Optimization of Shipping Routes for Container Ships from Indonesia to the Asia-Pacific Using Heuristic Algorithms

Abstract

Sea transportation such as that by container ships has an essential role in the economy both locally and internationally. Ships are a major commodity in distributing goods over long distances due to their relatively low price compared to air shipping. The study implemented an optimization method using heuristic algorithms with ship route selection to minimize operational costs based on the parameters of mileage between 12 ports in the Asia-Pacific region. The ship speed, engine power, and fuel prices at each port are processed using asymmetric traveling salesman problem modeling (ATSP). The research uses three different algorithms to compare with the performance of the traveling salesman problem, namely the nearest neighbor algorithm, simulated annealing, and a genetic algorithm, with an objective function of keeping fuel costs that ships will incur to a minimum. The results show that the genetic algorithm provides the route with the lowest fuel cost.

1. Introduction

Sea transportation has an essential role in the national and international economy. Ships have become a commodity to distribute goods over long distances due to their relatively low price compared to air shipments. Based on Hwang’s research [1], ships are already a means of transportation where 70% of the value of goods is transported between continents. The transportation costs result in moving goods in volume being more afford-able by using sea routes. The cost of transportation itself has a value of 66% of all logistics costs in the world [2]. Shipping companies should obtain transportation cost efficiency so that shipping companies can achieve operational benefits. The operational costs borne by a shipping company are divided into several aspects, namely, fuel costs, insurance, number of ship crews, and port service costs. An immense contribution, with a weight of 60% of all ship operating costs, is the cost of fuel [3]. In addition, fuel prices are also volatile and uncertain, so each port creates new challenges for shipping companies [4].

A container ship is a ship built to transport a container or containers. Each container ship is designed using units of TEUs with the aim that the ship’s hull can load the appropriate containers, with the number of containers requested by the owner. A boat container is slightly different from a general container ship, where a ship regularly has a specific route or is often referred to as a liner. Based on Kristensen [5], container ships can be divided into three types based on size as follows: feeder ships (less than 2900 TEUs), Panamax ships (between 1900 TEUs and 5300 TEUs), and Post-Panamax ships (less than 4000 TEUs). A boat container is a ship most often used in shipping intercontinental goods. The container size is also becoming more prominent along with the increasing demand in the market in terms of shipping goods using water transportation. Therefore, the world economy is highly dependent on the shipping industry, in which container ships carry non-bulk cargo.

These problems make it difficult for shipping companies to choose the best route. Therefore, this study aims to provide the best route with the minimum fuel cost and greatest efficiency for shipping companies. The solution is based on graph theory produced by Bondy and Murty’s research [6]. A graph with the notation G has a set consisting of a vertex. A graph is a collection of points. A line joining two points is an edge. Graph G can be represented by (V, E). Set V is a collection of points, and set E is a collection of edges. The points that exist can represent the location of a city. Meanwhile, the ribs symbolize the relationship between the two cities represented by the existing points. Below is an example of a graph G (V, E), with V = {v₁, v₂, v₃, v₄} and E = {e₁, e₂, e₃, e₄, e₅, e₆}, which can be explained through Figure 1.

The classical mathematical model that uses graph theory is the traveling salesman problem. The traveling salesman problem (TSP) is a classical mathematical model that requires an optimization algorithm to be used to search for solutions. This problem is an application of graph theory and is included in the scientific field of operations research. The traveling salesman problem originates from a problem where a salesperson must visit n cities to sell their products. The series of cities he visited would create a path, with the condition that these cities could only be visited once, and eventually return to the original city. According to Sengupta [7] and Çela [8], the traveling salesman problem can be used to choose the route with the lowest cost. The solution can be found by creating a traveling salesman problem matrix which contains the costs of traveling between cities through Equation (1); the matrix is below:

$${\left[D_{ij}\right]}_{nxn}=\left[\begin{array}{cc}\begin{array}{cc}{D}_{11}& D_{12}\\ D_{21}& D_{22}\end{array}& \begin{array}{cc}\cdots & D_{1n}\\ \cdots & D_{2n}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ D_{n1}& D_{n2}\end{array}& \begin{array}{cc}\ddots & ⋮\\ \cdots & D_{nn}\end{array}\end{array}\right]$$

(1)

where n is the number of ports to be visited, D_ij is the distance or cost from port-i to port-j, and i j, is the sequence of notation (1, 2, to n). Based on Hoffman’s research, such cases are often referred to as Hamiltonian cycles [9]. Thanks to its easy formulation, difficulty in solving, and wide range of real-life applications, the TSP is probably the most studied discrete optimization problem in Gross’s research [10]. A heuristic algorithm is an algorithm that belongs to the optimization algorithms; these algorithms can provide results that are close to the optimum value with fast computation time. Several models of heuristic algorithms can handle cases with large sample sizes with deviation values only ranging from 2–3% of the optimum solution [11]. Several examples of heuristic algorithms are the genetic, simulated annealing, ant colony, particle swarm, and nearest neighbor algorithms. In the case of the traveling salesman problem, the algorithm’s objective function is the same, namely, to find the best route with the minimum distance or cost.

The genetic algorithm is one of the best heuristic algorithms for combinatorial optimization problems [12,13]. This algorithm selects individuals randomly from a specified population to be parents to produce offspring in the next generation. Evolution will occur after several generations have passed. The scientific selection process will produce the best individuals because of the survival process. Continuous gene changes will occur in individuals to help them adapt or adjust to their living environment. Changes in genes in individuals are caused by mutation and crossover processes that occur in each individual. The mutation and crossover processes are the basis of the genetic algorithm, which provides offspring or a more optimal solution to solving a predetermined problem. A heuristic algorithm is a procedure that can provide a good or nearly optimum solution to a problem. This research aims to find the best route with the minimum fuel cost. This research hopes there is an easy process for selecting shipping routes for container ships so that operational cost efficiency can be achieved. This study uses three heuristic algorithms, namely, the nearest neighbor algorithm, simulated annealing, and the genetic algorithm, to find the most cost-optimal route.

Literature Review

Several studies have been conducted to analyze routing problem cases and heuristic algorithm methods. The manuscript reviews [14] examine the impact of genetic operators on algorithm efficiency, highlighting their significance in the search for solutions. The research focuses on applying metaheuristics, particularly genetic algorithms, to solve the vehicle-routing problem. The study includes the development of a prototype that incorporates various genetic operators tailored explicitly to the vehicle-routing problem. Through a series of experiments, the authors investigate the optimal combination of genetic operators for effectively solving the vehicle-routing problem. They also assess the level of involvement each genetic operator should have in the solution-generation process. They also identify the operator that facilitates the discovery of optimal solutions for large-scale, real-life instances of the vehicle-routing problem. Ilhan [15] experiments aims to improve the solution quality and convergence of a simulated annealing algorithm for the capacitated vehicle-routing problem (CVRP). They introduce several enhancements to the algorithm. Firstly, they improve the initial population generation and utilize advanced 2-opt algorithms to start the search process with better solutions. Unlike traditional approaches, they incorporate crossover operators and information exchange among solutions, leading to a higher convergence rate. They also employ a mixed selection procedure to sustain convergence in the later iterations. The authors use the Taguchi method to determine the optimal parameter values for their improved simulated annealing algorithm with crossover (ISA-CO). Through various performance measures, they demonstrate the competitiveness of ISA-CO as an alternative method for solving the CVRP. Overall, Ilhan’s study presents a comprehensive approach to enhancing the simulated annealing algorithm for the CVRP, showcasing its potential for effective and competitive problem solving. Zhena [16] researched route and speed optimization on liner ships by considering emission control policies. The study uses mixed-integer linear programming to minimize fuel costs and SO₂ emissions. Lashgari et al. [17] uses the stochastic linear-integer programming model to determine the ship speed and ship route by considering scenarios of differences in fuel prices. Wen et al. [18] investigate the case of multiple-ship routing and speed under various objectives, including time, cost, and environmental considerations. They develop a branch-and-price algorithm and a constraint-programming model to address this problem. Their models take into account several factors, including the fuel consumption based on payload, explicit fuel prices, freight rates, and costs associated with in-transit cargo inventory. The study considers alternative objective functions such as minimizing the total trip duration, minimizing total cost, and minimizing emissions. In their study, Moradi et al. [19] employ a novel method for optimizing routes using reinforcement learning (RL). The researchers first develop a generic ship model using artificial neural networks (ANNs) to accurately predict the fuel consumption of the ship. Subsequently, they apply different RL algorithms, including deep Q-network (DQN), deep deterministic policy gradient (DDPG), and proximal policy optimization (PPO), to optimize the routes.

2. Materials and Methods

David Goldberg [20] is the scientist who first introduced a cycle of genetic algorithms such that the research developed into a cycle that selects the best individuals. Furthermore, Zbigniew Michalewicz [21] developed a genetic algorithm cycle by changing the order of the selection process to be carried out after mutation and crossover and adding an elitism operator. The elitism operator in question provides a coding process by providing individual forming genes so that it results in encoding binary values, real numbers, and integers so that optimization occurs based on gene combinations. Then, the algorithm generates a fitness value that is useful for providing the best solution from the resulting combination of genes. This function is objective, so an initial population can be created by permuting genes. The initial population is finally selected so that the individual with the highest fitness value produces the best offspring or, in this case, the best container ship path from each port of origin and final destination based on the distance and the cost of fuel used during the trip. This process is explained further in the sub-chapters below.

2.1. Genetic Algorithm

The most important thing is the three processes that make the genetic algorithm function as it should, which is determined based on the fitness value as the objective function to minimize the total fuel cost spent, resulting in less cost and a better fitness value. The function can be summarized with the following fitness value of this problem case using the asymmetric traveling salesman problem mathematical model, which is defined in Equation (2). The F value is the fitness value, and f is the total fuel cost incurred, so the reference is an objective function that gives the minimum fuel cost value.

F = 1/f

(2)

First, in this study, the selection process used was the tournament method. The selection process is used to obtain the best individuals, which will then undergo crossover and mutation processes. This method can be described as a tournament in which n participants compete with each other to find the best one. The selection is based on the fitness value of each individual. Second, a crossover is a stage that will produce a new individual by involving the exchange of genes from two selected parents. The process is carried out with the results of the selection in the previous process with a predetermined probability. If the probability value of an individual is greater than the crossover, then the crossover will not be performed. The crossover process aims to provide new individuals who provide offspring with better values. Figure 2a illustrates this process based on research reviewed by Goldberg [20] by performing a one-to-many-point crossover, an arithmetic crossover, and a crossover for the representation of permuted chromosomes.

On the other hand, mutation is an advanced process that aims to produce an individual exchange of genes within an individual by performing the inversion of a gene. Mutations are carried out in individuals who pass the selection process in the previous step. Thus, the greater a mutation probability that is used, the more likely an individual is to lose the traits that it obtained from its parent. However, the mutation probability is small. Therefore, genes are never evaluated. This process can be seen in Figure 2b based on research conducted by Liu [22], which shows the process of gene transfer from parents to offspring.

2.2. Simulated Annealing

The simulated annealing method was discovered by observing a cooling process in a metal liquid that will form crystals; this process is often referred to as the annealing process. We can also define this process as slowly cooling an object until it reaches its freezing point. Liquid particles at high temperatures have a high energy level, which makes it easier for the particles to move around each other. When the object’s temperature is lowered slowly, it will make the particles try to arrange themselves to reach the minimum energy level and a stable particle arrangement [23]. This decreasing temperature, or annealing process, can be implemented as an approximation algorithm to solve simulated annealing problems. Simulated annealing has been proven to solve many real-life combinatorial optimization problems, including scheduling problems [24,25,26].

This algorithm depends on its parameters in generating a solution to the problem. Therefore, searching for optimal parameters such as the initial temperature, the cooling schedule, and the correct number of iterations will take much time in a trial-and-error process. Many studies have shown that the simulated annealing algorithm is susceptible to parameter settings [24]. Generally, the cooling schedule value of simulated annealing ranges from 0.8 to 0.99 [25]. The method can be clarified by the analogies related to simulated annealing based on the research of Vecchi et al. [23] in

Table 1. Analogical of Simulated Annealing Process [23].

Physics of Vecchi et al. [23] Research	Optimization of Simulated Annealing
The Initial State of Annealing	Initial Solution
Energy	Cost of Fuel
Physical Changes	Temporary Solution
Temperature	Controlling Parameter
Equilibrium State	Optimal Solution

below.

Simulated annealing can be used to solve the traveling salesman problem with the objective function of the minimum distance or minimum cost. According to Chibante [24], a solution can be found by inputting the required parameters, determining the initial route from an optimization process reference, and running iterations that are carried out by swapping neighbors of the route to be traversed randomly, where the algorithm generates a random number between values 0 and 1. Thus, a route evaluation can be carried out consisting of continued iterations based on the maximum number of iterations fulfilled. Another consideration is the current route and a whether it has a better value than the shipping route (current solution). Finally, suppose the resulting new route is not better than the current route. In that case, this will generate the value of r, which is between the values 0 and 1, which helps calculate the route’s probability value (p).

The expected final result has two possible cases regarding r. If the value is less than or equal to p, the resulting route can be the new current solution. However, if the value of r is greater than p, the new route is not allowed to become the new current solution. The process is carried out repeatedly until the desired value is achieved based on the parameters determined at the beginning. This process can be seen in Figure 3.

2.3. Nearest Neighbor Algorithm

The nearest neighbor algorithm is a greedy algorithm that can be used to solve the traveling salesman problem. Greedy is an algorithm paradigm that forms a solution based on the closest solution with the most significant benefit. The algorithm can be explained through the knapsack fractional problem by comparing a weighted object from each beam. The most negligible load will be put in the bag first compared to larger loads.

Route selection using nearest neighbor (NN) has an advantage because the route has only a few severe errors, but there are long segments that will connect the nodes with short edges [27]. Although this algorithm is straightforward to implement and can execute in a fast time, this algorithm performs route selection based on the distance of the nearest city from the city that is currently being visited. Therefore, this algorithm often cannot provide an optimal solution because it is often trapped in a local optimum. A flowchart of the nearest neighbor algorithm approach can be seen in Figure 4.

2.4. Experimental Method

At this stage, the data that have been collected are processed into a mathematical model of the asymmetric traveling salesman problem so that it can be solved with predetermined optimization algorithms. After successfully making a mathematical model of the existing problem, an optimization program is designed using three heuristic algorithms—a genetic algorithm, simulated annealing, and nearest neighbor—using the C++ programming language. The objective function of the three programs is to find the route with the lowest cost. From each optimization program with these three algorithms, three outputs will be issued: ship route selection, ship fuel operational costs, and optimization execution time.

The next step testing the program that had been made. At this stage, selection of input parameters was also carried out using the trial-and-error method to obtain the optimum output. The fuel used is VLSFO (very-low-sulfur fuel oil), a mixture of both fuel types with a sulfur content of 0.5% and an average main engine power of 30,900 kW. The average auxiliary engine power was 6800 kW, and the average ship speed by type was 21.6 knots for container ships. The trial was carried out with variations on the starting/ending destination, which became each route selection’s starting and ending points. An analysis of each optimization result was carried out. Comparisons were made based on operational costs and execution time. The nautical mile distance used between ports in this study uses data that can be seen in

Table 2. Nautical Mile Distance Between Ports; (a) Busan to Yantian, (b) Vung Tau to Colombo, and VLSFO Fuel Price 224 in USD/Ton [28].

(a)
Port Location	Busan	Shanghai	Qingdao	Tanjung Pelepas	Hong Kong		Yantian
Busan	0	535	616	3229	1475		1488
Shanghai	535	0	367	2713	955		968
Qingdao	616	367	0	3066	1307		1320
Tanjung Pelepas	3229	2713	3066	0	1851		1871
Hong Kong	1475	955	1307	1851	0		20
Yantian	1488	968	1320	1871	20		0
Vung Tau	2523	1996	2348	795	934		1085
Nansha New Port	1568	1048	1400	1842	100		80
Shekou	1486	966	1318	1869	18		25
Singapore	3208	1692	3044	57	1460		1851
Salalah	6787	6270	6623	3557	5410		5429
Colombo	4867	4350	4703	1637	3490		3510
(b)
Port Location	Vung Tau	Nansha New Port	Shekou	Singapore	Salalah	Colombo	USD/Ton
Busan	2523	1568	1486	3208	6787	4867	318
Shanghai	1996	1048	966	1692	6270	4350	318
Qingdao	2348	1400	1318	3044	6623	4703	328
Tanjung Pelepas	795	1842	1869	57	3557	1637	358.1
Hong Kong	934	100	18	1460	5410	3490	303
Yantian	1085	80	25	1851	5429	3510	401.4
Vung Tau	0	1056	1082	638	4353	2434	482
Nansha New Port	1056	0	82	1882	5400	3481	401.4
Shekou	1082	82	0	1849	5427	3507	401.4
Singapore	638	1882	1849	0	3614	1695	312
Salalah	4353	5400	5427	3614	0	1920	572
Colombo	2434	3481	3507	1695	1920	0	360

a,b.

The data regarding the distances between the 12 ports are made into a matrix [S_ij]_nxn, where n is the number of ports in 1 ship’s operating route that is in the study of 12 ports. J_ij represents the distance from port i to port j in nautical miles, and _i,j = 1,2,…,n. Calculation of the sailing time from between ports is performed with a matrix [T_ij]_nxn. The voyage time matrix is divided by the distance matrix between the ports and the container ship speed specified above. T_ij is a unit of time required by a ship sailing from port i to port j. The matrix relationship between T_ij and S_ij can be seen in Equation (3).

$${\left[D_{ij}\right]}_{nxn}=\frac{{\left[S_{ij}\right]}_{nxn}}{V}=\left[\begin{array}{cc}\begin{array}{cc}\frac{S_{11}}{V}& \frac{S_{12}}{V}\\ \frac{S_{21}}{V}& \frac{S_{22}}{V}\end{array}& \begin{array}{cc}\cdots & \frac{S_{1n}}{V}\\ \cdots & \frac{S_{2n}}{V}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ \frac{S_{n1}}{V}& \frac{S_{n2}}{V}\end{array}& \begin{array}{cc}\ddots & ⋮\\ \cdots & \frac{S_{nn}}{V}\end{array}\end{array}\right]= \left[\begin{array}{cc}\begin{array}{cc}{T}_{11}& T_{12}\\ T_{21}& T_{22}\end{array}& \begin{array}{cc}\cdots & T_{1n}\\ \cdots & T_{2n}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ T_{n1}& T_{n2}\end{array}& \begin{array}{cc}\ddots & ⋮\\ \cdots & T_{nn}\end{array}\end{array}\right]$$

(3)

$${\left[F_{ij}\right]}_{nxn}={\left[T_{ij}\right]}_{nxn}· K=\left[\begin{array}{cc}\begin{array}{cc}{T}_{11}·K& T_{12}·K\\ T_{21}·K& T_{22}·K\end{array}& \begin{array}{cc}\cdots & T_{1n}·K\\ \cdots & T_{2n}·K\end{array}\\ \begin{array}{cc}⋮& ⋮\\ T_{n1}·K& T_{n2}·K\end{array}& \begin{array}{cc}\ddots & ⋮\\ \cdots & T_{nn}·K\end{array}\end{array}\right]= \left[\begin{array}{cc}\begin{array}{cc}{F}_{11}& F_{12}\\ F_{21}& F_{22}\end{array}& \begin{array}{cc}\cdots & F_{1n}\\ \cdots & F_{2n}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ F_{n1}& F_{n2}\end{array}& \begin{array}{cc}\ddots & ⋮\\ \cdots & F_{nn}\end{array}\end{array}\right]$$

(4)

$${\left[C_{ij}\right]}_{nxn}=\left[\begin{array}{cc}\begin{array}{cc}{F}_{11}·f_1& F_{12}·f_1\\ F_{21}·f_2& F_{22}·f_2\end{array}& \begin{array}{cc}\cdots & F_{1n}·f_1\\ \cdots & F_{2n}·f_2\end{array}\\ \begin{array}{cc}⋮& ⋮\\ F_{n1}·f_n& F_{n2}·f_n\end{array}& \begin{array}{cc}\ddots & ⋮\\ \cdots & F_{nn}·f_n\end{array}\end{array}\right]= \left[\begin{array}{cc}\begin{array}{cc}{C}_{11}& C_{12}\\ C_{21}& C_{22}\end{array}& \begin{array}{cc}\cdots & C_{1n}\\ \cdots & C_{2n}\end{array}\\ \begin{array}{cc}⋮& ⋮\\ C_{n1}& C_{n2}\end{array}& \begin{array}{cc}\ddots & ⋮\\ \cdots & C_{nn}\end{array}\end{array}\right]$$

(5)

The weight of the fuel used by the ship can be calculated using Equation (4). It is assumed that the primary engine fuel consumption is 209 g/kWh and that the auxiliary engine fuel consumption is 211 g/kWh, according to the range written by Poehls [29]. The main engine power and auxiliary engine power of the container ship are taken from the average ship engine power table made by the Air Resources Board [30]. Next, the fuel weight required for each inter-port voyage is written in a matrix [F_ij]_nxn. First, F_ij represents the fuel the ship uses in tons to travel from port i to port j. After that, the fuel price from each port will be incorporated into a matrix to give the traveling salesman problem an asymmetrical shape that describes the fuel cost of each voyage. Finally, Cij describes the fuel costs incurred by the ship to sail from port i to port j, while fi describes the fuel price in USD/ton at port i, and i,j = 1,2,…,n. This C_ij matrix is the result of processing the data obtained into an asymmetric traveling salesman problem which will later be optimized for route selection.

$$\left[\left(Pe·Bme\right)+\left(Pae·Bae\right)\right]\frac{g}{v}·{10}^6$$

(6)

$$K=\left[\left(Pe·Bme\right)+\left(Pae·Bae\right)\right]·{10}^6$$

(7)

where Pe is the main engine power (kW), Bme is the primary engine fuel consumption (209 g/kWh), Pae is the auxiliary engine power (kW), and Bae is the auxiliary engine fuel consumption (211 g/kWh). After successfully performing optimization with the three heuristic algorithms used to solve the problem that has been posed, an analysis and comparison of the results issued from the three algorithms will be carried out. The outputs of the three algorithms are as follows: ship route selection and ship fuel operational costs.

The following is an individual example of the algorithm, along with the calculation of its fitness value, namely the fuel cost.

Individual (route): Tanjung Pelepas–Yantian–Nansha New Port–Vung Tau–Shekou–Hong Kong–Busan–Qing Dao–Shanghai–Singapore–Salalah–Colombo–Tanjung Pelepas.

Suppose that i = 1 for Tanjung Pelepas, i = 2 for Yantian, i = 3 for Nansha New Port, and so on. Then, the following is a formula for calculating the fuel cost (FC):

$$FC={\sum }_{i=1}^{12}\frac{S_{i,i+1}}{V} \times K \times f_i$$

(8)

Of the three outputs given, a comparison will be made regarding which algorithm provides the route with the most negligible operational costs. In addition to comparing the operational costs, this paper also analyzes the increase in program computing time with a decrease in the results of operating costs. After successfully comparing these things, the best algorithm to use in the problem can be determined.

3. Results and Discussion

Parameter variations in the optimization algorithm’s input parameters are carried out to maximize the performance of each heuristic algorithm used in solving the problem. The variation of the parameters in question is very different for each optimization algorithm because each algorithm has its own parameters. Therefore, each algorithm must perform parameter variations to obtain the ideal parameters. For example, parameter variations in the nearest neighbor algorithm are not carried out because there are no input parameters other than the ATSP matrix.

3.1. Genetic Algorithm Parametric Testing

In the genetic algorithm, parameter variations are carried out on the values of the population size, crossover rate, mutation rate, and generation size. The test aims to determine the ideal values of the four input parameters for this shipping route selection problem. This test aims to find the best population size. In the tests carried out, several test parameters were determined, namely: crossover rate = 0.1, mutation rate = 0.2, and generation size = 10.

The population size tested is in the range of 10 to 110. The test is performed 10 times to obtain the average fuel cost generated for these parameters. The following is a graph of population size against average fuel operating costs. Figure 5a shows that the lowest average fuel cost is obtained when the population size is 100. Meanwhile, the highest average fuel cost is obtained when the population size is 10. It can be concluded that the larger the population used, the more it is possible to find a better solution. Therefore, for the population size parameter, the value of 100 was chosen because this value managed to yield the lowest average fuel cost. Mutation and crossover rate tests were conducted to obtain the best ratio with the condition Cr + Mr = 1. In this test, several parameters were determined, namely, population size = 100 and generation size = 10.

The test is performed 10 times to obtain the average fuel cost of each combination of mutation rate and crossover rate. Above is a graph of the results of testing the combinations of crossover rate and mutation rate values regarding the average spent fuel costs. Figure 5b shows that the lowest average fuel cost is obtained when the mutation rate is 0.4 and the crossover rate is 0.6. Meanwhile, the highest average fuel cost is obtained when the mutation rate is 1 and the crossover rate is 0. A high mutation rate value with a low crossover rate indicates that the search for solutions is not optimal. Therefore, to obtain the best results, the optimization parameters of the genetic algorithm for this problem are chosen at the mutation rate and crossover rate values that give the lowest average fuel cost.

Generation size testing also aims to determine the best generation size. In the generation size test, the test parameters determined are population size = 100, mutation rate = 0.4, and crossover rate = 0.6. The generation size test is carried out from 10–100, and each generation size is tested 10 times to obtain the average fuel cost. This test aims to obtain the ideal generation size parameter value that can provide the result with the lowest average fuel cost. From Figure 5c above, it can be concluded that the larger the generation size used, the smaller the average value of fuel costs. Although the average cost of the resulting fuel will tend to become smaller as the generation size increases, the decrease in the average cost becomes less and less significant.

3.2. Simulated Annealing Parametric Testing

In the simulated annealing algorithm, there are two parameter variations carried out, namely, the initial temperature and the cooling rate value. This test is carried out to obtain the most ideal parameter values for the simulated annealing algorithm in the route-finding problem with the minimum fuel cost. This test was carried out in a temperature range of 10–200.

Tests on the initial temperature value are carried out with parameters at a final temperature of 0 °C and a cooling rate of 0.75. Figure 6a shows that the average fuel cost tends to decrease as the initial temperature value rises. The test proves that the greater the initial temperature value, the better the results. However, for the tested case, when the initial temperature passed 100 °C, the average fuel cost did not significantly decrease. The test was conducted with a cooling rate range of 0.3–0.99. The test was performed 10 times to obtain the average fuel cost value. Tests on the initial temperature value are carried out with the following parameters: a final temperature of 0 °C and an initial temperature of 200 °C. It can be seen from Figure 6b that the higher the cooling rate in the test, the lower the average fuel cost. The best average value for fuel costs is obtained when the cooling rate is 0.99. Therefore, the cooling rate value that gives the best average will be used as a simulated annealing parameter in selecting ship routes. From the results of the tests carried out on the parameters of the simulated annealing algorithm, it was found that the parameters that give the ideal value are a final temperature of 0 °C, an initial temperature of 200 °C, and a cooling rate of 0.99.

3.3. Optimum Solution for Asia-Pacific Cargo Shipping Route

Based on the optimizations carried out, several results will be given by the algorithm for the selection of the optimum route and the lowest fuel cost for each track that is traced. With this, a comparison can be made between the ship routes from each destination. In this discussion we use the results of the initial and final destinations at Tanjung Pelepas and compare them with those of the other three algorithms. Visualizations and other route selection results can be seen in separate results. The route selections obtained from each algorithm are presented in

Table 3. Route Selection from Each Algorithm via Tanjung Pelepas.

Algorithm	Route Selection
Genetic Algorithm	Tanjung Pelepas–Vung Tau–Nansha New Port–Yantian–Shekou–Hong Kong–Busan–Qingdao–Shanghai–Singapore–Salalah–Colombo–Tanjung Pelepas
Simulated Annealing	Tanjung Pelepas–Vung Tau–Nansha New Port–Hong Kong–Yantian–Shekou–Singapore–Salalah–Colombo–Shanghai–Qingdao–Busan–Tanjung Pelepas
Nearest Neighbor	Tanjung Pelepas–Singapore–Vung Tau–Hong Kong–Shekou–Yantian–Nansha New Port–Shanghai–Qingdao–Busan–Colombo–Salalah–Tanjung Pelepas

and Figure 7 below as visualizations of the route selection of each algorithm at each starting/ending destination via Tanjung Pelepas, Indonesia.

After testing each initial location/destination using each of the previously determined parameters, the three algorithms succeeded in conducting route searches. The proves that the three algorithms can find ship routes with the lowest fuel costs. The results of the three algorithms are in the forms of route selection and the cost of existing fuel. Therefore, we can compare the three algorithms in terms of the fuel costs incurred when transiting the route chosen in

Table 4. Fuel Cost (USD) of Route Selection Results for Each Algorithm.

Start-End Destinations	Genetic Algorithm	Simulated Annealing	Nearest Neighbor
Busan–Busan	1,812,235	1,958,933	2,437,987
Shanghai–Shanghai	1,812,235	1,971,558	2,356,335
Qingdao–Qingdao	1,812,235	1,841,853	2,419,796
Tanjung Pelepas–Tanjung Pelepas	1,812,235	2,346,975	2,093,880
Hong Kong–Hong Kong	1,812,235	1,882,814	2,297,187
Yantian–Yantian	1,812,235	1,842,789	2,300,718
Vung Tau–Vung Tau	1,812,235	2,221,920	2,282,336
Nansha New Port–Nansha New Port	1,812,235	1,882,814	2,282,336
Shekou–Shekou	1,812,235	1,983,543	2,299,934
Singapore–Singapore	1,812,235	1,944,683	2,136,125
Salalah–Salalah	1,812,235	1,841,853	1,937,599
Colombo–Colombo	1,812,235	1,841,853	1,937,599

and Figure 8.

The genetic algorithm always gives the best output from the three algorithms tested for each starting/ending destination (Figure 9). Meanwhile, the nearest neighbor algorithm provides the route with the worst fuel cost when compared to the other two algorithms. For the simulated annealing algorithm, in selecting the route, the fuel value is often close to the results of the genetic algorithm.

In the simulated annealing algorithm, the fuel cost of each selected route is highly variable. This is because simulated annealing is often trapped at a local optimum. Simulated annealing also cannot give the better results than the genetic algorithm because the simulated annealing algorithm is a single-solution-based algorithm. A single-solution-based algorithm starts with a random solution. Then, it obtains a new candidate solution by finding a random neighbor solution that depends on the current solution and that satisfies the existing criteria. Therefore, the best result of the route search carried out is very dependent on the random solution generated at the beginning of the running of the algorithm. In the graphs above, the search pattern of simulated annealing is very varied, which means it is very dependent on the initial solution chosen at random.

The nearest neighbor algorithm provides the worst route selection because the route search is very dependent on the initial destination. We can see in the graphs of fuel costs against the number of destinations that the nearest neighbor algorithm always starts with a low fuel cost at the beginning of the number of destinations visited. However, the higher the number of destinations visited, the more fuel costs tend to increase sharply. In the nearest neighbor algorithm, the selected destination is the destination with the lowest cost from the location at the stopover time. The pattern repeats itself until all destinations have been visited. This pattern means the nearest neighbor algorithm does not give ideal results. Due to the algorithm’s characteristics that do not search for the best route globally, the route search is very dependent on the nearest destination from the stopover location.

Genetic algorithms engaged in route searching find the shortest route. We can see in the graphs of fuel costs against the number of destinations above in the genetic algorithm that fuel costs look high when the number of destinations that have been visited is small. However, the genetic algorithm succeeded in providing the lowest fuel costs when all destinations were successfully visited. The analysis proves that the route selection made by the genetic algorithm considers the best route globally, coupled with the fact that the genetic algorithm is a population-based algorithm. The genetic algorithm starts with a solution of a specified population and then performs selection, mutation, and crossover. The genetic algorithm has succeeded in proving that algorithms of its type are the best for solving the problem of finding ship routes with the goal of the lowest material costs due to considering many solutions that are executed and compared.

4. Conclusions

From this study of optimizing the selection of cargo ship routes using heuristic algorithms, it can be concluded that the three algorithms can be used to solve the traveling salesman problem to find the best ship route. In this case, the genetic algorithm has the most optimal results with the lowest fuel cost of USD 1.5 million. The results are achieved by using the best route selection and minimizing the cost of fuel used. On the other hand, population-based algorithms such as genetic algorithms will be better applied than single-solution-based algorithms to solve multimodal problems such as the one in this study. This study can be applied to other sea areas as long as the characteristics of those seas are the same as, or at least similar to, those of the seas in the Asia-Pacific region. This study only considers the cost of ship fuel and the distance between ports, so the characteristics of the sea and the weather are not accounted for. The option of considering the sea characteristics could be used in future research.

Furthermore, the nearest neighbor algorithm cannot provide the optimum solution because the algorithm does not consider the global route search solution. Another suggestion that could be considered in further research is that other operational aspects that may affect route selection and fuel consumption should also be considered. Finally, artificial intelligence can consider environmental conditions so that route selection can effectively occur.