1. Introduction
Maritime transportation is the main mode of transportation in trade, and the shipping industry accounts for more than 80% of global trade. However, with the continuous development of shipping, the direct result is an increase in global greenhouse gas (GHG) emissions. To this end, the International Maritime Organization (IMO) has established that by 2050, the global shipping industry will reduce carbon by at least 50% compared with 2008, and other shipping organizations have proposed a series of measures [
1,
2]. The acceleration of the global economic process has put forward higher requirements for maritime transportation. It has become an urgent task to improve shipping efficiency and reduce shipping costs [
3]. However, due to the influence of the marine environment, the safety of ships has been greatly tested. Planning a route as economical and safe as possible has become a key factor in improving shipping efficiency.
The core of route optimization is to find an optimal route. Based on marine environment data, combined with ship performance and navigation tasks, a safe and economical route is selected. With the increasingly significant role of route optimization in maritime navigation, many methods for planning routes have been proposed. When navigating in complex weather conditions, the captain uses weather data to avoid potential dangers during the voyage and maximizes the safety of ship navigation. According to different theories, ship route planning methods can be divided into three categories: graph search method, meta-heuristic algorithm of dynamic path, and trajectory analysis method using big data [
4].
In the early days, the graph search method usually set the route at a fixed speed, and then adjusted the speed according to the navigation loop. The navigation area is discretized into nodes and edges, and the route planning problem is transformed into directed graph search, such as Dijkstra algorithm and A-star algorithm [
5], which are gradually applied to path planning to obtain the shortest time path and the shortest distance path. Although these algorithms can effectively avoid static obstacles, they are not suitable for dynamic and complex marine environments. In order to solve different problems, the improvement of these algorithms has been ongoing. Silveira et al. [
6] and Wang et al. [
7] made targeted improvements to the Dijkstra algorithm, although there is still more room for improvement. Liu et al. proposed a PE-A* algorithm, which uses the potential energy field to express the environmental field to achieve the effect of global planning and local collision avoidance [
8]. Addressing this problem, Shin et al. proposed an improved A-star algorithm, which can avoid the limitations of the initial A-star algorithm [
9]. Mannarini et al. proposed a new dynamic programming method (VISIR: Discovering Safe and Efficient Routes) [
10], which evaluates the optimal path in time by solving the basic differential equation of the optimal path in dynamic wind and waves.
At present, various meta-heuristic algorithms are applied to the field of route planning, such as the genetic algorithm (GA) [
11], simulated annealing algorithm, particle swarm optimization (PSO) [
12], and ant colony algorithm. Zhou et al. proposed a hybrid genetic algorithm, which combines a simulated annealing algorithm with a genetic algorithm to avoid falling into a local optimum [
13]. Zhang et al. combined the ant colony algorithm with the A-star algorithm [
14], and introduced the Bessel curve method to smooth the path to obtain the optimal path.
The big data mining method is based on the statistical analysis of a large number of ship route data from the automatic identification system (AIS) to improve shipping efficiency [
15,
16,
17,
18,
19]. This method usually considers historical ship trajectory data, using clustering methods for statistical analysis and establishing a route trajectory model to improve the safety of navigation.
Reasonable meteorological route arrangement can minimize the risks caused by total fuel consumption, sailing time, and adverse conditions, but in practice, it is difficult to pursue these three objectives at the same time. At present, the multi-objective optimization method is roughly divided into two kinds. One is to assign weights to the objective function and transform the multi-objective problem into a single-objective problem. This method simplifies the solution process, but the route involves conflicting objectives, such as reducing the total cost, emissions, and navigation time. The single-objective optimization method with heavy weight does not solve this conflict well. The latter finds the approximate optimal solution of the problem by finding the Pareto front and can complete the search in an acceptable time [
20,
21,
22,
23,
24]. Du et al. proposed an improved fractional order particle swarm optimization (FOPSO) algorithm [
25] that avoids problems such as falling into local optimization in solving multi-objective problems. Ma et al. used a non-dominated sorting genetic algorithm to optimize the cost and time of navigation [
26]. Considering carbon emissions and weather factors, Yuan et al. also proposed a weather uncertainty model [
27], and added the weather probability model as a penalty factor to the non-dominated sorting genetic algorithm for route optimization. Polar navigation is also an important part of ship navigation. Szlapczynski et al. added the information of into the multi-objective evolutionary algorithm to find a route that is more in line with the decision makers’ ideas [
28], and considered the uncertainty of weather forecasts.
Addressing the problem of ship route optimization, this paper focuses on finding a safe and low-carbon route-optimization method, which designs a route for ocean-going navigation in bad sea conditions, takes the safety of ship navigation, sailing time, and fuel consumption as variables to optimize the route, and can update the weather and meteorology in time, avoid rough seas, and reach the destination safely. To summarize our views, the work we have done mainly includes the following aspects.
- (1)
A model for sea conditions using weather data is established. The complex sea area exceeding the threshold is marked as a dangerous area as a danger zone according to the set safety threshold, and the calibrated dangerous area is updated by updating the environmental data.
- (2)
Considering the speed loss of ships in different marine environments, a speed-loss model of ships is established.
- (3)
On the basis of the above model, a route-optimization model considering both ship navigation time and fuel consumption is established by using a non-dominated sorting genetic algorithm (NSGA-II), and ship stability is added to the constraint factor to make the optimal path safer and more energy-saving.
The structure of the rest of this paper is as follows: In the second part, a complex marine environment model is proposed. By rasterizing the environmental data, a marine environment model is established to study the ship speed loss under different sea conditions. In the third part, the objective function of route optimization is designed, the constraints of navigation are proposed, and the NSGA-II is used to solve the route optimization problem. In the fourth part, the validity and reliability of the method are verified by comparing it to path optimization based on a genetic algorithm. Finally, the work done in this paper is summarized and discussed.
3. Multi-Objective Route Optimization Model of the Ship
3.1. Objective Functions
In addition to the need to ensure the safety of navigation, a voyage must also consider the economic benefits of shipping, such as the amount of time it takes and how much fuel is used, in addition to ensuring the safety of the route. Therefore, ship route optimization is a complex problem involving multiple objectives, variables, and factors. The impact of sudden situations such as gusts is not taken into account by meteorological conditions to simplify the problem and facilitate the establishment and resolution of the model. The mathematical expression of the multi-objective route optimization model is as follows:
Here, is the decision variable represented by the set of waypoints and the corresponding actual speed ; is the number of waypoints on the route. are the objective functions, where is the total time of leglines by the ship along the route, and is the total fuel consumption of the ship during navigation, which is related to the speed of the ship in still water. is the distance between waypoints and , is the still water speed of a ship sailing between waypoints and , is the minimum speed, is the maximum speed, and is the fuel consumption rate determined by the performance of the main engine of the ship and the speed, in which the baseline fuel consumption of the ship can be obtained by using the Lagrange method based on a large number of voyage data of the ship.
During the voyage, ships will encounter shoals, reefs, and other obstacles, which greatly limit the scope of ship activities. Therefore, in the optimization process, it is necessary to stay as far away from them as possible. Equation (9) represents the geographical constraints of the ship in navigation:
where
is the water depth of the ship at the
-th waypoint, and
and
respectively represent the average draft of the ship at this position and the gap under the keel.
During the voyage of the ship, under the combined action of external parameters such as wind, waves, and currents, the periodic change in the geometric shape of the immersed part of the hull and the shape of the waterline surface leads to the periodic change in the restoring moment of the ship, that is, the rolling phenomenon. This phenomenon can easily cause the ship to capsize and is a huge potential safety hazard. However, an important factor affecting rolling is the relationship between the encounter period and the natural period:
In Equation (10), ; is the width of the ship; is the waterline length; is the average water intake; and is the height of initial stable center.
In Equation (11),
is the encounter angle;
is the heading angle;
is the absolute wave direction angle;
is the natural frequency of waves;
is the ship speed; and
g is the acceleration due to gravity.
Equation (12) addresses the stability constraint of the ship in navigation, where
and
are the thresholds, taken here to be 0.5;
is the length of the ship;
is the wave length of the wave; and
is the wave period. Equation (13) represents the constraints of weather or sea condition that the ship can withstand:
These conditions are usually simply set to deterministic thresholds, such as maximum allowable wave height and maximum allowable wind speed.
and
represent the wave height and wind speed encountered by the ship at the
-th waypoint, which do not exceed the maximum allowable wave height
and the maximum allowable wind speed
.
3.2. Designing the Algorithm
By considering multiple objective functions and constraints at once, they can determine that a population approach is the most optimal solution in the feasible region. The behavior of organisms during evolution is simulated using natural selection, elimination, and reproduction mechanisms. Selecting a suitable multi-objective evolutionary algorithm (MOEA) is crucial to finding the optimal solution for the multi-objective and multi-constraint route optimization problem.
Currently, the non-dominated sorting genetic algorithm is a mature MOEA. It achieves efficient search, uniform distribution, and diversity maintenance of solutions through fast non-dominated sorting, congestion comparison, and elite strategy, and has high convergence and robustness. Compared to other algorithms, the NSGA-II is more stable and efficient, so it is chosen to solve the multi-objective route optimization problem.
In the MOEA, the initial population is randomly generated in the solution space, and the characteristics of individuals are represented by chromosomes, and then the population is updated by crossover, mutation, and selection. Similarly, when solving the multi-objective route optimization problem, the individual ship route can be composed of variables determined by waypoints and speed information, as shown in Equation (14):
where
and
are the departure position and destination position, respectively; and
is the number of individuals in the population.
In order to improve the calculation speed and accuracy of the algorithm, the initial population is generated randomly, that is, the routes around the great circle empirical routes are randomly generated as individual populations, and the navigation speed is also generated randomly. A new population Q0 is generated by selecting, crossing, and mutating the initial population I0 to ensure that the population sizes of I0 and Q0 are both N. It and Qt are merged into Ct, and after the Ct is quickly non-dominated, a partial ordered set is established by calculating the crowding distance of all individuals in a certain self, and then individuals are selected in turn to enter It+1 until the scale reaches N. Then, an iterative operation is performed to determine if the maximum number of iterations is satisfied. If not, the new generation of population Qt+1 is continued to be generated. If it is satisfied, the individual of It+1 is output, that is, the non-dominated optimal solution set.
Route optimization is an optimization problem to find the minimum value of the objective function; the smaller the value, the better the individual’s fitness and survival probability. In the process of population evolution, a penalty factor
ζ is added to the objective function to evaluate the fitness of the individual. The individual increases the elimination rate of the next generation by increasing the value of the cost function, but the individual can evolve through operations such as cross mutation, which not only ensures the randomness of the individual, but also improves the efficiency of algorithm optimization. As shown in Equation (15):
It is worth noting that the NSGA-II algorithm’s discovery of the Pareto solution set is not the only optimal solution. Therefore, solving the route optimization problem by evolutionary algorithm is to find a set of solution sets that make the objective function more complementary, and then sort the solution sets according to the user’s optimization criteria based on sorting method, and finally obtain the optimal route.