Multi-Objective Route Planning Model for Ocean-Going Ships Based on Bidirectional A-Star Algorithm Considering Meteorological Risk and IMO Guidelines

Abstract

In this study, a new route planning model is proposed to help ocean-going ships avoid dangerous weather conditions and ensure safe ship navigation. First, we integrate ocean-going ship vulnerability into the study of the influence of meteorological and oceanic factors on navigational risk. A multi-layer fuzzy comprehensive evaluation model for weather risk assessment is established. A multi-objective nonlinear route planning model is then constructed by comprehensively considering the challenges of fuel consumption, risk, and time during ship navigation. The International Maritime Organization (IMO) guidelines are highlighted as constraints in the calculations, and wind, wave, and calm water resistance to ships in the latest ITTC method is added to the fuel consumption and sailing time in the objective function. Finally, considering the large amount of data required for ocean voyages, the bidirectional A* algorithm is applied to solve the model and reduce the planning time. Furthermore, our model is applied to the case of an accident reported in the Singapore Maritime Investigation Report, and the results show that the model-planned route is very close to the original planned route using the Towing Manual, with an average fit of 98.22%, and the overall meteorological risk of the model-planned route is 11.19% smaller than the original route; our model can therefore be used to plan a safer route for the vessel. In addition, the importance of risk assessments and the IMO guidelines as well as the efficiency of the bidirectional A* algorithm were analyzed and discussed. The results show that the model effectively lowers the meteorological risk, is more efficient than the traditional route planning algorithm, and is 86.82% faster than the Dijkstra algorithm and 49.16% faster than the A* algorithm.

1. Introduction

In the past, maritime transportation research focused on ship safety [1,2] as it is closely related to real-world events involving ship navigation, and meteorological and marine environments have a significant influence in this regard. Recent maritime accidents include the following: in April 2014, the Sewol ferry capsized in strong winds and waves, killing about 300 people; in December 2014, the Norman Atlantic ferry faced strong winds and waves in the Adriatic Sea, resulting in 11 deaths; in October 2015, the cargo ship El Faro went missing in a hurricane, killing all 33 crew members; in March 2021, strong winds pushed the Ever Given off course in Egypt’s Suez Canal and it ran aground; and in November 2020, a container ship stack collapse occurred after the ONE Apus encountered adverse weather and sea conditions in the North Pacific Ocean. Moreover, the loss rate of ships, of which bulk carriers account for the majority, is increasing amidst changes in wave patterns in recent years [3]. Consequently, research on optimal route planning to help ships avoid dangerous regions is crucial for ship safety.

At present, route planning is the main topic of research in this field, but many studies have concentrated on weather and ocean conditions and have not quantitatively studied the risk-bearing capacity of ships when assessing risk values. For example, Ma et al. [4] calculated the sailing cost associated with weather conditions by using weather data and terrain data for each section, but they did not factor in ship vulnerability. Absolute danger, hazard factors, and influential factors were used for risk contour mapping as the structure of their route planning technique [5]. Pennino et al. [6] provided a model to optimize routes with the maximum property values according to weather forecast maps and studied the effect of ship speed on these optimal routes. Ship information including the under keel clearance (UKC) and draft was calculated for safety contours that were used to assess the probability of ships running aground [7]. Although these risk evaluation methods mentioned the ship parameters during navigation, they did not quantitatively analyze the ship vulnerability. Mamenko et al. [8] only qualitatively constructed the risk field of ships needed to plan optimal routes, without calculating the risk values of sailing ships in depth. Therefore, this study proposes a multi-layer fuzzy comprehensive evaluation method to integrate the external environment of weather and marine factors with ship vulnerability. The multi-layer fuzzy comprehensive evaluation method can effectively synthesize different evaluation indicators from different criteria.

When sailing in harsh weather conditions, ships may encounter a variety of dangerous situations including wind-induced waves and heavy swells. To assist ship captains in handling such situations, the International Maritime Organization (IMO) [9] set guidelines which apply to all types of ships. Weather routing has captured researchers’ attention in many studies, but only a few of them have taken the IMO guidelines into account. For example, a voyage optimization approach aimed to minimize fuel consumption and air emissions was tested in a harsh sea environment to prove that the approach was effective in avoiding storms [10]. Gkerekos and Lazakis [11] qualitatively used available weather forecast information when selecting the optimal routes via a data-driven decision support framework. Penalty costs were added to the fitness function of the algorithm for finding an optimal route with the minimum travel time and fuel consumption when the wave height was greater than the maximum acceptable wave height [12]. As Wang et al. [13] used a real-coded genetic algorithm to investigate how to plan minimum sailing time routes for ocean going vessels to avoid weather hazards such as wind and waves. if the wave height was significantly exceeded for the area. In a study on optimizing ship motion, Sotnikova and Veremey [14] marked areas in harsh weather conditions, including strong winds or high waves, as the alarm area as dynamic constraints. While these studies were effective in finding routes that avoided storms or high waves, they did not quantitatively discuss the effect of wind and waves according to the IMO guidelines. In this study, dangerous situations that may lead to capsizing or severe roll motions causing damage to ships are quantificationally used as constraints to plan less risky routes.

Furthermore, the Dijkstra and A* algorithms are often applied to search for an optimal route which comprehensively considers the sailing time, risk, terrain, and other criteria with minimal influence. Zhu et al. [15] improved the Dijkstra algorithm by adding the pheromone idea of the ant colony algorithm, thus reducing the moving cost of routes. Silveira et al. [16] combined the information from an Automatic Identification System with the Dijkstra algorithm to plan a safe route. Ship navigation rules were added to the A* algorithm, forming the dynamic collision avoidance route planning algorithm under the circumstances of a complicated multi-ship encounter [17]. Dong and Bian [18] proposed an optimized A* algorithm based on a genetic algorithm to solve the problem of ship pipe route design. In terms of algorithm selection, the Dijkstra algorithm finds the optimal solution by searching all directions around the current node, while the A* algorithm, adding the heuristic function to change the direction of the search towards the target, is more efficient than the Dijkstra algorithm [19]. However, the Dijkstra algorithm searches too many useless nodes, and the A* algorithm takes up too much space, both of which will affect the efficiency of finding the optimal route in the map with too large a size or too complex an environment [20]. In recent years, the A* algorithm has been optimized; for example, the bidirectional A* algorithm [21] can concurrently search from the start node to the end node and from the end node to the start node to reduce the number of nodes that need to be searched and improve the search efficiency. In view of the wide navigation range and long timespan of the ocean route, this study applies the bidirectional A* algorithm to solve our model.

Therefore, the main objectives of this study are as follows. First, the multi-layer fuzzy comprehensive evaluation method is applied to integrate the external environment of meteorological and marine factors with the vulnerability of ships so that the risk values are more accurate and comprehensive for the ships. Second, the dangerous situations which cause ships to capsize or suffer damage, as mentioned in the IMO guidelines, are quantificationally used as constraints in the algorithm. At the same time, the influence of wind, waves, and calm water on ship resistance is also considered. Then, a multi-objective nonlinear programming model is built, which comprehensively considers voyage time, voyage risk, fuel consumption, and constraints. Finally, the bidirectional A* algorithm is applied to solve the above model to obtain the optimal route. The remainder of this manuscript is organized as follows. Section 2 presents the methods of this study. The case of the capsizing of Singapore barge CB100-01 is studied to demonstrate the validity of our model in Section 3. Some discussions and conclusions are presented in Section 4.

2. Methods

2.1. Risk Assessment Model

When a ship is sailing in adverse weather conditions, it is prone to capsizing or other dangerous phenomena, thus resulting in damage. That means the safety of ships sailing at sea is influenced by the meteorological and marine environment. Moreover, different ships have different stability parameters, hull geometries, capacities, and speeds, which gives each ship a different sensitivity to dangerous phenomena and thus a different probability of their occurrence in the context of a particular sea state [9]. Considering these differences, risk values will be evaluated from two aspects according to the actual navigation circumstances: one is the hazard effect that the meteorological and marine environment has on ships [22], and the other is the risk-bearing capacity of ships during navigation, termed the vulnerability effect.

2.1.1. Risk Assessment Indicators

This study selects five meteorological and marine environment factors and then computes the absolute rate of information transfer from these factors to the damage caused by dangerous phenomena (based on historical marine data) to indicate any causal relationships [23].

$$T_{j\to i}={\displaystyle \frac{1}{\det C}}\cdot {\displaystyle \sum _{k=1}^d{\Delta }_{jk}{C}_{k,di}\cdot \frac{C_{ij}}{C_{ii}}}$$

(1)

where $T_{j\to i}$ is the absolute rate of information transfer from variable $X_j$ to variable $X_i$, $C$ is the covariance matrix, ${\Delta }_{jk}$ are the cofactors of $C$, $C_{k,di}$ is the sample covariance between all $X_k$ values and the Euler forward difference approximation of ${\displaystyle \raisebox{1ex}{$d{X}_i$}\!\left/ \!\raisebox{-1ex}{$dt$}\right.}$ ($t$ is time), $C_{ij}$ is the sample covariance between $X_i$ and $X_j$, and $C_{ii}$ is the sample variance in $X_i$.

According to the reliability test rule, the standardized $T_{x\to danger}$ values of variables $x$ (variables x are wind speed, wave height, ocean currents, and visibility) to the damage of dangerous phenomena are 0.1178, 0.2230, 0.0151, and 0.0521 in

Table 1. Absolute rate of information transfer from variable $X_j$ to variable danger.

$\mathit{j}$	Wind Speed	Wave Height	Ocean Currents	Sea Water Temperature	Visibility
$T_{j\to danger}$	0.1178	0.2230	0.0151	0.0036	0.0521

, respectively, all of which are greater than 1%, thus indicating that the results are statistically significant at a 95% confidence level. The standardized $T_{temperature\to danger}$ to the damage of dangerous phenomena is 0.0036, which means the causal link is weaker from sea water temperature to the damage of dangerous phenomena than the other four factors. Therefore, the variables of wind speed, wave height, ocean current, and visibility are selected to quantitatively calculate the hazard effect risk value.

The vulnerability of a ship represents the risk-bearing capacity of the ship; i.e., the higher the vulnerability value of a ship, the higher the risk of navigation and the lower the level of safety of the ship. Since different ships have different probabilities of encountering danger in specific sea conditions mentioned in the IMO, the four factors selected in Nivolianitou and Koromila’s probability model for predicting accidents are adopted in this study [24].

• Vessel type and vessel size: During ship construction and design, classification can usually be based on vessel type and vessel size due to the different purposes of the ships being designed;
• Vessel age: As one of the most important parameters, vessel age was mentioned in the Ship Risk Profile, and older age may increase the risk profile;
• Vessel flag: The “White, Grey and Black (WGB) list” in the Paris MoU ranked countries’ performance in different states from high to low as ‘White’, ‘Grey’, and ‘Black’ based on the total number of inspections and detentions over a 3-year rolling period for flags with at least 30 inspections in the period.

To summarize, the risk assessment indicators established in this study are shown in Figure 1.

2.1.2. Risk Grading Standards of Indicators

Grading methods are applied for risk classification to achieve the purpose of quantifying the risk and simplifying the data. Grading statistical analysis is the process of dividing a dataset into subsets, the attributes of which do not change, based on certain methods or standards, aiming to highlight individual differences in the dataset. Grading methods include the equal arithmetic interval, standard deviation, and normal percentile grading methods [25]. The equal arithmetic interval grading method was chosen for this study.

The equal arithmetic interval grading method grades data on the equal arithmetic interval and has the same width in each level, and the formula for calculating the arithmetic interval of the data is as follows:

$$\Delta ={\displaystyle \frac{X_{\max }-X_{\min }}{n}}$$

(2)

where $\Delta$ is the arithmetic interval, $X_{\max }\mathrm{and}{X}_{\min }$ are the maximum value and minimum value in the dataset, respectively, and $n$ is the number of levels required.

According to the National Oceanic and Atmospheric Administration (NOAA) indicator classification and the Beaufort scale wind and wave classification frameworks, the grading standards for each hazard indicator are obtained as shown in

Table 2. Risk grading standards.

Risk Value	Wind Speed	Wave Height	Ocean Currents	Visibility
High risk	[20.8, ∞)	[6, ∞)	[2, ∞)	[0,3.1)
Higher risk	[15.6, 20.8)	[4.5, 6)	[1.5, 2)	[3.1, 6.2)
Medium risk	[10.4, 15.6)	[3, 4.5)	[1, 1.5)	[6.2, 9.3)
Lower risk	[5.2, 10.4)	[1.5, 3)	[0.5, 1)	[9.3, 12.4)
Low risk	[0, 5.2)	[0, 1.5)	[0, 0.5)	[12.4, ∞)

.

2.1.3. Multi-Layer Fuzzy Comprehensive Evaluation Model

Fuzzy comprehensive evaluation is a method based on the principles of fuzzy mathematics, quantifying some fuzzy or difficult-to-quantify factors of problems by applying the membership theory of fuzzy mathematics and distinguishing the membership levels of evaluated objects from multiple factors [26]. This study uses the multi-layer fuzzy comprehensive evaluation method to evaluate the risks that affect the safety of sailing ships.

The main steps in a fuzzy comprehensive evaluation include establishing a set of evaluation factors and evaluation comments, determining the weights of evaluation indicators and membership functions, and selecting comprehensive operators. Firstly, this study determines the set of evaluation factors that we need to evaluate the risk of maritime navigation environment, namely establishing hazard assessment indicators and vulnerability assessment indicators. Then, the evaluation level is determined for each indicator, and the fuzzy evaluation matrix and the weight of the evaluation indicators are determined based on relevant data. Finally, the weight of the indicators and the fuzzy evaluation matrix are fuzzy-synthesized, obtaining the final risk assessment result. Multi-layer fuzzy comprehensive evaluation is an improved evaluation which adds a criterion layer based on the single-layer fuzzy comprehensive evaluation. The steps are as follows:

Step 1

Establish a set of evaluation factors and evaluation comments

We use the risk assessment indicators mentioned in Section 2.1.1 as the set of evaluation factors for the evaluation model, namely $U=\left\{U_1,U_2\right\}$, where $U_1=\left\{\mathrm{Wind}\mathrm{speed}\left(u_1\right),\mathrm{Wave}\mathrm{height}\left(u_2\right),\mathrm{Ocean}\mathrm{currents}\left(u_3\right),\mathrm{Visibility}\left(u_4\right)\right\}$ and $U_2=\left\{\mathrm{Vessel}\mathrm{size}\left(u_5\right),\mathrm{Vessel}\mathrm{age}\left(u_6\right),\mathrm{Vessel}\mathrm{flag}\left(u_7\right)\right\}$.

Based on Section 2.1.2, we divide the indicator data into five levels, which we mark as the set of evaluation comments for the set of evaluation factors, namely, $V=\left\{\mathrm{High}\mathrm{risk},\mathrm{Higher}\mathrm{risk},\mathrm{Medium}\mathrm{risk},\mathrm{Lower}\mathrm{risk},\mathrm{Low}\mathrm{risk}\right\}$.

Step 2

Constructing membership functions

Common membership functions include triangular distribution, trapezoidal distribution, parabolic or semi-parabolic distribution, Gaussian distribution, and sigmoidal distribution [27]. Considering the characteristics of the value range of the indicators selected in this study and the adaptability of the sigmoidal membership function, the sigmoidal membership function is the best choice. The basic formula is as follows.

$$f\left(x,a,c\right)={\displaystyle \frac{1}{1+e^{-a\left(x-c\right)}}}$$

(3)

where the magnitude of $a$ controls the width of the transition area at the relevant level, $c$ presents the center of the transition area, and return value $f$ is the degree of subordination of the indicators to risk levels.

Step 3

Determine the single-factor evaluation matrix of the sub-criteria layer

The single-factor evaluation matrices $E_k$ are constructed according to Equation (3), and the results are shown in Equations (4) and (5). The ${\epsilon }_{\mathrm{ij}}$ in matrices $E_k$ represents the membership degree of the $i$th indicator belonging to the $j$th evaluation level.

$$E_1=\left(\begin{array}{cccc}{\epsilon }_{11}& {\epsilon }_{12}& {\epsilon }_{13}& \begin{array}{cc}{\epsilon }_{14}& {\epsilon }_{15}\end{array}\\ {\epsilon }_{21}& {\epsilon }_{22}& {\epsilon }_{23}& \begin{array}{cc}{\epsilon }_{24}& {\epsilon }_{25}\end{array}\\ {\epsilon }_{31}& {\epsilon }_{32}& {\epsilon }_{33}& \begin{array}{cc}{\epsilon }_{34}& {\epsilon }_{35}\end{array}\\ {\epsilon }_{41}& {\epsilon }_{42}& {\epsilon }_{43}& \begin{array}{cc}{\epsilon }_{44}& {\epsilon }_{45}\end{array}\end{array}\right)$$

(4)

$$E_2=\left(\begin{array}{c}\begin{array}{cccc}{\epsilon }_{51}& {\epsilon }_{52}& {\epsilon }_{53}& \begin{array}{cc}{\epsilon }_{54}& {\epsilon }_{55}\end{array}\end{array}\\ \begin{array}{cccc}{\epsilon }_{61}& {\epsilon }_{62}& {\epsilon }_{63}& \begin{array}{cc}{\epsilon }_{64}& {\epsilon }_{65}\end{array}\end{array}\\ \begin{array}{cccc}\begin{array}{cc}{\epsilon }_{71}& {\epsilon }_{72}\end{array}& {\epsilon }_{73}& {\epsilon }_{74}& {\epsilon }_{75}\end{array}\end{array}\right)$$

(5)

Step 4

Determine the evaluation matrix of the criterion layer

After determining the weight vectors $W_k$ of the indicators in the sub-criterion layer and the single-factor evaluation matrices $E_k$, the evaluation matrix $B$ of the criterion layer is calculated as shown in Equations (6)–(8).

$$\begin{array}{l}{B}_1=W_1\times E_1\\ =\left(w_1,w_2,w_3,w_4\right)\times \left(\begin{array}{cccc}{\epsilon }_{11}& {\epsilon }_{12}& {\epsilon }_{13}& \begin{array}{cc}{\epsilon }_{14}& {\epsilon }_{15}\end{array}\\ {\epsilon }_{21}& {\epsilon }_{22}& {\epsilon }_{23}& \begin{array}{cc}{\epsilon }_{24}& {\epsilon }_{25}\end{array}\\ {\epsilon }_{31}& {\epsilon }_{32}& {\epsilon }_{33}& \begin{array}{cc}{\epsilon }_{34}& {\epsilon }_{35}\end{array}\\ {\epsilon }_{41}& {\epsilon }_{42}& {\epsilon }_{43}& \begin{array}{cc}{\epsilon }_{44}& {\epsilon }_{45}\end{array}\end{array}\right)\\ =\left(b_{11},b_{12},b_{13},b_{14},b_{15}\right)\end{array}$$

(6)

$$\begin{array}{l}{B}_2=W_2\times E_2\\ =\left(w_1,w_2,w_3\right)\times \left(\begin{array}{c}\begin{array}{cccc}{\epsilon }_{51}& {\epsilon }_{52}& {\epsilon }_{53}& \begin{array}{cc}{\epsilon }_{54}& {\epsilon }_{55}\end{array}\end{array}\\ \begin{array}{cccc}{\epsilon }_{61}& {\epsilon }_{62}& {\epsilon }_{63}& \begin{array}{cc}{\epsilon }_{64}& {\epsilon }_{65}\end{array}\end{array}\\ \begin{array}{cccc}\begin{array}{cc}{\epsilon }_{71}& {\epsilon }_{72}\end{array}& {\epsilon }_{73}& {\epsilon }_{74}& {\epsilon }_{75}\end{array}\end{array}\right)\\ =\left(b_{21},b_{22},b_{23},b_{24},b_{25}\right)\end{array}$$

(7)

$$B=\left(\begin{array}{c}{B}_1\\ B_2\end{array}\right)=\left(\begin{array}{c}\begin{array}{cccc}{b}_{11}& b_{12}& \cdots & b_{15}\end{array}\\ \begin{array}{cccc}{b}_{21}& b_{22}& \cdots & b_{25}\end{array}\end{array}\right)$$

(8)

Step 5

Comprehensive evaluation

$$I=B\times K=\left(\begin{array}{c}\begin{array}{cccc}{b}_{11}& b_{12}& \cdots & b_{15}\end{array}\\ \begin{array}{cccc}{b}_{21}& b_{22}& \cdots & b_{25}\end{array}\end{array}\right)\left(\begin{array}{cccc}0.2& 0.2& 0.2& \begin{array}{cc}0.2& 0.2\end{array}\end{array}\right)\prime =\left(\begin{array}{c}{I}_{haz}\\ I_{vul}\end{array}\right)$$

(9)

where $K$ is the corresponding weights for risk levels (low risk, lower risk, medium risk, higher risk, and high risk), and $I$ is the comprehensive evaluation of hazard effect and vulnerability effect. In this study, the evaluation value is obtained by weighting the membership values of the five risk levels and then accumulating them. Finally, the hazard effect and vulnerability effect evaluation values are synthesized according to the model of synthesizing risk values built by Balmat et al. [28]. The model is shown as follows.

$$R=I_{haz}(1+I_{vul})$$

(10)

The final risk value ranges from 0 to 1, and the closer the risk value is to 1, the higher the risk and the lower the safety of the sailing ship.

2.2. Route Planning Model

2.2.1. Mathematical Model

This study employs the bidirectional A* algorithm to solve the route planning model. By improving the cost function and adding constraints to the bidirectional A* algorithm, the problem of integrating the IMO guidelines, risk, sailing time, and fuel consumption is solved, ultimately achieving the goal of obtaining the optimal route.

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 Ming-jun 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 $\left(F\right)$ is as follows:

$$F=F_{water}+F_{wind}+F_{wave}$$

(11)

$$F=P\cdot S_{SFOC}$$

(12)

$$P={\displaystyle \frac{R\cdot V_S\cdot {10}^{-6}}{{\eta }_S\cdot {\eta }_D}}$$

(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 $\left(F_{water}\right)$:

$$F_{water}={\displaystyle \frac{R_f\cdot V_S\cdot P_{SFOC}\cdot {10}^{-6}}{{\eta }_S\cdot {\eta }_D}}$$

(14)

$$R_f=\left(C_f+\Delta C_f\right){\displaystyle \frac{1}{2}}{\rho }_S\cdot S_s\cdot {V_S}^2$$

(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:

$$C_f={\displaystyle \frac{0.4631}{{\left(\lg \mathrm{Re}\right)}^{2.6}}}$$

(16)

The Reynolds number $R_e$ is given by Equation (17).

$$R_e={\displaystyle \frac{V_S{L}_{pp}}{v}}$$

(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 $\left(F_{wind}\right)$

$$F_{wind}={\displaystyle \frac{R_{AA}\cdot V_S\cdot P_{SFOC}\cdot {10}^{-6}}{{\eta }_S\cdot {\eta }_D}}$$

(18)

$$R_{AA}={\displaystyle \frac{1}{2}}{\rho }_A{V_S}^2{A}_T\cos {\theta }_{wi}$$

(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 $\left(F_{wave}\right)$

$$F_{wave}={\displaystyle \frac{R_{AW}\cdot V_S\cdot P_{SFOC}\cdot {10}^{-6}}{{\eta }_S\cdot {\eta }_D}}$$

(20)

$$R_{AW}=2{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{R_{wave}(\omega ,\alpha ;V_S)}{{\zeta }_A^2}}}E\left(\omega ,\alpha \right)d\omega d\alpha$$

(21)

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:

$$E\left(\omega ,\alpha \right)=S_{\eta }\left(\omega \right)G\left(\alpha \right)$$

(22)

The frequency spectrum $S_{\eta }\left(\omega \right)$ and the angular distribution function $G\left(\alpha \right)$ are calculated according to Tsujimoto and Orihara [34].

$$S_{\eta }\left(\omega \right)={\displaystyle \frac{A_f}{{\omega }^5}}\left(-{\displaystyle \frac{B_f}{{\omega }^4}}\right)$$

(23)

$$A_f={\displaystyle \frac{1}{4\pi }}{\left({\displaystyle \frac{2\pi }{T_{02}}}\right)}^4{H}^2$$

(24)

$$B_f={\displaystyle \frac{1}{\pi }}{\left({\displaystyle \frac{2\pi }{T_{01}}}\right)}^4$$

(25)

$$T_{02}={\displaystyle \frac{\Gamma \left(3/4\right)}{{\pi }^{1/4}}}T\approx 0.9204T$$

(26)

where $H$ is the significant wave height, and $T$ is the mean wave period.

$$G\left(\alpha \right)={\displaystyle \frac{2^{2s}}{\pi }}{\displaystyle \frac{{\Gamma }^2\left(s+1\right)}{\Gamma \left(2s+1\right)}}{\cos }^{2s}\left(\alpha -{\theta }_m\right)$$

(27)

where ${\theta }_m$ is the primary wave direction (0 refers to the heading wave) and $-{\displaystyle \frac{\pi }{2}}\le \alpha -{\theta }_m\le {\displaystyle \frac{\pi }{2}}$, and $s$ is the directional spreading parameter.

$$R_{wave}(\omega ,\alpha ;V_S)=R_{AWM}+R_{AWR}$$

(28)

where $R_{AWR}$ is the wave reflection-induced component, and $R_{AWM}$ is the motion-induced component.

$$R_{AWR}={\displaystyle {\sum }_{i=1}^4{R}_{AWR,i}}$$

(29)

$$R_{AWM}=3859.2{\rho }_{s}g{\zeta }_A^2{\displaystyle \frac{B^2}{L_{pp}}}{C}_B^{1.34}{k}_{yy}^2{a}_1{a}_2{a}_3{\varpi }^{b_1}{e}^{{\displaystyle \frac{b_1}{d_1}}\left(1-{\varpi }^{d_1}\right)}$$

(30)

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, $\varpi$ 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 $135°\sim 225°$ and the actual speed $v$ is greater than the critical speed $v_l$, the ship should avoid this area. The critical speed is calculated as follows (Equation (31)):

$$v_l={\displaystyle \frac{1.8\sqrt{L}}{\cos \left(180-q\right)}}$$

(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)):

$$T_E={\displaystyle \frac{3T_W^2}{3T_w+V\cos \left(q\right)}}$$

(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.

Sailing Time

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 Chuang and Steen [36] as follows:

$${\displaystyle \frac{f_{water}v}{{\eta }_{DC}}}={\displaystyle \frac{\left(f_{wind}+f_{wave}\right)v_w}{{\eta }_{DW}}}$$

(33)

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.

$$T={\displaystyle \frac{2}{\pi }}\arctan {\displaystyle \frac{X}{v_w}}$$

(34)

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.

$$\min \mathrm{Cost}={\rho }_{1}R+{\rho }_{2}T+{\rho }_{3}F\cdot T$$

(35)

where $\mathrm{Cost}$ is the ultimate objective function, and ${\rho }_i\left(i=1,2,3\right)$ is the weight of each cost function and satisfies the following:

$$\{\begin{array}{l}0\le {\rho }_i\le 1,i=1,2,3\\ {\displaystyle \sum _{i=1}^3{\rho }_i=1}\end{array}$$

(36)

While satisfying the optimal solution of the objective function, this study takes the guidelines of IMO and the restrictions on wind and wave conditions in ship navigation manuals as constraint conditions. Assuming that the maximum wind speed and wave height that the ship can withstand are $W_{\max }$ and $H_{\max }$, respectively, the following constraint conditions can be constructed.

$$\{\begin{array}{c}135\le \mathrm{q}\le 180\mathrm{or}\mathrm{v}\le {\mathrm{v}}_l\\ {\displaystyle \frac{T_r}{T_E}}\text{<}0.7\mathrm{or}{\displaystyle \frac{T_r}{T_E}}\text{>}1.3\\ v_W\le W_{\max }\\ h\le H_{\max }\end{array}$$

(37)

2.2.2. Bidirectional A* Algorithm

The A* algorithm is a traditional static path-planning heuristic search algorithm that calculates the feasible shortest path node from the start point to the end point. The main body of the A* algorithm is OpenList and CloseList, which are used to represent the nodes to be traversed and the nodes that have been traversed, respectively. The A* algorithm is used to calculate the priority of each node through a heuristic distance cost function, as shown in Equation (31).

$$f(n)=g(n)+h(n)$$

(38)

where $g(n)$ is the cost of the distance from node $n$ to the start node, $h(n)$ is the estimated distance cost from node $n$ to the end node and also a heuristic function of the A* algorithm, and $f(n)$ is the final cost of node $n$ and also the basis for calculating the priority of the A* algorithm. The objective function used is $g(n)$, and the Euclidean distance calculation is used in $h(n)$ in this study.

The bidirectional A* algorithm is an optimization of the A* algorithm. Compared to the A* algorithm, the bidirectional A* algorithm searches from both the start node and end node simultaneously. When the two traversed nodes overlap, the optimal path is returned. This optimization can cover the entire state space, effectively avoiding the problems of traversing multiple nodes and a long computing time [37]. This study uses the bidirectional A* algorithm to solve our route planning model, and the algorithm process is shown in Figure 2.

3. Case Study

3.1. Case Description and Datasets

The case selected in this study is the final investigation report of a maritime accident from the Transport Safety Investigation Bureau of Singapore (https://www.mot.gov.sg, accessed on 28 June 2023), which classified it as a very serious maritime incident. The accident investigation report provides detailed records of the cause of the accident, information about and the status of the ship before and after the accident, and the planned route of navigation.

The ship in this accident was a barge, the CB100-01 (CB), with an overall length of 42.06 m, breadth of 15.24 m, depth of 3.05 m, and gross registered tonnage of 444 tons, and it was built in the PT ASL Shipyard, Indonesia, in 2016. The CB100-01, towed by the ASL Osprey (AO), departed the ASL Shipyard, Singapore, on 1 May 2022. The estimation from the AO’s captain was that the tug and tow would arrive at Djibouti on 3 June 2022, with an average towing speed of 5.5 knots. In addition, this voyage followed the planned route in the Towing Manual.

According to the investigation report, the planned route in the Towing Manual is divided into five parts: Singapore shipyard–North Sumatra–Sri Lanka–Mumbai, India–Salalah, and Oman–Djibouti. Due to the loss of the starboard side spud pole and the aft spud pole during the voyage to Sri Lanka around 12 May, arrangements were made for the maintenance, investigation, and fixation of the remaining port side spud pole after the tug and tow arrived in Colombo, Sri Lanka, on 18 May. One month later, two ships departed Colombo on 16 June and sailed along the original planned route. The tug and tow were expected to arrive in Djibouti on 7 July, with an average towing speed of five knots. On 2 July, due to a deviation from the planned route, the tug and tow entered an area with adverse weather, and ultimately, the CB capsized.

According to the average speed of the ships in the report and the planned route in the Towing Manual, the tug and tow could have reached Sri Lanka on the 12th day after departure and Djibouti on the 34th day after departure. Furthermore, according to the actual voyage, which was suspended in Colombo, Sri Lanka, for nearly a month, the original planned route, which is marked as TMroute in this manuscript, is divided into two sections to facilitate this study: the section from Singapore shipyard to Sri Lanka is recorded as Towing Manual route A (TMrouteA) from 1 to 12 May; and the section from Sri Lanka to Djibouti is recorded as Towing Manual route B (TMrouteB) from 16 June to 7 July, as shown in Figure 3. Taking the CB100-01 as our research object, our route planning model integrating weather conditions, speed loss, terrain risks, and constraint conditions is established for planning analysis under the constraints of the IMO guidelines and the meteorological requirements in the Towing Manual.

The range of the meteorological and marine environmental data used in this study is 5° S–25° N, 40° E–110° E, and the time ranges of the two routes are from 1 May 2022 to 18 May 2022 and from 16 June 2022 to 7 July 2022. The selected wind field ten meters above the sea surface is represented as three-dimensional grid data from the Copernicus Atmospheric Monitoring Service (CAMS). The wind field includes the two aspects of wind direction and wind speed, and the spatial resolution of the grid is 1/12$°$ × 1/12$°$. The selected sea surface wave data and sea surface temperature data are three-dimensional grid data from the Copernicus Marine Environmental Monitoring Service (CMEMS). The wave data include the effective wave height, average period, and wave direction, and the grid spatial resolution is 1/4$°$ × 1/4$°$. Due to the different sources of data, the spatial resolution of all data is consistent with the wave data. The type, size, flag, age, average speed, and other data of the ship are obtained from the maritime investigation report by the Transport Safety Investigation Bureau of Singapore.

3.2. Results and Analysis

3.2.1. Risk Assessment Results

Construction of Membership Function

According to the data of each indicator in the set of evaluation factors, the parameters of the sigmoidal membership function are fit to obtain the sigmoidal membership function and images of the indicators. We selected three indicators, namely, wave height, visibility, and vessel age, and their images are shown in Figure 4, Figure 5 and Figure 6.

The fitted parameters of the sigmoidal membership function and weights of risk indicators obtained via the analytic hierarchy process are shown in

Table 3. The parameters of the sigmoidal membership function.

Indicators	Risk Level	a	c
Wave height	High Risk	10	0.75
	Higher Risk	6	2.25
	Medium Risk	6	3.75
	Lower Risk	6	5.25
	Low Risk	6	6.75
Wind speed	High Risk	3	2.6
	Higher Risk	1.5	7.8
	Medium Risk	2	13
	Lower Risk	2	18.2
	Low Risk	2	23.4
Ocean Currents	High Risk	30	0.25
	Higher Risk	20	0.75
	Medium Risk	20	1.25
	Lower Risk	20	1.75
	Low Risk	20	2.25
Visibility	High Risk	4	1.55
	Higher Risk	3	4.65
	Medium Risk	4	7.75
	Lower Risk	3	10.85
	Low Risk	4	13.95
Vessel size	High Risk	0.002	22,500
	Higher Risk	0.002	17,500
	Medium Risk	0.002	12,500
	Lower Risk	0.002	7500
	Low Risk	0.002	2500
Vessel age	High Risk	2	3
	Higher Risk	1.5	9
	Medium Risk	1.5	15
	Lower Risk	1.5	21
	Low Risk	1.5	27
Vessel flag	High Risk	0.6	8
	Higher Risk	0.6	24
	Medium Risk	0.6	40
	Lower Risk	0.6	56
	Low Risk	0.6	72

,

Table 4. The weights of the vulnerability effect indicators.

	Vessel Size	Vessel Age	Vessel Flag
Weights	0.545	0.33	0.115

and

Table 5. The weights of the hazard effect indicators.

	Wave Heights	Wind Speed	Visibility	Ocean Currents
Weights	0.43	0.37	0.12	0.08

.

Risk Zoning Results

Following our risk assessment model in Section 2.1, the results of the risk assessments of the two sections in this case are shown in Figure 7 and Figure 8.

From Figure 7, during the time of the first section of the ship’s route, from 1 May to 12 May, the overall risk in the navigation area is low, and the level of safety of the sailing ship is mostly high.

From Figure 8, during the time of the second section of the ship’s route, from 16 June to 7 July, the risk of sailing in the navigation area gradually increased from 29 June, and the scope of the high-risk area gradually increased. According to an investigation report, the sea area entered a seasonal monsoon period at this time, which is consistent with the risk change in the results in Figure 8. We can see that the risk assessment based on the assessment model in this study is accurate.

3.2.2. Route Planning Results

The routes of the two sections are planned using our route planning model in Section 2.2. The route from the Singapore shipyard to Sri Lanka is recorded as IBA*routeA, and the route from Sri Lanka to Djibouti is recorded as IBA*routeB. The whole route from the Singapore shipyard to Djibouti is recorded as IBA*route. The route planning results are shown in Figure 9 and Figure 10.

According to the result in Figure 9, IBA*routeA basically coincides with TMrouteA. The average risk of TMrouteA is 0.3499, while the average risk of IBA*routeA is only 0.3388, which shows that IBA*routeA can avoid risk more effectively than TMrouteA.

According to the result in Figure 10, IBA*routeB coincides with TMrouteB for the most part, but it is better at avoiding high risks in the open sea than TMrouteB. The average risk of TMrouteB is 0.4424, while the average risk of IBA*routeB is only 0.3929, which shows that IBA*routeB can avoid risk more effectively overall.

3.2.3. Comparison and Verification of Routes

When there is little difference between the two planned routes visually, the similarity between TMroute and IBA*route can be compared using the goodness of fit $R_{NL}$ [38], and the calculation formula of $R_{NL}$ is shown in Equation (39).

$$R_{NL}=1-\sqrt{{\displaystyle \frac{{\displaystyle \sum {\left(y_i-{\widehat{y}}_i\right)}^2}}{{\displaystyle \sum {y_i}^2}}}}$$

(39)

The average relative error $R_E$ of TMroute and IBA*route is calculated as follows:

$$R_E={\displaystyle \frac{{\displaystyle \sum \left|y_i-{\widehat{y}}_i\right|}}{n(y_{\max }-y_{\min })}}$$

(40)

where $y_i\mathrm{and}{\widehat{y}}_i$ are the latitude grid values corresponding to the $i$th longitude grid value of TMroute and IBA*route, respectively, and $y_{\max }\mathrm{and}{y}_{\min }$ are the maximum latitude grid value and the minimum latitude grid value of TMroute, respectively.

TMrouteA and TMrouteB are compared with IBA*routeA and IBA*routeB, respectively. The goodness of fit and the average relative error are calculated, as shown in

Table 6. The comparison of $R_{NL}$ and $R_E$ between TMroute and IBA*route.

	From 1 to 12 May 2022	From 16 June to 7 July 2022
$R_{NL}$	0.9969	0.9675
$R_E$	0.0224	0.0302

.

In Table 6, the goodness-of-fit values of IBA*routeA and IBA*routeB separately compared with TMrouteA and TMrouteB are both higher than 95%, and their average relative errors are lower than 5%, indicating that IBA*route and TMroute are highly similar.

According to the investigation report, the ship deviated from the TMroute between 19 June and 2 July and entered an area of increasingly severe weather conditions, as shown in Figure 11.

In Figure 11, the route that the CB actually sailed, which is marked as Realroute in this manuscript, entered the high-risk area around 2 July after deviating from TMroute, while according to the ship’s planned arrival time in the investigation report, it can be calculated that TMrouteB had already left the open sea area on 30 June. IBA*routeB bypassed the high-risk area on 28 June and also left the open sea on 30 June.

To study the risks of Realroute, TMrouteB, and IBA*routeB during the period of the accident, the average risks of the three routes during the period from Colombo, Sri Lanka to Salalah were calculated, respectively, as shown in

Table 7. The comparison of risk between Realroute, TMroute, and IBA*route.

	Realroute	TMrouteB	IBA*routeB
Average risk	0.4262	0.4424	0.3929
Maximum risk	0.8373	0.7121	0.6592
Minimum risk	0.1180	0.1252	0.1252

, and the box plots of the three routes are shown in Figure 12.

In Table 7, Realroute is larger than TMrouteB and IBA*routeB in terms of the average risk, maximum risk, and minimum risk values. Moreover, the average risk value of IBA*routeB is only 0.3929, the maximum risk value is only 0.6592, and the minimum risk value is only 0.1252, all of which are the minimum values in the three routes, so it can be seen that IBA*routeB exhibits the minimum overall risk.

In Figure 12, IBA*routeB has the smallest maximum, lower quartile, and upper quartile values of risk, which indicates that our planning model has a strong risk avoidance ability.

3.3. Validation

To verify the validity of this model on other ship types, CELSIUSBRICKE, a container ship from the Marshall Islands, was selected for a voyage of 24 days in the Maritime Silk Sea (Hainan–Indian Ocean). The ship is 246.87 m long, 32.2 m wide, and 19.3 m deep, with a displacement of 61,521 tons, a draft of 9.8 m, and a maximum speed of 23.0 knots. The route starts in Mombasa (4°4.490 S/39°41.172 E) and ends in Shanghai, China (31°14.442 N/122°2.208 E). The voyage lasted 24 days, from 2 May 2023 to 25 May 2023, covering a total distance of 6476 nautical miles at an average speed of 13.32 knots. On May 13, 14, 21, and 24, ships slowly circled or stopped sailing in place in the sea area, so the simulated ships did not sail on these four days during the planning process. All information about the voyage comes from the Hifleet website (https://www.hifleet.com/, accessed on 30 May 2023). The actual route and planned route are shown in Figure 13.

The planned route is a red line segment, while the actual route is a yellow line segment. It can be seen from the figure that the distance between the planned route result and the actual result is relatively small. The calculation results of the three comparison models show that the average relative error $R_E$ is 1.62%, the goodness of fit $R_{NL}$ is 98.71%, and the risk is 4.27% lower than the actual route. According to the actual sailing situation, this voyage is safer, and the route planned by the track planning model is thus safer.

3.4. Sensitivity Analysis

In this section, we perform a sensitivity analysis on several parameters to analyze the results of the impact of different parameters on the planning avoidance risk of the model. We show how different propulsion efficiencies ${\eta }_D$ and the roughness allowance factors accounting for the roughness of the hull surface $\Delta C_f$ differ in the average risk of a planned route to be analyzed.

To analyze the effect of the ship’s propulsion efficiency on the risk caused by bad weather and the sea state during the ship’s voyage planning route, we applied the proposed model under different propulsion efficiency scenarios and calculated the average risk for the whole planned route. Figure 14 shows that the average risk value of the navigation will decrease with the increase in propulsion efficiency under the planned route. This means that the new model can avoid high risk in the route and further help the ship to better cope with the dangers of bad weather and the sea state in the ocean.

Second, we analyzed the average risk of the planned route under different roughness allowance factor scenarios. Figure 15 shows that the value of the navigational risk will increase with the increase in the roughness allowance factor under the planned route. This implies that the roughness subsidy coefficient of different hulls has an impact on the navigational risk.

4. Discussion

Meteorological and marine environment factors directly or indirectly cause damage to ships or the loss of ships [39]. In the field of maritime transport, in which cargo transport accounts for more than 90% of world trade [40], the relationship between ship safety and meteorological and marine environment factors has been broadly investigated. Recently, many researchers have studied the optimization of route planning to improve the safety of sailing ships [41,42,43,44]. However, these planning models do not consider the constraints of the IMO guidelines, and there are few studies on the impact of ship vulnerability on vessel safety.

The purpose of the revised IMO guidelines is to provide a basis for ship captains to make decisions about ship operations in bad weather and sea conditions, thereby helping them avoid the dangerous phenomena they may encounter in such conditions, including surf-riding and broaching-to phenomena, a reduction in intact stability when riding a wave crest amidships, synchronous rolling motion, parametric roll motions, and successive high-wave attacks. The empirical equation of the IMO guidelines is not sufficient and can only avoid most of the common risks to ships. Moreover, the sudden change in a ship’s state while sailing at sea is a highly chaotic nonlinear system, and the unfavorable combination of ocean parameters and the ship’s state may lead to dangerous situations. In this study, we take the empirical formula of the IMO guidelines as the basis of the principle of avoiding danger, while for dead ship conditions, excessive accelerations, and other situations that cause ship instability and damage, further research into and the prediction of these nonlinear dangerous situations are significant for the optimization of the weather route model. Therefore, based on this study, we can continue to study the influence of ship stability under dynamic constraints from the perspective of ship dynamics and add these constraints into the multi-objective programming model.

In this study, meteorological and marine environment factors and the vulnerability of ships are integrated by a multi-layer fuzzy comprehensive evaluation to obtain a risk assessment value. Meanwhile, the IMO guidelines are taken as hard constraints, and the bidirectional A* algorithm is used for route planning. If risk and the IMO guidelines are not considered during the voyage from Colombo, Sri Lanka to Djibouti, the bidirectional A* algorithm is used for route planning, the route of which is marked as WrouteB, compared with IBA*routeB, and the result is shown in Figure 16.

In Figure 16, when risk and the IMO guidelines are not considered, WrouteB enters the high-risk area around 27 June, while IBA*routeB bypasses the high-risk area. It can be understood that the consideration of risk and the IMO regulations is therefore very important for the safety of sailing ships.

In previous studies, common algorithms to solve the pathfinding problem have included the Dijkstra and A* algorithms. Compared with Dijkstra, the A* algorithm not only considers the distance from the start node to the current node, but also considers the estimated cost from the current node to the end node. However, the area involved in the ocean route is large and the timespan is long, so more efficient algorithms are needed for route planning. The bidirectional A* algorithm, an improvement on the A* algorithm, searches the intersection points from the start node and the end node at the same time, thus reducing the number of nodes to search and improving the search efficiency. We compare the running time of the Dijkstra, A*, and bidirectional A* algorithms when planning routes for the case chosen in this study, as shown in

Table 8. Running time of the Dijkstra, A*, and bidirectional A* algorithms.

Algorithms	Dijkstra	A*	Bidirectional A*
Running Time (s)	31.739	8.236	4.187

.

In Table 8, the Dijkstra algorithm has the longest running time of 31.739 s, which is much longer than the A* and bidirectional A* algorithms, and the bidirectional A* algorithm has the shortest running time of 4.187 s. It can thus be concluded that the operation efficiency of the bidirectional A* algorithm is greater than that of the Dijkstra and A* algorithms.

5. Conclusions

In view of the few studies involving quantitative considerations of ship vulnerability and IMO guidelines in ocean route planning models, this study proposes a multi-objective nonlinear programming model which integrates ship vulnerability and meteorological and marine environment factors to obtain a comprehensive risk assessment, and it implements the IMO guidelines as hard constraints to solve the model using the bidirectional A* algorithm. Firstly, this study uses the multi-layer fuzzy comprehensive evaluation model, which comprehensively assesses the risk of sailing ships by evaluating ship vulnerability and meteorological and marine environment factors. Secondly, the multi-objective nonlinear programming model is constructed by adding IMO constraints and comprehensively considering the navigation risk, sailing time, and fuel consumption in this study. This study then uses the bidirectional A* algorithm to solve the route planning model. Finally, an accident case in the Singapore Maritime Investigation Report was studied, and the experimental results show that the model planning route in this study is similar to the original route in the Towing Manual, with an average fit of 98.22%, and the average risk is 11.19% lower than that of the original route. Comparing algorithms, the bidirectional A* algorithm is 86.81% faster than the Dijkstra algorithm and 49.16% faster than the A* algorithm. In summary, our model realizes two goals: a safer ocean route and more efficient computing. In future works, the algorithms for ocean ship route planning can be improved, which will greatly improve search efficiency and further approach an optimal solution. Finally, we will continue studying the influence of ship stability under dynamic constraints from the perspective of ship dynamics, and these constraints will be added into the proposed multi-objective programming model.