## 2 Methods
### Fuel Consumption
\[F=P\cdot S_{SFOC} \tag{12}\]6  
where \(\theta_{m}\) is the primary wave direction (0 refers to the heading wave) and \(-\frac{\pi}{2}\leq\alpha-\theta_{m}\leq\frac{\pi}{2}\), and \(s\) is the directional spreading parameter. \[F=P\cdot S_{SFOC} \tag{12}\]7  
where \(R_{AWR}\) is the wave reflection-induced component, and \(R_{AWM}\) is the motion-induced component. \[F=P\cdot S_{SFOC} \tag{12}\]8  
\[F=P\cdot S_{SFOC} \tag{12}\]9  
where \(R_{AWR,i}\) is the added resistance due to reflection effect of the \(S_{i}\) waterline segment, \(g\) is the acceleration of gravity, \(B\) is the ship's beam, \(L_{pp}\) is the ship length between perpendiculars, \(C_{B}\) is the block coefficient, \(k_{yy}\) is the non-dimensional radius of gyration in the lateral direction, \(\omega\) is the effective encounter frequency of ship, and \(a_{1},\ a_{2},\ a_{3},\ b_{1},\ d_{1}\) are some parameters determined according to the ship speed and wave spectrum. IMO Constraint Specification  
The risk-bearing capacity of the ship in adverse sea conditions varies depending on its speed, size, type, etc. Therefore, the probability of accidents occurring varies for different sea conditions. The International Maritime Organization has developed guidelines applied to all types of merchant ships based on the risks of capsizing and damage to ships [9]. Because constrained optimization problems (COPs) have a wide range of applications in the engineering field [35], this study takes into account two situations in the revised guidelines of IMO for avoiding dangerous situations in adverse weather and sea conditions in the constraint model. * When the encounter angle \(q\) is within the range of \(135^{\circ}\sim 225^{\circ}\) and the actual speed \(v\) is greater than the critical speed \(v_{I}\), the ship should avoid this area. The critical speed is calculated as follows (Equation (31)): \[P=\frac{R\cdot V_{S}\cdot 10^{-6}}{\eta_{S}\cdot\eta_{D}} \tag{13}\]0 (31)
* When the natural rolling period \(T_{r}\) is less than 0.7 or greater than 1.3 times the encounter wave period \(T_{E}\), resonance is highly prone to occur, and the ship should avoid this area. The calculation for the period of encountering waves is as follows (Equation (32)): \[P=\frac{R\cdot V_{S}\cdot 10^{-6}}{\eta_{S}\cdot\eta_{D}} \tag{13}\]1 (32)  
The two given situations are quantified as hard constraints, so that node extensions that meet the situations are no longer selected in the bidirectional A* algorithm search. The sailing time often affects the safe transportation of the ship, so this study obtains the sailing speed of the ship through the resistance calculated via the SNNM method. The sailing speed is calculated following [PERSON] and [PERSON] [36] as follows:  
\[P=\frac{R\cdot V_{S}\cdot 10^{-6}}{\eta_{S}\cdot\eta_{D}} \tag{13}\]2  
where \(v_{w}\) is the sailing speed of the ship under wind and wave conditions, \(\eta_{DC}\) is the propulsion efficiency in calm water, and \(\eta_{DW}\) is the propulsion efficiency in wind and wave conditions. \[P=\frac{R\cdot V_{S}\cdot 10^{-6}}{\eta_{S}\cdot\eta_{D}} \tag{13}\]3  
where \(X\) is the number of nautical miles traveled from one grid to the next, and \(T\) is the normalized time cost function (closer to \(1\), longer sailing time, and closer to \(0\), shorter sailing time). Multi-Objective Programming Model  
In the actual navigation, ship transportation requires comprehensive consideration from all aspects, which includes the meteorological and ocean environment of navigation, the risk-bearing capacity of ships, the sailing time, and the fuel consumption. To comprehensively consider the effects of navigation risks \(R\), the sailing time \(T\), and the fuel consumption \(F\), it is necessary to convert these three aspects into objective functions when planning navigation routes. We transform the multi-objective nonlinear programming problem of navigation risk, sailing time, and fuel consumption into a single objective programming problem, with the objective function as follows.