted following ITTC-2021 [32; 33], \(\zeta_{A}\) is the wave amplitude, \(\omega\) is the circular frequency of regular waves, \(\alpha\) is the angle between the ship heading and component waves, and \(E\) is directional spectrum, calculated by the following equation:  
\[F=P\cdot S_{SFOC} \tag{12}\]1  
The frequency spectrum \(S_{\eta}(\omega)\) and the angular distribution function \(G(\alpha)\) are calculated according to Tsujimoto and Orihara [34]. \[F=P\cdot S_{SFOC} \tag{12}\]2  
\[F=P\cdot S_{SFOC} \tag{12}\]3  
\[F=P\cdot S_{SFOC} \tag{12}\]4  
\[F=P\cdot S_{SFOC} \tag{12}\]5  
where \(H\) is the significant wave height, and \(T\) is the mean wave period.