1. Introduction
The shipping industry, as a crucial pillar of global trade and economic development, is increasingly gaining attention and importance. As the primary means of maritime transportation, ships have their security, efficiency, and environmental impact recognized as key areas for research and improvement. Therefore, a safe, cost-effective, and environmentally friendly path has become a necessary requirement for green sailing. In this context, the maritime sector continuously faces new challenges and opportunities. Ship voyage times, fuel consumption, and voyage security have long been focal points of concern within the shipping industry. Under safe sailing conditions, decision-makers will pursue less energy consumption and shorter voyage times. However, if there are certain rigid requirements in tasks or trade, such as an estimated time of arrival (ETA), decision-makers must first consider the numerical value of the shortest voyage times and make different choices in different situations, which corresponds to multi-objective optimization problems. Therefore, the optimization of multi-objective problems is a critical aspect of selecting a reasonable path.
The multi-objective optimization problem is an important research area that exists in many practical applications. In real life, many decision-making problems need to consider multiple interconnected objectives simultaneously. For instance, there is a need to balance cost, performance, and reliability in engineering design [
1], resource allocation requires considering effectiveness and efficiency [
2], and so on. These problems often encompass multiple conflicting objectives; however, finding a globally optimal solution becomes extremely complex in this situation. Moreover, one of the difficulties in multi-objective optimization problems is the interrelationships and trade-offs between objectives. Improving one objective might lead to the degradation of others, so it is necessary to find a balance point in decision making so that multiple goals can be reasonably balanced and met. Furthermore, a multi-objective optimization problem typically has countless possible solutions that form a curve or surface known as a “Pareto front”, representing the optimal solutions. The solutions generated by a high-performance algorithm should effectively cover the Pareto front in a well-distributed manner [
3].
In order to solve multi-objective optimization problems, researchers have proposed many methods and techniques. For example, evolutionary algorithms [
4,
5], genetic algorithms [
6,
7], particle swarm optimization algorithms [
8,
9], and other biological population behavior simulation methods. These methods design efficient optimization strategies by imitating the mechanisms of species evolution and collective behavior in nature. However, as the scale and complexity of the problems increase, traditional methods may encounter challenges such as large search spaces and slow convergence rates [
3]. When tackling a challenging multi-objective optimization issue, the authors of [
10] proposed a novel nature-inspired optimization algorithm that utilizes the efficient evolution process and opposition-based learning (OBL) to enhance optimization performance. The Guided Population Archive Whale Optimization Algorithm (GPAWOA) was proposed in [
11]. It stores the non-dominated solutions found during the optimization process in an external archive. The authors of [
3] applied the Whale Optimization Algorithm (WOA) to balance the fuel consumption and voyage time of ships.
The Whale Optimization Algorithm (WOA) is an evolutionary algorithm inspired by the collective behavior of whales in nature. It is used to solve complex optimization problems. This algorithm originates from the observation of the predatory behavior of whale populations, aiming to simulate the cooperation and adaptability of whales in searching for food. The WOA propels the optimization process by emulating the behavioral strategies of individual whales, such as chasing prey, social behavior, and call signals. Among them, the authors of [
12] applied a chasing prey strategy to find the global optimal solutions, adjusting the search scope to enhance the algorithm’s global search capability. Moreover, the WOA has shown strong performance in solving continuous optimization [
13] and discrete problems [
14]. Its advantage lies in its ability to efficiently search in the solution space and in its faster convergence speed and better global convergence performance compared to traditional optimization algorithms. Although the WOA has been well applied in many problems such as PID controller parameter tuning [
15,
16], medical data analysis [
17], and image segmentation [
18,
19], there are still some problems, such as the selection of algorithm parameters and adaptability to certain complex problems. The authors of [
20] improved the WOA algorithm when facing large-scale global optimization (LSGO) problems.
Inspired by the above research, this paper proposes a scheme for ship multi-objective path planning based on the meteorological environment. This scheme applies meteorological data to the objective function of multi-objective problems and performs path planning in the established grid marine environment model through the Whale Optimization Algorithm. At the same time, the performance of the algorithm is tested by using the test functions Viennet3, DTLZ series functions, and UF8. The findings confirm the effectiveness of the suggested scheme. The primary contributions of this paper can be outlined as follows:
(1) In order to obtain more realistic ship fuel consumption, voyage time, and voyage security, environmental factors such as wind and waves were considered. Kwon and Lap–Keller methods were used to apply the meteorological data to the objective function of the multi-objective problem.
(2) The Whale Optimization Algorithm is utilized for addressing the multi-objective path-planning issue for a ship. To boost performance, the convergence factor is non-linearly changed to improve the algorithm’s global search and local refinement abilities.
(3) The effectiveness of the algorithm in the multi-objective problem (three objectives) is proved by the multi-objective test function. Compared with other advanced multi-objective algorithms, the effectiveness of the algorithm is also proven. In addition, in order to get a more realistic and reasonable path, the grid method is used to establish the marine environment model, and a more realistic and reasonable grid accuracy is designed.
The remaining sections of this paper are organized as follows:
Section 2 presents the mathematical model of the objective function for the ship multi-objective path-planning problem.
Section 3 introduces the multi-objective improved Whale Optimization Algorithm for path planning.
Section 4 compares the algorithm with several optimization algorithms and applies it to ship multi-objective path planning.
Section 5 summarizes the content of this paper and formulates the conclusion.
4. Algorithm Performance Test
This section tests the convergence, distribution, diversity, and other performance of the algorithm, and conducts research and verification in multi-objective testing functions. The specific process is to compare the improved WOA with multi-objective snake optimization (MOSO) [
32], multi-objective jellyfish search (MOJS) [
33], and multi-objective seagull optimization algorithm (MOSOA) [
34,
35] on seven test functions. Several comparison algorithms are similar to the framework of the improved WOA, but the stages of different algorithms are different, and the location update function and search mechanism are different. The test results were verified by the inverted generational distance (IGD), hypervolume (HV), and spacing metric (SP).
4.1. Test Functions
The test functions utilized in this section consist of Viennet3, DTLZ series functions, and UF8. Viennet3 is a commonly used multimodal optimization test function that can be used to evaluate the performance of the optimization algorithm. Viennet3 has a complex topological structure and nonlinear characteristics, which puts forward higher requirements for the convergence and global search ability of the algorithm [
36,
37]. Therefore, the study of Viennet3 can help us understand and improve the effect and performance of the optimization algorithm. In addition, many researchers use DTLZ series functions [
38,
39,
40] and UF8 [
41,
42] to compare and improve multi-objective optimization algorithms, improve the convergence, diversity, and adaptability of algorithms, and provide guidance for finding optimal solutions to practical decision-making problems. The details of the seven test functions are shown in
Table 6.
The meaning of the Pareto front is that there is no possibility of making an individual better by redistribution without making any other individual worse. In the two-objective optimization, it is represented as a curve, and in the three-objective optimization, it is represented as a surface or curve. The seven test functions used in this paper have clear Pareto front solutions, which makes it convenient to use evaluation metrics for evaluation.
4.2. Algorithm Performance Evaluation Metrics
In order to evaluate the algorithm’s performance in terms of convergence, distribution, and universality, researchers have proposed numerous methods for evaluating performance. The evaluation metrics of this paper include generational distance, hypervolume, and spacing metrics. At the same time, the algorithm is evaluated by the generational distance (GD). The meaning of the GD is similar to the IGD, so we analyze the IGD only. The results of GD are presented in
Table 7.
4.2.1. Inverted Generational Distance
The IGD calculates the sum of the distances between each point in the reference set (ideal Pareto front) and the non-dominant solution set generated by the algorithm (actual Pareto front approximation) to evaluate the convergence and distribution of the algorithm. A smaller IGD indicates greater convergence and a more distributed solution set distribution. The specific calculation is shown in Equation (37).
where
is the point set,
is the number of points distributed on the true Pareto surface,
is the optimal Pareto solution set, and
is the minimum Euclidean distance.
Table 8 shows the IGD of the multi-objective algorithms. It is evident that the IGD value obtained by the improved WOA is minimum, indicating that the convergence and distribution of the improved WOA are better compared to other comparative algorithms.
4.2.2. Hypervolume
In recent years, HV evaluation metrics have been favored by more and more scholars. EMO (Evolutionary Multi-Criterion Optimization) has articles related to hypervolume evaluation metrics in each session. HV evaluates the performance of the multi-objective algorithm by measuring the volume of hypercubes formed by individuals in the solution and true Pareto front solutions. If the comprehensive performance of the algorithm is better, the calculated HV must be high. The specific calculation is shown in Equation (38).
where
represents the Lebesgue measure.
represents the number of solutions, and
represents the hypervolume of the true Pareto front solution and the
ith solution.
Table 9 shows the tested result of hypervolume metric. The hypervolume metric can simultaneously measure both the convergence and distribution of the solution set. The greater the convergence and distribution of the solution set, the higher the hypervolume metric obtained. It is evident that the HV value obtained by improved WOA is the highest, and the convergence and distribution of the corresponding solutions are better than MOSO, MOJS, and MOSOA.
4.2.3. Spacing Metric
Spacing metric evaluates the distribution of multi-objective algorithms by measuring the distance between each solution in the non-dominant solution set obtained by the algorithm and its nearest neighbor solution. If the evenness of the distribution of the Pareto-optimal solutions is better, the calculated spacing value must be small. The specific calculation is shown in Equation (39).
where
is the minimum distance from the
ith solution to the other solutions in
, and
is the mean of
.
Table 10 shows the performance of the comparison algorithm. The value of a small spacing metric indicates the effective distribution of the algorithm. From the table, it can be seen that the solution generated by MOSO in the distribution is slightly better than other algorithms but inferior to the improved WOA.
4.3. Analysis of Performance Test Results
The test functions Viennet3, DTLZ series, and UF8 all have true Pareto front solutions, which appear as a curve or surface in the case of multi-objective optimization.
Figure 7,
Figure 8,
Figure 9,
Figure 10,
Figure 11,
Figure 12,
Figure 13,
Figure 14,
Figure 15,
Figure 16,
Figure 17,
Figure 18,
Figure 19,
Figure 20,
Figure 21,
Figure 22,
Figure 23,
Figure 24,
Figure 25,
Figure 26,
Figure 27,
Figure 28,
Figure 29,
Figure 30,
Figure 31,
Figure 32,
Figure 33 and
Figure 34 show the performance test results of each algorithm on the test functions. Different algorithms represent points with different colors applied, and the circles represent the solutions generated through algorithms on the test functions. The black curve (or black dotted surface) is the true Pareto front solution.
From the figure, the solutions generated by the improved WOA are uniformly distributed across the surface. The solutions generated by the comparison algorithms, with the exception of the improved WOA, either fail to fully cover the Pareto front or are unevenly distributed. Taking the most typical test function, UF8, as an example, the solutions generated by the comparison algorithms are mostly located on the surface. Moreover, the solutions generated by the comparison algorithm have some solutions that do not converge to the Pareto front. This indicates that the improved WOA has good distribution and convergence. For the test function Viennet3, the performances of the MOSOA, MOSO, and improved WOA are not significantly different, but the MOSOA and MOSO are not well distributed on the Pareto front, which can be clearly seen from
Table 8 and
Table 10. From the performance of the test function, regardless of whether it is a multimodal optimization problem or a uniform distribution problem, the solution set generated by the improved WOA can be well distributed to the Pareto front and has good convergence when the same parameters are set.
The analysis also includes the factors contributing to the underperformance of other algorithms. For the MOSOA, the algorithm is divided into two stages: migration and predation. This is similar to the improved WOA but lacks an adjustment strategy. Therefore, higher requirements are placed on the setting of the problem parameters. When the number of iterations is not sufficiently large, its ability is not reflected. For MOSO and MOJS, it is also an algorithm simulated by biological group behavior. The mechanism is reasonable, but there are still some gaps compared with whales. Therefore, the improved WOA has better robustness.
4.4. MASS Multi-Objective Path Planning
Planning a safe and cost-effective path in multi-objective path planning demands comprehensive consideration of environmental conditions and meteorological data. The data are based on historical reanalysis datasets from ECMWF and NASA. The non-sailing areas, such as land and obstacles, are composed of 49,435 latitude and longitude coordinate points.
The Belt and Road (B&R) is beneficial to many countries in the world. The path of this paper is from Guangzhou to Hanoi, with the starting and ending points described by latitude and longitude as (113.7, 22.63) and (106.7, 20.8), respectively. The spatial resolution is set to
, and the relevant literature that can provide a reference value is listed in
Table 11. The path is planned based on the established environmental model and meteorological data.
Figure 35 is the map of the corresponding area, and
Figure 36 is the case after setting the spatial resolution. Where brown dots represent rocks, black squares represent ocean platforms, and purple-red areas represent restricted areas. Moreover,
Figure 37 and
Figure 38 are the enlarged Qiongzhou Strait area. By comparing the image of the environmental model with the actual situation, the accuracy of the model we established is high enough, and the error is small.
In the MASS multi-objective path-planning problem, when considering the actual course from Tokyo to Oakland, sections are set to 20 [
3]. The environment considered in this paper is more detailed, and the sections are set to 25. The four paths planned by improved WOA are shown in
Figure 39 in yellow, orange, green, and blue, respectively. At the same time, the planning maps for the three most important areas of the entire voyage are also shown in
Figure 39.
Table 12 presents ship fuel consumption, voyage time, and voyage security of the corresponding paths. It can be seen that the improved WOA has four Pareto-optimal solutions. Improving them would weaken at least one objective function.
In the path planning from Guangzhou to Hanoi, we also apply other comparison algorithms. The results are shown in
Table 13. In ship multi-objective path-planning problems, the improved WOA and MOSO are superior to other comparison algorithms in fuel consumption, voyage time, and voyage security, indicating that these two algorithms have good applicability and robustness. However, due to the differences in behavioral mechanisms brought about by the living environment of snakes and whales, the results of the improved WOA are better than those of MOSO. The MOSOA only has two stages of migration and predation, so it is superior to all other algorithms in regard to voyage time. Due to the lack of adjustment strategies, the results of ship fuel consumption and voyage security are relatively general. Therefore, evaluating the performance of an algorithm needs to be conducted from multiple perspectives. Whether it is the performance results under the test function or the path-planning results under different meteorological conditions, the performance of the improved WOA in the MASS multi-objective path problem is better than that of other comparison algorithms.
5. Conclusions
Based on marine meteorological data, this paper studies the multi-objective path-planning problem for MASS. The mathematical model of the objective function of the multi-objective problem is introduced, and the problem is solved by the improved WOA. The multi-objective effectiveness of the improved WOA is validated through the use of multi-objective test functions. Furthermore, the algorithm is utilized for the purpose of planning the path from Guangzhou to Hanoi. The research in this paper has important theoretical and practical significance for the MASS’s meteorological sailing path planning ability. The main conclusions are as follows:
(1) The multi-objective problem will have a lot of solutions. When the consideration factor is greater than two, the simple size relationship is not enough to evaluate the comparative advantage among them. Non-dominated sorting can play a good role in this problem and help us make trade-off choices.
(2) In the aspect of algorithm performance testing, the predominant tests are focused on the convergence and distribution of the solutions. The true Pareto front solution can help us intuitively see the difference in the performance of algorithms. The performance of improved WOA outperforms MOSO, MOJS, and MOSOA. Moreover, obtained by the Improved WOA, the Pareto solution set is well covered and converges to the Pareto front.
(3) The improved WOA algorithm outperforms other comparison algorithms in regard to fuel consumption, voyage time, and voyage security when optimizing the route from Guangzhou to Hanoi.
(4) There are many factors affecting MASS sailing under marine environmental conditions, among which wind and waves are the main environmental factors that a ship’s voyage is affected by. Ship fuel consumption, voyage time, and voyage security are different due to different routes. In practical situations, the needs of each decision-maker are different, so it is essential to design a rational path.