ZHAO Wei ,WANG Hongbo, ,GENG Jianning ,HU Wenmei ,ZHANG Zhanshuo ,and ZHANG Guangyu

1) State Key Laboratory on Integrated Optoelectronics,College of Electronic Science and Engineering, Jilin University,Changchun 130000, China

2) China State Shipbuilding Corporation (CSSC) Marine Technology Co., Ltd., Beijing 100070,China

Abstract Maritime transportation has become an important part of the international trade system.To promote its sustainable development,it is necessary to reduce the fuel consumption of ships,decrease navigation risks,and shorten the navigation time.Accordingly,planning a multi-objective route for ships is an effective way to achieve these goals.In this paper,we propose a multi-objective optimal ship weather routing system framework.Based on this framework,a ship route model,ship fuel consumption model,and navigation risk model are established,and a non-dominated sorting and multi-objective ship weather routing algorithm based on particle swarm optimization is proposed.To fasten the convergence of the algorithm and improve the diversity of route solutions,a mutation operation and an elite selection operation are introduced in the algorithm.Based on the Pareto optimal front and Pareto optimal solution set obtained by the algorithm,a recommended route selection criterion is designed.Finally,two sets of simulated navigation simulation experiments on a container ship are conducted.The experimental results show that the proposed multi-objective optimal weather routing system can be used to plan a ship route with low navigation risk,short navigation time,and low fuel consumption,fulfilling the safety,efficiency,and economic goals.

Key words weather routing;particle swarm optimization;route planning;multi-objective optimization

1 Introduction

The maritime transportation environment is complex and dynamic.Severe weather conditions threaten the navigation safety of ships,and shipwrecks are common.Such conditions can not only cause damage to ships but also seriously threaten the lives of crewmembers.In addition,the rapid development of the maritime industry has resulted in large amounts of greenhouse gas emissions,which have severely affected the global climate.In 2018,the International Maritime Organization (IMO) adopted the Initial IMO Strategy on the Reduction of Greenhouse Gas Emissions from Ships (Chircop,2019).For the first time in the global maritime community,greenhouse gas emission targets have been formulated in response to shipping greenhouse gas emissions.Accordingly,in this study,we propose a multi-target ship route planning algorithm for weather routing to reduce ship fuel consumption and greenhouse gas emissions,improve ship navigation safety,and shorten ship navigation time.

The core of weather routing is the method used to design the best ship route.This method is based on shortand medium-term weather and marine forecasts and combines the ship performance,technical conditions,and navigation tasks to select the best route for ship navigation.The objectives of determining the best route generally include the following aspects:safety,efficiency,and economy (Zyczkowskiet al.,2019).With the increasingly significant role of weather routing in the maritime navigation of ships,many traditional and intelligent algorithms for the design of weather routing for ships have been proposed.Among them,the earliest traditional algorithm is the isochronous method (James,1957).However,this method is unsuitable for computer calculations.To solve such shortcomings,the modified isochrone method (Hagiwara and Spans,1987) was proposed,which is mainly used to calculate the shortest navigation time and least fuel consumption route,but it encounters the ‘isochronic loop’ problem.Accordingly,a three-dimensional (3D) modified isochrone method(Linet al.,2013;Fang and Lin,2015) was proposed,which achieves the optimal minimum fuel consumption and expected arrival time.Traditional dynamic programming methods have been applied to weather routing (Wit,1990).Shaoet al.(2012) proposed a forward 3D dynamic programming method to optimize routes while minimizing fuel consumption.The Dijkstra algorithm (Panigrahiet al.,2012;Sen and Padhy,2015;Mannariniet al.,2016) and A* algorithm (Xieet al.,2019) have also been applied to the weather routing problem,aiming to obtain the shortest voyage route,least time route,and lowest fuel consumption route.Zyczkowskiet al.(2018) proposed a deterministic algorithm aid to determine the route of a sailing vessel and reduce the navigation time and number of turns.With the continuous development of intelligent heuristic optimization algorithms,swarm intelligence algorithms,such as the ant colony algorithms,genetic algorithms,and particle swarms,have also been applied to the optimal weather routing design problem.For example,Tsou and Cheng (2013) used the ant colony algorithm to plan the route of ships with minimum fuel consumption.Makiet al.(2011) applied the real-coded genetic algorithm to search the trade-off ship route and achieve economic and safety goals.Kanget al.(2012) developed a metaheuristic algorithm based on a genetic algorithm to reduce transportation costs.Wanget al.(2018) presented a real-coded genetic algorithm to determine the minimum voyage route time for point-to-point problems in a dynamic environment.Vettor and Guedes Soares (2012) presented a robust multi-objective evolutionary algorithm to approximate the most favorable set of solutions.Joanna (2015) presented a weather routing algorithm utilizing a multi-objective evolutionary to determine Pareto-optimal transoceanic ship routes.Szlapczynski and Ghaemi (2019) applied an evolutionary multi-objective optimization approach to pursue three objectives:minimization of collision risk,minimization of fuel consumption,and minimization of navigation time.Tagliaferri and Viola (2017) applied a neural network to produce a short-term wind forecast and obtain the route closest to the shortest voyage.In this study,we propose combining the multi-objective particle swarm optimization (MPSO) algorithm with the genetic algorithm to solve the optimal weather routing problem.The research content and structure of this article are as follows:

First,a mathematical model for the ship route design is established,including a route mathematical model,fuel consumption mathematical model,and navigation risk mathematical model.Second,a hybrid non-dominated sorting MPSO (HNDS-MPSO) algorithm is proposed to solve the route planning problem in consideration of static obstacle constraints (i.e.,coastlines,islands,reefs,and shoals) and dynamic obstacle constraints (i.e.,high-wind and wild-wave areas) at sea.The proposed algorithm is used to plan an optimal route while ensuring safety,being economical,and saving time.Third,to improve the diversity of solutions and avoid premature convergence of the algorithm,elite selection and mutation operations are added.Finally,based on the Pareto optimal front and Pareto optimal solution set obtained by the algorithm,a recommended route selection criterion is proposed,and a simulation experiment is designed for a container ship.The experimental results show that the algorithm is feasible and effective.

2 Multi-Objective Weather Routing System Framework

The weather routing system proposed in this paper consists of six interconnected parts:parameter input,environment construction,ship route model,multi-objective weather route planning algorithm,route evaluation,and route recommendation.The overall structural framework of the weather routing system and the relationship among the six parts are shown in Fig.1 and detailed as follows.

Fig.1 Multi-objective weather routing system framework.

First,the parameter input includes the ship’s departure port position,arrival port position,preset speed,and fullload displacement.The parameter values are provided by the operator according to the navigation task and requirements.Second,the environment construction includes the navigation area map obtained from the electronic chart and the meteorological forecast information obtained by the meteorological center.In this part,the system analyzes the static and dynamic obstacle areas of the navigation area to identify the non-navigable and high-risk areas and to lay the foundation for the risk assessment of the best weather routing plan.Third,the ship route model is based on the particle swarm algorithm and the characteristics of the ship route.The routing model not only conforms to the characteristics of the actual ship route but also conveniently calculates the ship’s sailing time,navigation risk,and fuel consumption.The non-dominated sorting multi-objective particle swarm algorithm is applied to solve the ship’s weather routes.Fourth,the proposed algorithm is a multi-objective weather route planning algorithm based on the theory of particle swarm optimization,which combines the mutation operation and elite selection operation.Fifth,the route evaluation aims to obtain the Pareto optimal solution set and Pareto optimal front and achieve the shortest sailing time,lowest navigation risk,and least fuel consumption.Lastly,based on the obtained Pareto solution set data,the route recommendation is performed based on the idea that the total expected error is the smallest.

2.1 Ship Route Model

A ship route is defined by the following adjustable variables:the number of route segments,latitude and longitude of each waypoint,heading angle information of each route segment,and ship’s hydrostatic speed in each route segment.In this study,we consider that the shipping route is composed of multiple route segments and that every two waypoints are connected by a rhumb line.As shown in Fig.2,SandEare the starting point and target point,respectively;Ni(i∈N+) is theith waypoint of the ship’s route,whereN0=S,Nn=E;λiis the latitude coordinate of waypointNi;liis the longitude coordinate of waypointNi;Virepresents the set water speed of the vessel in theith section of the route;φirepresents the heading information of theith section of the route;and the direction of north is 0?,which increases in the clockwise direction.

Fig.2 Ship route diagram.

The number of route segmentsnin the route model is determined according to the ship’s hydrostatic speed and great circle route.In this study,the preset initial value ofnis determined by Eq.(1),whereLse(nmile) is the great circle route distance between the departure point and destination,V(kn) is the hydrostatic speed,the voyage of a ship sailing forH(h) is a route segment in which the ship sails along the rhumb line,and the ceil function requires rounding to positive infinity.

The position information of waypoints can be represented by a structureSTR.The position information of each waypoint in a route is shown in Eq.(2).

A ship route consists of multiple route segments.To characterize the course and speed of a ship in different route segments,the vectorφis used to represent the direction of the ship on each route segment,and the vectorvis used to represent the ship’s hydrostatic speed on each route segment,as shown in Eqs.(3) and (4):

whereφirepresents the heading of the ship in theith route segment,the value of which can be determined by the latitude and longitude coordinates of theith waypoint and the latitude-longitude coordinates of thei+1waypoint,φi∈ [0?,360?),andvirepresents the hydrostatic speed of the ship in theith route segment.

2.2 Ship Stall Characteristics

The ship’s stall will affect the ship’s sailing time and fuel consumption and will have an important impact on the results of the ship’s best weather routing plan.Therefore,speed loss is an important factor that must be considered in the design of ship weather routes.In this study,we use the ship stall calculation formula proposed by Liu(1992),which takes into account the effects of the sense wave height,wave direction,wind speed,and wind direction on the ship speed.Based on real ship observation data,the iterative method is used to obtain the ship stall equation based on the least-squares method:

whereVa(kn) is the actual speed of the ship against the wind and waves,V0(kn) is the hydrostatic speed of the ship,F(kn) is the wind speed,D(t) is the actual displacement of the ship,h(m) is the significant wave height,q(rad) is the relative angle between the ship’s heading and the wave direction,andα(rad) is the relative angle between the ship’s heading and the wind direction.This formula can be used for various types of ships with a hydrostatic speed between 9 kn and 20 kn and a ship displacement between 5000 t and 25000 t.

2.3 Ship Sailing Time

According to the established ship routing model,the length of a ship route can be obtained by adding up the lengths of all route segments:

whereLis the total route length,nis the total number of route segments,andLiis the length of the rhumb line for each segment.If the earth is regarded as an ellipsoid,then the formula for calculating the distance (length) between any two points on the Mercator projection map is as follows (Snyder,1984):

whereλ1andl1are the latitude and longitude coordinates of the first point,respectively;λ2andl2are the latitude and longitude coordinates of the second point,respectively;φrhis the direction of the rhumb line;Lrhis the distance between the two points (in radians);ande=3.355× 10-3is the eccentricity of the earth.The above formula applies to ships sailing along non-iso-latitude lines,whereas for iso-latitude lines,that is,when the ship’s heading is 90? or 270?,the following equation can be used:

Thus,the total time a ship sails along a route can be obtained by summing the time consumed for all route segments,as shown in Eq.(10):

whereTvoyageis the total sailing time of the ship,tiis the sailing time of the ship on each route section,andis the actual speed of the ship on theith route segment.

2.4 Ship Navigation Risk

This study considers the wave height and wind speed in view of the second type of risk factor and establishes a risk assessment formula for ship routes:

whererriskis the risk value of the ship route;nis the total number of route sections of the ship route;hiwaveis the wave height of theith section of the ship route;hmaxwaveis the maximum alert wave height,the default value of which is 5 m;viwindis the wind speed of theith section of the ship’s route;vmaxwindis the maximum warning wind speed,the default value of which is 15 m s-1;αandβare the coefficients of influence of the wave height and wind speed on the navigation risk of the ship,respectively;andTalarmis the alarm time,which represents the accumulated time of the ship sailing in the high-risk area.In high-risk areas,the wave height is greater than the maximum warning wave height,or the wind speed is greater than the warning wind speed.According to the formula,the risk of a ship sailing along a route will change with the wave height and wind speed.Eq.(11) indicates that when the average cumulative wave height and cumulative wind speed are larger,the navigation risk value is larger,and Eq.(12)indicates that the longer the navigation time in the warning area,the larger the navigation risk value.Here,when the risk value of a route is greater than 0.6,the route is not acceptable.When the risk value is less than 0.6,although it is within an acceptable range,the risk value is expected to be as small as possible to reduce the risk of navigation.

2.5 Ship Fuel Consumption

Each ship route consists of multiple route segments.The intersection of each route segment and the integer latitude and longitude is called the sub-waypoint.Fig.3 shows a navigation area map with a resolution of 1?×1?.The red square point are the waypoints optimized in the algorithm;the black round-point are the sub-waypoints;SandEare the starting and ending points,respectively;andN1andN2are the waypoints.The resolution of the meteorological data in this study is selected as 1?×1?.Thus,the total sailing time and total fuel consumption of the ship are the cumulative sum of the sailing time and fuel consumption between the two adjacent points (diamond and circle) in the figure.

Fig.3 Ship route decomposition diagram.

The total fuel consumption can be determined by the following formula:whereffuelis the total fuel consumption of the ship’s route,mis the total number of waypoints (diamond and circle),tiis the ship’s sailing time along theith route segment,andFCPHiis the average fuel consumption per unit hour of the ship along theith route segment.

A search conducted by Duet al.(2011) and Kontovas(2014) showed that the relationship between the sailing speed and bunker consumption is nonlinear and that the daily bunker consumption is approximately proportional to the sailing speed cubed.Therefore,based on the fuel consumption data obtained from the actual sailing of the S-175 ship,this study uses the least-squares fitting method to obtain the curve relationship between the actual fuel consumption and speed of the ship,as shown in Eq.(14):

whereva,irepresents the actual speed of the ship along theith route segment.To ensure that the fuel consumption results are as practical as possible,only the data with a speed of 10 kn to 20 kn are fitted.

3 Weather Routing Algorithm Based on Non-dominated Sorting Multi-object Particle Swarm Optimization

3.1 Ship Navigation Area Initialization

In this study,the space for ship route optimization is limited to a local sea area that includes the starting point and ending point.By taking the fixed route as the reference line and expanding a certain length in the direction of the angle bisector of the angle formed by two adjacent equal course lines,the space of the ship’s sailing area designed for the best weather route is established.This study uses the rhumb line between the starting point and ending point as the reference line.Fig.4 depicts an established navigation area.

Fig.4 Schematic diagram of the ship navigation area.

The light-blue area represents land,the light-gray area represents the route optimization area,the dark-blue line represents the reference route,the dark-blue dots represent the ship waypoints,the green dotted lines represent the direction of the route optimization area expansion,and the red line represents the upper and lower boundaries of the route optimization area.

3.2 Particle Population Initialization

This study encodes the waypoint position information of the route control variable into a vectorXto characterize particles,as shown in the following equation:

The positions of the red upper and lower borders in Fig.4 can be calculated from the reference route and extended distance.The two structuresUpperBoundandLowerBoundare used to represent the upper and lower boundaries,respectively,which are expressed in Eqs.(16) and (17):

Xirepresents the position information of theith waypoint,Upperirepresents the position information of theith upper boundary point,andLowerirepresents the position information of theith lower boundary point.

To improve the diversity of the population,the initial particle swarm should be distributed as evenly as possible throughout the solution space.Therefore,this study uses a randomly generated,uniformly distributed random number within the upper and lower boundaries of each waypoint to generate a route.According to Eq.(18),an initial value of one particle can be randomly generated.If the generated route passes through islands or land,then a new route is regenerated.If the route generated for five consecutive times passes through the land,the objective function values of the route are all set to infinity,and the subsequent crossover and mutation operations are performed.

When the number of particle swarms is 50,the distribution of the initial particle swarms in the entire navigation area is as shown in Fig.5,and the initial particle swarms are uniformly distributed throughout the entire optimization region.

Fig.5 Route position with 50 particle populations.

3.3 Method of Designing the Ship’s Best Weather Route Based on HNDS-MPSO

A particle swarm optimization algorithm is a random search algorithm based on swarm cooperation,which is developed through the foraging behavior of bird swarms(Vettor and Sobieszczanski-Sobieski,2003).The basic particle swarm algorithm has a better effect when solving single-objective problems,but it cannot optimize multiple targets at the same time.Therefore,this study combines multi-objective optimization with particle swarm optimization and introduces elite selection and mutation operations in the genetic algorithms.An HNDS-MPSO algorithm is proposed to solve the problem of optimal weather navigation for ships.

The steps of the algorithm are shown in Table 1.To introduce the operation steps of the algorithm more clearly,the process of the route optimization algorithm is herein analyzed in detail:

Table 1 HNDS-MPSO steps

In step 1,the particle swarm is initialized.The particle speed and position are represented by the position information of the waypoints along the route,where the particle speed determines the direction and distance of the current particle movement.

In step 2,there are three optimization goals:the minimum ship navigation risk,minimum ship navigation time,and minimum ship fuel consumption.The smaller the three target values of the ship route are,the better the route.However,in the optimal weather navigation problem for ships under multiple constraints,it is almost impossible to simultaneously find three routes with the smallest target value.This study introduces the Pareto idea.According to the three target values of the particles,the domination level of each particle is calculated,and the crowding distance of the particles in each domination level is calculated.The definitions of dominance and non-dominance are as follows:

wherep'(p'={p1,p2,···,}) andq'(q'={q1,q2,···,}) are the decision variable vectors and the position information of the two particles is represented in the algorithm.u(u={u1,u2,···,}) andu'(u'={,,···,}) are the optimization target vectors of two particlesp'andq',respectively.Because there are three optimization targets in this study,n=3.If the performance vectorsuandu'satisfy Eq.(21),then particlep'dominatesq'.If a particle neither dominates nor is dominated by other particles,then the particle is called a non-dominated solution,and the set of all particles that satisfy the non-dominated solution is called a non-dominated solution set.The division of dominance levels in the particle swarm is determined according to the following steps:

a) Perform non-dominated sorting of the initial particle swarm to obtain a set of Pareto optimal frontiers and Pareto optimal solutions.The particles in the solution set have the first domination level,that is,the highest domination level.Seti=2.

b) Exclude particles in the population that have been assigned a dominance level,and then perform non-dominated sorting among the remaining particles to obtain the Pareto optimal solution set,which has theith dominance level.Seti=i+1.

c) Repeat step b) until all particles have been assigned a dominance level.

To further evaluate the performance of the particles in each domination level,the crowding distance between the particles at each domination level is defined according to Eq.(22):

whereCrowd(m) is the crowding distance of themth particle (m=2,3,···,N'-1);fjis thejth objective function value;fjMaxandfjMinare the maximum and minimum values of thejth optimization target,respectively;N'is the number of particles at the same dominance level;and the crowding distance for edge particles is set to infinity.

Then,among the particles with the highest domination level and a crowding distance not equal to infinity,a particle is randomly selected as the global optimal particle.

Step 3 is a process of iterative optimization.In step I,the speed of the particles is updated according to Eq.(23),and the position of the particles is updated according to Eq.(24).In step III,the particle population is uniformly mutated at the mutation probabilityP,and the mutated population is taken as POP3.In step IV,the three populations are merged,andMparticles are selected as the next generation through non-dominated sorting based on the domination level and the crowding distance.The purpose is to retain the superior particles of the parent.In step V,the global optimal particle position is updated.In step VI,the new population POP1 is used as the parent population,and the next iteration is performed until the conditions for exiting the iteration are satisfied.

Finally,based on the Pareto optimal front and Pareto optimal solution set obtained by the algorithm,a recommended route is given according to Eq.(25):

whereNis the number of optimization targets,cijis thejth objective function value of theith particle after normalization,yjis the normalized expected target function value,Mis the number of particles in the Pareto solution,min is a function for selecting recommended routes in the Pareto solution set and the route in the solution set corresponding to the minimum value in parentheses is obtained,andZis the recommended route that satisfies the conditions on the right side of the equation.

4 Results and Discussion

4.1 Algorithm Parameter Initialization

The experimental ship in this study is an S-175 container ship.The ship parameters are given in Table 2.

Table 2 Parameters of the S-175 container ship

The starting point is near the port of New York,and the destination is near the port of Porto,Portugal.The sailing time of the ship is 00:00 on July 1,2016.The navigation area is obtained by expanding by 8 degrees along the angle bisector of the reference line.The ship’s still water speed is set at 16 kn.

In Table 3,Genindicates the number of iterations of the algorithm,andPopindicates the number of particle populations.c1,c2,andωare the parameters in the particle swarm algorithm,andPis the uniform mutation probability.

Table 3 MPSO algorithm parameters

4.2 Acquisition and Analysis of Meteorological Data

In this study,we obtained the meteorological weather data for a period of time in advance from the European Centre for Medium-Range Weather Forecasts.The meteorological data considered in this experiment mainly include wind and wave information,which change over time.The meteorological data files used in this article are of the Network Common Data Format (NetCDF).The meteorological data used include the ‘10 meter U wind component’,‘10 meter V wind component’,‘mean wave direction’,and ‘significant height of combined wind waves and swell’.The accuracy of the selected meteorological data is 1?×1?,and the meteorological data update interval is 6 hours,which corresponds to the meteorological data at 00:00,06:00,12:00,and 18:00 every day.Fig.6 shows the visual information diagrams of the waves and wind at two moments in a certain area of the Atlantic Ocean on July 8,2016.

Fig.6 Wave and wind data at two different times.

4.3 Experimental Results and Analysis

This study sets up two experiments:one is planning the optimal route under good sea conditions and the other is planning the optimal route under severe sea conditions.

4.3.1 Good sea conditions

According to the meteorological forecast,from July 1 to 7,2016,the sea conditions in the ship’s navigation area were good,and there were no extreme winds or waves.In this case,after applying the non-dominated sorting MPSO algorithm for ship route planning,the Pareto optimal solution set for the ship weather route is shown in Fig.7,and the Pareto optimal frontier is shown in Fig.8.Therefore,by sorting the objective function values of the ship routes in the Pareto solution,the minimum sailing time route,minimum safety risk route,and minimum fuel consumption route can be obtained.We sety1=200.0,y2=0.30 andy3=900,which respectively indicate the expected sailing time,navigation risk,and fuel consumption of the ship.The recommended route can be obtained according to Eq.(22).The trajectories of the four ship routes are shown in Fig.9.In Fig.8,the green point D indicates the route with the lowest navigation risk,the black point B indicates the route with the shortest sailing time,the magenta point A indicates the route with the lowest fuel consumption,and the red point C indicates the recommended route.The recommended ship route is shown in Fig.10,and the specific performance of the route is shown in Table 4.

Fig.7 Schematic diagram of all routes.

Fig.8 Pareto optimal front.

Fig.9 Schematic diagram of four ship routes.

Fig.10 Recommended route.

As shown in Table 4,although the route with the shortest sailing time is shorter than the recommended route,the navigation risk of the recommended route is lower.The safest route has the lowest safety risk,but the ship sailing time is too long.The route with the lowest fuel consumption consumes the least amount of fuel,but the ship is at high risk.In addition,we compared the great circle route with the recommended route.Although the great circle route has a shorter sailing distance,it is the route with the highest risk.The recommended route proposed in this paper can combine the three goals according to the expected target value to obtain an optimal weather route that meets the requirements.For the recommended route,the sailing positions of the ship at four different sailing moments are as shown in Fig.11.The figure shows that under good sea conditions,the ship can sail along the recommended route with less navigation risk,lower fuel consumption,and a shorter sailing time.

Table 4 Objective function values for the four routes under good sea conditions

Fig.11 Ship’s trajectory along the recommended route at four different times.

4.3.2 Severe sea conditions

To further illustrate the effectiveness of the algorithm,the best weather route under severe weather conditions in the ship’s navigation area was also designed.According to the meteorological forecasts from July 3 to 10,2016,the sea conditions in the ship’s navigation area were severe.As shown in Fig.12,under severe sea conditions,the algorithm was applied to obtain the Pareto optimal solution set.By sorting the objective function values of the ship routes in the Pareto solution,the minimum sailing time route,minimum safety risk route,and minimum fuel consumption route can be obtained.We sety1=225.0,y2=0.32 andy3=950,which respectively indicate the expected sailing time,navigation risk,and fuel consumption of the ship.The recommended route can be obtained according to Eq.(24).The Pareto optimal frontier is shown in Fig.13,the trajectories of the four ship routes are shown in Fig.14,and the specific performance of the route is shown in Table 5.The recommended ship route is shown in Fig.15.For the recommended route,the sailing positions of the ship at four different sailing moments are shown in Fig.16.

Fig.12 Schematic diagram of all routes.

Fig.13 Pareto optimal front.

Fig.14 Schematic diagram of four ship routes.

Fig.15 Recommended route.

Fig.16 Ship’s trajectory along the recommended route at four different times.

As shown in Fig.12,most of the routes in the Pareto optimal solution set can avoid high-risk areas near (37?N,35?W).In the design of the best weather route,there are almost no ship routes with the characteristics of the shortest sailing time,lowest sailing risk,and least fuel consumption.Therefore,in this study,based on the expected target value,a comprehensive optimization was performed among the three targets.The selected route can meet the requirements of shorter sailing time and less fuel consumptionviasafe navigation of the ship.As shown in Fig.12 and Table 5,the recommended route can avoid dangerous sea areas with large winds and waves and reach its destination safely.

Table 5 Objective function values for the four routes under severe sea conditions

Based on the above experiments,the routes in the Pareto solution set provide the captain with the possibility of choosing routes to meet different needs.To meet the safety,efficiency,and economy requirements in ship navigation,this paper presents a recommended route that has the shortest sailing time,lowest fuel consumption,and lowered navigation risk.The results show that the route has good performance.

5 Conclusions

This study establishes a system framework for ship multiobjective weather routing based on the characteristics of a ship’s ocean voyage.The system consists of the parameter input,environment construction,route model,multiobjective optimization algorithm,route evaluation,and route recommendation.Based on the system framework,this paper proposes a non-dominated sorting and multiobjective ship weather routing algorithm based on particle swarm optimization.This algorithm combines mutations and elite selection operations in genetic algorithms.Such an algorithm can obtain the lowest sailing risk,shortest sailing time,and lowest fuel consumption of ships and provide the best weather routes for ocean voyages,guaranteeing that ships can sail safely,efficiently,and economically.According to the experimental results,the recommended route can integrate three types of optimization objectives,avoiding high-risk sea areas and ensuring as little ship sailing time and fuel consumption as possible.Therefore,the multi-objective weather routing system and non-dominated multi-objective route planning algorithm proposed in this paper are feasible and effective when applied to ship weather routing problems.Although the results show that the algorithm is effective,some shortcomings need to be resolved.First,visibility and other ships are not considered when calculating the navigation risk.Second,the fuel consumption calculation is not accurate enough.

Acknowledgement

This study was funded by the Russian Foundation for Basic Research (RFBR) (No.20-07-00531).