# Multi-Objective Ship Route Optimisation Using Estimation of Distribution Algorithm
## 2 Related Research
### Methods for Ship Routing Problem  
Ship safety and energy efficiency are important factors for making ship operations more competitive and sustainable [4; 8]. Therefore, more and more researchers are studying how to optimize ship routes based on weather and sea conditions [2; 3; 4; 9; 10]. These factors affect how fast a ship moves, how much fuel it uses, and how much energy it emits. To find a solution, a ship routing problem is reduced to a two-dimensional subpath search problem, which considers only position and assumes that either the ship's speed or main engine output stays the same [4; 11]. The reduced ship weather routing problem is usually solved by using the modified isochrone method [12], the isotone method for the minimum-time problem [13], dynamic programming for ship route optimisation with minimising fuel oil consumption [14], and [PERSON]'s algorithm using the weather forecast data [15]. Metaheuristic approaches (for example, evolutionary algorithms) were also used to find the optimal ship route taking into account weather conditions [16] or the route length and fuel prices in visited ports [5]. Notwithstanding, in reality, the lengthy routes and significant fluctuations in weather and sea conditions render it impracticable to maintain a consistent sailing velocity and primary engine output for vessels traversing the ocean [4]. A number of three-dimensional optimisation algorithms have been proposed to better simulate fuel consumption. The sum of weighted criteria of fuel consumption and ship safety was used in a real-coded genetic algorithm for weather routing proposed in [17]. In another approach, time was taken into account as a third additional dimension in addition to two-dimensional space, and the three-dimensional problem formulated in this way was solved by adapting the three-dimensional isochrones method with weighting factors [18]. The more realistic model of fuel consumption, taking into account the effect of hull and propeller fouling, and ocean currents was proposed in [19]. In this work, a genetic algorithm was used to find the optimal ship's path, and the goal was to minimize the fuel consumption. The application of machine learning-based approaches included the use of models trained on real-world navigation data of ships in various weather conditions. The trained machine learning models were then used to predict fuel consumption and, consequently, optimise the ship's route [20; 21]. The Attention-Bidirectional Long Short-Term Memory model optimised by the Whale Optimisation Algorithm was used for ship trajectory prediction aimed at collision avoidance and safety increase [22]. Statistical models taking into account wave height, wave period, wind speed, and the engine's RPM data were also used for ship route planning [23]. The goals of path optimisation in the case of such methods included safety, fuel consumption reduction, and on time arrival at the destination. The ship weather routing problem was also solved using multi-objective optimisation algorithms. In [24], the two-objective ship routing problem was formulated as a non-linear integer programming problem. The objectives included minimising the fuel consumption and the total risk. Multi-objective evolutionary algorithms were used to solve the ship weather routing problem in [16; 25; 26; 27; 28]. In these works, various objectives were used, like estimated time of arrival, fuel consumption and safety. User preferences were taken into account in the multi-objective evolutionary algorithm with weight intervals proposed in [3]. In the presented approach, the ship's route consisted of nodes including information about the location, estimated time of arrival, engine settings, and ship heading. There were also attempts to solve the ship weather routing problem using three-dimensional dynamic programming algorithms. Minimising the ship fuel consumption was a goal of a 3D dynamic programming-based method proposed in [2].